ABSTRACT
Title of dissertation: RECONSTRUCTION, ANALYSIS AND
SYNTHESIS OF COLLECTIVE MOTION
Biswadip Dey, Doctor of Philosophy, 2015
Dissertation directed by: Professor P. S. Krishnaprasad
Dept. of Electrical & Computer Engineering
As collective motion plays a crucial role in modern day robotics and engi-
neering, it seems appealing to seek inspiration from nature, which abounds with
examples of collective motion (starling flocks, fish schools etc.). This approach to-
wards understanding and reverse-engineering a particular aspect of nature forms the
foundation of this dissertation, and its main contribution is threefold.
First we identify the importance of appropriate algorithms to extract param-
eters of motion from sampled observations of the trajectory, and then by assuming
an appropriate generative model we turn this into a regularized inversion problem
with the regularization term imposing smoothness of the reconstructed trajectory.
First we assume a linear triple-integrator model, and by penalizing high values of
the jerk path integral we reconstruct the trajectory through an analytical approach.
Alternatively, the evolution of a trajectory can be governed by natural Frenet frame
equations. Inadequacy of integrability theory for nonlinear systems poses the ut-
most challenge in having an analytic solution, and forces us to adopt a numerical
optimization approach. However, by noting the fact that the underlying dynamics
defines a left invariant vector field on a Lie group, we develop a framework based
on Pontryagin’s maximum principle. This approach toward data smoothing yields
a semi-analytic solution.
Equipped with appropriate algorithms for trajectory reconstruction we analyze
flight data for biological motions, and this marks the second contribution of this
dissertation. By analyzing the flight data of big brown bats in two different settings
(chasing a free-flying praying mantis and competing with a conspecific to catch a
tethered mealworm), we provide evidence to show the presence of a context specific
switch in flight strategy. Moreover, our approach provides a way to estimate the
behavioral latency associated with these foraging behaviors. On the other hand,
we have also analyzed the flight data of European starling flocks, and it can be
concluded from our analysis that the flock-averaged coherence (the average cosine of
the angle between the velocities of a focal bird and its neighborhood center of mass,
averaged over the entire flock) gets maximized by considering 5-7 nearest neighbors.
The analysis also sheds some light into the underlying feedback mechanism for
steering control.
The third and final contribution of this dissertation lies in the domain of control
law synthesis. Drawing inspiration from coherent movement of starling flocks, we
introduce a strategy (Topological Velocity Alignment) for collective motion, wherein
each agent aligns its velocity along the direction of motion of its neighborhood
center of mass. A feedback law has also been proposed for achieving this strategy,
and we have analyzed two special cases (two-body system; and an N -body system
with cyclic interaction) to show effectiveness of our proposed feedback law. It has
been observed through numerical simulation and robotic implementation that this
approach towards collective motion can give rise to a splitting behavior.
RECONSTRUCTION, ANALYSIS AND SYNTHESIS
OF COLLECTIVE MOTION
by
Biswadip Dey
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
2015
Advisory Committee:
Professor P. S. Krishnaprasad, Chair/Advisor
Professor S. I. Marcus
Professor A. L. Tits
Dr. E. W. Justh
Professor D. Levy, Dean’s Representative
c© Copyright by
Biswadip Dey
2015
Dedicated to Shawon and my parents
ii
Acknowledgments
First and foremost, I would like to thank my advisor, Prof. P. S. Krishnaprasad
for giving me an invaluable opportunity to work on challenging and extremely in-
teresting problems over the past five and half years. He has always made himself
available for help and advice, and there has never been an occasion when I have vis-
ited his office and he has not given me time. His attention to detail and dedication
to exceptional scientific writing have made each of our manuscripts a work of art.
Furthermore, his standards of excellence in research and teaching, and his passion
for science, have been a continuous source of inspiration. It has been a pleasure to
work with and learn from such an extraordinary individual.
I would also like to thank Prof. Steve Marcus, Prof. Andre Tits, Dr. Eric
Justh and Prof. Doron Levy for agreeing to serve on my dissertation committee
and for sparing their invaluable time reviewing the manuscript. I am particularly
thankful to Prof. Tits for providing me guidance in improving the quality of Chapter
4.
I am indebted to the research group of Prof. Andrea Cavagna and Prof.
Cynthia Moss, for sharing with me their experimental data on starling murmuration
and bat foraging. These data formed the foundation for our analysis (in Chapter 5
and 6) on flight strategies and steering control mechanism.
This acknowledgement will be incomplete without recognizing the support of
my teachers from the Bachelor’s and Master’s program at Jadavpur University and
iii
IIT Bombay, respectively. In particular, Prof. Tapan Kumar Ghoshal, Prof. Ravi
Banavar, Prof. Madhu Belur, Prof. Harish Pillai have played very instrumental
roles in shaping my interest in control theory and motivating me towards research.
My friends and colleagues at the Intelligent Servosystems Laboratory have
enriched my graduate life in many ways and they deserve a special mention. I
am particularly thankful to Kevin Galloway, Matteo Mischiati, Ben Flom, Yunlong
Huang, Vidya Raju, Udit Halder for all the stimulating discussions and feedback.
It has been a pleasure to collaborate with Kevin and Udit during the last leg of this
journey.
I would like to take this opportunity to acknowledge help and support from
the staff members of the Electrical and Computer Engineering Department and the
Institute for Systems Research. In particular, I would like to thank Ms. Pamela
White, Ms. Alexis Jenkins, Ms. Regina King, Ms. Beverly Dennis and Ms. Melanie
Prange. With Pam and Beverly’s assistance, I never had to think about my con-
ference registrations and travel arrangements. Also, the technical support from the
ECE Department computing facilities help desk is highly appreciated.
Many other friends outside the lab have been great sources of conversation,
inspiration, motivation and sometimes just pure fun. Agniv, Anup, Arijit, Arya,
Barna, Dipankar, Jiaul, Pritam, Rajibul, Ranchu, Ranjith, Shinkyu, Siddharth,
Soumyadip, Srimoyee and Udit - thank you all for making my time at College
Park such a memorable one. I am also fortunate for my enduring friendships with
Mrinmoy and Trishit, the best buddies one can ever ask for.
No accomplishment of mine has been made without the strong support of
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family. My parents are a source of inspiration and have always done everything
in their power to make my dreams a reality. My wife Shawon has been a great
companion on this journey, lovingly sharing with me all the sacrifices, bitterness
and happiness of our lives, and playing a leading role in sorting out our very own
2-body problem. I am grateful for all her support, care and love.
I gratefully acknowledge the financial supports from the Air Force Office of
Scientific Research under AFOSR Grant No. FA9550-10-1-0250 and FA2386-12-1-
3002 (FY2012 DURIP); from the Office of Naval Research under ODDR&E MURI
Program Grant No. N000140710734; from the Army Research Laboratory under
ARL/ARO MURI Program Grant No. W911NF-13-1-0390; and from the Kulkarni
Foundation Summer Research Fellowship.
v
Table of Contents
1 Introduction 1
1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.1 Self-steering Particle Model . . . . . . . . . . . . . . . . . . . 9
1.3.2 Pursuit Strategies and Feedback Laws . . . . . . . . . . . . . 12
I Reconstruction of Collectives 18
2 Data Smoothing through Nonlinear Optimization - Mathematical
Programming 19
2.1 Regularized Inversion Problem . . . . . . . . . . . . . . . . . . . . . . 22
2.2 Some Practical Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.1 Numerical Integration for the Group Dynamics . . . . . . . . 24
2.2.2 Parametrization of SOp3q Using Cayley Transform . . . . . . 27
2.2.3 Customization for Mathematical Programming . . . . . . . . . 29
2.2.4 Multi-stage Approach for Optimization . . . . . . . . . . . . . 31
2.3 Ordinary Cross Validation for the Regularized Inversion Problem . . 33
2.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
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3 Data Smoothing through Linear Quadratic Optimal Control 38
3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 A Control Theoretic Approach . . . . . . . . . . . . . . . . . . . . . . 42
3.2.1 Path Independence Lemma and Its Application to Data Smooth-
ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2.2 Linearity of Reconstruction . . . . . . . . . . . . . . . . . . . 59
3.3 An Alternative Co-state Based Approach . . . . . . . . . . . . . . . . 62
3.4 Ordinary Cross Validation for Optimal λ Selection . . . . . . . . . . . 67
3.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4 Data Smoothing through Nonlinear Optimization - Maximum
Principle 77
4.1 Data smoothing in a Euclidean setting . . . . . . . . . . . . . . . . . 78
4.2 Data smoothing problems in a Finite Dimensional Matrix Lie group
setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.3 A Quick Revisit to Lie-Poisson Reduction . . . . . . . . . . . . . . . 99
4.4 Example I: Data Smoothing on SOp3q . . . . . . . . . . . . . . . . . 102
4.4.1 Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . 104
4.4.2 Frechet Derivative of the Fit Error . . . . . . . . . . . . . . . 105
4.4.3 Reduced Dynamics and Jump Discontinuities on so˚p3q . . . . 107
4.4.4 Explicit Solution of the Reduced Dynamics . . . . . . . . . . . 109
4.5 Example II: Data Smoothing on SEp2q . . . . . . . . . . . . . . . . . 111
4.5.1 Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . 114
4.5.2 Frechet Derivative of the Fit Error . . . . . . . . . . . . . . . 115
4.5.3 Reduced Dynamics and Jump Discontinuities on se˚p2q . . . . 117
4.5.4 Explicit Solution of the Reduced Dynamics . . . . . . . . . . . 119
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
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II Analysis of Collectives 121
5 Analysis of Bat Foraging in Two Different Contexts 122
5.1 Experiment Details and Reconstruction of Trajectories . . . . . . . . 124
5.2 Pre-processing of Trajectory Data . . . . . . . . . . . . . . . . . . . . 127
5.3 Analysis of Flight Strategy . . . . . . . . . . . . . . . . . . . . . . . . 132
5.4 Analysis of Steering Control . . . . . . . . . . . . . . . . . . . . . . . 137
5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6 Analysis of Flocking in European Starlings 149
6.1 Experiment Details and Trajectory Reconstruction . . . . . . . . . . 151
6.2 Analysis of Flight Strategy and Underlying Steering Control . . . . . 154
6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
III Synthesis of Collectives 162
7 Flocking through Topological Velocity Alignment (TVA) 163
7.1 Topological Velocity Alignment (TVA) Strategy . . . . . . . . . . . . 164
7.2 TVA Strategy for a Planar 2-agent System . . . . . . . . . . . . . . . 168
7.2.1 State Space and Its Reduction onto Shape Space . . . . . . . . 169
7.2.2 Shape Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 171
7.2.3 Analysis of TVA Feedback Law . . . . . . . . . . . . . . . . . 174
7.2.4 Extension to a Three Dimensional Setting . . . . . . . . . . . 180
7.3 TVA Strategy for an N -agent System with Cyclic Interaction . . . . . 181
7.3.1 State Space and Its Reduction onto Shape Space . . . . . . . . 182
7.3.2 Shape Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 184
7.3.3 Analysis of TVA Feedback Law . . . . . . . . . . . . . . . . . 189
7.4 Algorithm for an N -agent System in a Three-Dimensional Setting . . 194
7.5 Implementation on Mobile Robot Testbed . . . . . . . . . . . . . . . 196
viii
7.5.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . 196
7.5.2 Experiment I . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
7.5.3 Experiment II . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
7.5.4 Experiment III . . . . . . . . . . . . . . . . . . . . . . . . . . 200
8 Conclusion and Future Works 203
8.1 Summary of Contributions and Future Directions of Research . . . . 203
8.1.1 Reconstruction of Collectives . . . . . . . . . . . . . . . . . . 203
8.1.2 Analysis of Collective Behavior in Nature . . . . . . . . . . . . 210
8.1.3 Synthesis of Collective Motion . . . . . . . . . . . . . . . . . . 213
ix
List of Figures
1 Natural Frenet frame for a curve in 3-dimensional space . . . . . . . . 11
2 Pursuit strategies: CP & MC . . . . . . . . . . . . . . . . . . . . . . 15
3 Multi resolution approach to solve optimization problem . . . . . . . 32
4 Inclusion of virtual points to take care of the missing data points . . 33
5 Reconstruction of a circular helix . . . . . . . . . . . . . . . . . . . . 36
6 Reconstruction of a curve on a sphere . . . . . . . . . . . . . . . . . . 72
7 Performance comparison: synthetic data . . . . . . . . . . . . . . . . 72
8 Reconstruction of a bat-insect trajectory pair . . . . . . . . . . . . . 74
9 Performance comparison: bat-insect trajectory pair (κ) . . . . . . . . 75
10 Performance comparison: bat-insect trajectory pair (ν) . . . . . . . . 75
11 Lie-Poisson Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 101
12 Unicycle Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
13 A typical example of trajectory reconstruction (bat-mantis) . . . . . . 125
14 This figure shows the distribution of optimal values of regularization
parameter (λ˚) used to reconstruct trajectories for bat-bat interactions.126
15 This figure shows the distribution of optimal values of regularization
parameter (λ˚) used to reconstruct trajectories for bat-mantis inter-
actions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
x
16 This figure shows the CDF for contiguous durations of following and
converging (Class I) flight segments. . . . . . . . . . . . . . . . . . . 131
17 Segmentation of flight behavior for the bat-bat interaction . . . . . . 131
18 Segmentation of flight behavior for the bat-mantis interaction . . . . 132
19 Strategy comparison for bat-bat interaction . . . . . . . . . . . . . . 133
20 Strategy comparison for bat-bat interaction (class I) . . . . . . . . . . 134
21 CDF comparison for bat-bat interaction . . . . . . . . . . . . . . . . 134
22 Strategy comparison for bat-mantis interaction . . . . . . . . . . . . . 135
23 Strategy comparison for bat-mantis interaction (class I) . . . . . . . . 135
24 CDF comparison for bat-mantis interaction . . . . . . . . . . . . . . . 136
25 Comparison of CATD and CP feedback laws for bat-bat interaction . 141
26 Residual analysis of CATD feedback law for bat-bat interaction . . . 142
27 Residual analysis of CP feedback law for bat-bat interaction . . . . . 142
28 Comparison of CATD and CP feedback laws for bat-mantis interaction143
29 Residual analysis of CATD feedback law for bat-mantis interaction . 144
30 Residual analysis of CP feedback law for bat-mantis interaction . . . 144
31 Cross validation (OCV) results for flocking events . . . . . . . . . . . 152
32 Reconstruction of flocking events . . . . . . . . . . . . . . . . . . . . 153
33 Variation of flock averaged coherence w.r.t. neighborhood size . . . . 157
34 Variation in correlation of curvatures w.r.t. neighborhood size and
behavioral latency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
35 Individual trajectories along with their frame vectors . . . . . . . . . 165
36 Variation of Θi with change in angle between xi and xNi . . . . . . . 167
37 Illustration of scalar shape variables for a 2-agent system . . . . . . . 171
38 Illustration of TVA strategy for a 2 agent system . . . . . . . . . . . 176
39 Phase portraits for the restricted dynamics . . . . . . . . . . . . . . . 179
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40 Illustration of an N -agent cyclic interaction system . . . . . . . . . . 182
41 Illustration of scalar shape variables for an N -agent cyclic interaction
system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
42 Pioneer 3-DX Mobile Robot . . . . . . . . . . . . . . . . . . . . . . . 197
43 Experimental Setup at ISL, University of Maryland, College Park . . 198
44 Experiment I: 8 agents, Flocking . . . . . . . . . . . . . . . . . . . . 199
45 Experiment II: 8 agents, Splitting . . . . . . . . . . . . . . . . . . . . 200
46 Experiment III: 6 agents, Perturbation . . . . . . . . . . . . . . . . . 201
xii
List of Tables
2.1 Variation of OCV estimate of λ with different values of signal-to-noise
ratio (SNR). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.1 Theoretically plausible feedback laws for constant absolute target di-
rection (CATD/MC) and classical pursuit (CP). . . . . . . . . . . . . 138
5.2 Summary of the statistical analysis of steering control laws for bat-bat
pursuit events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.3 Summary of the residual analysis of steering control laws for bat-bat
pursuit events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.4 Summary of the statistical analysis of steering control laws for bat-
mantis pursuit events . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.5 Summary of the residual analysis of steering control laws for bat-
mantis pursuit events . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.1 Summary of Flocking Events . . . . . . . . . . . . . . . . . . . . . . . 154
6.2 Summary of the statistical analysis of steering control laws for starling
flocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
xiii
List of Abbreviations
CATD Constant Absolute Target Direction
CP Classical Pursuit
DMI Directed Mutual Information
MC Motion Camouflage
OCV Ordinary Cross Validation
ODE Ordinary Differential Equation
PMP Pontryagin’s Maximum Principle
ROS Robot Operating System
SNR Signal to Noise Ratio
TVA Topological Velocity Alignment
UAV Unmanned Air Vehicle
Notations
SEpnq Special Euclidean group of dimension n
SOpnq Special orthogonal group of dimension n
Sn n-dimensional sphere
sepnq Lie algebra of SEpnq
sopnq Lie algebra of SOpnq
In Identity matrix of size nˆ n
SOp3q Lie-group of 3ˆ 3 orthogonal Matrices
sop3q Vector space of 3ˆ 3 skew-symmetric matrices
Opǫq Big O-notation (Infinitesimal asymptotics)
xiv
Chapter 1: Introduction
1.1 Background and Motivation
Collective motion plays a pivotal role in modern robotics and engineering, es-
pecially in the area of search and rescue missions (Andriluka et al. [2010]; Liu &
Nejat [2013]; Marques et al. [2006]), surveillance (Ahmadzadeh et al. [2006]; Bethke
et al. [2009]; Harmon [1987]; Kingston et al. [2008]; Li et al. [2008]; Quigley et al.
[2005]), environmental monitoring (Dunbabin & Marques [2012]; Elfes et al. [1998];
Leonard et al. [2010]; Pinto et al. [2013]; Tokekar et al. [2010, 2013]) etc. On the other
hand, examples of collective motion can be observed in a variety of natural settings.
Fish schools [Gautrais et al., 2009], locust migratory bands [Bazazi et al., 2008],
pigeon flights [Nagy et al., 2010], starling murmurations [Ballerini et al., 2008b] -
examples are ubiquitous in nature. The reasons for living in a group are relatively
well studied, and researchers have attributed a range of evolutionary factors be-
hind group motion, including better defense against predator attack [Carere et al.,
2009], energy efficient movement due to aerodynamic (Cutts & Speakman [1994];
Weimerskirch et al. [2001]) or hydrodynamic [Herskin & Steffensen, 1998] coupling,
information sharing [Miller et al., 2013], cooperative food collection [Beekman et al.,
2001], and others. However, the individual-level interaction mechanisms, that give
1
rise to group level collective behavior, are not yet well-understood. Therefore, it
seems to be a relevant effort to explore the underlying strategies and control laws
governing collective motion, because it is not only beneficial from the perspective
of engineering adaptation and exploitation, but it will also further our basic sci-
entific understanding of nature. Also, with recent advances in bio-inspired designs
for unmanned vehicles (Roberts et al. [2014]; Sfakiotakis et al. [2014]; Tan et al.
[2006]), the importance of understanding the interaction mechanism (governing col-
lective behavior) is becoming more and more prominent. The primary objective of
this dissertation is to contribute to both the analysis of collective motion arising
in natural settings, and the synthesis of biomimetic, decentralized algorithms for
collective motion.
As the first step towards our goal, we identify the need to recover a trajectory,
along with its higher order derivatives, from a finite set of discrete, probably noisy,
observations. Generally, in the field of neuroethology and bio-inspired robotics, the
research is often aimed towards exploration of the underlying strategies and steering
control laws governing pursuit (Chiu et al. [2010]; Olberg et al. [2000]) and collective
motion (Ballerini et al. [2008b]; Nagy et al. [2010]), and an integral part of this study
involves analysis of relevant parameters of motion (namely curvature, speed, lateral
acceleration etc.). Although the existing techniques for trajectory reconstruction
use cubic splines or smoothing filters, in many cases they lack appropriate physical
justification. From a broader perspective, the problem of recovering a smoothened
signal from noisy observations appears in many areas of science and engineering,
and belongs to a broader class of ill-posed (due to non-uniqueness, high sensitivity
2
to noise) inverse problems.
In our approach, we tackle the lack of well-posedness of this inverse problem
by the method of regularization (Tikhonov [1963]; Wahba [1990]), and embed this
problem into a proper hypothesis space. With an intention to leverage the tech-
niques from optimal control theory, we introduce generative models (governed by
differential equations) with inputs, states and outputs, and construct the hypothesis
space as output of this generative model (given an input). This framework allows
us to turn data smoothing into a continuous time optimal control problem with
intermediate state costs, and the choice for path cost specifies the regularization
imposed on the data.
Some initial works along a similar line have previously been carried out by
Magnus Egerstedt, Clyde Martin and their collaborators. Zhang et al. [1997] have
introduced the idea that spline functions can be constructed using techniques from
linear control theory, and developed polynomial interpolating splines using linear
generative models. Later, by adopting a variational approach for linear time in-
variant generative models, Shan et al. [2000] develops the necessary framework to
recover a smoothing input (scalar) from noisy samples of output data (scalar). How-
ever, this analysis does not provide much insight about extending the results for a
linear multi input multi output system. Later works (Kano et al. [2008]; Zhou et al.
[2005, 2006]) along this line address some related aspects, and a book by Egerstedt
& Martin [2010] provides a nice survey of the key developments along this line.
Alternatively one can describe the trajectory using a nonlinear generative
model (probably evolving on a Lie group), and by choosing appropriate control
3
inputs, this problem can be cast as a data smoothing problem in a sub-Riemannian
setting. Over the last two decades, Peter Crouch, Fatima Silva Leite and their col-
laborators, by adopting a variational approach, have been exploring the problem
of data smoothing in Riemannian contexts wherein the number of control inputs is
same as the dimension of the underlying state space (Crouch & Leite [1991, 1995];
Jakubiak et al. [2006]; Machado et al. [2010]). In a more recent work, Brody et al.
[2012] and Burnett et al. [2013], while studying the data smoothing problem in a
quantum state transfer context, have used higher order Lagrangian (involving higher
order derivatives of control inputs) to impose smoothness of the reconstructed tra-
jectory.
As recently highlighted in the work of Ben-Yosef & Ben-Shahar [2012], this
regularization based approach towards data smoothing can be used to address prob-
lems in computer vision. Drawing inspiration from neuro-physiological aspects of
the primary visual cortex (V1), in particular the ice-cube model [Hubel, 1995], this
recent work suggests that a curve completion problem can be formulated as a data
smoothing problem on SEp2q.
Equipped with an appropriate algorithm for trajectory reconstruction and
parameter extraction, we set our next goal as analyzing flight trajectory data and
exploring feedback laws underlying pursuit and collective motion in natural settings.
Although the history of studying collective motion in nature dates back to the first
half of the last century [Spooner, 1931], studying collective behavior from a reverse
engineering perspective did not start until very recently. Complete understanding
of the sensory perception and motor control governing collective motion (or even
4
pursuit) still remains an open problem. Individual members of the group are believed
to use simple and plausible rules (control laws) which can be represented in a co-
ordinate free manner, or in other words the laws should depend only on the relative
motion information. One of the primary aims of this dissertation is to study the
underlying mechanism of starling flocks. Although researchers have studied different
aspects of flocking behavior in European starlings Ballerini et al. [2008a,b]; Cavagna
et al. [2010]; Young et al. [2013], the interaction laws are not yet understood. By
modelling the individual members as self driven particles under gyroscopic control,
we aim to gather some insight for this relatively unexplored aspect.
The existing models for collective motion can be classified into two distinct
categories: (i) Eulerian or continuum; and (ii) Lagrangian or individualistic. The
Eulerian model uses a set of partial differential equations to describe the spatio-
temporal evolution of the group density of a collective. Although this modeling
approach does not appear to be very relevant in the context of robotic implementa-
tions, it has shown its effectiveness in analyzing densely packed collectives (Kudrolli
et al. [2008]; Topaz et al. [2006]; Zhang et al. [2010]).
On the other hand, in a more individualistic Lagrangian approach the dynam-
ics of each member is influenced by the state of the other members of group (this
survey paper by Vicsek & Zafeiris [2012] provides a comprehensive review of the
research along these lines). Although the first algorithm for collective motion was
developed by Aoki [1982] for simulating the motion of a fish school, it is not the
most well-known paper in the field. Boids algorithm, proposed in a later work by
Reynolds [1987], is more famous in the context of collective motion. However, both
5
of these approaches consider three different modes of interaction between the indi-
viduals, namely, (i) velocity alignment with the neighbors; (ii) approach towards the
flock center of mass; and (iii) collision avoidance. It was also noted that collective
behavior emerges in a leaderless manner.
Almost a decade later, Vicsek et al. [1995] introduced the notion of self driven
particles for collective motion. By assuming equal and constant speed for individual
agents, this work demonstrates the emergence of ordered motion in a planar setting.
This flocking behavior is achieved by updating an agent’s direction (at each time
step) by the average direction of motion of its neighbors (i.e., agents within a fixed
distance from the focal agent). Later, Jadbabaie et al. [2003] gave the much needed
theoretical explanation for this flocking behavior. However, a later note by Bertsekas
& Tsitsiklis [2007] has shown that the main convergence results of this work can
be perceived as a special case of some earlier results by Tsitsiklis et al. [1986]. A
more recent work by Cucker & Smale [2007] describes the evolution of a flock by
considering a polynomial decay of the influence between individuals in the flock as
they separate in space. A common theme in this line of works is that individual
agents are assumed to governed by first-order dynamics, which allows us to interpret
the consensus dynamics as a discrete analogue of heat equation.
However, a recent work by Attanasi et al. [2014] suggests that directional
information within a flock propagates with an almost constant speed, which is much
faster than the diffusive transport of information predicted by first-order models with
heat-like aspects. This observation provides sufficient justification to look beyond
first-order consensus algorithms. Interestingly, over the last few years a shift of
6
attention towards second-order consensus algorithms has been noticed in the control
community too (Olfati-Saber [2006]; Ren [2008]; Ren & Atkins [2005]; Yu et al. [2010,
2011]). In this dissertation we propose a strategy for collective motion which can
explain how local interactions between neighbors can influence the agents to align
their headings in a single common direction. Moreover, this strategy, which we refer
as topological velocity alignment (TVA), has wave-like aspects and can explain fast
propagation of information across the flock.
1.2 Thesis Outline
In Chapter 2, we define the problem of trajectory reconstruction as a nonlinear
optimization problem and approach it from a mathematical programming perspec-
tive. By assuming the natural Frenet frame equations as the underlying generative
model, and penalizing high rates of change in curvatures and speed, we turn the
trajectory reconstruction problem into a regularized inverse problem. This regular-
ized inverse problem can be viewed as an optimization over an infinite dimensional
function space. We introduce mathematical reparametrization (via Caley trans-
form, exponential mapping etc.) of the underlying variables to make the problem
tractable, and then use a numerical routine (fminunc [fminunc]) to solve the op-
timization. Also, by noting the importance of cross validation in this context, we
perform ordinary cross validation to compute an optimal amount of regularization.
As the previous approach was computationally very demanding, and doesn’t
guarantee minimization in a global sense, we develop an alternative linear formu-
7
lation of the trajectory reconstruction problem in Chapter 3. By using a triple
integrator as the underlying generative model for trajectory evolution, we regularize
the problem by trading total fit-error against high values of the jerk path inte-
gral. Then, by using techniques from linear quadratic optimal control theory, this
approach yields an analytic solution for the true minimum of this problem.
In Chapter 4, we develop a framework to address data smoothing problems
arising in a nonlinear setting. Then, by using a modified version of Pontryagin’s
maximum principle, we derive the necessary conditions for a solution of the regu-
larized inversion problem. Moreover, as the natural Frenet frame equations for a
trajectory can be viewed as a left-invariant dynamics on a Lie group, we extend this
result to address data smoothing in a matrix Lie group setting. We end this chapter
by discussing two example problem on (on SOp3q and SEp2q, respectively).
Then we delve into the analysis of natural occurrences of collective motion. In
Chapter 5, we consider the most basic form of collective motion, namely a dyadic in-
teraction between two individuals (conspecific and contraspecific). Here, we analyze
the flight trajectory data provided by our collaborators from Auditory Neuroethol-
ogy Laboratory, Department of Psychology, University of Maryland, and show ev-
idences in favor of context specific switch in bat flight strategy. Our analysis also
provides an estimate of the sensorimotor delay, associated with the pursuit behavior.
Chapter 6 describes the flight strategy analysis of flocking behavior in Euro-
pean starlings. Here, we begin by extracting speed and curvatures from the sampled
dataset of observed positions. Then we perform correlation analysis to investigate
the feedback mechanism for steering control governing coordinate motion of the
8
flocks.
Finally, drawing inspiration from our findings in the previous chapter, we pro-
pose a flight strategy, called topological velocity alignment (TVA), along with a
plausible control law in Chapter 7. This strategy can perceived as a restriction in
the underlying state space, wherein each member of the flock align its velocity along
the velocity of its neighborhood of center of mass. We complement our theoreti-
cal analysis of the proposed control law (in some special cases) by providing some
implementation results.
1.3 Mathematical Preliminaries
Now we briefly describe some mathematical concepts which we will use through-
out this thesis.
1.3.1 Self-steering Particle Model
The trajectory of a single particle moving in three dimensional space can
be described by a function r : r0, Tfinals Ñ R3, where Tfinal ą 0. If t “ tpsq
is a smooth, real valued function and rptq is a curve in R3, we call the curve
βpsq fi rptpsqq a reparametrization of the curve r. We assume rptq to be a reg-
ular curve, i.e. 9rptq ‰ 0 @t P r0, Tfinals. Now we define the arc length parameter as
sptq “
şt
0 ‖ 9rpσq ‖ dσ. Under the regularity assumption, sptq is a continuous, strict
monotone increasing function and hence invertible. Therefore a reparametrization
of a curve can be obtained by using the arc length as the parameter and this partic-
9
ular parametrization has a special feature. Any curve parametrized using arc length
parameter will have unit speed.
Let rmodpsq be the reparametrization of rptq using arc length parameter. The
evolution of rmodpsq in three dimensional space can be described using the Frenet-
Serret framing of this curve, as this is standard in differential geometry [do Carmo,
1976]. But this approach requires the curve to be at least thrice continuously dif-
ferentiable and we need the curvature (κpsq “‖ r2modpsq ‖) to be positive to avoid
degeneracy in defining the normal direction. The requirement of non-zero curvature
everywhere poses serious difficulties in this particular problem of our consideration
(as the trajectories, to be reconstructed, may have point of inflection) and hence the
Frenet-Serret framing is not the best choice for our purpose. So we use an alternate
framing of the curve, the Natural Frenet frame, which is also known as the Relatively
Parallel Adapted Frame [Bishop, 1975]. The natural Frenet frame is an alternative
approach to define a moving frame that is well-defined even when the curve has
vanishing second derivative.
The natural Frenet frame is based on the observation that, while the unit
tangent vector xptq for a given curve is unique, we can choose two unit vectors
pyptq, zptqq on the plane perpendicular to xptq such that txptq,yptq, zptqu defines a
right-handed orthogonal frame. The evolution of the frame along the length of the
10
xp0qyp0q
zp0q
xpt1q
ypt1q
zpt1qxpt2q
ypt2q
zpt2q
rptq
Figure 1: Natural Frenet frame for a curve in 3-dimensional space.
trajectory is governed by
9rptq “ νptqxptq
9xptq “ νptq puptqyptq ` vptqzptqq
9yptq “ ´νptquptqxptq
9zptq “ ´νptqvptqxptq, (1.1)
where ν is the speed and (u,v) are the natural curvatures of the trajectory. In this
approach for framing a regular curve, we can choose the initial orientation of the
frame at our will as yp0q and zp0q can be chosen arbitrarily, in contrary to the
Frenet-Serret frame equations where all three frame vectors are uniquely defined
11
once the curve is given (under a hypothesis of positive curvature). From uptq and
vptq (which can be viewed as Cartesian components of the curvature) we can obtain
the curvature κptq and the torsion τptq using the following set of relations:
κptq “
a
u2ptq ` v2ptq, θptq “ arctan
ˆ
vptq
uptq
˙
, τptq “ dθptqdt .
1.3.2 Pursuit Strategies and Feedback Laws
1.3.2.1 Classical pursuit
In classical pursuit (CP) the follower (pursuer) moves directly towards the
evader (target). By representing the evader position by re, pursuer position by rp,
and pursuer direction of motion by xp, we can define a contrast function (Λ) as
Λ “ xp ¨
r
|r| , (1.2)
to measure how closely the leader-follower relationship matches the CP strategy,
where r “ rp ´ re. This contrast function can be interpreted as the cosine of the
angle between the baseline vector and the velocity of the pursuer and Λ “ ´1 implies
that the pursuer is on the CP manifold.
Proposition 1.1 (Galloway et al. [2010], Proposition 1). Let us assume that the
dynamics of the pursuer (rp) and the evader (re) trajectories be governed by the
natural Frenet frame equations, with (νp, up, vp) and (νe, ue, ve) being the speed and
12
curvatures, respectively. Consider the feedback control law given by
up “
1
νp
”
´ µ
`
yp ¨
r
|r|
˘
´ 1|r|
´
zp ¨
`
9rˆ r|r|
˘¯ı
(1.3)
vp “
1
νp
”
´ µ
`
zp ¨
r
|r|
˘
` 1|r|
´
yp ¨
`
9rˆ r|r|
˘¯ı
where µ ą 0 is the feedback gain. Then Λptq Ñ ´1 as t Ñ 8, whenever Λp0q R
t1,´1u.
Proof. We begin by differentiating Λ along the trajectories of the generative model
for both pursuer and evader.
9Λ “ 9xp ¨
r
|r| ` xp ¨
d
dt
ˆ r
|r|
˙
“ νp pupyp ` vpzpq ¨
r
|r| `
1
|r|xp ¨
ˆ
9r´
ˆ
9r ¨ r|r|
˙ r
|r|
˙
. (1.4)
Now we define the transverse component of the relative velocity as
w “ 9r´
ˆ
9r ¨ r|r|
˙ r
|r| ,
and by using the BAC-CAB identity, a ˆ pb ˆ cq “ b ¨ pa ˆ cq ´ c ¨ pa ˆ bq, for
arbitrary vectors a, b and c, we note that
w “ 9r
ˆ r
|r| ¨
r
|r|
˙
´ r|r|
ˆ r
|r| ¨ 9r
˙
“ r|r| ˆ
ˆ
9rˆ r|r|
˙
. (1.5)
13
As the feedback law is governed by (1.3) and the transverse component of the
relative velocity can be expressed as a vector triple product (1.5), we can re-write
(1.4) as
9Λ “ ´µ
«ˆ
yp ¨
r
|r|
˙2
`
ˆ
zp ¨
r
|r|
˙2ff
` 1|r|xp ¨w
´ 1|r|
„ˆ
zp ¨
ˆ
9rˆ r|r|
˙˙ˆ
yp ¨
r
|r|
˙
´
ˆ
yp ¨
ˆ
9rˆ r|r|
˙˙ˆ
zp ¨
r
|r|
˙
“ ´µ
«ˆ r
|r| ¨
r
|r|
˙2
´
ˆ r
|r| ¨ xp
˙2ff
` 1|r|xp ¨w
´ 1|r| pzp ˆ ypq ¨
ˆˆ
9rˆ r|r|
˙
ˆ r|r|
˙
“ ´µ
`
1´ Λ2
˘
` 1|r| rxp ¨w ´ pyp ˆ zpq ¨ws . (1.6)
Now yp ˆ zp “ xp because txp,yp, zpu forms an orthonormal triad. Therefore, we
have
9Λ “ ´µ
`
1´ Λ2
˘
, (1.7)
and it is clear from (1.7) that whenever Λ P p´1, 1q it results in 9Λ ă 0. In fact it
can be concluded that the level sets of tΛ “ 1u and tΛ “ ´1u are two invariant
manifolds under the closed loop dynamics.
Moreover, by assuming Λp0q ‰ ˘1, we have
Λptq “ Ke
´2µt ` 1
Ke´2µt ´ 1 (1.8)
where the constant K is defined as K “ Λ0`1Λ0´1 . Since e
´2µt Ñ 0 as t Ñ 8, we have
Λptq Ñ ´1 as tÑ 8.
14
(a) Classical Pursuit (CP). (b) Motion Camouflage (MC).
Figure 2: These figures illustrate the strategies to be examined, namely (a) Classical Pursuit and
(b) Motion Camouflage. The red curve and arrow represents the evader trajectory, along with its
velocity. In a similar way, the blue curve and arrow represents the pursuer trajectory, along with
its velocity. The black lines corresponds to the baselines connecting the pursuer with the evader.
1.3.2.2 Motion camouflage
In motion camouflage (MC) (with respect to infinity) the follower approaches
the leader in such a way that the relative velocity does not have any transverse
component with respect to the baseline vector [Justh & Krishnaprasad, 2006]. Mo-
tion camouflage with respect to infinity is a stealthy pursuit because it nullifies the
motion parallax. Similar to CP one can define a contrast function to measure how
closely the leader-follower relationship matches the MC strategy. One such contrast
function is given by
Γ “ r|r| ¨
9r
| 9r| , (1.9)
where the baseline vector is defined as r “ rp ´ re. One can easily check that
Γ is the cosine of the angle between the baseline vector and the relative velocity
15
of the follower with respect to the leader. Moreover Γ “ ´1 implies that the
follower is on the MC manifold. MC is also referred as the constant absolute target
direction strategy (CATD) because the direction of the baseline vector remains fixed
throughout the pursuit. Now we recall the following result by Reddy et al. [2006]
to show finite time accessibility of motion camouflage in three dimensions.
Proposition 1.2 (Reddy et al. [2006]). Let us assume that the dynamics of the
pursuer (rp) and the evader (re) trajectories are governed by the natural Frenet
frame equations, with (νp, up, vp) and (νe, ue, ve) being the speed and curvatures,
respectively. Consider the feedback control law given by
up “ ´µ
´
zp ¨
`
9rˆ r|r|
˘¯
(1.10)
vp “ µ
´
yp ¨
`
9rˆ r|r|
˘¯
.
Moreover we assume the following hypotheses to be true:
(A1) 0 ă νlowp ď νp ď νhighp ă 8, where νlowp and νhighp are constants.
(A2) 0 ă νlowe ď νe ď νhighe ă 8, where νlowe and νhighe are constants.
(A3) νe{νp ď νMAX ă 1, where νMAX is a constant.
(A4) ue and ve are piecewise continuous, and
a
u2e ` v2e is bounded.
(A5) 9νe and 9νp are piecewise continuous; | 9νe| ă αe and | 9νp| ă αp where αe and αp
are finite constants.
(A6) Γp0q ă 1 and |rp0q| ą 0.
16
Then motion camouflage is accessible in finite time using high-gain feedback, i.e. by
choosing µ ą 0 to be sufficiently large.
In a later work Reddy et al. [2007] have shown that motion camouflage is acces-
sible in finite time even when some amount of delay is incorporated into the feedback
law (under some constraints on the feedback gain, delay and relative speed).
17
Part I
Reconstruction of Collectives
18
Chapter 2: Data Smoothing through Nonlinear Optimization - Math-
ematical Programming
The problem of recovering a smoothened signal from noisy observations ap-
pears in many areas of science and engineering, and belongs to a broader class of
inverse problems. By noting that naive solutions are non-unique and highly sensi-
tive to noise, it can be concluded that this inverse problem is ill-posed. However,
this lack of well-posedness can be tackled through a regularized approach (Tikhonov
[1963]; Wahba [1990]). The main idea behind regularization is to embed the prob-
lem of interest into a hypothesis space and minimize a cost functional expressed as
a sum of two terms: (i) misfit of a hypothesis to observed data; and (ii) a penalty
functional accounting for complexity of a hypothesis. In our context, we build the
hypothesis space by introducing generative models governed by ordinary differential
equations with inputs, states and outputs. This yields the hypothesis as output of
the generative model, given an input (control). This set-up allows us to turn data
smoothing into a continuous time optimal control problem with intermediate state
costs.
Regularization, in our context, necessitates penalizing sharp turns in the tra-
jectory. By assuming an appropriate generative model for trajectory evolution, this
19
problem of trajectory reconstruction can be formulated as an optimization prob-
lem, wherein the regularization term is treated as the Lagrangian, and the fit-errors
(between observed data and smoothened data obtained from the generative model)
constitute the intermediate and terminal costs. This problem becomes quite rele-
vant when we attempt to analyze the strategies and feedback mechanisms governing
biological collectives (starling flocks, foraging bats) because the analysis requires
studying parameters of motion (speed, curvatures etc.) and this information is
available only after the inverse problem has been solved.
An important aspect of this approach is to estimate an optimal value for the
relative weight of the regularization term with respect to the sum of fit-errors, and
then use this estimated value to reconstruct the smoothened signal by solving the
optimization problem. In our study, relative weight of the regularization term has
been represented by introducing a smoothing parameter (λ) into the optimization
cost. Clearly, the choice of this smoothing parameter is critical to the nature of a
reconstructed trajectory. It controls the balance between the goodness of fit and
the smoothness of the regression function. The optimization algorithm will produce
a very wiggly estimate for low values of the smoothing parameter, and at large
values the goodness of fit will deteriorate. So a trade-off is required to choose an
appropriate value for the smoothing parameter. As the value of λ varies from 0
to 8, the estimate transforms itself from an interpolant of given data points to a
geodesic best fitting the data points in a least square manner.
The estimation of the smoothing parameter has been done using ordinary
cross validation technique. The main idea behind the cross validation technique
20
is the partitioning of given dataset into two subsets, namely the estimation subset
and the validation subset. Once the signal is reconstructed using data from the
estimation subset, the validation subset is used to examine the performance of the
reconstruction. Cross validation attempts to minimize error between the predicted
signal and the original signal over the validation data.
The inherent nonlinearity of the underlying generative model results in the
following interesting features:
• System nonlinearity makes it very difficult to obtain a closed form solution for
the transition matrix of the underlying generative model, and hence we need
to use geometric integration methods.
• Lack of closed form solution for the cross validation cost function forces us to
compute the cost over a finite set of smoothing parameter values.
• The cross validation technique has an inherent parallel structure which can be
exploited to accelerate the computational process.
Although the main ideas that we pursue in this chapter are similar to the ones
developed by Reddy [2007], they differ in the following aspects:
• In contrast to the previous approach we have used the initial position (rp0q) as
an optimizing variable. Earlier work assumed the initial position to be same
as the first data point (r0).
• We have used different ways to parametrize the rotation matrix for initial
frame orientation (Cayley transform, instead of Euler angles) and the speed of
the trajectory (using an exponential function we have shown bijection between
R and R`).
21
• The most distinctive difference lies in the way we choose a smoothing param-
eter (i.e. λ in (2.1)). In comparison to previously adopted heuristic approach,
we implemented an algorithmic way (ordinary cross validation) to select an
optimal value for the smoothing parameter.
2.1 Regularized Inversion Problem
Treating the feature point as a self-steering particle in three dimensions, a
natural generative model for its position rp¨q is given by the natural Frenet frame
equations (1.1), and the existence and uniqueness of this generative model has pre-
viously been discussed in the work of Bishop [1975]. Now, drawing inspiration from
findings in biomechanics, we impose regularization by trading total fit-error against
high rates of change in speed and curvatures. Hence, by letting triuNi“0 denote the set
of observed positions, one can formulate trajectory reconstruction as the following
optimal control problem
Minimize
rx,y,zspt0q,
rpt0q,u,v,ν
˜
Nÿ
i“0
}rptiq ´ ri}2 ` λ
tNż
t0
`
9u2 ` 9v2 ` 9ν2
˘
dt
¸
subject to System dynamics: Natural Frenet Frame (1.1),
rx y zspt0q P SOp3q, rpt0q P R3,
u : rt0, tN s Ñ R, v : rt0, tN s Ñ R, ν : rt0, tN s Ñ R`,
(2.1)
where the smoothing parameter (λ ą 0) is evaluated through the method of cross
validation.
22
Alternatively, we can pack the position vector rptq, along with frame vectors
xptq, yptq and zptq, inside a 4ˆ 4 matrix gptq defined as
gptq “
»
——–
Rptq rptq
0 1
fi
ffiffifl P SEp3q, (2.2)
where Rptq fi rxptq yptq zptqs P SOp3q represents the frame moving along the
trajectory. Then, by letting ξ0 “ E14; ξ2 “ E13 ´ E31; ξ3 “ E21 ´ E12 represent the
standard basis elements of sep3q, and teiu4i“1 represent the set of standard basis vec-
tors in R4, the nonlinear generative model, i.e. the natural Frenet frame equations,
of a curve (1.1) can be expressed as
9g “ gξ
r “
“
e1 e2 e3
‰Tge4,
(2.3)
where ξ is given by
ξ “ ν
`
ξ0 ` uξ3 ´ vξ2
˘
“
»
——————————–
0 ´νu ´νv ν
νu 0 0 0
νv 0 0 0
0 0 0 0
fi
ffiffiffiffiffiffiffiffiffiffifl
. (2.4)
Clearly, this generative model (2.3) can be perceived as a left-invariant dy-
namics on SEp3q, and hence the problem of trajectory reconstruction (2.1) can
be treated as a data smoothing problem in a sub-Riemannian setting wherein the
23
number of controls (3) is strictly less than the state-space dimension (6). Although
some initial results along this line has been proposed by Brody et al. [2012] and Dey
& Krishnaprasad [2014a], further work is required before it leads us to a solution
for this problem (2.1). On the other hand, we can treat (2.1) as an optimization
problem with constraints (due to the nonlinear generative model) on SEp3q. Then,
by expressing the trajectory rp¨q as a function of speed (νp¨q) and curvatures (up¨q,
vp¨q), it can be turned into an unconstrained optimization problem. The difficul-
ties associated with obtaining a closed form solution of gp¨q forces us to use special
approximation techniques.
Remark 2.1. It should be noted that λ, the smoothing parameter in the optimization
problem (2.1), is not a unit-free quantity. Rather, it has a dimension of rL4T s where
L and T represents the dimension of length and time, respectively. Also, to avoid a
dimension mismatch inside the integrand of the cost function (2.1), we assume that
a unit scaling factor of dimension rL´4T 2s has been applied to 9ν2.
2.2 Some Practical Issues
2.2.1 Numerical Integration for the Group Dynamics
One can easily check that the solution for natural Frenet frame equation (2.3)
can be expressed as
gptq “ gp0qΦT pt, t0q, t P rt0, tN s, (2.5)
24
where the state transition matrix Φpt, t0q is computed as the limit point of the
following Peano-Baker series
Φpt, t0q “ I4 `
tż
t0
ξT pσ1qdσ1 `
tż
t0
ξT pσ1q
σ1ż
t0
ξT pσ2qdσ2dσ1 ` ¨ ¨ ¨ , t ě t0. (2.6)
However, finding a closed form solution for Φpt, t0q for any general ξp¨q is very
challenging, and this difficulty forces us to adopt a numerical approach. As the
dynamical constraint evolves over SEp3q, special care has to be taken while choosing
a suitable numerical approach because otherwise numerical computation might force
gptjq to leave the manifold for some j P t0, 1, 2, ¨ ¨ ¨ , Nu.
To address this issue we adopt a geometric integration method which ensures
that the solution will lie on SEp3q at every sample point. However, this advantage
is gained at the cost of some freedom on the choice of speed and curvatures, in
particular by assuming the speed and curvatures to be piecewise-constant functions.
Now we consider a refined partition of the time interval rt0, tN s, given by tt0 “ t0r ă
t1r ă t2r ă ¨ ¨ ¨ ă tKr “ tNu, where the length of these Kr number of equal intervals
is given as δ “ tN´t0Kr . By using the semi-group property of state transition matrices,
we have
Φptkr , t0q “ Φptkr , tpk´1qrqΦptpk´1qr , tpk´2qrq ¨ ¨ ¨Φpt2r , t1rqΦpt1r , t0q,
25
and this provides a recursive representation of gptkrq, given by
gptkrq “ gptpk´1qrqΦ
T ptkr , tpk´1qrq. (2.7)
Now, by exploiting the piecewise-constant nature of speed and curvatures, we assume
uptq “ uk, vptq “ vk and νptq “ νk over the interval rtk´1r , tkrq. These assumptions
make ξp¨q constant on rtk´1r , tkrq, and as a consequence the state-transition matrix
ΦT ptkr , tpk´1qrq can be represented as
Φptkr , tpk´1qrq “
»
——————————–
cospδαkνkq ukαk sinpδαkνkq
vk
αk sinpδαkνkq 0
´ukαk sinpδαkνkq
pv2k`u2k cospδαkνkqq
α2k
ukvkpcospδαkνkq´1q
α2k
0
´ vkαk sinpδαkνkq
ukvkpcospδαkνkq´1q
α2k
pu2k`v2k cospδαkνkqq
α2k
0
1
αk sinpδαkνkq
ukp1´cospδαkνkqq
α2k
vkp1´cospδαkνkqq
α2k
1
fi
ffiffiffiffiffiffiffiffiffiffifl
, (2.8)
where αk “
a
u2k ` v2k. Finally, by using the expressions from (2.2) and (2.8), (2.7)
yields the following recursive equations to represent evolution of the trajectory rp¨q,
along with the natural Frenet frame Rp¨q,
Rptkq “ Rptk´1q
»
——————–
cospδαkνkq ´ukαk sinpδαkνkq ´
vk
αk sinpδαkνkq
uk
αk sinpδαkνkq
pv2k`u2k cospδαkνkqq
α2k
ukvkpcospδαkνkq´1q
α2k
vk
αk sinpδαkνkq
ukvkpcospδαkνkq´1q
α2k
pu2k`v2k cospδαkνkqq
α2k
fi
ffiffiffiffiffiffifl
(2.9)
rptkq “ rptk´1q `
1
α2k
rαk sinpδαkνkqxptk´1q ` ukp1´ cospδαkνkqqyptk´1q
`vkp1´ cospδαkνkqqzptk´1qs . (2.10)
26
Thus we have have solved the differential equation and a closed form solution
for the trajectory has been achieved under the piecewise constant assumption.
2.2.2 Parametrization of SOp3q Using Cayley Transform
Moreover, we let initial orientation of the natural frame (Rpt0q) to be another
variable for optimization. But Rpt0q lies on a nonlinear manifold (SOp3q), and
an optimization on a manifold is not as straightforward as an one on a Euclidean
space Rn. This forces us to adopt a suitable way to parametrize SOp3q. Although
there are various methods for parametrizing SOp3q, we choose the approach using
Cayley transform. There is another popular approach using the Euler angles but
this approach suffers from the singularity issue which arises from the fact that SOp3q
and S1 ˆ S1 ˆ S1 are not equivalent topologically.
Clearly, for any Θ P SOp3q and any x P R3, we have
ă Θx,Θx ą“ă x,ΘTΘx ą
ñ ă Θx,Θx ą“ă x, x ą as Θ P SOp3q
ñ ă Θx,Θx ą ´ ă x, x ą“ 0
ñ ă pΘ` I3qx, pΘ´ I3qx ą“ 0. (2.11)
Now we introduce a new variable z defined as z “ pΘ` I3qx. By assuming ´1 not
to be an eigen-value of Θ, we can show that pΘ ` I3q is invertible and its column
27
space is the full space R3. Then (2.11) can be equivalently represented as
ă z, pΘ ´ I3qpΘ` I3q´1z ą“ 0, @z P R3. (2.12)
It can easily be concluded from (2.12), that pΘ´ I3qpΘ` I3q´1 is a skew-symmetric
matrix. Hence we introduce Ψ P sop3q defined as
Ψ “ pΘ´ I3qpΘ` I3q´1, (2.13)
and thus we have shown that almost every element in SOp3q (excluding a small set)
can be mapped to sop3q.
On the other hand, for every Ψ˜ P sop3q, we can define Θ˜ “ pI3 ´ Ψ˜q´1pI3 ` Ψ˜q
(as the eigen-values of Ψ˜ are pure imaginary). As Θ˜ doesn’t have any eigen value at
-1, Ψ˜ can be represented as Ψ˜ “ pΘ˜´ I3qpΘ˜` I3q´1. Then for any x P R3, we have
ă x, pΘ˜´ I3qpΘ˜` I3q´1x ą“ 0. (2.14)
Now we introduce z “ pΘ˜` I3q´1x, and this allows us to express (2.14) as
ă pΘ˜` I3qz, pΘ˜´ I3qz ą “ 0 for any z P R3,
or equivalently, Θ˜T Θ˜ “ I3. (2.15)
28
Moreover, we have detpΘ˜q “ 1. Thus we can conclude that the map
f : sop3q Ñ SOp3q
Ψ ÞÑ pI3 ´Ψq´1pI3 `Ψq
(2.16)
is injective, but not surjective. A thin set U on SOp3q, defined as U fi tΘ P
SOp3q : detpΘ` I3q “ 0u, does not have any inverse image under this f . Thus the
parametrization of SOp3q using Cayley transform does not suffer from singularity
issues. As each element in sop3q can be identified with an element in R3 (through the
inverse hat operator) and vice-versa, we can reformulate an optimization problem
over SOp3qzU as an optimization problem over R3.
2.2.3 Customization for Mathematical Programming
As the speed (νp¨q “ | 9rp¨q}) is allowed only to be positive, we introduce a new
variable ν˜ defined as ν˜p¨q “ lnpνp¨qq. In other words νp¨q “ eν˜p¨q. This parametriza-
tion along with the usage of Cayley transform and assumption for piecewise constant
control inputs allows us to transform the original optimization problem (2.1) into
an equivalent optimization problem over Rl, where l is a large integer of appropri-
ate value. Now we assume uniform sampling of the trajectory and let rtk´1, tks
be partitioned into M equal sub intervals, i.e. tk ´ tk´1 “ Mδ where δ is de-
fined as δ “ tN´t0MN . We define U to be the sequence of piecewise constant curvatures
tu1, u2, u3, ¨ ¨ ¨ , uNMu. In a similar fashion, V and Ξ represent the sequences of piece-
wise controls tv1, v2, v3, ¨ ¨ ¨ , vNMu and tν˜1, ν˜2, ν˜3, ¨ ¨ ¨ , ν˜NMu, respectively. Now, by
29
letting qRp0q P R3 define the initial orientation of the natural Frenet frame through
Cayley transform, the equivalent optimization problem can be written as:
min
U P RNM ,V P RNM ,Ξ P RNM
qRp0q P R3, rp0q P R3
˜
Nÿ
j“0
Cj
¸
(2.17)
subject to the constraints
Rptkq “ Rptk´1q
»
——————–
cospδkαkeν˜kq ´ukαk sinpδkαke
ν˜kq ´ vkαk sinpδkαke
ν˜kq
uk
αk sinpδkαke
ν˜kq pv
2
k`u2k cospδkαkeν˜k qq
α2k
ukvkpcospδkαkeν˜k q´1q
α2k
vk
αk sinpδkαke
ν˜kq ukvkpcospδkαkeν˜k q´1qα2k
pu2k`v2k cospδkαkeν˜k qq
α2k
fi
ffiffiffiffiffiffifl
rptkq “ rptk´1q `
1
α2k
“
αk sinpδkαkeν˜kqxptk´1q ` ukp1´ cospδkαkeν˜kqqyptk´1q
`vkp1´ cospδkαkeν˜kqqzptk´1q
‰
where αk “
a
u2k ` v2k. Moreover, the cost associated with each interval is given by
Cj “ }rptjq ´ rj}2 ` λ
jMÿ
i“pj´1qM`1
1
δ
“
pui ´ ui´1q2 ` pvi ´ vi´1q2 ` e2ν˜ipν˜i ´ ν˜i´1q2
‰
for any j P t1, 2, 3, ¨ ¨ ¨ , N ´ 1, Nu and
C0 “‖ rpt0q ´ r0 ‖2 .
30
Thus, as a consequence of these numerical adjustments (reformulation over a re-
stricted search space of piecewise constant functions) and various reparametriza-
tions (Cayley transform, exponential function), we end up solving an optimization
problem over R3NM`6.
2.2.4 Multi-stage Approach for Optimization
We adopt a multi resolution approach for achieving faster convergence in the
optimization problem. In this approach, we first sample the data points, i.e. rj ’s at
a coarse resolution and run the optimization routine to yield a better control input.
Then we use the midpoint rule to interpolate the optimized control inputs and use
this finer set as an initial search point for the next step optimization with higher
resolution. Another important fact requires attention while going into a finer data
set. If the finer data set consisted of even number of data points, then we have to
include an extra data point at the end. The curvature and speed value from the last
interval of the coarse data set are extended to handle this situation. This process
has been explained pictorially in Fig 3. We keep on repeating this process until all
the data points are used.
Once all the data points are taken into consideration we focus our attention
to the missing data points1 within a trajectory, and by introducing some virtual
points (as shown in Fig.(4)) we attempt to have uniformly sampled curvature and
speed data. These set of virtual points equipartition the whole duration of the
1As three-dimensional position data, in most of the cases, is obtained via stereo-reconstruction
from multiple planar images, occlusion gives rise to missing data points in the raw dataset. Also,
leaving-out-one strategy for OCV is another source of missing data points.
31





















































































































































































































































