ABSTRACT
Title of Dissertation: CHARACTERIZATION AND ANALYSIS
OF FLUIDIC ARTIFICIAL MUSCLES
Jonathan Chambers
Doctor of Philosophy, 2021
Dissertation Directed By: Professor Norman M. Wereley
Department of Aerospace Engineering
Fluidic artificial muscles (FAMs) are a form of soft actuator that have been
applied to an expanding number of applications, due to their unique characteristics
such as low weight, simple construction, inherent compliance, and high specific force
and specific work capabilities. With energy sourced from a pressurized fluid, con-
tractile FAMs provide a uniaxial contractile force, while their morphing geometry
allows them to contract in length. In a design environment where actuators have
tight spatial requirements and must provide precise force and position control, it is
becoming more important than ever to have accurate mathematical representations
of FAM actuation behavior and geometric characteristics to ensure their successful
implementation. However, geometric models and force analyses for FAMs are still
relatively crude. Geometric models of FAMs assume a cylindrical geometry, the
accuracy of which is suspect because there are no documented methods for effec-
tively measuring FAM shape. Actuation force analyses are also relatively inaccurate
unless they are adjusted to fit to experimental response data. Research has continu-
ally pursued methods of improving the predictive performance of these analyses by
investigating the complex working mechanisms of FAMs.
This research improves these analyses by first, making improvements to the
experimental characterization of a FAM?s actuation response, and then using the
more comprehensive data results to test long-held modeling assumptions. A quanti-
tative method of measuring FAM geometry is developed that provides 0.004 in/pixel
resolution measurements throughout a characterization test. These measurements
are then used to test common assumptions that serve as sources of uncertainty:
the cylindrical approximation of FAM geometry, and assumption that the FAM?s
braid is inelastic. Once these sources of modeling error are removed, the model?s
performance is then tested for potential improvements. Results from this research
showed that the cylindrical approximation of the FAM?s geometry resulted in over-
estimations of the FAM?s average diameter by 4.7%, and underestimations of the
FAM?s force by as much as 37%. The inelastic braid assumption resulted in a maxi-
mum 4% underestimation of average diameter and a subsequent 5% overestimation
in force, while the use of softer braid materials was found to have the potential
for much larger effects (30% underestimation in diameter, 70% overestimation in
force). With subsequent adjustments made to the force model, the model was able
to achieve a fit with a mean error of only 2.8 lbf (0.3% of maximum force).
This research demonstrates improvements to the characterization of a FAM?s
actuation response, and the use of this new data to improve the fidelity of existing
FAM models. The demonstrated characterization methods can be used to clearly
define a FAM?s geometry to aid in the effective design and implementation of a
FAM-actuated mechanism, or to serve as a foundation for further investigation into
the working mechanisms and development of FAMs.
CHARACTERIZATION AND ANALYSIS OF FLUIDIC
ARTIFICIAL MUSCLES
by
Jonathan Michael Chambers
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2021
Advisory Committee:
Dr. Norman Wereley, Chair/Advisor
Dr. David Akin
Dr. Craig Carignan
Dr. Ross Miller
Dr. Anindo Roy
?c Copyright by
Jonathan Michael Chambers
2021

Acknowledgments
I would first like to acknowledge the support of my advisor, Dr. Wereley, for his
guidance through a wide-range of projects. The experience I garnered through these
projects will surely prove to be invaluable. I would also like to thank Dr. Fourney
for the opportunities he provided me at the beginning of my graduate studies.
I would like to thank Dr. Akin, Dr. Carignan, Dr. Miller, and Dr. Roy
for their support as my doctoral committee members. I would like to acknowledge
some of the external funding that supported my research, including: NAVAIR?
Patuxent River (Exoskeleton project: Research and Educational Development Pro-
gram N00421-13-2-0001); the Office of Naval Research Basic Research Challenge
(SEASTAR project: ONR BAA Announcement N00014-16-S-BA10); and, the Army
(SBIR Topic A17-115, Contract Number: W56HZV-17-C-0178). I would also like
to thank Dr. Carignan, Dr. Kothera, and Dr. Choi for assisting me through these
projects, and Mike Perna for his machinist expertise and guidance.
I would also like to thank many of the fellow graduate students I spent time
with through the years. Special thanks to Ami, Andrew, Caitlin, Chinonso, Frank,
Jacek, Rebecca, Sai, and Steve, for their company in the lab, and for the insights
they provided me towards my research. I would like to especially thank Tom for
passing all of his knowledge of FAMs down to me.
And finally, I would like to thank my family and friends who have encouraged
me throughout this long and arduous process. I especially would like to thank my
Dad, my Sister, and Rebecca for their support.
ii
Table of Contents
Acknowledgements ii
List of Tables viii
List of Figures x
List of Abbreviations xv
1 Introduction 1
1.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Fluidic Artificial Muscle Actuators . . . . . . . . . . . . . . . . . . . 3
1.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1.2 Working Mechanism . . . . . . . . . . . . . . . . . . 10
1.2.2 Experimental Characterization . . . . . . . . . . . . . . . . . . 11
1.2.3 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.3.1 Geometric Model . . . . . . . . . . . . . . . . . . . . 15
1.2.3.2 Actuation Force Model . . . . . . . . . . . . . . . . . 17
1.2.3.3 Review of Literature . . . . . . . . . . . . . . . . . . 19
1.3 Contributions of Dissertation . . . . . . . . . . . . . . . . . . . . . . 22
1.4 Overview of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . 26
2 Design, Fabrication, and Failure Testing of a Swaged End-fitting FAM 29
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 Component Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2.1 Internal Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2.2 Swage Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2.3 Bladder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.2.4 Braid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2.5 Epoxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.3 Determination of FAM Component Sizing . . . . . . . . . . . . . . . 45
2.3.1 Sizing of FAM to Achieve Desired Response . . . . . . . . . . 45
iii
2.3.1.1 Models for Estimation of FAM Radial Expansion
and Force Response . . . . . . . . . . . . . . . . . . 47
2.3.1.2 FAM Force Response Approximation . . . . . . . . . 49
2.3.1.3 FAM Contraction Approximation . . . . . . . . . . . 50
2.3.1.4 FAM Component Sizing Constraints . . . . . . . . . 51
2.3.1.5 Selection of Target FAM Sizing . . . . . . . . . . . . 54
2.3.2 Sizing of Components for Swaged end-fittings . . . . . . . . . 55
2.3.2.1 Steps for Sizing Components of Swaged end-fitting . 56
2.3.3 Final Sizing of FAM . . . . . . . . . . . . . . . . . . . . . . . 61
2.4 FAM Method of Fabrication . . . . . . . . . . . . . . . . . . . . . . . 61
2.4.1 Material Preparation . . . . . . . . . . . . . . . . . . . . . . . 63
2.4.2 Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.4.3 Swaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.4.4 Finishing Process . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.4.5 Modification to swaging process . . . . . . . . . . . . . . . . . 72
2.5 Testing Swaged FAMs . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.5.1 Characterization Test . . . . . . . . . . . . . . . . . . . . . . . 77
2.5.2 Ultimate Tensile Failure Test . . . . . . . . . . . . . . . . . . 78
2.5.3 Burst Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.5.4 Extended Use Failure . . . . . . . . . . . . . . . . . . . . . . . 82
2.5.5 Subsequent Improvements to end-fitting . . . . . . . . . . . . 84
2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3 Improvements in FAM Characterization Through Accurate Control of Pres-
sure 88
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.1.1 Basis of Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.2 Experimental Test Setup . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.2.1 FAM Test Specimens . . . . . . . . . . . . . . . . . . . . . . . 94
3.2.2 Pressure Systems . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.2.2.1 Pneumatic Pressure System . . . . . . . . . . . . . . 99
3.2.2.2 Hydraulic Pressure System . . . . . . . . . . . . . . 99
3.2.3 Pressure System Control . . . . . . . . . . . . . . . . . . . . . 101
3.2.3.1 Pneumatic Pressure Control . . . . . . . . . . . . . . 103
3.2.3.2 Hydraulic Pressure Control . . . . . . . . . . . . . . 104
3.2.4 Controller Tuning . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.3.1 Pressure Control Results . . . . . . . . . . . . . . . . . . . . . 109
3.3.2 Characterization Results . . . . . . . . . . . . . . . . . . . . . 110
3.3.3 Modeling Results . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
iv
4 A Photogrammetric Method for Measuring FAM Shape 116
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.2 Test Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.2.1 Image Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.2.2 Image Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.2.2.1 Color Model . . . . . . . . . . . . . . . . . . . . . . . 125
4.2.2.2 Boundary Detection . . . . . . . . . . . . . . . . . . 128
4.3 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.3.1 Image Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.3.2 Image Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.3.2.1 Step 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 140
4.3.2.2 Step 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.3.2.3 Step 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.3.2.4 Step 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 142
4.3.2.5 Step 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 142
4.3.3 Characterization Test Parameters . . . . . . . . . . . . . . . . 143
4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
4.4.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
4.4.2 Force-Contraction Response . . . . . . . . . . . . . . . . . . . 147
4.4.3 Shape Profile Characterization . . . . . . . . . . . . . . . . . . 148
4.5 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
4.5.1 Shape Profile Analysis . . . . . . . . . . . . . . . . . . . . . . 152
4.5.1.1 Profile Average Diameter . . . . . . . . . . . . . . . . 152
4.5.1.2 Profile Fourier Series Fit . . . . . . . . . . . . . . . . 153
4.5.1.3 Comparison to Previous Cylindrical Profile Estimates 157
4.5.2 Volume Approximation . . . . . . . . . . . . . . . . . . . . . . 160
4.5.3 Testing of Different FAMs . . . . . . . . . . . . . . . . . . . . 164
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
5 Investigation of the Inelastic Braid Modeling Assumption 172
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
5.2 Experimental Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
5.2.1 Description of Tested FAM . . . . . . . . . . . . . . . . . . . . 175
5.2.2 Characterization of the FAM?s Actuation . . . . . . . . . . . . 176
5.2.3 Characterization of the FAM?s Braid . . . . . . . . . . . . . . 176
5.3 Analysis of Braid Elasticity Using Measurements of FAM . . . . . . . 179
5.3.1 Analysis of Measured Elasticity Values . . . . . . . . . . . . . 182
5.4 The Effect of Braid Elasticity on Geometry and Actuation Force . . . 185
5.4.1 Effects of Elasticity with Kevlar-49 Braid . . . . . . . . . . . . 186
5.4.2 Estimated Effects of Lower Stiffness Braid Materials . . . . . . 188
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
v
6 Investigation of the Cylindrical Approximation Assumption 194
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
6.2 Geometric Dimensions Definitions . . . . . . . . . . . . . . . . . . . . 198
6.2.1 Estimated Dimensions ? Cylindrical Approximation . . . . . . 200
6.2.2 Measured Dimensions . . . . . . . . . . . . . . . . . . . . . . . 202
6.2.3 Measured Dimensions ? Cylindrical Approximation . . . . . . 203
6.3 Analysis of the Cylindrical and Measured Geometries . . . . . . . . . 204
6.3.1 Comparison of Braid Dimensions . . . . . . . . . . . . . . . . 205
6.3.2 Comparison of Bladder Dimensions . . . . . . . . . . . . . . . 207
6.3.2.1 Comparison of Bladder Thickness . . . . . . . . . . . 209
6.3.2.2 Comparison of Longitudinal Strain . . . . . . . . . . 211
6.3.2.3 Comparison of Circumferential Strain . . . . . . . . . 214
6.4 Effect of Geometry Profile on Force Model Results . . . . . . . . . . . 217
6.4.1 Gaylord Actuation Force Term (FG) . . . . . . . . . . . . . . . 219
6.4.2 Bladder Actuation Force . . . . . . . . . . . . . . . . . . . . . 220
6.4.2.1 Bladder Stiffness Characterization . . . . . . . . . . 221
6.4.2.2 Longitudinal Bladder Force Term (Fbld,z) . . . . . . . 225
6.4.2.3 Circumferential Bladder Force Term (Fbld,c) . . . . . 227
6.4.2.4 Combined Effect of Bladder Forces . . . . . . . . . . 230
6.4.3 Overall Effect of Geometry on Estimated Force . . . . . . . . 231
6.4.4 Comparison To Experimental Data . . . . . . . . . . . . . . . 233
6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
7 Modeling Improvements 242
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
7.2 Background on Model . . . . . . . . . . . . . . . . . . . . . . . . . . 244
7.2.1 Baseline Modeling Results . . . . . . . . . . . . . . . . . . . . 249
7.3 Model Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
7.4 Model Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . 258
7.4.1 Comparison of Experimental and Model Values of En . . . . . 259
7.4.2 Parametric Analysis for Optimal Model Fit . . . . . . . . . . . 261
7.4.2.1 Effect of Number of En Terms Used . . . . . . . . . 262
7.4.2.2 Effect of Variation of En with Pressure . . . . . . . . 265
7.4.2.3 Best-fit Model Selection . . . . . . . . . . . . . . . . 266
7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
8 Conclusion 273
8.1 Summary of Key Work and Conclusions . . . . . . . . . . . . . . . . 273
8.1.0.1 FAM Design and Fabrication . . . . . . . . . . . . . 273
8.1.0.2 Characterization Improvements - Pressure Control . 274
8.1.0.3 Characterization Improvements ? Profile Measure-
ments . . . . . . . . . . . . . . . . . . . . . . . . . . 275
8.1.0.4 Testing the Inelastic Braid Assumption . . . . . . . . 276
8.1.0.5 Testing the Cylindrical Approximation . . . . . . . . 277
8.1.0.6 Analysis of Improved Actuation Model . . . . . . . . 278
vi
8.1.1 Contributions to Literature . . . . . . . . . . . . . . . . . . . 279
8.1.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
Bibliography 286
vii
List of Tables
1.1 Comparison of direct-drive actuator technologies . . . . . . . . . . . . 6
2.1 List of swaged and comparable Flexo crimped FAMs, and their spec-
ifications for comparison to the 7/8 inch FAM developed in this work 33
2.2 Braid thickness, t, and range of achievable diameters, D, and range of
braid angles, ?, for Kevlar braids of three nominal braid sizes (defined
in 45 deg. braid orientation) (Source: Soller Composites, LLC) . . . . 52
2.3 The initial braid angle for each combination of FAM size and nominal
braid diameter. Red entries are at the threshold of the achievable
range of the braid, while blank entries are non-feasible combinations
(Reference Table 2.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.4 Construction of the 7/8 in diameter swaged FAM . . . . . . . . . . . 61
2.5 Specifications of swaged FAMs used for testing . . . . . . . . . . . . . 78
3.1 Construction and specifications of the 3/16 in and 7/8 in diameter
FAMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.2 The gains used for testing each FAM in water and air (psi pressure
input, PWM control output) . . . . . . . . . . . . . . . . . . . . . . . 107
3.3 Average pressure and standard deviation for the force response char-
acterization test at each test pressure (3/16 in FAM) . . . . . . . . . 108
3.4 Average pressure and standard deviation for the force response char-
acterization test at each test pressure (7/8 in FAM) . . . . . . . . . . 108
3.5 The force response root-mean-square error (RMSE) at each test pres-
sure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.1 Construction of the 7/8 in diameter FAM . . . . . . . . . . . . . . . . 143
4.2 The effect of the number of Fourier series fit terms used for fitting the
profile data on the maximum and Root-Mean-Square Error (RMSE)
averaged between all tested pressures . . . . . . . . . . . . . . . . . . 155
4.3 Coefficient values for the three-term Fourier Series fit performed in
Figure 4.17(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.1 Geometric dimensions calculated for the FAM in the resting length
state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
viii
5.2 Comparison of effects of braid stretch for different braid materials . . 192
6.1 Longitudinal Bladder Strain, nz (Free-contraction, 100 psi) . . . . . . 213
6.2 Circumferential Bladder Strain, nc (Free-contraction, 100 psi) . . . . . 217
6.3 Gaylord Force Term, FG (100 psi) . . . . . . . . . . . . . . . . . . . . 220
6.4 Longitudinal Bladder Stress, ?z (100 psi) . . . . . . . . . . . . . . . . 226
6.5 Longitudinal Bladder Force Term, Fbld,z (100 psi) . . . . . . . . . . . 227
6.6 Circumferential Bladder Stress, ?c (100 psi) . . . . . . . . . . . . . . 229
6.7 Circumferential Bladder Force Term, Fbld,c (100 psi) . . . . . . . . . . 230
6.8 Combined Bladder Force, Fbld (100 psi) . . . . . . . . . . . . . . . . . 232
6.9 FAM Actuation Force (Model), Fact (100 psi) . . . . . . . . . . . . . . 234
6.10 Mean Absolute Error (MAE) - Experimental versus Model Results . . 237
6.11 Blocked Force - Experimental versus Model Results . . . . . . . . . . 237
6.12 Free-Contraction - Experimental versus Model Results . . . . . . . . 237
7.1 Model Error with Respective Stress-Strain Constraints ? 100 psi (Ex-
perimental Blocked Force: 1058 lbf, Experimental Free-Contraction:
0.354) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
7.2 Model Error with Respective Stress-Strain Constraints ? Averaged
Across All Pressures . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
7.3 Material Coefficient Values, En (Fixed with respect to pressure) . . . 259
7.4 Average Model Error Using Experimental Versus Optimized En Val-
ues (E1?3 Fixed With Respect to Pressure) . . . . . . . . . . . . . . . 260
7.5 Fit Coefficient Values, En, For ?Best-Fit? Model . . . . . . . . . . . . 268
7.6 Error of the ?Best-Fit? Model with Respect to Experimental Data
for All Tested Pressures . . . . . . . . . . . . . . . . . . . . . . . . . 270
ix
List of Figures
1.1 Actuation of a contractile FAM . . . . . . . . . . . . . . . . . . . . . 2
1.2 FAMs with 1/8 in, 7/8 in, and 2 1/4 in diameters . . . . . . . . . . . 4
1.3 Examples of FAM actuated mechanisms . . . . . . . . . . . . . . . . 7
1.4 Historical images of FAMs . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 The braid acting as the working surface of the FAM transforms the
working fluid pressure forces into an actuation force expressed at each
end-fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6 Braid angle ?, and its effect on actuation force . . . . . . . . . . . . . 12
1.7 Experimental characterization of FAMs . . . . . . . . . . . . . . . . . 13
1.8 Comparison between the FAM?s geometry, and the commonly used
cylindrical approximation of its geometry . . . . . . . . . . . . . . . . 14
1.9 Triangle relationship commonly used to define the relationship be-
tween the FAM?s geometric parameters . . . . . . . . . . . . . . . . . 15
1.10 Free-body diagram of forces acting on and within a FAM [1] . . . . . 18
1.11 Overview of the FAM?s actuation force model development through
the following representative literature: Gaylord (1958) [2], Schulte
(1961) [3], Chou and Hannaford (1994) [4], Chou and Hannaford
(1996) [5], Tondu (1997) [6], Klute and Hannaford (2000) [7], Ts-
garakis and Caldwell (2000) [8], Tondu (2000) [9], Ferraresi (2001) [1],
Colbrunn (2001) [10], Davis et al. (2003) [11], Davis and Cald-
well (2006) [12], Kothera (2009) [13], De Volder (2011) [14], Solano
(2011) [15], Tiwari (2012) [16], Tondu (2012) [17], Hocking (2012)
[18], Meller (2014) [19], Robinson (2015) [20], Wang (2015) [21] . . . 23
2.1 Comparison of the refinement of wire wound, hose clamped, crimped,
and swaged end-fittings . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2 Cross-Sectional view of a swaged FAM?s end-fitting . . . . . . . . . . 36
2.3 Side and isometric views of the internal fitting . . . . . . . . . . . . . 37
2.4 The effect of initial diameter and braid angle on estimates of maxi-
mum force and maximum diameter of the FAM. The surface repre-
sents the unconstrained design space, while points indicate options in
the constrained design space . . . . . . . . . . . . . . . . . . . . . . . 50
x
2.5 The estimated force and diameter of the FAM for an assumed con-
traction of 40%. Results compare the response of FAMs with different
combinations of available braid and bladder sizes . . . . . . . . . . . 50
2.6 Dimensioned cross-section of the die used for swaging . . . . . . . . . 58
2.7 Dimensions of internal fitting used for the 7/8 in swaged FAM . . . . 61
2.8 Swage tools (bolded) and FAM assembly in alignment prior to swaging 62
2.9 Components of FAM and swage tools prepared prior to assembly . . . 63
2.10 Steps of the assembly process prior to swaging . . . . . . . . . . . . . 68
2.11 Progression of producing a swaged FAM end-fitting . . . . . . . . . . 71
2.12 Buckled swage tube of failed swaging attempt . . . . . . . . . . . . . 73
2.13 Modified Swage Assembly . . . . . . . . . . . . . . . . . . . . . . . . 76
2.14 Labeled components of modified swage assembly . . . . . . . . . . . . 76
2.15 Swaging progression with modified assembly. Per methodology, the
tube support and the standoff extension have been removed between
frames 2 and 3, prior to continuation of extrusion. . . . . . . . . . . . 77
2.16 Tested 7/8 in swaged FAMs . . . . . . . . . . . . . . . . . . . . . . . 78
2.17 Isobaric force-contraction response of the three tested FAMs . . . . . 79
2.18 Ultimate tensile failure test conducted on FAM 2 . . . . . . . . . . . 80
2.19 Burst test conducted on FAM 3 . . . . . . . . . . . . . . . . . . . . . 83
2.20 FAMs with failure due to internal fitting slipping out of end-fitting
assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.1 Illustrations of the components of the force-balance model given in
Equation 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.2 The FAMs used in this study with outer diameters of 7/8 in (top) and
3/16 in (bottom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.3 3/16 in FAM characterization testing with (black lines) and without
(colored lines) the addition of a controllable valve for accurate pres-
sure control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.4 7/8 in FAM characterization testing with (black lines) and without
(colored lines) the addition of a controllable valve for accurate pres-
sure control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.5 Pneumatic pressure control system . . . . . . . . . . . . . . . . . . . 100
3.6 Hydraulic pressure control system . . . . . . . . . . . . . . . . . . . . 101
3.7 Controller block diagrams . . . . . . . . . . . . . . . . . . . . . . . . 103
3.8 Open loop input, v(Pset), for hydraulic control of each tested FAM . . 105
3.9 Measured pressure for each force response characterization . . . . . . 109
3.10 Measured actuation force for each characterization . . . . . . . . . . . 112
3.11 Comparison between the averaged characterization data, and the
force response model . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.1 FAM with its shape profile traced in green along the edge of the braid 117
4.2 Variation of the FAM?s shape profile for (a) a single FAM in different
states of contraction, and for (b) three different FAMs in their free-
contraction states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
xi
4.3 The actual profile that has a varying radius along its length (left) ver-
sus the cylindrical approximation of this shape profile with a constant
radius along its length (right) . . . . . . . . . . . . . . . . . . . . . . 120
4.4 Cylindrical HSV color space . . . . . . . . . . . . . . . . . . . . . . . 127
4.5 Example depiction of Moore-Neighbor tracing of a boundary . . . . . 132
4.6 FAM with sensors, MTS fixtures, and masking applied prior to mount-
ing in the MTS machine . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.7 Views of FAM mounted in the MTS machine with the photogram-
metric test setup (camera, lights, and backdrop) . . . . . . . . . . . . 134
4.8 Image captured by camera for photogrammetric analysis with cali-
bration ruler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.9 The method of identifying the distance (in pixels) between 1/8 inch
increments on the calibration ruler is illustrated. Each hatch mark
has its left (blue), right (green), and then center (red) located on a
segment of the ruler image (top), and respective brightness gradient
plot (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.10 Identification of the FAM in the image (Steps 1-2) . . . . . . . . . . . 144
4.11 Identification of the end-fittings in the image (Steps 3-4) . . . . . . . 145
4.12 The resulting shape profile of the braid along the FAM?s length . . . 146
4.13 Actuation diagram of the 9-9/32 in, 7/8 in diameter test FAM . . . . 147
4.14 Comparison between contraction of image data, and displacement
recorded by the MTS machine for a single contraction cycle . . . . . . 149
4.15 Evolution of the shape profile for a complete contraction cycle (100 psi)150
4.16 Average diameter of the FAM?s shape profile for all tested pressure-
contraction states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
4.17 Comparison of assumed profiles to the measured shape profiles of the
FAM at 0, 8.75, 17.5, 26.25, and 35 (maximum) percent contraction
at 100 psi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
4.18 Comparison of the average measured diameter values, Davg, against
estimated diameters, Dest,1 and Dest,2 . . . . . . . . . . . . . . . . . . 161
4.19 Internal FAM volume with respect to contraction for all tested pressures164
4.20 Tested 7/8 in FAMs in free contraction and blocked force (resting
length) states at 100 psi . . . . . . . . . . . . . . . . . . . . . . . . . 166
4.21 Shape profile comparison of the three tested 7/8 in diameter FAMs
at 0, 8.75, 17.5, 26.25, and 35 percent contraction . . . . . . . . . . . 166
4.22 Characterization of the 2-1/4 in diameter FAM at 26 psi . . . . . . . . 167
5.1 Tested 7/8 in diameter FAM . . . . . . . . . . . . . . . . . . . . . . . 175
5.2 Characterization testing of the FAM?s actuation and geometry . . . . 177
5.3 Testing of Kevlar-49 thread samples . . . . . . . . . . . . . . . . . . . 178
5.4 Triangle relationship of FAM dimensions, and subsequent effect with
consideration of braid elasticity . . . . . . . . . . . . . . . . . . . . . 179
5.5 Analysis of the braid threads through in situ analysis . . . . . . . . . 183
xii
5.6 Comparison of the stress-strain relationship calculated from (1) the
tensile test measurements of the Kevlar threads, and (2) estimates
from the in situ measurements of the FAM?s braid. . . . . . . . . . . 184
5.7 Thread tension and length with respect to contraction and internal
pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
5.8 Estimated braid dimensions for the Kevlar-49 braid (Ebrd = 11.6 ?
106 psi) ? Comparison between elastic (colored lines) and inelastic
(black dashed lines) braid assumptions . . . . . . . . . . . . . . . . . 189
5.9 Gaylord actuation force estimates for the Kevlar-49 braid (Ebrd =
11.6 ? 106 psi) ? Comparison between elastic (colored lines) and in-
elastic (black dashed lines) braid assumptions . . . . . . . . . . . . . 189
5.10 Estimated braid dimensions for a lower stiffness braid (Ebrd = 1.16?
106 psi) ? Comparison between elastic (colored lines) and inelastic
(black dashed lines) braid assumptions . . . . . . . . . . . . . . . . . 191
5.11 Gaylord actuation force estimates for a lower stiffness braid (Ebrd =
1.16 ? 106 psi) ? Comparison between elastic (colored lines) and in-
elastic (black dashed lines) braid assumptions . . . . . . . . . . . . . 191
6.1 Comparison between the profile of the FAM, and the profile assumed
with the commonly used cylindrical approximation . . . . . . . . . . 195
6.2 Various assumed geometry profiles used for modeling of FAMs . . . . 196
6.3 Definitions of dimensions for the analyzed geometries . . . . . . . . . 200
6.4 The Triangle Relationship, enabled by the cylindrical geometry ap-
proximation, used to estimate the dimensions of the FAM between
states of contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
6.5 Comparison between the profiles defined from acquired image data,
and the Fourier-series fit equations that are used to define the profiles
for analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
6.6 Comparison of dimensions obtained from the cylindrical approxima-
tion of the measured geometry (colored lines), estimated cylindrical
geometry with elastic braid (gray lines), and estimated cylindrical
geometry with inelastic braid assumption (black dashed lines) . . . . 206
6.7 Bladder longitudinal and circumferential stress directions . . . . . . . 208
6.8 Bladder thickness calculated for each geometry (100 psi) . . . . . . . 211
6.9 Geometry-dependent longitudinal length assumed for longitudinal blad-
der strain calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 212
6.10 Longitudinal bladder strain calculated for each geometry (100 psi) . . 213
6.11 Geometry-dependent radial profile assumed for circumferential blad-
der strain calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 215
6.12 Circumferential bladder strain calculated for each geometry (100 psi) 216
6.13 Comparison of the Gaylord force term estimates obtained for the
cylindrical approximation of the measured geometry (colored lines),
the estimated cylindrical geometry with elastic braid (gray lines), and
the estimated cylindrical geometry with inelastic braid assumption
(black dashed lines) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
xiii
6.14 Latex bladder tensile test specimen . . . . . . . . . . . . . . . . . . . 222
6.15 Tensile testing of the latex bladder used in FAM construction . . . . 223
6.16 Engineering and True Stress values acquired from tensile test data . . 224
6.17 Longitudinal bladder stress, and longitudinal force term, calculated
for each geometry (all tested pressures) . . . . . . . . . . . . . . . . . 227
6.18 Circumferential bladder stress, and circumferential bladder force term,
calculated for each geometry (all tested pressures) . . . . . . . . . . . 229
6.19 Combined bladder force calculated for each geometry (all test pressures)231
6.20 Combining individual actuation force model terms to obtain overall
model results for each geometry . . . . . . . . . . . . . . . . . . . . . 234
6.21 Comparison between model and experimental test results . . . . . . . 235
7.1 Free-body diagram of the FAM used in derivation of the force model [1]244
7.2 Model results (7/8 inch FAM) using the approach of Hocking (2012)
to define En values. E1?4 values are defined with respect to pressure,
and without any constraints on their resulting stress-strain relation-
ships. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
7.3 Model results (7/8 inch FAM) using the approach of Pillsbury (2015)
to define En values. E1?4 values are constrained to be positive, and
as a linear function of pressure. . . . . . . . . . . . . . . . . . . . . . 252
7.4 Comparison of model results using different stress-strain constraints . 257
7.5 Stress-strain curves with the use of the respective constraints for the
model at 100 psi (curves vary with pressure) . . . . . . . . . . . . . . 258
7.6 Comparison between E1?3 values obtained through experimental char-
acterization, and those found through optimization. Values of E1?3
are independent of pressure. . . . . . . . . . . . . . . . . . . . . . . . 260
7.7 Model fit error ? averaged over all tested pressures ? for all tested
permutations of defining the material coefficients, En . . . . . . . . . 262
7.8 Comparison of model fits with variation in the number of En terms
used (with respect to pressure) . . . . . . . . . . . . . . . . . . . . . 264
7.9 Comparison of model fits with variation in the relationship between
En and pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
7.10 Modeling results for the ?best-fit? model, which uses terms E1?3 that
are constrained to vary linearly with pressure, and constrained to
comply with the Drucker stability criterion. . . . . . . . . . . . . . . 269
8.1 Overview of the FAM?s actuation force model development through
the following representative literature: Gaylord (1958) [2], Schulte
(1961) [3], Chou and Hannaford (1994) [4], Chou and Hannaford
(1996) [5], Tondu (1997) [6], Klute and Hannaford (2000) [7], Ts-
garakis and Caldwell (2000) [8], Tondu (2000) [9], Ferraresi (2001) [1],
Colbrunn (2001) [10], Davis et al. (2003) [11], Davis and Cald-
well (2006) [12], Kothera (2009) [13], De Volder (2011) [14], Solano
(2011) [15], Tiwari (2012) [16], Tondu (2012) [17], Hocking (2012)
[18], Meller (2014) [19], Robinson (2015) [20], Wang (2015) [21] . . . 283
xiv
List of Abbreviations
a0 Horizontal shift term (Fourier series equation)
Abrd Cross-sectional area of braid
an, bn Terms that scale the magnitude of each harmonic of order n
(Fourier series equation)
B Length of each thread of the braid
B0 Resting length of each thread of the braid
BAM Braided Artificial Muscle
Belastic Stretched braid thread length (elastic braid)
Best Estimated braid thread length in estimated cylindrical geometry
Bmeas Adjusted braid thread length defined using average of measured
diameter D?meas
BPA Braided pneumatic actuators
bn Y-intercept of linear equation that defines En
D Diameter of a FAM
D0 Diameter of a FAM at rest
Dbrd Diameter of the braid
Dbld Diameter of the bladder
Dcyl Diameter of FAM with cylindrical geometry assumed
Delastic Braid diameter - cylindrical geometry with stretched elastic braid
Dest Estimated Braid diameter found using cylindrical approximation
Di Diameter of FAM at discrete pixel row i of an image
Dmeas(z) Measured braid diameter defined along length of FAM
D?meas Average of measured braid diameter Dmeas(z)
Dmeas,bld(z) Diameter of the bladder based on Dmeas(z)
D?meas,bld Average diameter of the bladder based on D?meas
DSLR Digital Single-lens Reflex (Camera)
e Error signal
Ebrd Elastic modulus of the braid thread material
En Material stiffness coefficient matched with polynomial order n
Fact Actuation force of a FAM
FAM Fluidic artificial muscle
Fbld,c Circumferential bladder force?s effect on actuation force
Fbld,z Longitudinal bladder force?s effect on actuation force
Fexp Experimentally measured actuation force of a FAM
Ffail FAM actuation force at point of failure
FG Gaylord force of a FAM
FG,est Gaylord force calculated using estimated cylindrical geometry
FG,meas Gaylord force calculated using measured geometry
Fi Force measurement within a data set for a single isobaric test
xv
HAM Hydraulic artificial muscle
HSV Color model - Hue, Saturation, Value
ISO International Standards Organization (camera sensitivity)
Kp Proportional control gain
Ki Integral control gain
L Length of braided section of a FAM
L0 Resting length of braided section of a FAM
Lbld Length of the bladder
Lbld,0 Length of the bladder at rest
Lcyl Length of bladder based on a cylindrical geometry approximation
Lmeas Length of bladder based on the measured geometry
MAE Mean Absolute Error
mn Slope of linear equation that defines En with respect to pressure
n Order of the bladder?s stress-strain relationship
N Number of turns that a single thread of the braid makes
about the circumference of a FAM
nbrd Number of threads the braid is composed of
Ni Total number of data points at each tested pressure
Np Total number of experimentally tested pressures
ODfit,brd Internal end-fitting outer diameter for braid compression surface
ODfit,bld Internal end-fitting outer diameter for bladder compression surface
ODswg Outer diameter of swage tube after being swaged
P Pressure applied to a FAM
PAM Pneumatic Artificial Muscle
PET Polyethylene terephthalate
Pfail FAM internal pressure at point of failure
PI Proportional-integral (control)
Pmeas Measured pressure
Pset Desired set pressure
PWM Pulse-width modulation
R Radius of a FAM
R(z) Braid radius defined along length of FAM
R0 Radius of a FAM at rest
Rbld Radius of the bladder
Rbld,0 Radius of the bladder at rest
Rcyl Radius of FAM with cylindrical geometry assumed
Rest Estimated braid radius found using cylindrical approximation
RGB Color model - Red, Green, Blue
Rmeas(z) Measured braid radius defined along length of FAM
Rmeas,bld(z) Radius of the bladder based on Rmeas(z)
xvi
RMSE Root mean square error
tbld Bladder thickness
tbld,0 Bladder thickness at rest
tbrd Braid thickness
Tbrd Tension in the each braid thread
tswg Thickness of swage tube
u Control input signal
v Open-loop control input
Vbld Bladder volume
Vbrd Braid volume
Vint Internal working fluid volume of bladder
Vprof Volume of FAM as estimated from measured shape profile of FAM
w Fundamental frequency of harmonic (Fourier series equation)
wi Probability of a image pixel value being in class i (Otsu?s Method)
wp Width of each pixel in a photographic image
? Force magnification factor
brd Strain in threads of braid
c Circumferential bladder strain
z Longitudinal bladder strain
d Strain increment tensor (Drucker Stability)
?i Mean value of image pixels in class i (Otsu?s Method)
?T Mean value of all image pixels values (Otsu?s Method)
?b Inter-class variance (Otsu?s Method)
?brd Stress in threads of braid
?c Circumferential bladder stress
?z Longitudinal bladder stress
d? Stress increment tensor (Drucker Stability)
? Angle of the FAM?s braid defined with respect to the normal
of the longitudinal axis of the FAM
?(z) Braid angle defined along length of FAM
?0 Angle of the FAM?s braid in the FAM?s resting length state
?elastic Braid angle - cylindrical geometry with stretched elastic braid
?est Estimated braid angle found using estimated cylindrical geometry
?meas Braid angle defined using average of measured diameter D?meas
?meas(z) Actual braid angle defined along length of FAM
%Compbrd Percent compression of braid for swaged end-fitting
%Compbld Percent compression of bladder for swaged end-fitting
xvii
Chapter 1: Introduction
1.1 Problem Statement
Today, robotic systems have continually-increasing requirements and expecta-
tions to execute demanding tasks with precision and seamless motion. The burden
of these expectations lies especially on the actuators that drive these systems, which
must consistently deliver exact control of their position and force outputs. Many of
the most commonly used actuator types, such as electric motors and hydraulic ac-
tuators, have predictable and well understood qualities, but their overall packaging
? especially when gearsets are required ? can add a lot of weight, inertia, complex-
ity, and impedance to a robotic system. High-impedance robots produced by those
actuators types are common in industrial settings that value precision, speed, and
task repeatability; however, the stiffness and bulk of these actuators is less desirable
for use in the increasing number of robotic systems designed for human interaction
or for bio-inspired mechanisms, where an actuator?s low profile, compliance, and low
weight, are prioritized.
An alternative to bulky and rigid actuators is the fluidic artificial muscle
(FAM) soft actuator. With energy supplied by a pressurized working fluid, FAMs
provide a uniaxial contractile force and stroking motion that is enabled by its mor-
1
(a) Resting length state (deflated)
(b) Maximum contraction state (inflated)
Figure 1.1: Actuation of a contractile FAM
phing inflatable structure (Fig. 1.1). Interest in FAMs has been continually ex-
panding due to their unique set of characteristics, including their low weight, simple
construction, inherent compliance, and high specific force and specific work capabil-
ities. However, there are remaining challenges that have inhibited their widespread
use in robot systems and mechanisms. Further improvements are needed in their
design, fabrication, characterization, and modeling.
Effective application of an actuator in a manipulator requires a clear under-
standing of actuator force, stroke, spatial requirements, and required energy supply.
For FAMs, these specifications are difficult to know a priori, and must be ascer-
tained by characterization testing. FAM characteristics can be hard to predict due
to the complex interactions and nonlinear material properties of their components,
as well as variance from FAM-to-FAM that arise during fabrication.
The interdependence of the FAM?s pressure, force, contraction, and geometry
dictates that they should all be measured to obtain a complete characterization of a
2
FAM. Geometric measurements, however, are typically not obtained due to the lack
of available methods for capturing the changing bulbous shape of the FAM during
testing. Almost all FAM research has, instead, relied on a cylindrical estimation of
the FAM?s geometry. The amount of error ensconced in these estimates is uncertain,
yet these estimates are relied upon as bases for design decisions, and for multiple
inputs in all actuation force models. The bulk of FAM research has worked to
improve the accuracy of these empirically-based actuation force models, and to
take steps towards making the model more predictive. Despite these focused efforts,
multiple geometry-based assumptions have remained uninvestigated while also being
frequently cited as likely sources of error.
The goal of this research, therefore, is to produce a method of providing a
complete and accurate characterization of FAMs. With the acquisition of this char-
acterization data, the error that arises from common FAM modeling assumptions is
tested, and improvements are made to the existing actuation force model. This dis-
sertation aims to provide an additional dimension to the characterization of FAMs
for improved model design, and provide quantitative geometric data of FAMs, in or-
der to improve the ability to design mechanisms and systems around FAM actuation
characteristics.
1.2 Fluidic Artificial Muscle Actuators
This section contains an overview of fluidic artificial muscles (FAMs). A gen-
eral background, history, and details on the experimental characterization and mod-
3
eling of FAMs, is discussed to provide context for the presented research.
1.2.1 Background
A fluidic artificial muscle (FAM) is a form of soft actuator that provides a
uniaxial force. Its simple construction consists of an elastomeric bladder, a sur-
rounding braided sleeve, and two end-fittings. Upon internal pressurization, a FAM
produces a contractile or extensile actuation force depending on the orientation of its
braided sleeve. Because contractile FAMs are able to produce higher forces and are
less prone to buckling than extensile FAMs, contractile FAMs are more commonly
used [22]. The term FAM, therefore, typically refers to the contractile actuator type
that is the focus of this work.
The simple construction of FAMs make them lightweight(< 0.5 lbm), low cost,
and tolerant to bending and axial misalignments. Their soft construction and in-
herent compliance provide FAMs with their muscle-like qualities that make them
especially attractive for use in applications with human interaction or bio-inspired
designs [10,23?26].
Figure 1.2: FAMs with 1/8 in, 7/8 in, and 2 1/4 in diameters
4
FAMs have been developed with diameters ranging from 1/16 inch to 2 1/4 inch
[15]. With typical operating pressures below 100 psi, they produce actuation forces
ranging from tens of pounds to thousands of pounds, while also being able to provide
length contractions of up to 30-40% (Fig. 1.2) [27]. Coupled with their low weight,
these high force and stroke capabilities give them high force-to-weight ratios (1000s
lbf/lbm), and specific work capabilities (1000s ft-lbf/lbm) [28]. Table 1.1 shows the
maximum strain (stroke divided by length), actuation stress (force divided by cross-
sectional area), and specific work (work divided by mass) of several types of direct-
drive actuators [29]. In comparison to the other actuator types, FAMs are able to
produce a good balance of actuation strain and stress (stroke and force) capabilities.
The direct-drive actuation of FAMs also provides them benefits over actuators that
require gearboxes, such as motors; FAMs are backdrivable, have zero backlash, and
are lower in weight and complexity. Researchers have reported specific power values
of FAMs ranging from 500-2000 W/kg (225-900 W/lbm) [30], while electric motors
are just on the order of 100 W/kg (50 W/lbm) [31]. With all of the aforementioned
positive attributes of FAMs, it should be noted that challenges remain in refining
the FAM?s durability, modeling, and control of position and pressure.
The FAM?s attributes make it well-suited for a wide-range of applications.
Typical applications include prosthetics and assistive technologies (Fig. 1.3(a))
which can take advantage of the low weight and compliance of FAMs [23,28]. Among
other attributes, their mechanical flexibility also makes them popular and uniquely
suited for use in bio-inspired mechanisms and appendages (Fig. 1.3(b)) [24]. Their
lack of moving parts, seals, lubrication, and electric power, makes them operable in
5
Actuator Maximum Actuation Specific Maximum
Type Strain Stress, Work, Frequency,
MPa (psi) J/kg (ft-lbf/lbm) Hz
Shape Memory 0.07 700 (101,500) 4,500 (1,500) 7
Alloy (SMA)
Piezoelectric 10?4 10 (1,450) 1 (0.33) 107
Magnetostrictive 0.002 10 (1,450) 20 (6.7) 107
Solenoid 0.4 0.1 (14.5) 5 (1.7) 80
Hydraulic 1 70 (10,150) 35,000 (11,700) 100
Cylinder
Pneumatic 1 0.9 (131) 1,200 (400) 100
Cylinder
FAMs 0.4 16 (2300) 4,400 (1,470) 100
(UMD)
Human Muscle 0.5 0.4 (58) 0.8 (0.27) 100
(Adapted from Huber et al., 1997) [29]
Table 1.1: Comparison of direct-drive actuator technologies
environments that are difficult for other types of actuators [32,33]. Unique examples
include aerospace [34?37] (Fig. 1.3(c)), subterranean (DARPA Underminer), and
subsea [24] applications.
1.2.1.1 History
The invention of the fluidic artificial muscle was reached through the evolution
of many similar technologies that preceded it. Reportedly, the first mechanisms
of this general type were developed in the 1930s by a Russian inventor named S.
Garasiev [39]. Similar devices can be found in patents from the era, including: a
rubber tube covered in a braided fiber exterior for use in mining (Pierce, 1940) [40];
a fluid-driven linear tension device (Haven, 1949) [41]; and an elastic diaphragm
6
(a) Exoskeleton with FAM actuated assistance [28]
(b) Brittle star inspired FAM actu- (c) FAM actuated trailing-edge flap
ated appendages [24] [38], and morphing wing concept [36]
Figure 1.3: Examples of FAM actuated mechanisms
covered with stiff threads (Morin in 1953) [42]. Richard Gaylord is often attributed
with the invention of the fluidic artificial muscle in 1958 with his patent titled Fluid
Actuated Motor System and Stroking Device (Fig. 1.4(a)) [2]. This patent included
the first equation to describe the force-pressure relationship of FAMs. In 1960,
Joseph McKibben used FAMs in an assistive device that he developed for his polio-
stricken daughter (Fig. 1.4(b)) [43]; Life magazine highlighted his device which
helped popularize FAM technology. However, FAM development stalled not long
7
after this due to needed improvements in pneumatic valve and storage technology,
along with the emergence of the miniaturized electric motor [17]. It was not until
the 1980s that industrial automation applications renewed an interest in FAMs.
Bridgestone developed the first commercialized FAM called the Rubbertuator, which
was used in the actuation of an industrial robotic arm (Fig. 1.4(c)) [44, 45]. Only
a few companies, including the Shadow Robot Group [46], and Festo [47], have
since developed commercialized versions of FAMs. Widespread adoption of FAMs
in industrial and commercial settings has been stymied by the need for further
refinement and maturation of the technology.
From the mid-1990s to the present, development of FAMs has occurred mainly
in research settings, and has been spurred on by rising interest in humanoid and
bioinspired robotics that FAMs are well-suited for. Research efforts have focused on
improvements in the modeling and control of FAMs, and on making adjustments to
the construction and selection of materials, in attempts to improve their performance
and durability.
Over time, slight variations in the construction, form, or application of FAMs,
has garnered them a variety of names in literature. While fluidic artificial muscle
(FAM) is the preferred nomenclature in this work, they are also often called McK-
ibben actuators. Other names in literature include: braided artificial muscle (BAM),
fluidic flexible matrix composite, rubber muscle actuator, and rubbertuator (trade-
marked by Bridgestone Company). Some names specify the working fluid used,
including: pneumatic artificial muscle (PAM), hydraulic artificial muscle (HAM),
braided pneumatic actuator (BPA), and air muscle.
8
(a) Patent submitted by R.H. Gaylord [2] (b) Device developed by McKibben to as-
sist his polio stricken daughter
(c) Rubbertuator robotic arm (d) Festo commercial FAMs
Figure 1.4: Historical images of FAMs
9
1.2.1.2 Working Mechanism
A FAM consists of an elastomeric bladder that contains the pressurized work-
ing fluid, a surrounding helically-braided sleeve that acts as the main load-bearing
component, and two end-fittings that act as mounting points at each end of the
FAM; one of the two end-fittings also acts as a conduit for the working fluid to flow
into and out of the FAM. Upon pressurization (by air, water, oil, etc.), the bladder
expands radially against the stiff braided sleeve. Radial expansion of the sleeve
causes the sleeve to contract axially, and the radial pressure force from the bladder
is transformed into an axial force which is transmitted through the sleeve and then
end-fittings (Fig. 1.5). Motion of the braid is enabled by the changing angle of its
threads that occurs as the FAM contracts in length; the braid angle ? ? defined with
respect to the normal of the longitudinal axis ? begins at a maximum in the resting
length state, and reduces as the FAM contracts in length and expands radially (Fig.
1.6(b)).
