ABSTRACT
Title of dissertation: PNEUMATIC ARTIFICIAL MUSCLE
ACTUATORS FOR COMPLIANT
ROBOTIC MANIPULATORS
Ryan M. Robinson
Doctor of Philosophy, 2014
Dissertation directed by: Professor Norman M. Wereley
Department of Aerospace Engineering
Robotic systems are increasingly being utilized in applications that require
interaction with humans. In order to enable safe physical human-robot interaction,
light weight and compliant manipulation are desirable. These requirements are
problematic for many conventional actuation systems, which are often heavy, and
typically use high stiffness to achieve high performance, leading to large impact forces
upon collision. However, pneumatic artificial muscles (PAMs) are actuators that can
satisfy these safety requirements while offering power-to-weight ratios comparable
to those of conventional actuators. PAMs are extremely lightweight actuators that
produce force in response to pressurization. These muscles demonstrate natural
compliance, but have a nonlinear force-contraction profile that complicates modeling
and control.
This body of research presents solutions to the challenges associated with the
implementation of PAMs as actuators in robotic manipulators, particularly with
regard to modeling, design, and control. An existing PAM force balance model
was modified to incorporate elliptic end geometry and a hyper-elastic constitutive
relationship, dramatically improving predictions of PAM behavior at high contraction.
Utilizing this improved model, two proof-of-concept PAM-driven manipulators were
designed and constructed; design features included parallel placement of actuators
and a tendon-link joint design. Genetic algorithm search heuristics were employed
to determine an optimal joint geometry; allowing a manipulator to achieve a desired
torque profile while minimizing the required PAM pressure. Performance of the
manipulators was evaluated in both simulation and experiment employing various
linear and nonlinear control strategies. These included output feedback techniques,
such as proportional-integral-derivative (PID) and fuzzy logic, a model-based control
for computed torque, and more advanced controllers, such as sliding mode, adaptive
sliding mode, and adaptive neural network control. Results demonstrated the benefits
of an accurate model in model-based control, and the advantages of adaptive neural
network control when a model is unavailable or variations in payload are expected.
Lastly, a variable recruitment strategy was applied to a group of parallel muscles
actuating a common joint. Increased manipulator efficiency was observed when fewer
PAMs were activated, justifying the use of variable recruitment strategies.
Overall, this research demonstrates the benefits of pneumatic artificial muscles
as actuators in robotics applications. It demonstrates that PAM-based manipulators
can be well-modeled and can achieve high tracking accuracy over a wide range of
payloads and inputs while maintaining natural compliance.
PNEUMATIC ARTIFICIAL MUSCLE ACTUATORS FOR
COMPLIANT ROBOTIC MANIPULATORS
by
Ryan Michael Robinson
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
2014
Advisory Committee:
Prof. Norman Wereley, Chair/Advisor
Prof. Robert Sanner
Prof. Inderjit Chopra
Prof. Derek Paley
Prof. Balakumar Balachandran
Dr. Curt Kothera
c© Copyright by
Ryan M. Robinson
2014
Acknowledgments
I would like to gratefully acknowledge the support of my advisors, Dr. Norman
Wereley and Dr. Curt Kothera. Dr. Wereley has been my advisor for nine years, and
I could not hope for a better mentor. He has been incredibly supportive, providing
guidance, inspiration, and ideas that were vital to my success. Dr. Wereley gave
me the freedom to be creative and explore topics that I found interesting. Even
with his responsibilities as aerospace chair, he continued to take a deep interest in
his students’ work and always worked tirelessly to keep us on track. Dr. Kothera
is a great mentor, colleague, and friend. It has been my pleasure to work with
him on numerous projects which now constitute the majority of this dissertation.
Dr. Kothera motivated me to make steady progress every week, and he was always
there to listen carefully and provide fresh insights. He is an excellent writer and
proofreader; I hope that I was able to absorb some of those skills.
I would also like to thank Dr. Sanner, Dr. Chopra, Dr. Paley, and Dr.
Balachandran for their support, patience, and flexibility as my doctoral committee
members. Their interest in my work was motivation to ???, and their willingness to
share their expertise has been greatly appreciated.
I must thank Dr. Wei (Peter) Hu for taking me on as an undergraduate lab
assistant in Dr. Wereley’s group seven years ago. From my humble beginnings
cleaning magnetorheological fluid dampers to my later work designing them, Dr. Hu
taught me a great deal about how high-level research is conducted. He was always
patient and more than willing to devote his time to support my work.
ii
To my colleagues and good friends in the lab, you all have given me so much
help over the years; I truly appreciate it. We’ve also had a lot of fun, which helped
keep me sane and motivated. Special thanks to Robbie, Harinder, Andrew, Ami,
Tom, Steve, Pablo, Ben and Dr. Choi. It’s crazy that five of us doctors are graduating
together–truly the end of an era. To my colleagues that still have a few more years,
all my best...also, I highly recommend my tranquil back cubicle, which kept me
insulated from distractions (with the exception of the occasional Nerf dart).
I would like to acknowledge the support of my family and friends. Mom, Dad,
Tim, Kelli, and Kerri, your encouragement, kindness, humor, and love are always
present in my life. To my lifelong friends, you guys are the best, thanks for believing
in me, listening to me babble on about my work, and for understanding when I had
to go into dissertation mode.
And to Janene: The past three years of my life have been amazing and
unforgettable because of you. I am ridiculously lucky to have found such a talented,
compassionate, and beautiful woman. You are my constant inspiration through the
incredible, hard work you do as you pursue your doctorate in psychology. Thank
you for being there for me through every step of my dissertation. I’m so glad that
finally, our lives, which have been on parallel tracks for so long, are beginning to
merge. I can’t wait to build our future together.
iii
Table of Contents
List of Tables vii
List of Figures viii
1 Introduction 1
1.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Compliant and Human-Friendly Robotics . . . . . . . . . . . . . . . . 4
1.2.1 Conventional Actuation Systems . . . . . . . . . . . . . . . . 5
1.2.2 Smart Material Actuators . . . . . . . . . . . . . . . . . . . . 7
1.2.3 Series Elastic Actuators . . . . . . . . . . . . . . . . . . . . . 8
1.2.4 Active Safety Control . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Pneumatic Artificial Muscle Actuators . . . . . . . . . . . . . . . . . 14
1.3.1 Background and Motivation . . . . . . . . . . . . . . . . . . . 14
1.3.2 PAM Literature Review . . . . . . . . . . . . . . . . . . . . . 17
1.3.2.1 Modeling of Pneumatic Artificial Muscles . . . . . . 17
1.3.2.2 Control of PAM-Actuated Systems . . . . . . . . . . 20
1.3.2.3 Robotics Applications . . . . . . . . . . . . . . . . . 24
1.3.2.4 Orthotics Applications . . . . . . . . . . . . . . . . . 31
1.3.2.5 Aerospace Applications . . . . . . . . . . . . . . . . 36
1.4 Contributions of Dissertation . . . . . . . . . . . . . . . . . . . . . . 39
1.5 Overview of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . 41
List of Abbreviations 1
2 Quasi-Static Modeling and Analysis of Pneumatic Artificial Muscles 45
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.2 Improvements to Force Balance Model . . . . . . . . . . . . . . . . . 49
2.2.1 Original Force Balance Model . . . . . . . . . . . . . . . . . . 49
2.2.2 Non-constant Bladder Thickness Modification . . . . . . . . . 54
2.2.3 PAM Extension . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.2.4 Elliptic Toroid Shape for Bladder Ends . . . . . . . . . . . . . 57
2.2.5 Braid Length along the Surface of the Elliptic Toroid . . . . . 60
iv
2.2.6 Linear Relationship between Hyperelastic Constants and Pressure 67
2.3 Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3 Structural Design of PAM-Based Robotic Manipulators 77
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.2 Design Requirements and Simulation . . . . . . . . . . . . . . . . . . 80
3.2.1 Design Requirements . . . . . . . . . . . . . . . . . . . . . . . 80
3.2.2 Actuator Development and PAM Characterization . . . . . . . 81
3.2.3 PAM Comparison . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.2.4 Robot Arm Simulation . . . . . . . . . . . . . . . . . . . . . . 85
3.2.5 Proof-of-Concept Manipulator Construction . . . . . . . . . . 95
3.3 Test Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.3.1 Fabrication of Test Setup . . . . . . . . . . . . . . . . . . . . . 97
3.3.2 Testing and Experimental Results . . . . . . . . . . . . . . . . 97
3.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.5 Optimization of Joint Geometry via Genetic Algorithms . . . . . . . 104
3.5.1 Optimization Problem . . . . . . . . . . . . . . . . . . . . . . 104
3.5.2 Optimization Results . . . . . . . . . . . . . . . . . . . . . . . 109
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4 Control of PAM-Based Robotic Manipulators 115
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.2 PAM-Actuated Robotic Manipulator . . . . . . . . . . . . . . . . . . 120
4.2.1 Design and Experimental Setup . . . . . . . . . . . . . . . . . 120
4.2.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.3 Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.3.1 Output Feedback Control . . . . . . . . . . . . . . . . . . . . 125
4.3.1.1 Proportional-Integral-Derivative Control . . . . . . . 125
4.3.1.2 Fuzzy Control . . . . . . . . . . . . . . . . . . . . . . 126
4.3.2 Model-Based Feedforward Control . . . . . . . . . . . . . . . . 128
4.3.2.1 Model-Based Control without Feedback . . . . . . . 129
4.3.2.2 Model-Based Control Augmented with Output Feed-
back . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.4 Control Analysis Via Simulation . . . . . . . . . . . . . . . . . . . . . 132
4.4.1 PID and Fuzzy Control . . . . . . . . . . . . . . . . . . . . . . 132
4.4.1.1 Gain Tuning Metrics . . . . . . . . . . . . . . . . . . 132
4.4.1.2 Simulated Trajectory Following . . . . . . . . . . . . 134
4.4.2 Model-Based Control with Output Feedback . . . . . . . . . . 135
4.5 Experimental Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.5.1 PID and Fuzzy Control . . . . . . . . . . . . . . . . . . . . . . 137
4.5.1.1 Experimental Validation . . . . . . . . . . . . . . . . 137
4.5.1.2 Experimental Gain Tuning . . . . . . . . . . . . . . . 138
4.5.1.3 Comparison of Output Feedback Controllers . . . . . 140
4.5.2 Model-Based Control with Output Feedback . . . . . . . . . . 143
v
4.5.2.1 Model Validation . . . . . . . . . . . . . . . . . . . . 143
4.5.2.2 Experimental Analysis of Feedforward Gain . . . . . 145
4.5.3 Discussion of Controllers . . . . . . . . . . . . . . . . . . . . . 146
4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5 Advanced Control of PAM-Based Robotic Manipulators 150
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
5.2 Robotic Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.3 Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5.3.1 Sliding Mode Control . . . . . . . . . . . . . . . . . . . . . . . 157
5.3.2 Adaptive Sliding Mode Control . . . . . . . . . . . . . . . . . 159
5.3.3 Adaptive Neural Network Control . . . . . . . . . . . . . . . . 161
5.4 Position Control Results . . . . . . . . . . . . . . . . . . . . . . . . . 164
5.4.1 Sliding Mode Control . . . . . . . . . . . . . . . . . . . . . . . 165
5.4.2 Adaptive Sliding Mode Control . . . . . . . . . . . . . . . . . 166
5.4.3 Adaptive Neural Network Control . . . . . . . . . . . . . . . . 170
5.5 Discussion and Comparison of Controllers . . . . . . . . . . . . . . . 175
5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
6 Variable Recruitment 186
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
6.2 Preliminary Characterization and Analysis of Bladder Effects . . . . . 189
6.3 Variable Recruitment on a Robotic Manipulator . . . . . . . . . . . . 192
6.4 Variable Recruitment Simulations . . . . . . . . . . . . . . . . . . . . 195
6.4.1 Dynamic Modeling . . . . . . . . . . . . . . . . . . . . . . . . 195
6.4.2 Simulation Procedure . . . . . . . . . . . . . . . . . . . . . . . 199
6.4.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 201
6.5 Variable Recruitment Experiments . . . . . . . . . . . . . . . . . . . 204
6.5.1 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . 204
6.5.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . 205
6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
7 Conclusion 213
7.1 Summary of Research and Key Conclusions . . . . . . . . . . . . . . 213
7.1.1 Pneumatic Artificial Muscle Modeling . . . . . . . . . . . . . . 213
7.1.2 Manipulator Proof-of-Concept and Joint Optimization . . . . 214
7.1.3 Proof-of-Concept Manipulator Control Study . . . . . . . . . . 215
7.1.4 Adaptive Control Study . . . . . . . . . . . . . . . . . . . . . 215
7.1.5 Variable Recruitment . . . . . . . . . . . . . . . . . . . . . . . 216
7.2 Contributions to Literature . . . . . . . . . . . . . . . . . . . . . . . 217
7.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
Bibliography 222
vi
List of Tables
1.1 Comparison of actuation technologies (adapted from [1–4]). . . . . . . 16
3.1 Prototype design goals. . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.2 Comparison of different-sized PAMs. . . . . . . . . . . . . . . . . . . 86
3.3 Geometric parameters for PAM force model. . . . . . . . . . . . . . . 91
3.4 Joint design requirements and PAM characteristics. . . . . . . . . . . 105
3.5 Optimal parameters of candidate designs, determined by genetic
algorithm-based optimization. . . . . . . . . . . . . . . . . . . . . . . 113
4.1 Fuzzy ruleset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.1 Parameter values of sinusoids in desired trajectory. . . . . . . . . . . 165
5.2 Adaptation gains for adaptive sliding mode control. . . . . . . . . . . 168
6.1 Geometric and material parameters for test PAMs. . . . . . . . . . . 191
6.2 Test matrix for variable recruitment, 50 deg max. angle. . . . . . . . 200
6.3 Test matrix for variable recruitment, 90 deg max. angle. . . . . . . . 200
vii
List of Figures
1.1 Geared motor (Maxon). . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Robot finger actuated by SMA (Bundhoo, 2009). . . . . . . . . . . . 9
1.3 Electroactive polymer legged robot (Pei, 2003). . . . . . . . . . . . . 9
1.4 Series elastic actuator concept. . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Baxter robot (Rethink Robotics). . . . . . . . . . . . . . . . . . . . . 14
1.6 Miniature PAM (Hocking and Wereley, 2013). . . . . . . . . . . . . . 17
1.7 Bridgestone “Rubbertuator” arm. . . . . . . . . . . . . . . . . . . . . 25
1.8 7 DOF robot (Tondu, 2005). . . . . . . . . . . . . . . . . . . . . . . . 25
1.9 FESTO AirArm (Eichorn, 2009). . . . . . . . . . . . . . . . . . . . . 26
1.10 “OctArm” continuum robot (Walker, 2005). . . . . . . . . . . . . . . 28
1.11 Shadow Robotic Hand (Shadow Robot Company). . . . . . . . . . . . 29
1.12 Walking Robots, (a) PAM-based leg (Colbrunn, 2000), (b) Lucy
bipedal walking robot (Verrelst et al., 2005). . . . . . . . . . . . . . . 30
1.13 Human-lifting robots (a) Robotic nursing assistant (HStar Technolo-
gies [5]), and (b) Battlefield extraction assist robot (Vecna Technolo-
gies [6]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.14 McKibben muscle orthosis with CO2 tank. . . . . . . . . . . . . . . . 32
1.15 7 DOF rehabilitation exoskeleton (Caldwell and Tsagarakis, 1994). . . 34
1.16 “Power assist wear” (Noritsugu, 2008). . . . . . . . . . . . . . . . . . 34
1.17 Ankle orthosis (Park et al., 2014). . . . . . . . . . . . . . . . . . . . . 35
1.18 PAM-actuated trailing edge flap (Woods et al., 2011). . . . . . . . . . 37
1.19 Active camber airfoil (Peel et al., 2009). . . . . . . . . . . . . . . . . 37
1.20 Morphing skin test article (Bubert, 2009). . . . . . . . . . . . . . . . 38
1.21 MiniPAM cartridge (Vocke et al., 2012). . . . . . . . . . . . . . . . . 38
2.1 PAMs grouped in an antagonistic pair, actuating a rotating joint. . . 49
2.2 Force balance model [25]. . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.3 Force-contraction profile comparing model and experiment; 1.9 cm
(0.75-in) diameter, 18.5 cm (7.3-in) length PAM with a 76.5 deg braid
angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.4 Nonlinear modulus values as determined by optimization procedure;
1.9 cm (0.75-in) diameter, 1.5 cm (7.3-in) length PAM with a 76.5
deg braid angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
viii
2.5 Blocked force vs. pressure; 1.9 cm (0.75-in) diameter, 18.5 cm (7.3-in)
length PAM with a 76.5 deg braid angle. . . . . . . . . . . . . . . . . 56
2.6 Geometric model of PAMs in (a) contraction, (b) extension. . . . . . 59
2.7 Cross-section showing estimated bladder thickness during (a) contrac-
tion and (b) extension; 1.9 cm (0.75-in) diameter, 18.5 cm (7.3-in)
length PAM with a 76.5 deg braid angle. . . . . . . . . . . . . . . . . 64
2.8 Comparison of cylindrical and elliptic tip models (a) radius vs. con-
traction and (b) thickness vs. contraction; 1.9 cm (0.75-in) diameter,
18.5 cm (7.3-in) length PAM with a 76.5 deg braid angle. . . . . . . . 65
2.9 Elliptic tip model parameters (a) tip eccentricity and (b) braid length
along the elliptic tip; 1.9 cm (0.75-in) diameter, 18.5 cm (7.3-in) length
PAM with a 76.5 deg braid angle. . . . . . . . . . . . . . . . . . . . . 66
2.10 Force vs. contraction, nonlinear force balance model with non-constant
thickness and elliptic tip geometry; 1.9 cm (0.75-in) diameter, 18.5
cm (7.3-in) length PAM with a 76.5 deg braid angle. . . . . . . . . . 68
2.11 Force vs. contraction with hysteresis estimates, nonlinear force balance
model with non-constant thickness and elliptic tip geometry; 1.9 cm
(0.75-in) diameter, 18.5 cm (7.3-in) length PAM with a 76.5 deg braid
angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.12 Modulus values optimized to vary linearly with pressure using nonlin-
ear force balance model with non-constant thickness, (a) first-order,
(b) second-order, (c) third-order, and (d) fourth-order polynomial
model for strain; 1.9 cm (0.75-in) diameter, 18.5 cm (7.3-in) length
PAM with a 76.5 deg braid angle. . . . . . . . . . . . . . . . . . . . . 70
2.13 Normalized mean squared error of model force predictions for Mth-
order polynomial strain models; 1.9 cm (0.75-in) diameter, 18.5 cm
(7.3-in) length PAM with a 76.5 deg braid angle. . . . . . . . . . . . . 71
2.14 Force vs. contraction, nonlinear force balance with non-constant
thickness, (a) first-order, (b) second-order, (c) third-order, and (d)
fourth-order polynomial model for strain; 1.9 cm (0.75-in) diameter,
18.5 cm (7.3-in) length PAM with a 76.5 deg braid angle. . . . . . . . 72
2.15 Stress-strain relationship in the (a) axial and (b) circumferential direc-
tions, as determined by polynomial fitting; 1.9 cm (0.75-in) diameter,
18.5 cm (7.3-in) length PAM with a 76.5 deg braid angle. . . . . . . . 73
2.16 Experiment vs. model using fourth-order polynomial model for strain
and Ek(P ) interpolated from data of three experimental pressure levels
; 1.9 cm (0.75-in) diameter, 18.5 cm (7.3-in) length PAM with a 76.5
deg braid angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.17 Normalized mean squared error of model predictions, comparing Ek(P )
interpolated from data of all eight experimental pressure levels vs.
only three experimental pressure levels (0.14, 0.41, and 0.83 MPa [20,
60, and 120 psi]); 1.9 cm (0.75-in) diameter, 18.5 cm (7.3-in) length
PAM with a 76.5 deg braid angle. . . . . . . . . . . . . . . . . . . . . 76
3.1 Characterization of a 2-in OD, 13-in long PAM. . . . . . . . . . . . . 83
ix
3.2 Testing of a 2-in OD, 13-in long PAM (a) resting, (b)contracted. . . . 84
3.3 Characterization of a 0.625-in OD, 7.8-in long PAM. . . . . . . . . . . 84
3.4 Parameters for design analysis. . . . . . . . . . . . . . . . . . . . . . 86
3.5 Concept for variable moment arm joint. . . . . . . . . . . . . . . . . . 87
3.6 Geometry of robotic arm joint. . . . . . . . . . . . . . . . . . . . . . . 88
3.7 Torque output vs. angle of rotation. . . . . . . . . . . . . . . . . . . . 90
3.8 Simulation of step input, elbow joint. . . . . . . . . . . . . . . . . . . 95
3.9 CAD model of PAM manipulator. . . . . . . . . . . . . . . . . . . . . 96
3.10 PAM arm extended during testing. . . . . . . . . . . . . . . . . . . . 98
3.11 PAM arm contracted during testing. . . . . . . . . . . . . . . . . . . 98
3.12 Elbow PAM pressure and rotation vs. time. . . . . . . . . . . . . . . 100
3.13 Elbow joint torque vs. pressure. . . . . . . . . . . . . . . . . . . . . . 100
3.14 Experimental and predicted force vs. displacement. . . . . . . . . . . 102
3.15 Experimental and predicted static torque vs. angle of rotation. . . . . 103
3.16 Dynamics of upper arm contraction. . . . . . . . . . . . . . . . . . . . 103
3.17 Second-generation, two degree-of-freedom manipulator. . . . . . . . . 106
3.18 Joint parameters for optimization. . . . . . . . . . . . . . . . . . . . . 107
3.19 Torque vs. range of motion, illustrating regions to be minimized. . . . 109
3.20 Fitness value vs. generation, genetic algorithm-based optimization. . 110
3.21 Joint dimensions vs. generation, genetic algorithm-based optimization.111
3.22 Torque vs. range of motion, initial and final generations. . . . . . . . 111
3.23 Torque vs. range of motion, shoulder joint. . . . . . . . . . . . . . . . 112
3.24 Torque vs. range of motion, elbow joint. . . . . . . . . . . . . . . . . 113
4.1 Schematic of PAM-Actuated Manipulator System. . . . . . . . . . . . 121
4.2 PAM-actuated joint holding 50 kg (110 lb). . . . . . . . . . . . . . . . 121
4.3 Fuzzy membership functions for input/output variables. . . . . . . . . 128
4.4 Model-based feedforward (a) without output feedback, (b) with PID
or fuzzy feedback control. . . . . . . . . . . . . . . . . . . . . . . . . 131
4.5 Performance metrics as g1 and h vary (g2 = 0.6) with a 23 kg (50 lb)
payload (a) error metric; (b) smoothness metric; (c) combined metric.135
4.6 Simulated control, feedforward with fuzzy feedback . . . . . . . . . . 136
4.7 Simulated control, feedforward with PID feedback . . . . . . . . . . . 137
4.8 PID control, integral gain selection with a 23 kg (50 lb) payload (a)
error metric; (b) smoothness metric; (c) combined metric; (d) time
response for best cases. . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.9 Experimental vs. simulated optimal gains (a) PID; (b) fuzzy. . . . . 141
4.10 PID control with optimized gains. . . . . . . . . . . . . . . . . . . . . 143
4.11 Fuzzy control with optimized gains. . . . . . . . . . . . . . . . . . . . 144
4.12 Comparison of optimized experimental PID and fuzzy control, 23 kg
(50 lb) payload. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
4.13 Simulated and experimental results using feedforward control with
fuzzy feedback, 45 kg (100 lb) payload. . . . . . . . . . . . . . . . . . 145
4.14 Lift-hold-return trajectory while varying model gain kM , 23 kg (50 lb)
payload. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
x
4.15 Comparison of lift-hold-return trajectories, 11 kg (25 lb) payload. . . 147
5.1 One degree-of-freedom robotic arm with 50 lb payload. . . . . . . . . 154
5.2 Torque vs. angle, comparison of models with experimental data. . . . 156
5.3 Diagram of a cascaded controller. . . . . . . . . . . . . . . . . . . . . 157
5.4 Diagram of sliding mode controller. . . . . . . . . . . . . . . . . . . . 159
5.5 Diagram of adaptive sliding mode controller. . . . . . . . . . . . . . . 161
5.6 Single layer neural network structure. . . . . . . . . . . . . . . . . . . 163
5.7 Diagram of adaptive neural network controller. . . . . . . . . . . . . . 164
5.8 Sliding mode control results: (a) angle, (b) angle error, and (c) mean
squared error. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
5.9 Adaptive sliding control results: (a) angle, (b) angle error, and (c)
mean squared error. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
5.10 Parameter adaptation for three experiments, varying adaptation gains.169
5.11 Adaptive neural network control for two distinct initial conditions; (a)
angle, (b) angle error, and (c) mean squared error. . . . . . . . . . . . 171
5.12 Neural network adaptation over 100 s, 50 lb payload, no prior infor-
mation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
5.13 Neural network adaptation over 100 s, 50 lb payload, with model. . . 173
5.14 Adaptive neural network control, varying payloads; (a) angle, (b)
angle error, and (c) mean squared error. . . . . . . . . . . . . . . . . 174
5.15 Neural network adaptation over 100 s, 15 lb payload, no prior infor-
mation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.16 Neural network adaptation over 100 s, 50 lb payload, with network
initialized from 100 s of training with 15 lb payload. . . . . . . . . . . 176
5.17 Adaptive neural network control, varying initial network structure;
(a) angle, (b) angle error, and (c) mean squared error. . . . . . . . . . 177
5.18 Diagram of a cascaded PID controller. . . . . . . . . . . . . . . . . . 177
5.19 Comparison of control strategies, compound sinusoidal trajectory; (a)
angle, (b) angle error, and (c) mean squared error. . . . . . . . . . . . 179
5.20 Comparison of control strategies, lift-hold-return trajectory; (a) angle,
(b) angle error, and (c) mean squared error. . . . . . . . . . . . . . . 180
5.21 Adaptive neural network control, 0 lb payload, Gaylord model vs. no
information. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
5.22 Neural network adaptation over 100 s, 0 lb payload, with Gaylord
model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
6.1 Comparison of single-PAM (full pressure) vs. dual-PAM (half pressure)
force, PAM A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
6.2 Comparison of single-PAM (full pressure) vs. dual-PAM (half pressure)
force, PAM B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
6.3 Robotic arm holding a 95 kg (210 lb) payload. . . . . . . . . . . . . . 194
6.4 Exploded view of shoulder joint. . . . . . . . . . . . . . . . . . . . . . 194
6.5 Diagram of PAM-based robotic system. . . . . . . . . . . . . . . . . . 196
6.6 Diagram of cascaded PI control system. . . . . . . . . . . . . . . . . . 200
xi
6.7 Efficiency of manipulator with variable recruitment, 50 deg max. angle.202
6.8 Efficiency of manipulator with variable recruitment, 90 deg max. angle.202
6.9 Maximizing efficiency through variable recruitment, simulations, 90
deg max. angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
6.10 Efficiency of variable recruitment with and without bladder elasticity,
simulations, 90 deg max. angle. . . . . . . . . . . . . . . . . . . . . . 204
6.11 Angle and torque vs. time, 2 PAMs, 18 kg (40 lb) payload, 90 deg
max. angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
6.12 Torque vs. pressure, experiment and simulation, 2 PAMs, 18 kg (40
lb) payload. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
6.13 Efficiency of manipulator with variable recruitment, 90 deg max. angle.207
6.14 Work output of manipulator with variable recruitment, 90 deg max.
angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
6.15 Energy input of manipulator with variable recruitment, 90 deg max.
angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
6.16 Two non-activated PAMs. . . . . . . . . . . . . . . . . . . . . . . . . 211
6.17 Four non-activated PAMs. . . . . . . . . . . . . . . . . . . . . . . . . 212
xii
Chapter 1: Introduction
1.1 Problem Statement
Although decades have passed since the adoption of robotic systems in industry,
the modern concept of the robot continues to evoke thoughts of rigid and clunky
machines, echoing the “mechanical knight” envisioned by Leonardo da Vinci at the
turn of the 16th century [7]. High impedance robots—systems that are heavy and
stiff—have secured a niche in industrial operations because of their precision, speed,
and task repeatability. However, these characteristics that improve accuracy also
prevent safe human-robot interaction. High impedance systems do not yield in the
presence of an obstacle or disturbance, and upon impact can damage the surrounding
environment or injure the operator. Often, “safety cages” must be erected, limiting
the robot to a highly structured, undisturbed environment [8].
Interest in robots outside of the industrial arena has begun to blossom, as
evidenced by the success of small, mobile domestic robots for tasks like floor cleaning.
Nevertheless, systems that perform manipulation tasks have not gained popularity
in domestic applications. Among problems such as prohibitive cost and limited
autonomy, safety hazards are a major obstacle to wide adoption that must be
overcome in order for robotic systems to interact with humans in unstructured
1
environments.
There is growing interest in biologically-inspired robotic systems that exhibit
lower impedance, and there is currently a research thrust in the new field of soft
robotics. These compliant systems will deform at the joints in the event of a
disturbance, allowing for greater safety. Compliant motion with minimal trade-offs
in other aspects of performance would pave the way for robots in the home and in a
wide variety of other applications.
The research described in this dissertation explored the concept of human-safe
robotic manipulation, specifically through the employment of lightweight, naturally
compliant, biologically-inspired actuators known as pneumatic artificial muscles.
These muscles utilize the energy stored in compressed air to produce mechanical
work. Pneumatic artificial muscles are capable of high specific force and force density.
This enables lifting of massive payloads, including humans themselves—something
that other soft actuators cannot achieve without prohibitively high mass, volume,
or pressure. Specific applications for lifting humans include patient placement in
hospitals, casualty extraction in combat situations or disaster relief efforts. Related
applications which could benefit from high-specific-work, compliant muscles include
orthotics for rehabilitation and exoskeletons for strength augmentation.
Challenges exist that currently inhibit the widespread use of PAMs in robotic
systems. Foremost is their inherently nonlinear static and dynamic behavior, a byprod-
uct of hyperelastic bladder deformation, bladder-braid interaction, and compressible
fluid dynamics. In order to effectively design manipulators to meet specifications
of torque, speed, volume, and range of motion, an accurate model of actuator force
2
behavior is required. Similarly, control of these nonlinear actuators must enable
accurate trajectory following and high power output while maintaining predictable,
smooth motion. The controller must also demonstrate robust, stable recovery in the
presence of disturbances.
Researchers, especially within the past two decades, have studied pneumatic
artificial muscle behavior and proposed numerous architectures for reliable actuator
control in a variety of applications. These applications include robotic manipulators
and legs, orthotics, exoskeletons, and morphing aerospace surfaces. However, certain
demanding applications have remained out of reach. One of these applications is
a lightweight PAM-based robotic manipulator that is able to lift a wide range of
payload weights (including weights over 150 lb) along a desired trajectory over a
wide range of motion ( 90 deg), while compensating for disturbances. The present
manipulators take full advantage of the high strength-to-weight ratio of PAMs,
maximizing the payload it can lift. The application is “demanding” for two major
reasons: (1) The limit to the mass flow rate of air into the actuators often introduces
time delay in reaching desired pressure, and (2) the system must be able to safely
react to large changes in inertia, from payloads that can reach several times the mass
of the manipulator itself.
The purpose of the research, therefore, is to investigate the use of PAMs as
actuators in robotic manipulators, specifically heavy-lift manipulators. The major
goals are to precisely model PAM behavior, maximize manipulator torque output,
and maximize trajectory tracking performance. Furthermore, a variable recruitment
strategy is evaluated to improve efficiency for lightweight payloads. More broadly, this
3
dissertation aims to complement and improve upon the existing knowledge of PAM-
based human-friendly robotics. As compliant manipulation technology improves and
proliferates, a new paradigm in human-robot interaction will be achievable.
1.2 Compliant and Human-Friendly Robotics
High inertial force is the result of a high-mass system being slowed down from
a high velocity. In order for a robotic manipulator to operate safely with humans,
the system must have low mass such that an impact with the robot moving at high
speed will not cause severe damage. Injury can also occur if a human (in motion
with his or her own inertia) impacts a robotic manipulator that is stiff and resists
deformation. Sharp edges and hard surfaces are problematic as well.
Some clues for the design of human-friendly manipulators can be found in
human arms themselves (which obviously interact with humans). The average adult
arm is soft, highly dexterous, and can manipulate payloads several times its own
weight, traits that are highly desirable in a manipulator. To achieve these feats,
the human arm employs lightweight, soft (biological) muscles that possess naturally
compliant characteristics. Hence, light weight and compliance are factors that would
be useful to emulate in the design of human friendly systems.
The remainder of this section overviews current actuation technology, including
“smart” actuators, compliant actuators, and active control algorithms meant to
improve the safety characteristics of a manipulator.
4
1.2.1 Conventional Actuation Systems
Conventional robotic actuation systems fall into two a major categories: elec-
tromechanical and hydraulic. Both technologies offer high overall efficiency and
static positioning accuracy. However, these systems fall short in many ways when
safe human interaction is a consideration. Hydraulics and geared electric motors
are naturally rigid and can only be made to act compliant through complex control
strategies (and even then, bandwidth is limited).
Electric motors produce mechanical force through the interaction of a magnetic
field with the current passed through wound coils. In practical robotics applications,
motors must be geared because of their poor torque density in a direct drive configu-
ration. Gearing leads to very high stiffness and backdrive friction, characteristics
which can cause injury to humans or damage the surrounding environment. Further
disadvantages of geared systems include backlash, torque ripple, and noise [9]. Shock
loads to the system can cause high stress on gear teeth, which could easily damage
the system.
Hydraulic systems are known for their superior specific work, primarily due to
the high operating pressure of the hydraulic fluid (commonly between 20-35 MPa
[3000-5000 psi]). The incompressibility of hydraulic fluid leads to high actuator
efficiency. Nevertheless, there are major drawbacks to using hydraulic systems in
certain applications. Hydraulic fluid is typically a synthetic oil, which is prone to
leakage at high pressures, and can be hazardous to humans and the environment.
Hydraulic systems are also inherently very stiff because of the incompressibility of
5
Figure 1.1: Geared motor (Maxon).
the fluid. The volumetric flow rate of a hydraulic pump dictates the bandwidth
at which a hydraulic system can successfully implement compliant control (which
will be discussed in Section 1.2.4), but in many instances, the system cannot react
quickly to sudden disturbances (impacts).
A widely used alternative to hydraulics and electric motors is the traditional
pneumatic cylinder. Pneumatic cylinders take advantage of the compressibility of
air to provide natural compliance. However, the cylinder is typically heavy in order
to ensure pressurized air cannot escape. The maximum pressure used in pneumatics
is typically an order of magnitude less than in hydraulics due to efficiency losses
and hazards associated with highly compressed gas. Subsequently, work density
is at least an order of magnitude lower than hydraulics. Furthermore, the friction
introduced by the piston seal can be problematic.
6
1.2.2 Smart Material Actuators
Several “smart” actuators exist that have gained major attention over the past
three decades. This category of actuators encompasses material alloys that exhibit
controllable induced strain under various types of fields. Smart actuators include
piezoelectrics, shape memory alloys, magnetostrictives, and electroactive polymers.
Despite their utility and popularity in certain applications, these materials each have
significant drawbacks that disqualify them in most robotic manipulator applications.
Piezoelectrics, such as lead zirconium titanate (PZT), strain when an electric
field is applied, and can react at a very high bandwidth. However, the maximum
induced strain from these materials is too low to be useful in a robot joint requiring
large range of motion. A miniature piezoelectric motor was devised that weighed
70 g, but could only reach torques of 0.07 Nm [?]. Rather than act as the primary
actuators for the robot, piezoelectric devices have been more successfully used to
control the vibrations of a flexible robotic manipulator [10].
Shape memory alloys (SMA), which extend or contract in response to thermal
fields (often by running current through the material to generate resistive heat) are
capable of high specific work and moderate strain, but exhibit low bandwidth and
very low thermal efficiency. In research by Bundhoo [11], a robot finger actuated by
SMA was developed, but there was a delay of 5 to 10 s before joint motion occurred.
Another study conducted by Dilibal created an SMA-actuated robotic hand; however,
the complete system is complex and bulky, requiring water or freon to dissipate heat
energy quickly [12]. Precise control of SMA is difficult, and many systems can only
7
reliably switch between “on” and “off” states [4].
Magnetostrictives strain in response to magnetic fields, yet require a pro-
hibitively large, heavy electromagnet around the material to generate the desired
field [4].
Electroactive polymer (EAP) muscles are soft, compliant actuators that mimic
the properties of muscle, but the technology has not reached maturity for high-force,
high-strain robotic manipulator applications [13]. Some versions of these muscles are
capable of high strains and stresses, but require high driving electric fields, which
could pose danger to humans. A set of dielectric elastomer muscles have been used
as robot legs in work by Pei et al. [14] that bend in response to a driving voltage of
5.5 kV.
1.2.3 Series Elastic Actuators
One method of implementing natural compliance is the addition of an elastic
component (i.e., a spring) in series with a high stiffness actuator. There are several
benefits to this configuration. The high stiffness actuator can be controlled to great
accuracy, while the elastic component deforms as a known function of the applied
force. The compound stiffness of the series results in a low impedance system
that can be tuned through proper selection of the spring. Furthermore, energy
can be temporarily stored in the elastic component for repetitive motions such as
walking [15].
There are some disadvantages to this configuration, however. First, the elastic
8
Figure 1.2: Robot finger actuated by SMA (Bundhoo, 2009).
Figure 1.3: Electroactive polymer legged robot (Pei, 2003).
9
spring is passive, and may not be tuned for all necessary functions. A very low
stiffness system under limited actuator authority will respond slower to torque inputs
and, below a certain stiffness, requires more time to reach a new desired position
than a moderately stiff system moving slowly enough to minimize injury. Therefore,
there is a minimum limit to achievable positioning time [16]. Second is low actuator
performance when the load inertia is small compared to the motor’s reflected inertia.
In this case, an impulse applied to the load can cause high frequency oscillations,
instability, and contact chatter. A damper is recommended to handle small load
inertias, and the system should be critically damped for the lowest expected load [17].
However, damping increases the impedance of the actuator. Additions to the actuator
will increase the overall system weight, decreasing specific work.
1.2.4 Active Safety Control
Compliant manipulation can be obtained with traditional actuators through the
application of an impedance control strategy. The term impedance control is widely
used, but in this strategy, position and orientation are actually being commanded,
while impedance is modulated. The objective of impedance control is to make the
system exhibit target dynamics, including stiffness and damping properties chosen
by the designer, about the commanded trajectory.
Injury criteria exist to estimate the consequences of a collision. The severity of
a head impact is related to the acceleration of the head and the pulse duration. To
measure collision severity in car accidents, Versace [18] proposed a metric known as
10
Figure 1.4: Series elastic actuator concept.
11
the Head Injury Criterion (HIC):
HIC = T
(
1
T
∫ t
0
a(τ)dτ
)2.5
(1.1)
where T is impact duration, a is head acceleration, and τ is a variable of integration
representing time. HIC values of 1000 or greater are associated with severe injury.
The HIC can be extended to other parts of the body [19]. In robotics research, this
metric has been employed to measure the severity of injury imparted by a robotic
manipulator [16] and to evaluate the safety of different actuation strategies.
Kazerooni [20] developed a controller with variable impedance, where the
robotic system was designed to provide compensation such that the end effector
position was proportional to the interaction force (within a given operating frequency
range). The stiffness of the system could be chosen by the designer to allow human-
friendly interaction. However, depending on the actuator and system properties,
high frequency external forces, such as sudden impacts, may not be possible to fully
compensate. Furthermore, an accurate model of the system is required for high
performance. In demanding applications, the designer must accept a certain trade-off
between performance and robustness to modeling uncertainty. Another issue with
employing a naturally stiff actuator is fail safety, in which the system can no longer
be controlled as desired. In this case, the system may revert to stiff, potentially
unsafe behavior.
Collision detection and reaction is another form of active safety control. Had-
dadin [21] demonstrated how reactive control strategies can significantly contribute
12
to safety. The system was able to distinguish between desired physical interaction
with the robot and unexpected collisions with the potential to cause injuries. Upon
detection of a dangerous collision, the robot would react with one of several strategies
tested, including zero-gravity compliance control and admittance control (which led
to the robot “fleeing” from the disturbance). The system can compensate when there
is enough time to categorize the disturbance, but in some cases, impacts with rigid
surfaces (e.g., the human head) may result in a hard shock before reactive strategies
can be implemented.
Bicchi and Tonietti [16] proposed a variable stiffness transmission (VST) concept
for enhanced safety. In this strategy, the joint actuation scheme consists of compliant
components that can be tuned on-line. The authors suggest tuning for high stiffness
during low velocity motion and lower stiffness during high velocity motion. In
this way, a variable stiffness “trajectory” can be implemented in tandem with a
motion trajectory planner. The authors state that pneumatic artificial muscle (PAM)
actuators arranged in an antagonistic pair are “among the best candidates” for this
task.
A combination of active and passive safety features are most likely to produce
the safest systems. Baxter, by Rethink Robotics [22] is advertised as a low cost
industrial robot that can be trained by humans and work in close proximity with
humans. Its passive safety features include series elastic actuators, back-drivable
motors, and padded surfaces. Active control features include visual object detection,
collision detection, and emergency stop mechanisms. However, Baxter is only rated
to handle payloads up to 5 lb. This limits the type of tasks that can be performed.
13
Figure 1.5: Baxter robot (Rethink Robotics).
Moreover, the system has a maximum speed below that which would injure humans,
limiting task completion times.
1.3 Pneumatic Artificial Muscle Actuators
The background and motivation for employing pneumatic artificial muscles
will be discussed in further detail in this section. Research performed to date and
the evolution of PAM-based design will be examined.
1.3.1 Background and Motivation
Pneumatic artificial muscles (PAMs) are lightweight, soft actuators that employ
pressurized air to generate axial force. The term “pneumatic artificial muscle”
encompasses a range of designs. The common components of these actuators are
an elastomeric “bladder” material and an inextensible fiber material. The fiber can
be embedded inside the elastomer, or can surround and move freely on the bladder
14
surface. The latter is traditionally known as a McKibben actuator, named after
Joseph McKibben, who popularized these devices as orthotics for polio patients in
the 1950s [23]. The invention of the McKibben-type pneumatic muscle is generally
attributed to R. H. Gaylord, who patented the “Fluid actuated motor system and
stroking device” in 1958 [24]. A similar “Elastic diaphragm” patented by A. H.
Morin [25], awarded in 1953, is cited in the Gaylord patent. Pressurization of a soft
bladder causes the stiff braid fibers to reorient, generating stroke in one direction.
Typically, McKibben-type PAMs are contractile, meaning that pressurization causes
the PAM to shorten axially, while expanding radially. Resistance to this contraction
generates a tensile force.
PAMs have several appealing features for use in robotic systems. They are
capable of excellent power-to-weight ratios and high specific work when highly
pressurized. Additionally, they are easily scaled to different sizes; in fact, many
researchers have focused on “miniature” PAMs, roughly denoting muscles with
diameters less than 0.25-in [26,27]. These compliant actuators are damage tolerant
[2,28], and have been endurance-tested for as many as 125 million cycles with minimal
wear [2] and little change in force characteristics. Table 1.1 displays important metrics
for comparison between types of actuators, including specific work, work density,
maximum frequency, and efficiency [1–4].
In some applications, PAMs are employed in antagonistic pairs. This arrange-
ment is widely observed in biological systems. A commonly cited agonist-antagonist
pair is the biceps-triceps pair of muscles on the human arm. As the agonist contracts,
the antagonist extends, preserving the stiffness of the joint.
15
Table 1.1: Comparison of actuation technologies (adapted from [1–4]).
Actuation Max Strain Actuation Specific
Technology Stress
(MPa)
Work
(J/kg)
Hydraulic 1.0 70 35000
Electromechanical 0.5 1 300
Pneumatic
Cylinder
1.0 0.9 1200
Piezoelectric 10−4 10 1
Shape Memory
Alloy (SMA)
0.1 700 4500
Magnetostrictive 0.002 10 20
Electroactive
Polymer (Gel)
0.4 0.3 1
Pneumatic Arti-
ficial Muscle
0.4 16 5000
Human Muscle 0.5 0.4 0.8
Actuation Work Max Efficiency
Technology Density
(kJ/m3)
Frequency
(Hz)
Hydraulic 105 100 0.95
Electromechanical 103 100 0.35
Pneumatic
Cylinder
250 100 0.40
Piezoelectric 10 107 0.95
Shape Memory
Alloy (SMA)
104 7 0.02
Magnetostrictive 15 107 0.80
Electroactive
Polymer (Gel)
0.4 100 0.30
Pneumatic Arti-
ficial Muscle
103 100 0.40
Human Muscle 0.8 100 0.25
16
Figure 1.6: Miniature PAM (Hocking and Wereley, 2013).
1.3.2 PAM Literature Review
A large body of literature exists on the subject of pneumatic artificial muscles.
Included are nonlinear modeling studies, design and characterization of systems
employing PAMs for actuation, and control studies of PAM-based systems. Thorough
reviews on PAM modeling and control are available [29, 30]. The following sections
will briefly overview the history and state-of-the-art in these areas, as well as provide
examples of PAM applications in robotics, orthotics, and aerospace.
1.3.2.1 Modeling of Pneumatic Artificial Muscles
Since the inception of PAM-actuated systems, researchers have relied upon
complex mathematical models to help characterize their nonlinear behavior. PAM
size, shape, and material characteristics affect force and strain characteristics. PAMs
also exhibit hysteresis; which implies force is dependent on the history of the actuator
state. For any PAM, the axial force of the actuator is dependent on the internal
pressure and the level of strain. As the PAM radius expands and length contracts,
17
braid fibers reorient, and the angle θb between the braid fibers changes. Gaylord [24]
first modeled the relationship between actuator force, pressure, and strain with a
simple geometric model that ignores bladder elasticity, hysteresis, and non-cylindrical
effects:
F =
PpiD245◦
2
(
3 cos2 θb − 1
)
=
P
4piN2
(
3L2 −B2
)
(1.2)
In this equation, P is pressure differential, D45◦ is the actuator diameter when braid
angle is 45 deg, N is number of turns a single braid makes around the circumference,
L is the strained actuator length, and B is braid length. Subsequent studies have
improved upon this model using energy balance, force balance, or large deformation
theory approaches that do incorporate the losses caused by bladder elasticity.
Schulte [23] improved upon Gaylord’s original equation to include effects of
bladder elasticity and friction, but the equation was not derived. Schulte postulated
that friction and elasticity were highly nonlinear. Thirty-five years later, Chou
and Hannaford [31] introduced a PAM modeling approach based on the principle of
virtual work, which results in a force equation similar to the formula in Schulte’s work.
Klute and Hannaford [32] then introduced a nonlinear Mooney-Rivlin mathematical
description of the bladder to this model, improving force estimates at high contraction.
The Mooney-Rivlin constants can be known a priori based on the material selected,
or determined empirically. Using the same methodology, Tsagarakis and Caldwell [33]
incorporated a conical surface profile at the actuator ends to compensate for the
non-cylindrical shape and accounted for the initial “activation” pressure required to
18
overcome the radial elasticity of the bladder.
Ferraresi et al. [34] developed an alternative approach to PAM modeling known
as force balancing. This model accounts for geometry and bladder elasticity by
summing forces in the axial and circumferential directions, but also assumes that the
PAM shape is always fully cylindrical. Interestingly, both force and energy balance
derivations uncover Gaylord’s original geometric force term. The force balance model
was modified by Kothera et al. [35] to incorporate non-constant bladder thickness
into the formulation, significantly improving results. Hocking and Wereley [27] used
the Ferraresi model to characterize miniature PAMs with stiff silicone bladders, but
replaced the elastic modulus of the bladder with a nonlinear hyperelastic function
that could be empirically fit to the data. However, these models could not accurately
obtain PAM force during extension past resting length because they assumed a fully
cylindrical PAM shape.
Large deformation theory can be used to precisely model the extreme shape
change undergone by a bladder during expansion. Rivlin [36] laid the groundwork
for this theory in the late 1940s. Adkins and Rivlin [37] modeled large elastic
deformations of thin shells composed of isotropic materials. Kydoniefs [38] used
this modeling approach for an elastomeric cylinder reinforced with inextensible
fibers. Further extending this work, Shan et al. [39] developed a finite axisymmetric
deformation model that predicts the changing shape of a flexible matrix composite
membrane under different conditions. However, large deformation theory requires a
complex numerical procedure for convergence. Likewise, it is assumed in these studies
that the inextensible braid fibers are embedded in the material, and thus cannot
19
fully account for free braid movement on the surface of McKibben-type actuators.
Despite decades of research [40], an all-encompassing model that reliably
estimates PAM nonlinear force-contraction behavior over the operational envelope
of actuator strain and pressure has been elusive. The existing models have not
demonstrated sufficient accuracy and scalability for demanding applications such as
model-based control of a variable-payload robotic manipulator. Accuracy in most
models decreases at high bladder strains (ε ≥ 1) due to the nonlinear stress-strain
relationship of these actuators. Most actuator models also assume a fully cylindrical
shape, and those that do consider shape change simplify the geometry of the rounded
ends and do not account for the new path that the braid must travel. Absent
from the literature is a model that can precisely capture the nonlinear stress-strain
relationship of the artificial muscle while accounting for shape change.
1.3.2.2 Control of PAM-Actuated Systems
The nonlinearities present in dynamic PAM-based systems lead to difficulties in
smooth, responsive control. These nonlinearities are not limited to PAM force behav-
ior; compressible airflow dynamics and the mechanics of the overall system must also
be considered in many applications. While simple output feedback strategies (such
as proportional-integral-derivative control) are often able to achieve stable trajectory
following with proper gain tuning, problems such as phase delay, undesired oscillation,
and overshoot are common. Therefore, several advanced control algorithms have
been studied for PAM-based systems.
20
Past efforts to control pneumatic artificial muscles span a wide range of es-
tablished techniques. Caldwell et al. [41] used adaptive control based on model
estimation, demonstrating accuracy for lightweight payloads (0.325 kg [0.7 lb]) over
9 deg of arm rotation. However, PAM-based systems do not reach their full potential
unless they can handle payloads exceeding their own weight and can operate over a
much larger range of motion. Ahn and Nguyen [42] developed an intelligent switching
algorithm using a learning vector quantization neural network for various payloads,
showing experimental data for step inputs. This controller requires training the
neural network for an extended amount of time before the system could reliably lift
different payloads. Wu et al. [43] developed a self-tuning fuzzy PID controller for a
hand exoskeleton actuated by PAMs. The experiments showed undesired oscillation
about the reference trajectory, though there was a time-varying disturbance due
to human input. Yeh et al. [44] designed an optimal controller using loop transfer
recovery (LTR) for a leg exoskeleton, which was deemed successful, but limited to
only 15 deg of rotation.
When manipulator joint trajectories require high speed and acceleration, the
tracking capabilities of pure output feedback control are degraded. Moreover, over-
simplified computed torque controllers or adaptive controllers with slow parameter
updating may not have the capability to track fast nonlinear dynamics. To address
this problem, several approaches to PAM-based control have incorporated highly
detailed models of the system. Zhu et al. [45] designed an adaptive robust controller
for a parallel manipulator with only a few degrees of movement, incorporating a model
of the flow dynamics and valve. Ganguly et al. [46] introduced static and dynamic
21
empirical models of PAM actuation and valve characteristics in a model-based PID
controller for a rotary joint. The adoption of feedforward compensation was shown
to improve system response in work by Nho and Meckl [47], who demonstrated
neuro-fuzzy and inverse dynamics feedforward control on a two-link manipulator.
Fateh and Izadbakhsh [48] employed a hybrid computed torque approach to a two-
link manipulator and found that feedforward control reduces tracking error. These
model-based feedforward strategies influenced the design of the control system in
the present work.
Control studies on PAM-based systems have often employed sliding mode
control, a robust nonlinear control strategy that drives the dynamics of the system
to that of an exponentially stable system [49]. Many sliding control algorithms
have been proven to be globally stable when model errors and system disturbances
are bounded [50]. Carbonell et al. [51], Cai and Dai [52], and Lilly [53] performed
simulations of PAM-actuated systems using sliding mode controllers. However, these
second-order controllers assume that the PAM pressure or force being used as a
control input is instantaneous, which cannot be assumed in a practical system with
large PAM volume. Similarly, Nouri et al. [54] designed an adaptive controller with a
sliding component which neglected airflow dynamics for low payloads (0.6 kg [1.3 lb]).
Xing et al. [55] applied sliding mode control with a disturbance observer to a PAM in
linear motion experiments with a 1 kg (2 lb) load. While the controller showed good
performance, airflow dynamics were neglected and commanded pressure was assumed
to be instantaneous, which is not realistic in more demanding applications. In order
to address such problems, Shen [56] designed a model-based sliding mode controller
22
including a dynamic airflow model, and applied it to a linear table actuated by
PAMs. Aschemann and Schindele [57] developed a cascaded sliding mode controller
for a high-speed linear axis with a detailed empirical model of valve dynamics
for pressure feedback and a model of the linear axis for position feedback. As in
other model-based control strategies, detailed physical models were necessary to
achieve satisfactory performance. Tondu et al. [58] applied sliding mode control with
twisting and super-twisting algorithms to two degrees-of-freedom of a PAM-based
manipulator, noting that the equivalent force control term was not helpful on the
link farther from the base because of model uncertainty. Overall, studies that have
successfully applied sliding mode control to experimental PAM-actuated systems
have used systems with high stiffness, low inertial loads, and minimal time delays.
However, the robotic manipulators in the present study exhibit relatively low stiffness,
highly variable inertial loads, and substantial time delay. Moreover, sliding mode
control can be negatively affected by measurement noise and low resolution sensors,
which are present in the current systems.
While past research has demonstrated smooth and accurate motion control,
the systems were simple and the payloads were much less than the mass of the
manipulator. The control of a PAM-actuated manipulator with both high range of
motion and high tip payload variations has not yet been attempted.
23
1.3.2.3 Robotics Applications
As a result of their functional similarity to natural muscles, numerous studies
have focused on applicability to robotic systems, including manipulation and joint
control of robotic arms and legs, or human assistance and rehabilitative devices.
PAMs are especially suitable for use in robots intended to physically interact with
human subjects because their inherent compliance increases safety.
One of the first PAM-actuated robotic manipualtors was developed by Bridge-
stone and Hitachi in 1984 [59]. The creators dubbed their actuator the “Rubbertuator”
and their system the “SoftArm.” Linear motion from the actuators was transmitted
through steel cables and pulleys to bend the arm at its joints. The arm had seven
degrees-of-freedom, and could lift up to 4.4 lb.
In 1994, Tondu et al. [60] employed McKibben muscles in a naturally compliant
two degree-of-freedom SCARA arm actuated by PAMs in an antagonistic configura-
tion, using a constant radius pulley to convert linear motion to rotational motion.
About a decade later, Tondu et al. extended their work to a seven degree-of-freedom
robot arm [40]. In this manipulator, gears were used to change the axis of rotation
and to allow the two shoulder joints to operate independently. By varying the
average pressure of the antagonistic pair, variable stiffness could be achieved. The
greatest peak torque produced by this robot was 67 Nm (50 ft-lb) in the shoulder
abduction joint, but because constant-radius pulleys were employed at the joints,
this torque quickly trailed off as the actuators contracted. The robot was able to lift
and maneuver delicate objects through tele-operation.
24
Figure 1.7: Bridgestone “Rubbertuator” arm.
Figure 1.8: 7 DOF robot (Tondu, 2005).
25
Figure 1.9: FESTO AirArm (Eichorn, 2009).
Eichorn et al. [61] designed and constructed Festo’s four degree-of-freedom PAM-
actuated manipulator called AirArm. Instead of attempting to transmit shoulder
abduction torque from a stationary frame to a rotating frame, as in Tondu’s work,
the designers placed the shoulder abduction PAMs in a cylindrical column that
rotates along with the arm when the other shoulder degree-of-freedom (extension) is
actuated. Experiments demonstrated that the arm could reach tip velocities of up
to 4 m/s. However, payload lifting was not studied.
Tuijthof and Herder designed a unique four-degree-of-freedom manipulator
that they claim is inherently safe because the arm uses pneumatic artificial muscles
and a gravity equilibrator system [62]. Springs and masses positioned and tuned so
as to keep the potential energy of the system constant.
Anthropomorphic manipulators have been constructed in order to study and
emulate human anatomy and to increase manipulator dexterity. Caldwell et al. [63]
created a robotic arm and tested PAM-actuated elbow control with adaptive control.
The manipulator was meant to replicate the layout and dimensions of the human
26
arm, with three degrees-of-freedom in the shoulder, one in the elbow, and three in
the wrist. However, the shoulder joint was actuated by motors. Departing from the
conventional method of single-degree-of-freedom joints, Yaegashi et al. [64] employed
a spherical (ball and socket) joint to demonstrate high torque and range of motion
within a compact volume. H-infinity motion tracking control was demonstrated with
fair results. A similar parallel manipulator was designed by Zhu et al. [45], which
used three PAMs to pivot a plate about a ball joint. This setup had limited range of
motion, but high accuracy and controllable stiffness.
Not all PAM-based robotic manipulators have used discrete joints. In fact,
their flexibility and light weight characteristics are ideal for non-discrete jointed,
“continuum” robots. Walker et al. [65] created a robotic arm modeled from an
octopus appendage. A cross-section of the robot reveals three PAMs surrounding a
central flexible rod. Pressure in each PAM can be varied independently, allowing
omni-directional bending along the main axis. Advantages to this configuration
are high dexterity and ability to conform to surroundings. The arm can curl and
constrict around objects of varying sizes and shapes in order to lift them. Trivedi [66]
determined an exact geometric model for this type of manipulator. This was useful
for calculating the controllable force throughout the workspace.
Other research groups have focused on PAM-actuated robotic hands. Shadow
Robot Company [67] created a robotic hand that has 20 degrees-of-freedom. The
PAMs and valves required for the many degrees-of-freedom reside outside the hand
itself, packed along the forearm. The inherent compliance in PAMs is useful in
delicate grasping tasks. Nagase [68] developed PAM-like actuators called pneumatic
27
Figure 1.10: “OctArm” continuum robot (Walker, 2005).
balloons for a compliant robotic hand, and used force sensing to grasp delicate
objects like paper cups (maximum force was less than 5 N [1 lbf]). The pneumatic
balloons were internal to the hand, but only four actuators were used.
Robotic legs intended for walking have also been actuated by pneumatic
artificial muscles. van der Linde [69] designed a PAM-based low power joint for
walking robots. PAM stiffness could be quickly varied in order to enhance ballistic
movement with cycle adjustability and orbital control. Colbrunn [70] designed a
robot leg intended for walking that could produce stable, repeatable forward walking
motion. The system was able to take advantage of the passive damping and elasticity
characteristics of PAMs—valves could be closed between 66% and 94% of the walking
motion. However, the full desired range of motion was not achievable with an
28
Figure 1.11: Shadow Robotic Hand (Shadow Robot Company).
antagonistic configuration. Additionally, tracking error was high. This may have
been a result of using a solenoid valve with pulse-width-modulation instead of a
more precise proportional valve. A similar walking design was created by Verrelst et
al. [71]. PAM leg muscles were connected to rods which rotated the joints, and the
attachment points could be varied to change the joint torque profile. The system
demonstrated slow walking motions with basic PID control.
As previously stated, pneumatic artificial muscles have many advantages that
make them desirable in portable, heavy-lift robotic systems intended for interaction
with humans [72,73]. Robots with the strength and dexterity to lift humans could
be used for nursing assistance [5] and in casualty extraction [6]. However, there is
very little research into PAM-based manipulators designed to lift heavy payloads
29
(a) (b)
Figure 1.12: Walking Robots, (a) PAM-based leg (Colbrunn, 2000), (b) Lucy bipedal
walking robot (Verrelst et al., 2005).
30
(a) (b)
Figure 1.13: Human-lifting robots (a) Robotic nursing assistant (HStar Technologies
[5]), and (b) Battlefield extraction assist robot (Vecna Technologies [6]).
(greater than 20 lb), or control studies where payloads have greater mass than
the manipulator itself. No known studies to date consider optimization of the
manipulator joint structures to maximize torque output over the full range of motion
with minimal operating pressure. This type of research would provide a more
accurate and valuable assessment of PAM-based manipulator abilities in comparison
to manipulators equipped with other actuation technologies such as electric motors
and hydraulics.
1.3.2.4 Orthotics Applications
The objective of many orthotic devices is to prevent muscular degeneration
and the development of abnormal motions resulting from neuromuscular disorders
such as cerebral palsy, amyotropic lateral sclerosis, multiple sclerosis, or stroke [74].
These systems not only prevent degeneration, but are often able to assist muscle
re-education. Active orthotic devices that interact with humans, especially wearable
systems, must be safe, compliant, and lightweight. Traditional systems using electric
31
Figure 1.14: McKibben muscle orthosis with CO2 tank.
motors are too inflexible, causing a loss of smooth motion [75].
Due to their light weight, flexibility, low friction, and inherently safe actuation,
PAMs have been employed in active orthotic devices. The first PAM orthotics
were developed in 1957 at Los Ranchos Hospital [76] in collaboration with Joseph L.
McKibben (coincidentally, the physicist who made the final connections to trigger the
first atomic bomb [77]). The impetus for the design was McKibben’s daughter, Karan,
who had been afflicted by polio and had lost the ability to move her fingers. The
pneumatic muscle was the perfect size and weight to actuate a wrist-hand orthotic
device, allowing a patient to pinch objects with three fingers. The pneumatic valve
supplying air from a CO2 tank was controlled by a manual lever.
While the concept was abandoned for some time because of poor valve tech-
nology and heavy air tanks, PAMs began to reappear in orthotics research during
the 1990s. Winters [78] carried out an investigation of McKibben and other types of
32
pneumatic artificial muscles for applications in orthotics. Prior et al. [79] described
the design of a wheelchair-mounted rehabilitation robot using their “Flexator” pneu-
matic muscle, although no information was given describing how rehabilitation would
be performed.
Caldwell and Tsagarakis [80] developed an arm exoskeleton orthosis with seven
degrees-of-freedom (Figure 1.15), each using an antagonistic PAM configuration.
The system could be used for strength augmentation or resistance exercise. The
overall mass of the manipulator, which was primarily composed of an aluminum
structure, was only 2 kg. However, maximum joint torque was only 2 Nm. Torque
feedback was studied in an elbow joint impedance control scheme. In a subsequent
study [81], torque was boosted to up to 30 Nm in the shoulder and 6 Nm in the
elbow. Although lightweight, the system was not designed to be wearable.
Noritsugu developed a power assist glove and upper body strength augmentation
suit based on “pneumatic rubber muscles” [82]. These actuators are somewhat
different than McKibben muscles because they are composed of an asymmetric fiber
bellows instead of a helical braid. Normally, the fiber is restricted on one side,
causing the muscle to extend and curl in one direction during pressurization. The
power assist glove is capable of 4 N pinch force. The upper body suit (Figure 1.16)
does not require a solid structure, and is therefore claimed to provide more comfort
than a cable and pulley system.
In order to provide enhanced rehabilitation, Ueda et al. [83] developed an
upper body PAM-actuated exoskeleton system that purposefully activates specific
muscles of the wearer in a systematic manner. The influence of exoskeleton torques
33
Figure 1.15: 7 DOF rehabilitation exoskeleton (Caldwell and Tsagarakis, 1994).
Figure 1.16: “Power assist wear” (Noritsugu, 2008).
34
was predicted using a human musculoskeletal model.
Wearable PAM systems have also been used in lower body rehabilitation. A
knee joint was constructed by Jordan et al. [84] for soft fluidic actuators with rotary
elastic chambers. Jamwal et al. [85] created a wearable parallel robot for three degrees-
of-freedom in an ankle orthosis. A multi-objective genetic algorithm was used to
find a family of optimal designs. Chandrapal [86] designed an intelligent PAM-based
knee orthosis that was capable of recognizing user intent from electromyography
(EMG) signals. The nonlinearities of the PAM system were compensated through an
intelligent self-organizing fuzzy controller.
The design and control of a wearable PAM-based ankle-foot rehabilitation
system was studied by Park et al. [74]. A unique feature was the absence of
rigid mechanical linkages. PAM placement was inspired by human anatomy; each
individual PAM was placed on top of the muscle it was intended to augment.
Figure 1.17: Ankle orthosis (Park et al., 2014).
35
The orthosis successfully assisted dorsiflexion/plantarflexion, and inversion/eversion
movements.
1.3.2.5 Aerospace Applications
Recently, the aerospace field has begun to explore unconventional, bio-inspired
methods for task-specific wing morphing and enhanced control of aerodynamic
surfaces. As a result, innovative research has been undertaken using PAMs as
actuators in aerospace applications.
Woods et al. [87] performed several experiments with a helicopter rotor trailing
edge flap using PAMs. A bellcrank linkage system was devised in order to place
PAMs near the root of the rotor blade, where the centrifugal force was lowest. Wind
tunnel tests determined that PAMs could produce the necessary torques to resist
aerodynamic loads and are controllable at the high frequencies necessary in rotor
applications. Tests in rotation proved that the pneumatic components could operate
normally in the rotating environment [88].
Peel et al. [89] developed an active camber airfoil actuated by lightweight
PAMs. The nose and tail of the structure were able to deform by 25 deg and 20 deg,
respectively. The wing structure was specifically designed to eliminate buckling of
the lower skin.
Bubert [90] designed a highly extensible skin elongation system for a morphing
aircraft wing that employed PAMs as actuators. Skin elongation of 100% was
achieved in the span direction while maintaining a constant chord, thereby doubling
36
Figure 1.18: PAM-actuated trailing edge flap (Woods et al., 2011).
Figure 1.19: Active camber airfoil (Peel et al., 2009).
37
Figure 1.20: Morphing skin test article (Bubert, 2009).
Figure 1.21: MiniPAM cartridge (Vocke et al., 2012).
the effect area, with PAMs operating an X-frame structure. A simple PID controller
was sufficient for the linear motion.
Vocke et al. [91] designed and tested a miniature PAM actuation package
(Figure 1.21) for use in a micro-air-vehicle (MAV). Findings included that well-
designed miniature PAM mechanisms can generate a greater static torque than
similarly-scaled conventional servo motors, leading to superior specific torque, while
torque density of the PAM device was on par with the servos.
38
1.4 Contributions of Dissertation
The objective of this research was to design and evaluate manipulators actuated
by lightweight, compliant pneumatic artificial muscles and intended to lift a large
range of payload masses over a large range of motion. Special emphasis was placed on
a compact design and smooth, accurate motion control. Several original contributions
were made to the state of the art in PAM modeling, design, and control.
Prior to this research, most PAM models assumed a fully cylindrical PAM shape
in contraction and extension when, in fact, the actuator ends form an elliptic shape
when deformed. Some models attempted to account for these tip effects by including
conical tip surfaces [63] or assuming a reduction in active length [35]. Furthermore,
existing models have not sufficiently accounted for the complex, nonlinear actuator
stress-strain relationship generated by the interaction between the bladder and braid.
In the present work, an existing PAM model (Ferarresi, 2001) was refined to improve
force estimates over the full range of actuator strain and operating pressure. First, a
new method was devised to account for the path a braid fiber travels around the
elliptical ends of the PAM, leading to a better approximation of geometric shape
change, and thus, PAM force behavior. Additionally, the actuator stress-strain
relationship was changed from linear to fourth order to account for nonlinear bladder-
braid interaction in both positive and negative strain. The empirically-fit constitutive
relationship was shown to accurately predict force-contraction curves at untested
pressures.
While dexterous multi-degree-of-freedom robotic manipulators have been de-
39
veloped in past work, the payload capacity of these systems is low. The highest
reported payloads on PAM-based manipulators working against gravity is 2 kg (4.5
lb) [40, 59]. These low maximum payloads are mainly a result of two design choices:
(1) employing antagonistic actuation, which allows bi-directional torque, but forces a
trade-off between torque and range of motion; and (2) ignoring optimization of joint
geometry. While one study has attempted to optimize antagonistic actuation on a
knee orthotic [92], the dual four-bar-linkage design is not optimal for a uni-directional
system. This work is the first known study to design PAM-based manipulators with
the specific intention of maximizing uni-directional torque over a large angular range
of motion (90+ deg) in order to lift payloads an order of magnitude greater than
current PAM-actuated manipulators (over 70 kg [150 lb]). To achieve this goal,
the manipulators were constructed with unique design features, including parallel
PAM arrangements and a novel joint design optimized for uni-directional actuation.
The genetic algorithm optimization procedure used for kinematic design of a PAM-
actuated manipulator is an original approach in PAM literature, adopted to ensure
that the joint can maintain high torque output from PAM actuators even as their
force reduces with contraction.
Numerous control strategies have been applied in PAM literature. Among these
are model-based controllers, which attempt to account for the complex pneumatic
and mechanical dynamics of the PAM-actuated system [56] but do not have the
ability to adapt to changes in system parameters. Alternatively, adaptive neural
network controllers that utilize a ”black box” model of the system [93] have also been
applied. A neural network structure does not take advantage of the known system
40
dynamics. In the present work, an ideal combination of modeling and adaptation is
determined for PAM-based manipulator control. The controller incorporates neural
networks within expressions of the known system dynamics to create a hybrid neural
network control law. In this controller, the neural networks are nonlinear functions
of angle involving the payload parameters and the quasi-static relationship between
angle and torque. The new model structure assumes a linear relationship between
pressure and joint torque, as well as a common equation of motion for an open-chain
manipulator. Including these relationships preserves controller structure without
requiring knowledge of nonlinear PAM dynamics, joint kinematics, or payload.
Lastly, this work investigates a novel variable recruitment strategy—activating
the minimum number of PAMs necessary to meet torque requirements—on a PAM-
actuated robotic manipulator. While previous research tested variable recruitment
to measure force, this is the first study to quantify the efficiency benefits of PAM
variable recruitment. The improved PAM force model was leveraged to accurately
simulate variable recruitment.
1.5 Overview of Dissertation
This dissertation is organized into seven chapters, each presenting a different
aspect of the research performed.
Chapter 1: Introduction. This chapter presents the motivation for soft ac-
tuation, specifically the use of pneumatic artificial muscles in robotic applications
that require human-robot interaction. Current technologies, including conventional
41
actuators, series elastic actuators, and compliant control strategies are discussed,
and their limitations are explained. Pneumatic artificial muscles are introduced as
a solution to these limitations, and a broad literature review on the modeling and
control of PAMs, as well as applications in robotics, orthotics, and other areas is
presented.
Chapter 2: Quasi-Static Modeling and Analysis of Pneumatic Artificial Muscles.
This chapter investigates modifications to an existing PAM modeling approach. The
first original modification is a detailed model of the elliptic ends of the actuator.
The second is a reformulation of the stress-strain relationship that accounts for
braid and bladder interaction, even in extension beyond resting length. These
improvements provide higher-accuracy force-contraction predictions than previous
models. The constitutive relationship is found to be nearly linear with pressure,
allowing straightforward extension to untested pressure levels.
Chapter 3: Structural Design of PAM-Based Robotic Manipulators. The design
of a two degree-of-freedom PAM-based manipulator is discussed. Important features
for compact design, such as multi-PAM actuator groups, tendon-based joints, and
pressure manifolds are described in detail. Simulations, which include dynamics of
the PAM actuators and airflow components, are conducted to predict performance,
followed by open-loop experiments to validate performance. Constrained optimization
is employed to determine joint geometry that minimizes the PAM pressure required
to satisfy a torque profile over a given range of motion.
Chapter 4: Control of a Heavy Lift Robotic Manipulator with Pneumatic
Artificial Muscles. Two output feedback controllers (proportional-integral-derivative
42
and fuzzy logic) are applied to a PAM-based manipulator in the pursuit of smooth and
accurate motion tracking. Due to the limitations of pure output feedback control, a
model-based feedforward controller is developed and combined with output feedback
to achieve improved closed-loop performance. Simulations are performed in order to
validate the control algorithms, guide controller design, and predict optimal gains.
Using real-time interface software and hardware, the controllers are implemented
and experimentally tested on the manipulator.
Chapter 5: Advanced Control of PAM-Based Robotic Manipulators. Building
upon the work in Chapter 4, advanced control strategies are implemented. A sliding
mode controller is employed in conjunction with a model-based sliding mode observer,
yielding the highest accuracy of all the control strategies when payload is known a
priori. To achieve high performance regardless of payload, an adaptive controller
based on neural networks is designed and implemented. Tests are conducted with
real-time payload modification under different waveforms, validating the effectiveness
of the controller.
Chapter 6: Variable Recruitment in PAM Groups. This chapter presents a
bio-inspired variable recruitment strategy to improve system efficiency. Efficiency
gains arise from a lower total bladder resistance and a diminished air “throttling”
effect when fewer PAMs are employed. Quasi-static characterization data is analyzed
to compare single vs. multiple PAM force output for a constant air mass, and the
effect of bladder material selection is explored. Dynamic tests are performed on a
PAM-based manipulator, and muscle recruitment is varied to determine its effect
on efficiency at different levels of payload and muscle contraction. Considerations
43
for the proper addition and subtraction of activated muscles in real-time control
applications are discussed.
Chapter 7: Conclusions. This final chapter summarizes the key conclusions
from this research. The original work contributed to literature is identified, and
several topics worthy of future work are discussed.
44
Chapter 2: Quasi-Static Modeling and Analysis of Pneumatic Artifi-
cial Muscles
2.1 Introduction
Pneumatic artificial muscles (PAMs) are lightweight, compliant actuators com-
posed of a helically braided sleeve surrounding an elastomeric bladder, which are held
together at each end by a fitting [29,40]. Pressurization of the soft bladder inflates
the muscle and causes the stiff braid fibers to reorient, generating a contractile
stroke and a pulling force. PAMs were originally devised by Joseph McKibben to
actuate orthotic devices for polio patients in the 1950’s [23]. Similar applications
have been studied more recently, including PAM-powered devices for orthotics and
rehabilitation [29, 30, 69] and bio-inspired humanoid robotic devices, such as robotic
hands [68, 94] and haptic devices [75]. These compliant actuators are particularly
desirable in portable, heavy-lift robotic systems intended for interaction with hu-
mans [72,73], specifically those envisioned for nursing assistance and in battlefield
casualty extraction, or in soft robotic applications [95]. In addition, these compliant
actuators are damage tolerant [2, 28], and have been endurance-tested for as many
as 125 million cycles with minimal wear [2].
45
In many applications, PAMs are employed in antagonistic pairs. This arrange-
ment, illustrated in Figure 2.1, is widely observed in biological systems. A commonly
cited agonist-antagonist pair is the biceps-triceps pair of muscles on the human
arm. As the agonist contracts, the antagonist extends past resting length, preserving
the stiffness of the joint. Antagonistic PAM configurations have been applied to
robotics [96], trailing edge flaps [87], and miniature servo-motors [91].
Although several models have been proposed to capture static and dynamic
PAM behavior as described in a review by Tondu [40], these models have not
demonstrated sufficient accuracy and scalability for demanding applications such
as model-based control of a robotic manipulator in high-speed conditions [97]. The
accuracy of most physical models is often poor at low pressures due to the pressure
deadband effect, a phenomenon in which the applied pressure must exceed a threshold
(activation) pressure before PAM contraction begins [27]. Additionally, accuracy
decreases at high bladder strains (ε ≥ 1) due to the nonlinear stress-strain relationship
in the bladder as it deforms from its resting state as a hollow cylinder to a shorter,
larger-diameter hollow cylinder with rounded ends. The internal pressure, actuator
strain, and geometry of the braid dictate this complex shape. However, most PAM
models assume a fully cylindrical shape, neglecting the boundary constraints that
produce rounded ends, or subtracting the rounded portions from the active length.
Gaylord [24] first modeled the relationship between actuator force, pressure,
and strain with a simple geometric model that completely ignores bladder effects.
Subsequent studies have improved upon this model with energy balance and force
balance approaches that incorporate the losses caused by bladder elasticity. Klute
46
and Hannaford [32] introduced a nonlinear Mooney-Rivlin term based on the principle
of virtual work, improving the force estimates at high contraction. Tsagarakis and
Caldwell [33] proposed a model that incorporates a simple model of tip deformation
and compensates for the activation pressure required to overcome the radial elasticity
of the bladder. Ferraresi et al. [34] developed the force balance approach, which was
modified by Kothera et al. [35] to account for bladder thickness change. Shan et
al. [39] developed a finite axisymmetric deformation model that could predict the
changing shape of a flexible matrix composite membrane under different conditions,
but force generation was not explored in detail. Despite decades of research [40], an
all-encompassing model that reliably estimates nonlinear force-contraction behavior
over the operational envelope of actuator strain and pressure for PAMs of different
sizes has not yet been developed.
While the behavior of an individual PAM can be determined through exhaustive
experimental tests, this is often impractical. In our prior work [27] with a miniature
PAM (4.16 cm [1.64-in] length, 0.41 cm [0.1625-in] diameter, composed of a silicone
bladder and polyethylene terephthalate [PET] braid), the nonlinear elasticity of the
bladder was captured by modeling stress as a polynomial function of the strain, in
which coefficients were determined empirically at many pressure levels. This model
was found to be highly accurate at the tested pressures, predicted blocked force with
high accuracy, and accounted for hysteresis in quasi-static experiments. However,
this model was not applicable to PAMs undergoing extension, that is, when a PAM
is stretched past its resting length. Inaccuracy of the extensile force predictions
resulted from assuming a fully cylindrical PAM shape and neglecting curvature at
47
the actuator ends. Furthermore, attempts to interpolate force-contraction profiles
to untested pressures often yielded low-accuracy results, rendering the model only
useful at or near the tested pressures. Other models, such as those presented by
Liu and Rahn [98] and Shan et al. [39] are able to capture extension beyond resting
state, but the models are complex and require a numerical procedure for convergence.
Additionally, these studies assume that the inextensible braid fibers are embedded
in the material, rather than allowing free movement on the surface of the bladder.
The present work addresses these issues by developing a new model that im-
proves blocked force predictions, characterizes PAMs in extension, and decreases
the amount of empirical data required to accurately characterize this type of actua-
tor. Several improvements to the current model were considered. First, this work
incorporates non-constant bladder thickness, a phenomenon resulting from volume
conservation as the bladder expands circumferentially. Then, a detailed geometric
model is introduced, which preserves the boundary conditions at the bladder ends
and better approximates the observed shape of the actuator. The improved model
is applied in extension, where it is found to well-predict the dramatic increase in
force. Finally, the constants of the stress-strain polynomial are constrained to vary
linearly with pressure during the curve-fitting procedure, dramatically improving
the predictive ability of the final model.
48
Figure 2.1: PAMs grouped in an antagonistic pair, actuating a rotating joint.
2.2 Improvements to Force Balance Model
2.2.1 Original Force Balance Model
A free-body diagram for the PAM force balance model that was originally
derived by Ferraresi et al. [34] and improved upon by Hocking and Wereley [27]
is shown in Figure 2.2. The actuator is modeled as a cylinder that expands and
contracts as the actuator strain is varied. The braided fibers which wind helically
around the bladder are assumed to have constant length B (a valid assumption as
the Kevlar fiber modulus is four to five orders of magnitude higher than the latex
bladder modulus), so that braid length is determined by:
B =
L
sinα
(2.1)
where L is the length of the actuator and α is the angle that the actuator braid makes
49
with the circumferential plane on the actuator surface. Geometric constraints also
dictate the number of turns, N , completed by the braid around the circumference of
the bladder, which is given by:
N =
L
2piRtanα
(2.2)
The outer radius of the PAM, R, is given by:
R =
√
B2 − L2
2piN
(2.3)
The thickness of the bladder, t, is calculated under the assumptions that
bladder volume VB, is a constant determined from the initial bladder geometry as a
uniform hollow cylinder, and that the bladder remains a uniform hollow cylinder but
with changing length and outer radius:
t = R−
√
R2 −
VB
piL
(2.4)
Applying the concept of force balance to a quasi-static PAM, we obtain equa-
tions in the circumferential (c) and axial (z) directions, respectively:
PRL = σctL+NT cosα (2.5)
F + piR2P = σzAB + T sinα (2.6)
50
Figure 2.2: Force balance model [25].
where P is internal pressure, T is tension in the braid, AB is bladder cross-sectional
area, σz is axial bladder stress, σc is circumferential bladder stress, and F is the
actuator force produced in the axial direction. By solving for T in Eqn. 2.5 and
substituting the result into Eqn. 2.6, we obtain an expression for actuator force:
F = −piR2P + σz
VB
L
+
PRL− σctL
N
tanα (2.7)
Substituting Eqns. 2.2 and 2.3 into Eqn. 2.6, the expression simplifies to:
F =
P
4piN
(3L2 −B2) + σz
VB
L
− σc
tL2
2piRN2
(2.8)
The total actuator force includes the Gaylord force term (the first term shown)
and terms dependent on bladder stress in the axial and circumferential directions.
51
While a number of methods can be used to determine these stresses, such
as finite elasticity theory [39, 99], we have elected to treat the nonlinear bladder
stress as a polynomial function of strain [27] and use system identification techniques
to determine the unknown coefficients. Yeoh []Yeoh1993 stated that the strain
energy density function of a nearly incompressible, nonlinear elastic material can be
expressed as a polynomial function; we use these findings as the inspiration for our
approach. Employing a polynomial stress-strain relationship relieves the restrictions
imposed by defining specific material properties or elasticity relations that would
not be able to account for the complex interactions that occur as the braid slides
freely along the outer surface of the bladder, including strain stiffening and softening
effects at high radial strains, as well as the pressure-dependent stiffness that was
observed experimentally.
The unknown coefficients of the constitutive polynomial function are the
modulus values Ek:
σ =
M∑
k=1
Ekε
k (2.9)
The axial and circumferential strains are given respectively by:
εz =
L
L0
− 1 (2.10)
εc =
R
R0
− 1 (2.11)
52
The curve-fitting optimization procedure from [27] was applied to experimental
data. At each pressure level, we choose Ek values in an M -th order nonlinear
force-balance model that minimize the mean squared error F between experimental
measurements and model estimates (using Matlab’s fmincon function) summed over
the n data points on the curve:
F =
n∑
i=1
|Fa,i − Fm,i|
2 (2.12)
Figure 2.3 displays the model and experimental data for a 1.9 cm (0.75-in)
diameter, 18.5 cm (7.3-in) length PAM with a 76.5 deg braid angle, using an optimized
4th-order polynomial stress-strain relation. Note that this particular PAM size will
be used throughout the remainder of the analysis as the illustrative example of the
model improvements. FAV G refers to the averaged data sets at each pressure level
(from 0.07 to 0.8 MPa [10 to 120 psi]), and FNLFB refers to the force predicted by
the nonlinear force balance model assuming constant bladder thickness. The model
matches very well with the data at high contraction; however, the blocked force is
overestimated at each pressure. Moreover, the model cannot accurately capture the
extensile behavior of the actuator that occurs during negative contraction. Figure 2.4
shows the values of Ek at each of the tested pressures. Note that there is a clear
linear trend between the pressure and the modulus values at pressures greater than
0.2 MPa (30 psi).
53
Figure 2.3: Force-contraction profile comparing model and experiment; 1.9 cm
(0.75-in) diameter, 18.5 cm (7.3-in) length PAM with a 76.5 deg braid angle.
Figure 2.4: Nonlinear modulus values as determined by optimization procedure; 1.9
cm (0.75-in) diameter, 1.5 cm (7.3-in) length PAM with a 76.5 deg braid angle.
2.2.2 Non-constant Bladder Thickness Modification
A modification to Ferraresi’s original force balance model was proposed by
Kothera et al. [35], incorporating the change in thickness of the bladder into the
54
formulation. Eqns. 2.5 and 2.6 are changed to reflect how the change in thickness of
the bladder affects static equilibrium:
P (R− t)L = σctL+NT cosα (2.13)
F + pi(R− t)2P = σzAB + T sinα (2.14)
Additionally, the average circumferential strain in the bladder is more accurately
modeled with thickness change:
εc =
R− t/2
R0 − t0/2
− 1 (2.15)
This leads to a modified equation for the actuator force (e.g., compare to Eqn. 2.8):
F =
P
4piN
(3L2 −B2) + P
(
VB
L
−
tL2
2piRN2
)
+ σz
VB
L
− σc
tL2
2piRN2
(2.16)
where we observe a new pressure-dependent term.
Figure 2.5 displays blocked force models assuming constant and non-constant
thickness for the sample PAM, compared with experimental measurements at tested
pressures. In the figure, FNLFBnc refers to the force predicted by assuming non-
constant bladder thickness. The model with the non-constant thickness assumption
clearly fits experimental data more closely than the model derived assuming constant
bladder thickness. Both models account for the fact that the PAM begins to produce
55
Figure 2.5: Blocked force vs. pressure; 1.9 cm (0.75-in) diameter, 18.5 cm (7.3-in)
length PAM with a 76.5 deg braid angle.
force at non-zero pressure. This pressure deadband is caused by bladder resistance,
and varies with bladder material and thickness. For the actuators tested in this work,
the activation pressure was found to be approximately 0.07 MPa (10 psi).
Note that the accuracy of both of these models is dependent upon braid angle,
which is somewhat difficult to measure and carries some uncertainty. Even small
variations in braid angle can produce large changes in force output. For instance, if
a lower braid angle was estimated on the PAM data shown in Figure 2.5 (74 deg
instead of 76.5 deg), the constant-thickness model would be more accurate than the
non-constant thickness model.
2.2.3 PAM Extension
As mentioned previously, PAMs are often used in antagonistic pairs. In some
cases, a bias contraction is applied and both PAMs are pressurized to continuously
56
maintain a desired hinge stiffness. However, in many systems, the agonist muscle
contracts while the antagonist extends past resting length (acting as a nonlinear
spring) to maintain high stiffness, eliminating the need for dual pressurization. In
such applications, an accurate model PAM extension is necessary for proper control
design and simulation.
Extension past resting length causes the braided sleeve to constrict around the
bladder until the PAM “necks down” at the ends. As mentioned earlier, previous
models have assumed a fully cylindrical shape. This assumption produces fairly
good estimates of PAM radius and bladder thickness (and hence, PAM force) during
contraction. However, during high bladder extension, this simple model calculates
bladder thickness to be greater than the predicted PAM radius, which cannot occur
as it is physically impossible. This inconsistency motivates a more detailed method
of calculating PAM radius and bladder thickness.
2.2.4 Elliptic Toroid Shape for Bladder Ends
McKibben-type PAMs display a physical shape characterized by a cylindrical
section in the axial center, encapsulated by two rounded ends. The braid and bladder
are clamped down at the ends so that the radius at each end of the active length
region is constant. When a pressurized PAM is in the contracted state, the ends of
the bladder exhibit convex curvature between the end fittings and the cylindrical
shape in the center. The outer area of the curved surface can be approximated
by the partial surface area of an elliptic toroid. An elliptic toroid is a surface of
57
revolution, similar to a ring torus, and is formed when an ellipse is fully revolved
around a central axis. The equation for such an ellipse is given by:
x(u, v) = cos u(a cos v + c)
y(u, v) = sinu(a cos v + c)
z(u, v) = b sin v,
(2.17)
where v is the angle inside the two-dimensional ellipse shape beginning at the minor
axis, and u is the angle as the ellipse rotates around the axis of revolution. As
illustrated in Figure 2.6(a), the cross section of a PAM between the end fitting and
the central cylindrical section can be closely approximated as a half-ellipse with
minor axis a and major axis b, offset from the longitudinal axis of the PAM by a
distance c. Experimental observations of multiple PAMs suggest that the part of the
braided surface at the edge where it protrudes from the end fitting is usually sloped.
The boundary angle of pi/3 rad (60 deg) with respect to the coordinate system of
the ellipse was selected as a reasonable approximation of the slope at the bladder
ends. Therefore, in this work we assume the surface of the ellipse shared by the
PAM surface runs from v ∈ [0, pi/3].
Alternatively, in extension, PAMs exhibit concave curvature near the ends
while maintaining the initial radius at the end fitting boundary. The radius of the
central cylindrical section is less than the resting radius, and the curvature at the
ends produces a different elliptic pattern. The outer surface of the PAM in the
curved region is now approximated by a part of the internal section of an elliptic
toroid. Figure 2.6(b) illustrates this cross-section.
58
(a)
(b)
Figure 2.6: Geometric model of PAMs in (a) contraction, (b) extension.
In both cases, the length from the center of the ellipse to the resting radius is
denoted as a′. The vertical distance from the end fitting edge to the center of the
ellipse is denoted as b′. These terms are a function of the ellipse radius r. Radius is
a function of the major and minor axes, and the angle with respect to the minor
axis; it is given by:
r =
√
a2b2
b2 cos2 v + a2 sin2 v
(2.18)
Substituting the proportionality constant k = b/a and solving for v = pi/3 or
v = 2pi/3, the radius is
r = a
√
4k2
k2 + 3
(2.19)
59
Using trigonometric relationships, we determine a′, b′, and c:
a′ = ±a
√
k2
k2 + 3
(2.20)
b′ = a
√
3k2
k2 + 3
(2.21)
c = R0 − a
′ = R− a (2.22)
By rearranging the expression for c and substituting for a′, we can find a:
a = ±
R0 −R
1−
√
k2/(k2 + 3)
(2.23)
2.2.5 Braid Length along the Surface of the Elliptic Toroid
The aforementioned force balance model requires a good approximation of
the central radius as PAM length varies to produce accurate results. Since braid
length B is constant, we can determine the radius R by finding a PAM shape for
which the braid length over the deformed PAM is equivalent to the resting braid
length. The total braid length is the sum of the braid length over each elliptic end
and the cylindrical center. First, the arc length s of the braid wrapping around the
elliptic ends is calculated. The length of a squared line element for a curve on a
60
three-dimensional surface is given by:
ds2 = dx2 + dy2 + dz2 (2.24)
Differentiating the equations of motion for an elliptic toroid, we determine the
infinitesimally small elements in rectangular coordinates as a function of those in
angular coordinates:
dx = − sinu(a cos v + c)du− a cosu sin vdv
dy = cosu(a cos v + c)du− a sinu sin vdv
dz = b cos vdv
(2.25)
leading to a total squared line element length in terms of angular coordinates:
ds2 = (a cos v + c)2du2 + (a2 sin2 v + b2 cos2 v)dv2 (2.26)
The slope of the braid as it wraps around the PAM bladder is assumed to
be constant, rotating a total of N times over the entire length. Hence, over one
elliptic end, the braid rotates N(b′/L) times, and the braid length in the u-direction
is 2piN(b′/L). Therefore, the average braid slope over the end arc (v = 0 to pi/3) is
given by:
du
dv
=
2piN(b′/L)
pi/3
=
6Nb′
L
(2.27)
In numerical simulations, v is varied over n small increments ∆v, and the
61
corresponding values of ∆u are determined from Eqn. 2.27. By calculating ∆s for
each increment from Eqn. 2.26, and summing over the curve, we obtain a good
approximation of s, the length of braid fiber over the curved surface:
s2 =
n∑
i=1
[
(a cos vi + c)
2(
6Nb′
L
)2 + (a2 sin2 vi + b
2 cos2 vi)
]
∆v (2.28)
The total braid length estimate, Best, is given by
Best =
Lc
L
√
L2 + (2piRN)2 + 2s (2.29)
where Lc = L− 2b′ is the length of the central cylindrical portion. By minimizing
the error B between the estimated braid length and the known resting braid length,
we numerically solve for PAM radius at a given strain:
B = |B −Best(R)| (2.30)
Due to the non-cylindrical shape of the PAM, the thickness of the bladder can
only be loosely approximated by Eqn. 2.4. Instead, we use numerical integration of
the curved surface to find the deformed bladder volume for a given thickness. From
the assumption of an incompressible bladder, the deformed thickness is the thickness
at which deformed volume is equal to resting volume. The integration is performed
using two curves corresponding to the inner and outer surface of the bladder, and
62
completing a solid of revolution about the longitudinal z-axis:
Vest = pi
L∫
0
∣
∣f 2(z)− g2(z)
∣
∣ dz (2.31)
where f(z) and g(z) are piecewise-continuous functions described by:
f(z) =
c+ a
√
1− ( z+b
′
b )
2
z ≤ b′
L ≤ L0
c+ a
√
1− ( z−(L−b
′)
b )
2
z > L− b′
L ≤ L0
c+ 2a′ + a
√
1− ( z+b
′
b )
2
z ≤ b′
L > L0
c+ 2a′ + a
√
1− ( z−(L−b
′)
b )
2
z > L− b′
L > L0
R
b′ < z ≤ L− b′
L ∈ R
(2.32)
g(z) = f(z)− t (2.33)
The error V between the resting volume and the deformed volume must be zero at
the exact deformed thickness. The error is given by:
V = |V − Vest(t)| (2.34)
Typical cross-sections of the bladder with calculated thicknesses are shown in
63
(a)
(b)
Figure 2.7: Cross-section showing estimated bladder thickness during (a) contraction
and (b) extension; 1.9 cm (0.75-in) diameter, 18.5 cm (7.3-in) length PAM with a
76.5 deg braid angle.
Figure 2.7, with the dashed black lines denoting the outer diameter at rest.
Using this procedure, the radius was calculated as a function of contraction,
shown in Figure 2.8(a). This new model is compared with radius estimates from
the fully cylindrical model and experimental data. To obtain experimental radius
measurements, the sample PAM was displaced (extended past resting length and
contracted) by a load frame and radius was measured along the central cylindrical
portion with calipers. The experimental data matches with the elliptic ends model
more accurately than the fully cylindrical model in both extension and contraction.
In extension, the central radius of the PAM under the elliptic tip model remains above
0.76 cm (0.3 in). This allows the bladder thicken, as shown in Figure 2.8(b), without
being physically restricted by the braid. On the other hand, the fully cylindrical
PAM model predicts that PAM radius collapses so quickly that at around 0.1 in of
extension past resting length, there is not enough space for the bladder to fit inside.
It is important to note that the bladder radius and thickness measurements
are dependent on the ellipse axis ratio k, which defines the eccentricity of the ellipse.
The most accurate results are produced when k is not simply a constant, but a
function of actuator strain. The optimal relationship was determined empirically,
64
(a) (b)
Figure 2.8: Comparison of cylindrical and elliptic tip models (a) radius vs. contraction
and (b) thickness vs. contraction; 1.9 cm (0.75-in) diameter, 18.5 cm (7.3-in) length
PAM with a 76.5 deg braid angle.
and was found to follow the piecewise linear relationship:
k =