               

 
 
 
 
 



 
 
 
 
 





 
 
 
 
 




    







 





 






 





 







Application of Midpoint Rule
Application of Midpoint Rule
Optimization Routine
Cinii`2
Copti`1
Cinii`1
Copti
Fine Data
Coarse Data
Figure 3: This figure illustrates the multi resolution approach used in solving the optimization
problem (2.17). The original data set (with 23 data points) is downsampled twice to yield a coarse
data set with 6 data points. Once the control inputs are optimized for the coarse data set, we apply
mid-point rule, along with extrapolation, to generate the initial search points for the intermediate
resolution data set with 12 data points. Then, after optimization has been carried out for this stage,
we apply mid-point rule to obtain the initial search points for the original data set.
trajectory without incurring any extra fitting cost, but we consider the smoothness
cost associated with them. As a result the optimization yields better result.
We also proceed beyond the given resolution by dividing the interval between
two consecutive data points (real or virtual) into smaller sub-intervals, i.e. our
approach is capable of up-sampling. Thus we obtain more finely interpolated values
of speed (ν) and curvatures (u and v).
32
Optimization Routine
Inclusion of Virtual Points
Copti`1
Cinii`1
Copti
Virtual Data Point
Real Data Point
Figure 4: This figure illustrates the inclusion of virtual points to take care of the missing data
points.
2.3 Ordinary Cross Validation for the Regularized Inversion Problem
Ordinary cross validation (OCV) was first proposed by Allen [1974] (in the
context of regression) and Wahba & Wold [1975] (for smoothing splines). The main
idea behind cross validation is to use a subset of the given dataset to obtain a
parameter estimate and to use the rest of the data to validate the performance
under that estimate. However, cross validation does not use one subset solely for
one purpose (estimation or validation); it allows each data point to be used for both
purposes. For instance, we can divide the data set into m subsets; compute an
estimate from all the subsets but one; and validate the estimate using the left-out
subset. Then, we perform the estimation-validation after leaving out a different
33
subset. This process is repeated multiple times until every possible subset has been
considered for validation. In our work we use “leaving-out-one” strategy for perform
OCV. Here, an estimate for the trajectory is obtained using all but one data points
(by solving an optimization problem), and then the prediction error is computed at
the left out data point. Once this process has been repeated for each data point,
the prediction errors are summed to yield the ordinary cross validation cost, and
an optimal value of the regularization parameter (λ) is chosen in such a way that it
minimizes the OCV cost.
In what follows we provide a brief description of the ordinary cross validation
procedure for the regularized inversion problem of our interest (2.1). Let rkλp¨q be a
minimizer of the following optimization problem:
min
¨
˚˝ Nÿ
j“0
j‰k
}rptjq ´ rj}2 ` λ
tNż
t0
`
9u2pσq ` 9v2pσq ` 9ν2pσq
˘
dσ
˛
‹‚ (2.18)
subject to the dynamical constraint given by (2.3), where gptq and ξptq are defined
in (2.2) and (2.4), respectively. It it worth mentioning here that we solve this
optimization problem (2.18) for ordinary cross validation, by transforming it into
an equivalent optimization over a high dimensional Euclidean space with discrete
constraints (following the guidelines described in Section 2.2). Then the ordinary
cross validation cost V0pλq is defined as
V0pλq “
1
N ` 1
Nÿ
j“0
}rjλptjq ´ rj}2, (2.19)
34
and the corresponding OCV estimate for λ is given by
λ˚ “ argmin
λą0
pV0pλqq . (2.20)
2.4 Numerical Results
As a demonstration of concept for our proposed approach, we devise six toy
problems related to reconstruction of a circular helix with radius r “ 0.2 and pitch
2πh with h “ 0.25. Hence the helix can be parametrized as pr sinpωtq, r cospωtq, hωtq
with ω “ 1?r2`h2 being a speed normalizing factor. 46 equi-spaced noisy observations
are made along the length of the helix (from t0 “ 0 to tN “ 3.6 time units). These
six toy problems under consideration vary only in terms of observation noise, which
is independent and identically distributed zero mean Gaussian in each of these six
cases. The variance varies from problem to problem which is equivalent to a changing
the Signal-to-Noise ratio (SNR).
As expected, our numerical experiments show that there is a strong relation-
ship between SNR and an OCV estimate for the smoothing parameter pλ˚q (Ta-
Prob. No. Noise Std. Deviation (σ) OCV Estimate of λ (λ˚) SNR
1 0.005 1.50ˆ 10´6 40
2 0.010 2.50ˆ 10´6 20
3 0.015 9.00ˆ 10´6 13.33
4 0.020 7.50ˆ 10´6 10
5 0.040 1.50ˆ 10´5 5
6 0.050 2.60ˆ 10´4 4
Table 2.1: Variation of OCV estimate of λ with different values of signal-to-noise ratio (SNR).
35
−0.2
−0.1
0
0.1
0.2
−0.2
−0.1
0
0.1
0.2
0
0.5
1
1.5
2
2.5
 
y−a
xis
x−axis
 
z
−
a
x
is
Reconstructed Curve Sampled Data Ground Truth
Figure 5: This figure illustrates the reconstruction of a circular helix (standard deviation of the
zero-mean observation noise 0.020).
ble 2.1). Moreover, from these observations, we can conclude that λ˚ varies almost
proportionally with the inverse of SNR for higher values of SNR.
Once we obtain an OCV estimate for the smoothing parameter (λ˚), we recon-
struct the trajectory from the observed data points. The reconstructed trajectory
for one of the problems is shown in Fig 5.
2.5 Conclusion
In this chapter, we have formalized the inverse problem of trajectory recon-
struction. Using a nonlinear generative model, we have defined this data smoothing
problem as an optimization problem, and treated it from a mathematical program-
ming perspective. Although lack of integrability refrains us from getting a semi-
36
analytic solution, we have been able to reformulate the problem over a restricted
search space of piecewise constant functions, and then, by exploiting numerical
reparametrization (Cayley transform, exponential function), we solved it numeri-
cally over a very high dimensional Cartesian space. However, this numerical op-
timization is non-convex, computationally very demanding, and can at best lead
to a local minimum. Finally, we would like to conclude this chapter by mentioning
that this algorithm has been applied to reconstruct flight trajectories of bat foraging
events (as discussed in Chapter 5).
37
Chapter 3: Data Smoothing through Linear Quadratic Optimal Con-
trol
Earlier we have noticed (in Chapter 2) that lack of an integrability theory
prevents us from solving the trajectory reconstruction problem (with natural Frenet
frame as the underlying generative model) in an analytic way. Although we have
solved it numerically over a restricted search space of piecewise constant functions,
this numerical optimization problem is non-convex, computationally very demand-
ing, and can at best lead to a local minimum.
On the other hand, by noting that the steering controls can be expressed in
terms of lateral acceleration 9x, we present an alternative linear generative model (in
Section 3.1), and exploit the well-developed integrability theory of linear-quadratic
optimal control to obtain an analytic solution. Our proposed generative model is
fundamentally a triple integrator driven by jerk, the derivative of acceleration, as
the control input, and we impose regularization by trading total fit-error against
high values of the jerk path integral.1
1A preliminary version of this work can be found in a previous paper by Dey & Krishnaprasad
[2012]. Also, a significant portion of this chapter has been taken verbatim from a pre-print by Dey
& Krishnaprasad [2014b].
38
3.1 Problem Formulation
The primary objective of this work is to reconstruct a trajectory and extract
relevant parameters of motion (namely speed, curvatures, etc.) from a given time
series of observed positions (from motion capture system, GPS data) in a three
dimensional space. To ensure smoothness of the reconstructed trajectory we penal-
ize high values of the jerk path integral which is very significant in the context of
physiological movement. As described in the literature of locomotion and manipu-
lation (Flash & Hogan [1985]; Todorov & Jordan [1998]), the 2/3-power law 2 can
be interpreted as a consequence of the minimization of jerk path integral.
Let triuNi“0 be the set of observed positions. Then we are interested in finding
a trajectory r : rt0, tN s Ñ R3 to minimize the following cost:
Nÿ
i“0
}rptiq ´ ri}2 ` λ
tNż
t0
}rp3qptq}2dt
where p¨qpkq implies the k-th derivative of a function, if it exits. Similar to the
nonlinear optimization problem (2.1) mentioned in Section 2.1, the regularization
parameter λ forces a balance between goodness of fit and smoothness of the trajec-
tory. The trajectory dynamics, or in other words the underlying generative model,
2The power law says that the speed of an endpoint is inversely proportional to the 1/3-rd power
of curvature of the end effector.
39
is given by
9rptq “ vptq
9vptq “ aptq (3.1)
9aptq “ uptq
where vp¨q, ap¨q and up¨q represent the velocity, acceleration and jerk respectively.
Then the cost can be expressed as:
Nÿ
i“0
}rptiq ´ ri}2 ` λ
tNż
t0
}uptq}2dt.
Now we define a state-vector x : rt0, tN s Ñ R9 as:
x fi
»
——————–
r
v
a
fi
ffiffiffiffiffiffifl
. (3.2)
Moreover, the input and output of the underlying dynamical system is represented
by u : rt0, tN s Ñ R3 and y : rt0, tN s Ñ R3, respectively. Clearly, u “ u and y “ r
for the problem under consideration. Therefore the underlying generative model for
a trajectory can be represented in the following compact form,
9xptq “ Axptq `Buptq
yptq “ Cxptq,
(3.3)
40
where
A “
»
——————–
0 I3 0
0 0 I3
0 0 0
fi
ffiffiffiffiffiffifl
;B “
»
——————–
0
0
I3
fi
ffiffiffiffiffiffifl
;C “
„
I3 0 0