The braid angle and concomitant radius of the braid ? both defined by the
length state ? largely determine the actuation characteristics of the FAM. Actuation
force can be estimated as a function of the FAM?s instantaneous internal pressure
P , braid angle ?, and radius R, using the following equation:
FG = P?R2(2tan2? ? 1) (1.1)
This equation, typically attributed to Gaylord [2], can be separated into two parts:
the P?R2 term defines the force of a cylinder actuator, and the (2tan2? ? 1) term
10
defines the force magnification (?) that the braid provides. The amplification of
actuation force that the FAM provides over a traditional cylinder actuator (where
? = 1), is enabled by the larger working area of the FAM (i.e., the braid surface)
in comparison to the working area of a traditional cylindrical actuator (i.e., the
piston area) (Fig. 1.5). As seen in Figure 1.6(a), the multiplication factor increases
dramatically at higher braid angles. A typical initial braid angle of 70 degrees
provides a 14.1 magnification of force over that of a cylindrical actuator with an
equivalent initial diameter. Also of interest is the braid angle value that returns a
magnification factor of zero; this braid angle (? = 35.3 deg.) demarcates whether a
FAM will contract or expand in length upon pressurization, and therefore, defines
the principle difference between contractile and extensile FAMs.
Figure 1.5: The braid acting as the working surface of the FAM transforms the
working fluid pressure forces into an actuation force expressed at each end-fitting
1.2.2 Experimental Characterization
The actuation of a FAM is characterized by defining its pressure-dependent
force-contraction response. The response is typically measured on a test apparatus
such as the material test machine shown in Figure 1.7(a). This machine directs a
controlled motion of the FAM, while pressure is controlled with a regulator, and
force is measured by a load cell. Testing is performed by quasi-statically cycling the
11
(a) Braid angle dependent magnification of actuation (b) Decrease in braid angle between rest-
force. Estimated actuation force equals P?R2? ing length and maximum contraction
states (threads marked on FAM)
Figure 1.6: Braid angle ?, and its effect on actuation force
FAM between its resting length and maximum contraction states at fixed pressures.
A complete characterization requires repeating this test for multiple discrete set-
pressures within the standard operating range of the FAM.
The experimental characterization results shown in Figure 1.7(b) illustrate the
typical actuation characteristics of FAMs. A FAM provides its maximum force when
in its resting length state, termed the blocked force state; its force decreases with its
contraction until a pressure-dependent maximum contraction state, also known as
a free-contraction state, is reached. Actuation force is nominally proportional with
pressure, as implied by the Gaylord force equation. Each isobaric test exhibits hys-
teresis due to friction within the FAM, with higher forces observed during eccentric
motion. The eccentric and concentric motion curves are typically averaged to create
a single curve at each pressure to simplify modeling and analysis.
While experimental characterization of a FAM typically entails defining its
isobaric force-contraction response, other useful characterizations include measur-
12
ing the change in volume with contraction [19], and failure testing to quantify the
maximum strength or pressure capabilities of a FAM [37].
1000 100 psi
80 psi
60 psi
800
40 psi
20 psi
600
400
200
0
0 0.1 0.2 0.3 0.4
Contraction, L /L
0
(a) Test machine often used for (b) Actuation response of a 7/8 inch diameter FAM
characterization testing of FAMs
Figure 1.7: Experimental characterization of FAMs
1.2.3 Modeling
Obtaining accurate models of a FAM?s actuation is pivotal to effectively design-
ing a mechanism for FAM actuation and for accurate control of the FAM. Because
of this, much of the research has focused on incremental improvements to the accu-
racy of FAM models. While acquiring models of the FAM?s actuation force response
is often the main focus of these studies, modeling the changing geometry of the
FAM ? which acts as an input into the actuation force model ? is also of primary
importance.
Despite their complex geometry, virtually all modeling and analysis efforts
have assumed a cylindrical geometry (Fig. 1.8) [1?3,5,8?10,13,16,18?21]. This sim-
13
Force, lb
plifying assumption assumes a constant diameter and braid angle along the length of
the FAM. By using a cylindrical approximation, cylindrical dimensions of the FAM
are easily estimated for all states of contraction, and have been used as an input in
almost every model formulated for FAMs. The cylindrical assumption has been used
in the derivation of models representing the FAM?s internal volume [19], friction [9],
bladder strain [13], and actuation force [17]. Researchers have long acknowledged
this inaccurate representation of FAM geometry ? inherent when using the cylindri-
cal approximation ? is a likely source of modeling error [13]. Few attempts, however,
have been made to address this issue.
An overview of the standard modeling practices of FAMs is provided in the
following subsections.
(a) Shape profile of a FAM?s geometry (b) Isometric view of a
FAM?s geometry
(c) Cylindrical approximation of a FAM?s geometry (d) Isometric view of a
cylindrical geometry
Figure 1.8: Comparison between the FAM?s geometry, and the commonly used
cylindrical approximation of its geometry
14
1.2.3.1 Geometric Model
An accurate measure of a FAM?s geometry is important for: (1) ensuring
proper clearances within an actuated mechanism, and (2) ensuring accurate inputs
for the actuation force model. FAM research has lacked a method of accurately
measuring a FAM?s geometry, and has therefore relied on a cylindrical approximation
for the model to extract estimates of the geometry in all length states.
The geometry of the FAM is defined by its instantaneous length L, diameter
D, braid angle ?, along with braid parameters defining the length of each braid
thread B (assumed fixed), and the number of turns of each braid thread about the
circumference of the FAM N (fixed) (Fig. 1.9). The braid diameter and angle are a
function of both the position along the FAM, and the contraction state of the FAM.
Notably, the geometry of the FAM unpredictably varies between otherwise identical
FAMs.
Figure 1.9: Triangle relationship commonly used to define the relationship between
the FAM?s geometric parameters
The cylindrical geometry assumption fixes D and ? along the length of the
15
FAM; the geometry parameters can then be related through a triangle relationship
[4] (Fig. 1.9), which allows for D and ? to be estimated for all length states L, using
the following equations:
?
B2= ? L
2
D (1.2)
?N( )
? = sin?1 L (1.3)
B
Values of B and N are typically estimated using the the resting length dimensions
L0, D0, and ?0, with the following equations:
= L0B ( ) (1.4)sin ?0
N = L0 ( ) (1.5)?D0tan ?0
Instantaneous bladder thickness tbld ? which varies with straining of the bladder ? is
also estimated using the cylindrical approximation through the following equation:
?
2 Vbldtbld = R? R ? (1.6)
?L
where R is the radius (R = D/2), and Vbld is the bladder volume which is assumed
incompressible (i.e., fixed) and is estimated in the resting length state configuration.
Bladder radius is assumed equivalent to braid radiusR due to the negligible thickness
of the braid.
Estimates of bladder internal volume (i.e., the working fluid volume) are also
made by assuming an estimated cylindrical geometry. Estimation of instantaneous
volume can be a required input for some actuation force models, but is also useful
for estimating fluid flow rate requirements to aid in pressure control of the FAM,
and for sizing fluid handling equipment and storage. Very few researchers have
16
experimentally characterized internal volume due to the specialized test equipment
required to do so [19].
1.2.3.2 Actuation Force Model
Models relating the force, contraction, and actuation pressure, have been de-
veloped using both force-balance [1] and energy method derivations [4]. Basic forms
of the developed models are largely independent of the derivation method used, how-
ever the force-balance method is more widely adopted because it has been shown to
provide more opportunities for model refinement.
The force-balance form of the actuation force model was originally derived
by Ferraresi (2001) [1]. Many authors have since improved upon this model by
adding or adjusting terms to better represent the FAM?s dimensions, or components?
interactions and material properties. The model takes a mechanistic approach, and
is based on a free-body diagram which accounts for pressure, bladder, and braid
forces (Fig. 1.10). From this, force-balance equations for equilibrium in the FAM?s
longitudinal (z) and circumferential (c) directions are obtained:
PRL = ?ctbldL+NTbrdcos(?) (1.7)
F 2act + ?R P = ?z(Vbld/L) + Tbrdsin(?) (1.8)
These equations include: unknown values for actuation force (Fact) and braid
thread tensions (Tbrd); a working fluid parameter for pressure (P ); bladder param-
eters for volume (Vbld), thickness (tbld), longitudinal stress (?z), and circumferential
stress (?c); and braid geometry-based parameters for active length (L), radius (R),
17
Figure 1.10: Free-body diagram of forces acting on and within a FAM [1]
braid angle (?), and number of turns of each thread about the circumference of
the FAM (N). Through substitution for the unknown value of Tbrd, the system of
equations is used to solve for Fact. The following expression for actuation force of
the FAM ? as a function of length L ? is then defined as follows:
( ) ( 2 )
F = P?R2(2 Vbld tbldLact tan2? ? 1) + ?z ? ?c 2 (1.9)L ?RN2
The model has three basic terms. The first term ? which is the Gaylord force equa-
tion ? accounts for the actuation force produced by the internal fluid pressure acting
on the braid. The second and third terms of Equation 1.9 are bladder force terms;
these terms account for actuation force losses due to the stretching of the bladder
during actuation. Attempts at making model improvements have often focused on
these bladder force terms, and specifically in accurately estimating bladder stresses
?c and ?z which rely on accurate and difficult to obtain models of the bladder?s hy-
perelastic stiffness and geometry-dependent strain estimates. Also note that many
of the force model?s inputs rely on the geometry estimation model, and therefore
inherits the uncertainty in the geometry model?s accuracy.
18
1.2.3.3 Review of Literature
A literature review provides insight on past developments of the FAM?s actua-
tion force model. Figure 1.11 provides specific actuation mechanisms or parameters,
and the history of their inclusion amongst twenty-one representative works dating
back to the first model, in 1958 (the Gaylord equation) [2]. Since the FAM?s in-
ception, all FAM models have used the Gaylord force equation, and the cylindrical
approximation of the FAM?s geometry that it is based on, as a foundation for model
development. Energy methods were used for model derivations until the develop-
ment of the force-balance approach by Ferraresi (2001) [1]; the force balance ap-
proach has dominated FAM model development ever since, due to its more flexible
and intuitive form that has provided better model accuracy.
Model development has focused on the addition of parameters and correction-
terms to better capture effects of the bladder or braid. As previously noted, most
additions to the model have been based on a cylindrical approximation of the FAM?s
geometry.
Energy losses due to bladder strain, and the effect that has on actuation force,
has been the focus of much of the research over the past decade [7, 13, 17, 18, 48].
Efforts have focused on accurately characterizing the stiffness of the hyperelastic
bladder; stiffness of the bladder can be difficult to capture due to the many depen-
dencies of the bladder?s stiffness, and because of the difficulty in experimentally test-
ing hyperelastic materials. Models have included bladder thickness parameters [13]
? along with a parameter that defines the bladder?s inflation initiation pressure (i.e.
19
deadband pressure) [14, 18] ? since their introduction in the 2000?s. The chart in
Figure 1.11 also shows that all models have assumed a cylindrical approximation of
the bladder?s dimensions to obtain the estimates of bladder strain, which are then
used to calculate bladder stress and force values. This assumption discounts the
variations in bladder strain that result from the varying radius of the FAM along
its length.
Researchers have paid less attention to better-representing the braid?s effects
within the model. Efforts to better model the braid?s effect on actuation force have
focused on friction from braid-braid and braid-bladder interactions. The effects of
braid elasticity have largely been ignored; the inelastic braid assumption is com-
monly assumed, despite past researchers having demonstrated the effect that braid
elasticity can have on the geometry and actuation force of the FAM [11, 13]. The
chart also shows how frequently the rounded geometry of the braid is assumed to
be cylindrical. This assumption is often made despite commonly being cited as a
probable source of modeling error [4,8,9,13,16,18?20,49,50]. More recently, a few re-
searchers have applied shape corrections to the cylindrical approximation (based on
assumed shape profiles near each end-fitting), however, the accuracy of those correc-
tions has gone relatively untested [8, 13, 16?18,20]. The geometry values calculated
using the cylindrical approximation have remained unconfirmed experimentally for
years, likely due to the lack of available methods for accurately measuring the FAM?s
geometry.
It should also be noted that the accuracy of the force model is only as good
as the accuracy of the experimental characterization that it is based on. Character-
20
ization testing of the FAM?s response often assumes an isobaric contraction of the
FAM. Therefore, accurately controlling a constant pressure to the FAM is impera-
tive to an effective characterization. Some past research has had inconsistencies in
characterizations or exhibited responses that did not match modeled results; these
instances have been attributed to inaccuracies in pressure control during charac-
terization testing [16]. The vast majority of research that has performed isobaric
characterization testing, as indicated in Figure 1.11, has not portrayed accurate
control of pressure (including pressure tolerances), as should be done for assurance
of the accuracy of their experimental findings.
Overall, the chart in Figure 1.11 highlights physical aspects of the FAM that
have been neglected in the model. Almost all FAM models, developed over the past
six decades, have relied on two modeling assumptions ? the cylindrical approxima-
tion, and the inelastic braid assumption ? as methods of simplifying analysis. Both
assumptions have been questioned on multiple occasions, but have yet to be com-
prehensively addressed, likely due to the lack of a method for accurately measuring
the FAM?s geometric profile.
This work acts to address areas where the current actuation force models are
lacking: the representation of the braid?s elasticity, and the effects of the bladder
and braid?s non-cylindrical geometry. Improvements are first made to the methods
of experimentally characterizing FAMs; accurate pressure control is assured, and
then a method of accurately measuring the geometric shape profile of the FAM is
developed. The acquired geometric measurements of the FAM are then used to test
the cylindrical approximation and inelastic braid assumption. Uncertainty in the
21
model ? due to inaccurate or incomplete characterizations, and the aforementioned
modeling assumptions ? is thereby substantially reduced, and an updated version
of the FAM model is then analyzed for potential improvements in accuracy.
1.3 Contributions of Dissertation
The objective of this research was to improve the characterization and model-
ing of FAMs. Special emphasis was placed on acquiring geometric measurements of
the FAM, and then using that data to test long-held modeling assumptions. Sev-
eral contributions were made to the state-of-the-art in FAM design, fabrication,
characterization, and modeling.
Previous work developed a swaged end-fitting design that improves a FAM?s
durability, working lifetime, and refinement. Swaged end-fitting FAMs had been
made in a limited number of diameter sizes (5/8, 3/4, and 1 inch diameters), which
limited the number of applications they could be applied to [51]. In this work, a
method of scaling the design and fabrication of swaged end-fitting FAMs is devel-
oped, and is used to produce a new 7/8 inch diameter FAM that improves upon the
force and stroke capabilities of previous swaged end-fitting FAMs. The developed
7/8 inch FAM was used in this research, and has since been applied to multiple
robotic applications.
The accuracy of a model is only as good as the experimental data that it
is based on. An accurate characterization of a FAM, therefore, is important to
producing an accurate model of its actuation response. Prior to this research, char-
22
23
Figure 1.11: Overview of the FAM?s actuation force model development through the following representative literature: Gaylord
(1958) [2], Schulte (1961) [3], Chou and Hannaford (1994) [4], Chou and Hannaford (1996) [5], Tondu (1997) [6], Klute and
Hannaford (2000) [7], Tsgarakis and Caldwell (2000) [8], Tondu (2000) [9], Ferraresi (2001) [1], Colbrunn (2001) [10], Davis et al.
(2003) [11], Davis and Caldwell (2006) [12], Kothera (2009) [13], De Volder (2011) [14], Solano (2011) [15], Tiwari (2012) [16],
Tondu (2012) [17], Hocking (2012) [18], Meller (2014) [19], Robinson (2015) [20], Wang (2015) [21]
acterization of a FAM?s response was done by setting a fixed feed-pressure to the
FAM, and measuring its actuation force at different states of contraction. This
method of characterization often had two faults that would lead to errors in subse-
quent modeling efforts: they (1) often lacked the assurance of accurate control of
the assumed isobaric feed-pressure during testing; and (2) did not include a mea-
sure of the FAM?s geometry. With standard pressure-regulators, the pressure to the
FAM can drift with working fluid flow into and out of the FAM; if the pressure is
assumed fixed, this can result in inaccurate predictions of the FAM?s force response.
Secondly, the geometry of the FAM is typically estimated based on a cylindrical
approximation; this can result in error of the geometric parameter inputs used for
the actuation force model, but also leaves uncertainty in design requirements such
as the required space allowance for the actuator within a mechanism, or the working
fluid flow-rate requirements for actuation of the FAM.
This present work provides improvements to the characterization of FAMs.
Accurate pressure control systems are developed ? for both air and water working
fluids ? that ensure accurate control of pressure as the volume of the FAM fluctu-
ates during eccentric and concentric motions of the FAM. These systems ensured
accurate characterization data for the modeling efforts performed in this research,
but were also used to prove the independence of the quasi-static actuation response
to changes in the compressibility of the working fluid used. This work also presents
a novel photogrammetric method to accurately measure FAM geometry during test-
ing. The photogrammetric method can be used to generate accurate geometric
parameter values for the force model, but also provides a non-intrusive method
24
of obtaining volume measurements of the FAM ? important for defining actuation
efficiency and flow-rate requirements. These geometric measurements provide an
opportunity to improve the understanding of FAM actuation, and aid in making
model improvements ? as is done in this work.
Previous FAM actuation models have long relied on simplifying assumptions.
Two of those simplifications ? the cylindrical approximation, and the inelastic braid
assumption ? have been used in actuation response models since the development of
the first FAM model in 1958. The cylindrical approximation neglects the curvature
of the FAM by approximating the geometry of the FAM as a cylinder. The inelastic
braid assumption neglects the compliance of the braid, and the effect that this
could have on the geometry and force of the FAM. Each of these assumptions have
remained ? despite often being cited as likely sources of error ? in part, due to the
lack of methods for accurately quantifying their effects on the actuation of FAMs.
These assumptions, therefore, have persisted as a source of uncertainty in FAM
models.
In the presented work, photogrammetric measurements of FAM geometry
make it possible to quantify the error in the cylindrical approximation and inelas-
tic braid assumption. Furthermore, these geometric measurements can be used in
place of cylindrical approximation values in geometry and actuation force models
to improve their accuracy. The geometric measurements also enable the magnitude
of braid stretch during actuation to be quantified in situ, and provide an avenue
for analyzing the effect of braid stretch on the geometry and actuation force of the
FAM. Removing the uncertainties instilled into FAM models by the assumptions
25
tested here improves the accuracy of the FAM model?s outputs, and also provides
new opportunities for further improvement.
1.4 Overview of Dissertation
This dissertation is organized into eight chapters, each presenting a different
aspect of the research performed.
Chapter 1: Introduction. This chapter presents the motivation for the research.
An overview of FAMs is provided, and current methods of characterization and
modeling them are introduced. Assumptions within current characterization and
modeling methods are presented. A literature review of past modeling improve-
ments is provided for context of this research.
Chapter 2: Development of the Swaged End-Fitting FAM. The design,
fabrication, and testing of a new 7/8 inch diameter swaged end-fitting FAM is pre-
sented. A standard procedure for scaling the design and fabrication of FAMs with
swaged end-fittings is introduced. Adjustments are made to the swaging process
to ensure efficient, safe, and repeatable fabrication. The 7/8 inch diameter FAM
developed in this chapter is then used for the research performed in the subsequent
chapters.
Chapter 3: Improvements in FAM Characterization Through Accurate
Control of Pressure. Pressure control systems ? for both air and water working
fluids ? are developed to ensure accurate pressure control during characterization
testing of FAMs. The assurance of fixed set-pressures during testing is paramount
26
to the production of accurate empirically-based force response models. A detailed
description of each pressure control system is presented, and are then used to char-
acterize both a large and small-scale FAM. The new pneumatic control system is
compared to the previous system, and subsequent improvements in the force char-
acterization are reported. Data from the air and water tests are compared to test
if the FAM?s quasi-static response is independent of working fluid. The acquired
characterization data is used for analysis in subsequent chapters.
Chapter 4: A Photogrammetric Method for Measuring FAM Shape. This
chapter presents a method of measuring a FAM?s geometry (i.e. shape profile).
The photogrammetric method developed for acquiring measurements is detailed in-
cluding the test setup, image analysis, and data analysis techniques. Geometric
measurement data of a FAM is acquired, and then used to perform a preliminary
analysis. Initial analysis of the data brings into question previous modeling assump-
tions ? including the cylindrical approximation and inelastic braid assumptions ?
which motivates the work of subsequent chapters.
Chapter 5: The Inelastic Braid Assumption. In this chapter, the validity
of the commonly used inelastic braid assumption is tested. Stretch in the braid is
quantified using the shape profile measurement data acquired in Chapter 4. This
data is used to approximate the stiffness of the braid?s threads while on the FAM,
and is compared to the stiffness measured through a tensile test performed on a
single braid thread. The effect of the braid?s elasticity on geometry and actuation
force is investigated for the tested Kevlar braid, and for a hypothetical braid with
lower stiffness.
27
Chapter 6: The Cylindrical Approximation Assumption. The error in the
commonly used cylindrical approximation of the FAM?s geometry is tested through
comparison with measured geometry data of a FAM. Since geometric parameters
serve as inputs to the actuation force model, the measured and cylindrically as-
sumed geometries are compared for their effect on estimated force. This comparison
is used to gauge potential improvements in model accuracy through the use of the
measured geometries.
Chapter 7: Model Performance with Reduction in Modeling Assump-
tions. In this chapter, the force model is updated ? based on the findings from
previous chapters ? and then its performance is analyzed. To ensure the model accu-
rately represents the FAM?s force response, the model uses optimized fit-parameters
? as used in many other recent works ? that are embedded into the model?s blad-
der material stiffness terms. Optimization constraints on the fit-parameters from
previous works are made more realistic while also improving model accuracy. The
optimized fit-parameter values are compared to the experimental material coefficient
values found in Chapter 6, and are also modified and analyzed for their effect on
model accuracy.
Chapter 8: Conclusion. This final chapter summarizes the key conclusions from
this research. The original work contribution to literature is identified, and future
work topics are discussed.
28
Chapter 2: Design, Fabrication, and Failure Testing of a Swaged
End-fitting FAM
2.1 Introduction
Despite the promising and unique performance capabilities of fluidic artifi-
cial muscles (FAMs), expanding their adoption hinges on improving their reliability,
their consistent and efficient fabrication, refinement in their design, and scalability
in size. The ability of FAMs to serve as a viable option for robotic and actuated
mechanism applications depends on whether they can prove to be reliable and ver-
satile in comparison to common forms of actuation such as pneumatic and hydraulic
cylinders, and electric motors. Much of the current work with FAMs remains in re-
search settings, where refinements in design, size, and cost are not often prioritized.
Refinements to FAM construction, especially in the end-fitting design, have
improved markedly through the years. Earlier FAMs ? including many used just
for research purposes ? have used end-fittings held together by wire winding or
hose clamps. Few options of FAMs exist today that offer a refined and low-profile
end-fitting design. The crimped fittings of Festo FAMs [47] have a refined ? but
somewhat bulky ? design, while the swaged end-fitting design developed jointly
29
through work done by the University of Maryland and InnoVital Systems Inc. [37,51]
has been shown to produce a refined and strong end-fitting construction (Fig. 2.1).
Research on swaged FAMs, resulting in a patent on the technology [51], has shown
that the swaged end-fitting design provides dramatic increases in the strength and
durability of FAMs while providing a functional and aesthetic form factor. Woods
(2012), for example, demonstrated the capabilities of a swaged 5/8 in diameter FAM
that could withstand pressures beyond 1000 psi, and forces of 3170 lb. The durability
of this FAM was also demonstrated by successfully cycling a 1 in swaged FAM for
120 million cycles with only minimal wear, compared to the 10 million cycle service
life of the Festo FAMs [47], and a maximum of 14,000 cycles for the next best FAM
after that [37].
Figure 2.1: Comparison of the refinement of wire wound, hose clamped, crimped,
and swaged end-fittings
FAMs with swaged end-fittings have been previously developed with nominal
30
diameters of 5/8 in, 3/4 in, and 1 in using the same approximate design formula
(Table 2.1). Their force capabilities increase with diameter, but are also largely
effected ? as are their contraction capabilities and radial expansion ? by the braid
angle of the braided sheath that is a principal component of the FAM. The sizing
of the components of the FAM, critical to the FAM?s resultant performance, is
also constrained by what is available through manufacturers and vendors. This
constraint limits the number of possible component combinations that could be
constructed to achieve a desired actuation response, and serves as a unique design
challenge of FAM development.
Numerous calls for research proposals and projects involving actuated mecha-
nisms have requirements that would necessitate the development of a FAM actuator
that improves upon previously developed swaged FAMs. We considered the follow-
ing design requirements:
? Actuation force of about 1,000 lbf
? Percent contraction greater than 30%
? Minimize radial expansion and respective change in internal volume
? Operating range of 0?100 psi
Secondary Considerations include:
? Minimize deadband pressure
? Maximize end-fitting internal airflow cross-sectional area
A newly-designed FAM should produce a maximum actuation force of about 1,000
lbs, and a contraction of greater than 30%. The FAM will be applied to applications
where the pressurized air will come from shop air supply systems that commonly
provide pressures of about 100 psi for use with pneumatic tools. Therefore, the
31
operating range of the designed FAM will be 0 to 100 psi. A factor-of-safety of at
least two is desired for the maximum pressure and actuation force to ensure safety
and robustness of the FAM. These requirements should be met while minimizing the
FAM?s radial expansion and respective volume required for actuation to facilitate the
application of FAMs in mechanisms with space constraints. The FAM is expected
to have refined end-fittings, and good reliability and durability traits as have been
provided by previous swaged FAMs. Manufacturing the FAM should be an efficient
and repeatable process, with the goal of producing FAMs having nearly identical
actuation characteristics to improve the interchangeability of FAMs, as well as the
efficiency and coherence of FAMs used in parallel such as for variable recruitment
schemes. Development of detailed methods for readily scaling the production of
swaged FAMs can also aid in future development.
This research details the development of a 7/8 in swaged FAM that improves
upon the capabilities of previous FAMs to achieve the above-stated objectives.
Through the development process, a methodical approach to FAM sizing and fabrica-
tion is presented, which includes improvements devised and applied to methods used
previously in the fabrication of swaged FAMs. A method of determining the required
sizing for a FAM for desired actuation attributes ? while considering practicalities
and constraints on sizing ? is provided as a general guide for FAM development, and
it also provides a background to the motivation of developing the 7/8 in FAM. A
breakdown of the swaged FAM?s components is provided. The process of fabricating
a swaged FAM is detailed as an enumerated sequence to aid in future swaged FAM
design efforts. Difficulties with both fabrication inconsistencies and failures led to
32
the development of improvements to the swaging process that are introduced and
detailed in this research. Finally, tests were performed on three FAMs ? fabricated
using these improvements ? to demonstrate the consistency and integrity of the
FAM?s performance, and to test their failure strength to determine safety factors.
The failure mechanisms discovered through these tests were investigated for possible
improvements.
FAM Max Max Percent Braid Weight, lbm
Diameter, in Force, lbf Contraction Angle
5/8 in, UMD [52] 600 (90 psi) 34 % (90 psi) 72.5? 0.17
3/4 in, UMD [20] 950 (100 psi) 36 % (100 psi) 76.5? 0.18
0.79 in, FESTO [47] 337 (87 psi) 25 % (87 psi) 64.5? (est.) 0.52 (est.)
7/8 in, UMD 1049 (100 psi) 35 % (100 psi) 71.6? 0.19
1 in, UMD [37] 450 (90 psi) 27 % (90 psi) 55.5? 0.25
1.58 in, FESTO [47] 1037 (87 psi) 25 % (87 psi) 62.0? (est.) 1.83 (est.)
Table 2.1: List of swaged and comparable Flexo crimped FAMs, and their specifi-
cations for comparison to the 7/8 inch FAM developed in this work
2.1.1 Background
The three essential components of a FAM are an elastomeric bladder, a sur-
rounding braided sheath, and some form of rigid fitting at each end. The bladder
serves to encapsulate the pressurized working fluid, while the braid serves to trans-
mit the radial pressure force into an axial force expressed at each end-fitting. These
rigid end-fittings serve multiple roles for the FAM: They provide a rigid point of
connection for the braid and bladder at each end; they serve as rigid mounting
points for external hardware connections; they provide an air tight seal, and; they
serve as a conduit for the working fluid to flow into and out of the FAM.
33
Various methods have been pursued in efforts to form a viable connection
between an end-fitting, and the braid and bladder. Most methods include a rigid
internal fitting that sits internal to the braid and bladder at each end, and an
external component that encircles the braid, bladder, and internal fitting, and acts
to compress and hold the bladder and braid against the internal fitting to provide
a non-slip and pressure-tight seal. The cylindrical internal fitting often has some
type of textured surface ? such as barbs ? to increase the resistance to slipping
of the braid and bladder. Examples of external clamping components used include
hose clamps [1,52], wire windings [38], compression fittings [38], barbed fittings [19],
crimped tubes [18], and swaged tubes.
The end-fitting construction must balance between providing sufficient clamp-
ing force while not creating stress concentrations on the surface of the braid and
bladder. Points of high stress concentrations can lead to premature wear and even-
tual failure of the FAM at or near the end-fittings. This can happen with sharp or
abrupt edges, non-uniform clamping action, and overall relatively small clamping
regions. It is desirable to have an end-fitting that can evenly distribute the clamping
force at all points of contact with the braid and bladder within the fitting.
Swaging is the method of extruding a piece of metal ? in this case the external
end-fitting tube ? through a die to reduce its diameter in what is commonly per-
formed as a cold-working process. For the construction of FAMs, the external tube
of the end-fitting, or swage tube, is forced through the die with the internal fitting,
and with the braid and bladder sandwiched in between them (Fig. 2.2). The reduc-
tion in diameter of the swage tube, therefore, produces a permanent clamping force
34
on the braid and bladder as its outer and inner diameter are reduced. The method
of swaging the external fitting component onto the internal fitting was introduced
through the work done by Woods [37,51].
Swaging of the end-fitting provides some key advantages over other end-fitting
designs. Swaging provides a continuous clamping force along the entire length and
circumference of the fitting. This increases the effective clamping area for a stronger
hold, and the uniform clamping force reduces the peak stress on the braid and blad-
der. The swaged end-fitting design also provides a very low-profile and clean fitting
design that does not create an obstruction beyond the diameter of the FAM. The
internal fitting of the swaged end-fitting provides two distinct regions of clamping
action. The difference in stiffness between the braid and bladder necessitates the
use of two separate clamping regions for an effective clamping action. The smaller
diameter of the internal fitting serves as a contact surface for the bladder to ensure
a pressure-tight seal. The larger diameter of the internal fitting serves as a contact
surface for only the braid. This ensures a strong connection between the major load
bearing elements.
2.2 Component Descriptions
FAMs are composed of four primary components: two internal fittings, two
swage tubes, braid, and a bladder (Fig. 2.2). At each end, the end-fitting, comprised
of the internal fitting and swage tube, is used to clamp the bladder and braid. The
end-fitting is designed to provide a maximum holding strength to ensure that the
35
FAM can withstand high loads and pressures. Additionally, epoxy adhesive is used
during fabrication of each end-fitting to further enhance their holding strength. The
braid and bladder are selected to be a balance between maximum actuation force and
stroke, and durability. The design attributes and material selection considerations
are detailed in the following subsections for each of these components.
Figure 2.2: Cross-Sectional view of a swaged FAM?s end-fitting
2.2.1 Internal Fitting
The internal fitting acts as a clamping surface for the bladder and braid to be
clamped to by the swage tube. Each internal fitting has three distinct features that
help it serve its unique role: a stepped clamping surface, annular grooves, and an
external connection interface.
Swaging the swage tube onto the internal fitting in a single pass requires
the swage tube to have a fixed diameter along the length of the end-fitting. The
36
difference in stiffness between the braided sleeve and the bladder requires different
amounts of compression to provide the required clamping force. Controlling the
amount of compression to the bladder and braid individually requires a stepped
design, with the bladder clamped in one region, and the braid in a second region.
The lower modulus of elasticity of the bladder requires a larger displacement to
achieve the required clamping force compared to the stiffer braid. This design also
enables the braid, which carries the actuation force of the FAM, to have its own
dedicated clamping region.
Figure 2.3: Side and isometric views of the internal fitting
The stepped design of the internal fitting is shown in Figure 2.3, while its
orientation with respect to the bladder and braid is shown in Figure 2.2. The small
diameter step is the contact surface for the bladder, while the large diameter step is
the contact surface for the braid. The diameter of each step is sized to dictate the
desired amount of compression for both the bladder and braid based on the inner
diameter of the swage tube. Each of the stepped sections are typically made equal
in length. Increasing the length of the entire end-fitting can increase its holding
37
strength, but at the expense of decreasing the percentage of the FAM?s total length
that is active for actuation.
Prior internal fitting designs with flat contact surfaces allowed the braid to
pull out of the end-fitting at relatively low loads. Woods et al., therefore, intro-
duced an internal fitting design with annular grooves and chamfered edges [37]. The
annular grooves provide intermittent recessed regions where the bladder or braid are
uncompressed, resulting in thickness variations that serve as a locking mechanism
which increases the pullout strength of the end-fitting. The grooves also provide
small reservoirs for the epoxy to bond to the bladder and braid, further increasing
the pullout strength. The chamfered edges aid in reducing stress concentrations at
the edge of each stepped surface of the fitting. The pullout strength of various end-
fitting designs were tested to failure; those that had the updated internal fittings ?
with the annular grooves and chamfered edges ? exhibited a 550% advantage in their
holding strength, compared to the former designs with internal fittings that did not
have these features. Through finite element analysis of the stress on the braid, a
40% decrease in the average stress on the braid, and a decrease in the maximum
stress of 30% was found with the updated internal fitting design [37].
The connection interface of the internal fitting provides a point of mechani-
cal connection to external components at each end of the FAM (Fig. 2.3). This
interface is typically a concentric tapped hole that accepts a threaded stud for ex-
ternal connection. At least one internal fitting must have a tapped through-hole
that extends through the entire length of the end-fitting. With a hollow threaded
stud, this end-fitting serves as a conduit for the working fluid to flow into and out of
38
the FAM. The hollow stud is connected to a tube fitting or manifold that channels
the pressurized working fluid from the fluid source to the inlet of the FAM. It is
desirable to maximize the cross-sectional area for the flow of fluid channeling into
the FAM to maximize flow rates to improve the dynamic response of the FAM. The
cross-sectional area of the through-hole is constrained by the small-diameter step of
the internal fitting.
The internal fittings are commonly made of a rigid material that can be easily
formed to have the required features. Aluminum 6061-T6 has commonly been used
for its combination of strength and machinability.
With a large percentage of the FAM?s weight in the internal fittings, weight
saving in this area can have a significant effect on the force-to-weight ration of the
FAM. Therefore, lightweight materials ? such as 3-D printed plastics ? have also
been considered for applications with lower force and pressure requirements.
2.2.2 Swage Tube
The swage tube is a thin-walled cylindrical tube that provides the clamping
force of the FAM?s end-fitting. They are commonly made of a ductile metal, such
as aluminum, that can easily be formed through the swaging process. In order to
provide a clamping force through swaging, the diameter of the tube is reduced by
extruding it through a confining die. The confining die is designed for use with the
specific outer diameter of the swage tube, and reduces the diameter of the swage
tube by a prescribed amount to provide the desired clamping action onto the braid
39
and bladder. The wall thickness of the swage tube is assumed to be fixed through
the swaging process. Therefore, it is assumed that the inner diameter of the swage
tube is reduced by the same amount as its outer diameter.
Sizing options of swage tube diameter and thickness are often constrained by
what is available through vendors. Typical swage tube thicknesses have ranged from
0.035 ? 0.065 in. Choice of wall thickness is often driven by what is required to get
the desired levels of compression of the bladder. The effect of the wall thickness
on the tube?s stiffness during swaging must also be considered. A thinner wall, for
example, requires less force to push the tube through the die, but also makes it more
susceptible to buckling in the process. The selected swage tube must also have tight
wall thickness and diameter tolerances from the manufacturer. A variation of just
a few thousandths of an inch can result in significant variations in the compression
applied to the bladder and braid resulting in a diminished strength of the overall
end-fitting.
2.2.3 Bladder
The bladder serves as an internal membrane that confines the pressurized
working fluid within the FAM. The bladder is typically a cylindrical tube that
helps define the dimensions of the FAM. The bladder is subjected to large strains
from diameter increases that are often greater than 2.5 times the initial diameter.
Energy consumed to strain the bladder is energy that would otherwise go towards
actuation of the FAM. The bladder, therefore, is often thin and is made of a low
40
modulus, highly elastic elastomer. Wear on the bladder over time, especially for
very thin wall thicknesses, can lead to pin-hole leaks and catastrophic ruptures of
the bladder. Therefore, care must be taken to choose a wall thickness that is not so
thin as to limit the operating life and resilience of the FAM.
Common bladder materials include silicone, polyurethane, latex, and natural
gum rubber [48, 51]. Material selection is often limited by the availability of ma-
terials from vendors in the desired dimensions. Some researchers, however, have
used molds to make custom bladders to the dimensions of their choice [48, 52]. As
discussed above, it is desirable to have a bladder with a low modulus of elasticity.
Unfortunately, the stiffness of elastomers is not often provided by vendors due to,
in part, their unpredictable, non-linear response. The Shore Hardness, however,
is often provided for elastomers and can be used as a relative measure of stiffness
of available bladder materials. Elastomeric tubes are often manufactured through
extrusion or mandrel formed techniques that provide a seamless design with con-
stant wall thickness. Some cylindrical bladders, especially with larger diameters,
have seams produced from manufacturing techniques such as mandrel wrapping.
These seams produce potential points of weakness and non-uniform thickness that
can lead to uneven inflation and premature failure of the bladder. Compatibility
of the bladder material with the working fluid to be used must also be considered.
For example, some bladder materials ? such as many natural rubbers ? can deterio-
rate when exposed to hydraulic oils. For petroleum based working fluids, alternate
materials such as Viton and a non-rubber, low-density polyethylene (LDPE) bag
bladder have been used [19].
41
Comparing the force response and free contraction of FAMs with bladders
made of different materials has been a popular topic of research [19, 48]. Latex
has commonly been used as a baseline of comparison. Latex is often available in a
wide range of sizes, while also providing a good combination of durability and low
stiffness. Silicone is a common alternative that is often used to make custom sizes
of bladders, using molds, but it has also been found to exhibit large amounts of
strain-stiffening which blunts the maximum contraction capability of the FAM [48].
Latex, therefore, is used for the development of the FAM presented in this research
with the opportunity to adjust the bladder material in future iterations if required.
2.2.4 Braid
The helical braid serves as the primary load-bearing element of FAMs. The
braid fibers transfer the radial pressure force of the internal working fluid into an
axial force expressed at each end-fitting, while they also enable axial contraction of
the FAM through the reorientation of its fibers. The braid material, therefore, must
have a high tensile strength while also being pliant to allow for the relative braid
movement.
The actuation characteristics of FAMs are highly dependent on the angle of
their braid. The braid angle ? defined with respect to the circumference of the FAM
? decreases as the FAM contracts in length and expands in diameter. For contractile
FAMs, the initial braid angle must be greater than 35.26 degrees [22]. FAMs with
a braid angle that is less than 35.26 degrees form an extensile FAM that extends
42
upon internal pressurization. This is proven through the Gaylord Force equation
(Eq. 2.5), which also demonstrates the strong relationship between braid angle and
the actuation force. A larger initial braid angle also results in a larger achievable
stroke of the FAM [13]. The maximum achievable initial braid angle, however, is
limited by the physical jamming constraint of the braid [53].
Examples of braid materials include Kevlar, polyester (PET), nylon, and fiber-
glass [11,37,51,52]. Braid materials with a high elastic modulus and tensile strength
are preferable to provide a braid that does not stretch or break under load. Elon-
gation of the braid fibers results in energy losses, and also limits the application of
rigid fiber assumptions used in many models. It is also desirable to have a braid with
low friction characteristics. Braid-on-braid friction, and potential braid-on-bladder
friction, have been shown to contribute to the hysteretic force response of FAMs
that is indicative of additional energy losses [5, 9]. As with other components of
the FAM, the choice of braid material is often limited by the sizing and materials
available through vendors.
Kevlar and PET braids are frequently used for FAMs, and are often the sub-
ject of comparison tests. Kevlar often provides better performance than PET with
greater stiffness (Young?s modulus of 24,650 kpsi (170 GPa) compared to 1,600 kpsi
(11 GPa)), and a higher failure stress (232,000 kpsi (1600 MPa) compared to 116,000 kpsi
(800 MPa)) [54]. PET, however, is less expensive and can provide better performance
than Kevlar with a few FAM diameters where PET braid can provide a higher braid
angle [48]. Kevlar is selected as the preferred material for this research due to its
durability and proven performance, as well as its availability in a range of sizes.
43
2.2.5 Epoxy
Epoxy is used to augment the holding strength of the end-fitting provided by
swaging. Epoxy is applied to the inside of the end-fitting prior to swaging to produce
a high-strength bond between the internal fitting, bladder, braid, and swage tube.
WEST SYSTEM 105 resin-based epoxy is used in conjunction with the slow-curing
WEST SYSTEM 206 hardener [55] in this research. The slow-curing hardener was
selected, among other options, for its cited thin-film working time of 90-110 minutes.
When the epoxy resin and hardener are mixed, self-heating of the mixed epoxy
occurs which reduces its working time down to about one hour. Since the complete
process of applying epoxy and swaging both ends of the FAM takes 30-60 minutes,
there is typically just enough time to make a FAM with each mixed batch of epoxy.
After swaging of the FAM is done, the epoxy requires 10-15 hours of curing time to
get to a solid state, and 1-4 days to reach working strength. When fully cured, the
epoxy becomes a high-strength plastic that has a tensile strength of about 7,300 psi.
By using epoxy in conjunction with swaging, improvement in the component?s
holding strength has been demonstrated [56]. A comparative tensile test was per-
formed on two 1/4 in FAMs that were crimped with and without the addition of
epoxy. Vocke et al. demonstrated that a FAM with epoxy had a three-fold improve-
ment in holding strength over a FAM without epoxy.
44
2.3 Determination of FAM Component Sizing
Sizing of FAM components is a methodical and iterative process that requires
consideration of not only the desired actuation response of the FAM, but also the
holding strength of the end-fittings. The braid and bladder sizing drive the charac-
teristic response of a FAM. Sizing of all the FAM components must be considered
together to ensure an acceptable level of compression of the bladder and braid within
the end-fitting for the required strength of the FAM?s construction.
2.3.1 Sizing of FAM to Achieve Desired Response
Preliminary sizing of the FAM?s diameter and braid angle are essential to esti-
mating the force, contraction, and radial expansion of the FAM prior to performing
detailed design efforts. Note that the diameter and length parameters of a FAM
are defined as the FAM?s outer diameter and the length of its bladder at rest, re-
spectively. While determining the approximate sizing of FAM components, it is
important to consider the limited and discrete availability of the bladder and braid
sizes from vendors, and how those constrained sizing options will affect the resultant
sizing configuration and respective response characteristics of the FAM.
The presented force and radial expansion models were used for initial sizing
of the FAM with the following design objectives (as stated in the introduction) in
mind:
? Actuation force of about 1,000 lbf
? Percent contraction greater than 30%
45
? Minimize radial expansion and respective change in internal volume
? Operating range of 0?100 psi
Secondary Considerations include:
? Minimize deadband pressure
? Maximize end-fitting internal airflow cross-sectional area
An actuation force of about 1,000 lb with a minimum of 30% contraction at
a maximum operating pressure of 100 psi was deemed sufficient for most projected
applications of the FAM. Many applications have space constraints that limit the
radial expansion allowed for the FAM, while larger values of radial expansion also
detrimentally increase the required volume and flow requirements of the source air.
Minimizing the pressure deadband ? the minimum pressure required to initiate ac-
tuation ? is desirable to ensure maximum responsiveness and efficiency of the FAM.
This objective can be met if the bladder and braid diameters are equivalent in the
resting length configuration. Maximizing the cross-sectional flow area within the
FAM?s end-fitting should be considered with the other design objectives to aid in
maximizing the dynamic response of the FAM.
Basic models of the expected force response and radial expansion of the FAM
are presented to aid in initial sizing of the FAM. Parameters that take into account
the stiffness of the bladder ? such as material modulus of elasticity and wall thickness
? are significant to the resulting actuation characteristics of the FAM. However,
consideration of the effects of the bladder on the response characteristics of the FAM
have been left out of initial FAM sizing exercises due to the modeling complications
they introduce. The presented models are based on the geometry of the FAM alone
46
without consideration of material properties of the FAM?s components. The outer
diameter of the bladder is assumed to be equivalent to the diameter of the braid for
all length states of the FAM. This is a reasonable modeling assumption, considering
that actuation of the FAM requires the bladder to be in contact with the braid to
transmit pressure forces to the braid. The models use the cylindrical approximation,
which assumes that the FAM takes on a perfectly cylindrical shape along its entire
length for all states of contraction. This cylindrical shape characteristic is used as
the basis for most force and geometric modeling efforts of FAMs [1,9, 13].
2.3.1.1 Models for Estimation of FAM Radial Expansion and Force
Response
The cylindrical approximation leads to a geometric relationship, termed the
triangle relationship, which correlates the diameter and respective braid angle to
the length of the FAM for all states of contraction. With an assumption made for
the maximum percent contraction of the FAM, this leads to estimations of radial
expansion and internal volume. The triangle relationship is found by first defining
the fixed length of each braid fiber, B, and the number of turns of each braid fiber
about the circumference of the FAM, N , defined by the following equations:
= L0B
sin(?0)
(2.1)
= L0N
D0?tan(
(2.2)
?0)
47
where L0 is the initial active length, D0 is the initial diameter, and ?0 is the initial
braid angle of the FAM. The FAM diameter, D, can then be found for any length
state, L, using the following equation:
?
( ) = B
2 ? L2
D L (2.3)
?N
For some calculations, such as force, it is necessary to have an approximation of the
instantaneous braid angle. The relationship between diameter and braid angle is
fixed in accordance with the following equation:
cos(?1) = cos(?2) (2.4)
D1 D2
where the braid angle, ?n, and diameter, Dn, are related between states 1 and 2.
Manufacturers commonly provide a nominal diameter with respective braid angle
for the delivered braid. This information, along with the diameter for any given
state as found from Equation 2.3 can be used to find the braid angle for the given
state.
Figure 2.4(b) illustrates the relationship between the initial braid angle, FAM
diameter, and the final diameter using Equations 2.2?2.3. The plot calculates radial
expansion of the FAM at an assumed 35% maximum contraction, and is independent
of the length of the FAM. Discrete points using braids of different nominal diameters
are included for later analysis of the radial expansion under constrained availability
of bladder and braid of sizing.
48
2.3.1.2 FAM Force Response Approximation
The cylindrical approximation also serves as the basis for most FAM force
response models. The Gaylord force, FG, provides an estimation of force based on
the geometry and internal pressure of the FAM; It uses the braid angle, ?, radius,
R, and internal pressure, P , to approximate the actuation force in the following
equation:
FG = P?R2(2tan2? ? 1) (2.5)
Values of R and ? can be found using Equations 2.3 and 2.4 respectively. Energy
expended through radial expansion of the bladder is not factored into the model, but
it would act to reduce the value of the actuation force estimated from the Gaylord
force equation. The Gaylord force calculation is, therefore, most accurate at low
levels of radial expansion that occur at low levels of FAM contraction.
Figure 2.4(a) illustrates the relationship between the initial braid angle, FAM
diameter, and the estimated maximum blocked force of the FAM at 100 psi using
Equations 2.4 and 2.5. Notable observations include the steep increase in force with
braid angle that is especially noticeable with larger diameter FAMs. Discrete points
indicating braids of different nominal braid diameters are included for later analysis
of force capabilities under constrained availability of bladder and braid sizing. A
red line indicating the 1000 lb mark is included as a reference for comparison to the
stated design objective.