1.64− 0.13∆L L ≤ L0
1.64− 0.33∆L L > L0
(2.35)
Figure 2.9(a) displays the eccentricity as a function of actuator strain for the
sample PAM. Note that the decreasing eccentricity with increasing contraction is
observed experimentally, with the ends becoming more hemispherical as the PAM
radius expands. Figure 2.9(b) illustrates the length of the braid following the elliptic
curve changing with PAM radius. The braid length on the ellipse is zero at resting
radius (0.95 cm [0.375 in]) because the PAM is assumed perfectly cylindrical. The
predicted braid length on the elliptic surface is higher during radius expansion than
radius contraction of an equal amount because the braid traverses a larger-area
inflated surface during radius expansion, as opposed to a necked down surface when
the radius contracts.
Applying this new PAM geometry to the force balance model derived by
65
(a) (b)
Figure 2.9: Elliptic tip model parameters (a) tip eccentricity and (b) braid length
along the elliptic tip; 1.9 cm (0.75-in) diameter, 18.5 cm (7.3-in) length PAM with a
76.5 deg braid angle.
Hocking and Wereley [27] produces a force vs. contraction profile that is much more
accurate in both extension and contraction. Figure 2.10 shows the modeled force vs.
contraction cycles for a range of pressures (0.07 to 0.8 MPa [10 to 120 psi]) compared
to experimental data of the averaged force. Figure 2.11 shows the same data, but
in its original, non-averaged form that contains hysteresis. The model can easily
capture this hysteresis with the addition of a friction term to the total force. The
friction force is given by:
FF = −cFF (2.36)
where cF is a friction coefficient determined empirically as in Hocking and Wereley [27].
The friction coefficient is a function of pressure, where for this example PAM
cF = 0.16 − 0.145P , P in MPa. It should be noted that other approaches for
modeling such hysteresis, based on viscoelastic mechanisms such as the Maxwell-slip
model [118], also add similar friction or slip terms to the analysis.
66
2.2.6 Linear Relationship between Hyperelastic Constants and Pres-
sure
The practicality of this analytical model is important. While pure interpolation
of results can lead to fairly good estimates when there is enough data available,
a great deal of characterization data and computational power is required. Even
after accounting for non-constant bladder thickness and complex PAM geometry,
the current force balance model is not suitable for interpolation without exhaustive
characterization data at small increments of pressure and contraction because the
relationship between elastic moduli Ek and pressure can be nonlinear. However, if
the optimization procedure is modified to guarantee a linear relationship between the
elastic moduli and pressure, smooth transitions between experimental pressure values
could be found, and interpolation would require less experimental data. Observation
of previous data (Figure 2.4) illustrates that a linear fit between the Ek constants
and pressure is nearly optimal at higher pressures. In the low pressure range, the
linear constraint will no longer produce optimal results; however, as the next section
illustrates, results were found to retain most of their accuracy.
A modified optimization procedure was adopted in order to achieve a linear
relationship between the elastic moduli and pressure. Instead of directly optimizing
Ek values for each pressure level and subsequently choosing a linear fit, a new set of
parameters were employed to optimize for all pressure levels simultaneously. The
parameters were the 2M constants mk and bk, which correspond to the slope and
y-intercept of the Ek = mkP + bk functions of an Mth-order polynomial. The
67
Figure 2.10: Force vs. contraction, nonlinear force balance model with non-constant
thickness and elliptic tip geometry; 1.9 cm (0.75-in) diameter, 18.5 cm (7.3-in) length
PAM with a 76.5 deg braid angle.
Figure 2.11: Force vs. contraction with hysteresis estimates, nonlinear force bal-
ance model with non-constant thickness and elliptic tip geometry; 1.9 cm (0.75-in)
diameter, 18.5 cm (7.3-in) length PAM with a 76.5 deg braid angle.
constants were varied using the fmincon optimization function in MATLAB until
the mean squared error was minimized for the entire set of experimental data.
68
2.3 Results and Analysis
Employing each of the improvements from the previous section, Model pre-
dictions of force vs. contraction using stress-strain relationships from first-order to
fourth-order are shown in Figure 2.12. The model is shown to accurately capture
all the extension behavior, while the predictive ability becomes increasingly more
accurate as the order of the stress-strain polynomial is increased. Figure 2.13 more
clearly illustrates the model error for each experimentally-tested pressure level. The
normalized mean squared error F,norm is given by:
F,norm(P ) =
√
1
n
n∑
i=1
Fa,i(P )− Fm,i(P )
F−2.75%(P )
(2.37)
where n=100 is the number of data points for each pressure level, Fa,i(P ) is the
averaged experimental force, Fm,i(P ) is the model-predicted force, and Fa,−2.75%(P )
is the experimental force at 2.75 % extension, i.e. the maximum force obtained,
for each pressure level. In nearly all cases, increasing polynomial order decreases
error. Furthermore, the fourth-order polynomial error is the lowest for each tested
pressure except 30 psi. Also notable is that the first- and second-order functions
exhibit higher error at high pressures, while the fourth-order model exhibits higher
model error at low pressures. Clearly, a fourth-order polynomial is much better at
capturing PAM force-contraction behavior at high pressure.
Figure 2.14 displays the empirically-determined modulus values, Ek, as a
function of pressure. The E1 term has a fairly consistent slope and y-intercept in
69
(a) M = 1 (b) M = 2
(c) M = 3 (d) M = 4
Figure 2.12: Modulus values optimized to vary linearly with pressure using nonlinear
force balance model with non-constant thickness, (a) first-order, (b) second-order,
(c) third-order, and (d) fourth-order polynomial model for strain; 1.9 cm (0.75-in)
diameter, 18.5 cm (7.3-in) length PAM with a 76.5 deg braid angle.
each of the plots, indicating the fundamental actuator characteristics are captured
with a linear stress-strain relationship. In fact, uniaxial tests of the bladder alone
indicated a linear stress-strain relationship at strains up to 200%. However, the
higher-order terms play an important role in accurately modeling high pressure and
biaxial high strain behavior. The improvement in force prediction gained by increasing
the order of the polynomial suggests there are complex bladder-braid interactions
being lumped into the bladder stress-strain relationship. These interactions are
separate from those that cause the hysteretic behavior displayed in Figure 2.11 and
warrant future investigation.
70
Figure 2.13: Normalized mean squared error of model force predictions for Mth-order
polynomial strain models; 1.9 cm (0.75-in) diameter, 18.5 cm (7.3-in) length PAM
with a 76.5 deg braid angle.
Figure 2.15 illustrates the stress-strain curves produced by the fourth-order
polynomial fitting of the example actuator at multiple pressure levels, showing that
increasing pressure increases the slope of the curve for low strains (ε < 2). The
intrinsic properties of the bladder material do not change in the presence of pressure;
however, this increased pressure, which increases contact force with the braid, may
cause the bladder to form a high-friction quasi-bond with the braid, stiffening the
actuator and resisting bladder expansion until high contraction is reached. Then,
when the PAM actuator is undergoing very large circumferential strains (ε ≥ 2),
a softening in the bladder material is observed, including a negative stiffness at
high pressure. Modeling this softening effect requires higher order Ek terms to
correctly match force-contraction data. This phenomenon is not expected in a
hyperelastic material, because while hyperelastic models often display a temporary
71
(a) M = 1 (b) M = 2
(c) M = 3 (d) M = 4
Figure 2.14: Force vs. contraction, nonlinear force balance with non-constant
thickness, (a) first-order, (b) second-order, (c) third-order, and (d) fourth-order
polynomial model for strain; 1.9 cm (0.75-in) diameter, 18.5 cm (7.3-in) length PAM
with a 76.5 deg braid angle.
plateau in stress levels, no known models predict stress decreasing with increasing
strain. Therefore it is likely that this effect is caused by the braid interacting with
the bladder. Further study is required to gain a better physical understanding of
this phenomenon.
Lastly, the improved model (including non-constant thickness, modified PAM
geometry, and a linear relationship between elastic moduli and pressure) can be
shown to accurately predict force vs. contraction curves at a wide range of pressures
using data at only a few pressure levels, thereby substantially reducing the experi-
72
(a) (b)
Figure 2.15: Stress-strain relationship in the (a) axial and (b) circumferential
directions, as determined by polynomial fitting; 1.9 cm (0.75-in) diameter, 18.5 cm
(7.3-in) length PAM with a 76.5 deg braid angle.
mental time required to parameterize an accurate model. Figure 2.16 illustrates this
predictive ability. The linearly-varying Ek values were calculated using experimental
data at 0.14, 0.41, and 0.83 MPa (20, 60, and 120 psi), shown in black. The model
was then interpolated to other pressures (0.07, 0.2, 0.28, 0.55, and 0.69 MPa [10,
30, 40, 80, and 100 psi]), shown in red. As the figure illustrates, the force is still
predicted extremely well for all pressures. Figure 2.17 compares the normalized mean
squared error, as calculated in Eqn. 2.37, for the model based on a linear Ek(P )
function from the full set of force-contraction data against the model based on a
linear Ek(P ) function from the reduced set. By reducing the set of experimental
pressures used in the fit, error only increases significantly at 0.07 MPa (10 psi) (likely
because the curve was extrapolated rather than interpolated) and shows little change
at the other tested pressure levels. This data suggests that the model can be used in
a predictive manner once a PAM has been initially characterized.
73
2.4 Conclusion
An improved model of pneumatic artificial muscle behavior was developed
for antagonistic actuation schemes. Several refinements were applied to the force
balance model in order to capture nonlinear PAM force behavior in contraction and
extension. The new model combines the effect of non-constant bladder thickness with
a hyperelastic bladder stress-strain relationship. A detailed geometric representation
of PAM shape incorporating elliptic deformation at the ends was developed to improve
model force predictions, particularly for PAM extension. Lastly, the polynomial
coefficients of the stress-strain relationship were constrained to vary linearly with
pressure, augmenting the ability to predict behavior at untested pressure levels while
preserving high model accuracy.
Analysis of the optimized PAM modulus terms suggests a stiffening effect as
pressure increases; however, the effect begins to weaken at around 200% circumfer-
ential strain, where a softening effect takes hold. While the original force balance
model accounts for the shape of the braid and energy storage in the bladder, it
does not attempt to model complex bladder-braid interactions (besides hysteresis
effects). Therefore, the nonlinear phenomena in the actuator stress-strain relationship
are likely attributable to these unmodeled interactions. Further studies must be
undertaken to better understand these effects and to pursue additional modeling
improvements.
Given that this improved model demonstrates high accuracy in quasi-static
experiments, it can be reliably employed into a model-based control strategy in
74
Figure 2.16: Experiment vs. model using fourth-order polynomial model for strain
and Ek(P ) interpolated from data of three experimental pressure levels ; 1.9 cm
(0.75-in) diameter, 18.5 cm (7.3-in) length PAM with a 76.5 deg braid angle.
PAM-based applications. While high frequency, demanding applications may include
unmodeled dynamics that were not captured by this model, such as nonlinear airflow
dynamics and damping behavior at high velocity, the model can be augmented with
additional terms, or the control strategy can incorporate closed-loop feedback.
75
Figure 2.17: Normalized mean squared error of model predictions, comparing Ek(P )
interpolated from data of all eight experimental pressure levels vs. only three
experimental pressure levels (0.14, 0.41, and 0.83 MPa [20, 60, and 120 psi]); 1.9 cm
(0.75-in) diameter, 18.5 cm (7.3-in) length PAM with a 76.5 deg braid angle.
76
Chapter 3: Structural Design of PAM-Based Robotic Manipulators
3.1 Introduction
Pneumatic artificial muscles (PAMs) are compliant actuators featuring an
elastomeric bladder surrounded by a braided sleeve. Applying pressure inside the
soft bladder (i.e., inflation) increases the diameter and decreases the length of the
actuator through reorientation of the stiff braid fibers, generating a contractile stroke
and pulling force similar to human muscle. Also known as McKibben actuators,
these were initially developed as orthotic devices for polio patients [23].
PAMs have several advantages over other forms of conventional actuation
technology. For example, they are extremely lightweight (10s of grams) and simple,
as they are only composed of a thin bladder, braid fibers, and small lightweight end
fittings. Accordingly, they are highly scalable to various physical sizes and force-
stroke requirements with a host of inexpensive, commercially available materials.
Additionally, PAMs can achieve much higher power-to-weight ratios than electrical
and hydraulic actuators [155]. They are capable of producing high forces at high
speed [162]. No gears are necessary, thus there is no inertia or backlash added to the
system. Other benefits of these devices include durability, reliability [2], operation
without precise mechanical alignment, and high force capability at comparably low
77
operational pressures (0 to 1.4 MPa [0 to 200 psi]). Operating with air also enables
use of an open fluid circuit, where return air is vented to the atmosphere and a
small compressor may be employed to extract air from the outside environment to
replenish the air lost.
As a result of their functional similarity to natural muscles, there have been
numerous studies that focused on applicability to robotic systems, including manipu-
lation and joint control of robotic arms [40, 157, 160] and legs [69, 132], or human
assistance and rehabilitative devices [75,80,152]. PAMs are especially suitable for use
in robots intended to physically interact with human subjects because their inherent
compliance increases safety.
The development of PAM-actuated systems required mathematical models
to help characterize PAM behavior. In 1958, Gaylord derived a simplified model
which considers only the geometric parameters of the actuator and assumes that
the actuator retains a cylindrical shape. More recently, studies have introduced a
nonlinear Mooney-Rivlin term to account for elastic energy storage [32], and other
terms to describe tip effects resulting from the non-cylindrical shape at the ends
of the actuator [33]. Other models employ a force-balance approach [34]. Due to
their compliance (from gas compressibility) and nonlinear behavior, PAMs have
the disadvantage of being difficult to accurately position in open-loop, but recent
advancements in modeling have enabled more reliable positioning and control [35,156].
Presently, there is significant interest in deploying robotic platforms to perform
duties that could be hazardous for humans. One particular application is casualty
extraction. In this case, the substitution of a robot for recovery personnel eliminates
78
the threat of harm to the rescuer(s). Some casualty extraction concepts employ
existing military robots to tow a casualty laid on a stretcher [154]. However, in some
situations, it may not be possible for the injured person to board the stretcher without
additional assistance. Furthermore, dragging the casualty across rugged terrain may
cause even more injury. A different approach is a robotic system that can lift and
carry the casualty with minimal or no required support from the casualty. The
benefits of using PAMs in this type of platform include the weight savings accrued
by utilizing lightweight PAM components as opposed to conventional hydraulic
actuators and systems, low pressure air (90-150 psi) as opposed to high pressure
(on the order of 20 MPa [3000 psi]), no hydraulic fluid, and reduced maintenance
requirements. As mentioned previously, the compliance of PAM actuators is also
beneficial for interacting with humans, such as lifting and carrying them to safety.
Other applications for such a robot include manipulating or moving heavy packages,
explosives, and hazardous substances, and patient placement in hospitals.
The goal of the present work is to identify requirements for a lightweight, high
force robotic manipulator actuated with pneumatic artificial muscles, design the
system for heavy lifting capability, and assemble a prototype arm that demonstrates
its abilities in a laboratory environment. Two actuation approaches are considered
using different sizes and numbers of PAMs for each, and they are compared to
determine the more efficient actuation scheme. A demonstrator arm is designed,
fabricated, and experimentally tested. A model characterizing PAM behavior is
coupled with a kinematic model for the manipulator and is validated by experimental
measurements in quasi-static and dynamic tests.
79
3.2 Design Requirements and Simulation
3.2.1 Design Requirements
Key requirements for a high-force robotic manipulator are torque, speed, and
range of motion. Many robotic arms currently used in military applications have
relatively low payload capacities, though they are typically smaller robotic platforms
than what is currently under consideration. An example is the Foster-Miller TALON
Engineer robot, a four degree-of-freedom bomb disposal robot that can handle
a maximum payload of 30 kg (65 lb) [161]. The latest version of the un-fielded
Battlefield Extraction Assist Robot (BEAR) by Vecna Robotics, whose primary
duty is casualty extraction, can lift over 225 kg (500 lb) using its two hydraulic
arms [6]. The arm designed in the present study was intended to be a proof-of-concept
demonstrator of similar lifting capabilities using PAM actuation. The manipulator
prototype was chosen to have two degrees-of-freedom (elbow pitch and wrist pitch)
across three links (upper arm, lower arm, hand). Elbow and wrist torque of 200
ft-lb per arm was estimated to be sufficient for two arms to lift a 95th percentile
male soldier [159] wearing an additional 100 lb of gear, totaling 316 lb. The range
of motion was chosen to be that of a full lifting motion, or 90 deg of rotation. The
desired speed is 57 deg/s (1 rad/s), or enough to complete this lifting motion in less
than 2 seconds. Design goals for both degrees-of-freedom are summarized in Table
3.1. The first two (arm) links are 13 inches long, and the hand manipulator is 7
inches long.
80
Table 3.1: Prototype design goals.
Torque at max pressure Range of Motion Angular Rate
200 ft-lb (271 Nm) 90 deg 57 deg/s
A secondary objective of the joint actuation system design was to minimize
volume, as space is a valuable commodity in robotic systems, and some PAMs can
inflate to more than triple their resting diameter during operation near their free
contraction limit. No specific values of arm cross-sectional area were chosen, but
design comparisons included minimizing volume as an important goal.
3.2.2 Actuator Development and PAM Characterization
Once the design goals were determined, PAMs of varying size were fabricated
and characterized in order to compare key actuation characteristics, where the
leading solution would be chosen to maximize force and speed capabilities while
minimizing weight and volume. It is commonly known that larger diameter PAMs
produce higher force capability than smaller diameter PAMs [35]. Likewise, design
guidelines from past experimental studies indicated that braid fibers aligned closer
to the longitudinal axis (high braid angle) lead to larger contraction and force [138].
Although this increased force does come at the expense of larger radial expansion,
which could lead to larger volume, the first approach considered implementation of
a few larger diameter PAM actuators. This is also due to fact that the heavy-lift
robot arm would demand higher forces than any previously constructed by the
research team or any that are available commercially. An outer diameter (OD) of
81
2-in was selected for this case, also having high braid angle. These PAMs were
fabricated in-house with commercial braid, but a custom bladder had to be fabricated
to match desired stiffness at that size. The bladder was constructed by pouring a
low-viscosity addition-cure silicone rubber (Bluestar V330) into a cylindrical tube
mold and allowing it to cure at room temperature. Two end fittings were integrated
onto the ends of the bladder, where one had a thru-hole to allow the passage of air
and the other was closed. The braided sleeve (fiberglass) was placed around the
bladder and end fittings and hose clamps were used to hold the bladder and braid
onto the end fittings. Epoxy was also applied to seal and strengthen the ends. Note
that this hose-clamp construction method is only used on preliminary laboratory
prototypes. The second approach was to consider several smaller diameter PAMs,
whose combined force would be on the same order as the fewer larger diameter PAMs
of the first approach. In this case, 0.625-in outer diameter PAMs were fabricated
using existing in-house techniques. Much higher braid angles were briefly considered
with these smaller diameter PAMs, but as studies have shown, this led to braid
instability and lower fatigue life [104]. Therefore, a more moderate braid angle was
selected to ensure actuator reliability.
Force vs. displacement cycles for one of the 2-in OD PAMs are shown in Figure
3.1. The behavior is nonlinear and increases with applied pressure. There is also a
distinct hysteresis; higher forces are produced during axial extension than during
contraction. This PAM, with a braid angle of approximately 77 deg, attained a
blocked force of over 4000 lb at 90 psi. The blocked force is defined as the force
generated in the actuator from the internally applied fluid pressure when the two ends
82
Figure 3.1: Characterization of a 2-in OD, 13-in long PAM.
are held in place at the resting length of the actuator; it is a common performance
metric. As the PAM is allowed to contract, force decreases nonlinearly until reaching
full contraction (where force equals zero). This point is defined as the free contraction
for a given actuator and is used as another performance metric. This 2-in OD PAM
contracted to 61 percent of its original resting length. Figure 3.2 shows the 2-in OD
PAM in (a) its blocked condition and (b) in a contracted/inflated state.
The 0.625-in OD PAMs were inflated to a maximum of 150 psi instead of 90 psi.
This was allowable because mature end fitting production technologies were available
with 0.625-in OD PAMs, and previous burst tests demonstrated no catastrophic
failure up to 1000 psi. To validate the robustness at 150 psi, one of these full-braid
PAMs was pre-tested at 150 psi with full contraction for over 3,000 cycles, and
exhibited no change in braid structure or force characteristics. The blocked force for
the 0.625-in OD PAM was approximately 820 lb at 150 psi. Free contraction length
was 63 percent of the resting length. These test results are shown in Figure 3.3.
83
(a) (b)
Figure 3.2: Testing of a 2-in OD, 13-in long PAM (a) resting, (b)contracted.
Figure 3.3: Characterization of a 0.625-in OD, 7.8-in long PAM.
84
3.2.3 PAM Comparison
A summary of the details for the two PAM actuator configurations considered
in this study are listed in Table 3.2. This is largely due to the weight of the end
fittings, which were not weight-optimized for the 2-in OD PAM. In order to meet
torque requirements, it was estimated that at least six 0.625-in OD PAMs and
at least two 2-in OD PAMs would be needed based on the force and contraction
profiles. Six 0.625-in OD PAMs have a lower total air volume (both resting and fully
inflated) than two 2-in OD PAMs, which implies a faster response time and a lower
volume requirement on the compressed air source during operation. When packed
in a circular arrangement around the center of the arm, six 0.625-in OD PAMs will
occupy a smaller volume than two 2-in OD PAMs, leading to a more compact design.
Increasing the number of PAMs also allows for more redundancy; if one PAM fails, an
emergency valve could close and the other PAMs would still be able to support their
proportion of the load. These factors help highlight the advantages of using multiple
0.625-in OD PAMs in the prototype device. Despite the increased complexity of
using a larger number of PAMs, the benefits of increased speed, better packaging,
and lower air requirements were significant enough to warrant the use of 0.625-in
OD PAMs in the prototype design.
3.2.4 Robot Arm Simulation
In order to approximate the speed and response capabilities from a particular
robot arm or joint, an analysis of arm kinematics was performed based on the design
85
Table 3.2: Comparison of different-sized PAMs.
Parameter 0.625-in OD PAM (150 psi) 2-in OD PAM (90 psi)
Braid Angle (deg) 72.5 78
Max Diameter (in) 1.75 4.5
Blocked Force (lbf) 820 4280
Mass (lbm) 0.17 2.2
Specific Blocked Force (lbf/lbm) 4280 1950
Figure 3.4: Parameters for design analysis.
shown in Figure 3.4. There are two links connected by a one degree-of-freedom joint
and the angular velocity of the extended forearm is of interest, as it is rotated at
the elbow joint by PAMs located in the upper arm region. The PAMs create the
force responsible for accelerating the forearm link as internal pressure builds up, and
this pressure increase is driven by the flow rate of air into the actuator. As pressure
increases in the PAMs, they contract and produce force.
A joint was devised in which an L-shaped member rotates around a hinge and
is connected to another L-shaped member by a “tendon,” as illustrated in Figure
86
Figure 3.5: Concept for variable moment arm joint.
3.5. The member on the same side as the PAM elongates by a distance of ∆l as the
PAM contracts by that same amount on the other side of the mobile plate. Force is
transmitted by the tendon from the mobile PAM side to the other immobile side
of the joint. From this geometry, the length of the moment arm increases as PAM
force decreases, which allows the joint to generate torque at a relatively constant
force level over the large range of angular motion.
From Figure 3.6, the hypotenuses of the right triangles formed between each
L-shaped member are given by:
AO =
√
(l0 + ∆l)2 + r20 (3.1)
BO =
√
l20 + r
2
0 (3.2)
87
Figure 3.6: Geometry of robotic arm joint.
where it should be noted that AO changes with ∆l as the PAM actuators contract
and BO is constant. The angle between AO and the upper link is:
θ1 = tan
−1
(
r0
l0 + ∆l
)
(3.3)
where l0 is the constant length between the joint pivot and the actuator plates at
rest and r0 is the linkage offset distance on the actuator plates from the aligned arm
link axis.
To determine the component of the PAM force that acts to rotate the joint,
we begin by using the above quantities to solve for the angle of the AOB triangle,
88
which is given by:
cos (θ1 + θf ) =
l2T + AO
2 −BO2
2lTAO
(3.4)
From here, the angle through which the PAM force acts is computed as:
θf = cos
−1
(
l2T + AO
2 −BO2
2lTAO
)
− θ1 (3.5)
which leads to the component of the PAM force that is available to rotate the joint:
Feff = Fp cos θf (3.6)
where Fp is the pulling force of the PAM actuators, and this force leads to the
effective radius about which the force component acts
reff = AO sin θ1 + θf (3.7)
This is then used to compute the torque in the joint:
τPAM = reffFeff (3.8)
Having established these expressions, a joint torque analysis was conducted.
Various geometries for each member dimension were simulated for the purpose of
maximizing the range of motion between 0 and 90 deg where torque is above 200
ft-lb (the typical range of motion of the robot under consideration here). The final
89
Figure 3.7: Torque output vs. angle of rotation.
lengths were chosen as l0 = 1.56-in, r0 = 0.42-in, and lT = 3.04-in. The analysis
estimated that this was achievable from 0 to 78 deg when using six 0.625-in diameter
PAMs, as shown in Figure 3.7. It was decided that this torque output was adequate
for the demonstrator. Using greater than six PAMs was considered, but it only
modestly increased joint range of motion.
For dynamic response analysis, the volume of the PAM was assumed to be of
cylindrical shape, with an increasing diameter and decreasing length as follows:
VPAM =
piD2L
4
(3.9)
L is the contracted length given by
L = λL0 (3.10)
90
Table 3.3: Geometric parameters for PAM force model.
L0 (in) D0 (in) N B (in) θb (deg)
7.785 0.625 1.25 8.16 72.5
where L0 is the resting length, and D is the inflated diameter, given by:
D =
√
B2 − L2
Npi
(3.11)
In this equation, N is the number of circumferential turns of a braid fiber around
the actuator, given by:
N =
L0
piD0 tan θb
(3.12)
where D0 is the resting diameter, θb is braid angle, and B, the length of a single
braid fiber, is given by:
B =
L0
sin θb
(3.13)
Values for the numerical parameters used in this model are listed in Table 3.3.
Note that PAM volume does not remain completely cylindrical as the PAM
contracts, so the values resulting from this first analysis were conservative estimates
of performance. The following equation predicts steady mass flow rate of the air
91
through a round orifice [163]:
m˙air =