. (3.4)
It is obvious that the system dynamics (3.3) is both controllable and observable.
Now we can pose our trajectory reconstruction problem as a special case of the
following linear-quadratic optimal control problem with intermediate state costs:
Minimize
xpt0q,u
Jpxpt0q, uq “
Nÿ
i“0
}yptiq ´ ri}2 ` λ
tNż
t0
uT ptquptqdt
subject to System dynamics (3.3),
xpt0q P R9, u P U ,
(3.5)
where U is the space of real-valued functions defined on the interval rt0, tN s.
Now we establish a relationship between the linear (3.1) and non-linear (1.1)
generative models for trajectory evolution. First we show that the velocity, acceler-
ation and jerk of a trajectory can be expressed in terms of the parameters of motion
obtained from the nonlinear generative model (2.1), namely speed, curvatures and
the frame vectors.
v “ νx
a “ 9νx` uν2y ` vν2z
u “ p:ν ´ ν3pu2 ` v2qqx` p3uν 9ν ` 9uν2qy ` p3vν 9ν ` 9vν2qz.
,
/////.
//////-
(3.6)
From (3.6) it becomes clear that the particular penalty term in (3.5) carries a similar,
41
although not the same, effect as the penalty term considered in (2.1). Alternatively,
speed and curvatures can be expressed in terms of velocity, acceleration and jerk
of the trajectory, and it enables us to use the results of (3.5) for curvature based
analysis of strategies and steering control laws. This inverse map is given by,
ν “ }v}
x “ v}v}
9x “ 1ν pa´ pa ¨ xqxq
κ “ } 9x}ν
τ “ v¨paˆuq}vˆa}2 ,
,
////////////.
////////////-
(3.7)
where κ and τ denote the classical curvature and torsion of the trajectory, respec-
tively. One can view (3.6) and (3.7) as a dictionary between two alternative view-
points for trajectory generation.
3.2 A Control Theoretic Approach
As λ has a positive value, the constrained optimization in (3.5) can be viewed
as a relaxed version of the well-studied fixed endpoint optimal control problem. We
begin by applying a standard tool from the theory of least squares, namely the
path independence lemma for trajectories of linear systems [Brockett, 1970]. Now
onwards, we will not show explicit time dependence for brevity of notation wherever
doing so does not create any ambiguity.
In this section we consider a broader class of generative models whose dynamics
42
is governed by a linear time invariant system (3.3), with A P Rnˆn, B P Rnˆm and
C P Rpˆn (i.e. an m-input, p-output system with n-states). We only assume that
the pair rA,Bs is controllable and the pair rA,Cs is observable.
3.2.1 Path Independence Lemma and Its Application to Data Smooth-
ing
Consider the quadratic form xT ptqKptqxptq, where K : rt0, tN s Ñ Rnˆn is a
symmetric matrix-valued function. Then, along any trajectory of the underlying
dynamical system (3.5), we have
t´i`1ż
t`i
d
`
xTKx
˘
“
t´i`1ż
t`i
ˆ
xTK pAx`Buq ` pAx`BuqT Kx` xT 9Kx
˙
dt
ñ
t´i`1ż
t`i
»
——–
x
u
fi
ffiffifl
T »
——–
ATK `KA ` 9K KB
BTK 0
fi
ffiffifl
»
——–
x
u
fi
ffiffifl dt
` xT ptiqKpt`i qxptiq ´ xT pti`1qKpt´i`1qxpti`1q “ 0. (3.8)
Adding (3.8) over pt`0 , t´1 q, ¨ ¨ ¨ , pt`N´1, t´Nq we obtain
tNż
t0
»
——–
x
u
fi
ffiffifl
T »
——–
ATK `KA` 9K KB
BTK 0
fi
ffiffifl
»
——–
x
u
fi
ffiffifl dt` x
T pt0qKpt´0 qxpt0q
`
Nÿ
i“0
xT ptiq
`
Kpt`i q ´Kpt´i q
˘
xptiq ´ xT ptNqKpt`NqxptN q “ 0. (3.9)
43
As the quantity given by (3.9) equals to zero for any u P U and any K differentiable
over pt`i , t´i`1q @i P t0, 1, 2, ¨ ¨ ¨ , N ´ 1u, a multiple of it can be added to the cost
Jpxpt0q, uq without incorporating further changes. Hence we have,
Jpxpt0q, uq “ λxT pt0qKpt´0 qxpt0q ´ λxT ptNqKpt`N qxptNq `
Nÿ
i“0
`
rTi ri ´ 2xT ptiqCT ri
˘
`
Nÿ
i“0
xT ptiq
“
λ
`
Kpt`i q ´Kpt´i q
˘
` CTC
‰
xptiq
` λ
tNż
t0
»
——–
x
u
fi
ffiffifl
T »
——–
ATK `KA` 9K KB
BTK Im
fi
ffiffifl
»
——–
x
u
fi
ffiffifl dt. (3.10)
As (3.10) holds true for any choice of K, at this point we make the following as-
sumptions on K,
9Kptq “ ´ATKptq ´KptqA `KptqBBTKptq,
Kpt`N q “ 0, (3.11)
Kpt`i q ´Kpt´i q “ ´
1
λC
TC.
With these assumptions (3.11), the cost Jpxpt0q, uq can be represented as
Jpxpt0q, uq “ λxT pt0qKpt´0 qxpt0q `
Nÿ
i“0
`
rTi ri ´ 2xT ptiqCT ri
˘
` λ
tNż
t0
}BTKptqxptq ` uptq}2dt. (3.12)
44
Now consider the linear functional xT ptqηptq, where η : rt0, tN s Ñ Rn is a
vector valued function. Then,
t´i`1ż
t`i
d
`
xTη
˘
“
t´i`1ż
t`i
`
xT 9η ` pAx`BuqTη
˘
dt
ñ
t´i`1ż
t`i
»
——–
x
u
fi
ffiffifl
T »
——–
AT η ` 9η
BT η
fi
ffiffifl dt` x
T ptiqηpt`i q ´ xT pti`1qηpt´i`1q “ 0. (3.13)
Adding (3.13) over pt`0 , t´1 q, pt`1 , t´2 q, ¨ ¨ ¨ , pt`N´1, t´N q we obtain
tNż
t0
»
——–
x
u
fi
ffiffifl
T »
——–
ATη ` 9η
BTη
fi
ffiffifl dt` x
T pt0qηpt´0 q `
Nÿ
i“0
xT ptiq
`
ηpt`i q ´ ηpt´i q
˘
´ xT ptNqηpt`Nq “ 0. (3.14)
As the quantity given by (3.14) equals to zero for any u P U and any η differentiable
over pt`i , t´i`1q @i P t0, 1, 2, ¨ ¨ ¨ , N ´ 1u, a multiple of it can be added to the cost
Jpxpt0q, uq in (3.12) without causing any change. Hence we have,
Jpxpt0q, uq “ λ
`
xT pt0qKpt´0 qxpt0q ` xT pt0qηpt´0 q
˘
´ λxT ptNqηpt`Nq
`
Nÿ
i“0
xT ptiq
“
λ
`
ηpt`i q ´ ηpt´i q
˘
´ 2CT ri
‰
`
Nÿ
i“0
rTi ri
` λ
tNż
t0
¨
˚˚
˝
»
——–
x
u
fi
ffiffifl
T »
——–
ATη ` 9η
BTη
fi
ffiffifl` }B
TKx` u}2
˛
‹‹‚dt. (3.15)
45
As (3.15) holds true for any choice of η, we make the following assumptions on η,
9ηptq “ ´
`
AT ´KptqBBT
˘
ηptq,
ηpt`N q “ 0, (3.16)
ηpt`i q ´ ηpt´i q “
2
λC
T ri.
With these assumptions (3.16), the cost Jpxpt0q, uq can be represented as
Jpxpt0q, uq “ λ
“
xT pt0qKpt´0 qxpt0q ` xT pt0qηpt´0 q
‰
´ λ4
tNż
t0
}BTηptq}2dt
`
Nÿ
i“0
rTi ri ` λ
tNż
t0
}uptq `BT
ˆ
Kptqxptq ` 12ηptq
˙
}2dt. (3.17)
From (3.17) it is clear, that by choosing
uptq “ uoptptq fi ´BT
ˆ
Kptqxptq ` 12ηptq
˙
, (3.18)
we have
Jpxpt0q, uoptq “ λ
`
xT pt0qKpt´0 qxpt0q ` xT pt0qηpt´0 q
˘
`
Nÿ
i“0
rTi ri ´
1
4λ
tNż
t0
}BTηptq}2dt.
(3.19)
As λ ą 0, it is apparent from (3.19), that the necessary and sufficient condition for
the cost to be minimized is,
u “ uopt and xT pt0qKpt´0 qxpt0q ` xT pt0qηpt´0 q be minimized over xpt0q P Rn.
46
Therefore,
xoptpt0q “ arg min
xpt0qPRn
`
xT pt0qKpt´0 qxpt0q ` xT pt0qηpt´0 q
˘
,
or in other words, the optimal initial state satisfies the following condition
“
Kpt´0 q
‰
xoptpt0q `
1
2ηpt
´
0 q “ 0. (3.20)
Hence, we have
Jmin “
Nÿ
i“0
rTi ri ´ λ
»
–xToptpt0qKpt´0 qxoptpt0q `
1
4
tNż
t0
}BTηptq}2dt
fi
fl . (3.21)
It is clear from definition (3.5) that the cost is never negative, and hence
Jmin ě 0
or in other words
xToptpt0qKpt´0 qxoptpt0q `
1
4
tNż
t0
}BTηptq}2dt ď 1λ
Nÿ
i“0
rTi ri.
Proposition 3.1 (Brockett [1970]). A Riccati equation of the form
9Kptq “ ´KptqA´ ATKptq `KptqBBTKptq (3.22)
KpT q “ Q,
has a symmetric, positive semi-definite solution Kptq for t ď T whenever the termi-
47
nal value Q is symmetric, positive semi-definite, and the pair rA,Bs is controllable.
Moreover, the solution can be represented as
Kptq “ e´AT pt´T q
˜
Q´Q
„`
Gpt, T q
˘´1 `Q
´1
Q
¸
e´Apt´T q (3.23)
where Gpt, T q is a controllability Grammian-like quantity.
Proof for Proposition.3.1. The adjoint system corresponding to the Riccati equation
(3.22) is
d
dt
»
——–
η1
η2
fi
ffiffifl “
»
——–
A ´BBT
0 ´AT
fi
ffiffifl
»
——–
η1
η2
fi
ffiffifl
and the associated transition matrix can be represented as
»
——–
φ11 φ12
φ21 φ22
fi
ffiffifl pt, T q “ φpt, T q “
»
——–
eApt´T q eApt´T qGpt, T q
0 e´AT pt´T q
fi
ffiffifl
where,
Gpt, T q fi ´
tż
T
eApT´σqBBT eAT pT´σqdσ
is positive definite for any t ă T because of controllability of the pair rA,Bs.
Hence the solution for (3.22) can be represented as,
Kptq “
„
φ21pt, T q ` φ22pt, T qQ
„
φ11pt, T q ` φ12pt, T qQ
´1
“ e´AT pt´T qQ
„
eApt´T q
`
In `Gpt, T qQ
˘´1
“ e´AT pt´T qQ
ˆ
In `Gpt, T qQ
˙´1
e´Apt´T q.
48
Now by applying the matrix inversion lemma,
`
E ´ FH´1G
˘´1 “ E´1 ` E´1F
`
H ´GE´1F
˘´1GE´1,
and by letting E “ In, F “ ´In, G “ Q and H “
`
Gpt, T q
˘´1 we obtain
ˆ
In `Gpt, T qQ
˙´1
“ In ´
ˆ`
Gpt, T q
˘´1 `Q
˙´1
Q.
As Gpt, T q ą 0 for any t ă T , its inverse is also positive definite for any t ă T .
Then positive definiteness of
“`
Gpt, T q
˘´1 ` Q
‰´1 is directly implied from the fact
that the terminal condition Q is positive semi-definite.
By defining Mptq fi
`
Gpt, T q
˘´1, we have
Kptq “ e´AT pt´T qQ
„
Q´Q
“
Mptq `Q
‰´1Q

Qe´Apt´T q.
As M ą 0 (implicit dependency on time t is not shown for the sake of clarity) and
Q “ QT ě 0, there exists a non-singular matrix P such that
P TQP “ Λ
P TMP “ In
where Λ is a diagonal matrix with non-negative entries. With the above expressions
49
from simultaneous diagonalization, we have
Q ´Q
“
M`Q
‰´1Q “ pP T q´1
„
Λ´ Λ
`
In ` Λ
˘´1Λ

P´1. (3.24)
Now, by assuming Λ “ diagpλ1, ¨ ¨ ¨ , λnq, λi ě 0, we obtain
`
In ` Λ
˘´1 “ diagp 11` λ1
, ¨ ¨ ¨ , 11` λn
q
ñΛ
`
In ` Λ
˘´1Λ “ diagp λ
2
1
1` λ1
, ¨ ¨ ¨ , λ
2
n
1` λn
q
ñΛ´ Λ
`
In ` Λ
˘´1Λ “ diagp λ11` λ1
, ¨ ¨ ¨ , λn1` λn
q.
Therefore Λ´ Λ
`
In ` Λ
˘´1Λ is a positive semi-definite diagonal matrix, and hence
from (3.24), Q ´Q
“
M`Q
‰´1Q is a symmetric positive semi-definite matrix.
Hence, Kptq is symmetric, positive semi-definite for any t ă T .
With assurance on the existence of solution, we make the following claim
regarding the form of the Riccati equation solution.
Lemma 3.2. The solution of the Riccati equation (3.11) assumes the form
Kpt´i q “
1
λ
Nÿ
k“i
ΦΣpti, tkqCTCΦTΣpti, tkq
for any i P t0, 1, ¨ ¨ ¨ , Nu where Σptq “ ´pA´ 12BBTKptqqT and ΦΣ is the transition
matrix for Σ.
Proof for Lemma.3.2. We will use mathematical induction to prove the above claim.
From the boundary and jump conditions in (3.11) it is obvious that the claim holds
50
true for i “ N .
Now we assume that it holds true for i “ m` 1, or in other words
Kpt´m`1q “
1
λ
Nÿ
k“m`1
ΦΣptm`1, tkqCTCΦTΣptm`1, tkq.
Using uniqueness of solution, one can easily verify that
Kptq “ ΦΣpt, tm`1qKpt´m`1qΦTΣpt, tm`1q
satisfies the Riccati differential equation
9Kptq “ ´ATKptq ´KptqA`KptqBBTKptq
for any t P ptm, tm`1q.
Therefore,
Kpt´mq “ Kpt`mq `
1
λC
TC
“ ΦΣptm, tm`1qKpt´m`1qΦTΣptm, tm`1q `
1
λC
TC
“ 1λ
Nÿ
k“m`1
ΦΣptm, tkqCTCΦTΣptm, tkq `
1
λΦΣptm, tmqC
TCΦTΣptm, tmq
“ 1λ
Nÿ
k“m
ΦΣptm, tkqCTCΦTΣptm, tkq.
Hence the claim is proved, as it holds true for i “ m.
Now we concentrate on the dynamics of η given by (3.16) and introduce a new
51
time-varying matrix
Σ˜ptq “ ´pA´BBTKptqqT .
Then the dynamics of η can be represented as
9ηptq “ Σ˜ptqηptq (3.25)
for any t P pti, ti`1q, i P t0, 1, ¨ ¨ ¨ , N´1u. By letting ΦΣ˜ denote the transition matrix
for (3.25), we can make the following claim regarding the solution for η variables.
Lemma 3.3.
ηpt`i q “ ´
2
λ
Nÿ
k“i`1
ΦΣ˜pti, tkqCT rk
ηpt´i q “ ´
2
λ
Nÿ
k“i
ΦΣ˜pti, tkqCT rk
Proof for Lemma.3.3. We will use mathematical induction to prove the above claim.
From the boundary and jump conditions in (3.16) it is obvious that the claim holds
true for i “ N as,
ηpt`Nq “ 0
ηpt´Nq “ ´
2
λC
T rN .
52
Now we assume that it holds true for i “ m` 1, or in other words
ηpt`m`1q “ ´
2
λ
Nÿ
k“m`2
ΦΣ˜ptm`1, tkqCT rk
ηpt´m`1q “ ´
2
λ
Nÿ
k“m`1
ΦΣ˜ptm`1, tkqCT rk.
Using the dynamics of η, given by (3.25), we have the following relationship
ηpt`mq “ ΦΣ˜ptm, tm`1qηpt´m`1q “ ´
2
λΦΣ˜ptm, tm`1q
Nÿ
k“m`1
ΦΣ˜ptm`1, tkqCT rk
“ ´2λ
Nÿ
k“m`1
ΦΣ˜ptm, tkqCT rk. (3.26)
Using the jump condition at tm, we obtain
ηpt´mq “ ηpt`mq ´
2
λC
T rm
“ ´2λ
Nÿ
k“m`1
ΦΣ˜ptm, tkqCT rk ´
2
λΦΣ˜ptm, tmqC
Trm
“ ´2λ
Nÿ
k“m
ΦΣ˜ptm, tkqCT rk. (3.27)
From (3.26) and (3.27) it is clear that the claim holds true for i “ m.
Hence the claim is proved.
Now we focus into the problem of our interest, i.e. the problem of trajectory
reconstruction through minimization of the jerk path integral. By exploiting the
particular structure of A, B and C (given by (3.4)), namely the triple-integrator
property, we claim observability for the p´ΣT , Cq pair.
53
Proposition 3.4. p´ΣT , Cq forms an observable pair for the trajectory reconstruc-
tion problem (3.3,3.4).
Proof for Proposition.3.4. K is a symmetric matrix by definition, and hence one
can assume the following block structure for K,
Kptq “
»
——————–
K11ptq K12ptq K13ptq
KT12ptq K22ptq K23ptq
KT13ptq KT23ptq K33ptq
fi
ffiffiffiffiffiffifl
.
With this particular structure for K, we have the following expression of ΣT ptq for
the jerk path integral minimization problem,
ΣT ptq “
»
——————–
0 ´I3 0
0 0 ´I3
1
2KT13ptq 12KT23ptq 12K33ptq
fi
ffiffiffiffiffiffifl
. (3.28)
Now, for the sake of convenience, we use Silverman-Meadows rank condition [Sil-
verman & Meadows, 1967] to prove our claim. To do so, we define the matrix Qobv
as
Qobvptq “
“
S0ptq S1ptq ¨ ¨ ¨ Sn´1ptq
‰
where Siptq’s are computed recursively using
Sk`1ptq “ ´ΣptqSkptq ` 9Skptq, S0ptq “ CT . (3.29)
54
The Siptq’s will assume the following form,
S0ptq “
»
——————–
I3
0
0
fi
ffiffiffiffiffiffifl
, S1ptq “
»
——————–
0
I3
0
fi
ffiffiffiffiffiffifl
, S2ptq “
»
——————–
0
0
I3
fi
ffiffiffiffiffiffifl
, S3ptq “ ´
1
2
»
——————–
K13ptq
K23ptq
K33ptq
fi
ffiffiffiffiffiffifl
,
and so on. Hence it can be immediately concluded that the pair p´ΣT , Cq is ob-
servable as the rank of Qobvptq is 9 for any t P R` Y t0u.
Theorem 3.5. For the trajectory reconstruction problem (3.3,3.4), the optimal ini-
tial condition (given by (3.20)) is uniquely solvable for almost any time index set
ttiuNi“0.
Proof for Theorem.3.5. From proposition 1 we have,
Kpt´0 q “
1
λ
Nÿ
k“0
ΦΣpt0, tkqCTCΦTΣpt0, tkq
“ 1λ
Nÿ
k“0
ΦT´ΣT ptk, t0qCTCΦ´ΣT ptk, t0q
“ 1λ
»
——————————–
C
CΦ´ΣT pt1, t0q
...
CΦ´ΣT ptN , t0q
fi
ffiffiffiffiffiffiffiffiffiffifl
T »
——————————–
C
CΦ´ΣT pt1, t0q
...
CΦ´ΣT ptN , t0q
fi
ffiffiffiffiffiffiffiffiffiffifl
“ 1λC
TC.
Now we investigate the rank of C because the solvability of (3.20) is equivalent to
55
the fact of C having full rank. To do so we consider the following system
9ξptq “ ´ΣT ptqξptq
γptq “ Cξptq,
(3.30)
which is observable (proposition 3.3 ). We can easily show that the j-th derivative
of its output can be represented as
γpjqptq “ STj ptqΦ´ΣT pt, tiniqξptiniq
where Sjptq’s are defined in (3.29).
Let ξ1 ‰ ξ2 be two different choice of initial state ξpt0q for the system (3.30) and
γiptq be its output corresponding to the initial condition ξpt0q “ ξi. Now we define,
Yi fi
»
——————————–
γipt0q
γipt1q
...
γiptNq
fi
ffiffiffiffiffiffiffiffiffiffifl
“ Cξi.
Now we claim that the outputs of (3.30), corresponding to two different initial
conditions ξ1 ‰ ξ2, do not match identically over any interval T Ă R` Y t0u, or in
other words, there is no such interval T Ă R` Y t0u such that γ1ptq “ γ2ptq for any
t P T.
56
We can prove our claim by contradiction. Let
CΦ´ΣT pt, t0qξ1 “ CΦ´ΣT pt, t0qξ2
for all t belonging to some interval T. Then the derivatives, when they exist, should
match for any t˚ in the interior of T, i.e.
dj
dtj
ˆ
CΦ´ΣT pt, t0qξ1
˙ˇˇ
ˇˇ
t˚
“ d
j
dtj
ˆ
CΦ´ΣT pt, t0qξ2
˙ˇˇ
ˇˇ
t˚
ñ
»
——————————–
ST0 ptq
ST1 ptq
...
STn´1ptq
fi
ffiffiffiffiffiffiffiffiffiffifl
Φ´ΣT pt˚, t0qξ1 “
»
——————————–
ST0 ptq
ST1 ptq
...
STn´1ptq
fi
ffiffiffiffiffiffiffiffiffiffifl
Φ´ΣT pt˚, t0qξ2
ñQTobvptqΦ´ΣT pt˚, t0q
ˆ
ξ1 ´ ξ2
˙
“ 0
ñξ1 “ ξ2.
But it contradicts our initial assumption about inequality of ξ1 and ξ2, thereby
proves the claim. Hence Cξ1 ‰ Cξ2 for almost any time index set ttiuNi“0. Therefore
Kpt´0 q is positive definite almost surely because C has full rank almost surely.
When the rank condition fails, i.e. Cξ1 “ Cξ2, we can consider an arbitrary close
perturbation of the original time index. For any given ǫ ą 0 we can choose a
57
perturbed time index set tt˜iuNi“0, such that the following conditions holds true,
t0 “ t˜0,
Nÿ
i“1
|ti ´ t˜i| ă ǫ,
and,
»
——————————–
C
CΦ´ΣT pt˜1, t˜0q
...
CΦ´ΣT pt˜N , t˜0q
fi
ffiffiffiffiffiffiffiffiffiffifl
has full rank.
Therefore (3.20) can be uniquely solved, for almost any time index set ttiuNi“0.
As Kpt´0 q is shown to be a symmetric, invertible and positive definite matrix,
for almost any time index set ttiuNi“0, the optimal initial condition can be represented
as
xoptpt0q “ ´
1
2
“
Kpt´0 q
‰´1
ηpt´0 q “
1
λ
“
Kpt´0 q
‰´1 Nÿ
k“0
ΦΣ˜pt0, tkqCT rk. (3.31)
Remark 3.1. The work by Magnus Egerstedt, Clyde Martin and their collabora-
tors (Egerstedt & Martin [2010]; Kano et al. [2008]; Shan et al. [2000]; Zhou et al.
[2005, 2006]) provides an alternative view for exploiting linear optimal control for
smooth interpolation. Their work provides a framework to recover a scalar input
from sampled observations of scalar output data by solving a regularized optimal con-
trol problem, similar to the one given in (3.5). However, this work uses a variation
approach and makes some extra assumption to ensure smoothness. This variational
approach can also be viewed from a learning theoretic perspective. On the other hand,
58
our approach exploits the integrability of linear-quadratic optimal control problems,
and the observations introduce jumps in the optimal control input. Moreover, our
results can be used to construct control theoretic splines for multi-input multi-output
systems if the optimal initial condition is uniquely solvable from (3.20).
Remark 3.2. Moreover, our results can be generalized to reconstruct any trajec-
tory whose evolution is governed by a linear time invariant generative model. Given
the pair rA,Bs (rA,Cs) is controllable (observable) and the optimal initial condition
(xopt) is uniquely solvable from 3.20, one can use our approach for data smoothing.
In particular, trajectory reconstruction by penalizing high values of the snap, crackle
or pop (4th, 5th or 6th derivative of position, respectively) path integrals, will not af-
fect the structure for higher order integrators, and hence it can be easily shown that
Prop 3.4 holds true for those cases. Therefore our approach has a natural exten-
sion to tackle trajectory reconstruction through penalizing higher order derivatives
of motion.
3.2.2 Linearity of Reconstruction
Under the action of an optimal control input uopt the system dynamics can be
represented as
9xptq “
“
A´BBTKptq
‰
xptq ´ 12BB
Tηptq “ ´Σ˜T ptqxptq ´ 12BB
Tηptq, (3.32)
59
or in other words it can be viewed as a time-varying linear system with η being the
input, and the state xptq can be expressed as
xptq “ Φ´Σ˜T pt, t0qxoptpt0q ´
1
2
tż
t0
Φ´Σ˜T pt, σqBBTηpσqdσ
“ ΦTΣ˜pt0, tqxoptpt0q ´
1
2
tż
t0
ΦTΣ˜pσ, tqBB
Tηpσqdσ. (3.33)
Therefore the reconstructed states can be represented (at sampling time instances
ttiuNi“0) as,
xptkq “ ΦTΣ˜pt0, tkqxoptpt0q ´
1
2
tkż
t0
ΦTΣ˜pσ, tkqBB
Tηpσqdσ
“ ΦTΣ˜pt0, tkqxoptpt0q ´
1
2
kÿ
i“1
»
–
tiż
ti´1
ΦTΣ˜pσ, tkqBB
Tηpσqdσ
fi
fl
“ ΦTΣ˜pt0, tkqxoptpt0q ´
1
2
kÿ
i“1
»
–
tiż
ti´1
ΦTΣ˜pσ, tkqBB
TΦΣ˜pσ, tiqηpt´i qdσ
fi
fl
“ ΦTΣ˜pt0, tkqxoptpt0q ´
1
2
kÿ
i“1
»
–
tiż
ti´1
ΦTΣ˜pσ, tkqBB
TΦΣ˜pσ, tiqdσ
fi
fl ηpt´i q
“ ΦTΣ˜pt0, tkqxoptpt0q
` 1λ
kÿ
i“1
« tiż
ti´1
ΦTΣ˜pσ, tkqBB
TΦΣ˜pσ, tiqdσ ˆ
˜
Nÿ
j“i
ΦΣ˜pti, tjqCT rj
¸ff
“ ΦTΣ˜pt0, tkqxoptpt0q
` 1λ
Nÿ
i“1
«minti,kuÿ
j“1
¨
˚˝
tjż
tj´1
ΦTΣ˜pσ, tkqBB
TΦΣ˜pσ, tjqdσ
˛
‹‚ˆ ΦΣ˜ptj , tiq
ff
CT ri.
60
As the optimal initial condition is linear in observed data points, the smoothened
position at time tk can also be expressed as a linear combination of observed posi-
tions.
rptkq “
1
λ
Nÿ
i“0
“
CFλpk, iqCT
‰
ri (3.34)
where
Fλpk, iq “ ΦTΣ˜pt0, tkq
“
Kpt´0 q
‰´1 ΦΣ˜pt0, tiq `
minti,kuÿ
j“1
¨
˚˝
tjż
tj´1
ΦTΣ˜pσ, tkqBB
TΦΣ˜pσ, tiqdσ
˛
‹‚
(3.35)
Remark 3.3. As the coefficients Fpk, iq’s depend only on the sampling time in-
stances, namely t0, ¨ ¨ ¨ , tN , and the underlying system dynamics, these coefficients
can be pre-computed.
Remark 3.4. This approach can be perceived as a global alternative to Savitzky-
Golay smoothing filters (Savitzky & Golay [1964]; Schafer [2011]), wherein the fil-
tered outputs are obtained by fitting a least square polynomial (locally) through the
observed data points. In our approach the local nature is absent, instead each of the
filtered outputs depends on the complete data set. But because of this global nature
our approach has its own drawback. This method, in its true form, cannot be used
in real-time as it requires all the observations together.
Remark 3.5. The significance of the word “smoothing” is twofold in this context.
Firstly this approach penalizes high values of jerk path integral and thereby yields a
smoothened trajectory. Moreover, it uses data from both past and future to estimate
the present position and thus justifies the usage of “smoothing” in estimation context.
61
Remark 3.6. The formulation of the problem is an example of fixed interval smooth-
ing. One can use this as a building block and proceed to obtain a fixed lag smoothing
algorithm. The path is quite intuitive.
3.3 An Alternative Co-state Based Approach
Although we have developed an analytic method to solve data smoothing, the
procedure is computationally demanding because it involves solving a differential
Riccati equation. Now, to make computations more tractable, we represent the
solution in terms of co-state variables, defined as,
pptq fi Kptqxptq ` 12ηptq. (3.36)
Then the optimal control input (3.18) and system dynamics (3.3) will have the form
uoptptq “ ´BTpptq
9xptq “ Axptq ´BBTpptq,
and the dynamics of the co-states is given by
9pptq “ 9Kptqxptq `Kptq 9xptq ` 12 9ηptq “ ´A
Tpptq. (3.37)
62
Therefore the optimal trajectory between two observation times can be viewed as
the base integral curve of the following system
d
dt
»
——–
xptq
pptq
fi
ffiffifl “
»
——–
A ´BBT
0 ´AT
fi
ffiffifl
»
——–
xptq
pptq
fi
ffiffifl . (3.38)
From (3.38) it is apparent that the dynamics of p is decoupled from that of x.
Now we’ll focus on the jump conditions for the co-states
ppt`i q ´ ppt´i q “
“
Kpt`i q ´Kpt´i q
‰
xptiq `
1
2
“
ηpt`i q ´ ηpt´i q
‰
“ 1λC
T pri ´ rptiqq .
(3.39)
We also have the following terminal condition
ppt`N q “ Kpt`NqxptN q `
1
2ηpt
`
Nq “ 0
as both Kpt`N q and ηpt`Nq are equal to zero, and by letting xpt0q “ xoptpt0q, (3.20)
yields,
ppt´0 q “ Kpt´0 qxpt0q `
1
2ηpt
´
0 q “ 0.
Now we introduce a new variable, namely incremental time, defined as ∆i fi
63
ti`1 ´ ti i P t0, 1, ¨ ¨ ¨ , N ´ 1u. From (3.38) we have
pptq “ e´AT pt´tiqppt`i q t P pti, ti`1q (3.40)
ppt`i`1q “ ppt´i`1q `
1
λC
T pri`1 ´ Cxpti`1qq
“ e´AT∆ippt`i q ´
1
λC
TCxpti`1q `
1
λC
T ri`1 (3.41)
for i P t0, 1, ¨ ¨ ¨ , N ´ 1u. From the dynamics of x in (3.38), we have
xpti`1q “ eApti`1´tiqxptiq ´
ti`1ż
ti
eApti`1´σqBBTppσqdσ
“ eA∆ixptiq ´
ti`1ż
ti
eApti`1´σqBBT e´AT pσ´tiqppt`i qdσ
“ eA∆ixptiq ´ eA∆i
»
–
ti`1ż
ti
eApti´σqBBT e´AT pσ´tiqdσ
fi
fl ppt`i q. (3.42)
From (3.41) and (3.42) we obtain the following matrix representation for forward-
propagation of xptiq and ppt`i q
»
——–
xpti`1q
ppt`i`1q
fi
ffiffifl “
»
——–
eA∆i ´eA∆iWi
´ 1λCTCeA∆i
”
e´AT∆i ` 1λCTCeA∆iWi
ı
fi
ffiffifl
»
——–
xptiq
ppt`i q
fi
ffiffifl
`
»
——–
0
1
λCT
fi
ffiffifl ri`1 (3.43)
64
where Wi is defined as
Wi “
ti`1ż
ti
eApti´σqBBT e´AT pσ´tiqdσ “
∆iż
0
e´AτBBT e´AT τdτ pτ “ σ ´ tiq. (3.44)
From (3.44) it is apparent that the controllability Gramian Wi depends only on
the inter-sample intervals, not explicitly on the sampling instances. Moreover,the
Gramian is invertible as the underlying system (3.3) is controllable.
By defining a discrete time state vector as zi “
“
xT ptiq pT pt`i q
‰T , (3.43) can
be represented as the following discrete time system
zi`1 “ Λizi ` Γri`1 (3.45)
where Λi and Γ are defined as
Λi “
»
——–
eA∆i ´eA∆iWi
´ 1λCTCeA∆i
”
e´AT∆i ` 1λCTCeA∆iWi
ı
fi
ffiffifl
Γ “
»
——–
0
1
λCT
fi
ffiffifl .
Lemma 3.6. Λi is invertible for any i P t0, 1, ¨ ¨ ¨ , N ´ 1u.
65
Proof for Lemma.3.6.
Λi “
»
——–
eA∆i ´eA∆iWi
´ 1λCTCeA∆i
”
e´AT∆i ` 1λCTCeA∆iWi
ı
fi
ffiffifl
“
»
——–
In 0
´ 1λCTC In
fi
ffiffifl
»
——–
eA∆i ´eA∆iWi
0 e´AT∆i
fi
ffiffifl
“MΥi (3.46)
(3.46) gives a block LU-factorization for Λi and both M and Υi are invertible for
any i.
Hence, Λi is invertible for any i.
From (3.45) we obtain
zk “
˜
k´1ź
i“0
Λi
¸
»
——–
xpt0q
1
λCT pr0 ´ Cxpt0qq
fi
ffiffifl`
kÿ
i“1
˜
k´1ź
j“i
Λj
¸
Γri
“
˜
k´1ź
i“0
Λi
¸´
»
——–
In
´ 1λCTC
fi
ffiffiflxpt0q ` Γr0
¯
`
kÿ
i“1
˜
k´1ź
j“i
Λj
¸
Γri
“
˜
k´1ź
i“0
Λi
¸
»
——–
In
´ 1λCTC
fi
ffiffiflxpt0q `
kÿ
i“0
˜
k´1ź
j“i
Λj
¸
Γri (3.47)
where ś represents left multiplication. As ppt`Nq “ 0, xpt0q can be obtained by
66
solving the following equation
r0 Ins
˜
N´1ź
i“0
Λi
¸
»
——–
In
´ 1λCTC
fi
ffiffifl xpt0q “ ´ r0 Ins
Nÿ
i“0
˜
N´1ź
j“i
Λj
¸
Γri. (3.48)
From the way (3.48) has been obtained, it can be inferred that (3.48) is an alternative
form of (3.20). Hence, it can be concluded from Theorem 3.4 that for the trajectory
reconstruction problem (3.3,3.4) of our interest, (3.48) yields a unique solution for
the optimal initial condition for almost any time index set ttiuNi“0. Once xpt0q is
obtained by solving (3.48), the trajectory can be reconstructed using (3.38) and the
jump conditions given by (3.39).
Remark 3.7. The limiting case of λ “ 0 signifies the exact fitting problem, and
hence can be represented as an optimal control problem with both initial and final
points lying on an affine space. Although this problem can be solved by applying
suitable transversality conditions, it will result in non-unique state trajectories.
Remark 3.8. The proposed algorithm for data smoothing is fast, with complexity
of the order of sample size (OpNq).
3.4 Ordinary Cross Validation for Optimal λ Selection
As discussed earlier (in Section 2.3), ordinary cross validation (OCV) performs
reconstruction by considering a subset of the whole data set, and then computes the
fit-error at left-out data points. After this step has been repeated for all possible
subsets, the fit errors are summed up. This sum of errors can be perceived as a
67
Algorithm 1 Algorithm for trajectory smoothing
Data: Time index - ttiuNi“0; Data - triuNi“0; Smoothing Parameter - λ ą 0
Define: A, B and C
for i = 0 to N ´ 1 do
∆tÐ ti`1 ´ ti
Compute the Gramian W “
∆tż
0
e´AσBBT e´ATσdσ
—Due to special structures in A and B, W have a closed form solution (involving
polynomial functions of ∆t).
Υi Ð
„
eA∆t ´eA∆tW
0 e´AT∆t