49
(a) Blocked force at 100 psi (b) Maximum FAM diameter at an assumed
35% contraction
Figure 2.4: The effect of initial diameter and braid angle on estimates of maximum
force and maximum diameter of the FAM. The surface represents the unconstrained
design space, while points indicate options in the constrained design space
(a) Force with respect to contraction at 100 psi (b) Diameter with respect to contraction
Figure 2.5: The estimated force and diameter of the FAM for an assumed contraction
of 40%. Results compare the response of FAMs with different combinations of
available braid and bladder sizes
2.3.1.3 FAM Contraction Approximation
The overall length of the FAM is sized based on the contraction required of
the FAM for a given application. Contraction of the FAM is defined in terms of
a percent contraction of the resting length of the FAM. The percent contraction is
commonly used to define the contraction of FAMs because the percent contraction
50
? for a given size and material construction ? has been shown to be independent
of the resting length of the FAM [13]. Absolute length contraction of a FAM is,
therefore, proportional to its length.
The percent contraction of a FAM is primarily dependent on the initial braid
angle, and the stiffness of the bladder. Unfortunately, without material properties of
the bladder a priori, there is no simple model that estimates the contraction of the
FAM prior to testing. FAMs have commonly attained between 25-40% contraction.
Previously tested FAMs with comparable bladder materials and braid angle can
be used to provide an estimate of the attainable percent contraction. The trend
between braid angle and percent contraction is evident from the FAMs of Table 2.1.
2.3.1.4 FAM Component Sizing Constraints
When determining target values of FAM diameter and braid angle, it is com-
mon to think in terms of a design space without constraints. When thinking in
terms of an unconstrained design space, it is easy to think, for example that using
a larger diameter would lead to an increase in force and inflated diameter of the
FAM. However, in a constrained design space, this is not necessarily the case. It is
important to take into account the constraint imposed by the limited availability of
components that are only offered in discrete sizes.
Availability of different sizes of components is dependent on the materials and
measurement system used (Metric or Imperial). The latex cylindrical tubes used for
the bladder are often available in 1/8 in increments for diameters that are around 1
51
in. For this study, we will consider bladder sizes of 1/2, 5/8, 3/4, 7/8, and 1 in. Braid
is commonly sold in 1/2 in diameter increments defined at a set braid angle. The
Kevlar used for the development of the FAM comes in nominal diameters of 1/2 in
increments with the diameters defined for the braid in the 45 degree orientation. The
braid can be stretched or compressed to change its diameter and respective angle
in correspondence to the relation defined in Equation 2.4. Each braid has a range
of achievable diameters that is limited by jamming of the braid at the extremes.
Information on three candidate Kevlar braids are provided in Table 2.2.
t Dmin?max ?max?min
1.5 in Braid Dia. 0.012 in 0.60 - 1.75 in 73.6? - 34.4?
2.0 in Braid Dia. 0.011 in 0.76 - 2.23 in 74.4? - 38.0?
2.5 in Braid Dia. 0.018 in 1.40 - 2.65 in 66.7? - 41.5?
Table 2.2: Braid thickness, t, and range of achievable diameters, D, and range of
braid angles, ?, for Kevlar braids of three nominal braid sizes (defined in 45 deg.
braid orientation) (Source: Soller Composites, LLC)
1 in FAM 7/8 in FAM 3/4 in FAM 5/8 in FAM
1.5 in Braid Dia. 61.9? 65.6? 69.3? 72.9?
2.0 in Braid Dia. 69.3? 72.0? 74.6? ?
2.5 in Braid Dia. 73.6? ? ? ?
Table 2.3: The initial braid angle for each combination of FAM size and nominal
braid diameter. Red entries are at the threshold of the achievable range of the braid,
while blank entries are non-feasible combinations (Reference Table 2.2)
The constructed FAM should have equivalent diameters for the bladder and
braid. Every bladder and braid combination, therefore, will have a defined initial
diameter and braid angle. Table 2.3 provides the braid angles that can be achieved
for each combination of bladder and braid sizing used. There is a clear correlation
between reduced FAM diameter and increased braid angle, however the extent of this
52
is constrained by the diameter limits of the braid indicated in Table 2.2. Therefore,
some combinations are not feasible, while other cases are barely feasible (indicated
in red) but would result in an undesirable pressure deadband due to the initial
looseness of the braid around the bladder.
Figure 2.4 shows where the combinations of bladder and braid sizing lie within
the unconstrained design space plots of estimated force and radial expansion. It is
apparent that with the constrained sizing options, changes in initial FAM diameter
result in relatively little change in the realized force and maximum diameter of the
FAM. Figure 2.5 takes a closer look at this observed trend. The sizing combina-
tions are plotted with respect to the percent contraction state of the FAM with an
assumed maximum contraction of 40%. A prominent observation for both plots is
the apparent grouping of the force and maximum contracted diameter values based
on the braid diameters. This makes it clear that in the constrained sizing design
space, the nominal diameter of the braid is the key driver of the FAM?s characteristic
response, with little effect associated with changing bladder diameter.
It is important to consider what effect the inclusion of bladder stiffness as
a parameter (which these model approximations do not consider) would have on
the results plotted in Figure 2.5. Bladder strain will act to decrease the force
and contraction of the FAM. The increase in radial expansion for smaller bladders,
observed in Figure 2.5(b), will result in higher levels of strain in the bladder which
would act to further reduce contraction and force magnitudes, when compared to
larger diameter bladders. This would likely negate some of the slight advantage in
force observed with the smaller bladder diameters in Figure 2.5(a).
53
2.3.1.5 Selection of Target FAM Sizing
To achieve the target force of about 1,000 lbs, Figure 2.5(a) shows that a nom-
inal braid diameter of 2 in should be selected. The 2.5 in diameter braid provides far
more force than is necessary with unfavorable increases in radial expansion and air
volume expenditure. Selection of the bladder diameter is more nuanced, with con-
siderations of deadband pressure, material strain, air inlet diameter, and component
availability.
A larger diameter bladder results in less bladder strain resulting in less energy
loss and increased contraction. A larger diameter also does not increase the maxi-
mum contracted diameter of the FAM which is beneficial for fitting into tight spaces
or for arranging multiple FAMs in parallel. A larger diameter also provides the
opportunity for a larger air inlet cross-sectional area to increase volume flow rates,
and decreases the expected change in inflation volume with contraction, which to-
gether act to improve the dynamic response time of the FAM. A sizing combination
of a 3/4 in bladder with the 2 in Kevlar braid was used for the FAM developed
by Robinson [20] which skirts the threshold of the achievable range of braid angle
according to Table 2.2. This slightly oversized braid resulted in a relatively high
pressure deadband of about 10 psi for that 3/4 FAM. As indicated in Table 2.3,
defendable choices for FAM size selection can then be pared down to a 1 in or 7/8 in
diameter bladder FAM, with little difference in their expected actuation responses.
This leaves flexibility for the sizing process required for the design of the swaged
end-fitting, as covered in subsequent sections. The 7/8 in FAM with a 2 in nominal
54
braid diameter configuration was ultimately selected for the FAM used.
2.3.2 Sizing of Components for Swaged end-fittings
The method of sizing a FAM?s components has been refined over the years
since the introduction of the swaged FAM by Woods [51]. The process of both
sizing and fabrication of a FAM is as much of an art as it is a science. That is to
say, much of the development of swaged FAMs has entailed a bit of trial and error to
produce the desired results. Development of the swaged FAM has been a process of
understanding the required compression of the bladder and braid to ensure sufficient
strength of the end-fitting, while also finding the limit of compression that the swage
tube can withstand without failing when attempting to swage the end-fitting. This
includes the limits of force that can be applied to the swage tube before it buckles,
and the force that the swage die can withstand, as well as the force that the extrusion
press can provide.
The following benchmarks have previously been set for the successful swaging
of a FAM. The swage tube experiences a reduction in diameter of 2.7% upon being
extruded through the die. The internal fitting is sized to provide 17% compression
of the braid, and 30% compression of the bladder. This formula has been used with
multiple sizes of FAMs, and has been adjusted as necessary. Subsequent to the
fabrication of the FAMs built for this research, the braid compression was increased
to 27% compression, upon completion of this work, to improve the holding strength
of the end-fitting. The components of the 7/8 in FAM are sized to satisfy these
55
criteria. It is important to note that one of the biggest challenges in sizing is the
limited availability of sizes ? especially of the swage tube and bladder ? that are
available through retailers.
2.3.2.1 Steps for Sizing Components of Swaged end-fitting
The method of sizing for the FAM end-fitting is outlined in the following steps.
A detailed explanation of each step follows with a focus on the 7/8 in FAM that is
developed for this research.
1. Set Bladder Outer Diameter - list available wall thickness options, tbld
2. Set Braid Size - to maximize braid angle
3. Set Swage Tube Outer Diameter - list available wall thickness options
4. Set Die Inner Diameter - based on swage tube diameter
5. Select Test Bladder and Swage Tube Wall Thickness - from listed options and
subject to constraints
6. Size Internal Fitting - size to achieve desired end-fitting compression
The steps should be performed in order. Steps 5 and 6 are iterated for each
thickness option in the lists established in Steps 1 and 3 until a feasible solution for
the sizing of the FAM is found. If no solution is found, subsequent iterations must
incorporate previous steps until a solution is found.
Step 1: Set Bladder Outer Diameter. The outer diameter of the bladder
sets the overall diameter of the FAM, and is selected based on the criteria set in
the previous section. Bladder diameters are commonly offered in 1/8 in diameter
increments. The targeted 7/8 in diameter was found available in wall thicknesses of
1/16 in, 1/8 in, and 3/16 in. Which thickness to use is determined in the subsequent
56
steps, but preference goes to using a thinner bladder that can help maximize the
force and contraction of the FAM.
Step 2: Braid Sizing. The diameter of the Kevlar braid was selected in the
previous section as the 2.0 in diameter sleeve, with the diameter defined in the 45
degree braid angle configuration. The relation between diameter and braid angle
can then be found using Equation 2.4 as previously mentioned. Relevant to the
sizing of the end-fitting components is the thickness of the braid, which is 0.011
in. At 0.011 in, with the braid being so thin, the compression of the braid as a
percentage of its thickness is very sensitive to the tolerance that the internal fitting
is manufactured to.
Step 3: Set Swage Tube OD. Sizing of the swage tube requires defining its
outer diameter and wall thickness. Selection of the wall thickness occurs in later
steps where the overall sizing of the end-fitting is iteratively refined. The thin-
walled Aluminum 6061-T6 tubing used for the FAM is commonly offered in 1/8 in
diameter increments. With the 7/8 in diameter bladder, this drives the swage tube
size to an outer diameter of 1 in. Attempting to use a 7/8 in outer diameter swage
tube would result in difficulty getting the tube over the bladder, braid, and internal
fitting assembly prior to swaging.
Previous swaged FAMs have used tubing with wall thicknesses of 0.035 to
0.065 in. Within this thickness range, 1 in aluminum tubing is offered with wall
thicknesses of 0.035, 0.049, 0.058, 0.065 in. It should be noted how the wall thickness
affects the swaging process. Buckling of the swage tube in the process of attempting
to extrude it through the die is a common issue of concern when swaging. A thinner
57
wall thickness requires less force to extrude through the die, but has a lower critical
buckling load. Conversely, a thicker wall requires more force to extrude, but also
has a higher critical buckling load. The thicker wall thickness is likely to be more
robust in providing a constant clamping force for the operational life of the FAM.
The 1 in outer diameter tube is set for initial sizing with the four wall thicknesses
offered serving as viable options for the refinement of the FAM?s sizing.
Step 4: Die Sizing. The die is used to reduce the diameter of the swage tube
through the swaging process. Previous work established the desired percent reduc-
tion in diameter as 2.7%. This requires the die to reduce the 1 in outer diameter
of the swage tube down to 0.973 in. The reduction in diameter takes place in the
tapered section of the die, as illustrated in Figure 2.6. Sizing of the end-fitting
components assumes that the swage tube maintains a constant thickness through
the process of being extruded through the die.
Figure 2.6: Dimensioned cross-section of the die used for swaging
Step 5: Select Bladder and Swage Tube Thickness. Settling on a swage tube
and bladder thickness is performed as an iterative process alongside the sizing of
the internal fitting. Sizing likely starts with the bladder that has the thinnest wall
thickness to provide a maximum force and stroke of the FAM.
58
Step 6: Size Internal Fitting. Sizing of the internal fitting is the most important
step for creating an effective end-fitting design. The outer diameter of each stepped
surface of the internal fitting (Fig. 2.3) is sized in conjunction with the swage tube,
bladder, and braid, to ensure that the desired compression of the braid and bladder
is achieved.
Benjamin Woods, who first made FAMs with swaged end-fittings [51], set the
precedent for the compression required of the bladder and braid to successfully swage
an end-fitting. The braid requires a compression of 17% of its thickness, while the
bladder requires a compression of 30% of its thickness. These criteria are used for
sizing the internal fitting in conjunction with the other components.
The equations that define the desired sizing of each stepped surface of the
internal fitting are as follows:
ODfit,brd = ODswg ? 2(tswg + tbrd(1?%Compbrd)) (2.6)
ODfit,bld = ODswg ? 2(tswg + tbrd + tbld(1?%Compbld)) (2.7)
where ODfit,brd is the large diameter of the end-fitting that serves as the contact
surface for the braid, and ODfit,bld is the small diameter of the end-fitting that serves
as the contract surface for the bladder.
Each equation starts with ODswg, which is the outer diameter of the swage
tube after having its diameter reduced by 2.7% through the swaging process. The
thickness of the swage tube, tswg, is assumed to remain constant through the ex-
trusion process. The bladder and braid lie in the space between the inner diameter
59
of the swage tube and the internal fitting. Only the compressed braid, with thick-
ness tbrd(1?%Compbrd)), lies in the space between swage tube and the larger step
of the internal fitting, ODfit,brd. For the smaller diameter step of the internal fit-
ting, ODfit,bld, the braid and bladder lie in the region between the internal fitting
and swage tube. With the braid considered rigid in comparison to the bladder,
this region is occupied by the uncompressed thickness of the braid, tbrd, and the
compressed bladder thickness, tbld(1?%Compbld).
While Equations 2.6 and 2.7 make sizing seem rather trivial, there are con-
straints on what is considered a feasible design. First, the solution found for ODfit,bld
must be a value that is similar to the internal diameter of the bladder. A value of
ODfit,bld that is much smaller than the bladder internal diameter would make the
bladder bunch up and not lay flat on the internal fitting?s surface. On the other
hand, if ODfit,bld is much larger than the internal diameter of the bladder, it may
not be possible to stretch the bladder over the internal fitting. It is also desirable
to have a thin bladder to ensure that ODfit,bld is large enough to enable a large flow
area through the internal fitting to maximize the dynamic response of the FAM,
among other reasons.
As previously mentioned, the discrete and limited sizing of the bladder and
swage tube diameters and thickness options makes sizing of a FAM a challenging
design problem with few combinations that can be made to achieve the desired braid
and bladder compression values. The sizing process, therefore, is an iterative process
that requires different combinations of bladder and swage tube sizes to be tested
for feasibility before a final combination of materials and sizing can be settled upon
60
that obeys the constraints and compression requirements of the end-fitting.
2.3.3 Final Sizing of FAM
The finalized sizing of the 7/8 in FAM?s components and respective materials
is provided in Table 2.4. A dimensioned drawing of the internal fitting is provided
in Figure 2.7. Three FAMs were made to these specifications and tested for this
research (Fig. 2.16).
Material Outer Diameter Thickness
Braid Biaxial Kevlar Sleeve 2 in 0.011 in
Bladder Latex Tube 7/8 in 1/16 in
Swage Tube Aluminum 6061-T6 Tubing 1 in 0.058 in
Internal Fitting Aluminum 6061-T6 0.838 in, 0.748 in -
Table 2.4: Construction of the 7/8 in diameter swaged FAM
Figure 2.7: Dimensions of internal fitting used for the 7/8 in swaged FAM
2.4 FAM Method of Fabrication
Fabrication of a swaged FAM can be broken down into four basic steps: ma-
terial preparation, assembly, swaging, and finishing. Material preparation requires
61
cutting all components of the FAM to length, and fabricating the tools needed for
swaging. The FAM is then assembled by putting the FAM?s components in place,
prior to fixing them in place in the next step through the swaging process. Finally,
the excess swage material and braid is removed in the finishing process to complete
fabrication of the FAM.
Figure 2.8: Swage tools (bolded) and FAM assembly in alignment prior to swaging
The main components of the swaging process are detailed in Figure 2.8. The
tools used for swaging ? the swage tube (with extended length), standoff tool, push
plate, centering guide, and die ? are labeled in the figure. The basic components
of the FAM ? the braid, bladder, internal fitting, along with the swage tube ? are
labeled in bold in the figure. It should be noted that the swage tube is initially
much longer than the internal fitting. This extra length is necessary for extrusion to
ensure that a pushing force can be applied to the top of the tube until the internal
62
fitting portion is completely extruded through the die. The excess tubing is pushed
through the die using a follower tool, and then the excess material is removed later
in the process to produce the finalized product.
2.4.1 Material Preparation
Prior to fabricating a FAM, the components of the FAM, as well as the tools
required for swaging, must be prepared. Figure 2.9 is an image of these components
as they are arranged prior to swaging the FAM. Once made and assembled, the
tools used for swaging ? the die, centering guide, standoff tool, follower tool, and
push plate ? can be used to fabricate multiple FAMs. However, the components
of the FAM - the internal fittings, braid, and bladder - must be prepared prior to
fabrication of every FAM.
Figure 2.9: Components of FAM and swage tools prepared prior to assembly
The two internal fittings are machined to the dimensions defined in Figure
2.7. A cylindrical piece of Aluminum 6061-T6 is reduced to the desired dimensions
using a lathe. It is critical that the tolerances on the diameter of the internal fitting
are kept within a few thousandths of an inch to ensure the correct compression
of the braid and bladder. A deviation of just one-thousandth of an inch from the
63
prescribed dimensions of the internal fitting would result in a significant 9% change
in the compression of the braid.
The length of the bladder dictates the final length of the FAM. The bladder
length must account for the active length of the FAM plus the material that covers
each lower step of the internal fittings. Approximately 1/8 in of space is left between
the edge of the bladder and the step of the internal fitting to allow for axial expansion
of the bladder upon compression. For example, with an internal fitting step length
of 1/2 in, the bladder is cut to a total length that is 0.75 in longer than the desired
active length of the FAM.
A generous amount of braid is used to ensure that there is excess braid to pull
on while aligning components prior to swaging. The braid is cut to a length that
equals the active length of the FAM, combined with the length of the uncut swage
tube and width of two gripped fists. This combined length ensures that there is
sufficient space to grab the braid above each swage tube before sliding the swage
tube up into place over the internal fitting.
The swage tube is cut to a length that enables the press to apply continual
pressure at the push plate until the entire internal fitting exits the bottom of the
die (Fig. 2.11(a)). With the 1 in long internal fitting, and the 2.1 in depth of the
die, the swage tube must be at least 3.1 in in length. The swage tube is then pushed
the rest of the way through the die with the follower tool.
The follower tool is used to push the remaining swage tube out of the die
subsequent to swaging the end-fitting (Fig. 2.11(b)). The tool has a diameter that
is slightly smaller than the inner diameter of the die, allowing it to push onto the
64
top of the swage tube while not getting stuck in the die. In cases where the inner
diameter of the swage tube is larger than the inner diameter of the die, multiple
follower tools with reducing diameters may need to be used. An alternate method
of removing the swag tube is to cut the swage tube above the fitting, straighten the
cut on a lathe, and then push the swage tube out the way it entered the die with
the follower tool. However, this alternate method is more time consuming, and the
required cutting can be dangerous with the limited space between the bottom of the
die and the fitting.
The standoff tool enables the press to apply force to the swage tube and
internal fitting together ? via the push plate ? to ensure that there is no relative
motion between the two components as the tube is extruded through the die. The
combined length of the internal fitting and standoff tool are made to be equivalent to
the length of the swage tube. The diameter of the standoff tool is reduced to enable
space for excess braid between the standoff tool and the inside of the swage tube.
The standoff tool is threaded at its bottom to screw into the 7/16-20 UNF tapped
hole of the internal fitting. This provides a rigid connection between the internal
fitting and standoff tool that ensures the correct alignment between components
during the swaging process.
The push plate ? alluded to above ? is used to provide a surface for the press to
apply force to the swage tube and standoff tool together. The push plate is simply
a rigid metal plate with dimensions that are greater than the outer diameter of the
swage tube.
The die was machined to the dimensions in Figure 2.6. As previously noted,
65
the inner diameter of the die?s bore is sized to provide the desired compression of
the swage tube. The die is made of hardened steel to ensure that it can withstand
multiple swaging processes. The hardness of the steel, along with the large ratio
of the die to bore diameter, help ensure that the bore does not flex and increase
in diameter during the swaging process. The bore has a 7 degree tapered section
followed by a straight section. A low taper angle ensures a gradual and progressive
swaging process. Too shallow of a taper angle, however, requires a taller die and a
longer swage tube which is more susceptible to buckling. The 1.35 in straight bore
section is used to ensure the final diameter of the end-fitting remains fixed to the
bore diameter. This section should be at least as long as the internal fitting to
reduce the chances of the swage tube from rebounding, the tube increasing slightly
in diameter as it exits the bottom of the die.
The centering guide is used to ensure that the swage tube enters the bore of
the die straight. Any slight misalignment of the swage tube and bore can easily
lead to failure of the end-fitting swage, ruining the entire FAM. The centering guide
can be a specially fabricated part, or it can be as simple as machinist V-blocks as
is used for swaging the 7/8 in FAMs here.
The last step prior to swaging is to prepare the epoxy. The epoxy is given
about 15 minutes to thicken slightly prior to beginning assembly, to facilitate its
application onto the end-fitting without excessive dripping.
66
2.4.2 Assembly
Assembly of the FAM prior to swaging is the most challenging part of making
a swaged FAM. To assemble the FAM, the braid is first pulled over the bladder with
the bladder centered along its length. A swage tube is then slipped over the braid
and bladder. One end of the braid is clamped below the bladder to hold it in place
while the other end is worked on (Fig. 2.10(a)). The braid is then folded back onto
itself to expose the bladder in preparation for insertion of the internal fitting.
The internal fitting, with the standoff tool attached, has its outer surface
coated with epoxy. The small diameter of the internal fitting is then positioned
inside of the bladder (Fig. 2.10(b)). While maintaining the positioning of the
internal fitting, the braid is then unfolded back over the internal fitting and standoff
tool. The braid in the area around the internal fitting is then saturated with epoxy.
Prior to applying the epoxy, the swage tube is positioned so its top is aligned with
the bottom of the internal fitting to keep epoxy from running down onto the actuated
portion of the FAM.
While holding the braid taught with one hand just above the standoff tool, the
other hand is used to pull the swage tube up and over the internal fitting until the
tops of the swage tube and standoff tube are level with one another (Fig. 2.10(c)).
The hand holding the braid is also used to keep the standoff tool and end-fitting
in place as the swage tube is pulled up over them. Pulling the swage tube over
the internal fitting can take a considerable amount of force, with the required force
depending on the amount of clearance allowed between the end-fitting components
67
prior to swaging.
(a) Braid clamped be- (b) Epoxy covered internal fitting placed inside
low bladder bladder. The Swage Tube is positioned just be-
low the internal fitting.
(c) Swage tube is pulled up (d) Braid is unclamped and funneled
over the internal fitting un- through die. Die and FAM are placed
til even with top of standoff in press
tool
Figure 2.10: Steps of the assembly process prior to swaging
At this point, the end-fitting components are in their final positions, and are
ready to be swaged. The braid is unclamped from the table clamp, and the braid
and bladder are fed through the die until the swage tube is resting in position at
the entrance of the die?s bore (Fig. 2.10(d)). The end-fitting is then swaged.
68
After swaging the first end, the other end-fitting is swaged following the same
steps as with the first end-fitting. But prior to putting the second swage tube on,
the die must be slipped onto the FAM. There is no way to get the die below the
swage tube after the second swage tube is put onto the FAM. The extra braid that
is left below the swaged end-fitting is then clamped into the table clamp, and the
process is then repeated for the second end-fitting. It is especially important to
pull the braid taught as the swage tube is pulled up over the second end-fitting to
maximize the final braid angle of the FAM while ensuring that there is no space
between the braid and bladder. Additionally, it is also important to pull on the
braid along its longitudinal axis to ensure that the braid fibers are symmetric before
the braid is fixed in place by swaging the end-fitting.
2.4.3 Swaging
Swaging is performed by extruding the end-fitting through the die. The extru-
sion force is provided by a 10-ton press that is actuated by hand through a hydraulic
lever-arm. The force of the press is applied to the top of both the standoff tool and
swage tube by way of the push plate. The swage components are aligned in-line
with the bore of the die ?as depicted in Figure 2.8 ? before swaging of an end-fitting
is performed.
Prior to swaging, a lubricant (e.g. White Lithium Grease) is applied to the
outside surface of the swage tube to reduce friction forces between the swage tube
and die while swaging. The push plate is then placed on the top of the swage tube
69
and standoff tool, and the press is lowered to a position that places it just above the
push plate. Using the centering guide(s), a final check of alignment is performed
to ensure that the swage tube is vertical and oriented in-line with the bore of the
die. The press is then lowered to contact the push plate and then begin the swaging
process.
The progression of the swaging process is shown in Figure 2.11(a). The FAM
assembly is extruded through the die until the push block comes into contact with
the die. At this point, with the entire fitting having exited the bottom of the die, the
end-fitting is swaged. Excess swage tube still remains in the die, and is extruded the
rest of the way through the die using the follower tool (Fig. 2.11(b)). The follower
tool is carefully aligned with the swage tube, and then driven down into the swage
tube until the swage tube exits the bottom of the die. The follower tool ? which is
loose in the die ? can then be pulled back out of the top of the die by hand. After
the swaging process is complete, the FAM is set aside for about 24 hours to let the
epoxy cure before the final steps are performed to complete the fabrication of the
FAM.
2.4.4 Finishing Process
The excess swage tube and braid that remains with the fitting after swaging
is done needs to be removed before the FAM is complete (Fig. 2.11(c)). Removal
of this excess material is performed using a lathe. Prior to putting the FAM in the
lathe, the midsection of the FAM is covered with protective material (e.g. tape) for
70
(a) FAM assembly extruded through die until internal fitting is completely
through die
(b) Swage tube that remains in die is driven out using the follower tool
(c) Extra length of swage tube and braid is cut off to produce finished product
Figure 2.11: Progression of producing a swaged FAM end-fitting
71
its protection and the standoff tool is removed if possible. If the standoff tool can
not be removed prior to cutting, it can be removed after carefully cutting away a
portion of the swage tube to improve access.
The FAM is clamped by the jaws of the lathe chuck at the bottom of the swage
tube, and with the top of the swage tube facing outwards. The rest of the FAM has
a cylindrical space behind the chuck where it can lie and be supported while the
end-fitting is being trimmed. The excess swage tube is gradually cut away up to the
face of the internal fitting. The soft braid is more easily cut near the internal fitting
where excess cured epoxy makes it more rigid. With a final pass of the cutting
tool across the face of the internal fitting, the braid breaks away cleanly, leaving a
nicely finished surface (Fig. 2.11(c)). Once the finishing process is performed on
each end-fitting, the fabrication of the FAM is complete.
2.4.5 Modification to swaging process
Initial successes and failures in swaging 7/8 in FAMs resulted in a few modifi-
cations to the swaging process. Sometimes, attempts to swage a FAM were unsuc-
cessful, resulting in the FAM being scrapped. The mode of failure was most often
due to the swage tube buckling while being forced through the die (Fig. 2.12). A
buckled swage tube immediately makes the swaging process impossible to continue
with no way to force the end-fitting the rest of the way through the die.
Techniques to reduce the chance of the swage tube buckling during swaging
have been learned incrementally through trial and error. Efforts focused on attempt-
72
Figure 2.12: Buckled swage tube of failed swaging attempt
ing to reduce the force required to extrude the swage tube through the die, and on
supporting the swage tube to increase its critical buckling load. As a note, some of
these improvements have already been described in previous sections.
Steps for reducing the required extrusion force were targeted first. A straight,
vertical alignment of the swage tube with the bore of the die was initially challenging
to do by eye. Any slight misalignment in the swage tube results in the tube and
end-fitting jamming in the die, and the resultant non-vertical force of the press on
the swage tube reduces the load required for the tube to buckle. The centering
guide, which has initially been machinist V-blocks, was devised and employed to
ensure a vertical orientation of the swage tube. The use of a lubricant ? white
lithium grease ? was also introduced to decrease the required extrusion force by
decreasing the frictional forces between the straight section of the die bore and the
swage tube. The grease is applied to the outside surface of the swage tube prior to
73
swaging. This has dramatically decreased the apparent force required by the press,
and greatly improved the smoothness and consistency of the motion of the swage
tube as it was extruded through the die.
Subsequent efforts focused on increasing the buckling load of the swage tube.
The dimensions of the swage tube ? constrained by the dimensions of the die, and
by the sizing required to provide the desired compression of the bladder and braid
? could not be altered to increase the load-bearing capabilities of the swage tube.
Therefore, a means of supporting the swage tube was devised within this research.
The developed solution entails the use of an additional tube that slides within
the swage tube for support, but can then be removed prior to entering the die. This
revised swage assembly is illustrated in Figure 2.13, and images of the individual
components are provided in Figure 2.14. The tube support is a 1 in outer diameter,
0.125 in wall thickness tube, that has 1.25 in of its 2 in length turned down to fit
within the swage tube with minimal clearance. The tube support increases the area
moment of inertia in the overlapped section by 75%, and leaves a maximum of only
1.85 in of unsupported swage tube below it. Overall, based on Euler?s buckling load
theory, this should approximately double the buckling load of the swage tube. The
tube support adds a little bit of height above the swage tube, 0.75 in, that requires a
standoff extension tool to ensure that force can still be applied to both the internal
fitting and swage tube at the push plate. The standoff tool was modified to have
a male end that receives the female end of the standoff extension for a secure, but
easily removable connection between the two parts.
The entire modified assembly is aligned the same as with the prior basic swage
74
assembly shown in Figure 2.8, and includes the addition of just the two new parts:
the tube support and the standoff extension. The progression of the swage process
using the modified swage assembly is provided in Figure 2.15. As the swage tube
is extruded through the die, its unsupported length decreases which beneficially
increases its critical buckling load. Extrusion of the swage tube is performed with
the tube support in place until the tube support begins to enter the die. At this
point, the tube support and standoff extension are removed to ensure that they
are not swaged within the swage tube as is being done with the internal fitting.
Extrusion then continues as is done with the more general assembly until the end-
fitting, and then the swage tube, are extruded completely through the die.
Notably, this modified swage assembly, in conjunction with the efforts made
to reduce the required extrusion force, has greatly improved the swaging success
rate. Furthermore, it has also facilitated an improvement to the strength of the
end-fitting. Work on future FAM iterations desired an increase in the compression
of the braid to improve the strength of the end-fitting. Targeted improvements
in FAM development has included an increase in the compression of the braid to
improve the strength of the end-fitting.
This increase in compression entails a slight increase in the diameter of the
internal fitting, which further increases the extrusion force required to swage the
FAM. The developed improvements to the swaging process have made this possible;
Compression of the braid was increased to 27% which further increases the strength
of the end-fitting.
75
Figure 2.13: Modified Swage Assembly
Figure 2.14: Labeled components of modified swage assembly
2.5 Testing Swaged FAMs
Three swaged 7/8 in FAMs were made for initial testing (Fig. 2.16). All three
FAMs are identical in their construction, and were all made to similar active lengths.
An isobaric force-contraction test was performed to characterize the actuation re-
sponse of each of the FAMs for comparison. Then, two of the FAMs were tested
to failure in an effort to test the strength of the swaged end-fittings. An ultimate
76
Figure 2.15: Swaging progression with modified assembly. Per methodology, the
tube support and the standoff extension have been removed between frames 2 and
3, prior to continuation of extrusion.
tensile test was performed to test the end-fitting?s holding strength of the braid. A
burst test was performed to test the end-fitting?s ability to contain the pressurized
internal fluid. A factor of safety is calculated by comparing the maximum force
and pressure at failure to the maximum values experienced in the normal range of
operation of the FAM.
2.5.1 Characterization Test
The three FAMs were first characterized by testing their characteristic pressure-
force-contraction responses. This test was performed on a material testing MTS
machine, which controls the length state of the FAM while recording displacement,
pressure, and force measurements. The FAM?s internal pressure is fixed at a set
pressure, and the FAM is then cycled between its maximum force, zero contrac-
tion state, called its blocked force state, and its zero force, maximum contraction
state. The FAMs were redundantly cycled between these bounds three times at each
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pressure. For a complete characterization of the FAM?s expected pressure operat-
ing range, they were tested at pressures of 0 to 100 psi in 10 psi increments. The
defining characteristics of the PAMs are provided in Table 2.5. The results of the
characterization testing are shown in Figure 2.17. The test results for each FAM
are overlayed onto the same plot for ease of their comparison. The consistency of
the fabrication method is demonstrated by the fact that the responses of all three
FAMs are nearly identical.
Figure 2.16: Tested 7/8 in swaged FAMs
FAM 1 FAM 2 FAM 3
Active Length 9?3/16 in 8?1/64 in 8?29/32 in
Weight 0.190 lb 0.183 lb 0.192 lb
Blocked Force (100 psi) 1037 lb 1029 lb 1049 lb
Free Contraction (100 psi) 34.57 % 35.08 % 35.03 %
Table 2.5: Specifications of swaged FAMs used for testing
2.5.2 Ultimate Tensile Failure Test
An ultimate tensile failure test was conducted on FAM 2 to provide a measure
of the maximum load that the FAM can support in tension. Although this is not
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Figure 2.17: Isobaric force-contraction response of the three tested FAMs
within the normal mode of operation, it provides a quantitative measure of the
strength of the FAM and its end-fittings. The test is conducted by leaving the FAM
unpressurized, and slowly stretching it up to the point of failure. Figure 2.18(a)
provides a plot of the test results, while Figure 2.18(b) provides images of the FAM
at different stages of the test. This FAM was able to support over 2,147 lb before
failing. The failure occured from the internal fitting slipping out of the end-fitting
assembly, as can be seen at the top of the FAM in the pictures of Figure 2.18(b).
The bladder and braid remained in their original positions within the swage tube
after failure (Fig. 2.20). This indicated that the braid clamping region was the
main point of failure. With increased compression of the braid, the failure load
could likely be increased. This test showed, however, that the FAM can handle
double the maximum load that occurs in the 0-100 psi operating range, for a safety
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factor of about 2.
(a) Test results with failure of FAM occurring at 2,147 lbf
(b) Progression of FAM being pulled apart
Figure 2.18: Ultimate tensile failure test conducted on FAM 2
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2.5.3 Burst Test
Additional testing was performed on the swaged FAM to determine the factor
of safety with respect to the operating pressure. A burst test was performed on
FAM 3. For this test, the FAM is held in its zero contraction state as its internal
pressure is slowly ramped up until failure. While similar in nature to the ultimate
tensile failure test, this test seeks to focus on the effectiveness of the bladder clamp-
ing region of the end-fitting.
The results of the burst test are shown in Figure 2.19(a), while Figure 2.19(b)
has images of the FAM before and after failure. The FAM failed at a pressure of
280 psi, which provides the FAM a pressure factor of safety of 2.8 for the 0-100 psi
operating range.
Figure 2.19(c) provides an image of the FAM after removal from the MTS
machine with the braid rolled back to expose the underlying bladder. As with the
ultimate tensile failure test, the mode of failure was separation of the internal fitting
from the rest of the end-fitting assembly. The split in the bladder appeared to have
occurred subsequent to the separation of the internal fitting from the end-fitting.
Although this test targeted failure of the bladder, it is clear that the holding strength
of the braid was the point of weakness as it had been with the ultimate tensile failure
test. Past tests have displayed an actuation force at the burst pressure that was
below that of the ultimate tensile force [37]. This is simply an indication of the fact
that every FAM design has different limiting factors in the strength of their design.
With FAM 2 and FAM 3 having the same mode of failure, there would be an
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expectation that they would fail at similar loads. It is interesting to note that the
actuation force for the burst test was 2,639 lb, 492 lb higher than the maximum force
achieved from the ultimate tensile failure test that left the FAM unpressurized. This
could be due to any number of reasons. The FAM has a fixed length ? and respective
fixed braid angle ? for the burst test, while the FAM experiences an increase in its
length and braid angle for the ultimate tensile failure test leading to reorientation
and eventual jamming of the braid prior to failure. It is possible that the relative
motion and orientation of the fibers at failure could have detrimental effects on the
holding strength of the end-fittings. It is otherwise unknown if the internal pressure
of the FAM in the burst test would have any other effects on increasing the strength
of the FAM. The discrepancy in failure load could more easily be attributed to
slight differences in the fabrication of the two FAMs. An internal fitting of FAM 3
had a braid compression diameter that was two-thousands of an inch larger ? due
to limits in machinist precision ? that resulted in braid compression estimates of
15.9% versus 13.6%. This is compared to the desired design compression of 17%.
This slight variation of the braid compression is unavoidable, but may also indicate
how a few thousandths of an inch can have a large effect on the holding strength of
the FAM?s end-fittings.
2.5.4 Extended Use Failure
Many 7/8 in swaged FAMs have been made within this research since the
development of the presented swaged fabrication method. Besides the three FAMs
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(a) Test Results with failure at 280 psi (2,639 lbf)
(b) Images of before and after failure of FAM
(c) Failed FAM with braid rolled back to expose bladder
Figure 2.19: Burst test conducted on FAM 3
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tested for this research, twelve other FAMs have been fabricated, characterized,
and then used for various projects and research. Of the thirteen FAMs that were
not intentionally taken to failure, only three of them have failed after months of
extensive use.
One of those FAMs failed due to a fabrication flaw; epoxy had unintentionally
located on the edge of the braid during the swaging process making it rigid and
brittle. Over repeated cycles, braid fibers in this small region near the end-fitting
began to sever. This weakness in the braid left the underlying bladder unsupported,
leading to a pinhole leak in the bladder that made the FAM unusable.
The other two FAMs displayed a failure mode that was identical in nature
to the failures of FAM 2 and FAM 3 (Fig. 2.20). After many cycles of testing, the
FAM?s internal fittings slipped out of the end-fitting at a point when each FAM
was in its blocked force state at 100 psi. This shows that this failure mode was
not unique to the methods used to failure test the FAM, but was instead indicative
of the realistic failure mode of the FAMs through regular usage. This also made
clear that future improvements to increase the strength of the FAM should focus
on increasing the compression of the end-fittings. Subsequent efforts were, in fact,
made to increase the compression of the braid in the end-fitting.
2.5.5 Subsequent Improvements to end-fitting
Subsequent to the tests performed for this research, the compression of the
braid has been incrementally increased in an effort to increase the strength of the
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Figure 2.20: FAMs with failure due to internal fitting slipping out of end-fitting
assembly
end-fitting. Increasing the compression of the end-fittings increases the extrusion
force required to swage them. Therefore, the ability to increase the end-fitting com-
pression is an opportunity now afforded by the modifications made to the swaging
process as detailed in Section 4.5.
The compression levels have been increased to a maximum of 27% compression
of the braid, and a slight increase to the percent compression of the bladder. At
these compression levels, it is difficult to apply any more force through the lever
arm of the 10-ton press. Instead of smoothly and continuously moving through the
die during the extrusion process as it does at lower levels of compression, the swage
tube catches and gives with audible and visual hesitations as the swage tube resists
continued motion through the die. At these levels of force, the swage tube is more
prone to buckling, and the 10-ton press appears to be approaching the maximum
force that it can apply. Attempting to increase the compression of the end-fitting
beyond these levels, therefore, would likely require revisiting the design and sizing of
the FAM components, as well as require the acquisition of an upgraded press along
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with the fabrication of new swaging tools. For fabrication of all future 7/8 in FAMs,
the new level of braid compression of 27 % supplants the previous design objective
of 17% compression.
2.6 Conclusion
This research works to provide the following contributions: The initial develop-
ment of swaged FAMs by Robinson and Woods has been extended by this research,
with improvements made to the methods of FAM design and fabrication which can
serve as a reference for further development. The improvements are demonstrated
through the results of the 7/8 in FAM that was developed in this work. This de-
velopment of the 7/8 in swaged FAM increases the capabilities of FAMs, as well as
demonstrates consistency in the manufacturing process, required for greater adop-
tion of FAMs. With a source pressure of 100 psi, the resulting 7/8 in FAMs produced
a maximum force of over 1,000 lbs, and a maximum contraction of 35% while pro-
viding a safety factor of greater than two. The production of multiple FAMs with
nearly identical actuation characteristics speaks to the consistency of the improved
swaging process developed here. The improvements to the FAM swaging process
yielded an increased success rate and enhanced the efficiency of the FAM manufac-
turing process, and furthermore, enabled increases in end-fitting compression from
17% to 27% which would yield increases in the failure strength of the FAM.
The 7/8 in FAM developed in this work will serve ? and is already serving ? as
a basis for multiple research efforts. This FAM will be used for research involving
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improvements to FAM modeling and characterization methods. This FAM will also
be applied to actuate mechanisms being developed in a research setting that have
promise to be adopted for commercial use. The fabrication methods developed
in this work, with improvements to efficiency and safety of the extant methods,
help prepare for the scaling up of production efforts. The sizing and design methods
presented in this work can serve as a blueprint for the fabrication of FAMs at different
size scales to tailor the characteristics of the FAM depending on the application.
FAMs provide unique actuator characteristics that gives them distinct advan-
tages ? in comparison to other common forms of actuation ? in an engineering land-
scape that increasingly sees the adoption of ?bioinspired? designs. It is hoped that
swaged FAMs, using the improvements presented in this research, will help FAMs
to be viewed as a practical and desirable form of actuator for increased adoption in
future work.
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Chapter 3: Improvements in FAM Characterization Through Accu-
rate Control of Pressure1
3.1 Introduction
McKibben actuators are being applied to an increasingly diverse number of
applications due to their many desirable characteristics including their high force-
to-weight ratio, simplicity, and inherent compliance. Previous work has established
McKibben actuators as an ideal form of actuation when specific work, specific power,
and power density are of primary importance in actuator selection [18]. With a
basic construction consisting of an elastomeric bladder, surrounding load-bearing
braided sleeve, and two end-fittings, an internally pressurized McKibben actuator
can provide a contractile force of more than ten-thousand times its own weight,
and achieve maximum strokes of up to 40% of its resting length [37]. This simple
construction makes them readily adaptable to various environments, highly durable
with a long fatigue life [37, 57], and easily scalable with diameters ranging from
4 mm (0.15 in) [18] to 40 mm (1.5 in) [27] and forces ranging from several pounds to
1Chambers, J.M., and Wereley, N.M., 2021. Influence of hydraulic versus pneumatic working
fluids on quasi-static force response of fluidic artificial muscles. Journal of Intelligent Material
Systems and Structures 32.3 (2021): 385-396.
88
several thousand pounds.
Initially conceived by Joseph L. McKibben in the early 1950?s as a basic fluid
driven stroking device for use with his daughter?s arm orthotic, the application of
McKibben actuators has expanded to numerous robotic and aerospace mechanisms
[6,58]. Since its inception, the McKibben actuator, known more generally as a fluidic
artificial muscle (FAM), has typically been operated pneumatically. Incremental
improvements in performance and durability have been made through adjustments
in FAM geometry, material selection, end-fitting design, and fabrication methods.
Research efforts have also been focused on the accuracy and fidelity of modeling
pneumatic FAMs for control purposes. Using empirical test data, highly accurate
semi-mechanistic models have been developed for pneumatic FAMs [48].
In recent years, research efforts for FAMs has expanded to the use of different
working fluids as the source of internal pressure. FAMs have typically been operated
pneumatically, however, as their range of applications has expanded to aqueous
[59,60] and aerospace environments [34,35,61], it has become necessary to investigate
working fluids that are perhaps better suited to these roles. As an example, for
typical use in atmospheric environments, surrounding air can easily be obtained,
compressed and stored, and vented to the surrounding environment making air a
sensible working fluid choice. The same logic can be applied for the use of hydraulic
actuation in aqueous environments.
The comparison of working fluid compressibility can also differentiate fluid
selection. Fluid compressibility affects the stiffness of FAMs, with air providing
more compliance than what hydraulic fluids provide. However, the incompressibility
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of hydraulic fluids results in better position control accuracy, a quicker response to
load variations, and higher force and pressure bandwidth [33]. Actuation efficiency
is also affected, with hydraulic efficiencies shown to be about 60%, compared to
about 30% for pneumatics [19, 62, 63]. Other considerations for fluid choice include
weight, viscosity, heat capacity, thermal conductivity, and corrosion effects [63].
Comparisons between hydraulic fluids such as water and hydraulic oil have also
been drawn. Hydraulic oils provide lubrication and corrosion resistance benefits to
the pressure system, but also require specialized FAM bladder materials that do not
deteriorate upon contact with oils [19].
Few studies have investigated hydraulically powered FAM actuation [15,27,32,
59,64,65]. Fewer still have examined the direct comparison between pneumatic and
hydraulic FAM actuation [16,19,33]. Focchi (2010) performed a comparative study
focusing on the dynamics and control of water versus air actuation by analyzing
the stiffness variation of water and air with respect to pressure, and experimentally
validated that pressure bandwidth, force bandwidth, and position accuracy can be
increased using water. Meller (2014) investigated the efficiency of air versus water
actuation with results showing 35-60% efficiency with hydraulic actuation, while
pneumatic actuation only achieved an efficiency of about 25%.
Tiwari (2012) attempted to back the assumption from past pneumatic FAM
(PAM) models that have indicated that pneumatic and hydraulic powered FAMs
should provide the same force output independent of working fluid choice in quasi-
static operation. With a 1/8 in FAM, Tiwari (2012) conducted constant pressure
tests with water and air, but results showed significant discrepancies between their
90
response curves. The force response discrepancies between tests of the two working
fluids was cited as likely being due to inaccurate pressure control. This left their
hypothesis of working fluid independence of the quasi-static force response with an
uncertain conclusion.
The research described in this chapter is an experimental investigation of the
effect of working fluid choice on the quasi-static pressure dependent force-contraction
response of FAMs. Using air or water as the working fluid, characteristic isobaric
force-contraction response curves are compared for both a large and small scale
FAM. Learning from Tiwari?s work where inaccurate pressure control resulted in an
inconclusive comparison between air and water actuation, in this research, hydraulic
and pneumatic pressure systems were developed to provide precise and accurate
control of pressure to ensure a reliable comparison of the two working fluids. Back-
to-back testing of FAMs using water and then air as the working fluid was conducted
to show the independence of working fluid choice on the quasi-static response of
FAMs.