ACdPus
√
2
RTC1,
Pds
Pus
< Pcr
ACdPus
√
2
RTC2,
Pds
Pus
≥ Pcr
(3.14)
where A is the valve orifice area, Pu and Pd are absolute pressures upstream and
downstream of the valve, respectively, R is the gas constant for the medium (air),
and T is the temperature upstream of the valve, though the process was assumed
to be adiabatic. The term Cd is a discharge coefficient, capturing the losses in the
orifice. Critical pressure is given by:
Pcr =
2
γ + 1
γ
γ−1
(3.15)
where C1 and C2 are pressure and medium-dependent flow terms for subsonic and
sonic flow, respectively:
C1 =
√
γ
γ−1
[(
Pds
Pus
) 2
γ
−
(
Pds
Pus
) γ−1
γ
]
C2 =
(
2
γ+1
) 1
γ−1
√
γ
γ+1
(3.16)
Richer and Hurmuzlu [158] derive an expression for pressure change in a
container of changing volume using the ideal gas law, conservation of mass, and
conservation of energy:
P˙ =
RT
Vair
(αinm˙in − αout ˙mout)− αP
V˙air
Vair
(3.17)
92
where V is the actuator volume, and the coefficients a, ain, and aout are specific heat
ratios for the different processes. Pressure at any time can then be calculated by
integration:
P (t) =
∫ t
0
P˙ (σ)dσ (3.18)
Knowledge of the pressure inside the PAM actuator is required to determine
the force. Various quasi-static models have been proposed in order to capture the
force-contraction behavior of PAMs, and a comprehensive review of these models
is available [35]. The model employed in the present work takes into account the
Gaylord force with additonal Mooney-Rivlin correction terms to represent elastic
energy storage in the elastomeric bladder. The model estimates force-contraction
behavior as a function of the material properties of the component parts (Kevlar braid
and latex bladder), PAM geometry, braid angle, and input pressure. As previously
mentioned, the Gaylord force only considers the kinematic relationships of the braid
and assumes there is no loss or energy storage:
FGaylord =
P
4piN
(
3 (λL0)
2 −B2
)
(3.19)
When the Mooney-Rivlin term is included, the equation for the PAM force is given
as:
F = FGaylord − FM−R (3.20)
93
F =
P
4piN
(
3 (λL0)
2 −B2
)
− Vb
dW
dL
(3.21)
where Vb is the volume of the bladder material and W is the nonlinear relationship
between stress and strain based on geometric and material properties of the rubber.
Angular acceleration is given by:
α(t) =
∑
i τi(t)
I
(3.22)
where τi are the torques and I is the moment of inertia. Once the angular
acceleration is known, the angular velocity can be computed through integration:
ω(t) =
∫ t
0
∑
i τi(t)
I
dt (3.23)
Using this speed analysis development, open-loop simulations were performed
on the elbow joint for the chosen robot arm configuration. In the present case,
the wrist joint was assumed to be non-rotating and elbow rotation was the only
active degree-of-freedom. Figure 3.8 is a simulated time history of a pressure step
response for the lower limb with a 100 lb tip payload. The limb rotated 90 deg with
a maximum pressure of 150 psi. The data indicates a maximum angular velocity of
200 deg/s, and the average angular velocity in completing the motion is 60 deg/s,
which meets the criterion for angular velocity greater than 57.3 deg/s (1 rad/s).
94
Figure 3.8: Simulation of step input, elbow joint.
3.2.5 Proof-of-Concept Manipulator Construction
Figure 3.9 depicts a computer-aided design (CAD) model of the manipulator
when the wrist joint is fully extended and the elbow joint is fully contracted. Only
one PAM is shown installed for each degree-of-freedom to highlight other details of
the design. The prototype consists of two steel main arm links that are connected
by hinge joints. Around the arm links are flat plates. The upper plate is stationary
(immobile), while the lower plate is mobile due to linear bearings that make contact
with the arm. PAMs are attached to each plate, so that, as the PAMs contract,
the mobile plate, which is connected to the next arm link by a “tendon,” pulls the
link and forces it to rotate. The plates have attachment areas for up to 10 PAMs
on each degree-of-freedom. Besides satisfying range of motion, velocity, and torque
requirements, the manipulator should be as compact as possible and must withstand
95
Figure 3.9: CAD model of PAM manipulator.
the stresses of lifting heavy payloads. The 0.625-in OD PAMs were packed in a tight
arrangement around the steel rods, but given enough space to safely inflate to their
maximum diameter (1.75-in). All of the links are sized to avoid over-stressing the
joints at maximum torque with a safety factor of over 1.5. Additionally, the main
arm links are sized so that the arm will not bend more than 0.5-in when extended
in the horizontal direction and under a 500 lb tip payload. The hand link has a
0.5-in diameter hole at the tip through which Kevlar rope can be passed, which can
connect a hanging payload to the arm for testing.
A simple, volume-efficient method for delivering air to the multiple PAMs was
achieved by fabricating manifolds. A series of holes was bored through the upper
stationary plates connecting each set of PAMs to the arm. All of the inlet holes except
one were sealed. The manifolds allowed for only one air hose per degree-of-freedom.
The shoulder was composed of two rotating plates and could be adjusted in the pitch
and yaw directions. The shoulder on this prototype allows for manual adjustments
in 45 deg increments, but remains stationary during arm manipulation.
96
3.3 Test Setup
3.3.1 Fabrication of Test Setup
A total of twelve PAMs were fabricated with six PAMs being installed around
each of the two arm links in a hexagonal arrangement. The fully assembled pro-
totype hardware is shown in Figures 3.10 and 3.11 in the deflated/extended and
inflated/contracted conditions, respectively. The arm was constructed such that
when fully extended, the upper arm link was hanging 0 deg from vertical, the lower
arm link was directed 10 deg forward from vertical, and the hand link was directed
20 deg forward from vertical. For testing purposes, additional control and sensor
components were added to the system. Angular rotation sensors (Midori, QPC series)
were attached to the elbow and wrist joints to measure the rotation angle, and a
pressure transducer (Omega, PX-209 series) was connected to the incoming air line
to measure PAM pressure. Voltage-controlled pressure regulators (Parker, P3HP
series and Festo, MPPES-3-1) were controlled separately for both degrees-of-freedom.
3.3.2 Testing and Experimental Results
Operational experiments were conducted to examine the manipulator perfor-
mancewithdifferentdegrees-of-freedomactive.Theseincludedactuatingthehand
about the wrist joint only, actuating the lower arm about the elbow joint only, and
actuating both links simultaneously. In these experiments, different loads were hung
from the tip of the hand manipulator. The maximum payload tested was 380 lb. In
97
Figure 3.10: PAM arm extended during testing.
Figure 3.11: PAM arm contracted during testing.
98
each test, rotation data was gathered as the pressure to the PAMs was increased.
At maximum pressure, the maximum rotation angle was observed. Torque was
determined at this angle. With multiple tests, torque was experimentally determined
over most of the range of motion for both joints. It should be noted that due to
limits on the pressure regulators and air tubing used in the test setup, the maximum
pressure available in quasi-static experiments was 150 psi and the maximum torque
was limited accordingly.
Figure 3.12 shows measured time histories for PAM pressure and joint rotation
angle with a tip payload of 135 lb. The pressure was controlled open-loop by a
triangular wave function of 0.05 Hz input to the corresponding pressure regulator.
As pressure increases to approximately 150 psi, joint rotation increases as well.
Note that the figure shows data collected from actuating the elbow joint only. The
rate of rotation increases significantly once the bladder begins to expand radially
(as the pressure overcomes its elastomeric stiffness) and decreases to zero as the
PAM reaches maximum contraction. Due to hysteresis present in the PAM force-
contraction behavior, it can also be noticed that the rotation between the two sides
of the triangular wave is slightly asymmetrical.
Figure 3.13 shows the torque generated as a function of the time-varying
pressure in the elbow joint. Static torque in the joint was calculated by
τ = Fpayloadrpayload sinφ (3.24)
where φ is the angle between the arm link direction and the gravity vector, Fpayload
99
Figure 3.12: Elbow PAM pressure and rotation vs. time.
Figure 3.13: Elbow joint torque vs. pressure.
100
is the force exerted on the hand link due to the payload mass, and rpayload is the
moment arm extending from the joint center to the point of payload attachment.
The figure shows that the available torque increases with pressure. As expected,
there is also hysteresis between the contraction and extension portions of this curve.
3.4 Analysis
The accuracy of the system model relies heavily on its ability to predict Pneu-
matic Artificial Muscle behavior. Thus, experimental results from PAM characteriza-
tion were compared with the actuator model. Figure 3.14 depicts the experimental
and model-predicted PAM force at various pressures. The force-contraction profile
of the model at 150 psi matched with the lower (contractile) part of the curve.
However, the model of PAM force at lower pressures (30, 60, 90 psi) followed or
slightly over-predicted the upper (extensile) part of the curve. This model does not
account for hysteresis in the experimental data, which exists because of increased
braid and bladder friction during the extension/reducing diameter phase of the test,
as well as reduced resistance in the bladder. A PAM undergoing inflation has more
elastic energy stored than a PAM undergoing deflation, a characteristic not accounted
for in the Mooney-Rivlin term, which predicts force only for inflation/contractile
motion. Therefore, the model is more accurate at 150 psi than at lower pressures. If
future applications for this manipulator demand higher accuracy at low pressure,
corrections to the model should be considered.
As validation of the kinematic model for the robot arm, the maximum torque
101
Figure 3.14: Experimental and predicted force vs. displacement.
output from the experiments was compared to model predictions. As shown in Figure
3.15, the experimentally measured values for torque in both the elbow and wrist
joints are well-predicted by the model. Note also that the difference between the
torque curves for the elbow and wrist joints arises from the resting angle in relation
to the vertical direction.
Experiments were also performed to validate the dynamics of the system model.
These tests consisted of a 4 s ramp input followed by a 6 s hold of maximum pressure.
Figure 3.16 shows the model-predicted and experimental results using a 100 lb
payload. In the first graph, “desired” pressure is the ramp/hold signal sent to the
pressure regulators in both simulated and experimental tests. PAM pressure vs.
time in the experiment matches well with the simulation. Pressure did not reach the
desired 150 psi because of the limitations of the pressure regulators in dynamic tests
(145 psi maximum).
The system attained a maximum torque of 190 ft-lb at 83 deg of rotation,
102
Figure 3.15: Experimental and predicted static torque vs. angle of rotation.
Figure 3.16: Dynamics of upper arm contraction.
103
which was in agreement with the model. However, experimental data showed a more
gradual increase to the maximum torque than was predicted because the actuator
model does not predict force as accurately at lower pressures. The discrepancy in the
torque curves led to a similar discrepancy in angular rotation curves. An undesired
characteristic of the experiment is the significant delay (0.5 - 1 s) before the PAM
pressure rises above zero, causing the arm to remain stationary. This is caused by
the deadband in the regulator, blocking the passage of air until a voltage threshold
(corresponding to about 10 psi) is reached. This is an undesired characteristic of the
regulator that will lead to replacement in future revisions and more precise control
studies.
3.5 Optimization of Joint Geometry via Genetic Algorithms
3.5.1 Optimization Problem
The manipulator joint design on the first-generation proof-of-concept robot arm
presented above (Section 3.2.4) was the product of human trial-and-error, and only
two independent joint parameters were varied to determine a satisfactory torque pro-
file at maximum pressure. However, for the development of a second-generation robot
arm with greater lifting capability (Figure 3.17), joint parameters were determined us-
ing constrained multi-variable optimization. Design requirements and pre-determined
PAM characteristics (limited by volume requirements and commercially-available
materials) are listed in Table 3.4.
In the two-dimensional tendon-link arrangement described previously, there
104
Table 3.4: Joint design requirements and PAM characteristics.
Joint Shoulder Elbow
Torque (ft-lb) 788 362
Min Angle (deg) 15 10
Max Angle (deg) 120 80
Number of PAMs 6 8
PAM Diameter (in) 1 0.75
are four geometric parameters available for variation, as shown in Figure 3.18: These
are the length along the upper limb axis lu, perpendicular offset from the upper limb
axis ru, length along the lower limb axis ll, and perpendicular offset from the lower
limb axis ru. It should be noted that when the initial arm angle is known (as is the
case in this study), the length of the tendon link lT is not an independent variable
in the joint design, but its length does change as the other parameters vary. Due to
the complex joint geometry, an analytical solution was not feasible.
A detailed optimization procedure was devised to ensure that the selected
configuration could satisfy torque requirements over the full range of motion without
excess pressure. In each individual optimization step, the goal was to minimize the
area of the region(s) in a torque vs. angle profile where predicted joint torque falls
below a desired torque threshold. An example of this type of region is shown in
Figure 3.19. The objective function J was a weighted sum of this area below the
curve Ab, the area above the curve At (weighted very low, used simply to help guide
the optimization), and a penalty function PF to heavily penalize configurations
which cannot rotate the full range of motion within 15% of PAM stroke (otherwise
105
Figure 3.17: Second-generation, two degree-of-freedom manipulator.
PAM inflation would be prohibitively high). The function is given by:
J = Ab + 0.01At + PF (3.25)
PF =