end
M Ð
„
I 0
´ 1λCTC I

Γ Ð
„
0
1
λCT

Initialize: P0 Ð I
Initialize: S0 Ð Γr0
for i = 1 to N do
Pi ÐM ˚ Υi´1 ˚ Pi´1
Si ÐM ˚Υi´1 ˚ Si´1 ` Γri
end
Define: A Ð
“
0 I
‰
˚ PN ˚Mp:, 1q
Define: B Ð
“
0 ´ I
‰
˚ SN
Solve optimal initial condition: xoptpt0q “ A´1B.
for i = 0 to N ´ 1 do
zi Ð Pi ˚Mp:, 1q ˚ xoptpt0q ` Si
xptiq Ð
“
I 0
‰
˚ zi
end
Result: Compute reconstructed position - rptiq “ CXptiq, i P t0, 1, . . . , Nu.
sampled variance of the estimator for that particular amount of regularization. Our
objective is to pick the amount of regularization (λ-parameter) which minimizes
this sample variance. In our case, we have adopted the leaving-out-one strategy for
OCV, wherein all-but-one data point is used for reconstruction.
Now we’ll briefly discuss the ordinary cross validation procedure for the tra-
jectory smoothing problem. Let
`
xkλpt0q, ukλp¨q
˘
be a minimizer of the following op-
68
timization problem:
Minimize
xpt0q,u
¨
˚˝ Nÿ
j“0
j‰k
}yptjq ´ rj}2 ` λ
tNż
t0
uT pσqupσqdσ
˛
‹‚ (3.49)
subject to the constraints given by (3.3). Then the ordinary cross validation cost
V0pλq is defined as
V0pλq “
1
N ` 1
Nÿ
k“0
}rk ´ Cxkλptkq}2. (3.50)
Finally we pick up an optimal value of the regularization parameter as
λ˚ “ argmin
λą0
pV0pλqq . (3.51)
For the problem under consideration, the special structure of the underlying dynam-
ical system yields a nice form for the ordinary cross validation cost.
Now we solve the optimization problem (3.49) by following the path described
in Section 3.3. By following the co-state approach we can conclude that the optimal
trajectory will be a base integral curve of the associated Hamiltonian vector field,
with suitable jump conditions on the co-state variables. It can be easily observed
that the co-state variables are continuous at the left-out point, without any jump.
Then with a little bit of algebra we can show that xkλp0q, an optimal initial condition,
69
will satisfy a modified form of (3.48), in particular
r0 Ins
˜
N´1ź
i“k
MΥi
¸
Υk´1
˜
k´2ź
i“0
MΥi
¸
»
——–
In
´ 1λCTC
fi
ffiffifl x
k
λp0q
“ ´ r0 Ins
k´1ÿ
i“0
˜
N´1ź
j“k
MΥj
¸
Υk´1
˜
k´2ź
j“i
MΥj
¸
Γri
´ r0 Ins
Nÿ
i“k`1
˜
N´1ź
j“i
MΥj
¸
Γri (3.52)
where Υi’s are obtained by factorization of Λi’s, as mentioned in Lemma 4.1. There-
fore the reconstruction error encountered at the k-th data point can be represented
as
rk ´ Cxkλptkq
“ rk ´ C
»
——–
In
0
fi
ffiffifl
T
Υk´1
¨
˚˚
˝
k´2ź
i“0
MΥi
»
——–
In
´ 1λCTC
fi
ffiffifl x
k
λp0q `
k´1ÿ
i“0
˜
k´2ź
j“i
MΥj
¸
Γri
˛
‹‹‚
(3.53)
when we start the trajectory from xkλpt0q and apply the optimal input ukλ. From
(3.52) it is quite clear that λ affects xkλpt0q through M , Γ and
“
In ´ 1λCTC
‰
, and
hence the reconstruction error is a vector of rational functions in λ. Now we can
represent the cross validation cost, V0pλq, associated with this particular problem as
V0pλq “
1
N ` 1
Nÿ
k“0
ˆ
rTk rk `
`
xkλptkq
˘T CTCxkλptkq ´ 2rTk Cxkλptkq
˙
. (3.54)
70
As we have the (somewhat) closed form for the OCV cost, given by (3.54), we are
now ready to write down the first order necessary condition for the optimality of
the regularization parameter λ. We can easily check that the optimal value, λ˚, will
satisfy the following first order condition
Nÿ
k“0
¨
˚˚
˝
`
Cxkλ˚ptkq ´ rk
˘T C
»
——–
In
0
fi
ffiffifl
T
Υk´1
˛
‹‹‚
ˆ
˜
B
Bλ
«
k´2ź
i“0
MΥi
»
——–
In
´ 1λCTC
fi
ffiffifl x
k
λp0q `
k´1ÿ
i“0
˜
k´2ź
j“i
MΥj
¸
Γri
ff
λ˚
¸
“ 0. (3.55)
Remark 3.9. The optimal value of the regularization parameter (λ˚) depends on
the signal-to-noise ratio (higher SNR will cause a lower value for λ˚). In many
practical applications SNR might not be constant during data capture (in case of
position measurement using multi-camera network this might occur because of poor
calibration around the edges of the capture volume), and in that case using the same
λ will result in an erroneous reconstruction. However our algorithm works fine for
piecewise constant values of λ, with transitions occurring at the sampling instances
ttiuNi“1 (this can be shown rigorously with a little modification in the application of
path independence lemmas). In that case the optimal values for λ (for different
segments of the trajectory) can be computed by minimizing the OCV cost over a grid
of possible λ-values (each dimension signifying a particular segment of the flight).
71
Figure 6: This figure illustrates the reconstruction of a curve on a sphere.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
4.75
4.8
4.85
4.9
4.95
5
5.05
5.1
5.15
5.2
5.25
Time (sec)
D
i
s
t
a
n
c
e
 
f
r
o
m
 
O
r
i
g
i
n
 
 
Linear Model
Nonlinear Model
(a) This subplot shows the distance of the re-
constructed curves from the center of the un-
derlying sphere. Jerk path integral minimiza-
tion based approach yields a curve which is
closer to the sphere (in an average sense).
0 0.4 0.8 1.2 1.6 2
0
.4
.8
1.2
Time (sec)
C
l
a
s
s
i
c
a
l
 
C
u
r
v
a
t
u
r
e
 
 
Linear Model
Nonlinear Model
(b) This subplot shows the classical curvature
of the reconstructed curves. The curve, ob-
tained by minimization of jerk path integral,
has a curvature value greater than (almost al-
ways) the theoretical limit, i.e. 0.2.
Figure 7: The proposed algorithm, when applied to reconstruct a spiral on a sphere, performs
better than the nonlinear optimization based algorithm.
72
3.5 Numerical Results
First we test our algorithm on a synthetic data set; the sampled trajectory data
is obtained by adding an i.i.d. Gaussian noise (mean = 0, and standard deviation
= 0.15) to a spiral on a sphere of radius 5, and the number of samples used was
N “ 201. The reconstruction (Fig 6) through nonlinear optimization yields an
average fit error of 0.0599, whereas the proposed approach (3.5) yields an average fit
error of 0.0684. However, we compute two other metrics for performance comparison,
namely the distance of the curve from the center of the sphere (Fig 7a) and the
curvature of the reconstructed trajectory (Fig 7b). By analyzing these two metrics
it can be concluded that the proposed approach does a better job in trajectory
reconstruction.
Next, with permission from Kaushik Ghose and Cynthia Moss at the Auditory
Neuroethology Laboratory (BATLAB), Department of Psychology, University of
Maryland, we apply our algorithm to reconstruct a bat-insect trajectory pair. The
trajectory data was collected by Kaushik Ghose, and has previously been reported
in the context of prey capture flight strategies by echolocating bats [Ghose et al.,
2006]. The particular event, that we consider, had a flight duration of around
1.93s, and the corresponding data capture rate was set at 240fps. In this case, the
average fit error for the reconstruction of bat trajectory (Fig 8) through nonlinear
optimization is 2.2401ˆ 10´4, whereas the proposed approach yields a smaller error
of 7.6142 ˆ 10´5. Similarly, for the insect trajectory, our approach gives a better
fit (2.3913 ˆ 10´5 compared to 1.2966 ˆ 10´4). From Fig 10a and Fig 10b we can
73
0 0.5 1 1.5 1
1.5
−0.5
0
0.5
1
1.5
2
2.5
 
y-D
irect
ion
(m)
x-Direction (m)
 
z-
D
ir
ec
ti
on
(m
)
Bat- Raw Data
Bat - Linear Model
Bat - Nonlinear Model
Insect - Raw Data
Insect - Linear Model
Insect - Noninear Model
Figure 8: This figure illustrates the reconstruction of a bat-insect trajectory pair.
notice that the reconstructed speeds from two different regularization approaches
are almost equal for both trajectories, while the same does not hold true for the
evolution of curvature (Fig 9a and Fig 9b).
3.6 Conclusion
Using a simple linear generative model for trajectories (a triple integrator, with
jerk as the control input), we have developed a tool to obtain analytic solutions to
the inverse problem of trajectory reconstruction. Our approach casts the problem
in a linear framework with quadratic cost, and solve it by using techniques from
linear quadratic optimal control theory. Moreover, it has been shown that the
74
0 0.5 1 1.5 20
0.5
1
1.5
2
2.5
3
3.5
Time (sec)
C
u
r
v
a
t
u
r
e
 
−
 
κ
 
 
Linear Model
Nonlinear Model
(a) It shows the evolution of curvature(κ) for
the bat trajectory..
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time (sec)
C
u
r
v
a
t
u
r
e
 
−
 
κ
 
 
Linear Model
Nonlinear Model
(b) It shows the evolution of curvature(κ) for
the insect trajectory.
Figure 9: There is noticeable difference in the curvature profile obtained from two different ap-
proaches.
0 0.5 1 1.5 21.5
2
2.5
3
3.5
4
4.5
Time (sec)
S
p
ee
d
(m
/
s)
 
 
Linear Model
Nonlinear Model
(a) It shows the evolution of speed(ν) for the
bat trajectory.
0 0.5 1 1.5 21.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
Time (sec)
S
p
ee
d
(m
/
s)
 
 
Linear Model
Nonlinear Model
(b) It shows the evolution of speed(ν) for the
insect trajectory.
Figure 10: The reconstructed speeds from two different approaches are almost the same.
reconstructed positions can be expressed as a linear combination of measured data.
After trying the algorithm on synthetic data, we have applied it to reconstruct flight
trajectories of European starling flocks (as discussed in Chapter 6). Although this
approach overcomes the issues associated with the numerical optimization based
75
technique, it should be noted here that not all problem can be cast in a linear
quadratic framework, and that provides motivation for the next chapter.
76
Chapter 4: Data Smoothing through Nonlinear Optimization - Max-
imum Principle
As discussed earlier, the problem of reconstructing an underlying smooth signal
from sampled noisy observations arises in many areas of science and engineering (e.g.
trajectory reconstruction [Dey & Krishnaprasad, 2012], control theoretic splines
[Egerstedt & Martin, 2010] and quantum splines [Brody et al., 2012]), and in many
cases the underlying generative model, along with the regularizing penalty term, do
not allow us to cast the data smoothing problem in a linear quadratic framework.
One such example is the nonlinear version of the trajectory reconstruction problem
[(2.1), discussed in Section 2.1]. A quick reference to (3.6) reveals that this problem
can never be cast in a linear quadratic framework, and one needs appropriate tools
for data smoothing in nonlinear settings. However, the previous results towards
this direction are restricted to problems in a Riemannian setting (Burnett et al.
[2013]; Crouch & Leite [1991, 1995]; Jakubiak et al. [2006]; Machado et al. [2010]).
Although these works use calculus of variation based techniques, our earlier works
on data smoothing in a linear quadratic framework (in particular, Section 3.3) has
provided some insight about the applicability of Pontryagin’s maximum principle
[Pontryagin et al., 1962; Sussmann & Willems, 1997] in a sub-Riemannian setting.
77
This chapter1 focuses its attention on the nonlinear aspects of data smoothing,
and develops a general framework to address data smoothing problems from a control
theoretic perspective. Section 4.1 of this chapter presents a modified version of the
maximum principle to solve data smoothing problems on Rn, and this modified
maximum principle introduces jump discontinuities in the costate variables. Later,
in Section 4.2 we extend our result to address data smoothing problems in finite
dimensional Lie group settings. This framework, being a modified (extended) version
of Pontryagin’s maximum principle, can easily be exploited to solve problems in sub-
Riemannian setting as well (as demonstrated in Sections 4.4 and 4.5).
4.1 Data smoothing in a Euclidean setting
In this section we propose the modified maximum principle to address data
smoothing problems in a Euclidean setting. By introducing a nonlinear generative
model as
9qptq “ f
`
t, qptq, uptq
˘
, (4.1)
1A significant portion of this chapter has been reproduced from a paper by Dey & Krishnaprasad
[2014a].
78
the data smoothing problem can be formulated as the following optimal control
problem on Rn:
Minimize
qpt0q,u
Jpqpt0q, uq “
tNż
t0
L
`
t, qptq, uptq
˘
dt`
Nÿ
i“0
F pqptiq, qiq
subject to System dynamics (4.1),
xpt0q P Rn, u P U ,
(4.2)
where U is the space of piecewise continuous functions defined on the interval rt0, tN s.
Moreover, F pqptiq, qiq denotes the fit-error incurred at the i-th data point qi (at time
ti), and the Lagrangian Lpt, q, uq introduces regularization into the data smoothing
problem.
Before going into the details of necessary conditions for a sub-Riemannian op-
timal control problem, we focus on a special case, namely the Riemannian dynamics
given by fpt, q, uq “ u. Clearly, for this special case, (4.2) can be perceived as a
calculus of variation problem. Clearly, first variation of the cost can be expressed as
δJ “
tNż
t0
«ˆBL
Bq
˙T
δq `
ˆBL
B 9q
˙T
δ 9q
ff
dt`
Nÿ
i“0
ˆ BF
Bqptiq
˙T
δqptiq. (4.3)
By summing up the identity (from exact differential)
t´i`1ż
t`i
«ˆBL
B 9q
˙T
δ 9q ` ddt
ˆBL
B 9q
˙T
δq
ff
dt “
„BL
B 9q pt
´
i`1q
T
δqpti`1q ´
„BL
B 9q pt
`
i q
T
δqptiq
79
over the intervals pt`0 , t´1 q, pt`1 , t´2 q, ¨ ¨ ¨ , pt`N´1, t´Nq, we obtain
tNż
t0
ˆBL
B 9q
˙T
δ 9qdt “
„BL
B 9q pt
´
Nq
T
δqptNq `
N´1ÿ
i“1
„BL
B 9q pt
´
i q ´
BL
B 9q pt
`
i q
T
δqptiq
´
„BL
B 9q pt
`
0 q
T
δqpt0q ´
tNż
t0
d
dt
ˆBL
B 9q
˙T
δqdt. (4.4)
Then, by using the relationship given by (4.4), the first variation (4.3) can be ex-
pressed as
δJ “
tNż
t0
„BL
Bq ´
d
dt
ˆBL
B 9q
˙T
δqdt`
„BL
B 9q pt
`
N q
T
δqptN q
`
Nÿ
i“0
„BL
B 9q pt
´
i q ´
BL
B 9q pt
`
i q `
BF
Bqptiq
T
δqptiq ´
„BL
B 9q pt
´
0 q
T
δqpt0q. (4.5)
Therefore, first order necessary conditions (δJ “ 0) for minimality (in this special
Riemannian case, fpt, q, uq “ u) can be expressed as
(EL) ddt
ˆBL
B 9q
˙
´ BLBq “ 0, t P pti, ti`1q, i “ 0, 1, ¨ ¨ ¨ , N ´ 1
(JC-1) BLB 9q pt
`
i q ´
BL
B 9q pt
´
i q “
BF
Bqptiq
, i “ 0, 1, ¨ ¨ ¨ , N (4.6)
(BC-1) BLB 9q pt
´
0 q “ 0, and
BL
B 9q pt
`
Nq “ 0.
Now we consider second order necessary conditions for optimality. From (4.3)
80
we can note that the second variation of the cost can be represented as
δ2J “
tNż
t0
„
pδqqT
ˆ B2L
BqBq
˙
pδqq ` 2pδqqT
ˆ B2L
BqB 9q
˙
pδ 9qq ` pδ 9qqT
ˆ B2L
B 9qB 9q
˙
pδ 9qq

dt
`
Nÿ
i“0
pδqptiqqT
ˆ B2F
BqptiqBqptiq
˙
pδqptiqq, (4.7)
and by summing up the identity (from exact differential)
t´i`1ż
t`i
„
2pδqqT
ˆ B2L
BqB 9q
˙
pδ 9qq ` pδqqT
ˆ
d
dt ¨
B2L
BqB 9q
˙
pδqq

dt
“
`
δqpti`1q
˘T
„ B2L
BqB 9q pt
´
i`1q
 `
δqpti`1q
˘
´
`
δqptiq
˘T
„ B2L
BqB 9q pt
`
i q
 `
δqptiq
˘
over the intervals pt`0 , t´1 q, pt`1 , t´2 q, ¨ ¨ ¨ , pt`N´1, t´Nq, we obtain
tNż
t0
2pδqqT
ˆ B2L
BqB 9q
˙
pδ 9qqdt
“
`
δqptNq
˘T
„ B2L
BqB 9q pt
´
N q
 `
δqptNq
˘
´
`
δqpt0q
˘T
„ B2L
BqB 9q pt
`
0 q
 `
δqpt0q
˘
`
N´1ÿ
i“1
`
δqptiq
˘T
„ B2L
BqB 9q pt
´
i q ´
B2L
BqB 9q pt
`
i q
 `
δqptiq
˘
´
tNż
t0
pδqqT
ˆ
d
dt ¨
B2L
BqB 9q
˙
pδqqdt. (4.8)
81
Hence, the second variation (4.7) can be expressed as
δ2J “
tNż
t0
pδqqT
„ B2L
BqBq ´
d
dt ¨
B2L
BqB 9q