3.1.1 Basis of Work
The hypothesis that the quasi-static force response of FAMs is independent
of working fluid choice can be explained through analysis of previously developed
models of PAMs. These models have been developed and refined over the years to
provide an accurate representation of a PAM?s quasi-static behavior. Most recently,
the fidelity of a semi-mechanistic model based on a force balance approach was im-
91
proved by including empirically fit coefficients that represent the material properties
of the elastic bladder [48]. For this model, force is a function of the working fluid
pressure, P , and the PAM actuator length, L, as:
F (P,L) = FG + Fbld,c + Fbld,z (3.1)
where
FG = P?R(2(2tan)2? ? 1) (3.2)
= VbldFbld,c ?z (L 2 ) (3.3)
= ? tLFbld,z ?c 2 (3.4)?RN2
where the number of turns of a single braid fiber about the actuator diameter, N ,
and the volume of the incompressible bladder, Vbld, are fixed values that describe the
construction of the actuator. Braid angle, ?, bladder thickness, t, actuator radius,
R, and bladder stress in the longitudinal and circumferential directions, ?z and ?c
respectively, are all values that are directly or indirectly functions of the current
actuator length, L.
Through examination of the terms of this force model, it can be seen that
actuation force is dependent only on PAM geometry (R, ?, Vbld, t, L, R, N), material
composition (?c, ?z), and on the working fluid pressure (P ). The first term of
the model, commonly known as the Gaylord force, takes into account the internal
pressure acting on the bladder, and how this pressure is transformed into an axial
force through the surrounding braid (Fig. 3.1(a)). The second and third terms of the
model equation account for energy losses due to straining the bladder during inflation
(Fig. 3.1(b)). With no dependencies on fluid properties other than pressure, it can
be assumed that this force response model is transferable to use for FAMs with
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(a) Components that describe transfer of pressure force to axial force
through the braid (Equation 3.2)
(b) Components that describe energy loss from stretching the bladder
(Eqs. 3.3 and 3.4)
Figure 3.1: Illustrations of the components of the force-balance model given in
Equation 3.1
working fluids other than air. This research was conducted to confirm the fidelity of
this assumption, and therefore extend the use of current PAM models to use with
FAMs as a whole independent of the applied working fluid.
3.2 Experimental Test Setup
To compare FAM actuation with different working fluids, the FAM was ex-
perimentally tested with both air and water as the working fluid. The method
of characterizing the actuators pressure-force-contraction response follows that of
previous PAM characterization testing [18]. The FAM is quasi-statically and iso-
barically stroked from its resting length to its pressure dependent free-contraction
93
(zero force) length. For this research, the FAM was cyclically tested at set pressures
of 20-100 psi in 20 psi increments. Test cycles were performed at 0.02 in/sec, which
is slow enough to ensure the quasi-static test assumption holds.
Testing for this research was conducted in a 22 kip MTS servo-hydraulic test
machine that directs the stroking of the FAM, while recording external inputs of
pressure (Omega PC209-200G5V), and force (Honeywell Model 31 Mid). Testing
of each working fluid was conducted successively without removal of the FAM from
the test machine in order to retain the FAMs reference length state and alignment.
Fixtures attaching the FAM to the test machine were designed to be readily adapted
to the hardware required to allow flow of each working fluid into the FAM without
removal of the fixtures from the test machine.
3.2.1 FAM Test Specimens
Both a large and small scale FAM were fabricated for comparison testing of
air and water driven actuation to ensure size independence of observed phenomenon
(Fig. 3.2). The small-scale FAM is a 3/16 in diameter FAM as is used in [18]. This
FAM typically produces a maximum force in the blocked force condition of about
50 lbf, and can typically achieve a maximum contraction of 25-35%. The large-scale
FAM is a 7/8 in diameter FAM as is used in [28], and typically produces a maximum
force of about 1000 lbf, and has typically achieved a maximum contraction of 30-
40%. The construction materials and sizing of each FAM is provided in Table 3.1.
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Figure 3.2: The FAMs used in this study with outer diameters of 7/8 in (top) and
3/16 in (bottom)
Table 3.1: Construction and specifications of the 3/16 in and 7/8 in diameter FAMs
3/16 in Dia. FAM 7/8 in Dia. FAM
Active Length 6-27/64 in 8-49/64 in
Braid Polyethylene Terephthalate (PET) Kevlar
End-Fitting Machined Alum. 10-32 Threaded Rod Machined Alum. 6061-T6
Swage Tube Alum. 6061-T6 Tubing Alum. 6061-T6 Tubing
Total Length 6-59/64 in 10-15/16 in
Weight 0.01 lbm 0.18 lbm
3.2.2 Pressure Systems
Direct comparison between the two working fluids requires precise and accurate
control of the input pressure to the FAM to maintain a true isobar for each cyclic
test. Careful regulation of pressure is imperative to ensure that any discrepancy
between the FAM force response for each working fluid can be positively attributed
to the working fluid and not to an imprecise feed pressure.
This requirement drove the development of pressure control systems for water
and air. For previous PAM testing, air pressure had been controlled by a variable
pressure regulator. While using a pressure regulator is commonly sufficient for
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holding a fixed volume system at a constant pressure, the inflow/outflow of air
required for internal volume changes of a stroking PAM often results in inconsistent
control of outlet pressure to the PAM. An inherent drawback to pressure regulators is
their tendency to exhibit an observed outlet pressure that is inlet pressure, flow rate,
and flow direction dependent [66]. This phenomenon has been apparent in previous
PAM tests, where the regulator has constricted air inflow/outflow resulting in higher
than set pressures when the PAM internal volume is decreasing (during eccentric
motion), and lower than set pressures when the PAM volume is increasing (during
concentric motion). If a person were to assume the test pressure was isobaric,
this would result in an increase in apparent hysteresis, a decrease of the apparent
contraction, and an increase in apparent maximum force. Figures 3.3 and 3.4 show
an example of this described phenomenon by comparing the results of testing with
(black lines) and without (colored lines) the addition of a controllable proportional
valve to accurately control pressure (as will be described in subsequent sections)
for a small and large scale PAM respectively. Figures 3.3(b) and 3.4(b) show a
clear discrepancy in measured pressure between the use of the pressure regulator
with and without the controllable valve. This pressure inaccuracy resulted in the
differences in force response as seen in Figures 3.3(a) and 3.4(a), with an appreciable
difference in magnitude of the hysteresis and maximum force for each tested pressure.
Although the use of a pressure regulator alone was previously deemed sufficient for
basic PAM characterization, the examples shown in Figures 3.3 and 3.4 demonstrate
how constant pressure characterization tests have not been truly isobaric. With
inconsistent pressure variations, this would be an insufficient system for the direct
96
comparison of different working fluids, and requires an improved pressure control
system for handling air.
(a) Force-contraction response
(b) Measured pressure during response testing
Figure 3.3: 3/16 in FAM characterization testing with (black lines) and without
(colored lines) the addition of a controllable valve for accurate pressure control
To perform water FAM testing, a hydraulic pressure system had to be devel-
oped. The incompressibility of water results in a hydraulic pressure system that is
very different in nature to that of the open-circuit air pressure system. There are
a few examples of pressure systems that have been developed for hydraulic FAM
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(a) Force-contraction response
(b) Measured pressure during response testing
Figure 3.4: 7/8 in FAM characterization testing with (black lines) and without (col-
ored lines) the addition of a controllable valve for accurate pressure control
testing by other researchers [15,33,65]. Methods for building water pressure include
using a pump with valve restricted flow [65], a pump with accumulator and water
compatible proportional valve [33], or a piston-driven, closed-volume system [15].
An overview of the air and water pressure systems used for this research are detailed
in subsequent sections.
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3.2.2.1 Pneumatic Pressure System
Shown in Figure 3.5, the air pressure system for pneumatic FAM testing is the
same as what has been used for previous PAM testing with the addition of a propor-
tional valve to more precisely regulate air flow. A portable 200 psi air compressor
(DeWalt D55146) was used to compress and store pressurized air. A pressure regu-
lator (Excelon R73G) with digital pressure transducer (SSI Technologies MGA-9V)
was used to set a constant supply pressure of 115 psi, 15 psi above the maximum
pressure to be tested. A 5/3-way proportional control valve (Enfield Technologies
LS-V05s) was then used to directly control FAM pressure. Airflow in and out of the
FAM was directed through the top of the FAM, while the bottom of the FAM was
closed. A controller (Arduino Uno) was used to direct proportional valve actuation
to control FAM pressure. The Arduino received instantaneous pressure state values
for controller feedback through a pressure sensor (Omega PX209-200G5V) mounted
to the fixture just above the air inlet of the FAM. A 1000 lbf load cell (Honeywell
Model 31 Mid) was fixed in line with the FAM to measure the contraction force
output of the FAM.
3.2.2.2 Hydraulic Pressure System
Testing of a water driven FAM required the development of the hydraulic pres-
sure system as shown in Figure 3.6. The incompressible nature of water, along with
the limited availability of water compatible components dictated that the hydraulic
system would be very different in nature from the pneumatic system. The hydraulic
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(a) Control system schematic (b) Experimental test setup
Figure 3.5: Pneumatic pressure control system
circuit uses a continuously running, variable-speed pump. The fixed displacement
gear pump (Fluid-o-Tech FG204) requires a fluid reservoir and inline filter to pro-
vide circuit pressure. To ensure that the pump does not have to sustain damaging
hydraulic hammer effects from constantly switching directions to control pressure
about a set-point, the FAM was made open at each end-fitting to allow a continuous
flow of water in a single direction. Therefore, for pressure to build up in the circuit,
flow is restricted downstream of the FAM by a needle valve (max orifice diameter of
0.175 in). The needle valve orifice diameter is reduced to the point where the pump
can provide enough dynamic range without saturating: constrained enough to en-
able the maximum test pressure to be obtained, yet open enough to allow outflow
of the system for FAM internal volume reductions without having to reverse pump
direction. As a precaution to avoid sudden spikes in pressure due to the incom-
pressibility of water, an adjustable relief valve (Watts LF 530-C STD) is positioned
between the pump and FAM to relieve pressure if it exceeds 110 psi. An Arduino is
100
used to control pump speed to achieve the desired isobaric pressure.
(a) Control system schematic (b) Experimental test setup
Figure 3.6: Hydraulic pressure control system
3.2.3 Pressure System Control
A proportional-integral (PI) closed-loop feedback controller was implemented
and tuned to regulate each pressure system to the desired set isobar. The control
input to the system acts to minimize the error, e(t), between the desired set pressure,
Pset, and the measured pressure, Pmeas. The controller receives a pressure sensor
voltage as input, and outputs a control signal in the form of a 5 V pulse-width
modulation (PWM) signal. The Arduino produces a control signal value, u(t), of
0-255 as a digital representation of the PWM output that translates to a 0-100%
duty cycle. This 0-100% duty cycle produces an approximation of the 0-5 V analog
input values required for control of the proportional valve and pump.
The control signal, u(t), is calculated using Equation 3.5 for each control loop
iteration as illustrated by the block diagrams in Figure 3.7. The feedback control
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signal is manipulated by two user-defined tuning gains: the proportional gain, Kp,
and the integral gain, Ki. To limit integral windup and saturation of the control
input to the proportional valve/pump, both the integral term and overall control
signal are constrained from commanding a value below 0 (0 V) or above 255 (5 V).
A derivative term was not included to avoid the effects of noise from the pressure
sensor.
The feedback controller is augmented by a system dependent open-loop control
component, v, as defined in Equation 3.6. This open-loop control component reduces
the role of the feedback controller to disturbance rejection, and results in improved
pressure set-point tracking.
?
u(t) = K tpe(t) +Ki 0 e(?)d? + v (3.5)
where
???????v(P? set
) if hydraulic
v = ?????
(3.6)
2.5V if pneumatic
Values for the tuning gains, Kp and Ki, as well as the models used for the open-
loop control component, v, are dependent on the applied pressure system and FAM,
and are detailed in subsequent sections below. Control signal values are presented in
the equivalent 0-5 V analog output of the PWM signal (opposed to the 0-255 signal
used by the Arduino).
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(a) Pneumatic control
(b) Hydraulic control
Figure 3.7: Controller block diagrams
3.2.3.1 Pneumatic Pressure Control
The proportional valve of the pneumatic system controls inflow/outflow of air
to the FAM by accepting a 5 V PWM signal from the controller that dictates the
position of its valve spool using a bi-directional linear force motor. Using the valve
control diagram in the schematic of Figure 3.5(a) as reference, a command signal of 0
(0 V) positions the valve spool closest to the motor, a signal of 128 (2.5 V) positions
the spool in the middle, and a 255 (5 V) signal positions the spool farthest from the
motor. For this system configuration, this results in the valve acting to exhaust air
from the FAM with a command signal of less than 2.5 V, the valve closed at 2.5 V
(with a small deadband), and the valve acting to add pressurized source air to the
FAM with command signal voltages greater than 2.5 V.
For the pneuamtic system, the open-loop control component, v, takes the form
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of a bias term equal to 2.5 V (Eq. 3.6), which is included to position the spool valve
in the middle (valve closed) position when pressure error is equal to zero. Figure
3.7(a) shows the controller block diagram for the pneumatic pressure control system.
3.2.3.2 Hydraulic Pressure Control
The pump providing pressure to the hydraulic system accepts a 5 V PWM
signal from the controller, u(t), that correlates to a 0-5000 rpm pump speed (Eqs.
3.5, 3.6). The hydraulic pressure system is a closed-circuit system that requires
the pump to continuously displace water to retain system pressure. With the needle
valve position fixed, each desired pressure set-point has a corresponding pump speed
in the steady-state condition. Therefore, to improve closed-loop feedback control
performance, an open-loop model, v(Pset), was formed by defining the required signal
input to the motor (and respective pump speed) to achieve the desired range of set
pressures of 0-100 psi in steady state conditions. The characterization results for the
large and small scale FAMs are shown in 3.8, where it can be seen that the empirical
data was fit to a third-order polynomial to serve as a phenomenological model of
the required controller input to achieve desired set pressures in the steady-state
condition. The resulting open-loop model equations fit to the data in Figure 3.8 are
given for the small and large FAMs in Equation 3.7 respectively.
This model could serve as an open-loop controller on its own, but alone would
not be able to rapidly correct for the volume changes of the FAM in the system.
Therefore, the controller input serves as a feedforward term for the PI feedback
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controller loop (Fig. 3.7(b)). This leaves the PI feedback controller with the sole
task of adjusting pump speed according to the pressure error, e(t), to compensate for
volume changes of the system as a result of the FAM shape change. It is important to
note that the needle valve orifice area was adjusted between testing of the large and
small scale FAMs, and that more generally, the characterizations shown in Figure
3.8 are dependent on the position setting of the needle valve.
5
3/16 in FAM
7/8 in FAM
4
3
2
1
0
0 20 40 60 80 100
Pressure, psi
Figure 3.8: Open loop input, v(Pset), for hydraulic control of each tested FAM
?? 3 2
( ) = ?(1.6e?6)Pset ? (3.9e?4)Pset + 0.06Pset + 0.48 if 3/16 in dia.v Pset (3.7)(1.6e?6)P 3set ? (3.6e?4)P 2set + 0.05Pset + 0.30 if 7/8 in dia.
3.2.4 Controller Tuning
The controller was tuned for the pneumatic and hydraulic pressure circuits for
both the large and small scale FAMs resulting in a total of four sets of gains. For
each of the four cases, the proportional gain, Kp, and integral gain, Ki, were tuned
through an iterative process. To get a baseline approximation of the required gains,
105
 v(P  ), V
set
the FAMs were initially tabletop tested with zero applied force. Tuning continued
with the FAM in the MTS machine, where the ability to hold set pressures was tested
in the blocked force state (resting length state) of the FAM. Finally, controller gains
were tuned to maintain pressure isobars while cycling the FAM between the blocked
force and free contraction states as would be done for the final characterization
testing.
To maintain an isobar for the performed cyclic FAM testing, the main task of
the controller is to adjust fluid flow rates to account for the change in volume of the
FAM. The internal volume of the pressurized bladder of the FAM has been shown
to be predominantly a function of the contraction state of the FAM independent
of fluid pressure [19]. This made the rates of change of FAM volume with each
contraction cycle somewhat predictable during each contraction cycle, and helped
clarify points of interest for tuning purposes.
The steady increase in FAM volume with concentric motion, and the steady
decrease in volume with eccentric motion, are handled primarily with the slower
moving integral term of the controller input. The rapid change in volume that occurs
near the resting length state of the FAM is handled primarily by the proportional
term of the controller input. Both gains were set to be high enough to react to
these changes quickly, but are also restrained to mitigate any detrimental effects
of their presence. The proportional gain setting is limited to ensure that it does
not overreact to pressure sensor noise, causing the observed pressure to pulsate and
bounce around the desired set-point pressure. The integral gain is limited to restrain
integral windup that can occur with the sudden change of volume near the resting
106
length state of the FAM, and where the FAM changes in direction at the maximum
contraction state of the FAM for each cycle.
The ratio of the proportional gain to the integral gain for each working fluid
was largely affected by the compressibility of each fluid. The compressibility of air
made the pressure less reactive to sudden controller inputs from the proportional
gain, while the incompressibility of water made it much more reactive to the pro-
portional controller inputs and sensor noise. This resulted in an emphasis on the
proportional gain for controlling air pressure, and an increase in the integral gain
for controlling water pressure. As expected, the larger changes in volume of the
7/8 in diameter FAM required higher gain values than those required for the 3/16 in
diameter FAM. The gains used for the final water and air characterization testing
are provided in Table 3.2. Integer value gains were chosen while using units of psi
for pressure, P , and the control signal, u(t), in the digital 0-255 PWM value form
required by the Arduino (reference Equation 3.5).
Table 3.2: The gains used for testing each FAM in water and air (psi pressure input,
PWM control output)
Controller Gains
Gain Working Fluid
FAM Air Water
3/16 in Dia. Kp 1 1
Ki 2 3
7/8 in Dia. Kp 4 5
Ki 2 5
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3.3 Results
Accurate pressure control was achieved for both the hydraulic and pneumatic
pressure systems, resulting in a high fidelity comparison of the response of the two
tested working fluids. The results of the pressure control, as well as the characteri-
zation of the FAM response comparing air and water working fluids are detailed in
the following sections. The characterization test data is then compared to the force
response model to confirm the data?s agreement with the model.
Table 3.3: Average pressure and standard deviation for the force response charac-
terization test at each test pressure (3/16 in FAM)
Commanded Set Pressure: 20 psi 40 psi 60 psi 80 psi 100 psi
Average Pressure, psi (Air) 21.06 41.24 61.25 81.39 101.50
Average Pressure, psi (Water) 21.05 41.22 61.22 81.30 101.41
Standard Deviation, psi (Air) 0.23 0.18 0.21 0.26 0.29
Standard Deviation, psi (Water) 0.15 0.15 0.22 0.29 0.2
Table 3.4: Average pressure and standard deviation for the force response charac-
terization test at each test pressure (7/8 in FAM)
Commanded Set Pressure: 20 psi 40 psi 60 psi 80 psi 100 psi
Average Pressure, psi (Air) 20.08 40.00 59.89 80.04 100.19
Average Pressure, psi (Water) 19.64 39.96 59.96 80.03 100.11
Standard Deviation, psi (Air) 0.28 0.34 0.31 0.27 0.29
Standard Deviation, psi (Water) 0.39 0.43 0.40 0.42 0.37
108
100
80
60
100 psi
40 80 psi
60 psi
20 40 psi
20 psi
Water
0
0 0.1 0.2 0.3 0.4
Contraction, L /L
0
(a) 3/16 in FAM
100
80
60
100 psi
40 80 psi
60 psi
20 40 psi
20 psi
Water
0
0 0.1 0.2 0.3 0.4
Contraction, L /L
0
(b) 7/8 in FAM
Figure 3.9: Measured pressure for each force response characterization
3.3.1 Pressure Control Results
Results of the measured pressure during characterization testing for each FAM
is given in Tables 3.3 and 3.4. Average pressure and standard deviation values are
given for both air and water testing. Together, these values provide a depiction of
how well the pressure systems provided a true isobar for each set pressure test to
enable a good comparison between each working fluid. Figure 3.9 illustrates the
pressure with respect to FAM contraction for each working fluid characterization
109
Pressure, psi Pressure, psi
test for each size FAM.
Pressure control for the 3/16 in FAM proved to be accurate, with a maximum
averaged pressure differential of 0.09 psi. Standard deviation values were all below
0.29 psi, indicating that the pressure didn?t vary much around the desired isobar.
Figure 3.9(a) provides a qualitative look at how well pressure of the two working
fluids corresponded throughout the cycling of the 3/16 in FAM.
Pressure control of the 7/8 in FAM also proved to be very accurate. With
exception to the 20 psi test (0.44 psi differential), the maximum averaged pressure
differential was 0.08 psi. Figure 3.9(b) provides a depiction of the pressure correla-
tion between the two working fluid tests. The only point of discrepancy in pressure
is when the concentric motion of the FAM begins while near the zero percent con-
traction point. At this point of each pressure test, there is a relatively sudden jump
in the required inflow of water due to an instantaneous jump in FAM volume. This
effect could be reduced by increasing the proportional gain of the controller at the
detriment of increased variation in the pressure about the set-point for the remain-
der of each test. Overall, this minor discrepancy near the resting length of the FAM
did not produce a noticeable effect on the force response results.
3.3.2 Characterization Results
Characterization testing of the two FAMs with each working fluid showed
nearly identical results between air and water for all tested pressures. Results from
each characterization test are plotted in Figure 3.10. To provide a quantitative
110
comparison of the force response difference between the two working fluids, the
root-mean-square error (RMSE) between the two responses was calculated for each
pressure. The RMSE was calculated using the following equation:
? ?
RMSE 1avg = ((Fi)air ? (Fi) 2water) (3.8)Ni i
where F is the experimental force for each test, and N is the total number of data
points of each test with i representing each respective measurement point from 1 to
N . RMSE values for both working fluids at each pressure are given in Table 3.5. The
resulting RMSE values are all relatively low, with higher values for the 7/8 in FAM
resulting from the overall higher magnitude of forces it achieves. The RMSE values
for the results shown in Figures 3.3(a) and 3.4(a) , where the new pressure control
system was compared to the previous method of just using a pressure regulator, is
provided for a relative point of comparison for RMSE values.
Table 3.5: The force response root-mean-square error (RMSE) at each test pressure.
20 psi 40 psi 60 psi 80 psi 100 psi Average
3/16 in FAM, lbf (Fig. 3.10(a)) 0.19 0.27 0.45 0.46 0.75 0.41
7/8 in FAM, lbf (Fig. 3.10(b)) 2.29 2.39 1.41 4.85 7.65 3.72
3/16 in FAM, lbf (Fig. 3.3(a)) 0.10 0.98 0.54 0.79 0.86 0.65
7/8 in FAM, lbf (Fig. 3.4(a)) 19.35 12.18 8.97 8.19 8.41 11.42
3.3.3 Modeling Results
The FAM force response model (Eq. 3.1) served as the basis for proving
the original hypothesis: the independence of the FAM?s quasi-static force response
111
60
100 psi
50 80 psi
60 psi
40 40 psi
20 psi
Water
30
20
10
0
0 0.1 0.2 0.3 0.4
Contraction, L /L
0
(a) 3/16 in FAM
1200
100 psi
1000 80 psi
60 psi
800 40 psi
20 psi
Water
600
400
200
0
0 0.1 0.2 0.3 0.4
Contraction, L /L
0
(b) 7/8 in FAM
Figure 3.10: Measured actuation force for each characterization
from the working fluid used. We have shown this independence of working fluid
through the results in the previous section. To come full circle, it is now necessary
to demonstrate agreement between the characterization data from this research, and
the original FAM model used for justification of the hypothesis.
The force response model is applied following the same methods as used in [48].
The longitudinal and circumferential bladder stresses, ?c and ?z, are defined using
a third-order stress-strain relation as follows:
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Force, lb Force, lb
?c = E1c + E 2 32c + E3c (3.9)
?z = E 21z + E2z + E33z (3.10)
where c and z are the strains in the bladder in the circumferential and longitudinal
directions respectively. As in [48], each of the modulus terms, E1?3, are solved
using a least-squares optimization function, and their values are constrained to be
a linearly increasing function of pressure.
60
100 psi
50 80 psi
60 psi
40 psi
40
20 psi
Model
30
20
10
0 0.1 0.2 0.3 0.4
Contraction, L /L
0
(a) 3/16 in FAM
1200
100 psi
1000 80 psi
60 psi
800 40 psi
20 psi
600 Model
400
200
0
0 0.1 0.2 0.3 0.4
Contraction, L /L
0
(b) 7/8 in FAM
Figure 3.11: Comparison between the averaged characterization data, and the force
response model
113
Force, lb Force, lb
The model is compared against the data from the hydraulic actuation char-
acterization tests for both the 3/16 in and 7/8 in FAMs. With the pneumatic and
hydraulic test data being virtually identical, it is assumed that the same results
would be achieved with application of the pneumatic test data. This form of the
model does not capture the hysteresis in the FAM?s force response due to friction,
so the test data is averaged for each isobaric cyclic test for comparison to the model.
The comparison between the averaged data and model, given in Figure 3.11, show
a close correspondence in their results. This proves the adherence of the model to
the characterization data acquired in this research.
3.4 Conclusion
This work investigated the effect of working fluid choice on the quasi-static
force response of FAMs. Past studies have shown how the working fluid used with
a FAM can affect its dynamic behavior and efficiency, but none had conclusively
demonstrated the assumption that the quasi-static force response is independent of
the working fluid used to power it.
To investigate this assumption, a large and small scale FAM was fabricated,
and had their quasi-static force response characterized using water and air as the
working fluid. To ensure an accurate comparison between the two working fluids,
pneumatic and hydraulic pressure control systems were developed to accurately and
precisely control fluid pressure for each isobaric force-contraction characterization
test. The pressure systems were then used to perform back-to-back air and water
114
characterization tests which resulted in nearly identical isobaric force-contraction
response curves for the two fluids for both FAMs. Finally, agreement between the
characterization data and the original force response model was confirmed, com-
pleting the adherence of the proven hypothesis to the model. This result confirmed
that the FAM?s quasi-static force response is independent of working fluid, allow-
ing previously developed pneumatic FAM force response models to be adopted to
FAMs using any working fluid, and motivates the use of FAMs with a wider range
of application dependent working fluids.
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Chapter 4: A Photogrammetric Method for Measuring FAM Shape1
4.1 Introduction
Fluidic artificial muscles (FAMs) are a form of soft actuator that have been
applied to an expanding number of applications because of their unique characteris-
tics such as low weight, simple construction, inherent compliance, and high specific
force and specific work capabilities. In a design environment where optimization
techniques are increasingly used for mechanism development and packaging, it is be-
coming more important than ever to have accurate model representations of device
components, including actuators, to accomplish effective design outcomes. Actuator
design requirements typically include force and stroke values, while common actu-
ator design constraints include actuator size, energy efficiency, and storage require-
ments. Models have been developed that address each of these design constraints
and requirements for FAMs, but many of those models lack the accuracy required
for their use in a world of increasingly constrained design spaces, and improvements
to those models have remained stagnant for years.
1Chambers, J.M., and Wereley, N.M., 2021. Photogrammetric Measurement and Analysis of
the Shape Profile of Pneumatic Artificial Muscles. Actuators. Vol. 10. No. 4. Multidisciplinary
Digital Publishing Institute
116
Figure 4.1: FAM with its shape profile traced in green along the edge of the braid
While continual improvements have been made in modeling the force response
of FAMs, there has been little recent progress in the development of methods for
characterizing the size and overall shape profile of FAMs during operation, param-
eters commonly used for calculating efficiency, fluid storage, and flow rate require-
ments. The FAM?s shape profile (Fig. 4.1) has long been approximated as a circular
cylinder based on visual observations, and has often been cited as a suspected source
of modeling error [13, 20]. This research seeks to remedy this by providing a new
method of accurately measuring the FAM?s shape profile.
With the simple construction of an elastomeric bladder, a surrounding heli-
cally braided load-bearing sleeve, and rigid end-fittings at each end, a pressurized
contractile FAM can provide a tensile force along its longitudinal axis. Upon internal
pressurization of the FAM, the radial pressure force of the internal fluid is trans-
ferred through the bladder to the braided sleeve. The braided sleeve transforms that
radial force into an axial force expressed at each end-fitting. The actuation force
117
produced by a FAM is a function of its active length and internal pressure states.
The actuation force is at a maximum at its resting length, when the braid angle,
defined with respect to the radial axis of the FAM, is at a maximum. As the FAM
contracts, the braid angle reduces and the bladder strain increases, resulting in a
reduction of actuation force until the FAM reaches a pressure-dependent maximum
free contraction in its zero-actuation-force state.
The shape profile of a FAM changes with contraction of its length (Fig. 4.2(a)).
At its resting length, the FAM has an apparent constant diameter and braid angle
along its length. As a FAM contracts in length, its diameter expands and the braid
angle reduces as a function of the position along the length of the FAM. The braid
geometry is constrained only at each end-fitting where the diameter is equal to the
initial diameter. The shape profile is also highly dependent on the FAM, with FAMs
of identical material construction having visible differences in their shape profile
(Fig. 4.2(b)). FAMs have had shape profiles that range from a more cylindrical
mid-section shape to ones that have constant variations in curvature along their
entire length. The bulbous shape of a FAM is often approximated as a cylindrical
profile with a constant diameter along its entire length (Fig. 4.3). Assuming the
braid is inextensible, the cylindrically approximated shape enables a simplification
of analysis through the so-called triangle relationship that correlates the FAM?s
diameter, braid angle, and active length. This cylindrical approximation serves as
the basis for many existing FAM models, including models for volume and actuation
efficiency, as well as for force response models that have been developed using both
force-balance [1] and principle of virtual work based approaches [5].
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(a) A FAM in its resting length state (left), 17.5% contraction state (center),
and 35% (max) contraction state (right)
(b) Three different FAMs of identical diameter and construction in their
free-contraction state that have noticeably different shape profiles
Figure 4.2: Variation of the FAM?s shape profile for (a) a single FAM in different
states of contraction, and for (b) three different FAMs in their free-contraction states
119
Figure 4.3: The actual profile that has a varying radius along its length (left) versus
the cylindrical approximation of this shape profile with a constant radius along its
length (right)
Past modeling efforts of FAMs have often centered around rudimentary repre-
sentations of their shape profile. While the cylindrical approximation of the shape
profile simplifies analysis and modeling, it has often been a suspected source of er-
ror, thus making it the subject of more detailed modeling efforts [8,9,13,16,18?20].
Many researchers have inferred that the reduction in diameter near each end-fitting
produces what is often termed as end-effects that act to decrease the force response
of the FAM. Modeling corrections that take end-effects into account for the prin-
ciple of virtual-work models often seek to more accurately calculate the internal
volume of the FAM. Correction efforts for the force-balance modeling method have
centered around the effective reduction of active length due to the non-cylindrical
shape profile. Tondu and Lopez [9] were the first to use an end-effect correction
factor by including a tuning parameter that effectively amplified the contraction
ratio in the force response equation. Kothera et al. [13] took a similar approach
by reducing the effective active length in the model by two-times the difference of
120
the current and initial radius. Most models that have taken the shape profile of
the FAM into account have maintained a cylindrical approximation for the FAM?s
midsection, while applying shape corrections to the regions near the end-fittings at
each end. Assumed shape profiles have included a stepped cylindrical approxima-
tion [8], a 90 deg circular arc [18, 67], a semi-ellipsoid [16, 19], a conic frustum [8],
and the partial surface area of an elliptic toroid [20]. These shape profiles are often
selected based on qualitative observations and ease of computation. Only Meller et
al. [19] has noted that volume measurements appear to show that the cylindrical
approximation may be sufficient for modeling the internal volume of the FAM.
Our experience with FAMs has found that the overall shape profile of a FAM is
variable, and often exhibits undulating curvature along the full length of the FAM.
Having a quantitative method of measuring the shape profile of FAMs would be
highly beneficial for improving the accuracy of models that rely on the geometry of
the FAM.
Having a more accurate measurement of the shape profile would also aid in
characterizing the FAM?s internal volume which is often used in defining an actu-
ator?s system requirements such as the working fluid flow rate and storage volume
requirements. First-cut estimations of internal volume are often made using the
cylindrical approximation of the FAM, while end-effects corrections are sometimes
included to account for the reduced volume near each end-fitting with unknown im-
provements to accuracy. Some researchers have turned to quantifying the internal
volume through experimental approaches [19]. While experimental methods should
provide accurate calculations of internal volume, they often require specialized test
121
equipment. To date, experimental methods have required either submerging the
FAM in water, or pressurizing the FAM using water from a closed graduated cylin-
der [19], and then tracking the change in water level as the FAM contracts in length.
The specialized equipment required for characterizing the volume using these meth-
ods not only subject the FAM to fluid that its material composition may not be
compatible with, but also requires adjustment, replacement, or even redesign of the
test setup components depending on the size and length of the FAM to be tested.
Conversely, an experimental method that can test FAMs of all sizes while not re-
quiring specialized equipment is desirable.
It is evident that previous FAM research efforts have relied on multiple differ-
ent assumed shape profiles without consensus on which ones best fit the observed
profiles, and without any conclusive measurement of their accuracy. This research
seeks to provide a experimental method that enables quantitative analysis of the
shape profile of FAMs for improved accuracy in modeling efforts. In this research,
photogrammetry, the science of making dimensional measurements of physical ob-
jects from photographic images [68], is adapted for use in the measurement of the
FAM?s shape profile. An overview of the photogrammetric methods applied in this
research are presented first. A test setup for image acquisition is introduced followed
by a method of image analysis to render the shape profile measurements from the
acquired images. The photogrammetric methods are demonstrated through example
by testing a 7/8 in diameter, 9-9/32 in long FAM. The results from testing the FAM
are presented, followed by analysis of the FAM using the photogrammetric data.
The obtained shape profile data is compared to the profiles assumed by the cylin-
122
drical approximation. The validity and accuracy of the cylindrical approximation
is tested. If more detailed and accurate modeling forms are required, the develop-
ment of a fit equation is presented for use that accurately replicates the curvature
of the FAMs shape profile. Potential improvements to current modeling efforts are
also presented, including a demonstration of how easily inaccurate initial geometric
conditions can be applied to models without having photogrammetric data avail-
able to reference. The profile data is then used to present a simple and accurate
method of characterizing the internal volume of FAMs. Finally, the ability of the
photogrammetric test method to be universally adopted is demonstrated through
shape profile characterizations of FAMs ranging in length from 9-9/32 in to 20 in,
and diameters of 7/8 in and 2-1/4 in. Overall, this research aims to provide a new
method of characterizing FAMs that has the potential to enlighten future modeling
efforts with findings that are not otherwise realizable, and with measurements that
have not previously been quantifiable.
4.2 Test Methods
Development of the test setup and methods focused on simplicity to ensure
that the photogrammetric characterization would require only an adaptation of cur-
rent FAM test setups without the need for specialized equipment or photography
expertise. The photogrammetric methods performed for this research are detailed
in this section to provide context for the test procedure that follows.
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4.2.1 Image Acquisition
There are multiple methods of photogrammetric image acquisition that are
capable of achieving various levels of image scene reconstruction. Both the number
of measurable dimensions of an imaged object, and the requirements for scaling the
object in the imaged scene are dependent on the number of cameras used and their
relative positioning (i.e. imaging configuration). Imaging configurations include
single-, stereo-, and multi-image photogrammetry methods [69]. Stereo-image and
multi-image methods provide three-dimensional scene reconstruction and require
either the use of multiple cameras or the controlled movement of a single camera.
Single-image photogrammetry requires just a single camera in a fixed position, but
only provides a two-dimensional planar view of the imaged object. This shortcoming
can be overcome, however, by enlisting additional prior knowledge of the imaged
object. Three-dimensional measurements can be achieved with just a single camera
if geometric relations, measurements, or symmetries of the object are linked to the
image.
The scale of an object in an image must be known to acquire its photogram-
metric measurements. Scale is defined as the ratio of the size of an object in the
imaged world (i.e. in pixels) to its size in the physical world. The scale of an ob-
ject in an image depends on five things: (a) the object?s distance and position with
respect to the camera, (b) the lens used by the camera, (c) the distance of the lens
from the image sensor plane, (d) optical distortion due to inherent imperfections in
the lens, and (e) perspective distortion (i.e. radial distortion) which is dependent on
124
the angle-of-view of the object in the image [68]. These camera state values can be
used for scaling the imaged object, but they can also be arduous and time consuming
for the layman to obtain. A simpler method of scaling employs the placement of an
object with known dimensions into the image scene to calibrate the scaling of the
object. This can be achieved by placing a ruler adjacent to the object of interest in
the image scene prior to testing, as is done in this research.
4.2.2 Image Analysis
The goal of the photogrammetric analysis is to obtain the shape profile di-
mensions of the FAM?s braid from photogrammetric images such as in Figure 4.2.
In order to do this, image analysis techniques must be applied to isolate the FAM?s
braid within each photographed image. Image analysis techniques include locating
object boundaries, or segmenting image pixels into groups, based on color informa-
tion such as color values (e.g. brightness, hue, etc.). With potentially thousands
of images to be analyzed, it is essential to have automated methods of identifying
and outlining objects of interest within images. The image segmentation and object
identification techniques used in this research are outlined in the following sections.
4.2.2.1 Color Model
Segmentation of an image is the process of grouping the pixels of an image
based on their color values. With color contrast between objects in an image, an
image can be segmented and reduced to isolate pixels in an area of interest. With
125
image segmentation based on the color component values of each pixel, an important
first step is the decision of what color model to use.
A color model is the mathematical representation of a color as a tuple of num-
bers. The RGB color model is commonly used to define pixel color in photographic
images, but is often problematic for use in image analysis. The RGB color space
is not a perceptual color space which makes it difficult to perceive how changes in
color components (R, G, and B) will alter the hue and brightness of the color. This
makes it difficult and imprecise to define a range of R, G, or B based on color, hue,
or brightness, that would enable a desired segmentation of an image [70]. Another
difficulty that arises for image segmentation when using the RGB color model re-
sults because normalized RGB component values are not invariant with changes in
lighting or shading [71]. This makes the RGB component values vary unpredictably
with fluctuations in brightness across an object?s surface from shadows, specular
reflections, or changes in illumination due to motion of an object.
A color model that is more suitable than RGB for use in image analysis is
the hue, saturation, and value (HSV) color space. The HSV color space provides
a more intuitive method of defining colors that is beneficial for performing image
segmentation [71?73]. As illustrated in Figure 4.4, the HSV color space can be
represented as a cylindrical space with hue, saturation, and value represented along
respective coordinate axes [70].
Hue is defined as a 0?360 degree value that correlates to an angular position
within the cylinder, and represents a color?s position within the color spectrum.
Hue is helpful in segmenting an image based on color in a way that is invariant to
126
Figure 4.4: Cylindrical HSV color space
changes in lighting conditions. Saturation, represented as a 0?100% value along the
radius of the cylindrical space, defines the vividness of a color?s hue. Desaturation
of a color occurs with added white light. Value, which is represented as a 0?100%
value along the height of the cylindrical space, defines the perceived brightness of
a hue. Sometimes described as relative brightness, value can be used to describe a
color in varying lighting conditions. Therefore, segmentation using the value color
component can be used to segment objects within an image based on their relative
brightness [74]. Although photographic image data is often defined using the RGB
color space, in can be converted to the HSV color space by using the following
equations [75]:
M = max(R,G,B) (4.1)
m = min(R,G,B) (4.2)
127
?
??????undefined if M = mG?B
H = 60? ? ???M?
mod6 if M = R
m
??? B?
(4.3)
R
? + 2 if M = GM m
??
R?G
? + 4 if M = BM m
= ?0 if ,M = 0S ? (4.4)M m , otherwise
M
V = M (4.5)
An image can be segmented based on the hue, saturation, or value component
of each pixel. The invariance of hue to lighting conditions makes it a good candidate
for color-based image segmentation. Care must be taken, however, to avoid use of
hue segmentation in areas where pixel colors are near the hue singularity condition.
Hue singularity occurs with colors that have saturation and value levels near zero
(i.e. grayscale colors). In this color region, imperceptible changes in color can
produce large fluctuations in hue. Attempts at hue segmentation near the singularity
condition will result in arbitrary and inaccurate segmentation [71]. Those regions
are better suited for segmenting based on saturation or value. To aid in avoiding
hue singularities, for this research, pixel colors are defined in the HSV color space
to segment each image based on relative brightness (i.e. value) and then hue.
4.2.2.2 Boundary Detection
Automated identification of objects within an image boils down to the prac-
tice of segmentation through boundary detection. Common methods of boundary
detection include edge detection and thresholding. Edge detection methods identify
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boundaries of objects by locating the sharp changes in color or light intensity that
are characteristic of object boundaries [74]. It is often performed using gradient-
based methods such as Canny edge detection [76]. Thresholding techniques, on the
other hand, segment an image by grouping image pixels by comparing their color
value to a set threshold value. Since thresholding locates areas of an image and not
edges, finding boundaries through thresholding requires an additional step of using
a method of locating and tracing object boundaries.
Edge detection methods are effective at detecting edges with sharp contrast,
but can have difficulty with random specular reflections, or with soft edges that have
small color gradients [74]. Identification of the edge of the FAM braid using edge
detection methods proved to be difficult due to the soft edges of the FAM?s bulbous
shape, and random spikes in brightness due to specular reflections off the surface of
the Kevlar braid. However, the ruler used for calibration, with its flat surface and
high-contrast hatch markings, was found to be an ideal candidate for edge detection
methods, as detailed in Section 4.3.2.
Thresholding techniques, when combined with a boundary tracing algorithm,
can serve as a more robust method of boundary detection for identification of the
FAM?s braid. Setting a color threshold can be performed manually through subjec-
tive analysis, but automated methods of thresholding have been developed that can
identify a suitable threshold in a consistent and time-efficient manner. Automated
methods of thresholding are typically based on color histogram analysis. The his-
togram defines the distribution of the number of instances of each color value in an
image. Setting a threshold divides the histogram into two classes (i.e. groups) with
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all pixels below the color value threshold grouped into one class, and all pixels above
the color value belonging to a second class. An image is commonly transformed into
a class-membership based binary image to provide a visual representation of the
threshold segmentation, and to serve as an input in many boundary tracing algo-
rithms. Otsu?s method is a common method of histogram-based thresholding that
is used in this research [74,77,78]. This method?s algorithm recursively searches for
a threshold value that can divide the histogram into classes that have a maximum
inter-class variance.
The algorithm is detailed as follows. Let us consider a certain color value (e.g.
H, S, or V) to set a threshold on. Each pixel of an image is represented by a color
value level of the histogram (e.g. 0?100% for S or V), and within the histogram, the
height of each color value bar represents the fraction of the total number of pixels at
each respective color value level. For a given threshold, the pixels are separated into
two classes (i = 1, 2). wi is equal to the probability of a pixel being in class i, and ?i
and ?T are the mean value of each class and the total, respectively. Otsu?s method
then finds the optimal threshold value by maximizing the inter-class variance, ?b,
using the following equation [77,78]:
?2b = w1(?1 ? ?T )2 + w2(?2 ? ?T )2 (4.6)
Otsu?s method is utilized in Section 4.3.2 for image segmentation, with Figures
4.10(d) and 4.11(d) providing illustrations of the relative brightness and hue his-
tograms, respectively, with thresholds placed using Otsu?s method.
With an area of interest identified through threshold segmentation, the bound-
130
ary of the area must then be traced with a contiguous line. A simple method of
boundary tracing used in this research is the Moore-Neighbor algorithm [79, 80].
With a binary image as the input, the Moore-Neighbor tracing algorithm locates
any clusters of pixels of the same class, and then traces their boundaries. Starting
at a pixel along the boundary of a cluster, the recursive Moore-Neighbor algorithm
steps along a boundary until it meets a defined end condition.
The basis of the Moore-Neighbor algorithm is the Moore-Neighborhood of a
pixel. A pixel?s Moore-Neighborhood (Fig. 4.5(a)) is the set of eight pixels that
surround a pixel of interest (pixel C). To advance along a boundary, the algorithm
examines the Moore-Neighborhood of a boundary pixel in the clockwise direction,
starting from the previously found boundary pixel (pixel P), until it comes upon the
next boundary pixel. This process continues recursively until a stopping condition
is met. A frequently used stopping condition is the Jacob?s stopping criterion. This
criterion states that the Moore-Neighbor algorithm shall stop after visiting the start
pixel at least twice, or after entering the start pixel from the same direction as it
was initially entered.
Figure 4.5(b) provides an example of Moore-Neighbor tracing. A cluster of
pixels of the same class (gray) is identified, and a start boundary pixel (pixel 1)
located at the boundary of the cluster is established. From pixel 1, the Moore-
Neighborhood search begins until the next boundary pixel (pixel 2) is reached. The
algorithm repeats until it arrives back at the start boundary pixel from the original
direction. The overall boundary traced for this cluster is identified as pixels 1?8.
Through the detection of the boundary of an area of interest (e.g. the braid)
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(a) Moore-Neighborhood of pixel C (b) Tracing of a pixel cluster by step-
ping through pixels 1-8
Figure 4.5: Example depiction of Moore-Neighbor tracing of a boundary
within an image realized, analysis of the size and shape of the boundary can then
commence as desired.
4.3 Procedure
The methods of image acquisition and analysis described in the previous sec-
tion are employed by this research, as detailed in this section. Photogrammetric
data is acquired and the image is then reduced down to a traced outline of the FAM
braid for further analysis of the shape profile.
Photogrammetric data of the shape profile is acquired while simultaneously
characterizing the isobaric force-contraction response of the FAM. Testing of the
force response is performed using the same methods as done in past FAM research
[18,22]. The FAM is mounted in a MTS servo-hydraulic test machine, and is quasi-
statically stroked between its minimum and maximum contraction length states
while maintaining a constant internal working fluid pressure. The MTS machine
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directs a controlled stroking motion of the FAM while recording external inputs of
pressure (Omega PC209-200G5V), and force (Honeywell Model 31 Mid 1000 lbf.
load cell), which provides a complete data set to characterize the isobaric force-
contraction response of the FAM. The FAM is mounted in the MTS machine at
each end through clamped fixtures (Fig. 4.6). The top fixture serves as both a
mounting point for the pressure sensor and as a conduit for air to travel into and
out of the FAM. The bottom fixture connects to the load cell that is mounted inline
with the FAM.
Figure 4.6: FAM with sensors, MTS fixtures, and masking applied prior to mounting
in the MTS machine
4.3.1 Image Acquisition
For simplicity of implementation, image acquisition is performed with the
single-image photogrammetry configuration. The test setup to perform the force
response and shape profile characterization is shown in Figure 4.7. Photogrammet-
ric data acquisition requires the addition of only a camera, a backdrop, lighting, and
masking of components. A camera is placed in a fixed position, and photographic
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(a) Front view
(b) Side view
Figure 4.7: Views of FAM mounted in the MTS machine with the photogrammetric
test setup (camera, lights, and backdrop)
images are taken at a regular time interval while the FAM is quasi-statically stroked
during force response characterization testing. The camera used is a Nikon D7500
DSLR Camera with a AF-P DX NIKKOR 18-55mm f/3.5-5.6G VR lens. This cam-
era provides a high-resolution image (5,568 ? 3712 pixels) along with manual control
of settings including focus and image exposure.
Careful placement of test setup components is required to obtain images that
are sufficient for photogrammetric analysis. The scene of the image should be set to
promote maximum contrast between the components of the image to be segmented.