0, θmax,a ≥ θmax
10000, otherwise
(3.26)
where θmax,a is the maximum available angle and θmax is the maximum required angle.
PAM size, PAM number, and pressure level were held constant (though manual
effort was made to minimize the size and number of PAMs needed prior to initiating
this procedure). Constraints were placed on the design variables in order to ensure
106
Figure 3.18: Joint parameters for optimization.
107
sufficient space for PAMs and to accommodate physical limits to the proximity of
each joint (parts were sized to withstand the expected stresses).
A genetic algorithm was selected to perform a heuristic search for the optimal
joint dimensions. Genetic algorithms emulate the process of natural evolution. The
general principle is selection of the “fittest” designs, and “mating” these designs so
that the next generation inherits the desirable characteristics of the former generation.
Mutation, the random changing of one gene (variable), prevents the algorithm from
remaining stuck at local minima. As a result, this structured process of learning by
trial-and-error is very robust. While achieving a global minimum is not necessarily
guaranteed with this approach, it is more probable than other standard optimization
routines. The MATLAB function “ga()” was employed to perform this optimization.
To begin the design process, an initial population of 20 designs (sets of the
variables lu, ru, ll, and ru) was selected at random, though the randomness was
limited to parameter values that fell within preset upper and lower bounds. Based on
the objective of meeting the required torque over the range of motion (with minimal
pressure and within volume constraints), a score was given to each of the initial
random starting points. The two sets of design parameters from the initial 20 with
the lowest scores were selected and converted to a binary “gene” form. Then these
sets were “mated” by cutting the genes at a random point and trading parts of
one design with parts of another to form a new design. This was done 18 times to
create a new population. This new population of 18, along with the 2 best designs
from the previous generation, then formed a new generation of 20 designs. This
evolution process was set to be repeated for either 100 generations or until the system
108
Figure 3.19: Torque vs. range of motion, illustrating regions to be minimized.
converged to within established error tolerance bounds.
After each optimization, the area under the torque threshold was evaluated.
PAM pressure was adjusted and the optimization was repeated until the area under
the curve was essentially zero; therefore, the design corresponded to the minimum
pressure necessary to meet torque requirements.
3.5.2 Optimization Results
Results demonstrated convergence to joint configurations that met optimization
goals when pressure levels were sufficient. Figure 3.20 illustrates the gradual im-
provement in minimizing the objective function as generation number increases and
109
Figure 3.20: Fitness value vs. generation, genetic algorithm-based optimization.
eventual convergence to an optimal solution. Figure 3.21 shows a typical optimization
history for the shoulder joint of the PAM-based manipulator. The desired joint
torque was 788 ft-lb between an initial angle of 15 deg and a final angle of 120 deg.
Figure 3.22 illustrates the difference between initial and final torque curves. Several
runs were performed with different initial conditions in order to validate that the
system consistently converged to the same final values.
Due to additional unforseen constraints on joint geometry, optimization for each
joint was performed a second time. Specifically, the joint shafts, tendon links, and
supporting structure were enlarged and redesigned to improve the structural integrity
(increase the safety factor) of the system. This forced the minimum allowable values
of some link parameters on each joint to increase. The final torque profiles are shown
110
Figure 3.21: Joint dimensions vs. generation, genetic algorithm-based optimization.
Figure 3.22: Torque vs. range of motion, initial and final generations.
111
Figure 3.23: Torque vs. range of motion, shoulder joint.
in Figures 3.23 and 3.24. The final parameter values are given in Table 3.5. Note
that the torque generated at the minimum angles are well above the desired torque;
this is a consequence of the new geometric constraints.
3.6 Conclusion
A proof-of-concept demonstrator of a pneumatic artificial muscle-actuated
robotic manipulator intended for lifting heavy payloads was built and tested, showing
the potential that PAM actuators have for this type of application. The two degree-
of-freedom system with links that approximate the lengths of a typical human arm
was able to achieve joint torques in excess of 200 ft-lb for each joint of the initial
proof-of-concept demonstrator, and nearly 800 and 400 ft-lb for the shoulder and
112
Figure 3.24: Torque vs. range of motion, elbow joint.
Table 3.5: Optimal parameters of candidate designs, determined by genetic algorithm-
based optimization.
Joint Shoulder Elbow
Pressure (psi) 190 150
PAM Length (in) 10.88 4.94
lu (in) 1.00 0.70
ru (in) 1.08 0.70
ll (in) 1.08 1.15
rl (in) 0.19 -0.20
113
elbow, respectively, of a refined manipulator. A system model was devised that
took into account PAM behavior, arm dynamics, and airflow through the pneumatic
subsystem. The actuator model considered the Gaylord force from pressure effects
on the braid, as well as the Mooney-Rivlin energy storage term.
Error between quasi-static torque experimental values and model predictions
at 150 psi was on the order of a few percent. Dynamic testing uncovered some
discrepancy at low pressure, suggesting that the model may need adjustment for
precise, low pressure applications. Although the effects of experimentally observed
hysteresis in the PAMs were neglected, this actuator model overall showed good
accuracy of the system model when simulated predictions were compared to exper-
imental measurements, indicating that the simplified PAM model is sufficient for
initial performance predictions and design analysis.
Lastly, design optimization was performed on the joints of a second-generation,
two degree-of-freedom manipulator. The dimensions of the joints were selected
using genetic algorithms with the objective of minimizing the joint torque under a
threshold torque over the desired range of motion, while minimizing the required
pressure. This design method was found to be superior to simple trial-and-error.
While the initial joint designs were found to be optimal, they did not fully account
for structural integrity requirements; therefore, a second optimization was performed
with strict constraints on joint dimensions. The resulting joints required higher
pressures than the original joints and demonstrated torque levels considerably higher
than the desired torque at low angles; however, this was necessary to reach the
desired torque at high angles.
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Chapter 4: Control of PAM-Based Robotic Manipulators
4.1 Introduction
Commercial and domestic applications for mobile robots continue to expand
into areas that require human-robot interaction. Often, the design objectives con-
sidered in mobile robotic applications are low weight, high range of motion, high
torque, and low power. However, interaction with humans warrants additional
safety measures, including compliant manipulation. This requirement is problematic
for many conventional actuators, which typically use high stiffness to achieve high
performance, thereby increasing inertia, and leading to large impact forces upon
collision [166].
One type of actuator that can satisfy these competing requirements is the
pneumatic artificial muscle (PAM). PAMs are extremely lightweight, compliant,
and capable of higher specific work than comparably-sized hydraulic actuators and
electric motors [155]. They are composed of a helically braided sleeve surrounding
an elastomeric bladder and are held together by end fittings. Pressurization of
the soft bladder inflates the muscle and causes the stiff braid fibers to reorient,
generating a contractile stroke and pulling force similar to human muscle. Also
known as McKibben actuators, PAMs were initially developed as orthotic devices
115
for polio patients [23]. Similar applications have dominated the field over the
years, with PAM-powered devices for orthotics and rehabilitation [75,80,152] and
biologically-inspired humanoid robotic devices [40,69]. Looking to other applications,
these actuators are particularly desirable in portable robotic systems intended for
interaction with humans [166], such as those envisioned for nursing assistance and in
casualty extraction.
While being considered in robotics for several decades, work still remains to
develop an all-encompassing PAM model formulation. Many modeling approaches
exist, including energy balance [32], force balance [34] and finite elasticity theory [39],
but precise modeling of the nonlinear behavior has proven difficult. Moreover,
scalability to different sizes, length-to-diameter ratios, and bladder thickness-to-
diameter ratios has seen limited success. Modeling error is compounded farther
in PAM systems when considerations for the flow and compressibility of air are
included, making precise control difficult. Control is especially difficult for a PAM-
based manipulator with payload weights ranging from zero to several times the mass
of the manipulator itself.
Past efforts to control pneumatic artificial muscles span a wide range of es-
tablished techniques. Caldwell et al. [41] used adaptive control based on model
estimation, demonstrating accuracy for lightweight payloads (0.325 kg [0.7 lb]) over
9 deg of arm rotation. However, it is important that manipulators intended to lift
humans can handle payloads exceeding their own weight and can operate over a
much larger range of motion. Ahn and Nguyen [42] developed an intelligent switching
algorithm using a learning vector quantization neural network for various payloads,
116
showing experimental data for step inputs. This controller requires training the
neural network for an extended amount of time before the system could reliably lift
different payloads. Wu et al. [43] developed a self-tuning fuzzy PID controller for a
hand exoskeleton actuated by PAMs. The experiments showed undesired oscillation
about the reference trajectory, though there was a time-varying disturbance due
to human input. Yeh et al. [44] designed an optimal controller using loop transfer
recovery (LTR) for a leg exoskeleton, which was deemed successful, but limited to
only 15 deg of rotation.
When manipulator joint trajectories require high speed and acceleration, the
tracking capabilities of pure feedback control are degraded [165]. Moreover, over-
simplified computed torque controllers or adaptive controllers with slow parameter
updating may not have the capability to track fast nonlinear dynamics. To address
this problem, several approaches to PAM-based control have incorporated highly
detailed models of the system. Zhu et al. [45] designed an adaptive robust controller
for a parallel manipulator with only a few degrees of movement, incorporating a model
of the flow dynamics and valve. Ganguly et al. [46] introduced static and dynamic
empirical models of PAM actuation and valve characteristics in a model-based PID
controller for a rotary joint. The adoption of feedforward compensation was shown
to improve system response in work by Nho and Meckl [47], who demonstrated
neuro-fuzzy and inverse dynamics feedforward control on a two-link manipulator.
Fateh and Izadbakhsh [48] employed a hybrid computed torque approach to a two-
link manipulator and found that feedforward control reduces tracking error. These
model-based feedforward strategies influenced the design of the control system in
117
the present study.
Control studies on PAM-based systems have often employed sliding mode
control, a robust nonlinear control strategy that drives the dynamics of the system
to that of an exponentially stable system [49]. This methodology has been extended
to adaptively control robot manipulators [151], improving the response to unmodeled
dynamics or payload variations. Many sliding control algorithms have been proven
to be globally stable when model errors and system disturbances are bounded [50].
Carbonell et al. [51], Cai and Dai [52], and Lilly [53] performed simulations of
PAM-actuated systems using sliding mode controllers. However, these second-order
controllers assume that the PAM pressure or force being used as a control input
is instantaneous, which cannot be assumed in a practical system with large PAM
volume. Similarly, Nouri et al. [54] designed an adaptive controller with a sliding
component that neglected airflow dynamics for low payloads (0.6 kg [1.3 lb]). Xing
et al. [55] applied sliding mode control with a disturbance observer to a PAM in
linear motion experiments with a 1 kg (2 lb) load. While the controller showed good
performance, airflow dynamics were neglected and commanded pressure was assumed
to be instantaneous, which is not realistic in more demanding applications. In order
to address such problems, Shen [56] designed a model-based sliding mode controller
including a dynamic airflow model, and applied it to a linear table actuated by
PAMs. Aschemann and Schindele [57] developed a cascaded sliding mode controller
for a high-speed linear axis with a detailed empirical model of valve dynamics for
pressure feedback and a model of the linear axis for position feedback. As in other
model-based control strategies, these detailed physical models were necessary to
118
achieve satisfactory performance. Tondu et al. [58] applied sliding mode control with
twisting and super-twisting algorithms to two degrees-of-freedom of a PAM-based
manipulator, noting that the equivalent force control term was not helpful on the
link farther from the base because of model uncertainty. Overall, studies that have
successfully applied sliding mode control to experimental PAM-actuated systems
have used systems with high stiffness, low inertial loads, and minimal time delays.
However, the robotic arm in the present study, as outlined in the following section,
exhibits relatively low stiffness, highly variable inertial loads, and substantial time
delay. Moreover, sliding mode control can be negatively affected by measurement
noise and low-resolution sensors, which are present in the system. The authors intend
to investigate the implementation of an adaptive sliding mode controller on a system
better suited to this strategy in future work.
While many of these studies have demonstrated smooth and accurate motion,
control of a PAM-actuated manipulator with both high range of motion (90 deg)
and high tip payload variations (0 to 45 kg [0 to 100 lb]) has not been attempted. It
should be noted that the manipulator arm in question weighs 7 kg (15 lb), several
times less than the maximum payload. The objective of the present work is to develop
a control algorithm for this manipulator that satisfies accuracy and smooth motion
requirements. This began first by considering proportional-integral-derivative (PID)
and fuzzy logic controllers, but moved toward a model-based feedforward structure
to achieve improved performance. In this study, the controllers were simulated and
experimentally tested on the manipulator. System performance was evaluated for
different trajectories and payloads. Additionally, the effect of varying the gain in the
119
feedforward model was analyzed.
4.2 PAM-Actuated Robotic Manipulator
4.2.1 Design and Experimental Setup
The heavy-lift, PAM-actuated, proof-of-concept manipulator was designed and
constructed in Chapter 3. Figure 4.1 illustrates the components of the actuator
system, including a high pressure air source, pneumatic valves, and manipulator. As
illustrated, the actuators provide torque in only the positive direction, and gravity
provides a restoring force in the opposing direction. In other words, the system
is not operating in an antagonistic configuration. The manipulator design allows
for a compression spring to provide antagonistic force; however, a spring was not
used in this study. Figure 4.2 shows the arm in operation, holding 50 kg (110 lb).
Throughout experimentation, angular Hall-effect sensors (Midori Precisions) were
attached to each joint to measure rotation angles that typically varied from 0 to 100
deg. Pressure transducers (Omegadyne Inc.) were attached to incoming air lines to
monitor PAM pressure. Each degree-of-freedom can be fed compressed air through a
proportional servo-valve (Festo MYPE-type). However, in these experiments, only
the elbow joint was actuated. The cross-sectional area of the valve inlet and outlet
are a function of the voltage sent from the controller. Hence, the controller was
capable of directly manipulating the flow rate of air.
The sensors and control valves were connected to a dSPACE R©real-time inter-
face, which allowed for quick implementation and modification of controllers modeled
120
Figure 4.1: Schematic of PAM-Actuated Manipulator System.
Figure 4.2: PAM-actuated joint holding 50 kg (110 lb).
in Simulink R©. In these single-degree-of-freedom (SDOF) control experiments, a
single-input, single-output (SISO) controller was employed for elbow rotation. Out-
put error between the measured (actual) angle and the reference (desired) angle was
used to compute a voltage to the control valve. The pressure supply was set to 0.83
MPa (120 psi).
121
4.2.2 System Model
A complete model of the elbow degree-of-freedom consists of three major
components: the airflow through the valve, the response of the PAM actuators,
and the manipulator dynamics. The control valve regulates air pressure in the
actuators, which is used to calculate PAM force. PAM force is then translated to
actuator motion. Airflow is regulated by the control voltage to the valve, whose
orifice area A is a nonlinear function of the control voltage. Orifice area is calculated
by interpolating values from a lookup table.
The mass flow rate of air depends on both upstream and downstream pressures,
Pus and Pds, and the orifice area:
m˙air =