pδqqdt`
tNż
t0
pδ 9qqT
„ B2L
B 9qB 9q

pδ 9qqdt
`
Nÿ
i“0
`
δqptiq
˘T
„ B2L
BqB 9q pt
´
i q ´
B2L
BqB 9q pt
`
i q `
B2F
BqptiqBqptiq
 `
δqptiq
˘
`
`
δqptNq
˘T
„ B2L
BqB 9q pt
`
Nq
 `
δqptNq
˘
´
`
δqpt0q
˘T
„ B2L
BqB 9q pt
´
0 q
 `
δqpt0q
˘
. (4.9)
This leads us to express second order necessary conditions (δ2J ě 0) as
(LE) B
2L
B 9qB 9q ě 0, t P pti, ti`1q, i “ 0, 1, ¨ ¨ ¨ , N ´ 1
(JC-2) B
2L
BqB 9q pt
`
i q ´
B2L
BqB 9q pt
´
i q “
B2F
BqptiqBqptiq
, i “ 0, 1, ¨ ¨ ¨ , N (4.10)
(BC-2) B
2L
BqB 9q pt
´
0 q “ 0, and
B2L
BqB 9q pt
`
Nq “ 0.
At this point we introduce the adjoint/co-state variable (p) and define the
pre-Hamiltonian as
Hpt, q, p, uq “ pTu´ Lpt, q, uq, (4.11)
and therefore
BH
Bp “ u,
BH
Bq “ ´
BL
Bq ,
BH
Bu “ p´
BL
Bu ,
B2H
Bu2 “ ´
B2L
Bu2 .
It is worth mentioning here that we are focusing on regular extremals, i.e. B2LB 9qB 9q ‰ 0
along any solution of the Euler-Lagrange (EL) equation (4.6). Now we assume that
82
t ÞÑ q˚ptq is a trajectory of the system 9q “ fpt, q, uq “ u which solves the optimal
control problem (4.2) with control given as u˚ptq “ 9q˚ptq, and define
pptq “ BLBu pt, q
˚ptq, 9q˚ptqq. (4.12)
Then by applying the Euler-Lagrange condition (EL) from (4.6) we obtain
9pptq “ BLBq pt, q
˚ptq, 9q˚ptqq “ ´BHBq pt, q
˚ptq, pptq, 9q˚ptqq, (4.13)
and clearly we have
9q˚ptq “ u˚ptq “ BHBp pt, q
˚ptq, pptq, 9q˚ptqq. (4.14)
Also from the definition of the adjoint variable we have
BH
Bu pt, q
˚ptq, pptq, 9q˚ptqq “ 0, (4.15)
and the Legendre condition (LE) yields
B2H
Bu2 pt, q
˚ptq, pptq, 9q˚ptqq ă 0 (4.16)
because of the regularity property. As, (4.15) and (4.16) maximize the pre-Hamiltonian,
we have
Hpt, q˚ptq, pptq, 9q˚ptqq “ Max
u
Hpt, q˚ptq, pptq, uq (4.17)
83
A generalization of the above results leads us to an alternative version of the Pon-
tryagin’s maximum principle, which is tailored for the data smoothing problem.
Theorem 4.1 (PMP for data smoothing). Consider an optimal control problem
on Rn, given as
Minimize
qpt0q;u
Jpqpt0q, uq “
tNż
t0
Lpt, qptq, uptqqdt`
Nÿ
i“0
F
`
qptiq, qi
˘
subject to: 9qptq “ fpt, qptq, uptqq,
q : rt0, tN s Ñ Rn,
u P U : rt0, tN s Ñ U Ă Rm, u´ piecewise continuous.
(4.18)
Now we assume that u˚ is an optimal control input for (4.18), and q˚ denotes the
corresponding state trajectory. Then there exists a costate trajectory p : rt0, tN s Ñ
Rn such that
9q˚ptq “ BHBp pt, q
˚ptq, pptq, u˚ptqq
9pptq “ ´BHBq pt, q
˚ptq, pptq, u˚ptqq
(4.19)
during t P pti, ti`1q, i “ 0, 1, . . . , N , and
Hpt, q˚, p, u˚q “ max
uPU
Hpt, q˚, p, uq (4.20)
for t P rt0, tN sztt0, t1, ¨ ¨ ¨ , tNu, where the pre-Hamiltonian is defined asHpt, q, p, uq “
xp, fpt, q, uqy ´ Lpt, q, uq. Moreover, the intermediate state cost terms require jump
discontinuities in the costate variables, and the jump conditions and the boundary
84
values are given as
ppt´0 q “ 0,
ppt`i q ´ ppt´i q “
BF
`
qptiq, qi
˘
Bqptiq
, i “ 0, 1, . . . , N,
ppt`Nq “ 0.
(4.21)
Remark 4.1. A quick revisit to the trajectory reconstruction problem [ (3.5), dis-
cussed in Section 3.1] illustrates the control theoretic formulation (4.18) of a data
smoothing problem on Rn. It is easy to verify that this trajectory reconstruction
problem can be treated as a special case of (4.18) by introducing the following cor-
respondences
Lagrangian: L “ λuTu
Generative Model: f
`
t, qptq, uptq
˘
“ Aqptq `Buptq, qptq P R9, uptq P R3
Fit Error: F pqptiq, riq “ }Cqptiq ´ ri}2
“ qT ptiqCTCqptiq ´ 2xT ptiqCT ri ` rTi ri,
where ri denotes the measured position at time ti and C P R3ˆ9 maps the states into
the outputs.
Remark 4.2. It is worth mentioning here that the fit cost enters the problem through
the jump conditions in the co-state variables; while the flow of the system between
two consecutive data points is dictated by the path cost (penalty term) only.
85
Now we venture into the detailed proof of Theorem 4.1, and by adopting an
approach similar to the one taken in the book by Liberzon [2011], we develop the
proof using needle (strong) variation.
Proof. At the outset we introduce a new state variable q˜ : rt0, tN s Ñ R with its
dynamics governed by
9˜qptq “ Lpt, qptq, uptqq, t P pti, ti`1q,
q˜pt`i q ´ q˜pt´i q “ F
`
qptiq, qi
˘
, i “ 0, 1, . . . , N,
q˜pt´0 q “ 0,
(4.22)
and this leads us to an augmented system. By introducing
yptq fi
¨
˚˚
˝
q˜ptq
qptq
˛
‹‹‚P R
n`1, (4.23)
the dynamics of this augmented system (4.23) can be represented as
9yptq “
¨
˚˚
˝
Lpt, qptq, uptqq
fpt, qptq, uptqq
˛
‹‹‚fi gpt, yptq, uptqq, (4.24)
with the initial condition ypt´0 q “
¨
˚˚
˝
0
qpt0q
˛
‹‹‚. The corresponding jump conditions
86
in y can be expressed as
ypt`i q ´ ypt´i q “
¨
˚˚
˝
F pqptiq, qiq
0
˛
‹‹‚.
As a consequence, the cost functional (in 4.18) can be expressed as
Jpqpt0q, uq “ q˜pt`Nq “ Jpypt0q, uq. (4.25)
Clearly, an optimal trajectory q˚ (generated by u˚) of the original system (4.18) can
be retrieved from an optimal trajectory y˚ of the augmented system (4.23) through
a projection onto Rn along the q˜-axis.
Now, let a ą 0, b P R be such that Iǫ “ pb´ ǫa, bs Ă pt0, tNq, ti R pb´ a, bs @i,
and u˚ is continuous on Iǫ, @ǫ P p0, 1s and at b. Next we introduce a needle variation
(a perturbation pulse of short duration) by defining the perturbed control as
uw,Iǫptq fi
$
’’&
’’%
u˚ptq if t R Iǫ
w if t P Iǫ
, (4.26)
where w P U . Then, by letting k denote the index such that Iǫ Ă ptk´1, tkq, we have
yptq “ y˚ptq ` ǫΦ˚pt, bqδpw, a, bq `Opǫ2q (4.27)
for b ď t ď t´k [Liberzon, 2011, Section 4.2.4]. Here, the perturbation term δpw, a, bq
87
is defined as
δpw, a, bq “ a
´
g
`
b, y˚pbq, w
˘
´ g
`
b, y˚pbq, u˚pbq
˘¯
, (4.28)
and Φ˚ denotes the state transition matrix for the linearized dynamics governed by
9ψptq “
»
——–
0
`
Lqpt, q˚ptq, u˚ptqq
˘T
0nˆ1 fqpt, q˚ptq, u˚ptqq
fi
ffiffiflψptq.
Now we introduce a matrix Γk, defined as
Γk “
»
——–
1
ˆBF
Bq
`
q˚ptkq, qk
˘˙T
0nˆ1 Inˆn
fi
ffiffifl ,
to capture the effect of jump discontinuities on the perturbed trajectory y, where
BF
Bq denotes the partial derivative of F with respect to its first argument. This
enables us to express ypt`k q as
ypt`k q “ y˚pt`k q ` ǫΓkΦ˚ptk, bqδpw, a, bq `Opǫ2q. (4.29)
Proceeding this way the terminal point of the perturbed trajectory can be expressed
as
ypt`Nq “ y˚pt`Nq ` ǫΓNΦ˚ptN , tN´1q ¨ ¨ ¨ΓkΦ˚ptk, bqδpw, a, bq `Opǫ2q. (4.30)
Now, ǫΓNΦ˚ptN , tN´1q ¨ ¨ ¨ΓkΦ˚ptk, bqδpw, a, bq can be interpreted as an infinites-
88
imal perturbation of the terminal state caused by the needle variation in the control
input (uw,Iǫ), and its direction depends only on b and w (4.28,4.30). By letting
ρpw, bq denote the ray in this direction originating at y˚pt`N q, we define ~P as the
union of the rays ρpw, bq for all possible values of w and b. It can be noticed that
the cone ~P is not convex in general, and hence we concatenate different needle vari-
ations to generate a larger cone with the same vertex. These concatenations yield
a larger cone which contains the convex combinations of the points in ~P [Liberzon,
2011, Section 4.2.5], and we call it the terminal cone (TC
`
y˚pt`N q
˘
). Therefore, there
exists a nonzero vector µ P Rn`1 such that
µT
`
ypt`Nq ´ y˚pt`N q
˘
ě 0 (4.31)
for any perturbed trajectory y such that ypt`Nq ´ y˚pt`Nq P TC
`
y˚pt`N q
˘
[Tits, 2013,
Theorem B.3]. Then, by following the arguments given in [Liberzon, 2011, Sec-
tion 4.2.6], we can claim that (4.31) must be satisfied for the choice of
µ˚ “
»
——–
1
0nˆ1
fi
ffiffifl , (4.32)
because otherwise there exists a ypt`Nq which would violate the optimality of y˚. By
using this choice of µ “ µ˚ for the perturbed trajectories of the form (4.30), (4.31)
can be represented as
“
ΦT˚ ptk, bqΓTk ¨ ¨ ¨ΦT˚ ptN , tN´1qΓTNµ˚
‰T δpw, a, bq ě 0. (4.33)
89
Now we introduce ξ : rt0, tN s ÞÑ Rn`1, and by letting the dynamics of ξ be
governed by
9ξptq “
»
——–
0 0nˆ1
´Lqpt, q˚ptq, u˚ptqq ´
`
fqpt, q˚ptq, u˚ptqq
˘T
fi
ffiffifl ξptq, (4.34)
we define an adjoint system on Rn`1. Moreover, we define ξpt`Nq “ µ˚, and introduce
the interface conditions given by
ξpt´i q “
»
——–
1 01ˆn
BF
Bq
`
q˚ptkq, qk
˘
Inˆn
fi
ffiffifl ξpt
`
i q (4.35)
at tN , tN´1,¨ ¨ ¨ ,t0. This adjoint system allows us to represent (4.33) as
ξT pbq
´
g
`
b, y˚pbq, w
˘
´ g
`
b, y˚pbq, u˚pbq
˘¯
ě 0 (4.36)
for all w P U and b P rt0, tN sztt0, ¨ ¨ ¨ , tNu.
It can be easily verified that the first component of 9ξ is identically zero (4.34),
and the jump discontinuity doesn’t exist for the first component of ξ (4.35). Then it
directly follows from (4.32) that the first component of ξ is set constant at 1. This
enables us to decompose ξptq into
ξptq “
»
——–
1
´pptq
fi
ffiffifl . (4.37)
90
This decomposition of ξ yields the dynamics, boundary values and jump conditions
for p : rt0, tN s ÞÑ Rn as:
9pptq “ ´Lqpt, q˚ptq, u˚ptqq `
`
fqpt, q˚ptq, u˚ptqq
˘Tpptq (4.38)
and
ppt`N q “ 0
ppt`i q ´ ppt´i q “
BF
Bq
`
q˚ptkq, qk
˘
.
(4.39)
Using the fact that the pre-Hamiltonian is defined as
Hpt, q, p, uq “ xp, fpt, q, uqy ´ Lpt, q, uq, (4.40)
the dynamics of p (4.38) can be expressed as
9pptq “ ´BHBq pt, q
˚, p, u˚q. (4.41)
Moreover, by using the fact that qpt0q is free and minimizes the cost, it can be
concluded that ppt´0 q “ 0 [Liberzon, 2011, Section 4.3.1]. Also, the dynamics of
the optimal state trajectory q˚ can be derived straightforward from (4.40). Now we
focus to (4.36) to show maximality of the Hamiltonian. By using the decomposition
of ξ, (4.36) can be represented as
»
——–
1
´pptq
fi
ffiffifl
T »
——–
L
`
t, q˚ptq, w
˘
´ L
`
t, q˚ptq, u˚ptq
˘
f
`
t, q˚ptq, w
˘
´ f
`
t, q˚ptq, u˚ptq
˘
fi
ffiffifl ě 0,
91
or equivalently
H
`
t, q˚ptq, pptq, u˚ptq
˘
ě H
`
t, q˚ptq, pptq, w
˘
(4.42)
for t P rt0, tN sztt0, ¨ ¨ ¨ , tNu. This concludes our proof for Theorem 4.1.
Remark 4.3. The maximum principle, when applied to the trajectory reconstruction
problem (referred in Remark 4.1), gives rise to the Hamiltonian dynamics given by
d
dt
»
——–
q˚ptq
pptq
fi
ffiffifl “
»
——–
A 12λBBT
0 ´AT
fi
ffiffifl
»
——–
q˚ptq
pptq
fi
ffiffifl ,
and the corresponding optimal control can be expressed as
u˚ptq “ 12λB
Tpptq.
Moreover, the boundary values and jump conditions for the costate p are computed
as
ppt´0 q “ ppt`Nq “ 0
ppt`i q ´ ppt´i q “ 2CT
“
Cqptiq ´ ri
‰
.
Although it appears slightly different from the solutions obtained in (3.38,3.39), this
discrepancy can be avoided by introducing the following change of variable
p˜ “ ´ 12λp.
92
Remark 4.4. It is worth mentioning here that jump discontinuities in optimal con-
trol problems are not new. In the context of calculus of variation, they can be traced
back to the time of Weierstrass (Weierstrass-Erdmann corner condition [Liberzon,
2011]). More recently, they arise in hybrid optimal control when the associated dy-
namics undergo switching (Shaikh & Caines [2007]; Taringoo & Caines [2013]).
4.2 Data smoothing problems in a Finite Dimensional Matrix Lie
group setting
In this section we extend our result to tackle data smoothing problems in finite
dimensional matrix Lie group settings. For example, by using natural-Frenet frame
equations [Bishop, 1975] as the underlying generative model and penalizing high
rates of change in speed and curvatures, trajectory reconstruction can be formu-
lated as a data smoothing problem on SEp3q ˆ R3 [Reddy, 2007]. Also, trajectory
smoothing on a sphere can be posed as a data smoothing problem on SOp3q (similar
to the problem discussed by Brody et al. [2012]).
We begin by considering a finite dimensional matrix Lie group G and a left
invariant vector field defined on G. Before defining left invariance we introduce left
action by letting Lg : G Ñ G, h ÞÑ gh denote the left translation by g P G and
ThLg : ThGÑ TghG denote its tangent map (linearization).
Definition 4.1 (Left invariant vector field). Given the left action Lg defined on a
93
Lie group G, a vector field X : GÑ TG, h ÞÑ ph, vhq will be called left invariant if
ThLgpvhq “ vgh @h P G.
This allows us to define a left invariant control system by letting ve (referred
as ξ henceforth) be a control curve in the Lie algebra gp“ TeG, tangent space to the
Lie group G at the group identity element e). Then the dynamics takes the form
9gptq “ TeLgptq ¨ ξuptq (4.43)
where each control input w defines an element ξw of g.
Now we consider the following optimal control problem on G:
Minimize
gpt0q;u
Jpgpt0q, uq “
tNż
t0
L
`
uptq
˘
dt`
Nÿ
i“0
F pgptiq, giq
subject to: 9g “ TeLg ¨ ξu “ gξu, (4.44)
g : rt0, tN s Ñ G, u P U ,
where F pgptiq, giq denotes the fit error at sampling instant ti and U is the space of
piecewise continuous functions over rt0, tN s. The control curve is defined as ξu “
X0 `
řk
i“1 uiXk where tXiuni“1 is a basis of the Lie algebra g, k ă n and X0 P
spantXk`1, Xk`2, ¨ ¨ ¨ , Xnu. Clearly, the Lagrangian is assumed to be left invariant.
94
We also assume that the fit cost F pgptiq, giq is invariant under the left action, i.e.
F pgptiq, giq “ F pLg ¨ gptiq, Lg ¨ giq
for any g P G, and uptq, t P rt0, tN s takes value in U Ă Rk.
Theorem 4.2 (PMP for data smoothing on a finite dimensional matrix
Lie Group). Consider an optimal control problem on a finite dimensional matrix
Lie group G, given as
Minimize
gpt0q;u
Jpgpt0q, uq “
tNż
t0
L
`
uptq
˘
dt`
Nÿ
i“0
F pgptiq, giq
subject to: 9gptq “ TeLgptq ¨ ξuptq “ gptq
`
X0 `
kÿ
i“1
uiptqXk
˘
,
g : rt0, tN s Ñ G,
u P U : rt0, tN s Ñ U Ă Rk, u´ piecewise continuous.
(4.45)
Now we assume that u˚ is an optimal control input for (4.45). Then, corresponding
state trajectory g˚ is the base integral curve pg˚, pq of a Hamiltonian vector field
XHpg˚,p,u˚q on T ˚G, where the pre-Hamiltonian is defined as
Hpg, p, uq “ xp, TeLg ¨ ξuy ´ Lpuq “ xp, 9gy ´ Lpuq,
and an optimal control input maximizes H (which clearly is G invariant), i.e.
Hpg˚, p, u˚q “ Max
u
Hpg˚, p, uq.
95
Moreover, intermediate state penalties require jump discontinuities in p and the
corresponding boundary values and jump conditions are given as
ppt´0 q “ ppt`N q “ 0
and, ppt`i q ´ ppt´i q “ Dg˚ptiqF, i “ 0, 1, ¨ ¨ ¨ , N
where Dg˚ptiqF represents the Frechet derivative of the fit-error at g˚ptiq P G.
Proof. Here we adopt a variational approach to derive necessary conditions for opti-
mality, and as a first step express the cost functional in terms of the pre-Hamiltonian.
Clearly, the cost can be represented as
Jpgpt0q, uq “
tNż
t0
`
xp, 9gy ´Hpg, p, uq
˘
dt`
Nÿ
i“0
F pgptiq, giq. (4.46)
As our focus is restricted to matrix Lie groups, the pairing x¨, ¨y should be interpreted
as a trace inner-product in an appropriate matrix space.
Let u˚ be an optimal control (piecewise continuous) and g˚ be the correspond-
ing optimal trajectory of the system. First we consider perturbed controls of the
form
uǫ “ u˚ ` ǫδu (4.47)
where δu is continuous in the intervals pti, ti`1q, and let ξǫ “ ξu˚ ` ǫδξu denote
the associated perturbed control curves on g. Then, the corresponding perturbed
96
trajectory can be represented as
gǫ “ g˚ ` ǫδg `Opǫ2q (4.48)
where δg “ g˚δξu.
We begin by considering the first variation of the cost functional J , and show
that the first variation δJ can be expressed as
δJ “
tNż
t0
´
xp, δ 9gy ´ x∇gH
`
g˚, p, u˚
˘
, δgy ´ x∇uH
`
g˚, p, u˚
˘
, δuy
¯
dt
`
Nÿ
i“0
xDg˚ptiqF, δgptiqy, (4.49)
where Dg˚ptiqF represents the Frechet derivative at g˚ptiq P G. Next we show that
the first integrand in the first variation (4.49) can be expressed as
tNż
t0
xp, δ 9gydt “
N´1ÿ
i“0
t´i`1ż
t`i
xp, δ 9gydt
“
N´1ÿ
i“0
´
xppt´i`1q, δgpt´i`1qy ´ xppt`i q, δgpt`i qy ´
t´i`1ż
t`i
x 9p, δgydt
¯
“
N´1ÿ
i“0
´
xppt´i`1q, δgpti`1qy ´ xppt`i q, δgptiqy
¯
´
tNż
t0
x 9p, δgydt
“ xppt`Nq, δgptNqy `
Nÿ
i“0
xppt´i q ´ ppt`i q, δgptiqy ´ xppt´0 q, δgpt0qy
´
tNż
t0
x 9p, δgydt. (4.50)
97
Then by replacing (4.50) into (4.49), the first variation can be represented as
δJ “ xppt`Nq, δgptNqy `
Nÿ
i“0
xppt´i q ´ ppt`i q `Dg˚ptiqF, δgptiqy ´ xppt´0 q, δgpt0qy
´
tNż
t0
x 9p`∇gH
`
g˚, p, u˚
˘
, δgydt´
tNż
t0
x∇uH
`
g˚, p, u˚
˘
, δuydt. (4.51)
Now, first-order necessary condition for optimality dictates that δJ (4.51) must
be zero for any perturbation in control (δu) or initial condition (δgpt0q), and this
condition holds true for every p. Now, we make a special choice and assume the
following structure on p:
‚ 9p “ ´∇gH
`
g˚, p, u˚
˘
‚ ppt`N q “ 0
‚ ppt`i q ´ ppt´i q “ Dg˚ptiqF, @i P t1, 2, ¨ ¨ ¨ , Nu.
(4.52)
These assumptions allow us to represent the first variation around an optimal tra-
jectory as
δJ “ ´xppt´0 q, δgpt0qy ´
tNż
t0
x∇uH
`
g˚, p, u˚
˘
, δuydt, (4.53)
and first-order necessary condition for optimality requires δJ “ 0 (4.53) for any
perturbation δu or δgpt0q. Thus, first-order necessary condition implies
‚ ∇uH
`
g˚, p, u˚
˘
“ 0
‚ ppt´0 q “ 0.
(4.54)
98
Moreover, the definition of the pre-Hamiltonian let us represent the dynamics of g˚
as
9g˚ “ ∇pH
`
g˚, p, u˚
˘
(4.55)
Next we focus on the second variation of J , show that a second order necessary
condition can be expressed as
∇uuH
`
g˚, p, u˚
˘
ď 0. (4.56)
Now, by narrowing our focus to regular extremals, i.e. ∇ 9g 9gL ‰ 0 along any solution
of the pg˚, pq-dynamics, (4.54) and (4.56) yield the following maximality condition
H
`
g˚, p, u˚
˘
“ Max
u
H
`
g˚, p, u
˘
(4.57)
This concludes our proof for Theorem 4.2.
4.3 A Quick Revisit to Lie-Poisson Reduction
This section provides a brief introduction to Lie-Poisson reduction. Interested
readers may refer the works of Krishnaprasad [1985, 1993] and Marsden & Ratiu
[2003] for further details.
As there exists a bundle isomorphism between the cotangent bundle (T ˚G) of a
Lie group G and the product Gˆg˚, we can easily introduce two bundle projections,
99
defined as
π : T ˚GÑ g˚
π˜ : T ˚GÑ G.
(4.58)
Moreover, any vector from a tangent space of the cotangent bundle T ˚G can be
translated to a vector in the Lie algebra (via consecutive action of appropriate
tangent lifts, as shown in Fig 11). These two facts provide a natural choice for a
one-form on T ˚G, namely the Poincare´ one-form (T ˚G Q a ÞÑ Θa P T ˚a pT ˚Gq),
defined as
Θa
´
v
¯
“
A
π
`
a
˘
, Tπ˜paqLπ˜paq´1 ¨
`
Taπ˜ ¨ v
˘E
, (4.59)
for any v P TapT ˚Gq, a P T ˚G. Then, by using exterior derivative of this one-form,
we can define a symplectic form on T ˚G (ω “ ´dΘ) [Marsden & Ratiu, 2003].
Clearly, this symplectic form associates a Hamiltonian vector field to each smooth
real-valued function (Hamiltonian) on T ˚G.
Now, by letting C8pT ˚Gq denote the space of smooth real-valued functions on
T ˚G, we can introduce a Poisson bracket t¨, ¨u : C8pT ˚Gq ˆ C8pT ˚Gq Ñ C8pT ˚Gq,
φ, ψ ÞÑ tφ, ψu “ ωpHφ,Hψq, where Hφ is the Hamiltonian vector field associated
with the smooth function φ. It can be easily verified that if φ,ψ P C8pT ˚Gq are
invariant under left translation, then tφ, ψu is also G-invariant. As the pullback of
any function on g˚ by π will define a G-invariant function on T ˚G, this enables us
to define the Lie-Poisson bracket t¨, ¨ug˚ : C8pg˚q ˆ C8pg˚q Ñ C8pg˚q as
π˚th1, h2ug˚ “ th1, h2ug˚ ˝ π “ tπ˚h1, π˚h2u (4.60)
100
D
ua
l
Figure 11: This figure illustrates Lie-Poisson Reduction.
where π˚ denotes the pullback by π and h1, h2 P C8pg˚q.
Next, we introduce µ “ TeL˚g˚ ¨p to represent the dual control curve on g˚ and
formalize the mapping of the integral curve pg˚, pq of a left invariant Hamiltonian
vector field onto the dual of the Lie algebra. By letting tX5i uni“1 denote the dual basis
for g˚, µ can be represented as µ “ řni“1 µiX5i . On the other hand, a left-invariant
Hamiltonian H
`
g˚, p, u˚
˘
projects to a reduced Hamiltonian (h) on g˚. Now, h
defines a Hamiltonian vector field through the Lie-Poisson bracket, thus defining
the dynamics for µ. Finally, through an explicit computation of the Lie-Poisson
101
bracket, we can derive the reduced dynamics as
9µi “ ´
nÿ
j“1
nÿ
k“1
µkΓkij
Bh
Bµj
, (4.61)
where Γkij denote the structure constants associated with the Lie algebra g.
4.4 Example I: Data Smoothing on SOp3q
First we consider a left-invariant dynamics on SOp3q governed by
9g “ g
`
u1X1 ´ u2X2
˘
“ gξu,
g P SOp3q, ξu P sop3q,
(4.62)
where u1, u2 denote curvature control inputs (for curvature), and
X1 “
»
——————–
0 ´1 0
1 0 0
0 0 0
fi
ffiffiffiffiffiffifl
, X2 “
»
——————–
0 0 1
0 0 0
´1 0 0
fi
ffiffiffiffiffiffifl
, X3 “
»
——————–
0 0 0
0 0 ´1
0 1 0
fi
ffiffiffiffiffiffifl
,
define a basis for the associated Lie algebra sop3q. Clearly, by assuming only two
controls for a system evolving on a three dimensional manifold, we have cast the
problem in a sub-Riemannian setting.
Now we consider a data smoothing problem on SOp3q which attempts to find
a curve g : rt0, tN s Ñ SOp3q to traverse (approximately) through the sequence of
targeted orientations g0 Ñ g1 Ñ ¨ ¨ ¨ Ñ gN at time t0, t1, t2, and so on, respectively.
102
Our approach imposes regularization to this inverse problem by trading total fit-
error against high values of the curvature path integral, and therefore the smoothing
problem can be expressed as the following optimal control problem:
Minimize
gpt0q,u1,u2
Nÿ
i“0
}I3 ´ gptiqgTi }2F ` λ
tNż
t0
`
u21 ` u22
˘
dt
subject to 9g “ g
`
u1X1 ´ u2X2
˘
,
g : rt0, tN s Ñ SOp3q, u1, u2 P U ,
(4.63)
where U is the space of real valued functions on rt0, tN s and I3 denotes a 3 ˆ 3
identity matrix. λ (ą 0) is the regularization parameter which maintains a balance
between goodness of fit and smoothness of the reconstructed trajectory on SOp3q.
By comparing this optimal control problem (4.63) with the one mentioned in the
statement of maximum principle (4.45) we have
Lpuq “ λpu21 ` u22q “ λxξu, ξuysop3q
F pgptiq, giq “ }I3 ´ gptiqgTi }2F
where the inner-product on sop3q is defined as xv1, v2ysop3q “ 12 TrpvT1 v2q “ 12 Trpv1vT2 q
for v1, v2 P sop3q.
103
4.4.1 Maximum Principle
Restricting our attention to normal extremals, we define the pre-Hamiltonian
as
Hpg, p, uq “ xp, TeLg ¨ ξuy ´ Lpuq (4.64)
where p P T ˚g SOp3q, and TeLg represents the tangent lift of the left translation by
a group element g on SOp3q. Now we introduce µ at the dual of the Lie algebra
(so˚p3q), defined as µ “ TeL˚g ¨ p. By letting X5i , i “ 1, 2, 3 denote a dual basis for
so˚p3q (corresponding to the primal basis tXiu3i“1), µ can be represented as
µ “
3ÿ
i“1
µiX5i .
Therefore, by exploiting left-invariance of the generative model (4.62), the pre-
Hamiltonian can be expressed as
Hpg, p, uq “ xTeL˚g ¨ p, ξuy ´ Lpuq “ x
3ÿ
i“1
µiX5i , pu1X1 ´ u2X2
˘
y ´ Lpuq
“ u1µ1 ´ u2µ2 ´ λpu21 ` u22q. (4.65)
As both ξu and Lpuq are differentiable with respect to u, an optimal control input
(u˚) can be obtained by solving
BH
Bui
ˇˇ
ˇˇ
ui“u˚i
“ 0, i “ 1, 2. (4.66)
104
Then (4.65) and (4.66) yield the optimal control inputs as
¨
˚˚
˝
u˚1
u˚2
˛
‹‹‚“
1
2λ
¨
˚˚
˝
µ1
´µ2
˛
‹‹‚, (4.67)
and by substituting the optimal controls into the pre-hamiltonian, (4.65) yields an
SOp3q-invariant hamiltonian. Hence we have the reduced hamiltonian on so˚p3q,
given by
h “ 14λpµ
2
1 ` µ22q. (4.68)
4.4.2 Frechet Derivative of the Fit Error
On the other hand, we need to evaluate the Frechet derivative of the fit-error
in order to compute the jump conditions for µ. By using the definition of Frobenius
norm, the fit error (4.63) can be expressed as
F pgptiq, giq “ }I3 ´ gptiqgTi }2F gptiq, gi P SOp3q
“ Tr
”`
I3 ´ gptiqgTi
˘T `I3 ´ gptiqgTi
˘ı
“ 2Tr
“
I3 ´ gigT ptiq
‰
. (4.69)
105
Now we assume h to be a tangent vector at gptiq P SOp3q, and hence h can be
parametrized as h “ gptiqφ where φT “ ´φ P sop3q. Therefore we have
F
`
gptiqeǫφ, gi
˘
“ 2Tr
“
I3 ´ gi
`
eǫφ
˘TgT ptiq
‰
“ 2Tr
“
I3 ´ gi
`
I3 ` ǫφT `Opǫ2q
˘
gT ptiq
‰
. (4.70)
From (4.69) and (4.70) we can compute the Frechet differential of F along h in the
following way
DgptiqF
`
h
˘
“ lim
ǫÑ0
1
ǫ
´
F
`
gptiqeǫφ, gi
˘
´ F
`
gptiq, gi
˘¯
“ lim
ǫÑ0
2
ǫ Tr
´
´ ǫgiφTgT ptiq `Opǫ2q
¯
“ ´2Tr
´
giφTgT ptiq
¯
“ 2Tr
´
gT ptiqgiφ
¯
. (4.71)
Now we recall the fact that for a skew-symmetric matrix B “ ´BT P Rnˆn
we have
Tr
`
AB
˘
“ 12 Tr
`
pA´ AT qB
˘
for any A P Rnˆn, and by using this fact the differential of F : SOp3q Ñ R (4.71)
106
can be represented as
DgptiqF
`
h
˘
“ Tr
´“
gT ptiqgi ´ gTi gptiq
‰
φ
¯
“ Tr
´“
gT ptiqgigT ptiq ´ gTi
‰
h
¯
“
@
2
`
gptiqgTi gptiq ´ gi
˘
, h
D
gptiq , (4.72)
and hence we have DgptiqF “ 2
`
gptiqgTi gptiq ´ gi
˘
. By noting that h P TgptiqSOp3q
can be expressed as TeLgptiq ¨ φ, we get
TeL˚gptiq ¨DgptiqF “ 2
`
gTi gptiq ´ gT ptiqgi
˘
. (4.73)
4.4.3 Reduced Dynamics and Jump Discontinuities on so˚p3q
Next we focus on the derivation of reduced dynamics and associated jump
conditions. By following the path laid out by Krishnaprasad [1993], the reduced
dynamics on so˚p3q can be computed as
9µiptq “ ´
3ÿ
j“1
3ÿ
k“1
µkptqΓkij
Bh
Bµj
ptq, i “ 1, 2, 3, (4.74)
where the temporal variable t lies in the open intervals ptl, tl`1q, l “ 0, ¨ ¨ ¨ , N´1, and
Γkij denote the structure constants associated with the Lie algebra sop3q. Moreover,
the corresponding jump conditions for µ can be obtained via Frechet derivative of
the fit-error.
107
By computing the corresponding Lie brackets on sop3q as
rX1, X2s “ ´X3, rX2, X3s “ ´X1, rX3, X1s “ ´X2, (4.75)
the associated structure constants (the nonzero ones) can be expressed in the fol-
lowing way
Γ312 “ ´1, Γ123 “ ´1, Γ231 “ ´1, (4.76)
and Γkij “ ´Γkji, 1 ď i, j, k ď 3. Then, by exploiting (4.96), the reduced dynamics
on the dual of Lie algebra can be expressed as
¨
˚˚
˚˚
˚˚
˝
9µ1
9µ2
9µ3
˛
‹‹‹‹‹‹‚
“ 12λ
¨
˚˚
˚˚
˚˚
˝
µ2µ3
´µ3µ1
0
˛
‹‹‹‹‹‹‚
, t P ptk, tk`1q, (4.77)
along with the jump conditions
µipt`k q ´ µipt´k q
“
@
TeL˚gptkq ¨DgptkqF,Xi
D
I3PSOp3q i “ 1, 2, 3
“
@
2
`
gTk gptkq ´ gT ptkqgk
˘
, Xi
D
I3PSOp3q
“ Tr
`
gT ptkqgkXi ´ gTk gptkqXi
˘
, (4.78)
where k “ 0, 1, ¨ ¨ ¨ , N ´ 1.
108
4.4.4 Explicit Solution of the Reduced Dynamics
In what follows, we develop a closed form solution (involving trigonometric
functions) for the reduced dynamics. Clearly, (4.77) yields an explicit solution of
the form
µ1ptq “ Ak sinpCkt` φkq
µ2ptq “ Ak cospCkt ` φkq
µ3ptq “ 2λCk
t P ptk, tk`1q, (4.79)
where the piecewise constant parameters Ak, Ck and φk can be computed using
the boundary values and jump conditions of µ. As optimality (ppt´0 q “ 0) causes
µpt´0 q to be equal to 0, we can compute the initial values of the solution parameters
(Ck, Ak, φk) as
C0 “
1
2λ Tr
´`
gT pt0qg0 ´ gT0 gpt0q
˘
X3
¯
A0 “ 2
c´
Tr
`
I3 ´ pgT pt0qg0q2
˘¯2
´ λ2C20 (4.80)
φ0 “ atan2
´
Tr
``
gT pt0qg0 ´ gT0 gpt0q
˘
X2
˘
,Tr
``
gT pt0qg0 ´ gT0 gpt0q
˘
X1
˘¯
´ C0t0.
Now we focus on the jump conditions for the solution parameters. Clearly, the
jumps in µ3 can be translated to an equivalent condition for Ck, given by
Ck ´ Ck´1 “
1
2λ Tr
´`
gT ptkqgk ´ gTk gptkq
˘
X3
¯
, (4.81)
109
where k P t1, 2, ¨ ¨ ¨ , N ´ 1u. However, there is no straightforward way to represent
the jump discontinuities in Ak and φk. Instead, the following equations should be
used to update these parameters
Ak sinpCktk ` φkq “ Tr
´`
gT ptkqgk ´ gTk gptkq
˘
X1
¯
` Ak´1 sinpCk´1tk ` φk´1q
Ak cospCktk ` φkq “ Tr
´`
gT ptkqgk ´ gTk gptkq
˘
X2
¯
` Ak´1 cospCk´1tk ` φk´1q.
Finally, the terminal value of the costate variable yields the following terminal con-
dition for the solution parameters:
CN´1 “ ´
1
2λ Tr
´`
gT ptN qgN ´ gTNgptNq
˘
X3
¯
AN´1 “ 2
c´
Tr
`
I3 ´ pgT ptNqgNq2
˘¯2
´ λ2C2N´1 (4.82)
φN´1 “ atan2
´
Tr
``
gT ptNqgN ´ gTNgptNq
˘
X2
˘
,Tr
``
gT ptNqgN ´ gTNgptNq
˘
X1
˘¯
´ CN´1tN .
It is clear at this point that the optimal control inputs (4.67), along with their
boundary values and intermediate jump conditions, can be evaluated using (4.79)-
(4.82).
As the sinusoidal optimal control inputs are in phase quadrature, they can
be interpreted as the natural curvatures for a circular helix, and hence it is pos-
sible to write down explicit solutions for the group dynamics on SOp3q [Justh &
Krishnaprasad, 2011]. Finally, an optimal initial condition is selected in such way
that the terminal values of the solution parameters (4.82) are consistent with their
110
initial values (4.80) and intermediate update rules (4.82). Thus, we have turned an
optimal control problem over an infinite dimensional space (SOp3q ˆ U ˆ U) into a
two-point boundary value problem, which can be tackled by adopting an appropriate
multiple-shooting method [Morrison et al., 1962].
4.5 Example II: Data Smoothing on SEp2q
We begin our discussion about data smoothing on SEp2q by considering the
dynamics of a unicycle moving on a plane. By letting px, yq P R2 and θ P S1
denote position and heading angle of the unicycle, the underlying dynamics can be
expressed as
9xptq “ u1ptq cos θptq
9yptq “ u1ptq sin θptq (4.83)
9θptq “ u2ptq
where u1 and u2 denote the speed and steering rate, respectively. These equations
pose a nonholonomic constraint on the system, namely 9x sin θ “ 9y cos θ, prohibiting
any side-slip of the unicycle. Alternatively, by packing the position vector px, yq,
along with the heading direction (disguised through cos θ, sin θ), inside a 3ˆ3 matrix
111
gptq defined as
gptq “
»
——————–
cos θptq ´ sin θptq xptq
sin θptq cos θptq yptq
0 0 1
fi
ffiffiffiffiffiffifl
,
the dynamics of the unicycle (4.83) can be expressed as a left-invariant dynamics
on SEp2q. Now, by letting
X1 “
»
——————–
0 ´1 0
1 0 0
0 0 0
fi
ffiffiffiffiffiffifl
, X2 “
»
——————–
0 0 1
0 0 0
0 0 0
fi
ffiffiffiffiffiffifl
, X3 “
»
——————–
0 0 0
0 0 1
0 0 0
fi
ffiffiffiffiffiffifl
,
denote a basis for the associated Lie algebra sep2q, the dynamics of the unicycle
Figure 12: This figure illustrates the state variables associated to the dynamics of an unicycle.
can be represented as
9g “ g
`
u2X1 ` u1X2
˘
“ gξu, g P SEp2q, (4.84)
where ξu defines a control curve on the Lie algebra.
112
Given a set of planar positions triuNi“0 Ă R2, we focus on finding a curve
g : rt0, tN s Ñ SEp2q which would traverse (approximately) through the sequence of
targeted positions r0 Ñ r1 Ñ ¨ ¨ ¨ Ñ rN at time t0, t1, t2, and so on, respectively.
Our approach imposes regularization to this inverse problem by trading total fit-error
against high values of the sum of speed and steering path integrals, and therefore the
data smoothing problem can be expressed as the following optimal control problem
on SEp2q:
Minimize
gpt0q,u1,u2
Nÿ
i“0
}rptiq ´ ri}2 ` λ
tNż
t0
`
u21 ` u22
˘
dt
subject to 9g “ g
`
u2X1 ` u1X2
˘
,
g : rt0, tN s Ñ SEp2q, u1, u2 P U ,
(4.85)
where U is the space of real valued functions on rt0, tN s, and λ (ą 0) maintains the
balance between goodness of fit and smoothness of the path. This data smoothing
problem can also be interpreted as a trajectory reconstruction problem for planar
curves, where regularization is imposed by penalizing high values of the speed and
steering rate path integrals.
Now, by comparing this data smoothing problem (4.85) with the optimal con-
trol problem mentioned in maximum principle statement (4.45), we have
Lpuq “ λpu21 ` u22q “ λxξu, ξuysep2q
F pgptiq, riq “ }Agptiqe3 ´ ri}2
where A “ re1 e2sT , and teiu3i“1 denotes a standard basis vector in R3. Moreover,
113
the inner-product on sep2q is defined as xv1, v2ysep2q “ Trpv1MvT2 q for v1, v2 P sep2q,
where
M “
»
——————–
1
2 0 0
0 12 0
0 0 1
fi
ffiffiffiffiffiffifl
is a symmetric, positive definite matrix.
4.5.1 Maximum Principle
Restricting our attention to normal extremals, we define the pre-Hamiltonian
as
Hpg, p, uq “ xp, TeLg ¨ ξuy ´ Lpuq (4.86)
where p P T ˚g SEp2q, and TeLg represents tangent lift of the left translation by a
group element g on SEp2q. Now we introduce µ at the dual of the Lie algebra
(se˚p2q), defined as µ “ TeL˚g ¨ p. By letting X5i , i “ 1, 2, 3 denote a dual basis for
se˚p2q (corresponding to the primal basis tXiu3i“1), µ can be represented as
µ “
3ÿ
i“1
µiX5i .
114
Therefore, by exploiting left-invariance of the underlying dynamics (4.84), the pre-
hamiltonian can be expressed as
Hpg, p, uq “ xTeL˚g ¨ p, ξuy ´ Lpuq “ x
3ÿ
i“1
µiX5i , pu2X1 ` u1X2
˘
y ´ Lpuq
“ u2µ1 ` u1µ2 ´ λpu21 ` u22q. (4.87)
As both ξu and Lpuq are differentiable with respect to u, an optimal control input
(u˚) can be obtained by solving
BH
Bui
ˇˇ
ˇˇ
ui“u˚i
“ 0, i “ 1, 2. (4.88)
Then, from (4.87) and (4.88), the optimal control inputs can be expressed as
¨
˚˚
˝
u˚1
u˚2
˛
‹‹‚“
1
2λ
¨
˚˚
˝
µ2
µ1
˛
‹‹‚, (4.89)
and by substituting the optimal controls into the pre-hamiltonian (4.87) yields an
SEp2q-invariant hamiltonian. Hence we have the reduced hamiltonian on se˚p2q,
given by
h “ 14λpµ
2
1 ` µ22q. (4.90)
4.5.2 Frechet Derivative of the Fit Error
Now we shift our attention towards computing the Frechet derivative of the
fit-error. It is easy to check, that the fit error between data and the reconstructed
115
position (4.85) can be expressed as
F pgptiq, riq “ }Agptiqe3 ´ ri}2 gptiq P SEp2q, ri P R2
“ eT3 gT ptiqATAgptiqe3 ´ 2rTi Agptiqe3 ` rTi ri. (4.91)
Now we assume h to be a tangent vector at gptiq P SEp2q, and hence it can be
parametrized as h “ gptiqφ where φ P sep2q. Therefore, a perturbed value of the
fit-error (at time ti) can be expressed as
F
`
gptiqeǫφ, gi
˘
“ eT3
`
eǫφ
˘TgT ptiqATAgptiq
`
eǫφ
˘
e3 ´ 2rTi Agptiq
`
eǫφ
˘
e3 ` rTi ri
“ eT3
`
I3 ` ǫφT `Opǫ2q
˘
gT ptiqATAgptiq
`
I3 ` ǫφ`Opǫ2q
˘
e3
´ 2rTi Agptiq
`
I3 ` ǫφ`Opǫ2q
˘
e3 ` rTi ri. (4.92)
From (4.91) and (4.92) we can compute the Frechet differential of F along h as
DgptiqF
`
h
˘
“ lim
ǫÑ0
1
ǫ
´
F
`
gptiqeǫφ, gi
˘
´ F
`
gptiq, gi
˘¯
“ lim
ǫÑ0
1
ǫ
“
2ǫeT3 φTgT ptiqATAgptiqe3 ´ 2ǫrTi Agptiqφe3 `Opǫ2q
‰
“ 2
“
Agptiqe3 ´ ri
‰T “Ahe3
‰
“ 2Tr
´“
Agptiqe3 ´ ri
‰“
Ahe3
‰T¯
“ 2Tr
´
AT
“
Agptiqe3 ´ ri
‰
eT3 hT
¯
“
@
2AT
“
Agptiqe3 ´ ri
‰
eT3M´1, h
D
gptiq , (4.93)
116
and hence we have DgptiqF “ 2AT
“
Agptiqe3 ´ ri
‰
eT3M´1. Alternatively, by noting
that the differential can be expressed as
DgptiqF
`
h
˘
“ 2Tr
´
AT
“
Agptiqe3 ´ ri
‰
eT3 φTgptiqT
¯
“ 2Tr
´
gptiqTAT
“
Agptiqe3 ´ ri
‰
eT3 φT
¯
“
@
2gptiqTAT
“
Agptiqe3 ´ ri
‰
eT3M´1, h
D
gptiq , (4.94)
it can be concluded that
TeL˚gptiq ¨DgptiqF “ 2gptiq
TAT
“
Agptiqe3 ´ ri
‰
eT3M´1. (4.95)
4.5.3 Reduced Dynamics and Jump Discontinuities on se˚p2q
Next we focus on the derivation of reduced dynamics and associated jump
conditions. By following the path laid out by Krishnaprasad [1993], the reduced
dynamics on se˚p2q can be computed as
9µiptq “ ´
3ÿ
j“1
3ÿ
k“1
µkptqΓkij
Bh
Bµj
ptq, i “ 1, 2, 3, (4.96)
where the temporal variable t lies in the open intervals ptl, tl`1q, l “ 0, ¨ ¨ ¨ , N´1, and
Γkij denote the structure constants associated with the Lie algebra sep2q. Moreover,
the corresponding jump conditions for µ can be obtained via Frechet derivative of
the fit-error.
117
By computing the Lie brackets on sep2q as
rX1, X2s “ X3, rX2, X3s “ 0, rX3, X1s “ X2, (4.97)
the associated structure constants can be expressed in the following way
Γ312 “ 1, Γ231 “ 1, (4.98)
with Γkij “ ´Γkji, 1 ď i, j, k ď 3, and the rest of the structure constants are zero. As
a result, the reduced dynamics on se˚p2q can be expressed as
¨
˚˚
˚˚
˚˚
˝
9µ1
9µ2
9µ3
˛
‹‹‹‹‹‹‚
“ 12λ
¨
˚˚
˚˚
˚˚
˝
´µ2µ3
µ3µ1
´µ1µ2
˛
‹‹‹‹‹‹‚
t P ptk, tk`1q, (4.99)
and the corresponding jump conditions are given by
µipt`k q ´ µipt´k q
“
@
TeL˚gptkq ¨DgptiqF,Xi
D
I3PSEp2q i “ 1, 2, 3
“
@
2gptkqTAT
“
Agptkqe3 ´ rk
‰
eT3M´1, Xi
D
I3PSEp2q
“ Tr
`
2gptkqTAT
“
Agptkqe3 ´ rk
‰
eT3XTi
˘
, (4.100)
where k “ 0, 1, ¨ ¨ ¨ , N ´ 1.
118
4.5.4 Explicit Solution of the Reduced Dynamics
Now we attempt to obtain a closed form solution for the reduced dynamics
(4.99). It is easy to check that the reduced Hamiltonian (4.90) is a conserved
quantity. Furthermore, by introducing
C “ 14λ
`
µ22 ` µ23
˘
, (4.101)
we can show that C is also conserved along the trajectories of (4.99). Then, by
exploiting the constants of motion, namely the reduced Hamiltonian h (4.99) and
the Casimir C (4.101), the dynamics of µ2 can represented as
9µ2 “
1
2λ
b
p4λh´ µ22qp4λC ´ µ22q
“ 2
?
hC
dˆ
1´ µ
2
2
4λh
˙ˆ
1´ hC
µ22
4λh
˙
. (4.102)
Then it is straightforward to show that (4.102) yields an explicit solution involving
Jacobi’s elliptic sine function. Whenever h ď C, the solution can be expressed as
µ2ptq “ 2
?
λh Sn
˜c
C
λ pt` φkq,
c
h
C
¸
, (4.103)
119
where Snp¨, ¨q denotes Jacobi’s elliptic sine function and t P ptk, tk`1q. Now, by
exploiting the standard identities for elliptic functions, we can express µ1 and µ3 as
µ21 “ 4λh Cn2
˜c
C
λ pt` φkq,
c
h
C
¸
µ23 “ 4λC Dn2
˜c
C
λ pt` φkq,
c
h
C
¸
,
(4.104)
and the appropriate signs will depend on the initial/boundary conditions. A similar
solution exists for the situation when h ą C. Now, by exploiting the fact that
both µpt´0 q and µpt`Nq are equal to zero, we turn it into a two-point boundary value
problem, and solve it via an appropriate multiple-shooting method [Morrison et al.,
1962]. Finally, an optimal state trajectory can be obtained by integrating the group
dynamics along with an optimal initial condition and optimal control inputs.
4.6 Conclusion
In this chapter, we have developed a framework (based on a modified version
of the maximum principle) to solve data smoothing in a semi-analytic way, and our
results are applicable to problems in both Euclidean and finite dimensional matrix
Lie group settings. We demonstrate the pertinence of this approach by solving an
example problem, wherein the generative model is governed by a left invariant vector
field on a matrix Lie group (SEp2q) and regularization is imposed through a left
invariant path cost (Lagrangian). In this special case Lie-Poisson reduction leading
to explicitly integrable dynamics brings in further simplification into the problem,
and we get closed-form solutions (in terms of Jacobi’s elliptic functions).
120
Part II
Analysis of Collectives
121
Chapter 5: Analysis of Bat Foraging in Two Different Contexts
Research in bat echolocation has yielded a rich trove of insight into how the
bat perceives the world through active acoustic probing (Griffin et al. [1960]; Sim-
mons et al. [1979]). Yet surprisingly little is known about how perception is turned
into action such as steering towards a target in foraging. Limited availability of
suitable high speed motion capture technology hindered earlier efforts to address
this question. More recently, trajectory analysis in the work by Ghose et al. [2006]
showed that the big brown bat, Eptesicus fuscus, essentially maintains constant ab-
solute target direction (CATD) while chasing a free flying target, a species of praying
mantis (Parasphendale agrionina). However, in a later study of competitive forag-
ing for a single tethered food source [Chiu et al., 2010], evidence emerged that a
big brown bat resorts to directing its flight towards a competitor, thus employing
classical pursuit (CP).
In this present work1, using geometric and statistical analysis and control
theory, we describe a comparative study of how foraging context shapes bat flight
strategy. The current study examines two different flight control strategies, namely
classical pursuit (CP) and constant absolute target direction (CATD). The CP strat-
1A significant portion of this chapter has been reproduced verbatim from a pre-print by Dey
et al. [2014].
122
egy refers to a configuration in which the follower always points its velocity vector
towards the position of a target (Galloway et al. [2010]; Wei et al. [2009]). The
CP strategy can also be viewed as a special case of constant bearing (CB) pursuit
strategy, which has gaze heuristic [McBeath et al., 1995] as one of its manifestation.
On the other hand, CATD is a stealth strategy in which the follower approaches
the target in such a manner that from the target’s point of view the follower always
appears to be at the same bearing [Srinivasan & Davey, 1995]. This strategy is also
known as motion camouflage (MC) strategy as it nullifies the transverse component
of the relative velocity, and therefore the follower’s optic flow vanishes to zero in the
target’s field of vision [Justh & Krishnaprasad, 2006]. This work also provides some
insight about the behavioral latency associated with a bat-flight.
In one of the earliest works by Ghose & Moss [2006], investigating sensori-
motor transformation in a foraging bat, it was possible to simultaneously record what
the bat perceived through its acoustic gaze, and its response through steering action.
The behavioral context here - a single bat trained to seek, localize and capture a
tethered mealworm hanging from the ceiling of a darkened flight room - is simpler
than that in either of the two later studies (Chiu et al. [2010]; Ghose et al. [2006])
that constitute the focus of the present paper. The resulting bat flight trajectories
are essentially planar. A main result of the study by Ghose & Moss [2006] is the
discovery of a feedback law that relates planar turning rate of the bat to acoustic
gaze angle, and the observation that the gain parameter in the law is modulated
by the echolocation pulse production rate (PPR) - higher the PPR, greater is the
gain. This observed adaptive linkage of flight motor output to spatial auditory
123
information about the target (stationary food item) was suggested as a compromise
between uncoupling gaze direction and flight control (possibly conserving energy)
and tight coupling to ensure accuracy needed for successful taking of the food item.
While not emphasized by Ghose & Moss [2006], the feedback law is consistent with
executing what we call the CP strategy in this paper, thus indicating that the
foraging bat has CP strategy in its repertoire of flight behavior. Besides CP and
CATD other strategies such as following a boundary may also be identifiable as
being part of the repertoire of an echolocating bat.
5.1 Experiment Details and Reconstruction of Trajectories
In this work we have analyzed the flight data collected from a series of experi-
ments conducted by students of Prof. Cynthia Moss at the Auditory Neuroethology
Laboratory (BATLAB), Department of Psychology, University of Maryland, and
performed a comparative analysis of the underlying flight strategies and feedback
mechanism for steering control. The bat flight experiments were carried out in a
large flight room with two high-speed cameras (Kodak MotionCorder CCD-based
cameras, running at 240 frames/s) placed in adjacent corners of the room. A com-
mercially available motion analysis software (Motus, Peak Performance Technolo-
gies, Englewood, CO) was used to recover 3-D position data from high-speed stereo
images. Finally we use the regularized inversion approach (Section 2.1) to smoothen
the individual trajectories and extract their speed and curvatures. Furthermore, all
animal care and experimental procedures were approved by the Institutional Animal
124
0 0.5 1 1.5 2 2.5 3 0
1
2
−0.5
0
0.5
1
1.5
2
2.5
3
 