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The foreground is separated from the background of each image using a threshold
of relative brightness, and the end-fittings and braid of the FAM are separated in
each image by using a hue threshold. Therefore, the goal for setting the image
scene is to aid segmentation by maximizing the brightness differential between the
foreground and background, and by maximizing the differential in hue between the
end-fittings and the braid of the FAM. To determine the distance from the FAM of
the camera placement, a balance must be realized between positioning the camera
close to the FAM for maximum resolution, versus maintaining enough distance away
from the FAM to minimize the angle-of-view and related perspective distortion. For
the tested 9-9/32 in long FAM, the camera was placed 5 ft from the FAM. The height
of the camera is set to a level that makes it level with the mid-height of the FAM in
its resting length state. This results in a relatively small angle-of-view of the FAM
of 8.84 degrees when the FAM is at its resting length (Fig. 4.7(b)). Contrast in
brightness between the foreground and background is heightened by maximizing the
distance between the foreground and background. A 5?5 ft black cloth backdrop is
placed 4 ft behind the FAM. The FAM is front-lit by two fluorescent lamps (Smith-
Victor 10 inch Adapta-Light) positioned to each side of the camera (Fig. 4.7(a)).
The lamps act to further increase brightness contrast between the foreground and
background, and ensure complete, shadow-free lighting of the half of the FAM that
is facing the camera. Black gaffers tape is applied to everything but the FAM in the
foreground to enable these objects to be segmented out with the background by the
brightness threshold (Fig. 4.6). To maximize the contrast of hue between the yellow
braid and the aluminum end-fittings of the FAM, the end-fittings are wrapped in
135
blue tape (yellow?s color complement). Non-reflective materials are selected for all
applied surfaces to minimize specular reflections that can affect the image brightness
and hinder hue identification of pixels (due to the aforementioned hue singularity
conditions).
Camera settings are adjusted to ensure accurate coloration and ideal exposure
of images. The lens is set to its longest focal length of 55 mm (2.2 in) to maximize
the size and resolution of the FAM?s image. The camera?s auto-focus functionality
is used to obtain a sharp focus of the FAM in the image plane. However, a change
in the focus would result in a change in the scaling of the image, so the focus must
be set prior to scale calibration and testing. Additionally, the camera?s auto-focus
functionality is turned off prior to testing to ensure that the camera does not change
focus by reacting to the changing shape of the FAM.
The aperture?s f-stop is set to a minimum to achieve a narrow depth-of-field
which enables the background to be out of focus [68]. The motion of the FAM will
be very slow (less than 0.1 in/sec), so the choice of shutter speeds is minimally
constrained. Therefore, exposures can be iteratively tested until the desired result
is achieved through adjustments of the shutter speed and ISO sensitivity. For the
scene setup in this research, desired exposure is achieved with the f-stop set at f/5.6,
the shutter speed of 1/160 sec, and the ISO sensitivity at ISO 3200. To provide an
accurate color representation in the artificially lit indoor laboratory, white balance
is set to the incandescent setting which equates to a color temperature of 3000 K.
Calibration of the scale of the FAM is acquired by taking a picture of a ruler in
the image frame prior to testing. The 12 in ruler is positioned in the image alongside
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the FAM as seen in Figure 4.8. Placement of the ruler in close proximity to the FAM
helps ensure accurate calibration of scale, and enables the ability to correct for any
optical distortion as a function of its position along the ruler. The ruler attaches
to the top MTS fixture by a rigid mounting plate that ensures a vertically fixed
orientation of the ruler. Since the ruler would obstruct the radial expansion of the
FAM during testing, it must be removed prior to testing after the calibration picture
is taken. It is very important that the calibration ruler is placed at the mid-plane
of the FAM to ensure that the scale is calibrated at the same depth as the shape
profile outline to be measured. Prior testing for this test setup revealed that scale
calibration would change by about 2% of the mid-plane calibration value per inch
of depth change from the FAM?s mid-plane. This indicates that performing the
scale calibration with the ruler at the mid-plane of the FAM is critical to acquiring
accurate shape profile measurements of the FAM.
4.3.2 Image Analysis
Prior to analysis of the photogrammetric data, the scale of the FAM in the
images must be determined for the camera settings and position used. First, a
photograph is taken with a ruler placed adjacent to the FAM in the image scene (Fig.
4.8). This image is rotated by 90 degrees and then cropped to retain only the 1/8 inch
hatch marks of the 12 in calibration ruler. The scale is then determined by finding
the number of pixels between each 1/8 inch hatch mark of the ruler. The location of
each hatch mark is pinpointed by using an edge detection method. The left and right
137
edges of each hatch mark are located as the maximum and minimum points of the
brightness gradient. The midpoint between the minimum and maximum gradient
points is then identified as the center of each hatch mark. Figure 4.9 provides an
example of this hatch mark identification; the maximum and minimum brightness
gradient points are marked in blue and green respectively, and the center of each
hatch mark is marked in red. The distance between each hatch mark, measured in
pixels, indicates the scale of the FAM in the image. The scale can be averaged along
the ruler for the length of the FAM, or can be found as a function of the height
along the ruler if distortion of the image is considered appreciable.
Analysis of the photogrammetric test data begins with the original 5568 x
3712 pixel images (Fig. 4.8(a)). All images are then cropped to a size that cuts
out the metallic surfaces of the MTS machine and fixtures while retaining the entire
length of the FAM (Figs. 4.8(b), 4.10(a)). What remains in each image is the FAM
(braid and end-fittings), the connecting rod (masked in black), and the background.
This image is then analyzed to obtain the originally stated goal of outlining and
measuring the shape profile of the FAM?s braid. A sample image of the tested FAM
is shown in Figure 4.10(a) to serve as an example of the developed method. The
braid?s profile is acquired from each image through the following five steps:
1. Isolate the FAM in the image using a threshold on relative brightness, V
2. Trace the boundary of the FAM (braid and end-fitting) using the threshold
segmented image
3. Within the traced outline of Step 2, isolate the end-fittings using a threshold
on hue, H
4. Trace the boundary of the end-fittings from the threshold segmented image of
Step 3
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(a) Original image view, 5568x3712 (b) Cropped image
pixels view, 2650x1100 pixels
Figure 4.8: Image captured by camera for photogrammetric analysis with calibration
ruler
0
6
12
20 40 60 80 100
0.5
0
-0.5
0 20 40 60 80 100
Pixels
Figure 4.9: The method of identifying the distance (in pixels) between 1/8 inch
increments on the calibration ruler is illustrated. Each hatch mark has its left
(blue), right (green), and then center (red) located on a segment of the ruler image
(top), and respective brightness gradient plot (bottom).
139
V / P
Pixels
5. Using the outline of the FAM from Step 2, retain only the region between the
outlined end-fittings from Step 4
Details of each step in this process are described below. Using a sample image
of the tested FAM, each step is accompanied by an illustration of the described
analysis. The process of isolating the braid?s profile from each image is detailed
below, and is accompanied with an illustration of each step of the analysis using a
sample image of the tested FAM (Figs. 4.10?4.12).
4.3.2.1 Step 1
The FAM is first isolated in the image from the connecting-rod and background
through segmentation using a threshold on relative brightness, V . The histogram
in Figure 4.10(d) illustrates the relative brightness distribution of the pixels in the
image, and includes the threshold that separates the pixels into two classes (i.e.
groups). The dark pixels of the black background and masked connecting-rod are
in the class that is to the left of the threshold, while the light pixels of the FAM
are to the right of the threshold. The brightness contrast in the image, illustrated
by the separation of the two pixel groupings to each side of the threshold, indicates
the effective segmentation of the image. A binary form of the segmented image is
created (Fig. 4.10(b)), with pixels in the class below the threshold value presented
in black, and pixels in the class above the threshold value presented in white.
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4.3.2.2 Step 2
Using the binary image created in Step 1 (Fig. 4.10(b)), the Moore-Neighborhood
algorithm is used to trace the boundary of the FAM (Fig. 4.10(c)). The Moore-
Neighborhood algorithm locates and bounds all pixels of the image that have a
relative brightness that is above the threshold. This may include small areas from
random specular reflections, or other objects that come into the scene of the image.
These areas are located and traced by the Moore-Neighborhood algorithm as well,
but are rejected on the grounds of being a smaller area than the traced boundary
of the FAM.
4.3.2.3 Step 3
The end-fittings of the FAM must be separated from the braid. A threshold
segmentation is performed on the area within the outline of the FAM from Step 2
(Fig. 4.11(a))). A hue threshold value is set using Otsu?s formula to separate the
blue hue of the end-fittings from the yellow hue of the braid in the image. The
histogram in Figure 4.11(d) displays the threshold?s effective separation of the blue
and yellow pixels of the FAM. Just as in Step 1, a binary form of the image is created
for use by the boundary tracing algorithm (Fig. 4.11(b)). Pixels in the class below
the threshold are black, while pixels in the class above the threshold are white and
can be found in the image in the area of the end-fittings.
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4.3.2.4 Step 4
Using the binary image created in Step 3 (Fig. 4.11(b)), the Moore-Neighborhood
algorithm is applied to trace the boundary of each end-fitting (Fig. 4.11(c)). Just
as in Step 2, any extraneous areas located and traced by the Moore-Neighborhood
that are smaller than the areas of the two end-fittings are not considered in further
analysis.
4.3.2.5 Step 5
The final step is to outline the outer edges of the braid. For analysis of the
shape profile of the FAM, we are only concerned with a single left and right boundary
of the FAM in the area between the two end-fittings.
The top and bottom of each end-fitting appear to be perfectly horizontal if
viewed head-on. However, due to perspective distortion in the image, the top and
bottom of each end-fitting can appear to be rounded. Since we are interested in
the shape profile of the FAM at its mid-plane, we only consider the outline of the
braid to the inner edge of each end-fitting. To do this, the outline of the FAM
obtained in Step 2 (Fig. 4.10(c)), and the outline of the end-fittings from Step 4
(Fig. 4.11(c)), are compared. The areas where the FAM and end-fitting outlines
overlap is subtracted from the outline of the FAM to retain only the outline of the
FAM that lies between the two end-fittings. What remains are two contiguous lines
that delineate the left and right edges of the braid (Fig. 4.12(a)). Figure 4.12(b)
provides a depiction of the shape profile data to be used in subsequent analysis of
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the FAM?s profile.
4.3.3 Characterization Test Parameters
As a proof-of-concept of the presented photogrammetric method, a 9-9/32 in
long, 7/8 in diameter FAM was selected for initial testing and analysis. The material
composition of this FAM, as detailed in Table 4.1, is identical to that of FAMs used
in other research [28] resulting in comparable actuation characteristics.
Table 4.1: Construction of the 7/8 in diameter FAM
Material
Braid 2 in Biaxial Kevlar Sleeve
Bladder 7/8 inch Latex Tube (1/16 in Thickness)
end-fitting Machined Aluminum 6061-T6
Swage Tube Aluminum 6061-T6 Tubing
Photogrammetric data was acquired while the isobaric force-contraction re-
sponse of the FAM was tested in the MTS machine. The FAM was stroked three
times between a maximum and minimum length state for each isobaric test to ensure
accuracy and consistency of the characterization data. The minimum tested length
state of each test is the pressure-dependent free contraction, zero force state, of the
FAM. To ensure a quasi-static characterization, the FAM was stroked at 0.04 in/sec.
For the photogrammetric analysis, photographs were taken at 1 second intervals for
the first cycle of testing, resulting in shape profile data for every 0.04 in change in
length of the FAM. Testing was performed at isobaric pressures of 10?100 psi in
10 psi increments, for a total of ten tests. Pressure was controlled to be constant
143
(a) Initial Image (b) Step 1b: V seg- (c) Step 2: Traced out-
mented binary image line of FAM
15
10
5
0
0 20 40 60 80 100
Brightness, 0-100%
(d) Step 1a: Brightness pixel segmentation using Otsu?s Method
Figure 4.10: Identification of the FAM in the image (Steps 1-2)
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Pixels, % of Total
Threshold
(a) Image with FAM only (b) Step 3b: H seg- (c) Step 4: Outline of
mented binary image end-fittings
20
15
10
5
0
0 90 180 270 360
Hue, 0-360?
(d) Step 3a: Hue pixel segmentation using Otsu?s Method
Figure 4.11: Identification of the end-fittings in the image (Steps 3-4)
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Pixels, % of Total
Threshold
2.5
2
1.5
1
0 2 4 6 8
Length, in
(a) Step 5: Profile (b) Diameter of shape profile with respect to length
Figure 4.12: The resulting shape profile of the braid along the FAM?s length
during testing using the air pressure control system developed in Chapter 3.
4.4 Results
4.4.1 Calibration
The first image to be processed is the calibration image (Fig. 4.8) used to
calibrate the scale of the FAM in the images. The scale of the FAM was found to be
226.63 pixels/in, resulting in high resolution measurements from each image. Care
was taken to not change any settings of the camera (e.g. focus, position) subsequent
to this calibration being performed to ensure that the scale remained constant for
each photographed image. It is worth noting that since the scale was found to be
uniform along the entire length of the FAM, there was no compensation of scaling
146
Diameter, in
due to image distortion used for this test.
4.4.2 Force-Contraction Response
The isobaric force-contraction response of the tested FAM is shown in Figure
4.13. These results closely replicate those of other FAMs of the same size and
construction. The characteristic hysteretic response of FAMs, as seen in Figure 4.13,
results in the FAM expressing higher forces in eccentric motion than in concentric
motion. Despite this fact, subsequent analysis of the shape profile of the FAM
has found no dependency of the shape profile on the direction of motion. When
comparing the shape profile of the FAM for the full contraction range at 100 psi,
there was found to be an average differential of only 0.00051 in between the profiles
for concentric and eccentric motion.
1000 100 psi
80 psi
60 psi
800
40 psi
20 psi
600
400
200
0
0 0.1 0.2 0.3 0.4
Contraction, L /L
0
Figure 4.13: Actuation diagram of the 9-9/32 in, 7/8 in diameter test FAM
147
Force, lb
4.4.3 Shape Profile Characterization
The developed image analysis methods are applied to get measurements of
the shape profile for every image acquired during the characterization testing of the
force-contraction response. As a first step, the timing of the photogrammetric data
must be synchronized with that of the force-contraction response data recorded by
the MTS machine. This was done in this research by comparing the contraction
of the FAM, as directed by the MTS machine, to the contraction as found through
the shape profile data. To do this, the recorded time of the MTS data was shifted
to produce a minimum error between the MTS and photogrammetric contraction
data. With measurements of the MTS machine recorded every 0.02 seconds, fine
adjustment of the time shift can be made to synchronize the time between the two
measurement systems accurately.
The comparison of the photogrammetric and MTS contraction data also pro-
vides a method of ensuring the fidelity of the photogrammetric results. The pho-
togrammetric measurements can be checked through comparison to manually ac-
quired measurements taken during testing. However, manually acquiring measure-
ments would be time consuming and difficult to perform accurately. The MTS
machine controls and records the contraction of the FAM length during testing to
the ten-thousandth of an inch. Therefore, if we consider these contraction values as
the true contraction of the FAM, a simpler and more accurate method of checking
the photogrammetric measurements would be to compare the contraction as found
through the photogrammetric data to that of the MTS machine. This comparison
148
of the time synchronized data is shown in Figure 4.14.
0.4
0.3
0.2
0.1
0
0 50 100 150 200
Time, sec
Figure 4.14: Comparison between contraction of image data, and displacement
recorded by the MTS machine for a single contraction cycle
Figure 4.14 shows that the photogrammetric data tracks that of the MTS
machine very well, with a maximum differential between the two of 0.02 in occurring
at the FAM?s resting length, which results in an error of the length measurement
of 0.25%. With a measurement resolution of 0.004 in/pixel, a portion of this error
is a limitation of the resolution of the image. A portion of this measurement error
can also be attributed to the slightest amount of distortion along the length of the
FAM. The diameter measurements of the shape profile, with an even narrower angle-
of-view, would be subject to even less perspective distortion which would result in
even less measurement error. Therefore, the 0.25% error of the photogrammetric
measurements can be treated as an absolute maximum. With this comparison, the
photogrammetric measurements of this research can be received as a high-fidelity
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Contraction, L /L
0
2.5
2
1.5
1
0 0.2 0.4 0.6 0.8 1
Normalized Length, L/L
0
Figure 4.15: Evolution of the shape profile for a complete contraction cycle (100 psi)
representation of the true dimensions of the FAM?s braid. The shape profile of
the FAM is now analyzed with confidence in the accuracy of the photogrammetric
measurements.
With the maximum contraction achieved at the highest test pressure, the
100 psi test serves as a good starting point for analysis of the FAM?s shape pro-
file. Figure 4.15 provides snapshots of the FAM?s shape profile in 1% contraction
increments up to the maximum contraction of about 35% at 100 psi.
There are a few observations to be made from the shape profiles in Figure
4.15. The distinct shape, with a maximum diameter at each end of the midsection,
resembles the shape of a peanut. The profile near each end-fitting is also of interest,
with a slope of the braid that dramatically changes as the FAM contracts. The slope
is close to zero at low levels of contraction, but rapidly increases with increased
contraction. This distinct change in the shape of the braid makes it difficult to have
150
Diameter, in
a single assumed shape for the profile near each end-fitting, as has been assumed
by other authors (e.g. circular arc, toroid, etc.), that can accurately represent the
shape profile for the entire contraction range. Observation of other FAMs of the
same material construction have shown that this characteristic shape is not unique
to this FAM. Some of those FAMs have exhibited a midsection with a more severe
peanut shape than others, but all have exhibited the change in profile slope in
proximity to each end-fitting that is observed here.
This section demonstrated the successful measurement of the FAM?s shape
profile using photogrammetric analysis. Subsequent sections provide a preliminary
analysis of these profiles that demonstrates the added value that photogrammetric
characterization of FAMs can provide.
4.5 Analysis
Photogrammetry provides the first quantitative measurements of the shape
profile of FAMs, and opens the door for new and improved modeling efforts. As
a first step towards analysis of these shape profiles, each shape profile ? which is
comprised of thousands of data points ? must be defined in simpler terms. One
method could be to define each shape profile by a single value such as the average
or maximum diameter of the profile. Defining the profile by a single diameter value
assumes a cylindrical profile, and enables the measured data to be used with current
modeling frameworks that have an input of a single approximated cylindrical value.
Now, if a more detailed analysis of the shape profile is desired, each profile dataset
151
can be defined by a fit equation. A Fourier series fit equation is presented as a good
candidate for accurate representation of the FAM?s shape profile.
The shape profile data can also be compared to the cylindrical diameters that
have been assumed in previous work. Prior to the quantitative analysis method-
ology presented here, the cylindrical diameter could only be estimated based on
approximations derived from the resting (uncontracted) geometry of the FAM. We
will term these the estimated cylindrical shape profiles. The accuracy of these shape
profile estimates has gone relatively unchecked. Using the measured shape profile
data, we now have the ability to test the accuracy of these estimated cylindrical
approximation shape profiles.
Finally, analysis of the measured shape profile provides the unique opportunity
to define the internal volume of FAMs. Using an assumption of axisymmetry along
the longitudinal axis of the FAM, two-dimensional photogrammetry measurements
can be used to approximate the internal volume through a non-intrusive method
that is much simpler than previous methods that have required costly and intrusive
measurement setups.
4.5.1 Shape Profile Analysis
4.5.1.1 Profile Average Diameter
The average diameter is easily calculated from the raw shape profile data, and
is the simplest representation of a shape profile for analysis. With the average defin-
ing a single diameter value for each profile, the resulting assumed cylindrical profile
152
is in contrast to the complex curvature of the measured profile (Fig. 4.17(b)). The
average profiles result in average diameters that are about 10% below the maxi-
mum diameter due to the dramatic decrease in diameter at each end of the FAM
in proximity to the end-fittings. The average diameter was calculated for the entire
contraction range of the FAM at each pressure (Fig. 4.16). The results suggest
that the radius is nominally independent of internal pressure. Closer inspection of
the average diameters, however, reveals small increases in diameter with pressure.
This is especially apparent in the zero-contraction state, where the diameter swells
by about 0.02 in from 20 psi to 100 psi. This appears to capture the affect of ten-
sile stretching of the braid, a phenomenon that had not previously been captured.
Stretching of the braid has often been ignored with a rigid braid assumption, but
has also been accounted for ? without the benefit of having quantitative data ? in
some modeling efforts [11,13]. Further investigation of the apparent increase of the
braid diameter at fixed length states with increases in pressure warrants further
investigation in future work.
4.5.1.2 Profile Fourier Series Fit
Previously assumed shape profiles have failed to accurately capture the com-
plex curvature of the FAM?s shape profile. Previously piece-wise profile definitions
used for attempted corrections on the cylindrical profile approximation ? which typ-
ically use separate equations to define the profile of the midsection and the region
near each end-fitting ? increase modeling complexity with unvalidated improvements
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2.5
2
1.5
1
0 0.1 0.2 0.3 0.4
Contraction, L /L
0
Figure 4.16: Average diameter of the FAM?s shape profile for all tested pressure-
contraction states
in accuracy. A single equation representation of the measured shape profiles is de-
sired for simplicity of analysis.
Curve fit equations are used to fit the measured shape profiles. With multiple
forms of fit functions to choose from (e.g. polynomial, exponential, etc.), it is
important to select a function with a characteristic shape that resembles the shape
of the observed profiles to enable an accurate and low-order fit equation. Since
the observed profiles draw a resemblance to a half-sine pulse with higher order
harmonics, a Fourier series curve fit equation was selected. The form of the fit
equation is given as:
?3
y = a0 + ancos(nxw) + bnsin(nxw) (4.7)
n=1
where x represents the position along the length of the shape profile, a0 provides
a horizontal shift, n indicates the order of the harmonic, and an and bn are terms
154
Diameter, in
that scale the magnitude of each harmonic. The number of terms used for the fit is
adjusted to achieve a good fit while also keeping the number of terms to a minimum
for simplicity. A third-order Fourier series fit was found to achieve a very good fit of
the shape profile data (Fig. 4.17(a)). The resulting fit has a low combined average
error for all tested states of contraction. The accuracy of the fit of this FAM begins
to plateau with the use of three or more terms. Table 4.2 provides a comparison of
the accuracy of the fit depending on the number of terms used. Values for each term
of the three-term Fourier series fit performed in Figure 4.17(a) are provided in Table
4.3. The profiles for all tested states of pressure and contraction can be fit for a
complete characterization of the shape profiles. As noted from previous observation
of Figure 4.16, the profile is predominantly a function of the contraction state with
little variation with changes in pressure. This enables a reduction in the number
of required fits to fully define the shape profile of the FAM by only needing to
characterize the shape profiles of the maximum pressure tested.
Table 4.2: The effect of the number of Fourier series fit terms used for fitting the pro-
file data on the maximum and Root-Mean-Square Error (RMSE) averaged between
all tested pressures
Number of Terms: 1 2 3 4 5
Max Error (avg.), in 0.509 0.069 0.036 0.035 0.021
RMSE (avg.), in 0.108 0.015 0.005 0.005 0.003
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Measured Data
2.5
Fit Equations
2
1.5
1
0 0.2 0.4 0.6 0.8 1
Normalized Length, L/L
0
(a) Comparison of the measured shape profiles to the three-term Fourier
Series fit equation profiles
Measured Data
2.5
Avg. Diameters
2
1.5
1
0 0.2 0.4 0.6 0.8 1
Normalized Length, L/L
0
(b) Comparison of the measured shape profiles to assumed cylindrical pro-
files that have a diameter equal to the average diameter of the measured
shape profile at each respective contraction state
Figure 4.17: Comparison of assumed profiles to the measured shape profiles of the
FAM at 0, 8.75, 17.5, 26.25, and 35 (maximum) percent contraction at 100 psi
156
Diameter, in Diameter, in
Table 4.3: Coefficient values for the three-term Fourier Series fit performed in Figure
4.17(a)
%L0 a0 a1 a2 a3 b1 b2 b3 w
0% 1.02 -0.09 -0.07 -0.01 0.03 0.06 0.02 5.62
8.75% 1.37 -0.35 -0.15 -0.01 -0.17 -0.19 -0.06 -5.86
17.5% 1.51 -0.56 -0.10 0.04 0.43 0.41 0.10 6.00
26.25% 1.27 -0.83 0.29 0.18 1.20 0.75 0.04 5.89
35% -32.74 24.82 12.82 -3.90 44.44 -18.47 -0.5 3.35
4.5.1.3 Comparison to Previous Cylindrical Profile Estimates
The shape profile of FAMs has traditionally been modeled as a cylindrical
profile. The diameter of the FAM can be estimated for all states of contraction by
using this cylindrical approximation and an assumed initial geometry. The estimated
profile diameter has long been assumed to be nominally accurate enough for use
in almost all modeling efforts of FAMs. However, with no previous method of
measuring the actual shape profile of FAMs, there has not been any quantifiable
data to check the accuracy of these estimated cylindrical profiles until now. With
the shape profile measurements acquired using photogrammetry, the accuracy of the
cylindrical shape profile approximation can now be tested.
The cylindrical approximation has served as the basis of most FAM modeling
since the inception of the FAM in 1958 [2]. The cylindrical approximation, prior to
the addition of any applied correction factors, makes the following assumptions:
? The shape profile is cylindrical (constant diameter with respect to length)
? The braid angle is constant with respect to length
157
? There is no strain in the braid fibers
With these qualifying assumptions, the geometry of the FAM can be defined using
the triangle relationship which defines a set trigonometric relationship between the
active length, L, diameter, D, and braid angle, ?, of the FAM through fixed values
that are the length of each braid fiber, B, and the number of turns of each braid
fiber around the circumference of the FAM, N [5]. The fixed values of N and B,
can then be found using the initial geometry of the FAM defined by the initial braid
angle, ?0, the initial diameter, D0, and the initial active length, L0, through the
following relations:
N = L0 (4.8)
D0?tan(?0)
B = L0
sin(?0)
(4.9)
Finally, the diameter of the FAM can be estimated as a function of active length
through the following relationship:
?
B2 ? L2
Dest(L) = (4.10)
?N
The established direct relationship between Dest and L produces estimated diameter
values ? commonly used as an input parameter to models ? through relatively simple
geometric relationships. However, it is also relatively sensitive to the assumed initial
geometry of the FAM. The value of D0 is commonly assumed to be equal to the
resting outer diameter of the bladder (assuming a thin braid), or the braid diameter
at its point of insertion into the end-fitting. The value of ?0 can then be determined
using the method covered in Pillsbury (2015) [48] which uses the linear relationship
158
between blocked force and pressure to solve for ?0. It should be noted that the value
of ?0 found through this method is a function of the estimated value of D0.
The average diameter of the measured profile data will be compared to the
estimated cylindrical profile. The assumed cylindrical profiles for both the estimated
and average of the measured data provide a basis of direct comparison. An accurate
estimated profile diameter should match the average of the measured profiles.
The value ofD0 for the tested FAM is equal to the outer diameter of the bladder
at 7/8 in. An estimated value of ?0 was found using the blocked force with respect to
pressure data to be 72.27 degrees. The resulting estimated diameter values, labeled
as Dest,1, are compared to the results of the measured data in Figure 4.18(a) for
a single stroke of the FAM. There is an observed error in the estimated value of
D0 which is about 0.15 in below the initial measured average diameter of about
1.025 in. This initial increase in diameter was not due to any noticeable slack in the
braid prior to pressurization, with the braid being taut prior in the resting length
state. The resulting discrepancy between the estimated and measured diameters is
especially large at low levels of contraction.
With knowledge of the actual value of D0, using the measured data we can
attempt to estimate the shape profile again with corrected values of D0 and ?0.
Since there is a slight range of the measured initial diameter values with respect
to pressure, the average of these measured values of 1.025 in can be used for this
comparison. This value of D0 results in a value of 69.04 degrees, a significant
difference from the previous value of 72.27 degrees. Estimated diameter values,
Dest,2, are calculated using the new initial geometry conditions (Fig. 4.18(b)). With
159
proper alignment of the curves at low levels of contraction, there is now a sizeable
error in the maximum contraction state. The resulting error between curves could be
the result of the estimated value of ?0 being too high. A lower value of ?0 would result
in lower values of estimated maximum contraction. If the value of ?0 is reduced to
68-68.5 degrees with D0 still equal to 1.02 in, the estimated and measured diameter
curves match very well. However, an accurate physical measurement of the value
of N for the tested FAM found that the braid angle should be around 68.9 - 69.7
degrees which coincides well with the value of ?0 used to define Dest,2. It is therefore
likely that the discrepancy between the values of Davg and Dest,2 are a result of
error from using the cylindrical approximation for the estimated diameter values.
The cylindrical approximation, in estimating a diameter from the initial geometry
state, neglects the length of the braid that is effectively lost to the curvature of
the FAM?s braid resulting in overestimations of diameter. This research serves as
an initial comparison of these estimated and measured diameter values, with the
resulting discrepancy between these values deserving of further investigation.
4.5.2 Volume Approximation
Measurement of a FAM?s internal volume is often required for actuation sys-
tem design and modeling efforts. Estimates of volume are necessary to determine
requirements of the system such as the working fluid flow rate and storage require-
ments. Modeling efforts also use values of internal volume for force response models
(e.g. virtual work method), as well as for providing estimates of actuation efficiency.
160
2.5
2
1.5
1
0 50 100 150 200
Time, sec
(a) Comparison of Davg against estimated diameters, Dest,1,
using initial geometry D0 = 7/8 in, and ?0 = 72.27 deg.
2.5
2
1.5
1
0 50 100 150 200
Time, sec
(b) Comparison of Davg against estimated diameters, Dest,2,
using initial geometry D0 = 1.02 in, and ?0 = 69.04 deg.
Figure 4.18: Comparison of the average measured diameter values, Davg, against
estimated diameters, Dest,1 and Dest,2
The acquisition of accurate volume measurements has often proved difficult for re-
searchers due to the specialized equipment that is often required to do so. As a
consequence, researchers have often settled for using dimensions provided by using
the estimated cylindrical approximation diameter.
The photogrammetric data acquired in this research provides the opportu-
161
Diameter, in Diameter, in
nity to easily obtain an accurate estimation of the internal volume of the FAM
for all tested pressure and contraction states. Since the profiles only provide a
two-dimensional depiction of the FAM, a three-dimensional measurement of volume
requires the assumption that the FAM is axisymmetric about its longitudinal axis.
The symmetry of many previously tested FAMs usually makes this a reasonable
assumption. Appreciable deviations from symmetry have only been observed for
FAMs with flaws in their fabrication.
Using the shape profile data, the internal volume of the FAM, Vint, can be
found using the following basic equation:
Vint = Vprof ? Vbld ? Vbrd (4.11)
where,
????? ?Lw p ? 2p i=1 4Di , discrete profile
Vprof = ???? ? ( ) (4.12)? L D(x) 20 2 dx, continuous profile
Vprof is the volume encased within the outline of the shape profile of the FAM,
Vbld is the volume consumed by the incompressible bladder, and Vbrd is the volume
of the braid. Therefore, equation 4.11 states that the internal volume of the FAM
is equal to the volume within the shape profile minus the volumes of the bladder
and braid that are also contained within the shape profile volume. The bladder
and braid volumes are fixed values that can easily be found using their cylindrical
dimensions in the FAM?s resting length configuration (i.e L0). For the FAM used in
this research, the braid has an assumed resting diameter of 0.863 in equal to that of
the end-fitting at each end, and a thickness of 0.011 in resulting in a value for Vbrd
162
of 0.273 in3. The incompressible bladder has a resting outer diameter of 7/8 in and
thickness of 1/16 in, resulting in a bladder volume, Vbld, of 1.481 in3.
The volume Vprof can be found through integration of the defined shape profile
(Eq. 4.12). The method of integration is dependent on how the shape profile is
defined. In this research, the diameter of the shape profile has been defined as a
discrete function Di, with the width of the FAM defined for each pixel i along the
length of the FAM, and as a continuous function, D(x), defined by the Fourier series
fit equations used to describe the shape profile of the FAM. To find Vprof for the
discrete definition of the shape profile, integration is performed as a summation of
the volume of each pixel-width slice of the FAM. This discrete integration is shown
in Equation 4.12, where wp is the width of each pixel, and Lp is the current length (in
pixels) of the FAM. The continuous function definition of the shape profile provides
a much simpler method of calculating the volume of the FAM. As shown in equation
4.12, disc integration is performed to calculate the volume Vprof using the continuous
function form of the shape profile, D(x). The cylindrical approximated diameter of
the FAM, Dcyl,fit, having been shown to accurately represent the average diameter
of the FAM, can also be used to calculate the volume Vprof of the FAM.
Using the discrete form of the shape profile data, the internal volume of the
tested FAM is calculated as shown in Figure 4.19. As is predicted from the average
shape profile diameter results of Figure 4.16, the volume is a function of the length
state of the FAM and is effectively independent of internal pressure. There is only
a slight deviation in volume with respect to pressure due to the assumed stretch in
the braid. These results can be easily adjusted to provide values for the change in
163
volume required for contraction of the FAM as would be needed for flow rate and
efficiency calculations.
20
15
10 100 psi
80 psi
60 psi
5
40 psi
20 psi
0
0 0.1 0.2 0.3 0.4
Contraction, L /L
0
Figure 4.19: Internal FAM volume with respect to contraction for all tested pressures
4.5.3 Testing of Different FAMs
A few other FAMs have been characterized thus far using the photogrammetric
methods as described above. Three 7/8 in diameter FAMs and one 2-1/4 in diameter
FAM have been characterized using photogrammetry. The three 7/8 in diameter
FAMs have identical material construction. The 9-9/32 in long FAM has been used
to demonstrate the photogrammetric methods up to this point, but FAMs with
lengths of 12 in and 20 in have also been tested (Fig. 4.20). These FAMs were
tested to compare their shape profiles while also demonstrating the ability of the
photogrammetric test setup to characterize FAMs of varying length scales. Analysis
of the 2-1/4 in diameter by 13-13/16 in long FAM (Fig. 4.22) further demonstrates
164
3
Volume, in
the capability of the test method to characterize FAMs of a wide range of diameters.
The shape profiles of the 7/8 in diameter FAMs are compared in Figure 4.21 for
the same five states of percent contraction. Despite their slight differences in shape,
their resulting diameters are very similar. The three FAMs were all able to achieve
nearly identical free contraction values at 100 psi of 35.43%, 35.55%, and 35.36%
for the 9-9/32 in, 12 in, and 20 in FAMs respectively. Each of their profiles has an
undulation at each end of the FAM that resembles the previously noted peanut-like
shape which seems to be a characteristic of many FAMs. The shape profile of the
20 in (longest) FAM has an additional small undulation at its mid-length that is
barely visible. A Fourier series fit was conducted on the three FAMs resulting in
an accurate representation of their profiles. A three-term Fourier series provides an
accurate fit for the 9-9/32 in and 12 in long FAMs, while the additional undulation
of the 20 in long FAM profile requires a four-term Fourier series to achieve a good fit.
The largest discrepancy between the profiles comes in the zero-percent contraction
state, where the 9-9/32 in FAM has the largest average diameter of the three. The
difference in initial diameters is reflected by the difference in their estimated initial
braid angles. The three FAMs had estimated initial braid angles of 69.4, 71.1,
and 71.9 degrees for the 9-9/32 in, 12 in, and 20 in FAMs respectively. With all
three FAMs using the same braid, the differences in initial braid angle and diameter
indicates slight differences in their fabrication or identification of their resting length.
As previously noted from observation of Figure 4.16, the diameter of the FAM is
much more sensitive to changes in length at lower states of contraction, so a small
deviation in length or braid angle near the resting length state results in a relatively
165
(a) L 90 = 9 32 in (b) L0 = 12 in (c) L0 = 20 in
Figure 4.20: Tested 7/8 in FAMs in free contraction and blocked force (resting
length) states at 100 psi
3 L  = 9-9/32 in
0
L  = 12 in
0
2.5 L  = 20 in
0
2
1.5
1
0 0.2 0.4 0.6 0.8 1
Normalized Length, L/L
0
Figure 4.21: Shape profile comparison of the three tested 7/8 in diameter FAMs at
0, 8.75, 17.5, 26.25, and 35 percent contraction
166
Diameter, in
4
3.5
3
2.5
2
0 0.2 0.4 0.6 0.8 1
Normalized Length, L/L
0
(a) Free contraction and (b) Evolution of shape profile
resting length states
Figure 4.22: Characterization of the 2-1/4 in diameter FAM at 26 psi
large change in the expected diameter.
The shape profiles of the 2-1/4 in diameter FAM, as captured in Figure 4.22(b),
are also accurately captured using the described photogrammetric methods. The
simpler rounded shape of its profile allows for an accurate Fourier series fit of the
profiles to be achieved with the use of only two terms.
Testing of both the 7/8 in diameter by 20 in long FAM, and the 2-1/4 in diame-
ter by 13-13/16 in FAM, represents the maximum extent of what has been produced
with regard to sizing of FAM lengths and diameters, respectively. Although the
photogrammetric measurement of FAMs of different sizes might seem trivial, many
previous test setups for FAM volume testing have required modifications or redesigns
in order to administer tests for FAMs of different sizes. Photogrammetric character-
ization is capable of testing FAMs with a large range of sizes by just repositioning
167
Diameter, in
the camera to capture and fill each image with the entire FAM. As mentioned ear-
lier, the positioning of the camera will effect the accuracy of the photogrammetric
measurements. The accuracy is dependent on the competing effects of attempting
to reduce the angle-of-view by distancing the camera from the FAM, while also keep-
ing the distance at a minimum to maximize measurement resolution. For the 7/8 in
diameter FAMs, the camera was setup at the distance of 60.0 in for the shorter 9-
9/32 in length FAM, and it was placed 3.0 in farther away for the longer 20 in FAM to
capture its entire length. Those two camera placements resulted in angle-of-views of
8.84 and 18.04 degrees respectively, and measurement resolutions of 216.35 pixels/in
and 226.63 pixels/in respectively. Both tests achieved a high degree of accuracy de-
spite the slight changes in resolution and angle-of-view. Comparing the contraction
measurements of the MTS machine and photogrammetric data, as performed in Fig-
ure 4.14 for the 9-9/32 in FAM, resulted in a maximum error of 0.042 in (0.21%) for
the 20 in long FAM. This is in comparison to the 0.024 in (0.25%) error previously
cited for the 9-9/32 in FAM. Similarly, measurement of the 2-1/4 in diameter FAM
achieved results with a maximum error in the length contraction measurement of
0.05 in (0.36 %). It is assumed that even with the larger diameter of this 2-1/4 in
FAM, its diameter is still significantly smaller than the length measurement that is
used here to cap the maximum expected error in the measurement.
The accurate characterization of the shape profiles of the four FAMs demon-
strates that the photogrammetric methods presented in this research can be readily
adapted to FAMs of different length and diameter scales. A new method of exper-
imentally characterizing the volume of FAMs has been presented that can be per-
168
formed without the need for specialized test equipment, and can be easily adapted
for the measurement of FAMs of different sizes.
4.6 Conclusion
This research presents an approach for characterizing the shape profile of FAMs
through photogrammetric measurement. The method of measuring the FAM?s shape
profile is described including a basic test setup, method of image acquisition, and a
process of abstracting measurements from the image data. A 7/8 in diameter FAM
served as an example for describing the presented test method and analysis.
The photogrammetric testing was performed in parallel with the isobaric force-
contraction characterization that is typically performed to define the actuation of
FAMs. The photogrammetric test setup requires the addition of only a camera, back-
drop, lighting, and masking to acquire measurement data. The complete method
of acquiring measurements from the image data, including a background on the
methods used, was provided in detail to ease the adoption of this characterization
method for future research efforts.
Some preliminary analysis of the shape profile data was then performed. The
average diameter and volume of the FAM were found to be primarily a function of
the contraction state. However, there was an indication of strain in the braid fibers
from the small but appreciable increase in FAM diameter with increased pressure.
This strain in the braid fibers had not been quantifiable prior to the characterization
performed in this testing.
169
The measured shape profiles were redefined from a set of data points to a
cylindrical approximation or fit equation. The average of each measured profile
was used to define each profile as a single value. This assumes a cylindrical shape
profile that can be readily applied to previously existing models. If the assumption
of a cylindrical shape profile is not sufficient for a high-fidelity modeling effort, the
complex curvature of the shape profiles can be defined to a high-degree of accuracy
using Fourier series fit equations. Working with equations instead of the raw shape
profile data greatly simplifies usage of the shape profile information garnered from
the photogrammetric testing.
The estimated diameters found using the cylindrical approximation and initial
geometry have long served as the basis for most FAM modeling. These estimates
were tested by comparing them to the measured average diameters of the tested
FAM. The comparisons resulted in discrepancies between the estimated and mea-
sured diameters, with the first comparison finding an inaccuracy in the initial esti-
mated diameter. After this initial offset in diameter was corrected, the estimated
diameters still did not match the measured diameters, warranting further future
investigation of the effect of the cylindrical approximation on these estimates.
A new method of characterizing the internal volume of FAMs using the ac-
quired shape profile data was also presented. This new non-contact method of
calculating the internal volume is desirable compared to current methods which of-
ten require specialized test equipment that must be altered in order to test FAMs
of different sizes.
Finally, the accuracy and adaptability of the presented test method is demon-
170
strated through the shape profile characterization of multiple FAMs of different sizes.
A comparison of three 7/8 in diameter FAMs was performed, demonstrating some
of the similarities in their contraction-diameter characteristics, and also the slight
differences in the undulations and curvature of their profiles. A wide 2-1/4 in FAM
was also accurately characterized with the same test setup without the need for any
modifications. The Fourier series fit was able to accurately replicate the profile data
for all tested FAMs with an observed correlation between the number of undulations
of the profile?s curvature, and the number of terms required to provide an accurate
fit.
The presented method was developed out of a need to be able to quantify
modeling unknowns and long used approximations of diameter, but also out of the
necessity to define the dimensions of FAMs for implementation into mechanism de-
signs, to define actuation system requirements, and develop accurate models for
open-loop control of FAMs. It is the authors hope that this method of characteriza-
tion can aid in the discovery, confirmation, and necessary adjustments for improved
modeling efforts, as well as enlighten and ease the adoption of FAMs by providing
a more complete characterization of their actuation.
171
Chapter 5: Investigation of the Inelastic Braid Modeling Assumption
5.1 Introduction
To date, most of the methods used to model the actuation force and geometry
of FAMs have been based on the original modeling strategies used from when FAMs
were first in invented back in 1958 [2]. These models have continued to rely on
assumptions while having little or no investigation of their accuracy. Many of these
assumptions have been retained due to the modeling simplifications they provide,
or due simply to an inability to make further improvements to the accuracy of the
models.
One of the most prevalent modeling assumptions used with FAMs is the in-
elastic braid assumption, which assumes that the fibers of the braid do not stretch
under the tensile loads that are applied to them during actuation. This assump-
tion ? commonly used in conjunction with the cylindrical geometry approximation
? enables a simplification of the geometric and actuation force models of FAMs.
A lack of investigation into the effects of braid elasticity has persisted, in part,
due to an inability to experimentally capture and quantify the stretch of the braid
during actuation of the FAM. Stretching of the braid is often visually imperceptible,
and has consequently often gone unnoticed or ignored. Previous methods of mea-
172
suring the dimensions of the braid ? such as using a ruler to measure the braid at
discrete points along the length of the FAM ? have been rudimentary and lack the
resolution and accuracy required for an effective investigation of braid stretch.
A wide-reaching survey of previous FAM modeling efforts was conducted, and
found that very few researchers have investigated the effects of braid elasticity on
actuation. A few papers have included a cursory investigation of braid elasticity
and, absent the use of supporting experimental data, have subsequently assumed
the effects to be negligible [81]. Just two authors were found to have included
factors that directly account for braid elasticity in their actuation force models, and
they have both noted subsequent improvements in the accuracy of their models.
Davis et al. [11] added a term to their force-balance based model to account for
the estimated braid stretch during actuation after observing the radial expansion
of a FAM in isometric states with changes in pressure. Similarly, Kothera [13]
used an energy-method modeling approach, adding a term that accounted for the
elastic energy storage due to the stretch of the braid. With the inclusion of their
braid-stretch terms, both studies demonstrated improvements in the accuracy of
their actuation force models. However, they both lacked experimental evidence to
support the magnitude of the braid stretch assumptions used in their models.
Now, development of a photogrammetric method for measuring the FAM?s
external dimensions provides the degree of resolution and accuracy required of mea-
surements to analyze the stretch of the braid during FAM actuation. Initial analysis
of the photogrammetric measurement data (Ch. 4) found a slight increase in the av-
erage diameter of the FAM as it was isometrically pressurized. It was hypothesized
173
that this radial expansion is primarily due to the stretch of the braid fibers, and
not simply attributable to a reorientation and initial tensioning of the braid fibers
as has been noted in previous testing of FAMs. Further investigation is necessary
to affirm this hypothesis.
The goal of this research, therefore, is to investigate the effects of the braid?s
elasticity on the resulting geometry and actuation force of FAMs. The stretch of
the braid is calculated using an accurate representation of the FAM?s dimensions
obtained through photogrammetric measurement. Calculations of braid stretch and
tension are then used to calculate an approximate modulus of elasticity of the
braid material which is then compared to cited and experimentally-obtained val-
ues. Agreement of all of the obtained elasticity values indicates that the braid
elasticity is, in fact, the source of the observed radial expansion. The result of that
elasticity is then tested by approximating its effects on the resulting dimensions and
actuation force of the FAM. An assessment is then made on whether the inelastic
braid assumption should be maintained as a baseline in future work.
5.2 Experimental Testing
To investigate the effects of braid elasticity, characterization of both the braid
and the FAM?s actuation are needed. An actuation response was characterized
by performing isobaric force-contraction tests on the FAM while simultaneously
measuring its geometric dimensions. The braid was characterized in isolation by
performing tensile tests on individual threads of the aramid braid.
174
5.2.1 Description of Tested FAM
The tested FAM is composed of a latex inner bladder, an aramid braided
sleeve, and swaged aluminum end-fittings (Ch. 2) (Fig. 5.1(a)); It has a diameter
of 7/8 in, and a length of 9-9/32 in. The braid is defined by two parameters that
are the resting length of each braid thread, B0, and the number of turns of each
thread about the circumference of the FAM, N . For the tested FAM, values for B0
and N were found ? through direct measurement of the FAM ? to be 9.89 in and
1.09 respectively (Fig. 5.1(b)). Direct measurement of those two values ensured
accuracies within ?1/32 in, and ?0.005, respectively.
(a) Constructed FAM
(b) FAM threads marked for measurement of N and B0
(c) Kevlar braid used in FAM construction
Figure 5.1: Tested 7/8 in diameter FAM
175
5.2.2 Characterization of the FAM?s Actuation
The actuation response of the FAM was characterized using a 22 kip MTS
servo-hydraulic test machine (Fig. 5.2(a)). The MTS machine records the displace-
ment of the cyclic motion, along with the external inputs of force (Honeywell Model
31 Mid) and pressure (Omega PC209-200G5V). This testing was performed by first
setting a fixed internal pressure of the FAM, and then cyclically stroking the FAM
between its resting length and its pressure-dependent zero force length states. That
test sequence was repeated for isobaric pressure tests of 10-100 psi at 10 psi incre-
ments. The pressure of the working fluid was set and maintained at fixed pressures
(i.e. isobars) using the air pressure control system developed in Chapter 3. The
FAM was cycled three times at each pressure at a quasi-static rate of 0.04 in/sec.
Throughout the testing sequence, measurement data of the FAM?s outer diameter
was simultaneously collected at 1 second intervals, by using the photogrammetric
acquisition and analysis methods developed in Chapter 4. Observation of the mea-
surements shows an increase in diameter with increases in pressure, that is indicative
of what would be expected to occur with stretching of the threads in the braid (Fig.