ACdPus
√
2
RTC1,
Pds
Pus
< Pcr
ACdPus
√
2
RTC2,
Pds
Pus
≥ Pcr
(4.1)
The term Cd is a discharge coefficient capturing the losses in the orifice,
Pcr = (2/(γ + 1))γ/(γ−1) is the critical pressure separating subsonic and supersonic
flow, and C1 and C2 are given by:
C1 =
√
γ
γ−1
[(
Pds
Pus
) 2
γ
−
(
Pds
Pus
) γ−1
γ
]
C2 =
(
2
γ+1
) 1
γ−1
√
γ
γ+1
(4.2)
where γ is the specific heat ratio of air.
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Richer and Hurmuzlu [158] derived an expression for pressure change in a
container of changing volume using the ideal gas law, conservation of mass, and
conservation of energy:
P˙ =
RT
Vair
(αinm˙in − αout ˙mout)− αP
V˙air
Vair
(4.3)
where R is the specific gas constant for air, T is air temperature, Vair is air volume
in the PAMs, αin = 1.4 and αout = 1 are the respective specific heat ratios of the
gas flow into and out of the PAMs, and α = 1.2 is the approximate specific heat
ratio due to volume change. From this expression of pressure rate, the time-varying
pressure inside the PAM actuators can be determined by integration.
Quasi-static PAM actuator models relating force to contracted length and
applied pressure can be derived using a number of approaches [35]. Experimental
data was found to best match the force balancing method derived initially by Ferraresi
et al. [34]. In this model, corrected for the effect of non-constant bladder thickness,
PAM force is given by:
F =
P
4piN2
[
3(L0 −∆L)
2 −B2
]
+ P
[
VB
L0 −∆L
−
(t0 −∆t)(L0 −∆L)
2pi(R0 −∆R)2N2
]
+ FL + FT
(4.4)
where L0 is resting length, ∆L is contraction, VB is bladder volume, B is braid
length, N is number of braid turns around the circumference of the PAM, t0 is
the resting bladder thickness, ∆t is the change in bladder thickness, R0 is the
123
resting bladder radius, and ∆R is the change in radius. The forces FL and FT are
pressure-independent terms governed by the elasticity of the bladder ER, and are as
follows:
FL = ERVB
(
1
L0
−
1
L0 −∆L
)
(4.5)
FT =
ER(L0 −∆L)
2pi(R0 −∆R)N2
[(t0 −∆t)(L0 −∆L)− t0L0] (4.6)
This formulation was found to be accurate for PAMs undergoing contraction,
but there is an inherent hysteresis that alters the force-contraction profile in extension.
In extension, the resistive force preventing the PAM bladder from expanding increases
the PAM force. Also, there is friction both between the braid and bladder and between
the braid fibers themselves. This hysteresis is captured in the model by adding
empirical constants to the terms:
F2 =
P
4piN2
[
3(L0 −∆L)
2 −B2
]
+0.3P
[
VB
L0 −∆L
−
(t0 −∆t)(L0 −∆L)
2pi(R0 −∆R)2N2
]
+0.8FL+0.8FT
(4.7)
Force from the PAM group is translated into an arm torque, defined as τ =
Freff cos θfnPAM , where reff is the effective length of the moment arm (nonlinear
function of joint angle), θf is the angle between the stationary upper link and the
joint tendon, and nPAM is the number of parallel PAMs in the muscle group.
PAM torque is resisted by the weight and inertia of the arm and payload. The
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arm dynamics for single degree-of-freedom motion are related to PAM torque by:
τ = Iz θ¨d +mpldl
2
armθ¨d + (marmlc,arm +mpldlarm)g sin θd (4.8)
where θd is desired angle, Iz is the link inertia, marm is the arm mass, lc,arm is the
distance between the joint and the link center of mass, mpld is the payload mass, larm
is the link length, and g is gravitational acceleration. With the previous equation,
joint angle can be determined and continuously monitored by the controller.
4.3 Control Strategies
4.3.1 Output Feedback Control
4.3.1.1 Proportional-Integral-Derivative Control
Proportional-integral-derivative (PID) control is a feedback strategy widely
used in industrial applications. The controller input, voltage to the pneumatic valve,
is given by:
u(t) = kP e(t) + kI
∫
e(t)dt+ kDe˙(t) (4.9)
The control signal can be manipulated by three user-defined gains: the propor-
tional gain, kP , integral gain, kI , and derivative gain, kD. Additional improvement
was achieved by adding a low-pass filter to the controller output to eliminate high-
frequency content and provide a smoother response, while only slightly increasing
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total error.
4.3.1.2 Fuzzy Control
As an alternative to PID control, a fuzzy logic controller was designed and
implemented based on the approach described by Passino and Yurkovich [164], which
is briefly described in this section. Fuzzy control employs membership functions and
a ruleset established by the designer to translate controller inputs into a desired
output in a smooth, non-discrete manner. A fuzzy controller categorizes a numerical
input value using linguistic variables such as “positive big” and “negative medium.”
Each linguistic variable i has its own membership function µi(x), which quantifies
the “certainty” that the numerical input x can be classified as that variable (on a
scale from 0 to 1). Triangular-shaped membership functions are most common and
were used here [164].
In this fuzzy controller, the input variables were angle error and error rate, and
the output variable was valve voltage. Each of the inputs was scaled by gains g1 and
g2, and the output was scaled by gain h. Note that in this study, the inputs were
divided by g1 and g2, meaning that higher values of g1 and g2 decrease the system
sensitivity. However, the output was multiplied by h; therefore, higher values of h
increased the system reaction.
Figure 4.3 shows the seven membership functions associated with each linguistic
variable with unity gains. It is clear from the overlapping membership functions that a
numerical input can be categorized as more than one linguistic variable simultaneously.
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Table 4.1: Fuzzy ruleset.
This leads to the “fuzzy” combinations that enable smooth transitions between
different outputs.
Table 4.1 displays the user-defined ruleset for the fuzzy controller. The control
logic combines the linguistic input variables that are given non-zero membership
values and prescribes the output based on this ruleset. For example, IF the angle
error is “negative small,” AND the error rate is “positive small,” THEN the output
will be “positive small.” However, the membership of a given numerical value is
often split between two membership functions. To determine the weight given to a
particular rule, we define:
µp,ij = min(µei , µe˙,j) (4.10)
as the “premise” membership to rule ij, where µe,i and µe˙,j are the membership
values of error with respect to membership function i and error rate with respect to
membership function j. Since multiple linguistic variables can be activated, multiple
127
Figure 4.3: Fuzzy membership functions for input/output variables.
outputs from the ruleset are combined and weighted using the “center of gravity”
defuzzification method. This final combination is then multiplied by gain h to
produce the total output. It should be noted that, as with PID control, a low-pass
filter was added to the output to reduce oscillatory content.
4.3.2 Model-Based Feedforward Control
Although output feedback controllers, such as PID and fuzzy logic, can be
tuned for smooth trajectory following and good disturbance rejection, neither take
advantage of the known system dynamics to improve control. Therefore, a model-
based feedforward controller was designed and implemented. The system model
described in the last section was used previously for control studies [97]. These
simulations accurately predicted dynamic behavior of the arm when subjected to an
128
arbitrary input. Thus, a controller was designed with a feedforward control element
based on an inverse of the system model.
4.3.2.1 Model-Based Control without Feedback
Figure 4.4(a) depicts the structure of the basic model-based feedforward con-
troller. The system contains an inverse model of the arm, pneumatic muscles, and
airflow dynamics to help negate the nonlinear dynamics of the plant. This feed-
forward element samples the commanded trajectory and calculates desired PAM
pressure at each time interval. If desired pressure is greater than actual pressure, the
system exhausts air until it reaches the desired pressure, and vice versa. Note that
actual PAM pressure must be known in order to calculate the desired airflow rate in
the valve, which could be considered a feedback component within the feedforward
controller. Given the desired state of the system, inverse dynamics are used to
calculate desired torque using Eqn. 6.6, though payload mass must be known a priori
to achieve accurate trajectory tracking.
By rearranging the PAM group torque equation, desired force can be determined
as:
Fd =
τd
reff cos θf
(4.11)
129
Then, desired pressure in each PAM is calculated by rearranging Eqn. 4.3 as:
Pd =
(
Fd
nPAM
− FL − FT
)[
(3(L0 −∆L)2 −B2
4piN2
+
VB
L0 −∆L
−
(t0 −∆t)(L0 −∆L)
2pi(R0 −∆R)2
]
(4.12)
(or alternatively, Eqn. 4.7 for PAM extension). Simply differentiating desired pressure
to find the desired pressure rate does not produce good results because initial pressure
error would not be compensated and model inaccuracy may cause the error to
compound. Therefore, actual PAM pressure P is monitored by the feedforward
controller and pressure error ∆P = Pd − P is determined. The goal of the controller
now is to minimize the pressure error, by proportionally adjusting desired pressure
rate, P˙d = −kM∆P , where gain kM is introduced to adjust the speed of convergence
of the actual pressure to the desired pressure. Therefore, m˙d, the desired mass flow
rate of air to the PAM group, is estimated as:
m˙d =