y−
ax
is
x−axis
 
z
−
a
x
i
s
Bat (Smooth)
Mantis (Smooth)
Bat (Raw)
Mantis (Raw)
(a) A typical example of trajectory reconstruc-
tion.
10−5 10−4 10−3 10−2
101
102
103
104
105
Regularization Parameter (λ)
C
r
o
s
s
 
V
a
l
i
d
a
t
i
o
n
 
C
o
s
t
 
 
Bat
Mantis
(b) The corresponding variation in OCV cost.
Figure 13: This figure shows reconstructed trajectories for a particular pursuit event along the
with raw data. It also shows the variation of cross validation cost as a function of the smoothing
parameter.
Care and Use Committee at the University of Maryland, College Park.
At the initial stage of our study of bat-mantis interactions, eight bats were
trained to catch both free flying and tethered insects, and the room walls and ceiling
of the indoor flight arena were covered with sound-absorbent foam. Later during the
experiments, the praying mantis was released by hand as the bat was flying around
in the room. Moreover the ultrasound triggered diving behavior of the mantises were
suppressed by plugging Vaseline into their ears. For these bat-mantis experiments
we analyzed 18 successful trials wherein the bat eventually catches the insect. As the
number of trials were not very high we performed ordinary cross-validation (OCV)
for each trajectory in each of the trials, and used the result of OCV in reconstructing
the corresponding trajectory.
125
On the other hand, five big brown bats, forming four pairs, were used in the
experiments for studying the bat-bat interactions. The experiments were conducted
between July and September in 2005 and 2006. After being trained to capture a
tethered mealworm individually, two bats were released simultaneously from the
same spot in the flight room to compete for a single tethered mealworm. The trials
ended whenever one bat made contact with the mealworm. For this set of bat-
bat experiments we had 154 trials at our disposal. As the computational time for
cross-validation and trajectory reconstruction is quite significant (around 2 days on
a 12-core workstation, for a trajectory with 400 data points and 50 discrete values
of λ-parameter), we decided to perform OCV and trajectory reconstruction over
a subset of the whole data-set. This required extra care in picking the individual
trials for reconstruction. So, we randomly selected 30 Bat-1 trajectories (through
10−10 10−8 10−6 10−4 10−2 100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Optimal λ-Values (m4s)
N
or
m
al
iz
ed
C
ou
nt
(a) Distribution of optimal values of λ for Bat-
2 trajectories.
10−10 10−8 10−6 10−4 10−2 100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Optimal λ-Values (m4s)
N
or
m
al
iz
ed
C
ou
nt
(b) Distribution of optimal values of λ for Bat-
1 trajectories.
Figure 14: This figure shows the distribution of optimal values of regularization parameter (λ˚)
used to reconstruct trajectories for bat-bat interactions.
126
a uniform random sampling) from a pool of 154 Bat-1 trajectories, and performed
OCV on these trajectories. Then we chose the most frequent value of optimal λ for
reconstructing Bat-1 trajectories. Similar approach was adopted to choose a λ-value
for reconstructing Bat-2 trajectories. Next, we randomly selected 30 trials (out of
154) through a uniform random sampling, and this sampling was independent to
the previous selections (for OCV). Finally we reconstructed the Bat-1 and Bat-2
trajectories for these 30 trials using the previously chosen λ-values.
10−6 10−5 10−4 10−3 10−2 10−1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Optimal λ-Values (m4s)
N
or
m
al
iz
ed
C
ou
nt
(a) Distribution of optimal values of λ for bat
trajectories.
10−6 10−5 10−4 10−3 10−2 10−1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Optimal λ-Values (m4s)
N
or
m
al
iz
ed
C
ou
nt
(b) Distribution of optimal values of λ for man-
tis trajectories.
Figure 15: This figure shows the distribution of optimal values of regularization parameter (λ˚)
used to reconstruct trajectories for bat-mantis interactions.
5.2 Pre-processing of Trajectory Data
In the work by Ghose & Moss [2006], key aspect of data analysis is to recognize
that flight behavior can be demarcated into segments. While the bat is initially
ignorant of the location of the target (presented by opening a ceiling trapdoor at
127
a random location at random times), it goes through a sequence of stages from
search/approach to tracking to attack of the mealworm, ending the trial. This
demarcation of stages of flight was suggested by examination of histograms of PPR
of the echolocation calls which show three clear peaks corresponding to the stages.
Further in the course of a trial, increasing PPR is associated to increasing accuracy
in localization of target prior to capture. The demarcation into stages was then used
in the temporal segmentation of trajectory properties (turning rate, acoustic gaze
angle etc.) for the purposes of statistical fitting of steering feedback laws in each
segment.
In the setting of a single bat engaged in three-dimensional pursuit of a free-
flying insect as in the work of Ghose et al. [2006], similar trajectory segmentation
is possible, guided by echolocation PPR. In his dissertation, Reddy [2007] has used
such segmentation in fitting feedback laws, focusing essentially on attack segments.
In multiple trials highly maneuvering behavior is exhibited by the insect prey (whose
sensitive hearing of bat sonar is (only partially) disabled to reduce the incidence of
evasive action). Restricting data fitting mainly to segments characterized by PPR as
attack segments misses important components of pursuit behavior taking place over
the full course of a trial. In the setting of competitive prey capture the interaction
between competing bats appears to be of great relevance to the outcome. During the
course of such competitive interactions vocalization information is confounded by
the presence of interspersed social calls to possibly communicate intent of one bat to
another [Wright et al., 2014]. There are also periods of silence (as observed by Chiu
et al. [2008]) of one or more of the bats. Thus there is a need for disambiguation
128
between echolocation and social calls, appropriate for use in flight segmentation,
whereas in the context of single bat prey capture flights this consideration does
not arise. In this paper we side-step this disambiguation problem, by adopting an
approach to segmentation that does not use vocalization patterns in either context,
and instead is based on a geometric criterion described below.
In preparation for segmentation of trajectories as needed in the analysis of
flight behavior, we introduce two geometric concepts, namely following and con-
vergence, to assist us in understanding the individual roles in a dyadic (bat-mantis
or bat-bat) interaction. By letting ri and xi (respectively rj and xj) denote the
position and normalized velocity of the individual i (respectively the individual j)
in a dyadic interaction, these two notions can be defined as:
Definition 5.1 (Following Property). The flight behavior of an individual i (inter-
acting with individual j) is called “following” when its velocity vector has a negative
projection on its relative position vector rji (“ ri ´ rj), i.e.
rji
|rji|
¨ xi ă 0. (5.1)
Definition 5.2 (Convergence). The flight behavior of an individual i is called “con-
verging towards the individual j” if the distance between the individuals is shrinking,
i.e.
rji
|rji|
¨ 9rji ă 0, (5.2)
where the time derivative 9rji denotes the relative velocity.
129
Clearly, these two notions split a flight into the following four different regimes:
• Class I : Following and Converging,
• Class II : Following but Not Converging,
• Class III : Not Following but Converging, and
• Class IV : Neither Following nor Converging,
depending on the angle between the baseline vector and the velocity vectors of
the individuals. We should note that the convergence property is symmetric with
respect to individual roles. Any particular flight trial will have a number of different
contiguous segments characterized by the properties of following and convergence.
In Fig 16, we display distribution functions across all trials of the durations of
contiguous segments which are simultaneously following and converging. Averaging
over multiple flight trials, we designate an individual as follower (pursuer) in a
dyadic interaction, if it obeys the following condition for a duration longer than
that for the other individual (which we designate as pursuee).
The pie-chart in Fig 17a shows the percentage of all the four classes in the
flight data collected for Bat-2. From this pie-chart one can notice that Bat-2 follows
Bat-1 for more than 66% of the time. On the contrary Bat-1 follows Bat-2 for
only 27% of the time. Therefore we consider Bat-2 to be the follower of Bat-1 for
further analysis. This methodology for classification reveals that in spite of the setup
being symmetric there is a strong evidence of leader-follower relationship when a
bat competes for a single food source with another conspecific.
On the other hand the pie-chart in Fig 18 shows the percentage of all four
classes in the flight data from bat-mantis trajectory pairs. From this pie-chart it is
130
0 200 400 600 800 1000 1200 1400
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Contiguous duration of Following & Converging regimes (ms)
C
D
F
 
 
(a) Bat-bat interactions (Pursuer: Bat-2, Pur-
suee: Bat-1).
0 200 400 600 800 1000 1200 1400 1600 1800
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Contiguous duration of Following & Converging regimes (ms)
C
D
F
 
 
(b) Bat-mantis interactions (Pursuer: Bat,
Pursuee: Mantis).
Figure 16: This figure shows the CDF for contiguous durations of following and converging (Class
I) flight segments.
38.1229
28.2881
10.1109
23.4781
 
 
Class I Class II Class III Class IV
(a) Percentage of four classes of flight behavior
for the competitive interaction between two bats.
17.8791
9.7152
30.3547
42.051
 
 
Class I Class II Class III Class IV
(b) Effect of role reversal on the segmentation
of flight behavior for the bat-bat interaction.
Figure 17: This figure illustrates the fact that in a competitive interaction between two big brown
bats one of the bats leads the other.
clear that the bat follows the target for around 75% of the time. It also converges
towards the praying mantis for more than 70% of the time.
131
5.3 Analysis of Flight Strategy
Once we have segmented each trajectory into four classes (following and con-
verging ; following, but not converging ; not following, but converging ; and neither
following nor converging), we examine the trajectory data to weigh support for or
against the pursuit strategies CP and CATD. This is done by computing contrast
function values (Λ for CP and Γ for CATD/MC, Section 1.3.2) associated with the
data, and comparing the value distributions. A value close to ´1 for a contrast
function means strong support for the associated strategy. On the other hand a
value of `1 denotes maximum departure from the associated strategy. In order to
determine how long a trailing bat remained in the CP or CATD state, we consider
the duration of the flight when the contrast function value goes below ´0.9, i.e.
it lies in the range r´1,´0.9s. This duration of interest can be easily computed
as a percentage of the total flight duration by paying attention to the cumulative
61.0468
13.1282
9.1452
16.6798
 
 
Class I Class II Class III Class IV
Figure 18: This figure shows the percentage of four flight behavior classes in bat-mantis pursuit
events.
132
distribution of contrast function values. The rationale behind choosing the thresh-
old at ´0.9 is based on the principle that outcome of a statistical hypothesis test
should not depend on a test parameter. It should be noted here that, as a means for
closer inspection, we also restrict our focus on those flight regimes which are both
following and converging (Class I), and recompute the statistics. This provides a
common base for comparison, ensuring that neither of the contrast function values
become non-negative.
Initially we analyze the bat-bat pursuit strategy for the complete set of re-
constructed flight data, and the corresponding histograms of Λ and Γ are shown in
Fig 19. The closer the contrast function is to -1, the more the bats’ flight behavior
approaches a particular pursuit strategy. The peak of the CP contrast function is
positioned around -1, which indicates that the following bat mostly relies on the CP
strategy to pursue the leader (another conspecific). On the other hand, the CATD
contrast function is more evenly distributed between ´1 and 1, indicating lack of ev-
−1 −0.5 0 0.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Λ Values
N
o
r
m
a
li
ze
d 
Co
un
t
(a) CP (Contrast Function: Lambda).
−1 −0.5 0 0.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Γ Values
N
o
r
m
a
li
ze
d 
Co
un
t
(b) CATD (Contrast Function: Gamma).
Figure 19: Distribution of contrast function values shows dominance of CP during bat-bat inter-
actions. The frequency (y-axis) is normalized by the maximum count (considering both strategies).
133
−1 −0.5 0 0.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Λ Values
N
o
r
m
a
li
ze
d 
Co
un
t
(a) CP (Contrast Function: Lambda).
−1 −0.5 0 0.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Γ Values
N
o
r
m
a
li
ze
d 
Co
un
t
(b) CATD (Contrast Function: Gamma).
Figure 20: Distribution of Λ and Γ values shows dominance of CP during the following and
converging segments (class I) of bat-bat pursuit events. The frequency (y-axis) is normalized by
the maximum count (considering both strategies)
idence in favor of the CATD pursuit strategy. Moreover, the data show that 25.32%
of the time the following bat stays in the CP state, while on contrary it stays in
CATD state for only 3.79% of the time. This difference between evidence for each of
the individual strategies becomes more prominent if we focus on the following and
converging flight behavior, i.e. if data points from only class I are considered for
−1 −0.5 0 0.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Values for Contrast Function
C
D
F
 
 
Λ Γ
(a) Comparison of CDFs of the contrast func-
tions for CP and CATD (whole data set).
−1 −0.5 0 0.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Values for Contrast Function
C
D
F
 
 
Λ Γ
(b) Comparison of CDFs of the contrast func-
tions for CP and CATD (during class I).
Figure 21: Comparison of CDFs for the contrast functions shows dominance of classical pursuit
during bat-bat interactions.
134
−1 −0.5 0 0.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Λ Values
N
o
r
m
a
li
ze
d 
Co
un
t
(a) CP (Contrast Function: Lambda).
−1 −0.5 0 0.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Γ Values
N
o
r
m
a
li
ze
d 
Co
un
t
(b) CATD (Contrast Function: Gamma).
Figure 22: Distribution of Λ and Γ shows dominance of CATD/MC during bat-mantis interac-
tions. The frequency is normalized by the maximum count (considering both strategies).
strategy analysis. In that case the following bat stays in the CP state for 47.83%
of the time, against 7.48% of the time spent in the CATD state. The associated
histograms are shown in Fig 20. Therefore our present work reconfirms the findings
by Chiu et al. [2010], i.e. bats do not apply CATD while pursuing conspecifics, rather
they use CP strategy to follow another bat.
On the other hand, Fig 22 shows the histograms of Λ and Γ for the complete
−1 −0.5 0 0.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Λ Values
N
o
r
m
a
li
ze
d 
Co
un
t
(a) CP (Contrast Function: Lambda).
−1 −0.5 0 0.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Γ Values
N
o
r
m
a
li
ze
d 
Co
un
t
(b) CATD (Contrast Function: Gamma).
Figure 23: Distribution of Λ and Γ shows dominance of CATD/MC during the following and
converging segments (class I) of bat-mantis pursuit events. The frequency is normalized by the
maximum count (considering both strategies)
135
−1 −0.5 0 0.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Values for Contrast Function
C
D
F
 
 
Λ Γ
(a) Comparison of CDFs of the contrast func-
tions for CP and CATD (whole data set).
−1 −0.5 0 0.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Values for Contrast Function
C
D
F
 
 
Λ Γ
(b) Comparison of CDFs of the contrast func-
tions for CP and CATD (during class I).
Figure 24: Comparison of CDFs for the contrast functions shows dominance of CATD/MC during
bat-mantis interactions.
set of bat-mantis trajectory pairs. From this figure one can notice that the data
indicates strong evidence in favor of CATD pursuit strategy. Quantitatively, the bat
stays in the CATD state for 22.05% of the time in comparison to 9.12% of the time
in the CP state. Now we narrow our focus to the following and converging segments
of the flight, i.e. data points from only class I are considered for strategy analysis.
In that case the bat stays in the CATD state for 32.72% of the time, against 14.42%
of the time in the CP state (Fig 23). Therefore our analysis provides support for
bats’ use of CATD for pursuing a free flying mantis.
However, at this level of flight strategy analysis (through studying the distri-
bution of associated contrast function values), the significance of distinction between
CP and CATD is less sharply delineated for bat-mantis interactions than for bat-bat
interactions. This leads us to the next part of our analysis, i.e. comparison at the
level of feedback laws for steering control.
136
5.4 Analysis of Steering Control
A bat executes a pursuit strategy by continually adjusting the curvature of its
trajectory based on its perception of the relative motion of the target. Such strategy-
specific steering feedback control laws have been a subject of applied mathematical
research (Galloway et al. [2010]; Justh & Krishnaprasad [2006]; Reddy [2007]; Reddy
et al. [2006]). Here we investigate steering controls that underlie a bat’s pursuit
strategy by comparing empirical values of trajectory curvature with predictions from
theoretically well-founded feedback control laws (Table 5.1), and provide evidence
at the level of the steering control mechanism as well.
By letting, puem, vemq denote the empirical curvatures obtained from trajec-
tory smoothing and puth, vthq denote the curvature values computed using feedback
laws indicated in Table 5.1, we formalize this analysis as the following mismatch
minimization problem
Minimize
µą0,δPN
¨
˚˝ 1ř
jPS
`
|Ej| ´ δ
˘
ÿ
jPS
ÿ
tkPEj
”`
uemptkq ´ uthptk ´ δ∆q
˘2 `
`
vemptkq ´ vthptk ´ δ∆q
˘2ı
˛
‹‚.
(5.3)
Here, N, Ej and S represent the set of natural numbers, the set of time indices
associated with reconstructed trajectories for the j-th trial, and the index set of all
trials under consideration, respectively. It is essential to incorporate delays (δ∆)
into the theoretical curvature terms in the mismatch minimization problem, to take
into account the latency present in the sensorimotor feedback loops and to estimate
it from the data. We should also note that the discreteness of delay values (in
137
Strategy Steering Feedback Law
CATD/MC: uth “ ´
µ
νp
´
zp ¨
`
9rˆ r|r|
˘¯
vth “
µ
νp
´
yp ¨
`
9rˆ r|r|
˘¯
CP: uth “ ´
µ
νp
´
yp ¨
r
|r|
¯
´ 1νp|r|
´
zp ¨
`
9rˆ r|r|
˘¯
vth “ ´
µ
νp
´
zp ¨
r
|r|
¯
` 1νp|r|
´
yp ¨
`
9rˆ r|r|
˘¯
Table 5.1: Theoretically plausible feedback laws for constant absolute target direction (CATD/MC)
and classical pursuit (CP).
(5.3)) arises only because of data availability at a finite sample rate (with sampling
interval ∆). Clearly, in addition to gathering evidence for a particular pursuit
strategy, this approach also yields an estimate of behavioral latency (sensorimotor
delay) associated with the pursuit events.
Proposition 5.1. Let takuNk“0 and ta˜kuNk“0 be two finite sequences, and consider the
following optimization problem:
Minimize
µą0,δPN,ηPR
˜
1
N ` 1´ δ
Nÿ
k“δ
}apkq ´ µa˜pk ´ δq ´ η}2
¸
. (5.4)
Then (5.4) can be approximated by the following optimization problem
Maximize
δPN
Corrpa, a˜δq, (5.5)
where a˜δ represents a δ-shifted copy of ta˜ku, i.e. a˜δpkq “ a˜pk ´ δq.
138
Moreover, the optimal values of µ and η are given by
µ “ Covpa, a˜
δq
Varpa˜δq (5.6)
η “ 1N ` 1´ δ
Nÿ
k“δ
´
apkq ´ µa˜pk ´ δq
¯
(5.7)
Proof. One can notice that (5.4) can be expressed as
Minimize
δPN
˜
Minimize
µą0,ηPR
˜
1
N ` 1´ δ
Nÿ
k“δ
}apkq ´ µa˜pk ´ δq ´ η}2
¸¸
, (5.8)
and this formulation enables us to solve the optimization problem through a two-
step process. Now, for a given δ, we define aavg “ 1N`1´δ
Nř
k“δ
apkq and a˜avg “
1
N`1´δ
N´δř
k“0
a˜pkq. Then, for that particular choice of δ, we have
Nÿ
k“δ
}apkq ´ µa˜pk ´ δq ´ η}2
“
Nÿ
k“δ
´`
apkq ´ aavg
˘
´ µ
`
a˜pk ´ δq ´ a˜avg
˘
`
`
aavg ´ µa˜avg ´ η
˘¯2
“
Nÿ
k“δ
´
bpkq ´ µb˜pk ´ δq `
`
aavg ´ µa˜avg ´ η
˘¯2
“
Nÿ
k“δ
`
b2pkq ´ 2µbpkqb˜pk ´ δq ` µ2b˜2pk ´ δq
˘
`
Nÿ
k“δ
`
aavg ´ µa˜avg ´ η
˘2
` 2
Nÿ
k“δ
`
bpkq ´ µb˜pk ´ δq
˘`
aavg ´ µa˜avg ´ η
˘
(5.9)
where bpkq fi apkq ´ aavg and b˜pkq fi a˜pkq ´ a˜avg (k P tδ, . . . , Nu) are two zero mean
139
sequences. As the third term of (5.9) ceases to zero, one can notice that (5.9) get
minimized by choosing
η “ aavg ´ µa˜avg “
1
N ` 1´ δ
Nÿ
k“δ
`
apkq ´ µa˜pk ´ δq
˘
(5.10)
µ “
Nř
k“δ
bpkqb˜pk ´ δq
Nř
k“δ
b˜2pk ´ δq
“ Covpa, a˜
δq
Varpa˜δq . (5.11)
Replacing µ and η with their optimal values, (5.9) can be expressed as
Nÿ
k“δ
˜
b2pkq ´ 2Covpa, a˜
δq
Varpa˜δq bpkqb˜pk ´ δq `
ˆCovpa, a˜δq
Varpa˜δq
˙2
b˜2pk ´ δq
¸
“ pN ` 1´ δq
˜
Varpaq ´ 2Covpa, a˜
δq
Varpa˜δq Covpa, a˜
δq `
ˆCovpa, a˜δq
Varpa˜δq
˙2
Varpa˜δq
¸
“ ´pN ` 1´ δqVarpaq
˜ `
Covpaa˜δq
˘2
VarpaqVarpa˜δq ´ 1
¸
(5.12)
as both b and b˜ zero mean sequences. By using the minimum value from the inner
optimization (5.12), the problem of our interest can be expressed as
Maximize
δPN
˜
Varpaq
˜ `
Covpa, a˜δq
˘2
VarpaqVarpa˜δq ´ 1
¸¸
ô Maximize
δPN
Varpaq
´`
Corrpa, a˜δq
˘2 ´ 1
¯
. (5.13)
Assuming the delay to be sufficiently small compared to the length of the sequence
(δ ! N), we can ignore the effect of δ on the empirical (sample) variance of a (first
factor of the cost). As a consequence, the original optimization problem can be
140
approximated as
Maximize
δPN
Corrpa, a˜δq. (5.14)
One can readily recognize that solving the minimization problem (5.3) is com-
putationally demanding. However, this computational complexity can be tackled by
approximating it with a correlation maximization problem (as shown via Proposi-
tion 5.1). In this alternative approach, we compute the correlation between empirical
data (natural curvatures uem, vem stacked in a single array) and the theoretically
predicted curvature values (uth, vth stacked in a single array), as a function of de-
lay. Then, the delay which maximizes this correlation provides an estimate for the
behavioral latency.
For the bat-bat pursuit events the variation of correlation is shown in Fig 25,
and the corresponding values of optimal gain (µ˚) and optimal delay (δ˚) are pro-
0 50 100 150 200 250 300 350 4000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Delay (ms)
C
o
r
r
e
la
ti
on
 V
al
ue
 
 
U
V
U−V
(a) Correlation between empirical curvatures
and the curvatures obtained from the CATD
feedback law.
0 50 100 150 200 250 300 350 4000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Delay (ms)
C
o
r
r
e
la
ti
on
 V
al
ue
 
 
U
V
U−V
(b) Correlation between empirical curvatures
and the curvatures obtained from the CP feed-
back law.
Figure 25: Variation of correlation between empirical and theoretical curvatures shows dominance
of classical pursuit during bat-bat pursuit events.
141
Figure 26: Variation of residual for a CATD feedback law (Bat-Bat pursuit events). The residual
values (color-coded) are shown in log 10 scale.
Figure 27: Variation of residual for a CP feedback law (Bat-Bat pursuit events). The residual
values (color-coded) are shown in log 10 scale.
142
CATD CP
Maximum Correlation (ρmax) 0.5307 0.6159
Delay (δ˚) [ms] 272.9167 166.6667
Linear Gain (µest) 0.7156 1.9792
λ1 0.6256 0.3378
λ2 0.1607 0.0188
σ1 79.5584 94.7238
σ2 20.4416 5.2762
Table 5.2: Summary of the statistical analysis of steering control laws for bat-bat pursuit events.
λ1 and λ2 represent the principal component variances, i.e., eigenvalues of the covariance matrix.
σ1 and σ2 represent the percentage of total variance explained by principle components.
CATD CP
Minimum Value of Mismatch 0.3429 0.2010
Normalized Mismatch 1.0396 0.6216
Linear Gain 0.7165 1.9658
Delay (δ˚) [ms] 277.0833 177.0833
Table 5.3: Summary of the residual analysis of steering control laws for bat-bat pursuit events.
The mismatch is normalized by the product of rms values of empirical and theoretical curvatures.
vided in Table 5.2. One can notice that the correlation between the theoretical
and empirical values of the curvatures (u-v stacked together) attains maximum (at
0 50 100 150 200 250 300 350 4000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Delay (ms)
C
o
r
r
e
la
ti
on
 V
al
ue
 
 
U
V
U−V
(a) Correlation between empirical curvatures
and the curvatures obtained from the CATD
feedback law.
0 50 100 150 200 250 300 350 4000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Delay (ms)
C
o
r
r
e
la
ti
on
 V
al
ue
 