5.2(b)).
5.2.3 Characterization of the FAM?s Braid
The braid and its material properties are of particular interest in this research.
The braid characterized in this research, is composed of 144 Kevlar-49 threads woven
into a double-helical tubular sleeve (Fig. 5.1(c)). The Kevlar-49 thread is cited as
176
1000 100 psi
80 psi
60 psi
800
40 psi
20 psi
600
400
200
0
0 0.1 0.2 0.3 0.4
Contraction, L /L
0
(a) Actuation Force of FAM with respect to (b) Average diameter of FAM with respect to
contraction at set isobars contraction
Figure 5.2: Characterization testing of the FAM?s actuation and geometry
having an elastic modulus of 16.3 ? 106 psi, a breaking strength of 59.3 lbf, and
an elongation at failure of 2.4% [82]. Each of those threads is composed of 768
fibers and each fiber has a diameter of 0.00047 in, resulting in an overall thread
cross-sectional area of 1.332? 10?4in2.
The material specifications of the Kevlar-49 thread were experimentally val-
idated by conducting a tensile failure test on thread samples from the braid that
was used on the FAM. This provides a more complete picture of the stiffness profile
of the Kevlar threads, but also ensures that the braid?s material properties did not
change due to any variation in manufacturing or storage (e.g., degradation due to
UV light exposure). Tensile testing of the thread was performed on an Instron 8841
servo-hydraulic testing machine (Fig. 5.3(a)). Thread samples of about 4 in length
were prepared by using epoxy to attach tabs for clamping at each end of the thread.
Those thread samples were then stretched at a rate of 0.004 in/sec (0.1 mm/sec)
until failure.
Test results for three threads are shown in Figure 5.3(b). The experimental
177
Force, lb
modulus of elasticity was approximated by identifying the slope of the linear portion
of each curve. The average elasticity was found to be 16.9? 106 psi which is within
5% of the cited value for Kevlar-49. Prior to the linear elastic region of each curve,
there is a notable nonlinear region that begins at the onset of loading the thread.
This nonlinear portion of each curve ? which is typically left out of material property
reference charts [83] ? is due to the initial straightening and uncrimping of the thread
fibers, and it continues until all of the fibers in the thread are tensioned and aligned
with the thread axis [84]. Note that subsequent analysis of the strain in the braid
will find this region of the stress-strain curve to be significant.
5
10
6 Test 1
Test 2
5 Test 3
4
3
2
1
0
0 0.01 0.02 0.03 0.04
Strain ( ), in/in
(a) Single thread (b) Resulting stress-strain relationships of tested thread samples
tested on Instron
machine
Figure 5.3: Testing of Kevlar-49 thread samples
178
Stress ( ), psi
5.3 Analysis of Braid Elasticity Using Measurements of FAM
Characterization data of the FAM?s actuation force and diameter is used for
in situ analysis of the braid?s elasticity during actuation of the FAM. The actuation
force data is used to estimate the tension in the braid; the diameter measurements of
the FAM are used to estimate the stretch in the braid. Typical methods of modeling
the braid?s dimensions and tension require a cylindrical approximation of the FAM?s
geometry. This approximation is commonly made despite the FAM?s increasingly
noncylindrical shape in contracted length states. To ensure the accuracy of our
initial investigation of braid tension and stretch, we will focus on the FAM in its
resting length state where the cylindrical approximation of the geometry is most
accurate.
Figure 5.4: Triangle relationship of FAM dimensions, and subsequent effect with
consideration of braid elasticity
The cylindrical approximation enables a description of the FAM?s geometry
179
for all contraction states of a FAM through the use of a defined triangle relationship
(Fig. 5.4). This relationship relates the FAM?s length, L, and diameter, D, with
variables that describe the braid including the braid angle, ?, the length of each braid
thread, B, and the number of turns of each thread about the FAM?s circumference,
N . With the values of either N or D, and any two of the other dimension variables,
the geometry of the FAM can be fully defined using the triangle relationship. For
this work, we have known values for L, for the average diameter D, and for N ,
thus enabling B and ? to be estimated for respective states of contraction. Figure
5.4 shows that an increase in Dmeas at a fixed-length state of L would result in a
stretched length of the braid threads Belastic, and a respective change in the braid
angle to ?elastic. The stretched thread length can be estimated using the following
equation: ?
B 2elastic = (L + (N?D? 2meas) ) (5.1)
Estimation of the tension in each thread is based on force equilibrium equations from
actuation force models as shown in previous work and as described in Chapter 1.
These equations are also based on an assumed cylindrical geometry. Manipulation
of the longitudinal force equilibrium equation results in the following equation for
the thread tension Tbrd:
= 1 1 ( 2 + ? VbldTbrd P?R Fact ?z ) (5.2)
nbrd sin? L
where nbrd is the number of threads in the braid (nbrd = 144), P is the internal
pressure of the FAM, and Fact is the actuation force. The last term accounts for the
portion of the longitudinal force of the FAM that is due to strain in the bladder; it is
180
a function of the longitudinal stress ?z, bladder volume Vbld, and the instantaneous
length L.
Braid stretch and tension are initially investigated in the resting length state
where estimates acquired from Equations 5.1 and 5.2 ? based on the cylindrical ap-
proximation ? are most accurate. Figure 5.2(b) showed a slight increase in diameter
with pressure in the resting length state. In this state, the average diameter of the
FAM increased by 0.043 in with an increase in pressure from 0 to 100 psi (5.5(a)).
This increase in diameter creates changes in thread length and braid angle, as seen
in Table 5.1. The nominally linear increase in diameter with pressure supports the
hypothesis that the increase in diameter is likely due to braid elasticity. There is
a notable initial jump in diameter from the 7/8 in resting diameter of the bladder,
which indicates some initial tensioning and realignment of the braid upon pressur-
ization of the FAM. The fiber length at each pressure is estimated by inserting
the average diameter values at resting length, into Equation 5.1. This results in a
calculated braid stretch of up to 0.051 in (0.5% stretch) (Fig. 5.5(b)).
FAM 100 psi 80 psi 60 psi 40 psi 20 psi 0 psi (pred.)
Dmeas, in 1.044 1.037 1.028 1.020 1.009 1.001
Belastic, in 9.945 9.937 9.926 9.916 9.903 9.894
?elastic, deg. 68.94 69.07 69.22 69.38 69.58 69.73
Table 5.1: Geometric dimensions calculated for the FAM in the resting length state
The tension in the braid threads is estimated using Equation 5.2. Because
we are only investigating the resting length state, the form of Equation 5.2 can
be simplified to eliminate the last term of the equation by considering that ?z is
negligible when the bladder is in the resting length state. The tension in each thread
181
of the braid linearly increases with pressure (Fig. 5.5(c)). The maximum force of
8.5 lbf for each thread is well below the cited failure strength of 59.3 lbf for the
Kevlar-49 threads.
With the thread stretch and tension estimates, the stress (Eq. 5.3) and strain
(Eq. 5.4) of the braid thread is then calculated to obtain a measure of the braid?s
elasticity while on the FAM. The equations for stress and strain are as follows:
Tbrd
?brd = (5.3)
Abrd
= Belasticbrd ? 1 (5.4)
B0
The resulting values produce a nominally linear stress-strain relationship (Fig.
5.5(d)). A linear fit of the data results in an estimated modulus of Ebrd = 11.6 ?
106psi which is 29% less than the cited value of Ebrd = 16.3?106 psi. Closer inspec-
tion of the stress-strain trend reveals a slight nonlinearity in the results. Separate
linear fits performed on the low pressure and high pressure data points found that
the high pressure fit value of Ebrd = 15.5? 106 psi is significantly closer (within 5%)
of the cited value.
5.3.1 Analysis of Measured Elasticity Values
The elasticity found through the in situ measurements of the FAM (Fig.
5.3(b)) is compared to the elasticity found from the tensile characterization test-
ing of the Kevlar thread (Fig. 5.5(d)), for agreement, in Figure 5.6. The tested
braid material in both cases should have the same stiffness.
The stress values from measurement of the FAM appear to be very close to
182
9.95
1.05
9.94
1 9.93
9.92
0.95
9.91
0.9
Initial Bladder O.D. (7/8 in) 9.9
0.85 9.89
0 20 40 60 80 100 0 20 40 60 80 100
Pressure, psi Pressure, psi
(a) Measured average diameter, D?meas (b) Estimated length of braid threads, Bmeas
4
10 10
7
6
6 E  = 15.5*108 brd
5
6 6
4 E  = 11.6*10
brd
4 3
2
2
1 6
E  = 9.2*10
brd
0 0
0 20 40 60 80 100 0 1 2 3 4 5
Pressure, psi Strain, in/in -310
(c) Estimated tension in braid threads, Tbrd (d) Estimated stress, ?brd, and strain, brd,
with linear fit analysis
Figure 5.5: Analysis of the braid threads through in situ analysis
those found in the tensile characterization testing (Fig. 5.6(a)). The maximum
braid strain observed on the FAM is brd = 0.0052 which places it in the low-
strain nonlinear region of the tensile test data. This coincides with the nonlinearity
observed in the stress-strain relationship found from measurements of the FAM.
Comparison of the two sets of data shows that the values are very close in magnitude
and have the same basic curvature (Fig. 5.6(b)). Comparing the stress from the
FAM measurements against the average curve of the three tensile tests, results in
a RMSE for the estimates from the FAM measurements of 3,867 psi. The slight
discrepancy between the measurements of the FAM and the tensile test entails the
183
Braid Tension, lbf Average Diameter, in
Braid Fiber Length, in
Stress, psi
possibility of a slight overestimation of the tensile loads of the threads on the FAM
by 0.57?1.92 lbf. This is much lower than the 1.48?4.50 lbf overestimation of the
tensile loads that would be assumed by using the cited linear elastic modulus in the
recorded range of strain values.
5 4
10 10
6 Test 1 10 Test 1
Test 2 Test 2
5 Test 3 8 Test 3
Image Data Image Data
4 6
3 4
2
2
1
0
0
0 0.01 0.02 0.03 0.04 0 1 2 3 4 5 6
Strain, in/in Strain, in/in -310
(a) Test data (b) Zoomed-in view of strain region of interest
Figure 5.6: Comparison of the stress-strain relationship calculated from (1) the
tensile test measurements of the Kevlar threads, and (2) estimates from the in situ
measurements of the FAM?s braid.
Overall, comparison of the elasticity values found from in situ measurements
of the FAM, and from tensile testing of individual Kevlar-49 threads, confirms that
the source of the fixed-length radial expansion with increases in pressure is, in fact,
due to stretching of the braid?s threads. The elasticity of the braid observed on the
FAM matched closely with what was expected from both cited and tested values.
With the elasticity of the braid now fully characterized, the effects of the braid?s
elasticity on geometry and actuation force can be investigated.
184
Stress, psi
Stress, psi
5.4 The Effect of Braid Elasticity on Geometry and Actuation Force
With the stiffness of the braid characterized, the effects of elasticity on the
parameters that define the FAM?s geometry (D, ?, and B) and actuation force
can now be analyzed. This analysis is performed by first estimating the stretch
of the braid threads for all pressure-contraction states, and then using the triangle
relationship, as shown in Figure 5.4, to calculate the corrected values of B, ?, and D
that take the braid?s elasticity into account. The effect of the corrected dimensions
on the actuation force is then calculated using the Gaylord force equation (Eq. 5.9).
The traditional approach of estimating the dimensions and actuation force of
the FAM has relied on the inelastic braid assumption (i.e. a fixed value for B) and a
fixed value of N , while assuming a cylindrical geometry for all states of contraction.
Using the triangle approximation, values of D and ? are approximated for all length
states L of the FAM, using the following basic relations:
?
= B
2 ? L2
D (5.5)
?N( )
? = sin?1 L (5.6)
B
With this work, we are no longer assuming a fixed value for B. The stretched length
of each thread is found through the following equation:
= (1? TbrdBelastic B0 )
AbrdEbrd
= B0 (5.7)
1? 2B Fact+P?R0LEbrdAbrdnbrd
where Tbrd is calculated using Equation 5.2; once again, the last term is excluded
due to a negligible value of ?z. For this form of the equation, the interdependence
185
between the braid?s geometry and tension requires an estimation of R and ? based
on either the rigid braid assumption, or experimental data. Alternatively, with the
insertion of Equation 5.5 into Equation 5.7, a nontrivial solution that is independent
of D and ? can be obtained, which requires numerical methods to solve:
( )
0 = 3 B0P?Belastic( 4N2?2LAbrdEb(rdnbrd 2 ))
+ B0 L P?Belastic Fact ? 4 2 2 ? 1 +B0 (5.8)LnbrdAbrdEbrd N ?
For this equation, F can be an input from experimental data, or estimated using
the Gaylord force. The braid thread tension and length can then be calculated
for all pressure-contraction states of the FAM (Fig. 5.7). In this case, the elastic
modulus of the braid, Ebrd, is assumed to be 11.6? 106 psi which is the value found
by performing a linear fit of the stress-strain data obtained from measurement of
the FAM (Fig. 5.5(d)). The results of Figure 5.7(b)) show that:
? Stretch of the braid increases with pressure
? Maximum stretch occurs in the block force (i.e. resting length) state where
the actuation force is largest
The results show that maximum braid stretch is 0.06 in (0.6%) which occurs at
100 psi.
5.4.1 Effects of Elasticity with Kevlar-49 Braid
With the stretch of the braid threads now understood for all states of FAM
length and pressure, the effect of the braid?s elasticity on the dimensions and actu-
186
(a) Tension in each thread of braid, Tbrd (b) Length of each thread in braid, Belastic
Figure 5.7: Thread tension and length with respect to contraction and internal
pressure
ation force can now be investigated.
Having known values for Belastic, N , and L, the dimensions Delastic and ?elastic
can be estimated using Equations 5.5 and 5.6 (Fig. 5.8(a)). Figures 5.8(b)-5.8(d)
compare the dimensions obtained from assuming the braid is elastic versus inelastic.
As with the trends shown for Belastic, the effects of braid elasticity are largest in
the resting length state, and decrease as the FAM contracts. The difference in
dimensions between assuming an elastic braid versus an inelastic braid is barely
perceptible. Elasticity of the braid resulted in a maximum decrease in braid angle
of 0.73 degrees (1.1%), and a maximum increase in diameter of 0.04 in (3.8%).
The effect of the braid?s elasticity on actuation force can now be investigated
with knowledge of the change in the FAM?s dimensions. The Gaylord force FG
? which approximates the actuation force of the FAM ? is used to calculate the
expected change in actuation force, and is defined as:
FG = P?R2(2tan2? ? 1) (5.9)
187
where inputs include internal pressure P , and the dimensions R and ?. The effect
of the braid?s elasticity on the Gaylord force is shown in Figure 5.9. The relatively
small change in the FAM?s dimensions with the inclusion of braid elasticity results
in a similarly small effect on the actuation force; inclusion of the effects of braid
elasticity resulted in a 6 lbf (0.6%) decrease in force in the resting length state, and
a 3 lbf (5%) decrease in force in the free contraction state.
The overall effect of the Kevlar braid?s elasticity is minimal with only small
effects on the resulting dimensions and actuation force of the FAM. Therefore, the
inelastic braid assumption could be reasonably assumed for basic modeling efforts of
this FAM. Future FAM-related research, however, should be careful to not use this
conclusion to universally assume that the inelastic braid assumption is sufficient
for other FAMs of different constructions. The Kevlar used in this work, with a
cited modulus of 16.3 ? 106 psi, has a very high degree of stiffness in comparison
to other braid materials that are commonly used such as Nylon and polyethylene
terephthalate (PET).
5.4.2 Estimated Effects of Lower Stiffness Braid Materials
FAMs have commonly used braid materials that have a much lower modulus
of elasticity than Kevlar-49 [54]. To investigate the effects of a lower stiffness braid
material, a case study is performed. This case study assumes a braid modulus that
is 10% of the stiffness of the tested Kevlar-49 (from linear fit of nonlinear region)
(1.16? 106 psi) while maintaining the same actuation forces and initial dimensions
188
70
65 elastic
100 psi
60 80 psi
60 psi
55 40 psi
20 psi
50
45
40
35
0 0.1 0.2 0.3 0.4
Contraction, L /L
0
(a) Elastic braid dimension definitions (b) Braid angle estimations, ?
10 2.25
9.9 2
B  = 9.89 in
0
9.8 B 1.75 D
elastic elastic
100 psi 100 psi
9.7
80 psi 1.5 80 psi
60 psi 60 psi
9.6 40 psi 1.25 40 psi
20 psi 20 psi
9.5 1
0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4
Contraction, L /L Contraction, L /L
0 0
(c) Braid thread length estimations, B (d) Braid diameter, D
Figure 5.8: Estimated braid dimensions for the Kevlar-49 braid (Ebrd = 11.6 ?
106 psi) ? Comparison between elastic (colored lines) and inelastic (black dashed
lines) braid assumptions
1200
F
G,elastic
1000
100 psi
80 psi
800
60 psi
40 psi
600 20 psi
400
200
0
0 0.1 0.2 0.3 0.4
Contraction, L /L
0
Figure 5.9: Gaylord actuation force estimates for the Kevlar-49 braid (Ebrd = 11.6?
106 psi) ? Comparison between elastic (colored lines) and inelastic (black dashed
lines) braid assumptions
189
Thread Length, in
Force, lb
Diameter, in Braid Angle, deg.
of the tested FAM. This assumed modulus value is much closer to that of commonly
used materials such as Nylon (0.5? 106 psi) [85], and PET (0.62? 106 psi) [13].
The effect of the braid?s elasticity on the FAM?s dimensions and actuation force
is investigated again for comparison (Table 5.1). In contrast with the previous results
? where a stiffer braid was used ? the magnitude of stretch estimated for the lower-
elasticity material is substantial. The estimated stretch of the braid threads is 0.57 in
(5.8%). This results in a substantial decrease in braid angle of 6.3 degrees (12.3%),
and an increase in diameter of up to 0.38 in (30.1%). These results are comparable
to the work of Davis, where 5% braid stretch and 33% diameter expansion were
observed with a FAM that used a Nylon braid [11]. The similarities between the
values observed by Davis and the estimated values in this research supports the
fidelity of this case study.
The elasticity of the braid, and respective changes in braid dimensions, results
in a substantial decrease in actuation force with the inclusion of braid elasticity
effects (Fig. 5.11). When compared to the inelastic braid assumption, the force in
the resting length state decreases by up to 83 lbf (8%) at 100 psi, while the force
at free contraction decreases by up to 47 lbf (70.4%). Through this example case
study, it is evident that the elasticity of the braid can have a substantial effect on
the resulting dimensions and actuation force of the FAM.
190
70
65 elastic
100 psi
60 80 psi
60 psi
55 40 psi
20 psi
50
45
40
35
0 0.1 0.2 0.3 0.4
Contraction, L /L
0
(a) Elastic braid definitions (b) Braid angle estimations, ?
2.5
10.5
2.25
2
10
B D
elastic 1.75 elastic
B  = 9.89 in
0 100 psi 100 psi
80 psi 1.5 80 psi
9.5 60 psi 60 psi
40 psi 1.25 40 psi
20 psi 20 psi
1
0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4
Contraction, L /L Contraction, L /L
0 0
(c) Braid thread length estimations, B (d) Braid diameter, D
Figure 5.10: Estimated braid dimensions for a lower stiffness braid (Ebrd = 1.16 ?
106 psi) ? Comparison between elastic (colored lines) and inelastic (black dashed
lines) braid assumptions
1200
F
G,elastic
1000
100 psi
80 psi
800
60 psi
40 psi
600 20 psi
400
200
0
0 0.1 0.2 0.3 0.4
Contraction, L /L
0
Figure 5.11: Gaylord actuation force estimates for a lower stiffness braid (Ebrd =
1.16 ? 106 psi) ? Comparison between elastic (colored lines) and inelastic (black
dashed lines) braid assumptions
191
Thread Length, in
Force, lb
Diameter, in Braid Angle, deg.
Max Values Kevlar-49 Reduced E Kevlar Nylon [11]
Elastic Modulus 16.3? 106 psi 1.16? 106 psi 0.5? 106 psi (est.)
Tested Pressure 100 psi 100 psi 65 psi
Diameter Change 3.8% (0.04 in) 30.1% (0.38 in) 33% (0.3 in)
Thread Stretch 0.5% (0.05 in) 5.8% (0.57 in) 5% (4 in)
Act. Force Change 0.6% (6 lbf) 8% (83 lbf) ?
Table 5.2: Comparison of effects of braid stretch for different braid materials
5.5 Conclusion
Prior to this work, very little research had been performed on the effects of
braid elasticity on FAM actuation characteristics. The inelastic braid assumption
has been widely used to simplify analysis. A lack of experimental data to test the
effects of braid elasticity has abetted continuation of the use of this assumption, and
its validity must be questioned.
The introduction of a photogrammetric method of measuring the dimensions
of the FAM opened the door for analysis of braid elasticity and itsg effects on FAM
actuation. Photogrammetric measurements showed that the 7/8 in diameter FAM
would expand in diameter in its resting length state by 0.043 in with an increase in
pressure from 0 to 100 psi. Using the triangle relationship enabled by the cylindrical
approximation, this indicated that the Kevlar braid was stretching by up to 0.5%.
Through investigation of the braid?s elasticity using measurements of the FAM
during actuation, the braid was found to operate primarily in the initial nonlinear
region of its stress-strain curve. Furthermore, the stress-strain curve developed from
the in situ measurements of the FAM were found to closely replicate the nonlinear
stress-strain relationship found through tensile testing of individual threads. The
192
linear modulus of elasticity obtained from measurements of the FAM were within
5% of the cited value of Kevlar-49 which aided in assuring the fidelity of the braid
elasticity analysis.
The effects of the braid stretch on the resulting dimensions and actuation
force of the FAM were quantified. It was found that the Kevlar braid of the tested
FAM had an appreciable, but minimal, effect on the dimensions (3.8% maximum
increase in diameter) and actuation force (5% maximum decrease). A case study
was performed using a modulus of elasticity value similar to that of other commonly
used braid materials, and the study found that these less-stiff braid materials would
have a significant effect on the dimensions (30.1% maximum increase in diameter)
and actuation force (70.4% maximum decrease) of the FAM. The effects of elasticity
were especially noticeable in the blocked force state where the actuation force, and
resulting tension on the braid threads, is highest in magnitude.
Overall, it is clear that braid stretch, especially with lower modulus of elasticity
braid materials, can have a substantial effect on the dimensions and actuation force
of the FAM. This research demonstrates that the inelastic braid assumption cannot
be blindly assumed as has commonly been done in previous work with FAMs. To
ensure the accuracy of modeling efforts, both the material and loading of the braid
must both be considered to ensure that the possible effects of the braid?s elasticity
on the FAM?s actuation are understood.
193
Chapter 6: Investigation of the Cylindrical Approximation Assump-
tion
6.1 Introduction
The braid of a FAM serves as its main load-bearing component, and it plays
a critical role in determining the actuation characteristics of the FAM through its
geometry. The braid transforms the pressure force of the internal working fluid into
a contractile actuation force, while also enabling a contractile motion of the FAM
through the reorientation of its helically wound threads. As the FAM contracts
in length, the reorientation of the braid?s threads results in a simultaneous radial
expansion.
The braid is fixed to end-fittings at each end of the FAM which have diameters
that are nominally equal to the initial resting diameter of the braid. As the braid
radially expands with contraction, its diameter is constrained in proximity to each
end-fitting. This results in a shape of the FAM that, instead of being cylindrical,
has a bulbous shape with rounded ends (Fig. 6.1(a)), and a diameter and braid
angle that varies along the length of the FAM.
Despite its complex geometry, the modeling and analysis of FAMs ? dating
194
(a) Actual profile of FAM (b) Assumed cylindrical profile
Figure 6.1: Comparison between the profile of the FAM, and the profile assumed
with the commonly used cylindrical approximation
back to the invention of the FAM in 1958 ? typically assumes a cylindrical shape
profile (Fig. 6.1(b)) [1?3, 5, 8?10, 13, 16, 18?21]. This is a simplifying assumption
that assumes a constant diameter and braid angle along the length of the FAM.
The cylindrical assumption has been used in the derivation of almost every FAM
modeling effort, including models of the FAM?s internal volume [19], friction [9],
bladder strain [13], and actuation force [17]. By using a cylindrical approximation
assumption, cylindrical dimensions of the FAM are easily estimated for all states of
contraction, and have been used as an input in almost every model formulated for
FAMs.
Researchers have long understood that the inaccurate representation of a
FAM?s geometry ? by using the cylindrical approximation ? is a likely source of
modeling error [13]. For actuation force models, the inaccurate representation of
the shape profile ? especially in the region near each end-fitting ? is said to result
195
in end effects or tip effects which are thought to reduce the actuation force of the
FAM [13, 17]. To correct for the error induced by the cylindrical approximation,
researchers have implemented correction factors that are often used to tune their
models to experimental data, but which are based on little phenomenological rea-
soning. Almost all correction approaches assume a reduction in the effective active
length of the FAM, corresponding roughly with the length of the FAM?s nominally
cylindrical midsection [9]. A few researchers have assumed shape corrections of the
cylindrical profile ? based only on qualitative observations ? with focus on adjust-
ing the cylindrical profile in the region near each end-fitting. Examples of such
shape corrections includes replacing the region near each end-fitting with 90 de-
gree [13, 18, 67] or elliptical [20] arcs. However, these geometry corrections only act
to reduce the value of active length L ? a modeling input ? and therefore, still fail
to capture any detailed effects of these regions.
(a) Cylindrical (b) 90 deg. circular arc (c) Elliptical
Figure 6.2: Various assumed geometry profiles used for modeling of FAMs
Ability of the cylindrical approximation to accurately replicate the FAM?s true
geometry ? with or without shape corrections ? has not been tested in prior research.
196
Furthermore, a lack of available methods to directly measure the shape profile of
FAMs has stagnated attempts at improving upon the cylindrical approximation; the
only method extant was to use a ruler to manually measure the FAM diameter at
discrete points along its length, and at a limited number of pressure-contraction
states. Now, the invention of the photogrammetric measurement method ? detailed
in Chapter 4? enables a complete characterization of the FAM?s shape profile to be
acquired for all states of contraction and pressure, and opens the door for further
analysis. Photogrammetric measurements provide the ability to quantify, and pos-
sibly eliminate, the degree of modeling uncertainty associated with the inaccurate
representation of a FAM?s shape profile.
The cylindrical approximation has long been incorporated with FAM modeling
work without an understanding of the magnitude of error that it introduces. Here,
we attempt to quantify the error induced when using the estimated dimensions de-
rived from use of the cylindrical approximation. The same 7/8 inch diameter, 9-9/32
inch length FAM used in previous chapters will be used here for this analysis. As
the first step, the dimensions acquired using cylindrical geometry estimates will be
compared to the measured dimensions. These estimated and measured dimensions
are then used for a comparison of the bladder?s estimated strain state values. The
bladder?s stiffness is experimentally characterized, and then used, along with the
dimensional and strain estimates, to approximate the effect of the cylindrically es-
timated and measured profiles on the actuation force of the FAM. Through these
comparisons, the modeling error assumed in using these estimated dimensions by
way of the cylindrical approximation can now be understood, and subsequent cor-
197
rections in the dimensions and models can then be made.
6.2 Geometric Dimensions Definitions
The geometry of the FAM?s braid is defined by the following set of dimen-
sions: length L, diameter D, angle ?, length of each braid thread B, and number of
turns of each thread about the circumference of the FAM, N . Of these dimensional
parameters, N is always a fixed value. The value of B is also commonly assumed
as fixed with the application of the inelastic braid assumption. As seen in Chapter
5, the value of B actually changes due to stretching of the threads when tensioned,
and will be taken into consideration in this research. Length L, also known as
the active length, is measured as the longitudinal length of the braided section of
the FAM, and is commonly treated as the main independent variable in models of
FAMs. Values of D and ? are left to be determined, based on an estimated or a
measured profile of the FAM. Previous research has obtained dimension estimates
by assuming a cylindrical geometry of the FAM. That assumption uses estimated
values for the diameter, Dest and the braid angle ?est, which are assumed to be
constant along the length of the FAM. The photogrammetry measurements, on the
other hand, define the diameter of the FAM Dmeas, along the length of the FAM. A
method for directly measuring the braid angle has not yet been developed, so the
braid angle ?meas, must be approximated by using the other known dimensions, as
will be discussed.
Estimations of bladder dimensions, which will also be investigated, are reliant
198
on the assumption that the bladder is in continual contact with the inner surface of
the braid. This assumption is often true unless the bladder is not initially in contact
with the braid in the FAM?s resting length state; in that case, a minimum activation
pressure is required before this contact condition is satisfied. With the assumption
that the bladder is in contact with the braid, the diameter of the bladder, at its
mid-thickness, can be written in the following general form:
Dbld = Dbrd ? 2tbrd ? tbld (6.1)
where Dbrd is the photogrammetry-measured outer diameter of the braid, tbrd is the
fixed thickness of the braid (tbrd = 0.011 in for the tested FAM), and tbld is the
instantaneous thickness of the bladder.
The estimated dimensions of the FAM, as used in prior studies, are compared
here to the dimensions that are measured using photogrammetry. Previous efforts
have estimated the dimensions of the FAM in all states of contraction, using the
cylindrical approximation together with the known initial geometry 6.3(a). For this
analysis, the measured profile 6.3(b), or a cylindrical approximation of the measured
profile 6.3(c), will be used for analysis.
This work uses the same 7/8 inch diameter FAM as in previous chapters for
analysis. This FAM has a length L, of 9?9/32 inches, and its dimensions measured
to be N = 1.09, and B = 9.89 inches. These values will be used in subsequent
sections to analyze the dimensions, actuation force, and bladder strain of the FAM
acquired through estimated and measured geometry profiles of the FAM. The geome-
tries to be analyzed, both estimated and measured, are defined in the subsections
199
that follow.
(a) Estimated cylindrical geometry dimensions
(b) Measured geometry dimensions
(c) Cylindrical approximation of measured geometry dimensions
Figure 6.3: Definitions of dimensions for the analyzed geometries
6.2.1 Estimated Dimensions ? Cylindrical Approximation
Previous modeling efforts have used estimated dimensions for all length states
of the FAM (Fig. 6.3(a)). These estimates are derived based on the initial dimen-
sions of the FAM, and the cylindrical approximation enables the use of the triangle
relationship to define the dimensions (Fig. 6.4). In this case, fixed values of N and
200
Figure 6.4: The Triangle Relationship, enabled by the cylindrical geometry approx-
imation, used to estimate the dimensions of the FAM between states of contraction
B are determined in the resting length state L0. The estimated diameter, Dest, and
braid angle, ?est, are then calculated as a function of L using Equations 6.2 and 6.3,
as follows:
?
B2( ) = ? L
2
Dest L (6.2)
?N( )
?1 L?est(L) = sin (6.3)
B
The value of B is either assumed fixed by application of the inelastic braid assump-
tion, or assumed to be elastic with the magnitude of stretch calculated through
the methods of Chapter 5. Both elastic and inelastic braid assumptions, and their
respective effects on estimated dimensions and actuation force, are also included in
this research.
201
6.2.2 Measured Dimensions
Through a new method of measuring the FAM?s diameter (Ch. 4), Dmeas(z) is
known along the entire length of the FAM for all states of contraction and pressure
(Fig. 6.3(b)). Just as with the method of estimating dimensions, values of N and
B can be calculated or measured directly in the resting length state of the FAM.
In contracted length states, N is a fixed value, while B can only be approximated
using the other geometric dimensions, or assumed inelastic and fixed. The measured
diameter Dmeas(z), and respective braid angle ?(z), vary along the length of the
FAM and are a function of contraction and pressure. There is no current method for
directly measuring the braid angle along the length of the FAM which makes further
detailed dimensional analysis of the braid difficult. Analysis of the bladder, however,
is enabled to a greater level of detail using the measured dimension, Dmeas(z), as
discussed in subsequent sections.
For the analysis performed using Dmeas(z) in this work, the raw photogram-
metry data is represented by Fourier series equation curve-fits of each profile (Fig.
6.5). As performed in Chapter 4, the radius of each profile, Rmeas(z), is defined by
the equation:
?n
Rmeas(z) = a0 + ancos(nzw) + bnsin(nzw) (6.4)
1
where a0, an, and bn are parameters used to fit the curve to each profile, and only
three terms (n = 3) are necessary for a good representation of all profiles of the
tested 7/8 inch diameter FAM.
202
Figure 6.5: Comparison between the profiles defined from acquired image data, and
the Fourier-series fit equations that are used to define the profiles for analysis
6.2.3 Measured Dimensions ? Cylindrical Approximation
Most established modeling and analysis frameworks for FAMs have been de-
veloped to accept only cylindrical dimensions of a FAM. Therefore, especially for
comparison to the estimated dimensions, it is more straightforward to approximate
the measured dimensions through a cylindrical approximation (Fig. 6.3(c)). The
cylindrical approximation of the measured dimensions is calculated by assuming a
constant braid diameter that is equal to the average diameter of the measured pro-
file, D?meas. Using known values of N , D?meas, and L, the cylindrical dimensions ?meas
and Bmeas are approximated for all states of contraction and pressure. As will be
noted subsequently, the value of Bmeas is no longer an accurate representation of
thread length in this case, but rather is an effective value here when used only to
define the cylindrical approximated geometry. With the assumption of a cylindrical
profile, the value of Bmeas is less than that of B due to the loss of curvature along
203
the FAM?s surface.
The key difference between the cylindrical approximation of the measured and
the estimated dimensions can be explained as follows. The cylindrical approximation
of the measured dimensions assures the accuracy of the measured average diameter,
and provides an assumed subsequent improvement in accuracy of the average braid
angle. To maintain the triangle approximation, the value of Bmeas must assume some
inaccuracy with respect to the actual value B. In contrast, the estimated cylindrical
dimensions use an accurate value of B, but use of the cylindrical approximation leads
to error in the estimated diameter and braid angle values. Notably, those dimensions
D and ?, are prioritized in this research due to their role in determination of the
actuation characteristics of the FAM; therefore, a cylindrical approximation of the
measured dimensions should provide the closest thing to a ?true? representation of
these values while fitting into the cylindrical frameworks that have been previously
used.
6.3 Analysis of the Cylindrical and Measured Geometries
Previous research has calculated the FAM?s dimensions, actuation force, and
bladder strain states using an estimation of its geometric dimensions in all states
of contraction. Now, with access to actual measurements of the geometry of the
FAM, improvements are made to the accuracy of the dimensions, actuation force,
and bladder strain state values. In doing this comparison, the degree of error and
uncertainty produced by using the cylindrical approximation of the FAM?s geometry
204
can be understood and greatly reduced, and subsequent modeling improvements can
be established.
6.3.1 Comparison of Braid Dimensions
The dimensions from previous estimations are compared to the dimensions
obtained through direct measurements of the FAM?s geometry profile. For a direct
comparison, the cylindrical dimensions of the estimated geometry (Fig. 6.3(a)) are
compared to a cylindrical approximation of the measured profile geometry (Fig.
6.3(c)). The measured dimensions (D?meas, ?meas, Bmeas) are compared to the esti-
mated dimensions (Dest, ?est, Best) at each tested pressure (Fig. 6.6). Estimated
dimensions with, and without, the use of the inelastic braid assumption are consid-
ered for comparison to the dimensions of the measured geometry.
Figure 6.6 includes the estimated geometry with the inelastic braid assump-
tion (black dashed lines), and with braid elasticity effects included (gray lines) (as
shown in Chapter 5). This analysis focuses on the comparison between measured
geometries, and the estimated geometry with braid elasticity effects included, in an
effort to isolate the effects of the cylindrical approximation.
The comparison between the dimensions of the estimated and the measured
geometries displays a common trend: the error in the estimated dimensions in-
creases with increases in the contraction of the FAM. This trend stems from the
fact that the estimated dimensions, which rely on the cylindrical approximation,
become increasingly inaccurate as the actual profile of the FAM becomes increas-
205
(a) Cylindrical approximation of measured di- (b) Braid angle estimations, ?
mensions (Red values measured directly)
(c) Braid thread length estimations, B (d) Braid diameter, D
Figure 6.6: Comparison of dimensions obtained from the cylindrical approxima-
tion of the measured geometry (colored lines), estimated cylindrical geometry with
elastic braid (gray lines), and estimated cylindrical geometry with inelastic braid
assumption (black dashed lines)
ingly noncylindrical with contraction. Error in the estimated dimensions peaks at
the free-contraction state, where the diameter is overestimated by 0.11 in (4.7%),
and braid angle is underestimated by 1.25 deg. (3.3%).
The difference in the value of B between the estimated and the measured
geometries is interesting because it depicts the effect of the FAM?s curvature on
their resulting dimensions (Fig. 6.6(c)). Best is an accurate value of the braid
thread length that is then wrapped around the surface of the inaccurate cylindrical
206
geometry that is assumed. This, in turn, results in inaccurate estimates of Dest and
?est. In contrast, the measured geometry Dmeas, represents an accurate depiction of
the geometric profile of the FAM. Simplification to a cylindrical geometry, by going
from Dmeas to D?meas, eliminates the curvature of the geometry resulting in a shorter
than actual value of B. Therefore, the difference between Best and Bmeas quantifies
the length of each thread that is ?lost? to the curvature along the longitudinal
surface of the braid. For the tested FAM, the reduction from Best to Bmeas is 0.29 in
(3%).
6.3.2 Comparison of Bladder Dimensions
Improvements to actuation force models have often focused on better char-
acterizing the energy lost to the radial expansion and strain of the bladder. An
accurate depiction of the bladder?s geometry, therefore, is important for yielding
an accurate depiction of the bladder?s strain state; this is, in turn, important for
producing estimates of the bladder?s stress state, and the respective forces applied
to the bladder. The error inherent in the estimated cylindrical geometry of a FAM
results in uncertainty of bladder strain estimations. It is hypothesized here that
improvements in accuracy of strain estimations, by accounting for the curvature of
the FAM, would act to improve the accuracy of actuation force models. Stress,
and respective strain values, are defined in the FAM?s longitudinal (?z, z), and
circumferential (?c, c) directions, as illustrated in Figure 6.7.
Relevant to the research conducted in this chapter is the form of the intrinsic
207
(a) Longitudinal stress (b) Circumferential stress
Figure 6.7: Bladder longitudinal and circumferential stress directions
stress-strain relationship used to define the bladder?s material properties. Hypere-
lastic material models are commonly either strain energy based ? such as Mooney-
Rivlin and Ogden methods [7, 13, 17, 21]? or based on a polynomial stress-strain
relationship [18,86,87]. The polynomial stress-strain relationship (Eqs. 7.6 and 7.5)
is the most intuitive stress-strain relationship, and when used in actuation force
models, has provided the most accurate fits of experimental data. The polynomial
stress-strain relationship is defined as:
?n
?c = Ennc (6.5)
?1n
?z = E nnz (6.6)
1
which requires inputs values for strain, z and c, and the terms En which define the
stiffness of the bladder material. Prior works have defined the polynomial stress-
strain relationship using up to four terms (4th-order polynomial) [18,48]. Therefore,
in this work, we investigate strains up to the fourth power.
In this section, the resultant error in the strain values calculated using the
estimated cylindrical geometries will be quantified through comparison with the
more accurate strain values calculated using the measured geometries. Strain will
be calculated by using the estimated diameter Dest, and compared with the strain
208
calculated using the measured diameter Dmeas(z). It should be noted that, because
the braid is usually very thin ? 0.011 inch wall-thickness for the tested FAM ? the
outer diameter of the bladder is commonly considered to be equivalent to Dmeas(z)
as a simplification. The profile, Dmeas(z), is defined using the Fourier series fit
equations (Eq. 6.4) as described earlier.
6.3.2.1 Comparison of Bladder Thickness
Recent improvements to the calculation of bladder strain take into account
the change in bladder thickness with its contraction [13]. Since the thickness of
the bladder varies with the amount that it is strained, an estimation of thickness is
dependent on the assumed geometry of the FAM. In this section, we investigate the
effect of using the measured geometry ? instead of the previously used estimated
cylindrical geometry ? on the resulting estimation of bladder thickness.
The instantaneous thickness of the bladder tbld, is calculated for all states of
contraction. The tested FAM has a resting bladder thickness of 1/16 inch. As a sim-
plifying assumption, the bladder is assumed to have a uniform thickness throughout
the bladder in all states of strain. Bladder thickness is calculated for the estimated
cylindrical geometry, and for the measured profile cases, using the following equa-
tions:
?
Vbld
t = R ? R2bld est est ? (Cylindrical Profile) (6.7)?L
tbld = ? ?Vbld (Measured Profile) (6.8)L
0 2?Rmeas(z) 1 + [R?meas(z)]2 dz
where R and tbld are the instantaneous radius, and thickness, respectively, of the
209
bladder at FAM length states L. Equation 6.7 is used for the cylindrical geometry
assumption, and assumes the bladder takes on the shape of a constant diameter
tube in all length states. This is the standard equation that has been used for the
calculation of bladder thickness in prior research. Equation 6.8, on the other hand,
uses the arc-length of the measured geometry?s profile to obtain a higher fidelity
bladder thickness value. Both equations rely on the assumption that the bladder
volume Vbld, is fixed; this assumption of incompressibility is commonly used for
elastomeric materials. Bladder volume is easily defined in the resting length state
using the following equation:
V = L ?(R2 2bld 0 bld,0 ? (Rbld,0 ? tbld,0) ) (6.9)
where Rbld,0 and tbld,0 are the initial outer radius and thickness of the bladder,
respectively, with the FAM in its resting length state L0.
Figure 6.8 shows the comparison of the thickness calculated using a cylindri-
cal or a measured geometry. The estimated cylindrical bladder geometry, as used in
prior research, produces a minimum bladder thickness value of 0.0339 in (inelastic
braid), and 0.0337 in (elastic braid). This is comparable to the measured geome-
try?s minimum thickness of 0.0334 in, which is 53% of the bladder?s 0.0625 in resting
thickness. These results indicate that the measured bladder thickness in contracted
length states is barely thinner than the thickness value obtained by use of the esti-
mated cylindrical geometry ? with a calculated decrease in the minimum thickness
of 0.0005 in (1.5%).
210
0.055
Estimated Geom.
Estimated Geom. (El. Braid)
0.05 Measured Geom. (Avg. Dia.)
Measured Geom.
0.045
0.04
0.035
0.03
0 0.1 0.2 0.3 0.4
Contraction, L /L
0
Figure 6.8: Bladder thickness calculated for each geometry (100 psi)
6.3.2.2 Comparison of Longitudinal Strain
The longitudinal strain of the bladder is calculated based on the change in its
length in the direction of the longitudinal axis of the FAM. Longitudinal strain is
calculated using the following equation:
( )n
n = Lbldz ? 1 (6.10)Lbld,0
where Lbld,0 is the resting length of the bladder, equal to that of the FAM?s resting
length, and n is the order of strain used in the polynomial form of the stress-strain
relationship. The instantaneous length of the bladder Lbld, is defined depending on
whether a cylindrical geometry or a measured geometry is assumed for the calcula-
tion of longitudinal stress, as follows:
Lbld = L? (Cylindrical Profile) (6.11)L?
Lbld = 1 + [R?meas,bld(z)]2 dz (Measured Profile) (6.12)
0
211
Bladder Thickness, in
where the difference between the two approaches of defining Lbld is illustrated in
Figure 6.9. For a cylindrical profile (Fig. 6.9(a)), the longitudinal length of the
bladder is assumed to be equivalent to the length of the FAM (Eq. 6.11). However,
the length of the bladder in the longitudinal direction is more accurately captured
by including the curvature of the actual measured profile of the FAM (Fig. 6.9(b)).
To include the bladder?s curvature in the longitudinal length, the arc length of the
profile equation for the bladder Rmeas,bld(z) is calculated (Eq. 6.12).
(a) Cylindrical Geometry (b) Measured Geometry
Figure 6.9: Geometry-dependent longitudinal length assumed for longitudinal blad-
der strain calculations
Analysis of the results show that the cylindrical geometry assumption pro-
duces a maximum strain value at free-contraction of about 35.3%, compared to the
measured geometry maximum strain value of 28.9%. Therefore, the strain calcula-
tion method used in previous works would overestimate the longitudinal strain of
this FAM by 0.064 (a 22.2% decrease in strain). Figure 6.10 provides a depiction
of the results comparing the two geometries for strain orders 1?4. It is clear that
the maximum error in the cylindrical geometry occurs in the free-contraction state.
Table 6.1 provides the maximum values for strain orders 1?4, calculated for each
geometry with the respective error of the cylindrical geometry assumption shown as
212
percentages. While the magnitude of that error decreases with increased order of
strains, the percentage error increases from 22% up to 129%.
0 0
-0.1 -0.1
-0.2 -0.2
Estimated Geom. Estimated Geom.
-0.3 Estimated Geom. (El. Braid) -0.3 Estimated Geom. (El. Braid)
Measured Geom. (Avg. Dia.) Measured Geom. (Avg. Dia.)
Measured Geom. Measured Geom.
-0.4 -0.4
0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4
Contraction, L /L Contraction, L /L
0 0
(a) First-order longitudinal strain (b) Second-order longitudinal strain
0 0
-0.1 -0.1
-0.2 -0.2
Estimated Geom. Estimated Geom.
-0.3 Estimated Geom. (El. Braid) -0.3 Estimated Geom. (El. Braid)
Measured Geom. (Avg. Dia.) Measured Geom. (Avg. Dia.)
Measured Geom. Measured Geom.
-0.4 -0.4
0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4
Contraction, L /L Contraction, L /L
0 0
(c) Third-order longitudinal strain (d) Fourth-order longitudinal strain
Figure 6.10: Longitudinal bladder strain calculated for each geometry (100 psi)
Table 6.1: Longitudinal Bladder Strain, nz (Free-contraction, 100 psi)
Strain Order Measured Cylindrical
(nz ) Geometry Geometries
in/in in/in, (error)
n = 1 0.289 0.353 (22.2%)
n = 2 0.084 0.125 (48.8%)
n = 3 0.024 0.044 (83.3%)
n = 4 0.007 0.016 (128.5%)
213
3
Third-order strain, First-order strain, 
z z
4
Fourth-order strain, 2
z Second-order strain, 
z
6.3.2.3 Comparison of Circumferential Strain
The circumferential strain of the bladder is a measure of the change in its
diameter along its length. To have a single value of strain for use in modeling
equations, the average of the circumferential strain for the entire length of the FAM
is of interest. The average circumferential strain equation can be written as:
? ( )L
n 1 Rbld ?
t nbld
 = 2c t ? 1 dz (6.13)L 0 R ? bld,0bld,0 2
where Rbld,0 is the initial resting diameter of the bladder (equal to 7/8 in for the
tested FAM), and tbld,0 is the initial resting thickness of the bladder (equal to 1/16 in
for the tested FAM). The instantaneous radius of the bladder Rbld, is defined de-
pending on whether a cylindrical geometry or a measured geometry is assumed for
the calculation of circumferential stress, as follows:
Rbld = Rest (Cylindrical Profile) (6.14)
Rbld = Rmeas,bld(z) (Actual Profile) (6.15)
where the difference between the two approaches of defining Rbld is illustrated in
Figure 6.11. Previous efforts have assumed that the diameter of the bladder is equal
to the estimated dimensions, Dest (Eq. 6.14). But, calculating the circumferential
strain using a cylindrically assumed profile neglects the variation of strain along the
length of the FAM with variation in diameter.