P˙dV+αPV˙
RTαout
P˙d < 0
P˙dV+αPV˙
RTαin
P˙d > 0
(4.13)
From this flow rate, a desired valve orifice size Ad is calculated by rearranging
130
Figure 4.4: Model-based feedforward (a) without output feedback, (b) with PID or
fuzzy feedback control.
Eqn. 4.1 as:
Ad =



m˙d
CdC1Pus
Pds
Pus
< Pcr
m˙d
CdC2Pus
Pds
Pus
≥ Pcr
(4.14)
Ideally, the gain would be very high in order to obtain the desired pressure
nearly instantaneously, but problems with oscillation and model inaccuracies require
that the gain be tuned for an optimal combination of accuracy and smoothness.
4.3.2.2 Model-Based Control Augmented with Output Feedback
If the joint is following a commanded trajectory with only model-based feedfor-
ward control, minor inaccuracies in the model can produce significant steady-state
error, and major inaccuracies may lead to instability. Therefore, the model-based
feedforward controller can be augmented with a stabilizing feedback controller, such
as a PID or fuzzy controller, as illustrated in Figure 4.4(b). A desired joint angle
131
is passed into the model and the feedback controller in parallel. Their outputs are
summed, creating a total control voltage that is sent to the PAM manipulator. The
arm dynamics are sensed and used in feedback to ensure the arm closely follows the
trajectory.
4.4 Control Analysis Via Simulation
4.4.1 PID and Fuzzy Control
The two output feedback control strategies that were implemented on the
manipulator use distinct approaches to provide smooth, stable, and accurate motion.
However, both rely on gain tuning to achieve optimal results. This section details
the metrics used to evaluate controller performance in trajectory-following exercises
as the gains were varied.
4.4.1.1 Gain Tuning Metrics
For simulations and experiments, performance metrics were established to
ensure that the best set of gains was chosen. The first was an integration of the
squared error in angular position as a function of time, giving a total error metric:
ν =
∫ Tf
0
(yd − y)
2dt (4.15)
where y is the measured joint angle, yd is the desired joint angle, and Tf is the
final time of the test run. Note that the input to the controller is the error signal,
132
e = yd − y. While it may seem that this metric is sufficient in tuning the gains since
it minimizes angle deviation, it became quickly apparent that some responses with a
lower value of ν also had substantial oscillation about the desired profile yd. Hence,
another error metric was considered to highlight smoothness of the response. This
metric was based on the local curvature of the measured angle:
κ =
y′′
(1 + y′2)3/2
(4.16)
where y is the first time derivative of the measured angle and y′′ is the second time
derivative. With the local curvature at each time step, an error metric was computed
by integrating κ with respect to time, giving the total curvature:
ω =
∫ Tf
0
κdt (4.17)
This integration provides a single value for comparison where the response
with minimal curvature, or oscillation in the response, is easily identifiable. Having
two error metrics now, one based on minimal overall error and one based on minimal
oscillation, a combined metric was established drawing on contributions from both
individual metrics. Each individual metric was normalized with respect to the
minimum value of that metric and weighted with Wi as:
J = Wν
ν
minν
+Wω
ω
minω
(4.18)
such that the minimum possible combined value was J = 1. The weights were chosen
133
as Wν = 0.25 and Wω = 0.75 to give more precedence to smoothness of the controlled
response than total error, which tended to be undesirably oscillatory alone.
4.4.1.2 Simulated Trajectory Following
Prior to experimental testing, simulations of the feedback controllers were
performed. In each simulation, the elbow joint was commanded to follow a lift-
hold-return trajectory with 5 s segments, the error and smoothness were monitored,
and a combined metric score was calculated. Each of the control gains was varied
iteratively, and the set of gains that produced the lowest score was determined to be
optimal. Both PID and fuzzy controllers were simulated for various payload weights
hung from the tip of the end effector.
Figure 4.5 shows surface contour plots of the performance metrics from fuzzy
gain tuning simulations (similar results were also obtained for the PID controller).
These plots allow g1 and h to vary, while g2 = 0.6 is held fixed at the value determined
to be optimal for some of the cases. The two individual metrics are shown, along
with the combined metric. It is clear that the optimal gain set according to the
error metric and smoothness metric are different, producing a combined metric
that includes characteristics of both individual metrics and returns a new optimal
value (g1 = 0.45, h = 0.65). Note that the area directly surrounding this minimum
is smooth, meaning that small deviations from these gains will not cause abrupt
changes in behavior. This is important because a future controller may incorporate
gain scheduling in order to account for changes in payload, and as the controller
134
Figure 4.5: Performance metrics as g1 and h vary (g2 = 0.6) with a 23 kg (50 lb)
payload (a) error metric; (b) smoothness metric; (c) combined metric.
transitions from one set of gains to another, gains may not be at their exact optima
at all times.
4.4.2 Model-Based Control with Output Feedback
Model-based feedforward control was also evaluated in simulation. Some plant
parameters (PAM force, friction, and arm inertia) were intentionally made slightly
different from the controller parameters so the model would not match perfectly.
Figure 4.6 shows the trajectory and pressure response for the feedforward controller
with fuzzy feedback. It can be seen that both the reference trajectory and pressure
are closely followed in all three cases. Control gains are identical for each simulation
(g1 = 0.4, g2 = 0.3, h = 0.7, kM = 1.0). As predicted, larger payloads require higher
135
Figure 4.6: Simulated control, feedforward with fuzzy feedback
pressure to achieve the maximum angle. Error during the lift ramp is seen to increase
with increasing payload because the maximum mass flow rate of air decreases as
upstream pressure and downstream pressure converge. This effect could be reduced
with higher source pressure.
Figure 4.7 shows the same trial simulations with PID feedback in place of fuzzy
feedback. Again, controller gains are identical for each simulation (kP = 5.0, kI =
2.0, kD = 0.0, kM = 1.0). Operation is similar to feedforward control with fuzzy
feedback; however, there is slightly increased error in the return stage, which is likely
an effect of the integral term delaying a change in direction because of accumulated
prior error.
In both trials, the operation of the model-based controller is smooth and
accurate across a large range of payloads, and no overshoot is present. The simulations
provided a good case for implementation and evaluation on the manipulator hardware.
136
Figure 4.7: Simulated control, feedforward with PID feedback
4.5 Experimental Evaluation
4.5.1 PID and Fuzzy Control
4.5.1.1 Experimental Validation
Guided by optimal results from simulation, experimental tests were performed
and compared with simulation data. There was generally good correlation between
model predictions and experimental measurements, providing validation that the
simulation model is sufficient for closed-loop design with the noted PID and fuzzy
controllers. Validation of the predicted joint behavior suggested that the simulation
model could be extended to other manipulator configurations and control strategies,
such as model-based approaches.
137
4.5.1.2 Experimental Gain Tuning
Although simulations with optimal gains showed good correlation with ex-
periments, the differences between the model and actual system indicated that
experimental gain tuning could improve performance. Unlike in simulation, eval-
uating an exhaustive set of gain combinations was not practical, so experimental
gain tuning began with the predicted optimal gains and varied these in search of
experimental optima. Tuning of the PID controller was performed for three payloads:
11, 23, 34 kg (25, 50, 75 lb). The process began with kP . Next, kI was varied while
a highly rated value of kP was held constant. Then, kD was added and incremental
adjustments were made again. Fuzzy controller gains were tuned similarly, in the
order of g1, h, and then g2.
Figure 4.8 shows an example of how the different error metrics guided the
selection of different gains as being “optimal” in the experiments. This example is
with 23 kg (50 lb) and considers variations in kI with two different values of kP ,
while kD = 0. As can be seen for the two individual metrics in Figure 4.8(a)-(b),
each gives a best response (minimum value) with a different value of kP and that
minimum metric value occurs at a different gain value of kI . This is the main purpose
for considering the combined (normalized and weighted) metric in Figure 4.8(c).
With the weights discussed above, this figure shows that the best overall closed-loop
response with respect to these metrics and weights occurs when kP = 1.4 and
kI = 2.5. To help show the differences in the actual measured time response of
the joint rotation angle, Figure 4.8(d) has been included to show the responses
138
for the best gains according to each metric in Figure 4.8(a)-(c). As shown, “Best
Error” follows the “Desired Angle” the closest, but it also has the largest oscillations.
“Best Smoothness” and “Best Combined” are much closer to each other, though
slightly less accurate than “Best Error,” but both have less oscillatory behavior.
This provides an example of the trade-offs considered in tuning the gains.
Two data points corresponding to kI = 2.5 in Figure 4.8(a) seem to be outliers,
potentially caused by factors that are specific to this system and the chosen reference
trajectory. For instance, this integral gain value could have led to a buildup of steady
state error that started to rebound just as reference angle began to increase/decrease,
causing the arm to closely follow the reference angle in its upward/downward
movement, in effect anticipating the systems motion and cancelling out time delay.
Another possibility is that this specific integral gain value causes swaying of the
payload in a manner that helps the controller follow the path. The most obvious
outlier (kP = 2.6, kI = 2.5) was weeded out because of less impressive performance
in terms of the “smoothness metric.” The use of a combined metric helps to reduce
anomalies in the data, especially because low error often corresponds with high
oscillation. It is possible that the point (kP = 1.4, kI = 2.5) would no longer
correspond to the optimal set of gains if a different trajectory was chosen, but
because it is close to the other local minimum in Figure 4.8(c), (kP = 1.4, kI = 3.0),
it should be safe to assume good performance for other similar lifting trajectories.
Figure 4.9 compares the optimal gains determined from simulation and experi-
ment. Figure 4.9(a) shows that the PID experimental gains are within a reasonable
range of those simulated. The derivative gain is zero or near zero at each payload
139
and the proportional gain starts just above kP = 1 at 11 kg (25 lb) and decreases
with increasing payload. At the minimum payload, the integral gains for both cases
are near kI = 4, but above this payload there is some deviation with the experiment
showing a downward trend and the simulation showing an upward trend initially
before turning downward. Optimal fuzzy control gains are displayed in Figure 4.9(b).
The g1 values and trend from 23 to 34 kg (50 lb to 75 lb) are close, but there is a large
difference in the preferred values at 11 kg (25 lb). Both g2 and h follow experimental
trends better, where g2 stays relatively constant with payload (the values differ up
to 2x), and h shows an increasing trend with payload, with nearly equivalent values
at 23 and 34 kg (50 lb and 75 lb). The difference with small payload may be caused
by system parameters involved with the dynamics that were not well-modeled, but
were not critical factors at heavy payloads or for PID control. These parameters
(damping, friction, or payload swinging) are difficult to estimate. The discrepancy in
modeling created an optimal region in simulated gain tuning that did not exist for
the physical system. Overall, it can be stated that the experimentally determined
optimal gains are fairly similar to the optimal gains predicted by the simulation
model, though future model refinement may be beneficial.
4.5.1.3 Comparison of Output Feedback Controllers
Figure 4.10 shows the time histories of lift-hold-return trajectory-following
experiments using the experimentally-optimized PID gains for three payload weights.
In each case, the joint initiated positive rotation with about 1 s of delay and quickly
140
Figure 4.8: PID control, integral gain selection with a 23 kg (50 lb) payload (a)
error metric; (b) smoothness metric; (c) combined metric; (d) time response for best
cases.
Figure 4.9: Experimental vs. simulated optimal gains (a) PID; (b) fuzzy.
converged to a steady-state during the hold stage. Additional payload prevented
contraction until pressure increased, which slightly added to the delay. Due to the
larger weighting on the smoothness metric, oscillations reached acceptable levels, but
the total error was slightly larger. The cumulative angle error over the trajectory
141
also increases with payload. Most visible with the PID controller is the undesirably
large error in the return stage (arm lowering). With 34 kg (75 lb), this is attributable
to integrator windup, when the integral term accumulates high error because the
arm cannot produce required torque in this case. Consequently, when the reference
changes, the integrator must unwind before the arm can change position. This also
causes a considerable overshoot past the rest position when the arm is lowered. This
problem could be mitigated by preventing the integral term from increasing over
certain bounds. However, an anti-integrator windup will not remove this overshoot
phenomenon completely unless the integral gain is set to zero. Some integral gain
is needed to allow PID control to remain smooth, reactive, and able to settle to
steady-state. Hence, this is a drawback of the PID control strategy for this system.
Figure 4.11 displays results of optimal gains using the fuzzy controller under the
same experimental conditions. All three joint angle trajectories are similar in shape,
implying minimal change in lifting behavior with changes in payload weight. There
is some undesirable low-frequency oscillation during the lift stage, and the arm, while
close, does not quite reach the desired hold angle. There is no overshoot, however,
and more gradual movement during the return stage, making the closed-loop tracking
appear better overall with the fuzzy controller than the PID controller.
Figure 4.12 presents a direct comparison of the PID and fuzzy experiments
with the 23 kg (50 lb) payload. In lifting the payload, fuzzy control produces more
oscillation, but the overshoot seen with the PID controller does not self-correct until
after the return stage has begun. The fuzzy controller lags during the lift stage more
than with PID, but PID generally reacts more slowly to the return stage (mainly
142
Figure 4.10: PID control with optimized gains.
due to integrator windup). The consistent lag of about 0.75 s in each fuzzy control
case indicates that introducing a time-shift in the “Desired Angle” to account for the
characteristic delay in the system as a type of feedforward correction could decrease
error, thereby artificially improving the tracking capability of the joint. It should
also be noted that this type of correction would likely be less beneficial to the PID
controller because it can better track the lift slope and its major delay is only at the
start of the return stage.
4.5.2 Model-Based Control with Output Feedback
4.5.2.1 Model Validation
Following successful simulations, the feedforward controller with PID/fuzzy
feedback was implemented on the PAM-based manipulator using dSPACE real-time
control hardware and software. Figure 4.13 displays both simulated and experimental
143
Figure 4.11: Fuzzy control with optimized gains.
Figure 4.12: Comparison of optimized experimental PID and fuzzy control, 23 kg
(50 lb) payload.
results for the same lift-hold-return trajectory with 45 kg (100 lb) and identical
system gains (g1 = 0.4, g2 = 0.3, h = 0.7, kM = 0.5). The only major difference
between simulation and experimental results is the final resting angle after the
return stage, which maintains a steady-state error even after 5 s. The major cause
of the discrepancy is that the PAM model is inaccurate at low pressures (< 40
144
Figure 4.13: Simulated and experimental results using feedforward control with fuzzy
feedback, 45 kg (100 lb) payload.
psi). Additionally, friction in the joint and hysteresis in the actuators that was not
accurately described by the model exaggerate the error.
4.5.2.2 Experimental Analysis of Feedforward Gain
Significant variations in the system response with changing feedforward gain
kM warrant a more detailed analysis of this relationship. Figure 4.14 shows desired
trajectory of several experimental trials varying kM , while the feedback controller
gains were fixed (g1 = 0.45, g2 = 0.3, h = 0.7). With kM = 0 (pure fuzzy feedback),
the responsiveness is low, and the lag increases error. However, this case has the
lowest steady-state error at the initial/final angle, where the model is inaccurate.
In terms of pressure, trials with higher kM follow the predicted pressure (dotted
line) more closely. This pressure trajectory, representing model calculation of needed
pressure, is accurate at high pressures, but inaccurate lower. As kM increases, the
model aids the system in quickly responding to changes in direction, but the steady-
state error at low angles (i.e., low pressures) increases. The optimal gain based on
145
Figure 4.14: Lift-hold-return trajectory while varying model gain kM , 23 kg (50 lb)
payload.
RMS error over the 25 s interval is kM = 0.5.
4.5.3 Discussion of Controllers
Figure 4.15 compares the response of the four controllers investigated. Two
are simple output feedback controllers (PID and fuzzy) and two are model-based
feedforward controllers augmented with PID and fuzzy feedback (kM = 0.5). It
is clear that the feedforward term increases accuracy in all stages of the response.
Feedforward with PID control also decreases oscillation in comparison to PID control
alone.
Responsiveness to change in direction is also improved with model-based
feedforward control. In this manner, the feedforward term is similar to an increase
in the feedback gains. Unlike increasing feedback gains, however, the system does
not exhibit more oscillation. While there is a large time delay in arm angle, there is
146
Figure 4.15: Comparison of lift-hold-return trajectories, 11 kg (25 lb) payload.
significantly less time delay in achieving desired pressure. Therefore, the convergence
of actual and predicted pressure occurs smoothly and without overshoot.
4.6 Conclusions
This study investigated the use of several controllers on the elbow joint of a
heavy-lift two degree-of-freedom manipulator actuated by pneumatic artificial muscles.
Each method was designed, simulated, and implemented on experimental hardware.
Output feedback (PID and fuzzy) controllers were examined, as well as model-based
feedforward controllers augmented with output feedback. A model of the manipulator,
actuators, and airflow dynamics was developed to enable simulation control studies
and gain optimization. Trajectory-following experiments were conducted through
simulation, as were preliminary gain optimizations, where it was determined that a
combined performance metric weighing angle error and smoothness was necessary to
achieve the desired response.
147
Using real-time interface software and hardware, the controllers were experimen-
tally tested. Refinement of each technique was performed through an experimental
gain tuning procedure, where it was learned that the predicted optimal gains were
fairly accurate for PID and fuzzy control. When optimized, output feedback con-
trollers demonstrated robustness to variations in payload weight over a large range of
motion, and were shown to follow basic lift-hold-return trajectories with low oscilla-
tion, lag, and overshoot. PID control demonstrated higher accuracy and smoothness
than fuzzy control during the lifting phase, but was more susceptible to overshoot
and lag while holding or lowering the weight. The overall performance with output
feedback alone suggested the need for improvement.
To improve closed-loop performance, model-based feedforward control aug-
mented with output feedback was implemented. This approach was shown to produce
smooth and precise motions with low error for varying payloads, even without chang-
ing control gains. The feedforward gain was also varied to demonstrate its effect
on the response, where it was shown that there is an optimal gain setting. There
is room for improvement in model accuracy at low pressures, which would allow
for higher feedforward gain values and further improve the tracking ability of this
control strategy.
Due to the complicated control structure and the unmodeled or unanticipated
dynamics in the system, it is difficult to formally establish the stability of the system.
When the open loop system is in static equilibrium at any point between the joint
limits, the system is stable because of the large passive damping component inherent
to pneumatic muscles. This damping helps to ensure that PAMs are stable in well-
148
tuned closed loop feedback systems. However, to guard against potential instability
when a controller is employed, one should perform a careful analysis to determine a
stable range of control gain values over the known range of payloads.
Future improvements to consider include a dynamic PAM model that better
accounts for nonlinearities such as hysteresis, an adaptive element that responds to
abrupt changes in payload in real-time, and automated trajectory planning. Another
challenge is to extend this system to multiple degrees-of-freedom, where coupling
between joints and three-dimensional motion increases complexity and small model
errors may propagate to links farther down the manipulator.
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Chapter 5: Advanced Control of PAM-Based Robotic Manipulators
5.1 Introduction
Pneumatic artificial muscles (PAMs), also known as McKibben muscles, are
lightweight, compliant actuators composed of an elastomeric bladder surrounded
by a helically-braided fiber sleeve. Internal air pressurization causes the bladder to
make contact with the braid fibers, typically generating a contractile stroke as the
bladder expands radially and fibers reorient. When prevented from freely contracting,
these actuators generate axial force. As a result of their natural compliance and
high specific force/power capabilities, PAMs are desirable in lightweight robotic
manipulators, particularly those intended for interaction with humans. However,
control of PAM-actuated systems, including PAM-based manipulators, has proven
difficult due to the highly nonlinear nature of the actuators and the pneumatic
systems supporting their actuation.
Past efforts to control pneumatic artificial muscles span a broad range of
established techniques. In order to address the nonlinear behavior of these PAM-
based systems, many approaches to PAM-based control have incorporated detailed
empirical models of the system. The adoption of feedforward compensation was
shown to improve the system response in Nho and Meckl [47] through the use of
150
inverse dynamics feedforward control. Fateh and Izadbakhsh [48] employed a similar
hybrid computed torque approach. Vo Minh et al. [169] developed cascade position
control with hysteresis compensation for a robotic manipulator, using a Maxwell
slip model to characterize hysteretic behavior. Pure model-based control strategies
rely heavily on the accuracy of the model to cancel nonlinearities, which can be a
detriment to controller implementation, particularly in regard to development time
and effort.
Several PAM control studies have employed model-based sliding mode control,
a robust nonlinear control strategy intended to drive the dynamics of the system to
that of an exponentially stable system [49]. Carbonell [51], Cai and Dai [52], and
Lilly [53] each performed simulations of PAM-actuated systems using model-based
sliding mode controllers. These above studies assumed that PAM pressure or force are
instantaneous control inputs and must be regulated. In most practical applications,
however, PAM pressure is subject to inherent delays from airflow dynamics (i.e., not
instantaneous). To this end, Aschemann and Schindele [57] developed a model-based
cascaded sliding mode controller with pressure feedback for a PAM-actuated rotary
joint in a linear table. Shen [56] designed a model-based sliding mode controller
including a dynamic airflow model, and applied it to a linear table actuated by
PAMs. While sliding mode control is able to compensate for some uncertainty, model
accuracy significantly impacts performance. Unfortunately, in some scenarios, the
PAM actuator dynamics, system dynamics, and payload characteristics are unknown
or time-varying, and the model employed is a poor approximation, which adversely
affects closed-loop performance.
151
To better accommodate uncertainties in system dynamics, including large varia-
tions in payload, adaptive techniques may be employed. Caldwell et al. [63] employed
adaptive control based on model estimation, but the closed-loop performance was
sensitive to errors in the feedforward term. Rezoug et al. [170] developed an adaptive
fuzzy nonsingular terminal sliding mode controller for a manipulator actuated by
PAMs that requires no knowledge of the system. However, this controller requires
torque sensing and an accurate estimate of acceleration, which may be difficult to
obtain due to sensor noise.
An alternative approach that requires little or no prior knowledge is control
with artificial neural networks. Neural networks are computational models that
consist of a set of adaptive weights that are tuned by a learning algorithm. These
models are capable of approximating a wide range of nonlinear functions of their
inputs. Ahn and Nguyen [42] developed an intelligent switching algorithm for a PAM-
based robot arm using a learning vector quantization neural network. The single
degree-of-freedom system was restricted to low speeds and slowly varying inertial
loads to remain stable. In a subsequent study, Ahn and Ahn [93] implemented an
adaptive recurrent neural network on a PAM-actuated manipulator that was capable
of compensating for payload mass and time-varying parameters on-line. The authors
reported a 2000% change in system inertia between minimum and maximum loading
(0.5 kg to 10 kg). However, in some high-loading cases, undesired oscillations were
present. Furthermore, the stiffness of the system was high due to the antagonistic
actuator arrangement, and was limited to only 20 deg of motion. Lastly, the system
was actuated in the horizontal plane and was not subject to gravitational effects.
152
The objective of the present work is to develop control algorithms that allow a
single degree-of-freedom PAM-based manipulator to smoothly and accurately track
desired motions regardless of model quality and prior development effort. Studies
that have successfully applied control to experimental PAM-actuated systems thus
far have exhibited high stiffness, which generally reduces tracking error. However,
the robotic manipulator employed in the present study is a uni-directionally actuated,
low-stiffness system; PAMs lift the arm, but gravity provides the return force.
Combined with highly variable inertial loads (an over 2500% increase from the
unloaded condition) and a large (90 deg) range of motion, the problem of accurate
position control becomes even more difficult.
In order to investigate the effects of different advanced control strategies on
a heavy-lift PAM-actuated robotic manipulator, three concepts are developed for
position control: sliding mode control, adaptive sliding mode control, and adaptive
neural network control. These controllers are all partially based upon work by J.-J.
Slotine concerning passivity and sliding control [50, 151]. Numerous simulations and
experiments are performed to evaluate and compare closed-loop performance.
5.2 Robotic Manipulator
The manipulator employed in this study is a one degree-of-freedom version
(shoulder joint only) of the two degree-of-freedom arm designed and tested in Section
3.5; the new configuration is displayed in Figure 5.1. Important features include uni-
directional actuation (gravity provided the return force), a parallel arrangement of
153
Figure 5.1: One degree-of-freedom robotic arm with 50 lb payload.
PAMs, and a nonlinear joint geometry that was optimized for torque generation over
the full range of motion. The sensors on the manipulator are only capable of measuring
angular position and PAM pressure signals, both of which have measurement noise.
Due to the PAMs’ nonlinear force behavior and the aforementioned nonlinear
joint geometry, the equation of motion relating PAM pressure to joint torque is
complex. While a complete, highly detailed model of the pneumatic system was
employed in Chapter 4, quasi-static manipulator torque may be approximated by a
third-order polynomial function of angle and a linear function of pressure to simplify
the controller structure:
τ(P, θ) = C1(θ)P + C2(θ) (5.1)
154
C1(θ) = k1θ
3 + k2θ
2 + k3θ + k4 (5.2)
C2(θ) = k5θ
3 + k6θ
2 + k7θ + k8 (5.3)
The coefficients ki were determined by a least-squares fit to best match the original
model.
Figure 5.2 displays the original and simplified models, as well as experimental
values of joint torque. The torque during contraction is different than during
extension, which is a result of PAM hysteresis. Both models are fairly accurate
within the range of expected manipulator angles and pressures (indicated by the
vertical dashed lines at 15 deg and 110 deg).
Using the simplified model, the equation of motion for the present one degree-
of-freedom manipulator is given by:
C1(θ)P + C2(θ) = Iθ¨ + bθ˙ +mgr sin θ (5.4)
where I is inertia of the rotating arm link, b is viscous damping, m is the mass of the
link, g is the gravitational constant, and r is the distance from the joint to the link
center of mass. This formulation was applied in all three of the control strategies
outlined in the next section.
155
0 20 40 60 80 100 1200
50
100
150
200
250
300
350
400
450
500
Angle (deg)
To
rqu
e (f
t−lb
)
 
 Original Model
Simplied Model
Experiment
30 psi
50 psi
70 psi
20 psi
90 psi
Figure 5.2: Torque vs. angle, comparison of models with experimental data.
5.3 Control Strategies
This section describes the design of three distinct nonlinear controllers. Each
control strategy takes pressure and angle as sensor inputs, and ultimately outputs
a voltage to the pneumatic valve controlling the airflow from a pressure source to
the PAMs. It was determined in preliminary experiments that an inner pressure
feedback loop is highly desirable; therefore, a cascaded structure was implemented.
In this arrangement, angle feedback outputs a desired pressure, which is then used
for pressure feedback. An example of a cascaded controller is shown in Figure 5.3.
For the purposes of comparison, the controllers outlined below have identical PI
controllers for pressure feedback. This work focuses on the angle feedback component,
156
Figure 5.3: Diagram of a cascaded controller.
which simplifies the control problem, but relies on a high pressure loop tracking
bandwidth.
5.3.1 Sliding Mode Control
Sliding mode control attempts to send the state of the system to a “sliding
surface,” from which the system will exponentially converge. This sliding surface is
typically defined as s = 0, where s is an error metric given by s = θ˙ + λθ. In order
to drive the dynamics of a given system to that of an exponentially stable system, a
signum function is applied. Slotine and Li [151] developed a control law specific to
robotic manipulators:
τd = Hˆ(θ)θ¨r + Cˆ(θ, θ˙)θ˙r + Gˆ−KDsgn(s) (5.5)
where H, C and G are the inertial, Coriolis, and gravitational components of a
general manipulator, respectively. The terms θ¨r = θ¨d − λ
˙˜θ and θ˙r = θ˙d − λθ˜ are
used to simplify the expression. This control law is guaranteed stable, assuming
instantaneous torque and a sufficiently high controller gain KD. Slotine and Li show
157
this by defining a Lyapunov function candidate:
V =
1
2
sTHs (5.6)
Differentiating this expression yields
V˙ = sTHs˙+
1
2
sT H˙s = sT (Hθ¨ +Hθ¨r) +
1
2
sT H˙s (5.7)
By substituting in the system dynamics from Eqn. 5.4 and the control law, Eqn.
5.5, rearranging terms, and noting that the matrix H˙ + 2C is skew-symmetric, the
equation [151]
V˙ = sT (H˜θ¨r + C˜θ˙ + G˜)−
n∑
1
KD,i|si| (5.8)
is obtained, where H˜,C˜, and G˜ represent model error in the corresponding matrices.
So long as the second term is greater than the first (i.e., KD is sufficiently high),
V˙ is negative, and the system reaches the sliding surface in finite time, and then
converges to θd exponentially.
For a single degree-of-freedom manipulator with damping, as is the case for
the present work, the control law Eqn. 5.5 simplifies to:
τd = Iˆ θ¨ + bˆθ˙r + mˆgr sin θ + C2(θ)−KDsgn(s) (5.9)
Torque cannot be commanded or monitored for direct feedback control in the present
system, but pressure can be regulated. Therefore, desired pressure is treated as the
158
Figure 5.4: Diagram of sliding mode controller.
controller output (system input). Eqn. 5.4 can be rearranged as:
P =
1
C1(θ)
{Iθ¨ + bθ˙ +mgr sin θ − C2(θ)} (5.10)
With the addition of a sliding term, the sliding mode control law then becomes:
Pd =
1
C1(θ)
{Iθ¨ + bθ˙ +mgr sin θ − C2(θ)} −KDsgn(s) (5.11)
Figure 5.4 illustrates the sliding mode control strategy. Note the inner pres-
sure feedback loop, which employs proportional-integral (PI) control. The desired
and actual pressure signals are filtered to ensure smoothness, which improves the
performance of the PI controller without reducing the tracking bandwidth.
5.3.2 Adaptive Sliding Mode Control
The sliding control methodology presented above can be extended to permit
adaptive updating of system parameters, potentially improving the response to
unmodeled dynamics and payload variation. Slotine and Li [50, 151] provide a
modification to the sliding controller to permit adaptation of certain parameters such
159
as inertia, damping, and gravitational terms. When the manipulator dynamics are
well known and equations are structured in a simple manner, parameter convergence
and good trajectory-following is possible.
In many applications, the model might be only partially known or the system
payload may change over time. We can define a matrix of “known” parameters
(functions of angular position and velocity) Y = Y (θ, θ˙, θr, θ˙r) such that the control
law becomes
τd = Hˆ(θ)θ¨ + Cˆ(θ, θ˙)θ˙ + Gˆ−KDsgn(s)
= Y (θ, θ˙, θr, θ˙r)aˆ−KDsgn(s)
(5.12)
where aˆ is the set of parameters to be updated. In the present case, we choose
aˆ = [Iˆ bˆ mˆgr].
Reformulating the equation in terms of a desired pressure, as in 5.11:
Pd =
1
C1(θ)
{Y (θ, θ˙, θr, θ˙r)aˆ−KDsgn(s)} (5.13)
The parameter estimates aˆ can be updated using a number of methods. In this study,
we choose a simple gradient-based method:
˙ˆa = −ΓY T s∆ (5.14)
where Γ is a user-defined (normally diagonal) adaptation gain matrix and s∆ is the
160
Figure 5.5: Diagram of adaptive sliding mode controller.
error metric modified to include a deadband:
s∆ =