 
U
V
U−V
(b) Correlation between empirical curvatures
and the curvatures obtained from the CP feed-
back law.
Figure 28: Variation of correlation between empirical and theoretical curvatures shows dominance
of CATD pursuit strategy during a bat-mantis chase.
143
Figure 29: Variation of residual for a CATD feedback law (Bat-Mantis pursuit events). The
residual values (color-coded) are shown in log 10 scale.
Figure 30: Variation of residual for a CP feedback law (Bat-Mantis pursuit events). The residual
values (color-coded) are shown in log 10 scale.
144
0.6159) with the CP feedback law, and the corresponding delay is 166.67ms. We
also perform a principal component analysis to confirm that the data is directional.
The results show that the variance of one of the components is very high compared
to the other. We also show the variation of the residual (mismatch defined in (5.3))
as a function of gain and delay. From Table 5.3 one can conclude that CP feedback
laws yield a better match, and the corresponding gain and delay are similar to the
ones obtained through correlation maximization.
On the other hand, we show the variation of correlation for bat-mantis pursuit
events in Fig 28, and the corresponding values of optimal gain (µ˚) and optimal
delay (δ˚) are mentioned in Table 5.4. It can be noticed from the table that the cor-
relation between the theoretical and empirical values of the curvatures (u-v stacked
together) gets maximized (at 0.7403) by choosing the CATD feedback law, and the
corresponding delay is 120ms. A principal component analysis of the curvature
data shows that the variance of one of the components is very high compared to the
other. In addition to computing the variation of correlation as a function of delay,
we also analyze the variation of the residual as a bi-variate function of gain and
delay (Fig 29 and Fig 30).
We can observe that the empirical curvature values of the trailing bat in a bat-
bat pursuit event are better correlated with a the CP feedback law; while on the
other hand the empirical curvature values of the bat in a bat-mantis chase are better
correlated with the theoretical feedback law which makes the interaction approach
the CATD state. We have also found that the latency associated with the pursuit
of a free flying target p« 120msq is significantly smaller than the latency associated
145
CATD CP
Maximum Correlation (ρmax) .7403 0.0314
Delay (δ˚) [ms] 120 64
Linear Gain (µest) 1.2457 8.2354ˆ 10´7
λ1 0.5480 374938355.1287
λ2 0.0586 0.25895
σ1 90.3439 100
σ2 9.6561 6.9066ˆ 10´8
Table 5.4: Summary of the statistical analysis of steering control laws for bat-mantis pursuit
events. λ1 and λ2 represent the principal component variances, i.e., eigenvalues of the covariance
matrix. σ1 and σ2 represent the percentage of total variance explained by principle components.
CATD CP
Minimum Value of Mismatch 0.2029 2.8964ˆ 104
Normalized Mismatch 0.0511 7.6092ˆ 103
Linear Gain 1.2457 0.0100
Delay (δ˚) [ms] 118 102
Table 5.5: Summary of the residual analysis of steering control laws for bat-mantis pursuit events.
The mismatch is normalized by the product of rms values of empirical and theoretical curvatures.
with a competitive prey capture p« 170msq. This delay can be attributed to the
combined effect of neural processing delay and motor action delay.
5.5 Discussion
The results of this study demonstrate that the echolocating bat allows room
for flexibility in its flight strategy. It adapts the underlying strategy and feedback
mechanism for steering control to the context and goal of the task. Our analysis,
built on geometric notions and control-theoretic methods, shows that single bats
employ CATD while pursuing insect prey. On the other hand, when tasked to
compete for a single food source, a bat does not apply CATD to chase a conspecific.
Rather, it uses CP strategy while pursuing a competitor.
146
This work introduces two important notions, namely following property and
convergence of a trajectory, to better understand the individual roles in a dyadic in-
teraction. One distinguishing feature of these notions, similar to the ones introduced
by Chiu et al. [2008], is that they are completely based on the geometry of flight
trajectories, in particular individual positions and velocities; our approach does not
consider the patterns of vocalization (for echolocation and warning conspecifics).
Here, we have adopted a two stage approach for analysis of the trajectory data.
Besides analyzing the distribution of relevant contrast function (Λ and Γ) values,
we have also compared the empirical values of trajectory curvatures to the ones
predicted by theoretically well-founded and biologically plausible feedback control
laws. While analyzing the histograms of contrast function values, we noticed that the
significance of distinction between CP and CATD is higher for bat-bat interactions
than for bat-insect interactions. On the other hand, by performing the analysis
at the level of steering control law, the dominance of CATD during a bat-insect
interaction can be concluded in a much stronger way than the dominance of CP
during a bat-bat interaction. This observation emphasizes the indispensability of
a two stage approach to achieve a better understanding of the interactions and to
evaluate which strategy prevails.
During the trials involving two competing bats, our data shows emergence of
a pursuer-pursuee relationship. The approach taken here does not address how this
shift to asymmetry arises out of a symmetric setup. It has been known for some
time that food-associated vocalization may be responsible for the emergence of such
asymmetry (either by attracting or repelling conspecifics). Recent work by Wright
147
et al. [2014] suggests that this occurs in competitive foraging, when one of the bats
(exclusively male) emits a type of social call - frequency-modulated bout (FMB) -
to signify territoriality and food claiming. It would be of interest to find signatures
in flight steering of such FMB-induced shifts.
148
Chapter 6: Analysis of Flocking in European Starlings
Collective behavior in animal groups is quite fascinating and ubiquitous in
nature, and over the decades had drawn attention of researchers from various fields
of science. However, until recently it was very difficult to conduct any quantitative
analysis for large groups of animals which are quite common in natural settings.
This lack of studies, primarily caused by inadequacy of appropriate motion capture
techniques to track common flocks, used to pose a serious problem as an analysis
for a small group cannot be generalized for larger groups. The boundary effect is
much more dominant for smaller groups.
Equipped with advanced tools from stereometry and statistical analysis, Cav-
agna et al. [2008a,b] developed a way to collect three-dimensional position and
velocity data for large flocks of starling with number of birds varying from couple
of hundreds to couple of thousands. One of their initial contribution was to show
that local interactions, the building block for group level flocking behavior, does
not depend on metric distance, rather on the topological distance [Ballerini et al.,
2008a]. Based on the empirical data, they discovered that individual starlings in-
teract with six/seven nearest neighbors (on average), rather than with all neighbors
within a fixed radius. A more recent work, by suggesting that the flock maximizes
149
its robustness to uncertainty by interacting with six or seven neighbors, provides
a justification for individual starlings to interact with six/seven nearest neighbors
[Young et al., 2013]. Later, by measuring the correlation between velocity fluctua-
tions of different birds, Cavagna et al. [2010] have shown that behavioral correlations
are scale free, i.e. the behavioral change of an individual affects and is affected by
that of all other members of the flock, independent of flock size. In the most recent
work from this research group, Attanasi et al. [2014] have shown that directional
information within a flock propagates with an almost constant speed, and this linear
growth of information can be explained by models with wave-like aspects. However,
these studies do not provide much insight about the agent-level steering control laws
which give rise to flocking behavior.
In this work of ours we attempt to uncover the flight strategies and underlying
control laws by analyzing different parameters of motion, namely velocity, speed,
curvatures etc. For that purpose we assume each starling to be a point particle,
and apply the tools developed in Chapter 3 to extract speed and curvatures from
the sampled dataset of observed positions. Then we perform correlation analysis
to investigate the feedback mechanism for steering control governing coordinate
motion of the flocks. Our analysis also provides estimates of the sensorimotor delay
associated with the flocking behavior.
150
6.1 Experiment Details and Trajectory Reconstruction
This work analyzes the flight data from a series of data collection events con-
ducted by Dr. Andrea Cavagna and his collaborators from the Collective Behaviour
in Biological Systems (COBBS) group at the Institute for Complex Systems (ISC-
CNR), University of Rome “La Sapienza”. These time-sampled flight data were
taken from the roof of Palazzo Massimo, Museo Nazionale Romano, in the city cen-
ter of Rome, in front of one of the major roosting sites used by starlings during
winter. Starlings spend the day feeding in the countryside, and before settling on
the trees for the night they gather in flocks to perform aerial display, an apparently
purposeless dance where flocks move and swirl in a remarkable way. Interested read-
ers can refer to the work by Attanasi, A. and Cavagna, A. and Del Castello, L. and
Giardina, I. and Jelic, A. and Melillo, S. and Parisi, L. and Shen, E. and Silvestri,
E. and Viale, M. [2013] for further details about the experimental setup and the
sophisticated algorithm for stereo reconstruction.
151
10−7 10−6 10−5 10−4 10−3 10−2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Optimal λ−value (λ*)
N
om
al
iz
ed
 C
ou
nt
 
 
02−08−ACQ3
11−24−ACQ1
11−25−ACQ1
12−01−ACQ3
12−07−ACQ1
12−14−ACQ4
12−15−ACQ1
12−20−ACQ2
Figure 31: This figure shows the distribution of optimal regularization parameters (λ˚) for dif-
ferent flocks. The distribution corresponding to a particular flocking event is sharply peaked at a
distinct value, and therefore emphasizes the strong dependence of λ˚ on the signal-to-noise ratio.
Our work involves analysis of eight distinct flocking events captured during
the winter months of 2011. We begin our analysis by reconstructing the flight tra-
jectories using the smoothing algorithm (Algorithm 1) developed in Chapter 3. In
contrast to our earlier work with bat trajectories, here we have performed cross-
validation for individual trajectories and the corresponding optimal value of regu-
larization parameter (λ˚) has been used for trajectory reconstruction. Fig 31 shows
the distribution of λ˚, and thereby emphasizes that λ˚ has a strong dependence on
the signal-to-noise ratio. The flocking events that we analyze are quite distinct in
nature; while some of them (Fig 32d, Fig 32h) are minimally maneuvering flights,
some involve coordinated turning (Fig 32a, Fig 32f, Fig 32g). The following table
(Table 6.1) enlists the events under consideration along with the associated details
(duration, flock size and frame-rate of data capture).
152
(a) Flock: 2011 02 08 ACQ3 (N=180) (b) Flock: 2011 11 24 ACQ1 (N=125)
(c) Flock: 2011 11 25 ACQ1 (N=50) (d) Flock: 2011 12 01 ACQ3 (N=489)
(e) Flock: 2011 12 07 ACQ1 (N=109) (f) Flock: 2011 12 14 ACQ4 (N=162)
(g) Flock: 2011 12 15 ACQ1 (N=401) (h) Flock: 2011 12 20 ACQ2 (N=200)
Figure 32: Within the scope of this current work we have analyzed the flight data of eight flocks.
This figure shows the reconstructed trajectories for all these flocking events. Here “N” represents
the number of birds in a particular flock.
153
Flocking Event Flock Size Duration Data Capture Rate
(N) (seconds) (frames/second)
2011 02 08 ACQ3 180 5.4875 80
2011 11 24 ACQ1 125 1.8176 170
2011 11 25 ACQ1 50 5.6118 170
2011 12 01 ACQ3 489 2.3471 170
2011 12 07 ACQ1 109 3.8824 170
2011 12 14 ACQ4 162 4.1588 170
2011 12 15 ACQ1 401 5.7353 170
2011 12 20 ACQ2 200 1.7588 170
Table 6.1: Our study analyzes eight particular flocking events. This table enlists the details of
the individual events.
6.2 Analysis of Flight Strategy and Underlying Steering Control
One can easily notice (from Fig 32) that individuals in a starling flock fly
in such a way that there is not much variation between the individual directions
of motion (at least when they are involved in a coordinated turn or undergoing a
minimally maneuvering path). This perception forms the basis for our analysis, and
we introduce appropriate quantitative notions to investigate the applicability of this
idea.
We begin the analysis by computing the average cosine of the angle between
the velocity of a focal bird i and velocity of the center of mass of its neighborhood
(Ni). By letting vi denote the velocity of the i-th individual, the direction of motion
of its neighborhood at time tk can be defined as
xNiptkq “
ř
jPNiptkq
vjptkq
| ř
jPNiptkq
vjptkq|
, (6.1)
whenever the neighborhood center of mass velocity does not vanishes to zero. Then
154
the average cosine of the angle (between vi and xNi) can be computed as
Ci “
1
|E |
ÿ
kPE
viptkq
|viptkq|
¨ xNiptkq, (6.2)
where E represents the set of time indices associated with the flight duration. As
Ci can be interpreted as a measure of how coherently a bird is moving with respect
to its neighbors, Ci is named as the coherence for bird i.
This work constructs the neighborhood of a focal bird i by considering its K-
nearest neighbors. Then, by definition, Ni has a cardinality of K, and each node in
the underlying attention graph1 has an out-degree of value K. For this particular
choice for the neighborhood structure, we evaluate Ci for every member of the flock,
and compute the flock averaged coherence by averaging it over the flock members.
Clearly, this quantity is a property of the whole flock, and it depends only on the
neighborhood size (K). Fig 33 shows the variation of flock averaged coherence as a
function of the neighborhood size (K).
It can readily be noticed from this set of figures that the flock averaged coher-
ence gets maximized by considering 5-7 nearest neighbors. While some of the flocks
(Fig 33c, Fig 33e, Fig 33f) attain the maximum coherence by picking K-values in the
range (5-7), for rest of them, the gain/increase in flock averaged coherence becomes
insignificant if one goes beyond this range of nearest neighbors. Thus our analysis
1As the nearest neighborhood relationship is not symmetric, it can be illustrated by a directed
graph, wherein an edge exists from node-i to node-j if the individual j is one of the K-nearest
neighbors of individual i (i.e., individual i is paying attention to individual j). This graph is called
the underlying attention graph. Clearly, this graph changes over time, depending on the relative
position of the group members.
155
reconfirms the findings by Ballerini et al. [2008a].
Next we focus on investigating the underlying steering control mechanism by
comparing empirical values of trajectory curvature with predictions from theoreti-
cally justifiable feedback control laws (discussed in detail in Chapter 7). By letting,
puem, vemq denote the empirical curvatures obtained from trajectory smoothing and
puth, vthq denote the curvature values computed using theoretical feedback laws, we
formalize this analysis as the following mismatch minimization problem
Minimize
µą0,δPN
¨
˚˝ 1ř
jPS
`
|Ej| ´ δ
˘
ÿ
jPS
ÿ
tkPEj
”`
uemptkq ´ uthptk ´ δ∆q
˘2 `
`
vemptkq ´ vthptk ´ δ∆q
˘2ı
˛
‹‚,
(6.3)
Here, N, Ej and S represent the set of natural numbers, the set of time indices
associated with the flight duration of individual j, and the index set of all starlings
belonging to a particular flock, respectively. This mismatch minimization problem
incorporates delays into the theoretical curvature terms to account for the behav-
ioral latency present in the sensorimotor feedback loops. Moreover this approach
also provides an estimate of the sensorimotor delay. We should also note that the
discreteness of delay values (in (5.3)) arises only because of data availability at a
finite sample rate (with sampling interval ∆). Now, before going into the details of
analysis, it is worth mentioning the theoretically plausible feedback laws (although
the detailed analysis will be discussed later). By letting νi denote the speed of the
156
0 5 10 15 20 25
0.9915
0.992
0.9925
0.993
0.9935
0.994
0.9945
0.995
Number of Nearest Neighbors
Av
era
ge
 Va
lue
 of
 Di
rec
tion
 Co
sin
e
(a) Flock: 2011 02 08 ACQ3
0 5 10 15 20 25
0.995
0.9955
0.996
0.9965
0.997
0.9975
Number of Nearest Neighbors
Av
era
ge
 Va
lue
 of
 Di
rec
tion
 Co
sin
e
(b) Flock: 2011 11 24 ACQ1
0 5 10 15 20 25
0.989
0.99
0.991
0.992
0.993
0.994
0.995
0.996
0.997
Number of Nearest Neighbors
Av
era
ge
 Va
lue
 of
 Di
rec
tion
 Co
sin
e
(c) Flock: 2011 11 25 ACQ1
0 5 10 15 20 25
0.978
0.98
0.982
0.984
0.986
0.988
0.99
Number of Nearest Neighbors
Av
era
ge
 Va
lue
 of
 Di
rec
tion
 Co
sin
e
(d) Flock: 2011 12 01 ACQ3
0 5 10 15 20 25
0.978
0.979
0.98
0.981
0.982
0.983
0.984
0.985
0.986
Number of Nearest Neighbors
Av
era
ge
 Va
lue
 of
 Di
rec
tion
 Co
sin
e
(e) Flock: 2011 12 07 ACQ1
0 5 10 15 20 25
0.982
0.983
0.984
0.985
0.986
0.987
0.988
Number of Nearest Neighbors
Av
era
ge
 Va
lue
 of
 Di
rec
tion
 Co
sin
e
(f) Flock: 2011 12 14 ACQ4
0 5 10 15 20 25
0.9905
0.991
0.9915
0.992
0.9925
0.993
0.9935
0.994
0.9945
0.995
Number of Nearest Neighbors
Av
era
ge
 Va
lue
 of
 Di
rec
tion
 Co
sin
e
(g) Flock: 2011 12 15 ACQ1
0 5 10 15 20 25
0.9895
0.99
0.9905
0.991
0.9915
0.992
0.9925
0.993
0.9935
0.994
Number of Nearest Neighbors
Av
era
ge
 Va
lue
 of
 Di
rec
tion
 Co
sin
e
(h) Flock: 2011 12 20 ACQ2
Figure 33: This figure illustrates the variation of flock averaged coherence as a function of the
neighborhood size (K), i.e. the number of individuals a focal bird is paying attention to.
157
(a) Flock: 2011 11 24 ACQ1
(b) Flock: 2011 12 01 ACQ3
(c) Flock: 2011 12 20 ACQ2
Figure 34: This figure, through a heat map, illustrates the variation in correlation between the
empirically observed curvature of a focal bird and the curvature values predicted by a theoretically
plausible feedback mechanism (6.4), as a bivariate function of neighborhood size and delay.
158
i-th individual, the feedback law is given by
ui “ µ
„xNi ¨ yi
νi

vi “ µ
„xNi ¨ zi
νi

,
(6.4)
where xNi denotes the normalized velocity vector of the neighborhood center of
mass, and yi, zi carry their usual meaning of frame vectors associated with the
individual. As discussed earlier (in Section 5.4), the computational complexity
involved in solving the mismatch minimization problem (6.3) can be avoided by
approximating it with an equivalent correlation maximization problem (as shown
via Proposition 5.1). In this alternative paradigm, we compute the correlation be-
tween empirical data (natural curvatures uem, vem stacked in a single array) and
the theoretically predicted curvature values (uth, vth stacked in a single array), as
a function of neighborhood size and (sensorimotor) delay. Then, the neighborhood
size and delay which maximizes this correlation provides an estimate for the number
of nearest neighbors (an individual is paying attention to) and behavioral latency.
Event ID 2011 11 24 ACQ1 2011 12 01 ACQ3 2011 12 20 ACQ2
Max. Correlation .2797 0.4103 0.1278
Delay [ms] 138.8235 141.1765 135.2941
Neighborhood Size 12 40 16
Linear Gain (µest) 2.4462 4.1689 1.33
λ1 1271.1725 57592.5342 3505.8485
λ2 15.2880 462.4669 31.8121
σ1 98.8116 99.2034 99.1008
σ2 1.1884 0.7966 0.8992
Table 6.2: Summary of the statistical analysis of steering control laws for starling flocks. λ1 and
λ2 represent the principal component variances, i.e., eigenvalues of the covariance matrix. σ1 and
σ2 represent the percentage of total variance explained by the principle components.
159
The variation of correlation as a bivariate function of neighborhood size and
delay is shown in Fig 34, and the associated values of relevant parameters (optimal
gain, neighborhood size and delay, along with correlation values) are mentioned in
Table 6.2. It can be noticed from the three subfigures that, although the correlation
value (between theoretical prediction and empirical observation) is itself not very
high, there is an unmistakable consistency in the neighborhood size and delay values
which maximize the correlation. These three subfigures shows the correlation vari-
ation for the minimally maneuvering flocks (Flock ID: 2011 11 24 ACQ1, Flock ID:
2011 12 01 ACQ3, and Flock ID: 2011 12 20 ACQ2), and hence it can concluded
that the proposed feedback mechanism (6.4) plays an important role in governing
the steering control for flocking behavior
6.3 Discussion
In this study of ours, we have analyzed the flight data of European starling
flocks, and have provided some insight into the underlying flight strategies and steer-
ing control mechanism. Here, the attention graph has been constructed by ranking
the neighbors according to their distances from the focal bird and then considering
the first K-nearest neighbors. However, recent studies by Bhagavatula et al. [2011]
and Kane & Zamani [2014] have shown that optic flow plays an important role in
the context of perception in avian flights. Therefore, it seems reasonable to extend
the current work by considering attention graphs constructed by ranking the neigh-
bors according to the magnitude of optic flow at the focal bird’s eyes caused by
160
their motion. Also, it is worth mentioning here that the feedback law, that we have
introduced here, is responsible for aligning the individual velocities. A more general
form of the control law (although restricted to planar settings and uniformly moving
individuals) with three different components for attraction, repulsion and alignment
have been proposed earlier by Justh & Krishnaprasad [2002, 2004]. It would be in-
teresting to investigate the applicability of that control law in this context. Another
key aspect of this study is that the proposed control law (6.4) carries an wave-like
aspect which is necessary to explain a linear growth of information within a flock.
161
Part III
Synthesis of Collectives
162
Chapter 7: Flocking through Topological Velocity Alignment (TVA)
Now, drawing inspiration from our analysis of starling murmuration events, we
consider the problem of synthesizing a collective motion wherein each of the individ-
uals (birds in a flock, UAVs in swarm) controls its steering action in such a way that
their directions of motion remain parallel. It is worth mentioning that earlier works
of Justh & Krishnaprasad [2003, 2004] have considered a very similar form of the
proposed control law with three components for attraction (while the agents are far
away), repulsion (to avoid collision) and velocity alignment. However, the control
law introduced in this current study considers only velocity alignment, but extends
the scope by considering multiple agents in a three dimensional environment inter-
acting via a state-dependent (nearest neighbors based) attention graph. Moreover,
it relaxes the assumption on uniform speed of the collective by allowing the agents
non-uniform and time-varying speed profiles. This relaxation plays an important
role in the context of applying this control law to a group of heterogeneous agents.
This chapter introduces a strategy, named topological velocity alignment (TVA),
wherein each agent aligns its velocity along the direction of motion of its neighbor-
hood center of mass [Halder & Dey, 2015]. After introducing this strategy and the
associated steering feedback law, Section 7.2 and Section 7.3 provide a theoretical
163
analysis of this strategy in two special cases (two-agents, and N -agents interacting
in a cyclic way, respectively). Later, in Section 7.4 we propose a discrete-time al-
gorithm to implement the TVA strategy on a group of robotic agents. Finally we
conclude this chapter by showing some implementation results in Section 7.5.
7.1 Topological Velocity Alignment (TVA) Strategy
Here we treat the agents (of the collectives of size N) as self steering particles,
and use natural Frenet frame equations (Section 1.3.1) to model their dynamics.
Therefore, by letting ri denote the instantaneous position of the i-th agent, its
dynamics can be expressed as
9riptq “ νiptqxiptq
9xiptq “ νiptq
`
uiptqyiptq ` viptqziptq
˘
9yiptq “ ´νiptquiptqxiptq
9ziptq “ ´νiptqviptqxiptq
(7.1)
where xi is the unit tangent vector to its trajectory, and pyi, ziq form an orthonor-
mal basis for the plane perpendicular to xi. Moreover the natural curvatures pui, viq
are the steering controls, and the speed pνiq is a time function dictated by propul-
sive/lift/drag mechanisms. It can also be noted that pxi,yi, ziq form a local or-
thonormal frame which evolves along the length of the trajectory. Furthermore,
this model also carries a nice geometric interpretation because (7.1) can alterna-
tively be viewed as a left invariant control system evolving on the Lie group SEp3q
164
of rigid motion in three dimension.
x1
y1
z1
r1
x2
y2
z2
r2
x3
y3
z3
r3
Trajectory # 1
Trajectory # 2
Trajectory # 3
Figure 35: Individual trajectories along with their frame vectors.
By assuming the steering control pui, viq of an agent-i (or the focal bird in a
flock) to be dependent on the motion of its neighbors, we introduce the notion of
topological velocity alignment (TVA), wherein each agent moves parallel to the center
of mass of its neighborhood. We also derive the steering control law necessary to
achieve this goal. We begin by letting Ni denote the neighborhood of the i-th agent.
Then, by assuming identical mass for every agent, the velocity of the neighborhood
center of mass (COM) can be expressed as
viCOM “
1
|Ni|
ÿ
jPNi
vj “
1
|Ni|
ÿ
jPNi
νjxj . (7.2)
Moreover, we assume that viCOM does not vanish to zero1. Then, with this assump-
1It should be noted that viCOM becomes zero over a thin set in the underlying state space. As
the chance of getting into this thin set is essentially zero, we can overlook this situation for all
practical purposes.
165
tion, we can define the direction of the center of mass motion as
xNi “
viCOM
|viCOM |
“
ř
jPNi
vj
ˇˇ
ˇˇ
ˇ
ř
jPNi
vj
ˇˇ
ˇˇ
ˇ
. (7.3)
Now, we define the following contrast function associated with the i-th agent
Θi “
1
2 pxNi ´ xiq ¨ pxNi ´ xiq , (7.4)
and use it as a quantitative measure of the difference between heading of the i-th
agent and direction of the center of mass motion for its neighborhood Ni. If the
i-th agent moves in the same direction as the center of mass of its neighborhood Ni,
then the contrast function assumes its minimum value, i.e. Θi “ 0. On the other
hand, it assumes the maximum value (Θi “ 2) whenever the i-th agent moves in
a direction opposite to that of the center of mass motion. And, if the i-th agent
moves perpendicular with respect to direction of motion of the center of mass of
its neighborhood, then we have Θi “ 1. Fig 36 shows the variation of Θi as a
function of the angle between heading of the i-th agent pxiq and the direction of the
center of mass motion for its neighborhood pxNiq. As we can observe, Θi increases
monotonically with increase in the angle between xi and xNi. Alternatively, it can
also be expressed as
Θi “ 1´ xi ¨ xNi , (7.5)
as both xi and xNi are unit vectors. xi ¨xNi , being a dot product of two unit vectors,
166
essentially represents the cosine of the angle between them, and gets maximized
when they are aligned in the same direction. Thus, Θiptq can be interpreted as a
quantitative measure of departure from our goal of achieving alignment between
heading of the i-th agent (xiptq) and the direction of motion of its neighborhood
center of mass (xNiptq).
 
 2
1.5
1
0.5
0
0
´0.5
´1 π{4 π{2 3π{4 π
Θixi ¨ xNi
Angle between xi and xNi
Figure 36: Variation of the contrast function (Θi) with change in angle between xi and xNi .
Assuming a well-defined xNi , we propose a steering control law of the form
uiptq “ µ
„xNiptq ¨ yiptq
νiptq

viptq “ µ
„xNiptq ¨ ziptq
νiptq

,
(7.6)
where µ ą 0 denotes a feedback gain. A physical intuition for the steering control law
(7.6) can be obtained by shifting focus to the corresponding feedback law for lateral
acceleration. With this particular choice of control laws, the lateral acceleration can
167
be expressed as
alati ptq “ νiptq 9xiptq
“ µνiptq
“
xNiptq ¨ yiptq
‰
yiptq ` µνiptq
“
xNiptq ¨ ziptq
‰
ziptq
“ µνiptq
´
xNiptq ´
“
xNiptq ¨ xiptq
‰
xiptq
¯
. (7.7)
From (7.7) it is quite apparent that the lateral acceleration is proportional to the
difference between direction of motion of the center of mass and the component of
its own tangent vector along the desired direction.
Remark 7.1. Although there has been a long history of control algorithms for flock-
ing, almost every model of collective motion (Reynolds [1987]; Vicsek et al. [1995])
predicts diffusive transport of information. But, contrary to the existing models,
recent findings by Attanasi et al. [2014] from starling flocks suggest that directional
information within a flock propagates with an almost constant speed, and this linear
growth of information can be explained by models with wave-like aspects. Our pro-
poses strategy, i.e. topological velocity alignment (TVA), conforms to this criterion
and can explain how information about local neighbors can influence the agents in a
flock to align their headings in a single common direction.
7.2 TVA Strategy for a Planar 2-agent System
Before going into the analysis of an N -agent system we consider a very special
case for a 2-agent system wherein the motion is restricted onto a two dimensional
168
Cartesian plane. As we are considering a 2-agent system the neighborhood of a
particular agent comprises of the other agent, and hence the speed of neighborhood
center of mass never vanishes to zero (due to regularity of the model).
7.2.1 State Space and Its Reduction onto Shape Space
By exploiting natural Frenet frame equations restricted to a planar setting,
the dynamics of the two agent system can be expressed as
9riptq “ νiptqxiptq
9xiptq “ νiptquiptqyiptq
9yiptq “ ´νiptquiptqxiptq i P t1, 2u
(7.8)
where ri and xi denote the position and normalized velocity of the i-th agent, re-
spectively. yi is the unit frame vector normal to xi (uniqueness is guaranteed by
defining yi as the orthogonal rotation of xi in the counter-clockwise direction). We
also assume r1 ‰ r2 to avoid singularity of the shape variables (to be introduced
later). Moreover, a planar equivalent of (7.6) is chosen as the underlying feedback
law for steering control.
Alternatively, by packing ri,xi,yi inside a matrix gi P SEp2q, the special
Euclidean group of rigid motions in a plane, the state space for the 2-agent system
(7.8) can be defined as
Mstate “
!
g1, g2 P SEp2q ˆ SEp2q
ˇˇ
ˇg1e3 ‰ g2e3
)
(7.9)
169
where e3 “ r0 0 1sT and gi “
»
——–
xi yi ri
0 0 1
fi
ffiffifl. In terms of the lie-group formulation
the system dynamics can be represented as
9gi “ giξipuiq “ gi
`
A1uiνi ` A2νi
˘
(7.10)
where A1 “
»
——————–
0 ´1 0
1 0 0
0 0 0
fi
ffiffiffiffiffiffifl
and A2 “
»
——————–
0 0 1
0 0 0
0 0 0
fi
ffiffiffiffiffiffifl
are two linearly independent
elements of sep2q, the Lie algebra of SEp2q. Moreover A1 and A2 can generate the
lie-algebra under bracketing. As we are interested in steering laws which leave our
system dynamics invariant under rigid motion, we can formulate a reduction to the
shape space, a quotient manifold Mstate{SEp2q of relative positions and velocities
of the agents. By defining g P SEp2q as
g “ g´11 g2 “
»
——————–
x1 ¨ x2 x1 ¨ y2 x1 ¨ r
y1 ¨ x2 y1 ¨ y2 y1 ¨ r
0 0 1
fi
ffiffiffiffiffiffifl
, (7.11)
the shape space for the planar two-agent system can be defined as
Mshape “ Mstate{SEp2q “
!
g P SEp2q
ˇˇ
ˇg213 ` g223 ‰ 0
)
, (7.12)
where r fi r2 ´ r1 denotes the baseline vector. Moreover, g assumes a left-invariant
170
dynamics on SEp2q as
9g “ gξ (7.13)
where ξ “ ξ2pu2q ´ g´1ξ1pu1qg P sep2q, and the proposed control law (7.6) depends
only on the shape variable g as
ui “ µ
„
gi¯i
νi

, i, i¯ P t1, 2u, i ‰ i¯. (7.14)
Therefore it can be concluded that the reduced dynamics (7.13) evolves on the shape
space Mshape [Justh & Krishnaprasad, 2004].
r “ r2 ´ r1
y1
x1
r1
y2
x2
r2
θ1
θ2
ψ
φ
ϑ
Figure 37: Illustration of scalar shape variables (ρ, ψ, φ) used to parametrize the shape space
Mshape.
7.2.2 Shape Dynamics
Now we introduce some geometrically meaningful scalar variables to parametrize
the shape space. By identifying punctured R2 with the punctured complex plane,
we define
r “ r2 ´ r1 “ ρeiϑ. (7.15)
171
Moreover the unit vectors x1 and x2 are represented as
x1 “ eiθ1, x2 “ eiθ2.
These scalar shape variables are illustrated in Fig 37. Now we introduce ψ and
φ to represent the relative orientation of the velocity vectors. ψ “ π ´ ϑ ` θ1
represents the relative orientation of x1 with respect to the baseline vector r, and
φ “ θ1´θ2 represent the mismatch in velocity direction. From (7.11) one can notice
that g P Mshape can be represented in terms of the scalar shape variables as
g “
»
——————–
cos φ sinφ ´ρ cosψ
´ sinφ cosφ ρ sinψ
0 0 1
fi
ffiffiffiffiffiffifl
. (7.16)
Differentiating both sides, (7.15) yields
9ρ cosϑ´ ρ 9ϑ sin ϑ` i
`
9ρ sinϑ` ρ 9ϑ cos ϑ
˘
“ 9r
“ ν2x2 ´ ν1x1
“ ν2
`
cos θ2 ` i sin θ2
˘
´ ν1
`
cos θ1 ` i sin θ1
˘
, (7.17)
172
and then by equating the real and imaginary parts of (7.17) we have
9ρ cosϑ´ ρ 9ϑ sin ϑ “ ν2 cos θ2 ´ ν1 cos θ1 (7.18)
9ρ sin ϑ` ρ 9ϑ cosϑ “ ν2 sin θ2 ´ ν1 sin θ1. (7.19)
Solving (7.18) and (7.19) we obtain
9ρ “ ν2 cospϑ´ θ2q ´ ν1 cospϑ´ θ1q
“ ν1 cosψ ´ ν2 cospψ ´ φq, (7.20)
and
ρ 9ϑ “ ν2 sinpθ2 ´ ϑq ´ nu1 sinpθ1 ´ ϑq
“ ν1 sinψ ´ ν2 sinpψ ´ φq. (7.21)
On the other hand, yk is obtained by a 90˝ rotation of xk in the counter
clockwise direction, i.e. yk “ ei
π
2 xk. Therefore by using the fact that 9xk “ ieiθk 9θk “
eipθk`π2 q 9θk, the dynamics for the scalar variables θ1 and θ2 can be represented as
9θ1 “ ν1u1
9θ2 “ ν2u2,
(7.22)
173
and hence the associated shape dynamics on Mshape are given by
9ρ “ ν1 cosψ ´ ν2 cospψ ´ φq
9ψ “ ν1u1 ´
1
ρ
“
ν1 sinψ ´ ν2 sinpψ ´ φq
‰
(7.23)
9φ “ ν1u1 ´ ν2u2.
7.2.3 Analysis of TVA Feedback Law
Here we consider a particular context of the two-agent planar system wherein
each agent employs the strategy for topological velocity alignment (TVA), i.e. each
agent keeps moving in the same direction as the other. In terms of the original state
variables the contrast functions take the form
Θi “
1
2 pxi¯ ´ xiq ¨ pxi¯ ´ xiq , i P t1, 2u (7.24)
and the i-th agent is declared to attain the TVA strategy if Θi “ 0. From (7.24) one
can notice equality of the contrast functions for both agents, and hence we define
a common contrast function Θ “ Θ1 “ Θ2. Noting x1 ¨ x2 “ cosφ, the common
contrast function can be represented in terms of scalar shape variables as
Θ “ 12
`
x2 ´ x1
˘
¨
`
x2 ´ x1
˘
“ 1´ x1 ¨ x2
“ 1´ cosφ, (7.25)
174
and hence we have
Θ “ 0 ô φ “ 0. (7.26)
Therefore for a two-agent planar system wherein each agent employs the strat-
egy for topological velocity alignment (TVA), we define the 2-dimensional topolog-
ical velocity alignment manifold MTV A Ă Mshape as
MTV A “
!
ρ, ψ, φ P Mshape
ˇˇ
ˇφ “ 0
)
. (7.27)
A steering control law designed to attain the TVA strategy has been proposed (7.6)
in earlier sections of this chapter. Moreover from (7.14) one can observe that this
feedback law can be expressed in terms of shape variables, taking the form
u1 “ ´
ˆ
µ
ν1
˙
sinφ
u2 “
ˆ
µ
ν2
˙
sinφ.
(7.28)
From (7.26) we can observe that once the TVA strategy has been attained, i.e.
Θ “ 0, the steering control becomes identically zero, and as a consequence the
mismatch in velocity direction remains identically zero (7.23). Now we will formally
introduce the notion of invariance.
Definition 7.1 (Invariant Manifold). A manifold M is said to be invariant under
the flow of a vector field X on M if for any m P M, Ftpmq P M for small t ą 0,
where Ftp¨q is the flow of X. One can show that this condition is equivalent to X
being tangent to the manifold to M.
175
ry1
x1
r1
y2
x2
r2
ψ
ψ
Figure 38: Illustration of topological velocity alignment (TVA) strategy for a 2-agent system.
If both agents employ a steering control of the form (7.28), the closed loop
dynamics for a two-agent planar system can be represented as
9ρ “ ν1 cosψ ´ ν2 cospψ ´ φq
9ψ “ ´µ sinφ´ 1ρ
“
ν1 sinψ ´ ν2 sinpψ ´ φq
‰
(7.29)
9φ “ ´2µ sinφ.
We should note that prohibition on collocation, i.e. ρ ą 0, is not enforced by these
dynamics.
Proposition 7.1. The topological velocity alignment manifold MTV A Ă Mshape is
invariant under the closed loop shape dynamics (7.29). Moreover if γptq P Mshape is
a trajectory of (7.29) which does not have a finite escape time, and Θp0q ‰ 2, then
Θptq Ñ 0 as tÑ 8, (7.30)
i.e. γptq converges asymptotically to MTV A.
176
Proof. From (7.25) and (7.29) we have
9Θ “ 9φ sinφ “ ´2µ sin2 φ “ ´2µΘ
`
2´Θ
˘
, (7.31)
and hence MFL Ă Mshape is an invariant manifold under the shape dynamics. In
fact, from (7.31) it can be concluded that Θp0q “ 2 ñ Θptq “ 2 @t ě 0.
By assuming Θp0q ‰ 0, 2, we have
Θptq “ 2e
´4µt
K ` e´4µt
where the constant K is defined as K “ 2Θp0q ´ 1. Since e´4µt Ñ 0 as t Ñ 8, we
have Θptq Ñ 0 as tÑ 8.
We can formulate the restricted dynamics on the flocking manifold MTV A by
substituting φ “ 0 into (7.29) to obtain
9ρ “
`
ν1 ´ ν2
˘
cosψ
9ψ “ ´1ρ
`
ν1 ´ ν2
˘
sinψ.
(7.32)
Now we define f “ ν1 ´ ν2. By assuming f to be non-zero and differentiable
177
(7.32) can be alternatively represented as
:ρ “ 9f cosψ ´ f 9ψ sinψ
“ 9f cosψ ` 1ρf
2 sin2 ψ
“
˜
9f
f
¸
9ρ` 1ρ
`
f 2 ´ 9ρ2
˘
. (7.33)
Remark 7.2. If f is a non-zero constant then we can show that a Lagrangian
function
LTV A “
1
2ρ
2 ` 9ρ2 ` f 2
˘
has (7.33) as its Euler-Lagrange equation. Hence it can be interpreted as a spring-
mass system with a negative spring constant.
By assuming ν1´ν2 ą 0 we define, τ “
tş
0
`
ν1pσq´ν2pσq
˘
dσ to introduce a non-
uniform time-scaling. As a result the restricted dynamics (7.32) can alternatively
be represented as
dρ
dτ “ cosψ
dψ
dτ “ ´
1
ρ sinψ.
(7.34)
Fig 39 shows the phase portrait of the restricted dynamics with a non-uniform time
scaling. From this figure one can notice that the region {ρ ą 0, π ą ψ ą 0} (or
similarly {ρ ą 0, 0 ą ψ ą ´π}) is closed under the restricted dynamics (7.34), i.e.
no trajectory can enter or leave the region. One can notice that the trajectories
of (7.32) and (7.34) are essentially the same on the phase plane, they differ only
on the speed of system evolution along any particular trajectory. Hence it can be
178
concluded that the trajectories of (7.32) are also closed within {ρ ą 0, π ą ψ ą 0}
(or similarly within {ρ ą 0, 0 ą ψ ą ´π}).
 