The strain can instead be calculated using an accurate representation of the
geometry Dmeas,bld (Eq. 6.15). If the bladder material were a linear-elastic ma-
terial, the distinction between using a measured profile, versus using a cylindrical
214
(a) Cylindrical Geometry (b) Measured Geometry
Figure 6.11: Geometry-dependent radial profile assumed for circumferential bladder
strain calculations
approximation by calculating the average of the measured profile, would be irrel-
evant when calculating the average strain. However, that distinction is important
with the nonlinear stiffness properties of the bladder used for most FAMs.
A comparison of the circumferential strain calculated using the estimated cylin-
drical geometry (elastic and inelastic braid), the measured geometry Dmeas,bld(z),
and the average of the measured geometry D?meas,bld is illustrated in Figure 6.12.
For first-order strains, there is very little difference in the calculated circumferential
strain values. The differences in magnitude between the tested geometries increase
as the strain orders increase. The estimated cylindrical geometry underestimates
the circumferential strain at low levels of contraction, and overestimates at high
levels of contraction. The difference in circumferential strain magnitude between
the different geometries increases as the strain order increases.
Comparison between the measured geometry, and the cylindrical geometry
defined using the average of the measured profiles, illustrates the effect of taking
the changing radius of the FAM along its length into account (Fig. 6.12) . With
exception of first-order strains, neglecting to take into account the actual measured
215
profile results in an underestimate of strain magnitude for the other orders of strain.
This underscores the importance of calculating the actual profile rather than the
cylindrical profile for use with the nonlinear bladder materials used.
10 10
Estimated Geom. Estimated Geom.
Estimated Geom. (El. Braid) Estimated Geom. (El. Braid)
8 Measured Geom. (Avg. Dia.) 8 Measured Geom. (Avg. Dia.)
Measured Geom. Measured Geom.
6 6
4 4
2 2
0 0
0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4
Contraction, L /L Contraction, L /L
0 0
(a) First-order circumferential strain (b) Second-order circumferential strain
10 10
Estimated Geom. Estimated Geom.
Estimated Geom. (El. Braid) Estimated Geom. (El. Braid)
8 Measured Geom. (Avg. Dia.) 8 Measured Geom. (Avg. Dia.)
Measured Geom. Measured Geom.
6 6
4 4
2 2
0 0
0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4
Contraction, L /L Contraction, L /L
0 0
(c) Third-order circumferential strain (d) Fourth-order circumferential strain
Figure 6.12: Circumferential bladder strain calculated for each geometry (100 psi)
Table 6.2 provides the maximum value of strain calculated for each geometry
at strain orders 1?4, and are accompanied by the respective error of each cylindrical
geometry expressed as a percentage error through comparison to the measured ge-
ometry strain values. The cylindrical assumption of both the estimated geometry,
and the measured geometry, are considered to illustrate the effect of using the actual
profile in the measurement of circumferential strain. Maximum error induced by us-
216
3
Third-order strain,  First-order strain, 
4
Fourth-order strain,  2Second-order strain,  
ing the cylindrical approximation of the measured geometry ranges from 0?17.5%,
while the estimated cylindrical geometry results in maximum errors of 6.6?8.7%
(elastic braid). It is evident that correction of the strain calculation by using the
measured profile Dmeas,bld, has a significant effect on the magnitude of values for all
strain orders, but especially for the higher orders.
Table 6.2: Circumferential Bladder Strain, nc (Free-contraction, 100 psi)
Cylindrical Bladder Geometries
Strain Order Measured Measured Estimated Estimated
(nc ) Geometry (Avg. Dia.) (El. Braid) (InEl. Braid)
in/in in/in, (error) in/in, (error) in/in, (error)
n = 1 1.65 1.65 (0.0%) 1.77 (7.3%) 1.76 (6.7%)
n = 2 2.84 2.70 (-4.9%) 3.14 (10.6%) 3.09 (8.7%)
n = 3 4.99 4.40 (-11.9%) 5.58 (11.8%) 5.43 (8.7%)
n = 4 8.86 7.31 (-17.5%) 9.89 (11.6%) 9.53 (7.6%)
6.4 Effect of Geometry Profile on Force Model Results
The geometry and respective dimensions assumed for a FAM will have an effect
on its predicted actuation force. In this section, we will compare the actuation force
predictions for the estimated and the measured geometries. The model used to
estimate actuation force is as follows:
Fact = FG + Fbld,z + Fbld,c (6.16)
217
where,
F 2 2G = P?R((2tan) ? ? 1) (6.17)
Vbld
Fbld,z = ?z ( ) (6.18)L
= ? tL
2
Fbld,c ?c 2 (6.19)?RN2
This actuation force model (Eq. 6.16) has three basic terms. FG is the Gaylord force
? which serves as the original and most basic form of the actuation force model ? and
it accounts for the actuation force produced by the internal fluid pressure acting on
the braid. The Gaylord force is comprised of variables that represent pressure P , and
geometric dimensions for radius R and braid angle ?. The second and third terms
of Equation 6.16 are bladder force terms Fbld,z and Fbld,c; these terms account for
actuation force losses that are due to the stretching of the bladder during actuation.
They are a function of the bladder stresses ?z and ?c, and dimensions R, N , t, and
L. Values of ?z and ?c are based on the geometry-dependent bladder strain ? as
estimated in Section 6.3.2 ? and the intrinsic stiffness of the bladder. Defining the
nonlinear stiffness of the bladder is nontrivial, but it is required to obtain estimations
of Fbld,z and Fbld,c.
In the following sections, the effect of the assumed geometry on the predicted
actuation force is analyzed. Analysis of FG is covered first, followed by an investiga-
tion of terms Fbld,z and Fbld,c. The combined effect of these terms is then investigated
for each geometry. This will aid in quantifying the force model uncertainty and error
that comes with assuming a cylindrical approximation and estimated dimensions of
the FAM?s geometry.
218
6.4.1 Gaylord Actuation Force Term (FG)
The estimated and measured dimension sets are input into the Gaylord force
term to obtain a comparison of their respective effects on the actuation force. Be-
cause the Gaylord force term requires dimensional inputs in cylindrical coordinates,
the measured dimensions of the FAM must be defined in cylindrical terms. The
average measured diameter, therefore, will be used to define the measured profile
for analysis of the Gaylord force predictions.
A comparison of the Gaylord force term values for each geometry is provided
in Figure 6.13. The same basic trends seen in previous geometry comparisons are
observed here; the difference in Gaylord force values between the measured and
estimated geometries increases with increases in contraction and pressure. The esti-
mated dimensions (elastic braid) result in an underestimation of the FAM?s Gaylord
force, with a mean absolute error (MAE) of 19.9 lbf, and a maximum difference of
36.5 lbf (37% difference) observed in the free-contraction state at 100 psi. This un-
derestimation of the Gaylord force also leads to an underestimation of the FAM?s
maximum contraction capabilities by an estimated 1-2% for all pressures. Table
6.3 provides a comparison of the Gaylord force term values ? for the each assumed
geometry ? for 100 psi at free-contraction. Note that the maximum difference in
force between each geometry is observed at 100 psi, and decreases with a decrease
in pressure due to the respective reduction in geometric differences at lower levels
of contraction.
219
Figure 6.13: Comparison of the Gaylord force term estimates obtained for the cylin-
drical approximation of the measured geometry (colored lines), the estimated cylin-
drical geometry with elastic braid (gray lines), and the estimated cylindrical geom-
etry with inelastic braid assumption (black dashed lines)
Table 6.3: Gaylord Force Term, FG (100 psi)
Cylindrical Bladder Geometries
Length State Measured Estimated Estimated
(100 psi) (Avg. Dia.) (El. Braid) (InEl. Braid)
lbf lbf, (error) lbf, (error)
Resting Length 1068.3 1067.9 (0.0%) 1074.3 (0.6%)
Free-Contraction 98.6 62.1 (-37.0%) 65.4 (-33.7%)
Mean Abs. Error ? 19.9 16.8
6.4.2 Bladder Actuation Force
Energy lost to stretching of the bladder during actuation of the FAM acts to
reduce the FAM?s actuation force. As previously noted, obtaining values of Fbld,c and
Fbld,z requires estimations of the stress state of the bladder (?c, ?z), and a measure
or estimation of the bladder material?s stiffness.
Characterization of the bladder?s nonlinear stiffness is nontrivial, and often
220
requires either direct testing of the material using complex testing procedures and
equipment, or estimations based on empirical analysis of the FAM?s actuation re-
sponse data. Direct characterization is difficult for multiple reasons: hyperelastic
materials can withstand large deformations, so specialized clamping techniques are
required to ensure the test sample does not slip or tear in the test fixtures [88];
material stiffness is stress mode dependent, so multiple different tests are often re-
quired for a full characterization [89,90]; and, the material?s stiffness can vary from
sample-to-sample based on its strain history and environment [37]. These challenges
have led many FAM researchers to characterize the FAM?s bladder stiffness in situ
through analysis of the FAM?s experimental actuation response. This often involves
fitting the actuation force model to experimental data through adjustment of the
bladder?s material stiffness properties [7, 13, 17, 18, 21]. The model?s accuracy with
experimental data often still exhibits appreciable error; an unrealistic lack of mate-
rial property constraints will render the defined stiffness parameters questionable.
For these reasons, a direct characterization of the Latex bladder?s stiffness ? despite
the experimental challenges ? is attempted in this research to obtain values for Fbld,c
and Fbld,z.
6.4.2.1 Bladder Stiffness Characterization
A tensile test is performed on the same latex bladder that was used in the
construction of the tested FAM. A tensile test of the bladder has been deemed
sufficient to characterize the bladder?s stiffness [88, 91]. This bladder measures 7/8
221
inches in diameter, and has a 1/16 inch wall thickness (Fig. 6.14). Test fixtures
were fabricated to hold the bladder in place; these fixtures are cylindrical and have
grooves ? similar in design to the end-fittings of the FAM ? and use hose clamps
and epoxy to fix the bladder in place during testing. The resulting bladder sample
had a test length between the fixtures of 3 3/32 inches.
Figure 6.14: Latex bladder tensile test specimen
The test specimen was tested using a material testing machine (MTS), and
was stretched by 6.5 inches (2.1 strain) at a rate of 0.05 in/sec for a total of five
consecutive stretch and relax cycles (Fig. 6.15). Mullins effect, which results in an
initial relaxation of the bladder stiffness [87,92], was observed between the first and
second cycles. This resulted in a permanent stretch of the bladder by 3/32 inches
to a total resting length of 3 3/16 inches. The response of the bladder stabilized
after two cycles. A significant degree of necking of the bladder is observed, with a
reduction in diameter to about 0.5 inches in the maximum tested strain state (Figs.
6.15(a) and 6.15(b)).
Engineering stress versus strain was calculated from the test data for com-
parison to the stress-strain relationship defined in a referenced work by J.G. Niem-
czura [87] (Fig. 6.16(a)). In that study, a tensile test was performed on a 0.48 mm
222
40
30
20
Cycle 1
Cycle 2
10 Cycle 3
Cycle 4
Cycle 5
0
0 2 4 6 8
Stretch, in
(a) Zero strain (b) Max strain (c) Test results for eccentric-concentric cycling
of bladder
Figure 6.15: Tensile testing of the latex bladder used in FAM construction
(0.019 in) thick, 5 mm (0.20 in) wide, 76.2 mm (3.0 in) long latex sample, and its
stiffness was then defined by a third-order polynomial fit of the data. A third-order
fit of the tensile test data acquired in this research resulted in an average difference
of only 5.54 psi (6.24 psi RMSE) from the referenced data, and supported assurance
of the accuracy of the acquired experimental stiffness values.
The force model requires an input of true stress, opposed to engineering stress.
True stress references the instantaneous (deformed state) cross-sectional area, in-
stead of the resting cross-sectional area. True stress is commonly used to define
hyperelastic materials because of their non-negligible change in cross-sectional area
as they achieve large deformations. A true stress calculation, therefore, requires
an estimate of the instantaneous cross-sectional area of the bladder sample during
the experimental test. The cross-sectional area of the bladder is approximated to
be uniform along the deformed length of the bladder based on visual observations
during testing; each end of the specimen rapidly decreases to a relatively constant
223
Force, lbf
width for most of the deformed length (Fig. 6.15(b)). With a fixed bladder volume
of 0.49 in3 for the incompressible material in all length states, the cross-sectional
area is calculated by dividing the bladder volume by the instantaneous length. This
results in a cross-sectional area reduction from 0.16 in2 at zero strain, to 0.05 in2 at
the maximum tested strain. The true stress-strain relationship is then calculated, as
shown in Figure 6.16(b). The approximately 200% increase in maximum stress be-
tween the engineering and true stress values emphasizes the substantial effect of the
changing cross-sectional area and bladder thickness on the calculated stress values.
600
200
500
150
400
Cycle 1
100 300 Cycle 1
Cycle 2 Cycle 2
Cycle 3 200 Cycle 3
50 Cycle 4 Cycle 4
Cycle 5 100 Cycle 5
Cited 3rd-Order Fit 3rd-Order Fit
0 0
0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5
Strain, in/in Strain, in/in
(a) Engineering Stress versus Strain (b) True Stress versus Strain
Figure 6.16: Engineering and True Stress values acquired from tensile test data
As previously noted, hyperelastic material stiffness is commonly defined using
a polynomial fit of experimental data. A third-order polynomial is often used to fit
the distinct S-shape that is common for the stress-strain relationship of hyperelastic
materials, as seen in Figure 6.16(a). A least-squares polynomial fit was performed
on cycles 3-5 of the tensile test (Fig. 6.16(b)), resulting in the following definition
224
Stress (Eng.), psi
Stress (True), psi
of the Latex?s stress-strain relationship:
? = E1+ E 22 + E33 (6.20)
E1 = 257.8 psi
E2 = ?57.9 psi
E3 = 32.7 psi
The resulting fit accurately depicts the stiffness of the Latex bladder, as shown in
Figure 6.16(b), with a mean absolute error of only 4.4 psi (5.5 psi RMSE). Subse-
quent analysis will utilize this equation to define the Latex bladder?s stiffness.
6.4.2.2 Longitudinal Bladder Force Term (Fbld,z)
The longitudinal bladder force term Fbld,z, represents the effect of the bladder?s
changing longitudinal length on the actuation force of the FAM. Fbld,z, defined in
Equation 7.2, requires geometry-dependent inputs of longitudinal bladder stress ?z,
and FAM length L. The longitudinal bladder stress is calculated using the stress-
strain relationship defined for the Latex (Eq. 6.20), and the strain values defined for
each geometric profile in Section 6.3.2.2. The resulting bladder stress and respective
force values are presented in Figures 6.17(a) and 6.17(b), and are tabulated in Tables
6.4 and 6.5, respectively.
Analysis of the longitudinal stress values shows that the bladder is in tension
at the resting length state of the FAM, before shifting to a state of compression that
monotonically increases in magnitude with FAM contraction. The initial tension
in the bladder is a result of the slight radial expansion of the bladder in the fixed
225
resting length state of the FAM. The longitudinal force term Fbld,z, is then calculated
as the product of the instantaneous stress state and cross-section area of the bladder
(normal to FAM?s longitudinal axis).
Comparison of the results for different geometric profiles shows the difference
between the cylindrical and measured geometric profiles. As with their strain values,
the cylindrical geometric profiles all result in identical values of stress and force;
differences in the free-contraction values between cylindrical geometries are a result
of slight differences in their other force terms.
Longitudinal force term values are tabulated in Table 6.5 for each geometry
(100 psi fluid pressure). The longitudinal force for the measured geometry profile
ranges in magnitude from 0.1 lbf (tension) in the resting length state, to -17.9 lbf
(compression) in the free-contraction state. Error of the cylindrical geometry profiles
is minimal in the resting length state (0.1 lbf), and has a much larger impact in the
free-contraction state where the longitudinal force is overestimated by up to 2.2 - 3.7
lbf (12.2 - 20.4%). The mean absolute error of the force calculated for the cylindrical
geometry profiles (at 100 psi) is relatively low, at 1.4 lbf.
Table 6.4: Longitudinal Bladder Stress, ?z (100 psi)
Cylindrical Bladder Geometries
Length State Measured Measured Estimated Estimated
(100 psi) Geometry (Avg. Dia.) (El. Braid) InEl. Braid
psi psi, (error) psi, (error) psi, (error)
Resting Length 0.6 0.0 (100%) 0.0 (100%) 0.0 (100%)
Free-Contraction -71.6 -86.7 (-21.0%) -82.8 (-15.6%) -83.3 (-16.3%)
226
20 5
Estimated Geom. Estimated Geom.
0 Estimated Geom. (El. Braid) 0 Estimated Geom. (El. Braid)
Measured Geom. (Avg. Dia.) Measured Geom. (Avg. Dia.)
Measured Geom. Measured Geom.
-20 -5
-40 -10
-60 -15
-80 -20
-100 -25
0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4
Contraction, L /L  Contraction, L /L  
0 0
(a) Longitudinal bladder stress, ?z (b) Longitudinal bladder force term, Fbld,z
Figure 6.17: Longitudinal bladder stress, and longitudinal force term, calculated for
each geometry (all tested pressures)
Table 6.5: Longitudinal Bladder Force Term, Fbld,z (100 psi)
Cylindrical Bladder Geometries
Length State Measured Measured Estimated Estimated
(100 psi) Geometry (Avg. Dia.) (El. Braid) (InEl. Braid)
lbf lbf, (error) lbf, (error) lbf, (error)
Resting Length 0.1 0.0 (100%) 0.0 (100%) 0.0 (100%)
Free-Contraction -17.9 -21.6 (20.4%) -20.1 (12.2%) -20.3 (13.3%)
Mean Abs. Error ? 1.4 1.4 1.4
6.4.2.3 Circumferential Bladder Force Term (Fbld,c)
The circumferential bladder force term Fbld,c, represents the effect of the blad-
der?s radial expansion, and respective stretch, on the actuation force of the FAM.
Fbld,c, defined in Equation 6.19, requires geometry-dependent inputs of circumfer-
ential bladder stress ?c, and the instantaneous bladder dimensions of thickness t,
radius R, and length L. The circumferential bladder stress is calculated using the
stress-strain relationship defined for the Latex (Eq. 6.20), with the strain values
227
Longitudinal Stress, psi
Long. Force Term, lbf
defined for each geometric profile in Section 6.3.2.3.
The resulting stress and force values are shown in Figures 6.18(a) and 6.18(b),
and tabulated in Tables 6.6 and 6.7, respectively (100 psi fluid pressure). The cir-
cumferential stress monotonically increases in magnitude from a non-zero minimum
value in the resting length state; stress is non-zero in the resting length state due
to the initial unconstrained radial expansion of the bladder prior to it coming into
contact with the braid. In comparison to the stress curves, the circumferential force
term values increase in magnitude only up until about 8% contraction of the FAM.
The subsequent decrease in force at higher FAM contractions - despite the increase
in respective stress values - is primarily due to the effect of the bladder?s reducing
cross-sectional area that comes with increases in FAM contraction.
Comparison of the circumferential force term values for different geometric
profiles shows a distinct difference between the measured geometry profile, and the
cylindrical geometry profiles. The measured geometry profile predicts that the cir-
cumferential strain acts to reduce the FAM?s actuation force by up to 85.8 lbf. The
cylindrical geometries - using either measured or estimated dimensions ? tend to
over-predict the circumferential stress and force values. The estimated geometry
used in past research overestimated the circumferential force term by as much as
10.2 lbf (19.0% error) in the resting length state, while using a cylindrical approxi-
mation of the bladder?s measured dimensions resulted in a slightly lower magnitude
of maximum error of 3.4 lbf (5.5%) in the free-contraction length state. Overall,
the use of any cylindrical geometry in place of the measured geometry resulted in a
mean absolute error of about 3 lbf.
228
Note that in comparison to the values of the measured geometry, the cylindrical
approximation of the measured geometry results in a general underestimation of
stress, but an overestimation in force; this occurs despite the fact that the two
geometries share the same values for R, L, and N for the calculation of force, and
only differ in their value for t. This emphasizes the relatively large effect that the
value of t has on the predicted circumferential force term.
500 -40
Estimated Geom.
-50 Estimated Geom. (El. Braid)
400 Measured Geom. (Avg. Dia.)
Measured Geom.
-60
300
-70
200
-80
Estimated Geom.
100 Estimated Geom. (El. Braid)
Measured Geom. (Avg. Dia.) -90
Measured Geom.
0 -100
0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4
Contraction, L /L  Contraction, L /L  
0 0
(a) Circumferential bladder stress, ?c (b) Circumferential bladder force term, Fbld,c
Figure 6.18: Circumferential bladder stress, and circumferential bladder force term,
calculated for each geometry (all tested pressures)
Table 6.6: Circumferential Bladder Stress, ?c (100 psi)
Cylindrical Bladder Geometries
Length State Measured Measured Estimated Estimated
(100 psi) Geometry (Avg. Dia.) (El. Braid) (InEl. Braid)
psi psi, (error) psi, (error) psi, (error)
Resting Length 49.2 48.2 (-2.0%) 48.7 (-0.9%) 34.5 (-29.9%)
Free-Contraction 431.9 417.4 (-3.4%) 446.9 (3.5%) 443.8 (2.8%)
229
Circumferential Stress, psi
Circum. Force Term, lbf
Table 6.7: Circumferential Bladder Force Term, Fbld,c (100 psi)
Cylindrical Bladder Geometries
Length State Measured Measured Estimated Estimated
(100 psi) Geometry (Avg. Dia.) (El. Braid) (InEl. Braid)
lbf lbf, (error) lbf, (error) lbf, (error)
Resting Length -53.9 -55.7 (-3.3%) -56.1 (-4.0%) -43.7 (19.0%)
Free-Contraction -62.0 -65.4 (-5.5%) -67.6 (-9.0%) -67.3 (-8.5%)
Maximum Value -85.8 -88.6 (-3.2%) -89.3 (-4.1%) -88.8 (-3.5%)
Mean Abs. Error ? 2.6 3.1 2.7
6.4.2.4 Combined Effect of Bladder Forces
The combined values of Fbld,z and Fbld,c represent the overall actuation force
losses due to stretch of the bladder during actuation of the FAM. As seen in Figure
6.19, the measured geometry predicts that bladder forces reduce the actuation force
at 100 psi by about 53.8 lbf in the resting length state, by a maximum of 89.5 lbf at
10.4% contraction, and by 80.0 lbf in the free-contraction state. The overall trend
reflects a substantial increase in bladder force effects from 0% to 10% contraction,
and then a relatively consistent effect on actuation force from 10% contraction to
free-contraction. Figure 6.19 includes the results for pressures of 20-100 psi with
each geometry; the relative width of the curves indicates that the variation in force
values is slightly noticeable in the resting length state, and decreases with contrac-
tion. Comparing the circumferential and longitudinal bladder force terms shows the
dominance of the circumferential term. The maximum circumferential force term
value is 85.8 lbf, which is 4.8 times the maximum magnitude of the longitudinal force
term value of 17.9 lbf.
230
Table 6.8 provides a comparison between the measured geometry profile, and
the cylindrical approximation profiles. The cylindrical bladder geometries overpre-
dict the combined bladder force effects, and are all nearly identical in magnitude
with a mean absolute error of about 4 lbf. The estimated geometry with the elas-
tic braid assumption generally has the largest error in comparison to the measured
geometry profile. However, the inclusion of braid elasticity does reduce error in the
resting length state from 18.9% down to 4.3% (12.4 lbf difference). The cylindri-
cal approximations of the measured geometry and the estimated geometry (elastic
braid) never differ by more than 1 lbf for all contraction states. Cylindrical blad-
der geometries differ from the measured geometry profile by 1.9 lbf (3.5%) to 7.7 lbf
(9.7%) for all states of contraction.
-40
Estimated Geom.
-50 Estimated Geom. (El. Braid)
Measured Geom. (Avg. Dia.)
Measured Geom.
-60
-70
-80
-90
-100
0 0.1 0.2 0.3 0.4
Contraction, L /L  
0
Figure 6.19: Combined bladder force calculated for each geometry (all test pressures)
6.4.3 Overall Effect of Geometry on Estimated Force
Values for the Gaylord and bladder force terms obtained in the preceding sec-
tions are combined here to form the overall FAM actuation force model predictions
231
Combined Bladder Force, lbf
Table 6.8: Combined Bladder Force, Fbld (100 psi)
Cylindrical Bladder Geometries
Length State Measured Measured Estimated Estimated
(100 psi) Geometry (Avg. Dia.) (El. Braid) (InEl. Braid)
lbf lbf, (error) lbf, (error) lbf, (error)
Resting Length -53.8 -55.7 (-3.5%) -56.1 (-4.3%) -43.7 (18.9%)
Free-Contraction -80.0 -87.0 (-8.8%) -87.7 (-9.7%) -87.6 (-9.6%)
Maximum Value -89.5 -92.7 (-3.5%) -93.2 (-4.2%) -93.0 (-3.8%)
Mean Abs. Error ? 4.0 4.5 4.1
(Eq. 6.16). Figure 6.20 compares the investigated geometries. Figure 6.20(a) pro-
vides a comparison between the Gaylord force term, and combined bladder force
terms, for all pressures and geometric profiles. As previously noted, the bulk of the
FAM?s actuation force is accounted for with the Gaylord force term. The effects of
differences in geometric profiles, therefore, are much more noticeable in the Gaylord
force term than they are with the bladder force terms. The noticeable differences
in force between geometries in Figure 6.19 appear to to be almost negligible when
viewed in Figure 6.20(a).
Figure 6.20(b) shows the actuation force model results (Gaylord and blad-
der force terms combined). Table 6.9 provides mean absolute error, resting length
(blocked force) and free-contraction values of the resulting actuation force curves
for the 100 psi case. The measured geometry profile produces a blocked force that
is only 2 lbf greater (0.3%) than the cylindrical approximation of the measured ge-
ometry, and the estimated geometry (elastic braid). This result is not surprising
considering the FAM has a relatively cylindrical shape and minimal bladder strain
232
in the resting length state. The estimated geometry (inelastic braid) has the largest
magnitude of error in the resting length state out of all the geometries (17 lbf, 1.7%
error). Assuming an inelastic braid neglects the effect of braid stretch on the radial
expansion, braid angle, and the respective effects on actuation force.
Larger differences in the resulting values between geometries are seen in their
predicted maximum contraction values. The measured geometry predicts the largest
free-contraction value of 36.2% contraction; the cylindrical approximation of the
measured geometry ? which only differs in its calculated bladder force values ? pre-
dicts a free-contraction that is only 0.3% less. The estimated geometry profiles pre-
dict free-contraction values that are substantially lower (close to 2% lower) than the
measured geometry. Overall, we can see that the differences between the measured
geometry and the cylindrical geometries increase with contraction as the cylindrical
approximation of the FAM?s geometry becomes increasingly non-representative of
the FAM?s actual geometry. In comparison to the results from the measured geome-
try (at 100 psi), the cylindrical assumption of the measured geometry has the lowest
mean absolute error (4.0 lbf), while the estimated geometry (elastic braid) has the
highest mean absolute error (23.8 lbf).
6.4.4 Comparison To Experimental Data
Model results for each geometry are compared to experimental data of the
tested FAM (Fig. 6.21). Comparison of the model results and experimental data
show that the model overpredicts the actuation force over much of the FAM?s con-
233
1200
100 psi Estimated Geom. 100 psi1000 Estimated Geom.
1000 Estimated Geom. (El. Braid) Estimated Geom. (El. Braid)
80 psi Measured Geom. (Avg. Dia.) Measured Geom. (Avg. Dia.)
800 80 psi800 Measured Geom. Measured Geom.
60 psi
600 600 60 psi
40 psi
400
400 40 psi
20 psi
200
Gaylord Force 200 20 psi
0 20 - 100 psi
Bladder Force
-200 0
0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4
Contraction, L /L  Contraction, L /L  
0 0
(a) Components of actuation force model, Gay- (b) Actuation force model results (Fact)
lord Force (FG) and bladder force terms (Fbld)
Figure 6.20: Combining individual actuation force model terms to obtain overall
model results for each geometry
Table 6.9: FAM Actuation Force (Model), Fact (100 psi)
Cylindrical Bladder Geometries
Length State Measured Measured Estimated Estimated
(100 psi) Geometry (Avg. Dia.) (El. Braid) (InEl. Braid)
lbf lbf, (error) lbf, (error) lbf, (error)
Resting Length 1015 1013 (-0.2%) 1012 (-0.3%) 1032 (1.7%)
Free-Contraction 36.2% 35.9% (-0.9%) 34.3% (-5.5%) 34.5% (-4.9%)
Mean Abs. Error ? 4.0 23.8 19.6
traction range. The model results are most accurate for the regions near the resting
and free-contraction length states. Error is greatest at around 50% contraction.
Model error is also pressure-dependent, with error increasing as working fluid pres-
sure is increased. The mean absolute error (MAE) of the force model values at each
tested pressure are compared to the experimental force values in Table 6.10. The
measured geometries have larger MAE values than the estimated geometries at high
pressures, but lower MAE values than the estimated geometries at low pressures
(< 40 psi). The RMSE values of the model are similar in magnitude to the MAE
234
Force, lbf
Force, lbf
100 psi
1000 Estimated Geom.
Estimated Geom. (El. Braid)
Measured Geom. (Avg. Dia.)
800 80 psi
Measured Geom.
Experimental Data
600 60 psi
400 40 psi
200 20 psi
0
0 0.1 0.2 0.3 0.4
Contraction, L /L  
0
(a) Actuation response comparison
1200 0.4
Estimated Geom.
1000 Estimated Geom. (El. Braid)
Measured Geom. (Avg. Dia.) 0.3
Measured Geom.
800
Experimental Data
600 0.2
Estimated Geom.
400
Estimated Geom. (El. Braid)
0.1 Measured Geom. (Avg. Dia.)
200 Measured Geom.
Experimental Data
0 0
0 20 40 60 80 100 0 20 40 60 80 100
Pressure, psi Pressure, psi
(b) Blocked force results (c) Free-contraction results
Figure 6.21: Comparison between model and experimental test results
values (as expected); for the measured geometries, the RMSE of the model is 61.7 lbf
at 100 psi; 44.3 lbf at 80 psi; 28.4 lbf at 60 psi; 13.7 lbf at 40 psi; and 4.5 lbf at 20 psi.
Analysis of the model predictions for the blocked force state is given in Figure
6.21(b), and detailed in Table 6.11. The estimated geometry resulted in the lowest
error in the blocked force state, with an average blocked force error for all pressures
of 22.8 lbf (3.9% average error). The remaining three geometries all resulted in
similar accuracies, with the measured geometry performing slightly better than the
235
Blocked Force, lbf
Force, lbf
Free Contraction, L /L
max 0
others with an average blocked force error of 31.4 lbf (5.2% average error) . Measured
geometry error ranged from 45 lbf (-5.2% error) at 80 psi, to 11 lbf (-5.9% error) at
20 psi. With increases in pressure, the geometries each showed an increase in their
magnitude of error, and a decrease in their percent error.
Analysis of the model predictions in the free-contraction state are provided
in Figure 6.21(c), and detailed in Table 6.12. Improvements in modeling accu-
racy due to improvements in geometric accuracy are most apparent in the free-
contraction state. The measured geometry profile resulted in the lowest error in the
free-contraction state, with an average absolute error of 0.8% (3.0% error). Mea-
sured geometry error ranged from 0.2% contraction (0.7% error) at 60 psi, to 2.1%
contraction (-9.0% error) at 20 psi. The estimated geometries did not perform as
well; with an inelastic braid, they had an average free-contraction error of 2.0%
(7.1% error). The error at free-contraction had the overall trend of increasing with
decreasing pressure.
The error that remains between the experimental force and model predictions
indicates that there is still room for modeling improvements. Improvements to
the model?s accuracy ? through the use of the accurate measured geometries ? is
most apparent near the free-contraction states. These improvements are mostly
attributed to the geometry?s effect on the Gaylord force term; improvements in
dimensional accuracy had only a small impact on the results of the bladder force
terms. This observation, combined with the pressure-dependent nature of the force
error, suggests it is unlikely that further improvements to the model?s bladder force
terms would yield significant improvements in the model?s accuracy.
236
Table 6.10: Mean Absolute Error (MAE) - Experimental versus Model Results
Test Tested Measured Measured Estimated Estimated
Pressure (Exp.) Geometry (Avg. Dia.) (El. Braid) (InEl. Braid)
lbf lbf lbf lbf lbf
100 psi ? 58.4 54.5 37.9 41.8
80 psi ? 41.7 37.7 27.0 29.1
60 psi ? 26.0 22.6 16.9 17.8
40 psi ? 12.3 10.0 8.8 9.0
20 psi ? 3.1 5.1 7.0 6.7
Table 6.11: Blocked Force - Experimental versus Model Results
Test Tested Measured Measured Estimated Estimated
Pressure (Exp.) Geometry (Avg. Dia.) (El. Braid) (InEl. Braid)
lbf lbf, (error) lbf, (error) lbf, (error) lbf, (error)
100 psi 1058 1015 (-4.1%) 1013 (-4.2%) 1012(-4.3%) 1032 (-2.4%)
80 psi 850 805 (-5.2%) 802 (-5.5%) 802 (-5.5%) 817 (-3.7%)
60 psi 628 592 (-5.5%) 590 (-5.7%) 591 (-5.7%) 601 (-4.0%)
40 psi 403 381 (-5.1%) 379 (-5.7%) 380 (-5.5%) 386 (-3.9%)
20 psi 180 169 (-5.9%) 166 (-7.3%) 167 (-7.2%) 169 (-5.7%)
Table 6.12: Free-Contraction - Experimental versus Model Results
Test Tested Measured Measured Estimated Estimated
Pressure (Exp.) Geometry (Avg. Dia.) (El. Braid) (InEl. Braid)
% %, (error) %, (error) %, (error) %, (error)
100 psi 35.4 36.2 (2.3%) 35.9 (1.4%) 34.3 (-3.3%) 34.5 (-2.7%)
80 psi 34.8 35.3 (1.5%) 34.9 (0.4%) 33.4 (-4.0%) 33.5 (-3.6%)
60 psi 33.6 33.8 (0.7%) 33.3 (-0.8%) 31.8 (-5.2%) 31.9 (-4.9%)
40 psi 31.2 30.8 (-1.4%) 30.1 (-3.5%) 28.8 (-7.9%) 28.8 (-7.7%)
20 psi 23.7 21.6 (-9.0%) 20.7 (-12.7%) 19.8 (-16.6%) 19.8 (-16.5%)
237
These results emphasize the fact that further improvements should be made
to the Gaylord force term in the model. Analysis of the braid?s effect on actuation
force is still constrained to the assumption of a cylindrical geometry by the Gaylord
force term of the model, and the current framework of the model is unable to
take full advantage of the high-fidelity geometry measurements provided by the
photogrammetric method. Future studies, therefore, should focus on investigating
improvements to the model that do a better job of accounting for the complex
geometry of the braid.
6.5 Conclusion
Previous research on the modeling of FAMs has relied on a cylindrical ap-
proximation of the FAM?s geometry for estimations of dimensions and actuation
force. It has long been understood that dimensions estimated using the cylindrical
approximation misrepresent the actual geometry of the FAM, and have served as a
major source of error and uncertainty in the modeling of FAMs. In this research,
the results of using measured FAM dimensions were compared to the results pro-
duced with the estimated dimensions used in previous FAM modeling efforts. This
comparison was used to quantify the accuracy improvements in geometric and ac-
tuation force modeling by using an accurate representation of the FAM?s geometry.
The photogrammetric method of measuring the geometric profile of FAMs is a tool
that can provide a high-fidelity representation of the FAM?s geometric dimensions.
This research showed how the photogrammetric method can now be used to improve
238
upon previous modeling efforts by reducing modeling error previously introduced by
using the cylindrical approximation.
Initial analysis compared the dimensions of the estimated and measured ge-
ometries. The estimated dimensions are based on a cylindrical approximation of
the FAM?s geometry, however, the FAM?s geometric shape becomes increasingly
non-cylindrical with contraction. This fosters a consequent increase of the error ?
inherent in the estimated dimensions ? with increases in contraction of the FAM.
Accordingly, by neglecting the curvature of the FAM, the estimated dimensions were
found to overestimate the diameter of the FAM by up to 4.7%, and underestimate
the braid angle by up to 3.3%. These dimensions were then used to compare es-
timates of bladder strain. Use of the estimated cylindrical geometry was found to
overestimate the longitudinal strain by up to 22?129%, and overestimate circumfer-
ential strain values by up to 7?12% (maximum error varies with strain order).
The effect of the FAM?s geometry on the model?s actuation force values was
investigated piece-wise through isolation of each term of the model. For the Gaylord
force term, when compared to the measured geometries (at 100 psi), the estimated
geometries underestimated force by as much as 37%, and had a mean absolute error
of 19.9 lbf. For the bladder force terms, when compared to the measured geometries
(at 100 psi), the estimated geometries overestimated the bladder effects by 4.2% -
9.7% (elastic braid), and all cylindrical geometries resulted in a mean absolute error
of about 4 lbf ? relatively less than the magnitude of error in the Gaylord force term.
When the force terms were combined to observe their cumulative effects, variations
in force between geometries were minimal in the resting length state (3 lbf, 0.3%
239
maximum difference between tested geometries), but variations were relatively large
with the predicted maximum contractions (1.9% contraction, a 5.5% difference).
The mean absolute error for the cylindrical geometries (at 100 psi) ranged from
4.0 lbf (cylindrical approximation of measured geometries) to 23.8 lbf (estimated
geometry with elastic braid).
The results of the force model were then compared to experimental force re-
sponse data of the tested FAM, with a focus on the performance of the force model
when using the measured geometry profiles. In the blocked force state, the model
error using the measured geometries ranged from 45 lbf (-5.2% error) at 80 psi, to
11 lbf (-5.9% error) at 20 psi. Improvements in modeling accuracy by using the mea-
sured geometries was most apparent in the free-contraction state with an average
contraction error of 0.8% (3.0% error), which was much better than the average
free-contraction error of 2.0% (7.1% error) for the estimated geometries. Analysis
of the mean absolute error of the model fit to experimental data found that the
measured geometries had larger mean absolute error values than the estimated ge-
ometries at high pressures, but lower MAE values than the estimated geometries at
low pressures (< 40 psi). From this analysis, the error that remains in the model ?
even with the use of the measured geometric dimensions ? motivates the need for
future improvements to the model.
Overall, this research has quantified much of the uncertainty associated with
the use of a cylindrical approximation in acquiring dimension and actuation force
estimates of the tested FAM. The previously used estimated dimensions (acquired
through application of the cylindrical approximation) were replaced by measured
240
dimensions, and the error and uncertainty induced by the cylindrical approxima-
tion in the previous force models was largely quantified. Moreover, error due to
the cylindrical approximation was also fully quantified for the bladder force terms,
however, analysis of the Gaylord force term was limited by the term?s requirement
for dimensional inputs to be presented in cylindrical coordinates. Nonetheless, the
error in the Gaylord force was partially quantified by using a cylindrical approxima-
tion of the accurate measured dimensions in place of the estimated dimensions. The
current form of the actuation force model still has room for further improvements
in accuracy. This work presents evidence that future improvements should focus on
modification of the Gaylord force term to better represent the effects of the braid?s
complex curvature on actuation force.
241
Chapter 7: Modeling Improvements
7.1 Introduction
In the previous chapter, we found an undesirable amount of error in the actu-
ation force model despite the use of accurate inputs for the geometric and material
properties. Additional modification to the model is required to improve the fit and
predictive nature of the model. Until this happens, fit-parameters ? terms used to
adjust the model to match the FAM?s physical force response ? are needed. These
fit-parameters are typically identified through optimization techniques. While fit-
parameters are phenomenological in nature, they can be defined and constrained
in ways that reflect the FAM?s underlying physical mechanisms. The current chal-
lenge, therefore, is defining the fit-parameters in a way that reflects the physical
mechanisms of the FAM, while also ensuring the model accurately represents the
FAM?s actuation response.
Previous FAM modeling research has developed modeling approaches that em-
bed fit-parameters within the bladder?s material stiffness term of the force model
[18]. Placement of the fit-parameters within the material?s stiffness definition alle-
viates the need for a difficult-to-obtain experimental characterization of the blad-
der?s material stiffness. However, initial implementations of this form of the model
242
allowed for unrealistic bladder stiffness definitions that were free of the actual mate-
rial?s physical constraints. Subsequent research has moved towards applying realistic
constraints to these material-based fit-parameters, while still attempting to retain
model accuracy.
A model using this approach will be investigated in this research. While past
research has used this model approach with dimensional inputs that were estimated
using a cylindrical approximation of the FAM?s geometry, we will improve the fi-
delity of the model here, by instead utilizing the geometric measurements acquired
using photogrammetric methods (established in Chapters 4-6) to ensure accurate
dimensional inputs to the model. This work will then aim to improve the fidelity
and fit of the model by: (1) improving the material stiffness constraints that are
applied to candidate fit-parameter values; and, (2) identifying the necessary number
of fit-parameters to use along with their dependencies, if any, on the internal pres-
sure of the FAM (as used in previous works). As a gauge of the physical accuracy
of obtained fit-parameters, they will be compared to the analogous bladder material
stiffness coefficients obtained in Chapter 6. The 7/8 inch diameter, 9 9/32 inch
length FAM used for analysis in previous chapters is also used here. The model
is updated to make it more mechanistic in form, while also striving to provide a
model fit that is true to the experimental response of the FAM; the computational
complexity of the model will also be considered for its application in control schemes.
243
7.2 Background on Model
The force-balance form of the actuation force model used in this work was
originally derived by Ferraresi (2001) [1]. Many authors have since improved upon
that model by adding or adjusting terms to better represent the dimensions, inter-
actions, or material properties of the braid or bladder. The specific form of the
model used in this research was originally developed by Hocking (2012) [18]. The
Figure 7.1: Free-body diagram of the FAM used in derivation of the force model [1]
model takes a mechanistic approach, and is based on a free-body diagram (shown in
Figure 7.1) which accounts for pressure, bladder, and braid forces. From this, force-
balance equations for equilibrium in the FAM?s longitudinal (z) and circumferential
(c) directions are obtained:
PRL = ?ctbldL+NTbrdcos(?) (7.1)
Fact + ?R2P = ?z(Vbld/L) + Tbrdsin(?) (7.2)
These equations include: unknown values of actuation force (Fact) and braid thread
tensions (Tbrd); working fluid parameters for pressure (P ); bladder parameters for
244
volume (Vbld), bladder thickness (tbld), longitudinal stress (?z), and circumferential
stress (?c); and braid geometry-based parameters for active length (L), radius (R),
braid angle (?), and number of turns of each thread about the circumference of
the FAM (N). Through substitution of the unknown value of Tbrd, the system of
equations is used to solve for Fact. The following expression for actuation force of
the FAM ? as a function of length L ? is then defined as follows:
( ) ( 2 )
= 2(2 2 ? 1) + Vbld tbldLFact P?R tan ? ?z ? ?c
L 2?RN2 (7.3)
The model requires an estimation or measurement of the FAM?s geometric dimen-
sions in each length-state to define many of its input parameters. These dimensions
are commonly estimated using the triangle approximation [5], but can also be de-
fined based on photogrammetric measurements of the FAM?s geometry, as covered
in Chapter 4.
The model also requires an estimation of bladder stress in the longitudinal
(?z) and circumferential (?c) directions. As covered in Chapter 6, obtaining these
values requires an estimation of the bladder?s strain state (commonly based on
geometric dimensions) and its material stiffness. The bladder?s stiffness, however,
is difficult to accurately characterize for multiple reasons; testing to the high strain-
states of hyperelastic materials can require specialized clamping fixtures, testing
equipment, and multi-directional testing. Furthermore, even after a hyperelastic
bladder material is fully characterized, its stiffness can vary depending on storage
and working conditions (e.g. temperature, light or fluid exposure), strain history,
or variations in manufacturing methods [19,92].
245
Because of those challenges, the bladder?s stiffness is commonly defined using
indirect empirically-based methods instead of through direct characterization. These
methods typically involve fitting the force model to experimental force-response data
by adjusting the bladder material?s stiffness coefficients, and in effect, turning them
into fit-parameters [18,20,34]. These fit-parameters are typically identified through
optimization techniques.
Initial efforts used a single fit-parameter ? often connected to the Gaylord
force term ? to account for unmodeled phenomena, and inaccuracies in the blad-
der stiffness term [9]. The first applications to place the fit-parameter within the
bladder stiffness terms used a single fit-parameter; this simplistic form defined the
hyperelastic stiffness of the bladder as a linear-elastic material [1,13], and struggled
in matching the model to the nonlinear response of the FAM.
Hocking (2012) developed a method of defining the bladder stress input to the
model as a polynomial function of strain, as follows [18]:
?n
?c = Ennc (7.4)
1
?n
?z = Ennz (7.5)
1
where n defines the order of the polynomial ? and the corresponding number of ma-
terial stiffness coefficients En ? used to define the nonlinear stress-strain relationship
of the bladder. Values of En can be experimentally defined or, in this case, used as
fit-parameters that are defined by an optimization function. This method of model
fitting was developed to improve model accuracy by not only better representing
the bladder?s stiffness, but by also capturing unmodeled physical mechanisms of
246
the FAM. The multiple degrees-of-freedom that these fit-parameters provide aid in
ensuring a good fit of the model to the experimental force response of the FAM.
When used as fit-parameters, the stiffness coefficients (En) are design variables
that are identified by fitting the model to empirical force-response data of the FAM;
En values are identified to minimize the error between the model and experimental
results. In this work, the built-in MATLAB optimization function fmincon is used
to determine En values that minimize the error between the model output and the
average force data from each experimentally obtained response curve. The objective
function that the optimizer is tasked with minimizing is a least-squares regression
error, defined as:
???? ?[ ]2
f(x) = 1 ? 1 (Fact)p,i + (Fexp)p,i (7.6)
NP p Ni i
where Fact is the model force, Fexp is the experimental force, Np is the number of
experimentally tested pressures, and Ni is the number of data points at each tested
pressure.
The initial modeling effort by Hocking provided a very good fit of her exper-
imental data by using a fourth-order polynomial stress-strain relationship (E1?4).
Unique values of En were defined to fit the experimental data at each of the ten
tested pressures; material coefficient values were defined at each discrete pressure
without any imposed constraints, resulting in a total of forty unique En values.
This variation with respect to pressure is not representative of the material?s in-
trinsic properties, and also requires assumptions to be made to obtain En values at
experimentally untested pressures (as needed in control schemes).
247
The lack of optimization constraints imposed on values of En motivated sub-
sequent research. Both Pillsbury (2015) and Robinson (2015) used the same basic
form of the model, but with the values of En constrained to be defined as a linear
function of pressure [20, 34]; while still not representative of the material?s intrin-
sic properties, this approach does reduce the computation required to identify the
pressure-dependant En values (through interpolation) for use in control schemes.
The stress-strain relationship of materials has a true-stress state that increases with
increases in strain; this results in a curve that monotonically increases up to a point
of failure. Pillsbury (2015) was the first to introduce an additional constraint to the
model to ensure that the value(s) of En do not violate this inherent stress-strain
trend. In his work, this was implemented by requiring all optimized values of En
to be positive. This constraint makes the model more physically realistic, but is
also rather strict. The research presented in this chapter will improve upon this
constraint.