s s > db
0 otherwise
(5.15)
db is a constant denoting the deadband width, and was set to 0.02. A diagram
of the complete system is shown in Figure 5.5. Note that this approach does not
necessarily estimate the unknown parameters exactly, but simply produces values
that allow the system to follow the desired trajectory.
5.3.3 Adaptive Neural Network Control
The adaptive control strategy presented above cannot adapt the highly nonlin-
ear, manipulator-specific equations relating pressure input to PAM force, or from
PAM force to manipulator torque (due to the complex joint geometry). Parts of the
manipulator model must be determined a priori, and cannot be adaptively updated
using Slotine’s framework. While it is possible to update several parameters, such
as Iˆ, bˆ, mˆgr, and even the coefficients of Cˆ2(θ) if desired using the above adaptive
method, the above strategy does not permit the adaptation of Cˆ1(θ). Using adaptive
161
sliding mode control, we must assume that Cˆ1(θ) is accurate when in fact, from Figure
5.2, we know there is some error between model and experiment. This motivates
the use of neural networks, which employ a flexible mathematical framework to
conform to the nonlinear system dynamics. In many cases, the application of neural
networks renders a detailed system model unnecessary, which is typically a great
benefit, though it comes at the expense of physical understanding and intuition of
the system.
The neural network control law is composed of several terms relating the known
functions of angle (and its derivatives) to unknown functions of angle:
Pd = Hˆ(θ)θ¨ + Cˆ(θ)θ˙ + Gˆ(θ)−KDsgn(s) (5.16)
where the “effective” inertia, damping, and gravity terms are given respectively by:
Hˆ(θ) = Cˆ1(θ)
−1Iˆ (5.17)
Cˆ(θ) = Cˆ1(θ)
−1bˆ (5.18)
Gˆ(θ) = Cˆ1(θ)
−1(mˆgr sin θ − Cˆ2(θ)) (5.19)
Using the above equations, we can approximate H(θ), C(θ), and G(θ) as nonlinear
functions of angle, and adapt the estimates in response to system excitation.
The neural network architecture used to capture nonlinear plant behavior
162
Figure 5.6: Single layer neural network structure.
was composed of a single layer of basis functions with corresponding weighting
functions, similar to the approaches in Liu [174] and Sanner and Kosha [173]. A
visual representation of the network is shown in Figure 5.6. Mathematically, the
network approximation function for H(θ) is given by a summation of weighted basis
function nodes:
Hˆ(θ) =
N∑
k=1
cˆk,Hφ(hθ − k) (5.20)
where k is the node index, h is the scale factor between nodes, N is the total number
of nodes, ck,H is the kth weighting coefficient denoting the “height” of the kth node,
and φ(hθ−k) is the kth basis function. Identical formulas are used for Cˆ(θ) and Gˆ(θ).
Triangular basis functions were chosen for their simplicity and ease of computation.
Weighting coefficients ck,H were adjusted at each time step so that the network would
converge to the desired function. The update law for the weighting function is given
163
Figure 5.7: Diagram of adaptive neural network controller.
by:
˙ˆck,H = −γHφ(hθ − k)θ¨rs∆ (5.21)
˙ˆck,C = −γCφ(hθ − k)θ˙rs∆ (5.22)
˙ˆck,G = −γGφ(hθ − k)s∆ (5.23)
where the γ terms are gains.
Figure 5.7 illustrates the neural network controller in detail. Its overall structure
is very similar to that of the adaptive sliding controller, except the neural network
outputs arrays corresponding to angle-dependent functions, rather than a few updated
parameters.
5.4 Position Control Results
The controllers were implemented in MATLAB/Simulink, which interfaced
with dSPACE software and hardware for data acquisition and real-time control.
164
Table 5.1: Parameter values of sinusoids in desired trajectory.
Index i Amplitude Ai (deg) Frequency ωi (rad/s)
1 20 0.70
2 13 0.95
3 7 1.55
4 5 0.80
Results from the numerous trajectory-following experiments performed on the robotic
manipulator demonstrate the utility of these distinct strategies.
5.4.1 Sliding Mode Control
The sliding mode controller was implemented first in experiments. The arm
was commanded to follow a trajectory composed of several sinusoids and a small
ramp. This desired trajectory (in degrees) is given by:
θd =
4∑
i=1
Ai cosωit+ ytri + 50 (5.24)
where Ai and ωi are the amplitude and frequency, respectively, of each sinusoid,
and ytri is a repeating triangular wave with a frequency of 0.16 Hz, a peak-to-peak
amplitude of 20 deg, and a bias of 10 deg. Table 5.1 displays the parameter values
of each sinusoid.
Figure 5.8 displays results from multiple sliding mode control experiments.
Figure 5.8(a) illustrates the desired and acutal angle trajectories, Figure 5.8(b)
shows angle error, and Figure 5.8(c) depicts the mean squared error (MSE) of each
experiment as time progresses. Overall, results demonstrate that this controller
165
is capable of smooth, chatter-free motion. When the system model is accurate,
including parameters Iˆ, bˆ, and mˆgr, then the controller cancels out most of the
nonlinear dynamics. This leads to high accuracy trajectory-following. However, when
these values are inaccurate, the system tends to over- or under-estimate the required
pressure. For instance, in Figure 5.8, the blue curve corresponds to a controller
with accurate measurements of joint inertia, damping, and mass. The red curve
represents these values decreased by 90% (0.1x), and the green curve represents these
values increased by 900% (10x). Surprisingly, the trajectory-following remains stable
despite these extreme parameter estimates. However, it is clear that the experiment
exhibiting the lowest error corresponds to the controller with accurate parameter
measurements.
While sliding mode control is capable of smooth, accurate performance, the
potential for poor estimation (which may be exacerbated with poor choice of controller
gains KD and λ) or dynamic variations in payload motivates the design of a controller
that can adaptively update.
5.4.2 Adaptive Sliding Mode Control
As shown in Section 5.3.1, sliding mode control can be easily modified for
parameter adaptation using the formulation developed by Slotine, with modifications
for the additional nonlinearities in this PAM-actuated system (C1(θ) and C2(θ)).
The adaptive controller was tested several times for the same trajectory and utilizing
the same controller gains as in non-adaptive sliding control. Each trial employed a
166
0 10 20 30 40 50 60 70 80 90 1000
50
100
An
gle
 (de
g)
 
 
0 10 20 30 40 50 60 70 80 90 100
−20
0
20
40
An
gle
 Err
or (d
eg)
0 10 20 30 40 50 60 70 80 90 1000
100
200
300
400
Time (s)M
ean
 Sq
. Er
ror
 (de
g^2
)
Desired
SMC
SMC, Underestimated
SMC, Overestimated
Figure 5.8: Sliding mode control results: (a) angle, (b) angle error, and (c) mean
squared error.
different set of adaptation parameter gains, listed in Table 5.2. Note that the trajec-
tory is sufficiently complex to satisfy the “persistence of excitation” requirements for
an adaptive controller of this type [151].
Figure 5.9 displays the angle trajectories and error metrics associated with
each trial. The results demonstrate that this adaptive controller is somewhat useful
for improving trajectory-following after a “learning” period. However, there are
underlying problems with the adaptation process.
The time-dependent adaptive parameter values obtained in each trial are shown
167
Table 5.2: Adaptation gains for adaptive sliding mode control.
γIˆ γbˆ γmˆgr
Trial 1 10 10 0.01
Trial 2 100 100 0.001
Trial 3 100 100 0.0001
0 10 20 30 40 50 60 70 80 90 1000
50
100
An
gle
 (de
g)
 
 
0 10 20 30 40 50 60 70 80 90 100−20
−10
0
10
An
gle
 Err
or (d
eg)
0 10 20 30 40 50 60 70 80 90 1000
50
100
150
Time (s)M
ean
 Sq
. Er
ror
 (de
g^2
)
Desired
Trial 1
Trial 2
Trial 3
Figure 5.9: Adaptive sliding control results: (a) angle, (b) angle error, and (c) mean
squared error.
in Figure 5.10. It is clear that parameter estimates are dependent upon choice of
adaptive gains γIˆ , γbˆ, and γmˆgr. Moreover, in most cases, the Iˆ and bˆ terms do not
converge. Inertia and damping play only a minor role when the system is moving
slowly, so the trajectory is not affected; however, increasing the excitation frequency
168
0 10 20 30 40 50 60 70 80 90 1000
500
1000
1500
I Es
tim
ate
 
 
Trial 1
Trial 2
Trial 3
0 10 20 30 40 50 60 70 80 90 1000
50
100
150
200
b E
stim
ate
0 10 20 30 40 50 60 70 80 90 1000
0.2
0.4
0.6
0.8
Time (s)
mg
r Es
tim
ate
Figure 5.10: Parameter adaptation for three experiments, varying adaptation gains.
or further increasing the gains led to an unstable system. When the mˆgr update
gain is high (blue curve), it is also unable to reach a steady-state. In fact, the mˆgr
estimate exhibits angle dependence, most notably in Trial 1. This suggests that the
joint nonlinearities, which were supposedly negated by the C1(θ) and C2(θ) terms,
have not been fully accounted for. One potential solution to this problem is the use
of adaptive neural networks, as was described previously.
169
5.4.3 Adaptive Neural Network Control
Figure 5.11 displays a trajectory-following experiment using the neural network
approach. Again, the same trajectory is used; the complex waveform was sufficient to
induce network learning. The network was initialized in two ways, with and without
knowledge of an initial model. The blue curve displays the trajectory when the neural
network has no initial knowledge of the system structure, Hˆ(θ) = Cˆ(θ) = Gˆ(θ) = 0,
while the red curve corresponds to a network initialized with an educated guess using
the simplified model. The results show that the angle error is consistently reduced
as the neural network is trained on-line, regardless of initial parameterization.
Additional insight can be gained from observing the evolution of the adaptive
functions Hˆ(θ), Cˆ(θ), and Gˆ(θ) as time elapses. Figure 5.12 displays the adaptive
functions that begin with no a priori information. Note that this system initially
assumes zero inertia, damping, and mass, which is clearly an underestimate. As
the experiment progresses, the functions are constructed. By 100 s, the adaptive
functions have nearly converged because the trajectory is being followed with very
low error. Figure 5.13 shows a similar test where a priori estimates of Hˆ(θ), Cˆ(θ),
and Gˆ(θ) are given. The final structures of these three functions are very similar to
those gained from using no prior information.
The power of neural networks is clear: even with no knowledge of the PAM
actuators or joint kinematics, functions that model the system dynamics may be
quickly assembled and yield a smooth, accurate closed-loop response. However, when
the networks are initially zero-valued, we observe that the ability to construct the
170
0 10 20 30 40 50 60 70 80 90 1000
50
100
An
gle
 (de
g)
 
 
Desired
No Prior Data
Overestimated Data
0 10 20 30 40 50 60 70 80 90 100−20
−10
0
10
20
An
gle
 Err
or (d
eg)
0 10 20 30 40 50 60 70 80 90 1000
50
100
150
Time (s)Me
an 
Squ
are
d E
rro
r (de
g^2
)
Figure 5.11: Adaptive neural network control for two distinct initial conditions; (a)
angle, (b) angle error, and (c) mean squared error.
function at angles above 90 deg is limited. Sharp “peaks” and “valleys” often form
on the cusp of the region where the network has been trained and where it has not,
as shown in the G estimate of Figure 5.12, between 80 and 100 deg. When the
desired arm trajectory reaches an angle corresponding to a “valley,” the controller
assumes low torque requirements, and gravity causes the arm to drop so that it does
not reach the desired angle and cannot be tuned. In this manner, the system never
builds up the capacity to learn high angles. The structure of the networks may be
partially to blame: the triangular basis functions employed in the present work have
171
0 20 40 60 80 100 120−20
−10
0
10
20
H E
stim
ate
 
 
0 s
20 s
40 s
60 s
80 s
100 s
0 20 40 60 80 100 1200
5
10
15
20
C E
stim
ate
0 20 40 60 80 100 1200
10
20
30
40
50
Angle (deg)
G E
stim
ate
Figure 5.12: Neural network adaptation over 100 s, 50 lb payload, no prior informa-
tion.
narrow support and tend to create sharp transitions as angle increases. This effect
may be mitigated through the use of Gaussian basis functions, though this would be
at the expense of increased complexity and computational load.
The neural network is able to learn the structure of systems with high payload
variation. Figure 5.14 displays data from tests with 0 lb, 25 lb, and 50 lb payloads.
From the 0 lb configuration to the 50 lb configuration, the inertia increases by a
factor of 30. Generally, error increases with increasing payload. This trend partially
arises because the network requires more time to “learn” functions with higher values.
172
0 20 40 60 80 100 120−20
−10
0
10
20
Angle (deg)
H E
stim
ate
 
 0 s
20 s
40 s
60 s
80 s
100 s
0 20 40 60 80 100 1200
5
10
15
20
Angle (deg)
C E
stim
ate
0 20 40 60 80 100 1200
50
100
Angle (deg)
G E
stim
ate
Figure 5.13: Neural network adaptation over 100 s, 50 lb payload, with model.
Additionally, as torque requirements increase, required pressure also increases, which
is more difficult for the system to maintain. It is important to note that the internal
pressure loop is a limiting factor in the system bandwidth.
Following adaptation of the neural networks to a particular payload, the payload
can change, and adaptation to the new payload will occur. In order to demonstrate
effects due to payload change, the controller was first tested with a light payload (15
lb), and the neural networks were initialized at zero (no prior information). After 100
s, the neural network structure in the system was saved. Subsequently the payload
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50
100
An
gle
 (de
g)
0 10 20 30 40 50 60 70 80 90 100−30
−20
−10
0
10
An
gle
 Err
or (d
eg)
 
 
Desired
0 lb
25 lb
50 lb
0 10 20 30 40 50 60 70 80 90 1000
100
200
300
400
Time (s)M
ean
 Sq
. Er
ror
 (de
g^2
)
Figure 5.14: Adaptive neural network control, varying payloads; (a) angle, (b) angle
error, and (c) mean squared error.
was increased to 50 lb, and the initial structure of the neural network was chosen to
be the saved data from the 15 lb test.
Figure 5.15 displays the construction of the initial neural network from the 15
lb payload test. Figure 5.16 shows the adaptation from the neural network which
starts with the final structure from Figure 5.15 and ends with a newly adapted
network. As expected, the cG function has an overall greater value, indicating that
payload weight has increased.
Figure 5.17 shows the trajectory-following of the neural network controller
174
0 20 40 60 80 100 120−2
−1
0
1
2
Angle (deg)
H E
stim
ate
 
 
0 s
20 s
40 s
60 s
80 s
100 s
0 20 40 60 80 100 1200
1
2
3
Angle (deg)
C E
stim
ate
0 20 40 60 80 100 1200
10
20
30
40
50
Angle (deg)
G E
stim
ate
Figure 5.15: Neural network adaptation over 100 s, 15 lb payload, no prior informa-
tion.
with a 50 lb payload, comparing the response from a neural network initialized with
no prior data with a network pre-trained with a 15 lb payload. As expected, the
mean-squared error is less for the system that has been pre-trained.
5.5 Discussion and Comparison of Controllers
Figures 5.19 and 5.20 compare four controllers following a complex sinusoidal
trajectory and a lift-hold-return trajectory, respectively. In addition to the three
controllers designed in this work, a cascaded PID controller (with angle and pressure
175
0 20 40 60 80 100 120−2
0
2
Angle (deg)
H E
stim
ate
 
 
0 s
20 s
40 s
60 s
80 s
100 s
0 20 40 60 80 100 1200
1
2
3
Angle (deg)
C E
stim
ate
0 20 40 60 80 100 1200
20
40
Angle (deg)
G E
stim
ate
Figure 5.16: Neural network adaptation over 100 s, 50 lb payload, with network
initialized from 100 s of training with 15 lb payload.
feedback) was added for comparison. A diagram of the controller is shown in Figure
5.18. For these comparisons, the two adaptive controllers were trained in previous
experiments for 1000 s, and the resulting adaptive functions were saved. Therefore,
the adaptive controllers begin with the ability to compensate for the system dynamics.
The mean squared error plot, Figure 5.19(c), illustrates that the trained
adaptive controllers exhibit the highest cumulative accuracy by the time of trajectory
completion. With accurate initial parameters, sliding mode control is more accurate
than cascaded PID control, but less accurate than the two adaptive controllers. The
176
0 10 20 30 40 50 60 70 80 90 1000
50
100
An
gle
 (de
g)
0 10 20 30 40 50 60 70 80 90 100−30
−20
−10
0
10
20
An
gle
 Err
or (d
eg)
 
 
Desired
Trained w/ 10 lb
No Prior Training
0 10 20 30 40 50 60 70 80 90 1000
100
200
300
Time (s)
Me
an 
Sq.
 Err
or (d
eg 
2 )
Figure 5.17: Adaptive neural network control, varying initial network structure; (a)
angle, (b) angle error, and (c) mean squared error.
Figure 5.18: Diagram of a cascaded PID controller.
sliding mode controller and adaptive sliding mode controller trajectories both display
low-amplitude, high-frequency content, which can be observed in Figure 5.19(b).
Depending on the intended application, the vibration transmitted to the payload
(e.g., an injured person) may be undesirable.
177
In Figure 5.20 similar results are observed. The adaptive sliding mode controller
again provides the greatest accuracy, but sliding mode control is slightly more accurate
than neural network control. However, this is primarily an artifact of the error caused
by the neuro-controller’s long rise time to the initial “resting” angle of 15 deg. After
that initial rise, the steady-state error of the neural network control trajectory is
considerably lower than both forms of sliding mode control. Also note that the
adaptive and non-adaptive sliding mode controllers again sacrifice smoothness for
accuracy during the return phase, illustrated by the minor oscillations between 15
and 20 s in Figure 5.20(a), which is not preferred for the intended application of
human interaction, as stated previously. Overall, all of the advanced controllers
provide good performance.
Considering the results of the previous section, it is clear that the proper
selection of a control strategy depends upon the level of prior knowledge of the
system and the intended application. If the system has been well characterized
and parameters do not vary, then standard sliding mode control will likely suffice.
Experiments employing the sliding mode controller were able to follow a complex
waveform, and demonstrated robustness to large changes in model parameters.
If the system is poorly characterized, or is expected to confront dynamic
changes in payload, an adaptive strategy is likely preferable. In many lifting tasks,
particularly lifting humans for casualty extraction applications, the payload may
shift. For this reason, it is important that an adaptive capability be added to
the robot. The modified version of Slotine’s adaptive sliding controller displayed
some improvement over non-adaptive (standard) sliding mode control. However, the
178
0 10 20 30 40 50 60 70 80 90 1000
50
100
An
gle
 (de
g)
0 10 20 30 40 50 60 70 80 90 100−30
−20
−10
0
10
20
An
gle
 Err
or (d
eg)
 
 
Desired
PID
SMC
ASMC
NN
0 10 20 30 40 50 60 70 80 90 1000
100
200
300
Time (s)M
ean
 Sq
. Er
ror
 (de
g^2
)
Figure 5.19: Comparison of control strategies, compound sinusoidal trajectory; (a)
angle, (b) angle error, and (c) mean squared error.
system parameters were unrealistic and appeared to diverge; if the system runs for
an extended duration, instability may occur. The neural network controller was
designed to capture the dynamics of the PAMs and joint geometry that could not
be captured in the adaptive sliding mode controller. In experimentation, the neural
network controller was nearly as accurate as the adaptive sliding mode controller
and showed strong signs of convergence, but required much less knowledge of the
system structure than the adaptive sliding mode controller.
Although the neural controller is capable of adapting to nonlinear functions
179
0 5 10 15 20 250
50
100
An
gle
 (de
g)
0 5 10 15 20 25
−10
0
10
An
gle
 Err
or (d
eg)
 
 
Desired
PID
SMC
ASMC
NN
0 5 10 15 20 250
50
100
150
Time (s)M
ean
 Sq
. Er
ror
 (de
g^2
)
Figure 5.20: Comparison of control strategies, lift-hold-return trajectory; (a) angle,
(b) angle error, and (c) mean squared error.
of joint angle, some nonlinearities which are not accounted for may persist. A
key example of one such nonlinearity that will impact controller performance and
convergence properties is hysteresis. Hysteresis is dependent on the displacement
history, and implementing compensation in either of the present adaptive frameworks
is difficult. Major improvements in trajectory-following performance are expected if
hysteresis can be properly compensated. Hence, this will be the subject of future
work.
Neural networks are an extremely useful form of control, particularly when
180
much of the system dynamics are unknown. The above results demonstrate that
a network can be trained fairly quickly to recognize nonlinear functions of angle
without any prior information to estimate the functions. Although the detailed
kinematic and dynamic nonlinearities of a system may not be fully known, in some
cases, it may be useful to initialize the neural controller with a rough estimate of
Hˆ(θ), Cˆ(θ), and Gˆ(θ). This can be achieved by assuming that PAM force behavior
is characterized by the simple Gaylord model [23], and that the effective radius
providing the actuation moment arm is constant or a linear function of angle. To
this end, a supplementary experiment was conducted with initial estimates of the
control law terms, Eqns. 5.17, 5.18, and 5.19. The approximation for C1(θ) based
on the Gaylord model is given by:
C1(θ) =
3L(θ)2 −B2
4piN2
reff (5.25)
where L = 5 in is the length of the actuator (approximated as a linear function of
θ), B = 11.23 in is braid length, N = 0.9 is the number of turns a single braid fiber
makes around the PAM circumference, and reff = 0.7 in is the approximate effective
radius from which PAM force is translated into joint torque.
Figure 5.21 displays results from two trajectory-following experiments, com-
paring the controller initialized with the Gaylord model to a controller with no prior
information, as shown previously. The use of the Gaylord model slightly reduces the
overall error.
Figure 5.22 illustrates the neural network adaptation of Hˆ(θ), Cˆ(θ), and Gˆ(θ)
181
0 10 20 30 40 50 60 70 80 90 1000
50
100
An
gle
 (de
g)
0 10 20 30 40 50 60 70 80 90 100−30
−20
−10
0
10
20
An
gle
 Err
or (d
eg)
 
 
0 10 20 30 40 50 60 70 80 90 1000
50
100
150
200
Time (s)M
ean
 Sq
. Er
ror
 (de
g^2
)
Desired
Gaylord Estimation
No Prior Info
Figure 5.21: Adaptive neural network control, 0 lb payload, Gaylord model vs. no
information.
using the Gaylord model to construct initial estimates. Adaptation of the Cˆ(θ) and
Gˆ(θ) terms is clearly in the positive direction, signaling that the initial estimate
provided by this model is low. Note that the Gaylord model typically overestimates
force (especially at large contractions); thus, C1(θ) is expected to be overestimated
and Cˆ(θ) and Gˆ(θ) are underestimated as a result.
In order to perform complex manipulation tasks, potentially in three-dimensional
workspaces, systems with multiple degrees-of-freedom are required. When extended
to multi-dimensional systems, the principles of the three controllers will remain
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0 20 40 60 80 100 120−5
0
5
Angle (deg)
H E
stim
ate
 
 0 s
20 s
40 s
60 s
80 s
100 s
0 20 40 60 80 100 1200
5
10
Angle (deg)
C E
stim
ate
0 20 40 60 80 100 1200
10
20
30
40
Angle (deg)
G E
stim
ate
Figure 5.22: Neural network adaptation over 100 s, 0 lb payload, with Gaylord
model.
applicable, but their complexity will increase (e.g., Coriolis forces will appear, inertia
matrix will expand). In the case of neural network control, the networks themselves
must become multi-dimensional functions of the various angle and angular veloc-
ity terms. The addition of a multi-dimensional network structure may impart a
significant computational burden on the controller.
Lastly, it is important to note that, despite using a complex sinusoid, the
frequency of actuation in the chosen trajectory was relatively low (maximum of 0.25
Hz), due to the intended application of lifting and carrying humans. In many other
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manipulator applications, higher frequencies are required. High frequency motion
will cause the inertia and damping to play a greater role in the dynamics, and will
rely on the bandwidth of pressure feedback. Furthermore, additional high-frequency
dynamics may be excited in the manipulator or the PAMs.
5.6 Conclusion
Three advanced control strategies were applied to the shoulder joint of a single
degree-of-freedom PAM-actuated robotic manipulator. The controllers were evaluated
in trajectory-following experiments, demonstrating different levels of accuracy for
different situations. The standard sliding mode controller demonstrated robust
trajectory-following, but to guarantee low error, parameter estimates such as arm
inertia, damping, and mass were required to be known a priori. Because large
payload variations are expected in the present manipulator application, the standard
sliding mode controller was modified to include adaptation of payload-dependent
parameters. While trajectory-following was improved, the underlying nonlinearities
in the PAMs and joint kinematics were not fully addressed with this control structure.
Therefore, a neural network controller was implemented to determine angle-dependent
nonlinearities in the system while maintaining the ability to adapt to payload
variations.
Results demonstrate the utility of all three of the nonlinear controllers, including
improvements over cascaded PID control. Neural network control has great potential
for learning and negating the complex nonlinearities present in a PAM-actuated
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system. Future work may include hysteresis compensation, multiple degree-of-freedom
experiments, and high-frequency motion experiments.
185
Chapter 6: Variable Recruitment
6.1 Introduction
Pneumatic artificial muscles (PAMs) are actuators known for their high specific
work, natural compliance, flexibility, and extremely light weight. Also known as
McKibben muscles, PAMs were developed in the 1950s to actuate orthotic devices for
polio patients [23]. PAMs are composed of an elastomeric bladder surrounded by a
helically-braided fiber sleeve. Pressurization of the soft bladder causes the braid fibers
to reorient, generating contractile stroke in their typical configuration. As a result of
their functional similarity to human muscle (decreasing length, increasing diameter),
PAMs have been employed as the muscles in robotic arms [40,160], legs [132,137],
and hands [178,181].
Most PAM-based anthropomorphic robotic systems are designed with a single
PAM working as an analog to a single human muscle. However, force buildup in
human muscle is a complex phenomenon, developing on the level of small groups
of muscle fibers called motor units [189]. In the human neuromuscular system,
muscle force is adjusted by: (1) varying the firing rate of motor neurons, and (2)
selective recruitment of motor units [191]. The activation of a single motor unit will
create a distributed but weak signal; as force requirements increase, the number of
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active motor units also increases. Motor units are generally recruited in order of
smallest (slow, low-force, fatigue resistant) to largest (fast, high force, less fatigue
resistant) [180]. Motor unit firing rate typically increases with increasing muscular
effort, which helps smooth out the incremental force change as motor units are
added [175].
Analogously, PAMs can be easily arranged in parallel and activated individually
as torque requirements dictate. Although PAM research to date has mostly ignored
variable recruitment, there are advantages to matching the form and function of
natural muscles on the level of individual motor units. The main benefit of variable
recruitment is the potential for improved efficiency over the entire range of anticipated
force. When force requirements are low in a PAM-based system, a subset of the PAMs
in a muscle “group” can be set to a relatively high pressure, while not pressurizing
others, rather than activating all of the actuators at a lower pressure. Efficiency in a
parallel PAM-based system is consequently improved for two major reasons. First,
the energy required to overcome elastic bladder resistance is proportional to the
number of PAMs employed. Therefore, if fewer PAMs are required, less energy is
“wasted” to overcome bladder resistance. Second, pressure-dependent losses associated
with compressible airflow in the pneumatic system can be exacerbated because high
recruitment levels correspond to a larger gradient between source and actuator
pressure as well as higher actuator “dead space” volume.
Despite the potential advantages of variable recruitment, research regarding
the topic is limited. Odhner and Asada [194] explored the variable recruitment of
multiple shape memory alloy (SMA) wires in a robotic arm to modulate force and
187
stiffness. By using multiple thin wires instead of one large wire, faster heat diffusion
was achieved, and bandwidth was increased. However, efficiency was not quantified.
Lorussi et al. [192] developed simulations of a dielectric elastomer muscle bundle.
Huston et al. [190] explored concepts for hierarchical actuator systems, including
designs that can improve precision, speed, efficiency, smooth motion, and redundancy.
Repperger et al. [193] applied a bio-inspired control strategy to a single pneumatic
muscle. The authors claim that individual “force twitches” mimic the control aspect
of recruitment, but a single pneumatic muscle does not physically emulate variable
recruitment.
In research highly relevant to the current study, Bryant et al. [179] constructed
an actuator consisting of three small-diameter PAMs and three large-diameter PAMs
in a parallel arrangement. Their results demonstrated that the two motor units
can be activated separately or in unison to produce varying levels of force. The
researchers hypothesized that efficiency improvements could result from variable
recruitment, but measurements of actuator efficiency (ratio of output work to input
energy) were left for future work.
To date, no known studies have investigated variable recruitment of pneumatic
artificial muscles in practical robotics applications. The goal of the present research
is to determine the practical value of a variable recruitment scheme in a robotic
actuation system featuring a parallel arrangement of multiple equivalently sized PAM
actuators. The exemplary robotic system chosen for this study was the shoulder joint
of a robotic manipulator with heavy lifting capacity. First, a comparison of quasi-
static actuator force behavior is presented for the candidate PAMs, showing that
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fewer PAMs can produce greater force output for a given compressed air mass. Then,
the efficiency of the manipulator joint is experimentally evaluated for different levels
of recruitment. Simulations of the joint are also performed, validating experimental
trends. Results demonstrate the benefits of variable recruitment and point toward
the existence of an optimal recruitment strategy.
6.2 Preliminary Characterization and Analysis of Bladder Effects
A considerable source of efficiency loss during PAM force generation is the
elastic resistance of the internal bladder. Previous studies have shown that the
force behavior of a PAM actuator can be highly nonlinear as a result of bladder
hyper-elasticity [27, 30]. Kothera et al. [35] modified an existing force balance model
originally formulated by Ferraresi et al. [34] to better account for bladder effects and
provide more accurate force estimates. Using this approach, PAM force is given by:
F =
P
4piN
(3L2 −B2) + P
(
VB
L
−
tL2
2piRN2
)
+ σz
VB
L
− σc
tL2
2piRN2
(6.1)
where P is PAM pressure, L is strained actuator length, R is strained actuator radius,
t is strained bladder thickness, B is (constant) braid length, N is number of turns a
braid fiber makes around the bladder, VB is (constant) bladder volume, σz is axial
stress, and σc is circumferential stress. The first term, known as the Gaylord force,
is a function of actuator geometry. The second term accounts for the loss in force
due to bladder thickness subtracting from actuator volume. The third and fourth
189
terms quantify the change in force due to bladder resistance.
In order to determine the effects of bladder resistance and motivate a variable
recruitment strategy, force behavior was compared between “energy equivalent”
actuator configurations. In these experiments, the force behavior of one PAM
pressurized to a constant value was compared to two identical PAMs pressurized to
half that value. The PAM(s) were contracted at constant pressure, and quasi-static
force data was recorded. Following the ideal gas law, and assuming temperature is
constant and PAM volume is independent of pressure (only a function of contraction),
both configurations hold the same air mass for the same contraction. Therefore, the
energy input between the two configurations can be considered equal and provide a
baseline for comparison.
Measurements, however, show differences in force output. Figure 6.1 displays
comparisons between single- and dual-PAM configurations at “low” pressure levels (2
PAMs @ 2 bar vs. 1 PAM @ 4 bar) and “high” pressure levels (2 PAMs @ 4 bar vs.
1 PAM @ 8 bar) using sample PAM A, with geometric and material properties listed
in Table 6.1. Although quasi-static PAM force data exhibits direction-dependent
hysteresis, contraction is the only motion in which positive work is being done by
the PAM. Therefore, only the contractile portion of each curve is displayed. For
any substantial amount of contraction, the force from a single PAM is greater than
the force of two PAMs (at half pressure). Free contraction of the single PAM is
also greater. These phenomena are mainly caused by PAM elasticity that resists
circumferential expansion and axial contraction in a nonlinear manner. At low
pressure, a high proportion of the stored energy is being spent to overcome bladder
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Table 6.1: Geometric and material parameters for test PAMs.
Parameter PAM A PAM B
Diameter (cm [in]) 2.5 [1.0] 1.9 [0.75]
Length (cm [in]) 27.6 [10.9] 12.5 [4.9]
Braid Angle (deg) 73.5 75
Bladder Thickness (cm [in]) 0.32 [0.125] 0.32 [0.125]
Bladder Material Latex Silicone
Shore Hardness 35A 25A
Figure 6.1: Comparison of single-PAM (full pressure) vs. dual-PAM (half pressure)
force, PAM A.
resistance; as pressure increases, this proportion decreases, and the PAM becomes
more efficient. Hocking and Wereley [27] noted that initiation pressure is required for
the bladder to make substantial contact with the surrounding braid. This pressure
“deadband” accounts for some of the force reduction in the dual-PAM configuration
because two deadbands must be overcome instead of one. Results support using
as few PAMs as possible to lift a given payload to maximize efficiency. As force
requirements exceed the maximum force of the active PAM group, additional PAMs
can be activated as needed. In other words, a variable recruitment scheme should be
used.
191
Elasticity losses may be combated by fabricating PAMs with an extra-soft
bladder. To illustrate the effects of a softer bladder, a second set of experiments
were performed, with sample PAM B (see Table 6.1). This PAM has a silicone
rubber bladder (Bluestar Bluesil V-330), with a lower Shore Hardness than the latex
bladder used in PAM A [185]. Figure 6.2 shows force characterization data for two
recruitment configurations at “high” and “low” pressure levels as in Figure 6.1. The
reduction in force associated with using two PAMs is not nearly as dramatic for the
silicone PAM (B) as was observed with the latex PAM (A). In fact, force loss does
not occur until contraction reaches 5%. Although the different dimensions of these
two sample PAMs make a direct comparison misleading, one would expect the PAM
with a higher bladder thickness-to-diameter ratio (PAM B) to incur more losses
if both PAMs used the same bladder material. Nevertheless, PAM B has better
low-pressure performance because the bladder is softer. The minor difference in force
output suggests that a softer bladder reduces the benefit of variable recruitment.
However, in practical applications, one must consider not only quasi-static PAM
force, but the efficiency of the dynamic system as a whole.
6.3 Variable Recruitment on a Robotic Manipulator
Several factors contribute to efficiency losses in PAM-actuated dynamic systems,
many of which may affect a variable recruitment strategy. In addition to bladder
elasticity, losses caused by airflow dynamics, excess actuator and tubing volume,
friction, and damping may change with recruitment level. This section describes
192
Figure 6.2: Comparison of single-PAM (full pressure) vs. dual-PAM (half pressure)
force, PAM B.
the design and operation of a PAM-based robotic manipulator. In simulations and
experiments, the efficiency of the entire system (pneumatics and arm mechanics)
is measured at each level of motor unit activation in order to holistically evaluate
variable recruitment.
The robotic actuation system employed in this work was the shoulder joint of
the two degree-of-freedom planar manipulator shown in Figure 6.3. At maximum
pressure (13 bar), the shoulder joint is capable of 850 Nm (630 ft-lb) of torque up
to 90 deg, enough to lift a 95 kg (210 lb) tip payload with the 0.9 m (3 ft) arm
fully outstretched. This joint is actuated by six latex bladder PAMs, all identical
to PAM A (recall Table 6.1). The shoulder joint and actuators are shown in an
exploded-view CAD model in Figure 6.4. The PAMs pull downward on an aluminum
plate which slides vertically on linear bearings. Tendon links connect the plate to
the shoulder joint shaft, which allows the PAMs to generate rotation. The PAMs
act unidirectionally to lift the arm, while gravity acts in the opposing direction to
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Figure 6.3: Robotic arm holding a 95 kg (210 lb) payload.
Figure 6.4: Exploded view of shoulder joint.
restore the arm to its resting angle when pressure is removed.
In the variable recruitment study by Bryant et al. [179], the total force of
six PAMs actuated together was less than the sum of the forces produced by the
individual motor units. The reason for this force reduction (mostly at low contraction)
was unclear; however, the tightly packed configuration may have been a factor, as
friction between PAMs can adversely affect force output, as can limitations on radial
expansion imposed by the tight packing. In order to eliminate the effects of friction
and resistance between PAMs in the present study, the actuators were given ample
space (6.35 cm [2.5 in] diameter) to radially expand without interaction.
194
The PAMs are connected to a pressure source through a single proportional
valve. This valve varies its orifice area in response to a voltage input (0-10 V). Voltage
signals were produced in response to measured angle and pressure by dSPACE control
hardware and software [186]. The PAMs were recruited in groups of two, and the
number of active PAMs was varied off-line between (not during) tests by manually
adjusting the tubing connections.
6.4 Variable Recruitment Simulations
Simulations were performed to predict the effects of variable recruitment and
to justify experimental work. These simulations employed a detailed model of the
robotic manipulator described in Section 6.3, including the pneumatic components
used to channel pressurized air to the PAMs. Each simulation consisted of repeated
lifting cycles along a prescribed trajectory. Payload weight and level of recruitment
were varied, and system efficiency was determined for each test. This chapter
describes the dynamic model used in simulation, the procedure completed to obtain
reliable values of efficiency, and simulation results.
6.4.1 Dynamic Modeling
Figure 6.5 illustrates the system model used in simulation. This model describes
the dynamics of the airflow and the dynamic response of the manipulator. First,
voltage ε is input to the proportional valve, which controls the mass flow rate of air
m˙ from air source to PAMs (and in the case of deflation, from PAMs to the ambient
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Figure 6.5: Diagram of PAM-based robotic system.
environment) by changing the size of the valve orifice. Mass flow into the PAM
leads to pressure buildup, which generates PAM force. Pressurized PAMs, which are
indirectly connected to the joint shaft, produce shoulder torque. Shoulder torque is
a function of pressure and PAM contraction.
The system experiences compressible, potentially choked air flow through the
variable-area valve orifice. A well-founded dynamic model of the pneumatic system
must account for the compressibility of air, the heat transfer associated with mass
flow into and out of the valve, changing PAM volume, and dead space in the PAMs.
Dead space is the initial volume of the PAM that must be pressurized before motion
begins. The air that occupies this space does no work but nonetheless requires energy
to maintain pressure.
Some known but complex effects were neglected in this simulation. The Joule-
Thomson effect, better known as “throttling,” is an isenthalpic process (i.e. a process
which is adiabatic, irreversible, and extracts no work) that causes the temperature of
a non-ideal gas to drop as a function of the pressure gradient through an orifice [184].
While some cooling may occur as air passes through the valve, temperature change
is minimal at the experimental range of pressures and is believed to have very little
effect on overall efficiency. Therefore, we have opted to treat the air as an ideal gas.
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Losses from tubing friction were also neglected.
In a pneumatic system with a sharp-edged variable-area orifice [187], the
relationship between mass flow rate, pressure, and area is given by:
m˙air =