The order of the polynomial stress-strain equation required to obtain a good
model fit has also previously been investigated. The stress-strain curve of a hyper-
elastic material typically produces a S-curve which is characteristic of a third-order
polynomial (as seen in Chapter 6). To improve the degrees-of-freedom for fitting
the model to experimental data, researchers have typically used up to a fifth-order
polynomial relationship.
In this research, the model?s fit will be investigated using experimental data
of the tested 7/8 inch diameter FAM. Geometric dimensional inputs of R, N , ?, t,
for all length-states L are obtained from the previously acquired photogrammetric
248
measurements of the FAM (from Chapter 4). These dimensions account for the
actual geometry of the bladder, and use a cylindrical approximation of the braid
geometry (required for use in the model). The experimentally obtained bladder
stiffness coefficients from Chapter 6 are compared to, and replaced by, the material
fit coefficients obtained by the optimization function. The model is then investigated
with En values that are: a function of pressure; a linear function of pressure; or are
fixed with respect to pressure.
7.2.1 Baseline Modeling Results
To pinpoint areas of potential improvement, the model?s fit is first tested using
the model constraints established by Hocking (2012) and Pillsbury (2015) [18,34]. In
both works, a fourth-order stress-strain relationship was used to define the bladder
stiffness.
Hocking defined unique values of E1?4 for each experimentally tested pres-
sure. Having E1?4 values that vary arbitrarily with pressure does not represent the
intrinsic fixed stiffness properties of the bladder that are independent of pressure,
but does allow them to compensate for unmodeled effects that vary with pressure.
The values of E1?4 found using the optimization function are also not bound by
realistic stress-strain relations of hyperelastic materials; these materials should, at
the least, be defined by a monotonically increasing stress-strain relationship. For
the 7/8 inch FAM, the model fit achieved by using the model which lacks a stress-
strain constraint is shown in Figure 7.2(a). The resulting model fit is very accurate.
249
Averaged across pressures, the model has a root-mean-square error (RMSE) of only
2.0 lbf, a mean absolute error (MAE) of 2.7 lbf, blocked force error of 16.0 lbf, and
a free contraction error of only 0.003 in/in. The values of E1?4 identified by the
optimization function to produce this model fit are shown in Figure 7.2(b). With
the exception of the 10 psi case, many of the En values nominally trend as a linear
function of pressure; this entails that a linear constraint of En with pressure would
result in a minimal penalty in model accuracy. The pressure-dependent stress-strain
relationship curves produced by these E1?4 values, shown in Figure 7.2(c), are tested
up to the maximum estimated circumferential bladder strain (c) at each respective
pressure. Circumferential strain of the bladder is much larger than the maximum
magnitude of longitudinal strain, and therefore is the focus of the stress-strain anal-
ysis. Many of these curves violate the positive-monotonic stress-strain relationship
requirement by having points that have a negative slope. Constraints should be
applied to avoid obtaining En values that create this condition.
Pillsbury improved the model by defining values of En as a linear function
of pressure. This constraint aids in interpolating stiffness coefficients for untested
pressures. While this results in a model that is slightly more constrained, it still
enables the values of En to compensate for unmodeled effects of pressure. In an effort
to ensure a positive-monotonic stress-strain relationship, the model was constrained
to allow only positive values of En. The resulting model fit ? which uses a fourth-
order polynomial fit ? is shown in Figure 7.3(a). The resulting model fits fairly well,
but exhibits appreciable pressure-dependent error in the resting, mid-contraction,
and free contraction length states. Averaged across pressures, the model has a RMSE
250
100 psi
1000 Experimental Data
Model
800 80 psi
600 60 psi
400 40 psi
200 20 psi
0
0 0.1 0.2 0.3 0.4
Contraction, L /L  
0
(a) Comparison between model and experimental data
(shown at pressure increments of 20 psi for clarity)
1000 1000
800
500
100 psi
600 80 psi
0 60 psi
E 400 40 psi1
E
2 20 psi
-500
E
3 200
E
4
-1000 0
0 20 40 60 80 100 120 0 0.5 1 1.5 2
Pressure, psi Strain, in/in
(b) Optimized E1?4 values used in the model (c) Stress-strain curves using E1?4 values
Figure 7.2: Model results (7/8 inch FAM) using the approach of Hocking (2012) to
define En values. E1?4 values are defined with respect to pressure, and without any
constraints on their resulting stress-strain relationships.
of 10.3 lbf, MAE of 8.3 lbf, blocked force error of 40.9 lbf, and a free-contraction error
of 0.013 in/in. Error increases with increases in pressure in the blocked force state by
about 0.8 lbf/psi. The E1?4 values used to obtain the model fits are shown in Figure
7.3(b). It is evident that constraining the optimization to allow for only positive
values of E1?4 severely limited the domain of possible fit values; only the E1 term
was used, while E2?4 were effectively inactive. This results in a linear-elastic model
251
E , psi
n
Force, lbf
Stress, psi
that varies in stiffness with pressure. The constraint on the bladder?s stress-strain
relationship needs adjustment (as covered in this chapter) to improve the model fit
by enabling higher-order terms to be active.
100 psi
1000 Experimental Data
Model
800 80 psi
600 60 psi
400 40 psi
200 20 psi
0
0 0.1 0.2 0.3 0.4
Contraction, L /L  
0
(a) Comparison between model and experimental data
(shown at pressure increments of 20 psi for clarity)
1000 1000
800
500
100 psi
600 80 psi
0 60 psi
E
1 400
40 psi
E
2 20 psi
-500
E
3 200
E
4
-1000 0
0 20 40 60 80 100 120 0 0.5 1 1.5 2
Pressure, psi Strain, in/in
(b) Optimized E1?4 values used in the model (c) Stress-strain curves using E1?4 values
Figure 7.3: Model results (7/8 inch FAM) using the approach of Pillsbury (2015) to
define En values. E1?4 values are constrained to be positive, and as a linear function
of pressure.
252
E , psi
n
Force, lbf
Stress, psi
7.3 Model Improvements
In this section, improvements to the model are made by adjusting the opti-
mization constraints on candidate En values. Constraints are imposed on En values
to ensure that they produce a stress-strain relationship that obeys the constitutive
laws of hyperelastic materials. A feasible nonlinear stress-strain relationship should
ensure material stability [93]; this means that, in defining a stress-strain relation-
ship, an applied load to the material does not result in an arbitrary deformation
state. Material stability can be established by: (1) requiring a stress-strain rela-
tionship that monotonically increases with strain (slope > 0); or, (2) satisfying the
Drucker Stability criterion.
The fit of the model without constraints will be compared to the model with
added constraints (Fig. 7.4). In this section, the optimization function fits the
model to the experimental data using a bladder material defined by four stiffness
coefficients (fourth-order polynomial), and are defined to vary linearly with pressure.
The stiffness coefficients are defined with respect to pressure, in these cases, to
enable the degrees of freedom required to obtain a model fit that is sufficiently
accurate for comparison of the constraints. Without any constraints imposed on
the allowable stress-strain relationship, the model (labeled ?Free?) provides a good
fit of the experimental data, with a low degree of average error in the blocked force
state (16.6 lbf) and the free-contraction state (0.003 in/in). The average RMSE and
MAE across tested pressures is 3.1 lbf and 2.3 lbf, respectively.
As previously described, Pillsbury (2015) ensured a positive-monotonic stress-
253
strain relationship of the bladder within the model by constraining the optimization
function to allow for only positive values of E1?4. Although that constraint is simple,
it is overly strict because it is possible to have some negative En values while still
obeying one of the stability conditions as stated above. The results of using that
constraint on model fit are provided in Figure 7.4 with the label ?Pos. E-Vals?.
Because the optimization function is overconstrained, there is a significant drop-off
in model accuracy that is especially visible in the blocked force and free-contraction
states; the average respective errors are much higher at 40.9 lbf and 0.013 in/in,
respectively. RMSE averaged across pressures is 10.3 lbf, while MAE is 8.3 lbf.
In this work, we first modify the constraint to ensure a positive-monotonic
stress-strain relationship (stability condition 1). The developed inequality constraint
calculates the differential (i.e. slope) of the polynomial stress-strain equation, and
ensures that it is greater than or equal to zero in all strain states ( ? max). This
nonlinear constraint is applied to the optimization function, and rejects sets of En
values that produce a stress-strain curve with a negative slope at any point. The
resultant model, shown in Figure 7.4 (labeled ?Pos. Slope?), improves the fit of the
model in comparison to the previous ?Pos. E-Vals.? constraint. The average blocked
force error is reduced by 14.1 lbf (down to 26.9 lbf error), average free-contraction
error is reduced down by 0.007 in/in (down to 0.006 in/in error), overall RMSE is
reduced by 5.3 lbf (down to 5.0 lbf error), and the overall MAE is reduced by 4.8 lbf
(down to 3.5 lbf error). Distinct regions of error near the blocked force and free-
contraction states still remain however.
In an attempt to further reduce the error in the blocked force and free-
254
contraction states, a third constraint approach is introduced. In this iteration, the
constraint was developed in accordance with the Drucker Stability criterion (sta-
bility condition 2). This criterion ensures stability by requiring that a material?s
incremental energy only increases with increases in strain state magnitude, for the
developed stress-strain relationship. This is defined through the following equation:
d? : d ? 0 (7.7)
,where d? is the stress increment tensor associated with the strain increment tensor
d. This condition is identical to the previous slope constraint in the uniaxial strain
case. For the case of the FAM bladder, this postulate considers the combined ef-
fects of the longitudinal and circumferential stress-strain states together. This has
the effect of slightly loosening the constraint in comparison to the previous slope
constraint; a slightly negative slope at a point is allowed where the overall internal
energy of the bladder still increases. The incremental improvement in model fit
afforded by this constraint is visible in Figure 7.4, where it is labeled as ?Drucker?.
In comparison to the slope constraint, the Drucker Stability constraint further re-
duces the average error in the blocked force state by 12.1 lbf (down to 14.8 lbf error),
reduces average free-contraction error by 0.003 in/in (down to 0.003 in/in error), re-
duces overall RMSE by 1.7 lbf (down to 3.3 lbf error), and reduces overall MAE
by 0.9 lbf (down to 2.5 lbf error). At this point, the stress-strain constraint has a
minimal effect on model fit, while still ensuring the fidelity of the model.
The effect of each stress-strain constraint on model accuracy is tabulated for
the model fit at 100 psi (Table 7.1), and as an average over all tested pressures (Table
255
7.2). The constraints have the largest effect on the model at higher strains, and
therefore, have the most prominent effect on model fit at the highest tested pressure
of 100 psi. Figure 7.5 shows the difference between the stress-strain curves using
each constraint (at 100 psi). The effect of loosening the stress-strain constraint on
the resulting curves is evident: moving from the positive En values constraint, to the
slope constraint, and then to the Drucker Stability constraint, the curvature of the
FAM gradually approaches the curvature of the model that is free from constraints.
Overall, the stress-strain constraint is important in ensuring material stability for
the developed stress-strain curves, but often does not need to be very active in the
optimization.
Table 7.1: Model Error with Respective Stress-Strain Constraints ? 100 psi (Exper-
imental Blocked Force: 1058 lbf, Experimental Free-Contraction: 0.354)
Free Pos. E-Vals. Pos. Slope Drucker
Blocked Force, lbf 17.1 (1.6%) 71.4 (6.8%) 43.3 (4.1%) 11.9 (1.1%)
Free-Contr. 0.003 (0.8%) 0.013 (3.7%) 0.010 (2.7%) 0.006 (1.7%)
MAE, lbf 3.7 15.6 7.0 4.2
RMSE, lbf 4.5 18.7 9.2 5.2
Table 7.2: Model Error with Respective Stress-Strain Constraints ? Averaged Across
All Pressures
Free Pos. E-Vals. Pos. Slope Drucker
Blocked Force, lbf 16.6 (5.0%) 40.9 (7.9%) 26.9 (5.7%) 14.8 (5.0%)
Free-Contr. 0.003 (1.5%) 0.013 (4.2%) 0.006 (2.8%) 0.003 (1.4%)
MAE, lbf 2.3 8.3 3.5 2.5
RMSE, lbf 3.1 10.3 5.0 3.3
256
100 psi
1000 Experimental Data
Free
800 Pos. E-Vals.
Pos. Slope
Drucker Stability
600 60 psi
400
200 20 psi
0
0 0.1 0.2 0.3 0.4
Contraction, L /L  
0
(a) Force Response
500 200
1000
400 100
900
300 0
0 0.02 0.04 0.06 0.14 0.16 0.18 0.2 0.3 0.32 0.34
(b) Zoomed-in views of results in subfigure (a) (100 psi test). Regions
of focus are near (left) zero-contraction/blocked force, (middle) mid-
contraction, and (right) free-contraction
(c) Blocked force results (100 psi) (d) Free-contraction results (100 psi)
(e) MAE results (100 psi) (f) MAE results (avg.)
Figure 7.4: Comparison of model results using different stress-strain constraints
257
Force, lbf
1000
Free
Pos. E-Vals.
800
Pos. Slope
100 psi
Drucker Stability
600
400
200
0
0 0.5 1 1.5 2
Strain, in/in
Figure 7.5: Stress-strain curves with the use of the respective constraints for the
model at 100 psi (curves vary with pressure)
7.4 Model Results and Analysis
The performance of the updated model is now analyzed. In comparison to
the form of the model used in prior studies [18, 20, 34], this model includes the
updated Drucker Stability stress-strain constraint and accurate geometry inputs
acquired from photogrammetric measurements. The model fit is first tested using
En values that are constrained to be fixed with respect to pressure; this is the
most realistic case, and allows for a direct comparison between the experimental
stiffness coefficient values obtained in Chapter 6 and the En values obtained using
the optimization function. Fixed En values result in an unsatisfactory model fit,
so subsequent analysis focuses on improvements to the model with En values that
vary with pressure. The performance of the model fit is analyzed while varying:
(1) the number of En terms used to define the bladder?s stress-strain relationship;
and, (2) the relationship of En with respect to pressure. The model fit is analyzed
using up to four material coefficient terms, with the terms defined as: a free function
258
Stress, psi
of pressure; a linear function of pressure; and fixed with respect to pressure. The
model is fit to experimental data acquired for the 7/8 inch diameter, 9 9/32 inch
length FAM used in the previous chapters, which was tested at pressures from 10
to 100 psi in 10 psi increments.
7.4.1 Comparison of Experimental and Model Values of En
The optimization is first set to define values of En that are fixed with respect
to pressure. These values are then compared to the experimentally obtained values
from Chapter 6. An accurate fit of the model using optimized En values, together
with correspondence to the experimentally obtained En values, would provide as-
surance of the model?s fidelity in its current form.
The optimized and experimental En values are compared using a third-order
polynomial stress-strain relationship; the respective values are provided in Table 7.3.
The two sets of En values appear to be very different. To clarify the comparison, the
stress-strain curves produced by the two sets of En are provided in Figure 7.6(b).
The curves are not as different as the En values would entail, but they do deviate
from one another at strains above 0.75.
Table 7.3: Material Coefficient Values, En (Fixed with respect to pressure)
E1 E2 E3
Experimental Values, psi 257.8 -57.9 32.7
Optimized Values, psi 131.9 340.7 -146.0
The resulting model fits are compared in Figure 7.6(a). Neither model fit
259
matches the nonlinearity of the FAM?s experimental response very well, especially
at higher pressures. Table 7.4 provides a clear comparison of the average error
in each model fit. The optimized En values provide a slightly better fit than the
experimental values. Error of each model also increases with increases in pressure;
this observation emphasizes the need for optimal En values that vary with pressure
to obtain an accurate model fit.
Table 7.4: Average Model Error Using Experimental Versus Optimized En Values
(E1?3 Fixed With Respect to Pressure)
Experimental Values Optimized Values
Blocked Force, lbf 31.1 (7.5%) 18.7 (3.9%)
Free-Contr. 0.008 (4.5%) 0.020 (10.5%)
MAE, lbf 25.4 18.1
RMSE, lbf 27.7 19.7
1000
100 psi
1000 Experimental Data Opt. E-Vals. (Model)
Opt. E-Vals. (Model) Exp. E-Vals. (Model)
800
800 80 psi Exp. E-Vals. (Model)
600
600 60 psi
400 40 psi 400
200 20 psi 200
0
0 0.1 0.2 0.3 0.4 0
0 0.5 1 1.5 2
Contraction, L /L  
0 Strain, in/in
(a) Model comparison to experimental data (b) Comparison of stress-strain curves
Figure 7.6: Comparison between E1?3 values obtained through experimental char-
acterization, and those found through optimization. Values of E1?3 are independent
of pressure.
260
Force, lbf
Stress, psi
7.4.2 Parametric Analysis for Optimal Model Fit
Prior studies have adjusted the bladder?s polynomial stress-strain relationship
to improve the fidelity and computational ease of the model; both the order of
the stress-strain relationship ? defined by the number of En terms used ? and the
relationship of the En terms with respect to pressure, have been adjusted. Using
the updated form of the model developed in this research, different permutations of
the material coefficient definitions are analyzed for their effect on the accuracy of
the model.
The model is tested using the following polynomial stress-strain relation-
ship definitions: First-order (i.e. linear) (E1), second-order (E1, E2), third-order
(E1, E2, E3), and fourth-order (E1, E2, E3, E4). Each material coefficient value is
defined by the optimization function to be: (1) a function of pressure (f(P )); (2) a
linear function of pressure (f(P )(linear)); or, (3) a fixed value that is independent
of pressure (Fixed).
Figure 7.7 provides a comparison between the experimental data, and each
permutation of the model; the average error over all tested pressures is provided.
These results are accompanied by figures that illustrate the change in model fit by
varying the number of En terms used (Fig. 7.8), and the effect of changing the
relationship of En with respect to pressure (Fig. 7.9).
261
50
 f(P) 0.03  f(P)
 f(P) (linear)  f(P) (linear)
40
Fixed 0.025 Fixed
Fixed (Meas. E-vals) Fixed (Meas. E-vals)
30 0.02
0.015
20
0.01
10
0.005
0 0
E E E E E E E E
1 1-2 1-3 1-4 1 1-2 1-3 1-4
Model Terms, E Model Terms, E
n n
(a) Average force error in blocked force state (b) Average error in free-contraction states
50 50
 f(P)  f(P)
 f(P) (linear)  f(P) (linear)
40 40
Fixed Fixed
Fixed (Meas. E-vals) Fixed (Meas. E-vals)
30 30
20 20
10 10
0 0
E E E E E E E E
1 1-2 1-3 1-4 1 1-2 1-3 1-4
Model Terms, E Model Terms, E
n n
(c) Mean absolute error (MAE) over entire (d) Root-mean-square error (RMSE) over
contraction range entire contraction range
Figure 7.7: Model fit error ? averaged over all tested pressures ? for all tested
permutations of defining the material coefficients, En
7.4.2.1 Effect of Number of En Terms Used
Increasing the number of En terms generally acts to improve model fit by
increasing the degrees of freedom that the optimization function has to fit the model
to experimental data. The results in Figure 7.7 show the effects of varying the
number of En terms. The resultant model fit accuracy is dependent on the freedom
of En to vary with respect to pressure. Here, increasing the number of En terms
while also fixing their values with respect to pressure results in a minimal and
262
Model MAE, lbf Blocked Force Error, lbf
Model RMSE, lbf Free Contr. Error
inconsistent effect on model accuracy. However, when the En terms are allowed to
vary with respect to pressure, there is a clear improvement in model fit by increasing
the number of En terms; a distinct jump in accuracy is observed when a third-order
or higher (n > 2) stress-strain relationship is used.
Figure 7.8 illustrates the effect that the number of En terms has on model
accuracy. For this comparison, the values of En are defined as a linear function
of pressure. The results show a visible step increase in accuracy when n > 2, as
observed in Figure 7.7; between E1?2 and E1?3, there is a 64% decrease in MAE, 49%
decrease in blocked force error, and a 53% decrease in free-contraction error. Figure
7.8(c) shows a comparison of the stress-strain curves with variations in En terms.
Note that multiple pressure-dependent stress-strain curves are produced by defining
the En terms as a function of pressure. A distinct change in strain-strain curvature
is observed when n > 2 which coincides with the respective improvement in model
accuracy. Also of interest in Figure 7.8(c) is the magnitude that the stress increases
with respect to pressure (as allowed by En varying with pressure). This provides
an indication of how much the model?s bladder stiffness term ? and respective En
terms that vary with pressure ? are compensating for unmodeled effects of pressure
in the FAM?s force response. The value of E1 increases by about 2 psi per each psi
increase of internal pressure.
263
100 psi
1000 Experimental Data
E
1
800 E
1-2
E
600 60 psi
1-3
E
1-4
400
200 20 psi
0
0 0.1 0.2 0.3 0.4
Contraction, L /L  
0
(a) Comparison of model fits and experimental data
500 200
1000
400 100
900
300 0
0 0.02 0.04 0.06 0.14 0.16 0.18 0.2 0.3 0.32 0.34
(b) Zoomed-in views of results in subfigure (a) (100 psi test). Regions of focus are near
(left) zero-contraction/blocked force, (middle) mid-contraction, and (right) free-contraction
1000
E
1
800 E1-2
E 100 psi
1-3
600 E
1-4
60 psi
400
20 psi
200
0
0 0.5 1 1.5 2
Strain, in/in
(c) Stress-strain curves produced by optimized En
values
Figure 7.8: Comparison of model fits with variation in the number of En terms used
(with respect to pressure)
264
Stress, psi Force, lbf
7.4.2.2 Effect of Variation of En with Pressure
As noted earlier, the physical bladder material should not have stiffness prop-
erties ? and respective En values ? that vary with respect to pressure. However,
unmodeled effects of pressure within the model require the alleviation of this ma-
terial constraint to obtain an accurate model fit. Allowing the En terms to vary
with pressure provides a marked improvement in model accuracy; alleviating this
constraint reduces the MAE of the model by up to 86% (Figure 7.7). Requiring En
values to be a linear function of pressure ? opposed to being completely free with
respect to pressure ? carries only a minimal penalty in model accuracy. The con-
comitant decrease in required computation effort, occasioned by making En terms
vary linearly with pressure, is likely a worthwhile approach for applications of the
model in control schemes. Note that Figure 7.7 indicates that when using En val-
ues that are fixed with respect to pressure, the effect is the lowest error for the
blocked force and free-contraction states; however, these improvements in accuracy
are misleading, and are only local to these two length-state regions.
Figure 7.9 presents a direct illustration of the effect of varying the pressure-
dependencies of the En terms. These model fits use a fourth-order stress-strain
relationship (E1?4) while varying the relationship of En with respect to pressure.
Here, the improvement in accuracy, by allowing En values to vary with pressure, is
evident. It is interesting to note how the accuracy of the model fit for each case
varies with respect to pressure. When En values are allowed to vary with respect
to pressure, the model fit is very accurate in all pressure states (0.6-4.1 lbf MAE).
265
Notably, the model that uses the experimentally measured En values (fixed with
respect to pressure) is also able to provide a very good fit at low pressures (3.5 lbf
MAE at 20 psi), but the accuracy of the fit decreases with increases in pressure
(58.2 lbf MAE at 100 psi). This observation supports the assertion that the model is
missing a pressure-dependent term that, if added, could eliminate the need for En
to vary with pressure.
7.4.2.3 Best-fit Model Selection
From the parametric study above, a ?best fit? form of the model is selected
as the final demonstration of the model?s accuracy in its current form. This model
uses a stress-strain relationship defined by a third-order polynomial, and with fit-
coefficient terms (E1?3) that are both constrained to vary linearly with pressure, and
constrained to obey the Drucker Stability criterion. Previous analysis showed that
the relatively small decreases in modeling error that would come with adding an
additional material fit coefficient (0.3 lbf MAE improvement), removing the linear
pressure-dependence of the En terms (0.3 lbf MAE improvement), and removing any
stress-strain stability constraint (0.5 lbf MAE improvement), are likely not worth the
additional cost in model complexity, or the departure from accurately representing
the intrinsic properties of the bladder material.
A detailed illustration of the resulting model is provided in Figure 7.10. Figure
7.10(a) shows the model fit to the averaged response of each isobaric test. These
averaged experimental curves represent the complete set of data that the model
266
100 psi
1000 Experimental Data
 f(P)
 f(P) (linear)
800
Fixed
Fixed (Meas. E-vals)
600 60 psi
400
200 20 psi
0
0 0.1 0.2 0.3 0.4
Contraction, L /L  
0
(a) Comparison of model fits and experimental data
500 200
1000
400 100
900
300 0
0 0.02 0.04 0.06 0.14 0.16 0.18 0.2 0.3 0.32 0.34 0.36
(b) Zoomed-in views of results in subfigure (a) (100 psi test). Regions of focus are near (left)
zero-contraction/blocked force, (middle) mid-contraction, and (right) free-contraction
1000
 f(P)
 f(P) (linear)
800 Fixed
Fixed (Meas. E-vals) 100 psi
600
60 psi
400
20 psi
200
0
0 0.5 1 1.5 2
Strain, in/in
(c) Stress-strain curves produced by optimized En
values
Figure 7.9: Comparison of model fits with variation in the relationship between En
and pressure
is fit to, and they provide a more complete depiction of the model?s fit (figures
are commonly reduced to show curves in only 20 psi increments for visual clarity).
267
Stress, psi Force, lbf
Figure 7.10(b) shows how the model fit compares to the raw experimental data. The
model developed in this work can be adapted to take into account the hysteresis of
the FAM?s response through the use of an additional modeling term, as detailed in
Hocking (2012) [18]. Figure 7.10(c) provides a depiction of the optimized values of
E1?3 that are constrained to vary linearly with pressure; slope and y-intercept values
for each of these linear curves are provided in Table 7.5. The pressure-dependent
stress-strain relationships produced by values of E1?3(P ), and constrained by the
Drucker stability criterion, are shown in Figure 7.10(d). Again, note that the same
stress-strain relationship is used for the material in both tensile and compressive
stress states; however, the compressive strain due to longitudinal stresses ranges up
to a maximum of only 0.28 in/in (for 10 to 100 psi, respectively).
A detailed report of the model?s accuracy with respect to the experimental
data is provided in Table 7.6. Averaged over all pressures, the MAE is low at 2.8
lbf. Average error in the blocked force and free-contraction states is also fairly
good, at 20.8 lbf and 0.005 in/in, respectively. The optimization function has the
most apparent difficulty in fitting the model near the resting length and maximum
contraction states. If further improvements in accuracy are desired in certain regions
of the contraction range, the least-squares cost function of the optimizer can be
weighted towards these regions at the expense of reduced model fit in other regions.
Table 7.5: Fit Coefficient Values, En, For ?Best-Fit? Model
En = mn ? P + bn E1 E2 E3
Slope (mn) 257.8 -57.9 32.7
y-intercept (bn), psi 131.9 340.7 -146.0
268
1000 Model 1000 Model
100 psi 100 psi
90 psi 80 psi
800 80 psi 800 60 psi
70 psi 40 psi
600 60 psi 600 20 psi
50 psi
40 psi
400 30 psi 400
20 psi
200 10 psi 200
0 0
0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4
Contraction, L /L  Contraction, L /L  
0 0
(a) Model comparison to experimental data (b) Model comparison to raw experimental
(averaged for each pressure) data (includes hysteresis in force response)
1000 1000
E
1
E
2 800
500
E 100 psi
3
600 80 psi
0 60 psi
400 40 psi
20 psi
-500
200
-1000 0
0 20 40 60 80 100 0 0.5 1 1.5 2
Pressure, psi Strain, in/in
(c) Fit optimized E1?3 values used for the (d) Pressure-dependent stress-strain curves
model produced by E1?3(P )
Figure 7.10: Modeling results for the ?best-fit? model, which uses terms E1?3 that
are constrained to vary linearly with pressure, and constrained to comply with the
Drucker stability criterion.
7.5 Conclusion
The research presented in this chapter works to improve the model by bet-
ter representing the mechanisms of the FAM while also improving the accuracy of
the model. Previous chapters in this work have set a foundation for this effort,
providing an accurate characterization of the FAM by ensuring true isobars during
testing, and then developing a photogrammetric method for acquiring accurate di-
269
E , psi Force, lbf
n
Stress, psi
Force, lbf
Table 7.6: Error of the ?Best-Fit? Model with Respect to Experimental Data for
All Tested Pressures
Test Blocked Force Free-Contraction RMSE MAE
Pressure Error Error
psi lbf , (%) | ?Lmax | /L0, (%) lbf lbf
100 -28.0 (-2.6%) -0.009 (-2.6%) 7.36 5.58
90 -30.5 (-3.2%) -0.008 (-2.4%) 6.42 4.24
80 -31.5 (-3.7%) -0.008 (-2.4%) 5.75 3.65
70 -27.8 (-3.8%) -0.007 (-2.0%) 4.64 2.82
60 -24.4 (-3.9%) -0.005 (-1.6%) 3.58 2.08
50 -18.5 (-3.6%) -0.003 (-1.0%) 2.56 1.68
40 -13.1 (-3.2%) -0.001 (-0.3%) 1.82 1.26
30 -8.5 (-2.9%) 0.001 (0.5%) 1.43 1.09
20 -5.5 (-3.0%) 0.002 (1.0%) 1.70 1.14
10 -20.1 (-22.8%) -0.012 (-14.1%) 5.75 4.55
Total Avg. -20.8 (-5.3%) 0.005 (-2.5%) 4.1 2.8
mensional measurements of the FAM. By employing the developed photogrammetric
method, accurate dimensional inputs to the model can now replace the previously
used cylindrical approximations.
With those previous improvements in-hand, the model was tested here for
associated improvements in its accuracy. The results illuminated distinct regions
of modeling error near the blocked force and free-contraction states of the FAM.
The stress-strain relationship constraint placed on the model?s fit-parameters (i.e.
En values) to ensure material stability ? which in its prior form was thought to be
over-constraining the model ? was investigated for its potential to improve model
accuracy. Two approaches were taken to ensure material stability: a constraint
that ensures a positive monotonic stress-strain relationship, and a constraint that
270
ensures Drucker Stability. Replacing the prior constraint with either of these new
constraints resulted in a mean reduction in error from 8.3 lbf down to 3.5 lbf, or
2.5 lbf, respectively.
With the optimization function updated to include the accurate dimensional
inputs to the model ? along with the addition of the Drucker Stability constraint ?
the overall accuracy of the model fit was then tested. For the first test of potential
improvements to the model, En values were constrained to be fixed with respect
to pressure. These results were then compared to the experimentally obtained En
values from Chapter 6. While this fixed-value case represents the most physically
realistic definition of the bladder?s material properties, the resulting fit of the model
was not very accurate, with a mean model error of 25.4 lbf using the experimental
values, and 18.1 lbf for the optimized values. The inability of the model to match
the FAM?s response when using fixed material coefficient values demonstrates that
unmodeled phenomena still exist, and suggests that the model is not ready to be
used in a predictive manner. Values of En, and the respective bladder stiffness, must
still be allowed to vary with respect to pressure in order to obtain an accurate fit of
the model.
With no further model improvements to be made in this study, it was in our
best interest to determine how to provide the best performance of the model in its
current form. The model was tested for fit while adjusting the number of En terms
used (polynomial order), and by varying the relationship of the En values with
respect to pressure. A parametric study was performed to find the combination
of parameters that provides the most accurate fit of the model. The number of
271
En terms was varied from one to four terms, and they were allowed to vary as
a free function of pressure, as a linear function of pressure, or fixed with respect
to pressure. Increasing the number of En terms generally improved the fit of the
model by providing it with more degrees of freedom; a distinct jump in model fit
improvement was observed when more than two En terms were used. An accurate
fit required the En terms to vary with respect to pressure, but only a minimal
0.3 lbf average error penalty came with constraining them to be a linear function of
pressure. Finally, a best-fit model was produced which used three En terms that
were each defined as a linear function of pressure. The resultant model provided an
accurate fit with an average error across all pressures of only 2.8 lbf.
So, in its current form, this model is able to provide a very good fit of ex-
perimental data, overall. Even though this model still relies on fit-parameters (En)
that cannot produce a very accurate model fit when they are fixed ? as is physi-
cally expected of the bladder material ? steps were accomplished here in improving
the overall fidelity of the model through the use of accurate inputs and constraints.
Reliance on pressure-dependent En values for an accurate model fit points to an
opening for future works to develop an additional pressure-dependent term within
the model.
272
Chapter 8: Conclusion
8.1 Summary of Key Work and Conclusions
In this research, significant contributions were made towards the characteriza-
tion and modeling of FAMs. A 7/8 inch diameter FAM was developed, and in the
process, the design and fabrication methods for making swaged end-fitting FAMs
were improved. Experimental characterization data was improved by (1) ensuring
accurate control of pressure during testing, and by (2) developing a method of mea-
suring the FAM?s shape profile during testing. This characterization data was used
to test the accuracy of long-held modeling assumptions ? the cylindrical geometry
approximation, and the inelastic braid assumption. Subsequent adjustments were
made to the actuation force model, and the model was then tested for improvements
in its accuracy.
8.1.0.1 FAM Design and Fabrication
A 7/8 inch diameter FAM with a swaged end-fitting design was produced
in this work. Through the development of this FAM, a standard method of scal-
ing swaged end-fitting design was developed that considers the desired actuation
273
characteristics, the compression required in each end-fitting, as well as constraints
imposed by the limited availability of component sizing. Difficulties with the safety
and consistency of the existing fabrication method led to modifications that resulted
in strengthening of the FAM through a 10% increase in braid compression in the
end-fittings. The resulting 7/8 inch diameter FAM improved upon the capabili-
ties of previous swaged end-fitting FAMs; with a feed pressure of 100 psi, the new
FAM provides a maximum force of 1050 lbf (a 100 lbf increase over previous swaged
FAMs), a maximum contraction of 35%, and a safety factor of greater than two.
Since the completion of this work, the developed methods have been used in the
design and fabrication of additional swaged end-fitting FAMs with diameters of 1.5
inches, 2.25 inches, and a 7/8 inch extensile FAM ? the first extensile FAM with
swaged end-fittings.
8.1.0.2 Characterization Improvements - Pressure Control
Pressure control systems for air and water working fluids were developed to
provide accurate feed-pressures to the FAM during characterization testing. The
previous pneumatic system could only control set-pressures within 8 psi. The new
pneumatic and hydraulic systems provided average feed-pressure errors of only
0.1 psi for a small scale FAM (3/16 inch dia.), and 0.4 psi for a large scale FAM
(7/8 inch dia.). Subsequent analysis of characterization testing results found that
inaccuracies in the old pressure control system would result in overestimations of the
FAM?s maximum force by 5-30%, underestimations of maximum contraction by up
274
to 3%, and overestimations of hysteresis (resulting in overestimations of the FAM?s
internal friction). The developed pneumatic pressure control system was used in all
of the FAM testing performed (as detailed in the subsequent chapters), to ensure
the accuracy of the characterization data used for model development.
The new pressure systems were used to test the quasi-static response of FAMs
when driven by working fluids with different compressibilities (e.g. air and water).
Nominally identical responses of FAMs with air or water working fluids resulted in a
confirmation of the independence of the FAM?s quasi-static response to working fluid
compressibility. This conclusion elevates the ability to transition between different
working fluids for FAM actuation.
8.1.0.3 Characterization Improvements ? Profile Measurements
A method for measuring a FAM?s shape profile was developed to enable the ac-
quisition of geometric measurements during characterization testing. The photogrammetry-
based method was designed to be inexpensive and easy to implement. Photographs
are taken during testing at a regular interval, and measurements are then extracted
from the images using automated analysis techniques. This measurement system
was tested during the characterization of a 7/8 inch diameter FAM. While the FAM
was stroked during testing, photographs were taken at the rate of once-per-second
to obtain measurements for every 0.04 inches of the FAM?s contraction range. The
resulting measurements had a 0.004 in/pixel resolution. For analysis of the acquired
data, the profile measurements were represented as either an average diameter, or as
275
a highly-accurate Fourier-series fit of the profile. Initial analysis found evidence of
inaccuracies in the cylindrical approximation and inelastic braid assumption, which
motivated the work documented in subsequent chapters. A new method of acquir-
ing volume measurements was also presented, which can now replace prior methods
that were intrusive to the FAM and required elaborate test setups.
8.1.0.4 Testing the Inelastic Braid Assumption
The inelastic braid assumption, commonly used in the prediction of a FAM?s
geometric dimensions ? was investigated in this chapter. This assumption came
into question when measurements of the FAM?s shape profile captured a 0.043 inch
(4%) increase in braid diameter (in the FAM?s resting length state) as feed-pressure
was increased from 0 to 100 psi. Upon further investigation, it was estimated that
this increase in diameter was a result of the braid?s threads stretching by 0.5%
(while seemingly minuscule, even small changes in diameter and braid angle can
have a magnified effect on actuation force). A Kevlar thread ? as used in the FAM?s
braid ? was tensile tested in isolation for comparison to the observed measurements.
Agreement between the stress-strain curves obtained from the photographic mea-
surements, the tensile test, and the cited elastic modulus of the Kevlar, confirmed
that the observed changes in diameter at fixed length-states were, in fact, due to
stretching of the braid. The effect of braid elasticity on the actuation force response
of FAMs was then investigated. The effects of the Kevlar?s elasticity was found to
be greatest in the resting length state, and was estimated to cause a 5% decrease in
276
force (braid elasticity has typically been accounted for with the common use of the
inelastic braid assumption). To test the response of FAMs with softer braid materi-
als (e.g. PET, Nylon), a case study was performed using a braid with a hypothetical
stiffness that was closer to that of PET or Nylon (about 10% of Kevlar?s stiffness).
For these cases, the study estimated a decrease in actuation force of up to 70%,
and a fixed-length increase in diameter of up to 30%. This exemplifies the large
impact potential that braid elasticity has on the actuation of FAMs, and supports
the assertion that there are cases where an inelastic braid should not be assumed.
8.1.0.5 Testing the Cylindrical Approximation
The cylindrical approximation ? an assumption used in almost every model
of FAMs ? was tested in this chapter. First, estimated dimensions provided by the
cylindrical approximation were compared to the photographic measurements of the
FAM?s dimensions. Errors in the estimated dimensions were found to increase as
a FAM becomes increasingly less cylindrical in shape with increases in contraction.
Estimated geometries overestimated the FAM?s diameter by up to 4.7%, and un-
derestimated braid angle by up to 3.3%. These geometric errors of the estimated
dimensions resulted in force model estimates that had a mean absolute error of up
to 23.8 lbf (for all contraction states at 100 psi) when compared to force estimates
found using the measured dimensions. However, when the force model with mea-
sured dimensional inputs was compared to the experimental response of the FAM,
the model still had appreciable mean absolute errors ranging from 3 lbf (at 20 psi) to
277
58 lbf (at 100 psi). While improvements were surely made to improving the accuracy
of the model by replacing the cylindrical approximation dimensions with measured
dimensions, there was still a clear need for further corrections to the model.
8.1.0.6 Analysis of Improved Actuation Model
Further improvements were made to the model to improve its accuracy and
fidelity. To improve the model?s accuracy, the bladder?s material stiffness was de-
fined using fit-coefficients found through least-squares-fit optimization techniques
(in place of empirically-based constants), as has been used in previous works. An
optimization constraint in previous renditions of the model, used to ensure stability
of the bladder?s optimized stress-strain curve, was modified in this research in accor-
dance with the Drucker stability criterion. Modification of this constraint improved
the fidelity of the model, and was shown to provide a 70% reduction in the model?s
mean absolute error. The force model now included the following improvements:
accurate dimensional inputs (from measurements); removal of the cylindrical ap-
proximation from the model?s bladder force terms, with partial removal from the
Gaylord force term (by using measured dimensions); and, addition of the Drucker
stability criterion constraint to ensure the stability of the bladder?s optimized ma-
terial stiffness. To test this new version of the model, the optimized material coeffi-
cient values were compared to the empirical material coefficients found in Chapter
6. The optimized model did not provide sufficient accuracy, with a mean absolute
error of 18.1 lbf that was only marginally better than the 25.4 lbf mean absolute
278
error achieved by the model when using the empirically-based material coefficient
values. To improve the model?s fit, the fit-parameters (i.e. the material coefficient
values), were allowed to vary with respect to pressure ? as has been allowed in pre-
vious FAM modeling work ? while the number of material coefficients needed for a
good model fit was also identified through a parametric sweep. The selected form
of the model used three optimized material coefficients (a third-order stress-strain
polynomial relationship), that were each allowed to vary linearly with respect to
pressure. This form of the model provided a very accurate fit of experimental data,
with a mean absolute error of only 2.8 lbf. The overall final form of the model
proved to be very accurate while also improving the representation of the FAM?s
physical mechanisms within the model. However, the model?s continuing reliance
on pressure-dependent material coefficient terms, which is not mechanistically accu-
rate, emphasizes the need for further modification of the model to directly account
for this pressure dependency.
8.1.1 Contributions to Literature
This dissertation makes a number contributions to fluidic artificial muscle
literature. The major contributions of this work are as follows:
1) Developed method for measuring the FAM?s geometry: A novel method
of accurately measuring a FAM?s geometry during its actuation was developed in
this research. The method provides geometric measurements of the FAM that were
not previously available, and can now be used as an asset in the research and anal-
279
ysis of FAMs, or in defining design requirements for the implementation of FAMs
in robotic mechanisms. The photogrammetry-based measurement method was de-
signed to be easily adopted for future work, with minimal equipment or expertise
requirements, and the ability to measure a wide-range of FAM sizes. The measure-
ments were shown to provide very high resolution and accuracy measurements that
enable detailed analyses of a FAM?s actuation. Many of the findings for this research
were enabled by access to this advancement in the measurement of FAMs.
2) Assessment of the inelastic braid assumption: Elasticity of the braid can
affect the geometry and actuation force of a FAM. However, the braid is often
assumed as being inelastic in FAM models because of the modeling simplifications
provided by that assumption, and because its effects are often visually imperceptible,
and hence assumed to be inconsequential. The effects of braid elasticity became
apparent through analysis of the detailed geometric measurements of a tested FAM.
Analysis showed that the effects of braid elasticity, even for very stiff materials such
as Kevlar ? can affect the actuation force by as much as 5%. Commonly used braids
are often more compliant, and could affect actuation by as much as 70%. Overall,
the research demonstrated that the braid?s elasticity can have a significant effect on
a FAM?s actuation.
3) Removal of model uncertainty due to the cylindrical approximation:
Since the initial inception of the FAM, the cylindrical approximation of the FAM?s
geometry has been relied upon for simplified dimensional and actuation force es-
timates. This assumption has long been understood as a source of model error,
and could not be quantified due to a lack of dimensional measurement methods.
280
The photogrammetric measurement method enables a comparison between the mea-
sured dimensions, and the estimated cylindrical geometry dimensions. Error in the
estimated geometry was found to increase as the FAM becomes increasingly less
cylindrical in shape with contraction of the FAM. For a 7/8 inch diameter FAM,
the estimated geometry underestimated diameter by as much as 4.7%, and under-
estimated the Gaylord actuation force by as much as 37%. Due to its method
of derivation, the Gaylord force term of the model cannot be completely divorced
from the cylindrical approximation, however, the assumption has been completely
removed from the bladder force terms of the model. Removal of the cylindrical
approximation from FAM models results in a significant reduction in the degree of
uncertainty within them.
Additional contributions of this work include:
? Further development of methods for the design and fabrication of swaged end-
fitting FAMs
? Highlights the importance of accurate pressure control during characterization
testing
? Proof of the FAM?s quasi-static force independence from working fluid
? Introduction of a Drucker stability constraint into the force model optimization
of material stiffness coefficients
The chart in Figure 8.1 (which is an updated version of 1.11), exhibits the
actuation mechanisms within the force model that this dissertation addresses, in
281
comparison to previous works (column header: Chambers Dissertation, 2021). The
updated form of the actuation force model includes terms and dimensional inputs
that accurately capture both the elasticity and geometry of the bladder and braid.
FAM characterization data that uses accurate pressure control ensures that the
model is based on an accurate representation of the FAM. The deadband pressure
term was also effectively accounted for by capturing the initial strain of the bladder.
The current model does not include a term for friction, however, the model can be
easily appended with one of the many friction models developed in previous works.
8.1.2 Future Work
This dissertation serves as a foundation for subsequent research to build upon.
A natural extension of this research would include making further improvements to
the actuation force model. That would likely involve removing the current pressure-
dependency within the bladder?s optimized material coefficient terms (that are cur-
rently required for a good model fit) through modification of the model?s Gaylord
force term, and would require further investigation into the FAM?s actuation mech-
anisms. Next steps towards this goal would likely require further improvements in
the characterization and fabrication of FAMs, as described below.
In this research, dimensional parameters D, L, N , and B were all accurately
defined. However, braid angle (?) was never directly measured, and had to be
assumed as fixed along the length of the FAM. Future research could adapt the
photogrammetric measurement method to also capture braid angle measurements
282
283
Figure 8.1: Overview of the FAM?s actuation force model development through the following representative literature: Gaylord
(1958) [2], Schulte (1961) [3], Chou and Hannaford (1994) [4], Chou and Hannaford (1996) [5], Tondu (1997) [6], Klute and
Hannaford (2000) [7], Tsgarakis and Caldwell (2000) [8], Tondu (2000) [9], Ferraresi (2001) [1], Colbrunn (2001) [10], Davis et al.
(2003) [11], Davis and Caldwell (2006) [12], Kothera (2009) [13], De Volder (2011) [14], Solano (2011) [15], Tiwari (2012) [16],
Tondu (2012) [17], Hocking (2012) [18], Meller (2014) [19], Robinson (2015) [20], Wang (2015) [21]
along the length of the FAM. These measurements would ensure accurate braid
angle values for model inputs, but would also enable discretized analysis of the
braid?s forces along its length; sections of the FAM with distinct diameter-braid angle
dimensions could then be analyzed individually and as an interconnected whole.
In the Chapter 4 work, variation in the shape profiles of otherwise identically
constructed FAMs was observed. These shape profiles also appeared to transform
through time before eventually settling into a consistent shape. It is assumed that
this phenomenon is due to the subtle migration of braid threads that occurs until
the braid and bladder forces within the FAM equilibrate. Analysis of the FAM?s
dimensions over repeated cycles could help uncover why this occurs, to what extent
it occurs, and how it may effect the actuation force of the FAM.
Finally, the development and analysis of FAMs could be boosted with the
ability to better control their actuation characteristics through their construction
process. The FAM?s characteristics are driven by their braid angle and diameter,
but control of these parameters is currently limited by the availability of component
sizes through vendors. The inability to directly control braid angle often results in
an incoherence between the braid and bladder?s sizing that can convolute the FAM?s
actuation mechanisms and characteristics. Preliminary research at the University
of Maryland has investigated a filament-winding fabrication process of FAMs; this
process weaves braid fibers directly onto the bladder, with the ability to control the
angle of the braid threads. This offers many opportunities, including the ability to:
control the actuation characteristics of the FAM; reduce or eliminate actuation (i.e.
deadband) pressure by eliminating initial slack in the braid; have accurate knowledge
284
of braid parameters B, N , D0, and ?0 as model inputs; and, select from a wider-
range of braid materials for preferred properties. This fabrication method could
also allow for the braid to be contained within a bladder mold to produce a single
composite component. which would protect the braid from external abrasion, and
keep the braids from shifting on the surface of the bladder. The maturation of this
fabrication technique could aid in the modeling and analysis of FAMs by reducing
the number of complex interactions between the bladder and braid. Investigation
of individual actuation mechanisms would benefit from the ability to better control
individual aspects of the FAM?s construction.
285
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