ACdPus
√
2
RTC1,
Pds
Pus
< Pcr
ACdPus
√
2
RTC2,
Pds
Pus
≥ Pcr
(6.2)
In the above expression, m˙ is mass flow rate, A is orifice area, and Pus and Pds are the
upstream and downstream pressures, respectively. Note that during PAM deflation,
the upstream pressure corresponds to the PAM pressure, and the downstream pressure
is the ambient room pressure. A lookup table provided by the valve manufacturer
was used to determine orifice area for a given voltage input. The term Cd is a
discharge coefficient capturing the losses in the orifice, Pcr = (2/(γ + 1))γ/(γ−1) is the
critical pressure separating subsonic and supersonic flow, and C1 and C2 are given
by:
C1 =
√
γ
γ−1
[(
Pds
Pus
) 2
γ
−
(
Pds
Pus
) γ−1
γ
]
C2 =
(
2
γ+1
) 1
γ−1
√
γ
γ+1
(6.3)
where γ is the specific heat ratio of air.
Richer and Hurmuzlu [158] derived an expression for pressure change in a
container of changing volume using the ideal gas law, conservation of mass, and
197
conservation of energy:
P˙ =
RairT
V
(αinm˙in − αoutm˙out) +
PV˙
V
α (6.4)
where Rair is the specific gas constant for air, T is air temperature, and Vair is air
volume in the PAMs. The coefficients αin = 1.4 and αout = 1 are estimates of specific
heat ratio based on heat transfer as mass flows into and out of the valve, respectively.
The parameter α = 1.2 is the specific heat ratio corresponding to PAM volume
change. From the above expression of pressure rate, time-varying pressure inside the
PAMs can be determined by integration.
The mass flow exiting the air source implies decreasing source pressure. Using
prior knowledge of the mass flow rate exiting the constant-volume tank at constant
temperature, source pressure rate is given by:
P˙tank =
RairT
Vtank
m˙ (6.5)
Actuator force is given by Eqn. 6.1. For enhanced accuracy, it is assumed that
the PAM exhibits a hyperelastic stress-strain relationship. Axial and circumferential
stresses are empirically fit as fourth-order polynomial functions of axial and circum-
ferential strain to well match the experimental data of PAM A, as in our earlier
work [188].
PAM force is translated into joint torque through a complex, custom-designed
joint geometry intended to maintain a fairly constant torque for a given pressure
within the shoulder joint angle limits (15 deg to 120 deg). The details of the joint
198
geometry are inconsequential for this study, and are therefore omitted. Torque, like
PAM force, is a function of pressure and actuator strain.
The dynamics of the one-degree-of-freedom manipulator are given by:
τ = Iθ¨ +G(θ) + cθ˙ + fsgnθ (6.6)
where I is the total (manipulator and payload) inertia, G is torque due to gravity, c
is velocity-dependent damping, and f is torque due to Coulomb friction. Friction
and damping in the joint were estimated empirically from test data.
6.4.2 Simulation Procedure
A detailed procedure was implemented to simulate manipulator motion and
extract values of system efficiency. Simulations were designed such that they could
be reproduced in actual experiments. In each simulation, the manipulator followed a
time-periodic trajectory consisting of eight smooth lifting cycles. Each lifting cycle
was composed of a 6 s cosine ramp from minimum to maximum angle, a 5 s hold at
maximum angle, a 6 s cosine ramp back to the minimum angle, and a 5 s hold at
minimum angle. This was regulated by a closed-loop cascaded proportional-integral
(PI) controller with pressure and angle feedback. Figure 6.6 depicts the control
architecture used in the simulations. Controller gains were set to ensure that the
arm would reach the maximum desired angle smoothly and with negligible overshoot.
This process was repeated for four payload weights and two maximum arm angles
(50 deg and 90 deg). Test matrices for the two trajectories are shown respectively in
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Table 6.2: Test matrix for variable recruitment, 50 deg max. angle.
Payload Weight 2 PAMs 4 PAMs 6 PAMs
0 kg (0 lb) X X X
9 kg (20 lb) X X X
18 kg (40 lb) X X X
27 kg (60 lb) X X X
Table 6.3: Test matrix for variable recruitment, 90 deg max. angle.
Payload Weight 2 PAMs 4 PAMs 6 PAMs
0 kg (0 lb) X X X
9 kg (20 lb) X X X
18 kg (40 lb) - X X
27 kg (60 lb) - X X
Figure 6.6: Diagram of cascaded PI control system.
Tables 6.2 and 6.3.
Total work output was calculated by summing torque multiplied by increments
of angle. Because cycling the arm back to the initial angle yields a net work of zero
and thus masks information about efficiency, increments of work were only summed
if angle was increasing (∆θi > 0):
Wout =
Nθ+∑
i=1
τi∆θ+,i (6.7)
where Nθ+ is number of positive work increments, τi is torque, and ∆θ+,i is (positive)
change in angle.
Input energy was approximated by observing the change in initial and final
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stored energy in the air source:
Ein = (Pi − Pf )Vtank (6.8)
where Pi and Pf are initial and final pressure, and Vtank is the constant volume of
the pressure source (15 L).
The efficiency of the cycling process was then determined by dividing output
work by input energy:
η =
Wout
Ein
(6.9)
It should be noted that the electrical efficiency of the compressor itself was
not simulated and was excluded from the efficiency calculation. The compressed air
source was set to the same initial pressure regardless of PAM number; therefore, in
the present study, compressor efficiency had no bearing on a variable recruitment
strategy.
6.4.3 Simulation Results
Figures 6.7 and 6.8 plot system efficiency for each set of simulations, illustrating
the effects of different payload weights and numbers of PAMs. For each level of
recruitment, system efficiency increases with payload mass. It is also clear that using
fewer PAMs (when possible) increases system efficiency. The efficiency improvements
are most beneficial at the lowest payloads. Overall, experimental results provide
strong justification for a variable recruitment strategy.
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Figure 6.7: Efficiency of manipulator with variable recruitment, 50 deg max. angle.
Figure 6.8: Efficiency of manipulator with variable recruitment, 90 deg max. angle.
The results from simulations suggest a strategy for adjusting recruitment level.
At the lowest payloads, the fewest number of PAMs (a single motor unit) should be
recruited to maximize efficiency. When payload weight requires torque greater than
the capacity of a single motor unit, an additional motor unit can be recruited, and
so on as torque requirements continue to increase, until all motor units are recruited.
Figure 6.9 illustrates the most efficient number of motor units as payload changes,
with simulations of the 90 deg trajectory extended to 63 kg (140 lb).
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Figure 6.9: Maximizing efficiency through variable recruitment, simulations, 90 deg
max. angle.
Motivated by the observations in Section 6.2, the implications of bladder
stiffness were also explored. Simulations provide an opportunity to ignore the bladder
effects by using only the Gaylord force term to model PAM force. Comparing
simulations with and without elasticity in Figure 6.10, it is clear that bladder
resistance plays a major role in reducing efficiency. Without axial or circumferential
bladder stress (which is physically impossible, but helpful for illustration), the
efficiency range jumps from 10-30% to 52-66%. Moreover, when elasticity is ignored
in simulation, the importance of recruitment becomes less clear. For example, the
2-PAM configuration displays the greatest efficiency at zero payload, but becomes the
least efficient at the maximum payload. As explained earlier, however, simulations
treated air as an ideal gas, and unmodeled losses such as throttling may affect results.
It is important to note, however, that softer bladders are more prone to fatigue and
failure. A practical PAM-actuated system must strike a balance between efficiency
and durability.
203
Figure 6.10: Efficiency of variable recruitment with and without bladder elasticity,
simulations, 90 deg max. angle.
6.5 Variable Recruitment Experiments
Motivated by the results of simulations, experimental testing was performed
on the manipulator. A comparison revealed similar trends between simulations and
experiments. Using the combined results, the relationship between recruitment and
efficiency was evaluated.
6.5.1 Experimental Procedure
In order to accurately measure arm efficiency, a detailed experimental procedure
was developed and implemented. Experiments were performed in a manner that
mirrored simulations. First, a specified payload mass (0, 9, 18, or 27 kg [0, 20, 40
or 60 lb]) was securely fastened to the end of the arm. Next, the compressed air
tank was filled until an automatic stop was triggered (at 13.8 bar [200 psi]), and
this “initial” pressure was recorded. At that point, the tank auto-start trigger was
204
turned off and arm cycling was begun immediately (to minimize the amount of
air leakage) with a predetermined number of active PAMs. As in simulation, the
arm trajectory was composed of eight smooth lift-hold-drop cycles regulated by a
closed-loop cascaded proportional-integral (PI) controller, with pressure and angle
feedback. Controller gains were set in a pre-test to ensure that the arm would reach
the maximum desired angle smoothly and with negligible overshoot. During each
test, pressure, angle, and valve voltage data were recorded. When the cycling was
completed after 200 s, final tank pressure was recorded. Input energy, output work,
and efficiency were calculated, as in simulation (see Section 6.4.2).
While the arm is capable of lifting payloads over 90 kg (200 lb), source pressure
drops substantially during the cycling tests, causing the system to improperly follow
the desired trajectory. This limited the maximum test payload to 27 kg (60 lb).
6.5.2 Experimental Results
Figure 6.11 depicts the time history of arm angle and estimated torque during
an example experiment (18 kg [40 lb] payload, 2 PAMs activated), highlighting in
red the areas where positive work is being produced. The lifting motions are smooth
and repeatable despite the source pressure dropping during the tests. Applying Eqns.
6.7, 6.8, and 6.9 to each data set yields an approximation of manipulator efficiency.
Figure 6.12 displays simulation and experimental data for output torque vs.
pressure for a sample test case (18 kg [40 lb] payload, 2 PAMs). The strong
similarity between these hysteretic curves provides evidence that the joint and
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Figure 6.11: Angle and torque vs. time, 2 PAMs, 18 kg (40 lb) payload, 90 deg max.
angle.
Figure 6.12: Torque vs. pressure, experiment and simulation, 2 PAMs, 18 kg (40 lb)
payload.
actuator components of the system are modeled correctly.
Efficiency results from experiments display similarities with simulations. In
Figure 6.13, we see that the use of fewer PAMs improves efficiency as in simulation,
and efficiency increases with payload mass at about the same rate as in simulation.
However, there is an offset between simulation and experimental data, likely caused
by unmodeled losses.
206
Figure 6.13: Efficiency of manipulator with variable recruitment, 90 deg max. angle.
Figure 6.14 shows that work output remains essentially constant for all levels
of recruitment in both simulations and experiments. The discrepancy in efficiency
between the two is caused by the difference in input energy consumed, as shown in
Figure 6.15. The required energy for experiments is greater than simulations in each
test case, suggesting that unmodeled losses which are not present in simulation can
contribute to the efficiency. These unmodeled factors, which could include leakage,
tubing losses, and temperature change, are difficult to accurately simulate but could
be added in future work.
6.6 Discussion
Simulated and experimental results suggest a simple algorithm for achieving
high efficiency via variable recruitment: Employ as few motor units as possible to
satisfy torque requirements, and add/subtract motor units as torque requirements
change. Implementation of the variable recruitment strategy will require a controller
207
Figure 6.14: Work output of manipulator with variable recruitment, 90 deg max.
angle.
Figure 6.15: Energy input of manipulator with variable recruitment, 90 deg max.
angle.
that activates motor units intelligently. The controller must be able to sense or
predict torque requirements, and respond by creating smooth transitions when the
discrete units are activated.
If there is a priori knowledge of expected payload, recruitment level can be
planned. For instance, in a heavy-lifting scenario that consists of approaching and
lifting a payload of known mass, only a small subset of motor units would require
208
activation during approach. Once contact is made with the payload, additional
PAMs would then be recruited for lifting and manipulation. This strategy may
benefit from a slight over-activation of motor units to compensate for unanticipated
disturbances.
Alternatively, real-time adjustment of recruitment level may be desired to
manipulate payloads of unknown weight or improve robustness to disturbances. In
order to maintain stability and prevent shock to the payload while the system is in
motion, transitions in recruitment level must be performed smoothly. One method of
smoothly adding a motor unit is to use individual proportional valves for each motor
unit and slowly ramp up the orifice area. This strategy is analogous to increasing the
firing rate of individual motor units in human muscle. Furthermore, it is important
to minimize “chatter” between recruitment levels, because each time a pressurized
unit is deactivated, the air inside those PAMs will be lost to the environment.
One drawback to employing fewer PAMs at higher pressures is pressure satu-
ration between the source and the PAMs. In the present system, this becomes an
issue when the pressure gradient is less than 2 bar (30 psi). Saturation leads to
limited manipulator power and speed, and degrades control. To reduce this effect, a
higher source pressure and/or larger valve orifice(s) can be used. Alternatively, more
PAMs can be recruited when a saturation threshold is reached. In this case, a precise
control strategy must be implemented to ensure that transitions in recruitment can
be performed smoothly, in concert with other recruitment algorithms, and without
additional losses in efficiency.
Another consequence of a variable recruitment strategy is the buckling of
209
non-activated PAMs when the activated PAMs contract. Figures 6.16 and 6.17 show
that PAMs which are not recruited will bend in arbitrary directions. Buckling can
be eliminated by pre-tensioning the bladder during fabrication; however, braid will
remain loose, carrying the risk of accelerated actuator fatigue. Bryant et al. [179]
were able to keep PAM braid in place by embedding their PAM bundle in a silicone
matrix; however, this dramatically decreased force output and free contraction for
all levels of recruitment, and is not advisable if efficiency is important. In order to
counteract fatigue while preserving high efficiency, alternative methods for containing
the braid must be found.
6.7 Conclusion
A biologically-inspired variable recruitment architecture was explored in order
to improve the efficiency of a robotic manipulator actuated by pneumatic artificial
muscles. The actuators were arranged in a parallel bundle, and independently-
controlled motor units were employed to emulate the structure and operation of
human skeletal muscle. Simulations combining the pneumatic, mechanical, and
control components of the robotic system were conducted, predicting efficiency
increases as the number of activated motor units decreases. Nevertheless, motor
units must be added as payload increases to satisfy torque requirements, implying the
need to transition between recruitment levels. Experimental testing was performed
on a robotic manipulator to validate simulations and to further evaluate a variable
recruitment strategy. The experiments reproduced simulation trends across all tested
210
Figure 6.16: Two non-activated PAMs.
payloads. The offset error between experiment and simulation was attributed to
unmodeled pneumatic losses such as leakage or throttling.
Further discussion regarding a variable recruitment strategy highlighted several
challenges for future work. Full implementation of variable recruitment will require
a control strategy that can select recruitment level, either via pre-planning or in
real time. A real-time control strategy will require special attention to smooth
transitions between recruitment levels and minimizing chatter. Additionally, the
system should be designed to avoid or minimize the effects of pressure saturation
211
Figure 6.17: Four non-activated PAMs.
either through control or increasing source pressure. A variable recruitment control
study is a natural extension of this work, and will be examined in the future. Lastly,
a PAM design which eliminates bladder buckling and braid loosening due to variable
recruitment is likely to improve actuator durability, and will also be considered in
future work.
212
Chapter 7: Conclusion
7.1 Summary of Research and Key Conclusions
In this research, significant contributions were made toward the realization
of robotic manipulation using lightweight, compliant pneumatic artificial muscles.
Nonlinear PAM behavior, which has traditionally impeded the decision to use
PAMs, was modeled accurately and in a manner that allows accurate predictions for
simulation and motion control. Prototype robotic arms with several novel design
features were designed, constructed, and experimentally evaluated. Substantial
inroads have been made for improving arm control, including two types of adaptive
control to improve the response to variations in payload. Lastly, a bio-inspired
variable recruitment scheme was shown to increase manipulator efficiency, providing
quantitative evidence that supports its use in control applications.
7.1.1 Pneumatic Artificial Muscle Modeling
An improved model of pneumatic artificial muscle behavior was developed
that is applicable to both uni-directional and bi-directional (antagonistic) actuation
schemes. Several refinements were applied to the well-known force balance derivation
213
in order to capture nonlinear PAM force behavior over the full range of actuator
strain. This was achieved by combining a detailed model of the elliptic ends of the
PAM in deformation with a hyperelastic stress-strain system identification procedure
for nonlinear bladder effects. While the empirical modeling results are unique to each
actuator and are not generally scalable, the model maintains predictive ability for
untested pressure levels, which is very useful for model-based control applications.
7.1.2 Manipulator Proof-of-Concept and Joint Optimization
A two degree-of-freedom proof-of-concept demonstrator of a pneumatic arti-
ficial muscle-actuated robotic manipulator intended for lifting heavy payloads was
designed, constructed, and experimentally tested. The manipulator featured a com-
pact structure, natural compliance, and high specific force. Specific design choices
included a tendon link-based joint structure that provided a direct connection from
the PAMs to the arm linkage. As a consequence, the system exhibited virtually none
of the slack typical of cable-driven systems, nor the backlash usually observed in
geared systems. Moreover, the robotic joints featured bundles of small muscles in
parallel. This was the result of an actuator sizing and scaling study that determined
that multiple small muscles provided higher specific force capability than a single
large muscle, as well as faster response and higher efficiency due to a reduced initial
volume.
Following success with the proof-of-concept system, a second heavy-lift manip-
ulator was designed by implementing a detailed joint optimization procedure for the
214
kinematics. The optimization minimized the area of the region where the available
torque falls below a desired torque threshold, and heavily penalized joint geometries
that could not meet the full range of motion. The genetic algorithm was run at
multiple incremental pressure levels, until the optimization was able to meet torque
requirements. In this way, torque performance was achieved with minimal pressure.
Experimental data from the manipulator matched expected results, supporting the
use of this kinematic design procedure.
7.1.3 Proof-of-Concept Manipulator Control Study
Several controllers were tested on the proof-of-concept manipulator and evalu-
ated using a combined weighted metric to measure accuracy and smoothness. PID
and fuzzy logic control were initially tested; however, these output feedback ap-
proaches were found to have unsatisfactory performance due to phase lag, oscillation,
and overshoot. As a result, the PAM force model developed in Chapter 2 was
combined with the kinematic and structural mechanics of the manipulator to form an
inverse model feedforward component. When applied in parallel with PID or fuzzy
feedback, the model-based controller was found to be considerably more accurate
than output feedback alone, while maintaining smoothness.
7.1.4 Adaptive Control Study
In order to improve upon results from the initial control study, three advanced
control strategies were developed and applied to the shoulder joint of a PAM-
215
actuated robotic manipulator for demonstration. The controllers were evaluated
in trajectory-following experiments, demonstrating different levels of accuracy for
different situations. The initial sliding mode controller showed robust trajectory-
following, but to guarantee low error, parameter estimates such as arm inertia,
damping, and mass were required to be known a priori. With the expectation of
high payload variations in most manipulator applications, the sliding mode controller
was modified to include adaptive capability of payload-dependent parameters. While
trajectory-following was improved, the underlying nonlinearities in the PAMs and
joint kinematics were not fully addressed by this adaptive control structure. Therefore,
an adaptive neural network controller was applied to determine angle-dependent
nonlinearities in the system while maintaining the ability to adapt to payload
variations. Results demonstrated the utility of all three of the controllers, including
improvements over cascaded PID control. In particular, neural network control
has great potential for learning (and compensating for) the complex nonlinearities
present in a PAM-actuated system.
7.1.5 Variable Recruitment
A biologically-inspired variable recruitment architecture was explored in order
to improve the efficiency of a robotic manipulator actuated by PAMs. The actuators
were arranged in a parallel bundle, and independently-controlled motor units were
employed to emulate the structure and operation of human skeletal muscle. Simu-
lations that combined the pneumatic, mechanical, and control components of the
216
robotic system were conducted, predicting an increase in efficiency as the number of
activated motor units decreases. Motor units must be added as payload increases to
satisfy torque requirements, implying the need to transition between recruitment
levels. Experimental testing was performed on a robotic manipulator to validate
simulations and to further evaluate a variable recruitment strategy. The experiments
reproduced simulation trends across all tested payloads. The offset error between
experiment and simulation was attributed to unmodeled pneumatic losses such as
leakage, tubing friction, and throttling, though this error was minor.
7.2 Contributions to Literature
Improved actuator model: The PAM force model developed in this work com-
bines model terms existing in literature with original contributions. Specifically, a
completely new method was devised to account for the elliptical ends of the PAM,
leading to a better estimation of geometric shape change and PAM force behavior.
The empirically fit stress-strain relationship was extended to account for bladder-
braid interaction in both positive and negative strain. When these two modifications
were combined, force prediction accuracy was greatly improved.
Original manipulator design: This work is also the first known study to design a
PAM-based manipulator with the intention of maximizing uni-directional torque over
a large angular range of motion (90+ deg). To achieve this goal, the manipulator
was constructed with unique design features such as parallel PAM arrangements, a
tendon-link joint design, and pressure manifolds. Furthermore, the joint optimization
217
procedure used for kinematic design is an original approach adopted ensure that
the joint can maintain high torque output from PAM actuators even as their force
reduces as they contract.
Demonstration of novel nonlinear controller: Numerous control strategies have
been applied in PAM literature; however, in the present work, a hybrid combination
of modeling and adaptation is determined for PAM-based manipulator control. The
new controller incorporates neural networks to determine unknown functions within
expressions of the known system dynamics to create a hybrid neural network control
law. This type of controller preserves some knowledge of the system structure without
requiring knowledge of nonlinear PAM dynamics, joint kinematics, or payload.
Variable recruitment strategy: This is the first study to quantify the efficiency
benefits of a variable recruitment strategy on a quasi-static PAM-actuated robotic
manipulator. In addition, this study leveraged original work from the improved PAM
actuator model to simulate variable recruitment, and to compare the simulations
with corresponding experimental results with good accuracy.
7.3 Future Work
Several initiatives related to PAM-based manipulation would represent a natural
extension of the work presented here. The work in this dissertation solves only a
small portion of the challenges that need attention in order to propel the soft robotics
field into a commercially viable industry. Even as soft actuation technologies mature,
increasingly complex research will be necessary to expand human-robot interaction.
218
First, although the model introduced in Chapter 2 generates accurate PAM
force predictions for untested pressure levels given some a priori data at other
pressure levels, it cannot be used to predict PAM force for an entirely untested PAM.
Such a model will require knowledge of the complex physical interaction between the
bladder and braid, as well as the behavior of the bladder material itself. This has not
been fully investigated in the literature, and is worthy of attention in future work.
In regards to Chapter 3, while the PAMs themselves are incredibly lightweight,
little effort was made to minimize the mass of the manipulator itself. Several
straightforward improvements could be made to increase the power-to-weight of the
complete manipulator structure, or a structural optimization procedure could be
designed. In order to lighten the skeletal structure of the arm, a lighter material such
as carbon fiber composite or plastic could be used in place of solid metal; or perhaps
inflatable segments (even PAMs) could be used as structural linkages, creating a
fully-inflatable, highly portable, extremely human-friendly arm.
To improve upon the structural design in this research, a highly anthropomor-
phic PAM-based arm could be developed. For instance, the shoulder joint could
be of ball-and-socket type, providing high dexterity without the singularities that
arise from using multiple single-axis joints. Muscles might be placed and config-
ured as they appear in the human body. In this way, a PAM-based robot could
likely perform better in human-centric environments, for tasks like opening doors or
reaching into tight spaces. Human-like “ballistic” motions such as throwing could
also be attempted. A PAM-actuated hand could be added to enable grasping and
manipulation.
219
Future control work with PAM-actuated systems could investigate hysteresis
compensation, multi-degree-of-freedom control, high-frequency motion, and coordi-
nation with a visual system. Adaptive neural network control in Chapter 5 could be
augmented with a more complex network that takes pressure and angular velocity
as inputs in addition to angular position to improve learning and modeling ability,
especially for high-frequency movements.
Full implementation of the variable recruitment strategy presented in Chapter
6 will require a control strategy that can select recruitment level, either via pre-
planning or in real-time. A real-time control strategy will require special attention to
smooth transitions between recruitment levels and minimizing chatter. Additionally,
the system should be designed to avoid or minimize the effects of pressure saturation
either through control or increasing source pressure. A variable recruitment control
study is a natural extension of this work, and will be examined in the future. Lastly,
a PAM design that eliminates bladder buckling and braid loosening due to variable
recruitment is likely to improve actuator durability, and will also be considered in
future work.
Expanding upon the variable recruitment strategy, it should be possible to
employ large numbers (hundreds) of miniature PAMs in muscle bundles. In addition
to the complexity of regulating air to a large number of PAMs, the bundle approach
was not taken in this work because of the weak performance of miniature PAMs
using commercially available bladders. Commerically available bladders had high
thickness-to-diameter ratios which restrict motion and degrade force. However, recent
advancements in custom-bladder molding have produced thin, soft bladders, yielding
220
force-to-weight ratios that are comparable to the relatively larger PAMs employed in
the present work [196].
Finally, PAMs could be employed in a safety-conscious, variable stiffness control
strategy. This approach relies on an antagonistic actuator setup to increase stiffness
when low velocity, fine motion is desired, and reduce stiffness to perform high
velocity, low accuracy motions in scenarios where a stiff system would be unsafe. By
intelligently selecting these characteristics, such a manipulator would be considerably
safer to operate in the vicinity of humans or in direct contact with humans.
221
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