 
 
 
 
 
0 1 2 3 4 5 6 7 8 9 10
−3
−2
−1
0
1
2
3
ρ
ψ
Figure 39: Phase portraits for the restricted dynamics (7.34).
Alternatively, by assuming ν1´ν2 ‰ 0, the evolution of a phase plane trajectory
can be represented as
dρ
dψ “ ´
ρ cosψ
sinψ , (7.35)
and through integration with appropriate initial condition (7.35) yields
ρptq sinψptq “ ρpt0q sinψpt0q. (7.36)
From (7.36) one can notice that the phase plane trajectories are level sets of “ρ sinψ”
and hence it is clear that the region {ρ ą 0, π ą ψ ą 0} (or similarly {ρ ą 0, 0 ą
ψ ą ´π}) is closed under the restricted dynamics (7.32).
179
7.2.4 Extension to a Three Dimensional Setting
The analysis of topological velocity alignment for a 2-agent system has a nat-
ural extension from planar to a three-dimensional setting, with the underlying state
space being
Mstate “ SEp3q ˆ SEp3q. (7.37)
For a 2-agent system, the neighborhood of both the agents contains only the other
one (N1 “ t2u and N2 “ t1u), and as a consequence we have Θ1 ” Θ2, because
Θ1ptq “
1
2 pxN1ptq ´ x1ptqq ¨ pxN1ptq ´ x1ptqq
“ 12 px2ptq ´ x1ptqq ¨ px2ptq ´ x1ptqq (7.38)
and, Θ2ptq “
1
2 pxN2ptq ´ x2ptqq ¨ pxN2ptq ´ x2ptqq
“ 12 px1ptq ´ x2ptqq ¨ px1ptq ´ x2ptqq . (7.39)
By choosing the steering the control law as the one prescribed in (7.6), we
have
9Θ1 “ px2 ´ x1q ¨ p 9x2 ´ 9x1q
“ ´ν2 pu2y2 ¨ x1 ` v2z2 ¨ x1q ´ ν1 pu1y1 ¨ x2 ` v1z1 ¨ x2q
“ ´µ
`
py2 ¨ x1q2 ` pz2 ¨ x1q2
˘
´ µ
`
py1 ¨ x2q2 ` pz1 ¨ x2q2
˘
“ ´µ
`
1´ px1 ¨ x2q2
˘
´ µ
`
1´ px2 ¨ x1q2
˘
“ ´2µΘ1p1´Θ1q, (7.40)
180
and similarly the dynamics of Θ2 is given by
9Θ2 “ ´2µΘ2p1´Θ2q. (7.41)
From (7.40) and (7.41) it becomes obvious that 9Θi ă 0 whenever Θi P p0, 2q, i “ 1, 2.
For this flock of 2-agents, wherein each agent employs the TVA strategy, we define
the p2 ˚ 6´ 1)-dimensional TVA manifold MTV A Ă Mstate as
MTV A “ tr1, rx1,y1, z1s , r2, rx2,y2, z2s P Mstate|Θ1 ” Θ2 “ 0u
“ tr1, rx1,y1, z1s , r2, rx2,y2, z2s P Mstate|x1 “ x2u. (7.42)
Then, through a way similar to Proposition 7.1, it can be concluded that Θiptq Ñ 0
as tÑ 8 whenever Θip0q ‰ 2, for i “ 1, 2. Or in other words, almost any trajectory
in Mstate converges to MTV A asymptotically.
7.3 TVA Strategy for an N -agent System with Cyclic Interaction
Now we intend to consider the particular context of N -agent cyclic interaction
(illustrated in Fig 40) systems in which each agent employs the topological veloc-
ity alignment (TVA) strategy. In this setup agent i interacts with agent pi ` 1q
mod N , or in other words the neighborhood of agent-i (Ni) is a singleton set. As
a consequence the neighborhood center of mass velocity never ceases to zero (due
to regularity of individual trajectories), and each agent aligns its velocity along the
velocity of the next agent.
181
12
3
4
5
N
Figure 40: Illustration of an N -agent cyclic interaction system. The direction of influence is
conveyed through the direction of arrow-heads.
7.3.1 State Space and Its Reduction onto Shape Space
In an approach similar to the one in Section 7.2.1, we assume our agents to
be unit-mass particles tracing out twice continuously-differentiable curves in R2,
and model our dynamics using the natural-Frenet frame equations. Therefore our
N -agent system can be thought of as evolving on the product space of N copies
of the Lie group SEp2q. As we are interested in implementing topological velocity
alignment (TVA) strategy under a cyclic interaction framework, it is necessary to
define the neighbor for agent N . This is done through introduction of an additional
element gN`1 P SEp2q to our system state and imposing the closure constraint
gN`1 “ g1. Therefore the state space for our N -agent system can be defined as
Mstate “
!
g1, g2, . . . , gN`1 P SEp2q ˆ SEp2q ˆ ¨ ¨ ¨ ˆ SEp2ql jh n
pN ` 1q-copies
ˇˇ
ˇ
gN`1 “ g1; gie3 ‰ gi`1e3, i “ 1, 2, . . . , N
)
, (7.43)
182
where e3 “ r0 0 1sT and gi “
»
——–
xi yi ri
0 0 1
fi
ffiffifl. In this sense, one can think of our
system as a G-snake which bites its tail [Krishnaprasad & Tsakiris, 1994].
As we are interested in steering laws (7.6) which leave our system dynamics
invariant under rigid motion, we can formulate a reduction to the shape space by
defining g˜i P SEp2q as
g˜i “ g´1i gi`1 “
»
——————–
xi ¨ xi`1 xi ¨ yi`1 xi ¨ pri`1 ´ riq
yi ¨ xi`1 yi ¨ yi`1 yi ¨ pri`1 ´ riq
0 0 1
fi
ffiffiffiffiffiffifl
, (7.44)
and the closure constraint “gN`1 “ g1” can be expressed in the shape space repre-
sentation as
Nź
i“1
g˜i fi g˜1g˜2 . . . g˜N “ 1. (7.45)
Therefore the shape space (of relative positions and velocities of the agents) for our
N -agent system can be defined as
Mshape “ Mstate{SEp2q “
!
g˜1, g˜2, . . . , g˜N P SEp2q ˆ SEp2q ˆ ¨ ¨ ¨ ˆ SEp2ql jh n
N-copies
ˇˇ
ˇ
Nź
i“1
g˜i “ 1; pg˜iq213 ` pg˜iq223 ‰ 0, i “ 1, 2, . . . , N
)
.
(7.46)
It has been shown by Justh & Krishnaprasad [2004] that the shape variable g˜i
183
assumes a left-invariant dynamics on SEp2q as
9˜gi “ g˜iξ˜i (7.47)
where ξ˜i “ ξi`1pui`1q ´ g˜´1i ξipuiqg˜i P sep2q, and the proposed control law (7.6)
depends only on the shape variable g˜i as
ui “ µ
„pg˜iq21
νi

. (7.48)
The following result from the work of Galloway [2011] allows us to analyze the
shape space dynamics (7.47) as a system of unconstrained dynamics on the product
space of N copies of the Lie group SEp2q with the closure constraint (7.45) viewed
as a constraint on the initial conditions.
Proposition 7.2 (Proposition 2.2.1, Galloway [2011]). The constraint
śN
i“1 g˜i “ 1
is preserved by the shape dynamics (7.47).
7.3.2 Shape Dynamics
Similar to the analysis for a two-agent system (Section 7.2.2), we introduce
geometrically meaningful scalar variables by identifying punctured R2 with the punc-
tured complex plane. First, we define ri`1 ´ ri “ ρieiϑi and xi “ eiθi to parametrize
the shape space. Now one can notice that g˜i P Mshape can be represented in terms
184
of the scalar shape variables as
g˜i “
»
——————–
cosφi sinφi ´ρi cosψi
´ sinφi cosφi ρi sinψi
0 0 1
fi
ffiffiffiffiffiffifl
, (7.49)
where φi “ θi ´ θi`1 and ψi “ π ´ ϑi ` θi (illustrated in Fig 41).
ρi−1 = |ri − ri−1|
xi
φi
φi−1
ψi
φi
ψi+1
xi+1
xi−1
ri−1
ri
ψi−1ri+1
φi−1
ρi = |ri+1 − ri|
Figure 41: Illustration of scalar shape variables (ρ, ψ, φ) used to parametrize the shape space
Mshape.
Proposition 7.3 (in a spirit similar to Proposition 2.2.3, Galloway [2011]). Using
shape variables (ρi, ψi, φi) the constraint equation (7.45) can be represented as
R
´ Nÿ
i“1
φi
¯
“ 1 (7.50)
Nÿ
i“1
ρiR
´
ψi `
i´1ÿ
j“1
φj
¯
“ 0, (7.51)
where Rp¨q is a 2ˆ 2 rotation matrix.
185
Proof. The closure constraint (7.45) can alternatively be represented as
g˜N g˜1g˜2 . . . g˜N´1 “ 1. (7.52)
By defining
R
`
α
˘
“
»
——–
cosα ´ sinα
sinα cosα
fi
ffiffifl , α P r0, 2πq
we can represent g˜i P Mshape as
g˜i “
»
——–
R
`
´ φi
˘
ρiR
`
π ´ ψi
˘
e1
0 1
fi
ffiffifl .
Therefore by letting Bi “ R
`
´ φi
˘
and qi “ ρiR
`
π ´ ψi
˘
e1, we have
g˜N
N´1ź
i“1
g˜i “
»
——–
BN qN
0 1
fi
ffiffifl
»
——–
śN´1
i“1 Bi q1 `
řN´2
i“1
´śi
j“1Bj
¯
qi`1
0 1
fi
ffiffifl
“
»
——–
BN
śN´1
i“1 Bi qN `BNq1 `BN
řN´2
i“1
´śi
j“1Bj
¯
qi`1
0 1
fi
ffiffifl . (7.53)
186
Noting that
BN
N´2ÿ
i“1
´ iź
j“1
Bj
¯
qi`1 “ R
`
´ φN
˘N´2ÿ
i“1
´ iź
j“1
R
`
´ φj
˘¯
ρi`1R
`
π ´ ψi`1
˘
e1
“
N´2ÿ
i“1
ρi`1R
`
´ φN
˘
R
`
π ´ ψi`1 ´
iÿ
j“1
φj
˘
e1
“
N´2ÿ
i“1
ρi`1R
`
ϑi`1 ´ θi`1 ´ pθ1 ´ θi`1q
˘
R
`
´ φN
˘
e1
“
N´1ÿ
i“2
ρiR
`
ϑi ´ θ1
˘
R
`
´ φN
˘
e1
and
BNq1 “ R
`
´ φN
˘
ρ1R
`
π ´ ψ1
˘
e1 “ ρ1R
`
ϑ1 ´ θ1
˘
R
`
´ φN
˘
e1,
we have
qN `BNq1 `BN
N´2ÿ
i“1
´ iź
j“1
Bj
¯
qi`1
“ ρNR
`
π ´ ψN
˘
e1 ` ρ1R
`
ϑ1 ´ θ1
˘
R
`
´ φN
˘
e1 `
N´1ÿ
i“2
ρiR
`
ϑi ´ θ1
˘
R
`
´ φN
˘
e1
“ ρNR
`
ϑN ´ θ1 ´ φN
˘
e1 `
N´1ÿ
i“1
ρiR
`
ϑi ´ θ1
˘
R
`
´ φN
˘
e1
“
Nÿ
i“1
”
ρiR
`
π ´ ψi ` θi ´ θ1
˘ı
R
`
´ φN
˘
e1
“ ´
Nÿ
i“1
”
ρiRT
`
ψi `
i´1ÿ
j“1
φj
˘ı
R
`
´ φN
˘
e1. (7.54)
We also have
BN
N´1ź
i“1
Bi “ R
`
´ φN
˘N´1ź
i“1
R
`
´ φi
˘
“ R
`
´
Nÿ
i“1
φi
˘
“ RT
` Nÿ
i“1
φi
˘
. (7.55)
187
Using (7.54) and (7.55), (7.53) can be simplified into
»
——–
RT
`řN
i“1 φi
˘
´řNi“1
”
ρiRT
`
ψi `
ři´1
j“1 φj
˘ı
R
`
´ φN
˘
e1
0 1
fi
ffiffifl ,
and the closure constraint (7.45) can alternatively be represented as
R
` Nÿ
i“1
φi
˘
“ 1 (7.56)
Nÿ
i“1
”
ρiRT
`
ψi `
i´1ÿ
j“1
φj
˘ı
R
`
´ φN
˘
e1 “ 0. (7.57)
Now we recall the fact that a linear combination of SOp2q matrices is singular if
and only if it is a zero matrix. By using this fact one can conclude from that
Nÿ
i“1
ρiR
`
ψi `
i´1ÿ
j“1
φj
˘
“ 0, (7.58)
as R
`
´ φN
˘
e1 ‰ 0.
Now we adopt an approach similar to the one in Section 7.2.2, and derive the
dynamics of the scalar shape variables as
9ρi “ νi cosψi ´ νi`1 cospψi ´ φiq
9ψi “ νiui ´
1
ρi
“
νi sinψi ´ νi`1 sinpψi ´ φiq
‰
i P t1, 2, ¨ ¨ ¨ , Nu (7.59)
9φi “ νiui ´ νi`1ui`1.
188
7.3.3 Analysis of TVA Feedback Law
In this particular context of an N -agent planar, cyclic interaction system,
individual contrast functions take the form
Θi “
1
2 pxi`1ptq ´ xiq ¨ pxi`1 ´ xiq
“ 1´ xi ¨ xi`1
“ 1´ cosφi, (7.60)
and the i-th agent is considered to attain the TVA strategy if Θi “ 0. From (7.60)
one can notice that
Θi P r0, 2s
and, Θi “ 0 ô φi “ 0.
(7.61)
Therefore for this N -agent planar system wherein each agent employs the strat-
egy for topological velocity alignment (TVA), we define the p2N ´ 3q-dimensional
topological velocity alignment manifold MTV A Ă Mshape as
MTV A “
!
ρ1, ψ1, φ1, . . . , ρN , ψN , φN P Mshape
ˇˇ
ˇΘi “ 0, i P t1, 2, ¨ ¨ ¨ , Nu
)
. (7.62)
Now we introduce an alternative (holistic) contrast function defined as
Θtotal “
Nÿ
i“0
Θi “
Nÿ
i“0
`
1´ cosφi
˘
. (7.63)
189
Using the fact given by (7.61), it can be concluded that Θtotal “ 0 if and only if
individual contrast functions are equal to zero. Therefore we can represent the TVA
manifold (MTV A) in terms of the holistic contrast function as
MTV A “
!
ρ1, ψ1, φ1, . . . , ρN , ψN , φN P Mshape
ˇˇ
ˇΘtotal “ 0
)
. (7.64)
Now we focus on a particular steering control law, a planar equivalent of
the one given by (7.6), to analyze its effectiveness in attaining the TVA strategy.
Moreover from (7.48) we can notice that the feedback law can be expressed in terms
of shape variables as
ui “ ´
ˆ
µ
νi
˙
sinφi, i P t1, 2, ¨ ¨ ¨ , Nu. (7.65)
From (7.61) we can observe that once the TVA strategy has been attained, i.e.
Θtotal “ 0, the steering control becomes identically zero, and as a consequence the
mismatch in velocity direction remains identically zero (7.59). If each agent employs
a steering feedback of the form (7.65), the closed loop dynamics can be expressed
as
9ρi “ νi cosψi ´ νi`1 cospψi ´ φiq
9ψi “ ´µ sinφi ´
1
ρi
“
νi sinψi ´ νi`1 sinpψi ´ φiq
‰
i P t1, 2, ¨ ¨ ¨ , Nu (7.66)
9φi “ µ
`
sinφi`1 ´ sin φi
˘
.
190
Proposition 7.4. The topological velocity alignment manifold MTV A Ă Mshape is
invariant under the closed loop shape dynamics (7.66).
Proof. From (7.63) and (7.66) we have
9Θtotal “ µ
Nÿ
i“0
sinφi
`
sin φi`1 ´ sinφi
˘
“ ´µ2
Nÿ
i“0
`
sin2 φi ´ 2 sinφi sinφi`1 ` sin2 φi`1
˘
“ ´µ2
Nÿ
i“0
`
sinφi ´ sin φi`1
˘2. (7.67)
Therefore it can be concluded that MTV A is invariant under the shape dynamics
because φi “ 0 @i on MTV A Ă Mshape. In fact, from (7.67) it is clear that 9Θtotal ď
0.
Proposition 7.5. Consider the set E defined as
E “
!
pφ1, . . . , φNq P S1 ˆ ¨ ¨ ¨ ˆ S1l jh n
N-copies
ˇˇ
ˇ 9Θtotal “ 0, R
` Nÿ
i“1
φi
˘
“ 1
)
, (7.68)
and assume N not to be a multiple of 4. Then E has a finite cardinality.
Proof. From the closure constraint (7.50) we have
Nÿ
i“1
φi “ 2kπ, k P Z. (7.69)
191
On the other hand, 9Θtotal “ 0 if and only if
sinφi “ sinφi`1, i P t1, 2, ¨ ¨ ¨ , Nu, (7.70)
and (7.70) holds true if and only if either of the conditions is satisfied
φi`1 “ φi, or φi`1 “ π ´ φi. (7.71)
Therefore by assuming φ1 “ φ˚, we can construct the following tree wherein each
branch correspond to a solution for 9Θtotal “ 0.
φ˚
φ˚
φ˚
...
φ˚ÑφN
...
φ3 Ñ
φ2 Ñ
φ1 Ñ
π ´ φ˚
...
π ´ φ˚
... ...
π ´ φ˚
φ˚
... ...
π ´ φ˚
... ...
φ˚ π ´ φ˚
Now we assume that along a particular branch of the tree there are M number of
φ˚-nodes and pN ´Mq number of pπ ´ φ˚q-nodes. Therefore from (7.69) we have
Mφ˚ ` pN ´Mqpπ´ φ˚q “ 2kπ, k P Z, φ˚ P r0, 2πq, M P t1, ¨ ¨ ¨ , Nu. (7.72)
192
Assuming 2M ´N ‰ 0 we solve (7.72) to obtain
φ˚ “
ˆ2k `M ´N
2M ´N
˙
π, (7.73)
and as φ˚ P r0, 2πq we have,
k P
$
’’’’’’&
’’’’’’%
„
N ´M
2 ,
3M ´N
2
˙
X Z if, M ą N2
„3M ´N
2 ,
N ´M
2
˙
X Z if, M ă N2 .
(7.74)
Now if N is a odd number then M can never be equal to N2 , and hence for
each M P t1, ¨ ¨ ¨ , Nu, we will have finite number choices for φ˚. On the other hand,
it can be noted from the tree that a finite number of branches contain M number
of φ˚-nodes. Combining these two facts we can conclude that E has finite number
of elements.
Now we assume N to be a multiple of 2, but not a multiple of 4. In that case the
closure condition (7.69) gets violated forM “ N2 because it causes a contradiction in
(7.72), and thereby yields no solution for that particular M . For every other value
of M , (7.72) yields finite number of φ˚. Therefore, the corresponding E has finite
cardinality.
However, if N is assumed to be a multiple of 4, then for M “ N2 , (7.72)
holds true over a continuum of φ˚. As a consequence E will have uncountably many
elements.
193
Remark 7.3. Narrowing our attention to the shape dynamics on MTV A, we can
show that the restricted dynamics can be expressed as
9ρi “ pνi ´ νi`1q cosψi
9ψi “ ´
1
ρi
pνi ´ νi`1q sinψi
for every i P t1, 2, ¨ ¨ ¨ , Nu. This dynamics is essentially same as the restricted
dynamics for a two-agent system (7.32), and hence, by following the discussion in
Section 7.2.3, we can show that the phase plane trajectories of the reduced dynamics
are level sets of “ρi sinψi”.
7.4 Algorithm for anN -agent System in a Three-Dimensional Setting
In this section we focus toward topological velocity alignment in its true sense,
and assume that each agent pays attention to its K-nearest neighbors. However, in
this context of state-dependent attention graph, the possibility of viCOM becoming
zero cannot be ruled out, and we tackle this issue by bringing in an additional
neighbor (for the i-th agent) into consideration whenever viCOM becomes zero. As
each of the agents has non-zero speed, inclusion of an additional agent into the
neighborhood ensures that viCOM no longer remains zero. Moreover, it doesn’t
affect connectivity of the underlying attention graph. The following discrete time
algorithm provides a methodical way to implement TVA in a multi-agent robotic
system.
It is worth mentioning here that this algorithmic way towards flocking can
194
Algorithm 2 Topological Velocity Alignment (3D, Nearest Neighbors)
Data:
Initial Time ´ tinitial;
Final Time ´ tfinal;
Sampling Interval ´ ∆;
Number of Agents ´ N ;
Initial Position and Orientation ´ tgiuni“1;
Neighborhood Size ´ K
begin
Initialize: tcurrent ÐÝ tinitial
for i = 1 to n do
Initialize: State - Xi ÐÝ gi
while tcurrent ď tfinal do
for i = 1 to n do
Define: Ni - the set of K-nearest neighbors
Compute: Neighborhood center of mass velocity - viCOM
if viCOM “ 0 then
Define: Ni - the set of K ` 1-nearest neighbors
Compute: Neighborhood center of mass velocity - viCOM
Compute: Steering control - ui, vi
Implement: Steering Control - tui, viuni“1
Update: State - tXiuni“1
Update: Time - tcurrent ÐÝ tcurrent `∆
easily be modified to implement TVA in a planar setting, and this is done by con-
sidering the natural curvature ui alone (vi is ignored). By restricting (7.6) to a
planar setting, ui can be expressed as
uiptq “ µ
„xNiptq ¨ yiptq
νiptq

, (7.75)
where all the variables carry their usual meaning.
195
7.5 Implementation on Mobile Robot Testbed
In what follows we present implementation results of the TVA control law in
a planar setting. Our implementation avoids the singularity issue (i.e. viCOM “ 0)
by following Algorithm 2.
7.5.1 Experimental Setup
Our experimental test-bed is comprised of Pioneer 3 DX wheeled robots (Fig 42)
from Adept MobileRobots [P3-DX]. These compact differential-drive mobile robots
are equipped with reversible DC motors, high-resolution motion encoders and 19cm
wheels. The onboard computation is done via a 32-bit Renesas SH2-7144 RISC mi-
croprocessor, including the P3-SH microcontroller with ARCOS. The sensors on the
robot include eight forward-facing ultrasonic (sonar) sensors. ARIA [Aria] provides
an interface for controlling and receiving data from the robot, and communication
with the robot for sending control commands (forward velocity and turning rate) is
done via 802.11-b/g/n networking. The width of the robot is 380 mm and it has a
swing radius of 260 mm.
Algorithm implementation (i.e, feedback law computation) has been done in
C++ using ROS along with ROS-ARIA, as the interfacing robotics middleware.
The experiments have been carried out in a laboratory environment equipped with
a sub-millimeter accurate Vicon motion capture system [ViCoN]. We use a Dell
workstation to run ROS, and this computer is connected to the Vicon server via a
dedicated Ethernet connection (Fig 43).
196
Figure 42: Pioneer 3-DX Mobile Robot with Two-wheel Differential and Caster.
The Vicon system captures the motion of the robots and sends out the position
and heading data to the computer running ROS. The control law program listens to
this data, computes the curvature values, and finally transmits the individual turning
rates over a Wi-Fi network. All these operations are carried out at a frequency of 25
Hz. As the robot velocity ( 9rki , with k denoting the time index) is directed along the
robot heading, xki and yki can be directly computed from the heading data. Then,
the curvature variable uki is evaluated from the corresponding control laws (7.75),
and the turning rate ωki pi “ 1, 2, . . . , Nq (in degrees/sec) is computed as:
ωki “
ˆ180
π
˙
νki uki , (7.76)
where νki is the speed of the i-th agent at the k-th time instant.
Next we will present our implementation results from three different exper-
iments (refer [YouTube-Video] for implementation videos). In these experiments,
the sonar sensors on the robots were activated to sense any obstacle in the direction
of motion of the robots and if any robot can sense such an obstacle, it will simply
197
1ARIA
Robot Operating System (ROS)
Pioneer 3-DX
Vicon Motion  Capture
kŒ‹Šˆ›Œ‹Gl›Œ™•Œ›
zˆ™Œ‹G~Tm
Figure 43: Illustration of the Experimental Setup at Intelligent Servosystems Laboratory, Uni-
versity of Maryland, College Park.
apply a maximum turning rate (ωsat) to avoid collision. The sonars are programmed
to detect an obstacle only in close proximity („ 300 mm) of the robots. In all our
experiments ωsat is set at be 50 rad/sec, and the value of the feedback gain µ is cho-
sen to be 1 Hz. It should be noted that although the control law allows non-uniform
and time-varying forward speed of the robots, here we are presenting sample runs
for which the speeds of individual agents are same (60 mm/sec).
7.5.2 Experiment I
A system eight agents is considered, and we apply the same TVA law to all
of them. The neighborhood size is taken to be three (i.e. K “ 3). The robots are
initially placed in arbitrary positions and directions. The footprints of the robots
198
are shown in Fig 44a. The initial and final directions of the robots are shown using
arrows and the final positions of the robots are denoted using dots. It can be seen
from Fig 44b that the contrast function decays to zero very quickly which indicates
perfect velocity alignment within the swarm.
−3000 −2000 −1000 0 1000 2000
−2000
−1000
0
1000
2000
x (in mm)
y 
(in
 m
m)
(a) Robot Trajectories
0 10 20 30 40 500
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Time (in sec)
Θ
(t
)
(b) TVA Contrast Function
Figure 44: Results from Experiment I (8 agents, Flocking).
7.5.3 Experiment II
Next we decreased the neighborhood size, and set it at K “ 1, so that each
robot ‘communicates’ only with its closest neighbor. We chose the initial positions
in such a way that they may form sub-clusters instead of moving as a single swarm.
This behavior is called ‘splitting ’ in a swarm. From Fig 45a, we can clearly see that
the swarm of eight robots gradually split from each other and form three different
clusters. It is to be noted that even if all the agents are not going in the same
direction, the contrast function still converges to zero (Fig 45b). This happens
because each of the robots are aligned with their nearest neighbors, and hence each
199
of the individual contrast functions (Θiptq) are zero. This experiment may explain
the splitting phenomenon observable in nature.
−2000 −1000 0 1000 2000
−2500
−2000
−1500
−1000
−500
0
500
1000
x (in mm)
y 
(in
 m
m)
(a) Robot Trajectories
0 5 10 15 200
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (in sec)
Θ
(t)
(b) TVA Contrast Function
Figure 45: Results from Experiment II (8 agents, Splitting).
7.5.4 Experiment III
Lastly, we combined the above two experiments, and conducted an experiment
using six robots in a swarm and another robot as a predator. A separate computer
was used for manual control of the ‘predator’ robot.
At the beginning, neighborhood size is set at K “ 3, such that the ‘communi-
cation’ graph among the robots stays connected and they move as an entire swarm
in a common direction. When the swarm comes close to the predator, the neigh-
borhood size is decreased to one. As we are not using any onboard visual sensing
and the sonar sensing is done only in very close region („ 300 mm), the change in
neighborhood size is made manually. From Fig 46b, we can see that the change in
neighborhood size takes place at around 20 seconds and we can also see a tiny jump
200
in the contrast function value at that time. The predator then slowly approaches
to one of the agents in the swarm, which abiding to its collision avoidance rule,
turns to avoid the predator. In Fig 46a, the trajectories of the agents are drawn
in dashed lines before the occurrence of this event and in solid lines afterwards.
The trajectory of the predator robot in not shown in the figure. After creating
the initial perturbation, the predator is slowly moved through the swarm causing
some subsequent disturbances. These perturbations create a noticeable impact in
the swarm. As the attacked agent turns, its neighbor also tries to align itself with
that agent and so does its neighbor. This goes on until the communication graph
becomes disconnected, and then a split in the swarm is observed (refer [YouTube-
Video]) similar to the one in Experiment 2. As we can see in Fig 46a, the swarm is
divided in two groups after the attack of the predator. The jumps in the contrast
function plot (Fig 46b) symbolize the perturbations caused by the external agent.
The contrast function eventually converges to zero after the members are aligned
−4000 −2000 0 2000
−4000
−3000
−2000
−1000
0
1000
x (in mm)
y 
(in
 m
m)
(a) Robot Trajectories
0 20 40 60 800
2
4
Time (in sec)
Θ
(t)
0
1
2
3
4
5
N
um
be
r o
f n
ea
re
st
 n
ei
gh
bo
rs
(b) TVA Contrast Function
Figure 46: Results from Experiment III (6 agents, Perturbation).
201
with their neighbors within each subgroup.
202
Chapter 8: Conclusion and Future Works
The key focus area of this dissertation has been to demonstrate how compre-
hensive knowledge about the underlying mechanism behind pursuit and collective
motion in natural settings can be leveraged to synthesize decentralized control al-
gorithms for collective motion (to be applied to a group of robotic agents). Along
the way, we have also developed appropriate algorithms to extract parameters of
motion, namely speed, curvatures, lateral acceleration etc., from a discrete set of
observed (perhaps noisy) position data.
In the following section, we summarize the key contributions of this disserta-
tion and propose some topics for further research along these lines.
8.1 Summary of Contributions and Future Directions of Research
8.1.1 Reconstruction of Collectives
As access to the parameters of motion constitutes a necessary step towards
uncovering flight strategies and control laws behind collective motion in nature (for-
aging bats, starling flocks), in “part I” of this dissertation, we identified the need for
appropriate algorithms to reconstruct trajectories from a data-set of observed posi-
tions. As this problem of recovering a smoothened signal from noisy observations is
203
an ill-posed one, we introduce regularization to tackle lack of well-posedness. First,
to govern the evolution of a trajectory, we introduce generative models (expressed
via ODEs) with inputs, states and outputs. Then we impose regularization onto the
problem by penalizing appropriate functionals of the control input. Thus we have
turned this into a continuous time optimal control problem with intermediate state
costs (as shown in Chapter 2). A distinctive feature of this approach lies in the fact
that our choice of penalty term has been influenced by findings in bio-mechanics.
In Chapter 2, we have tackled this data smoothing problem from a mathemat-
ical programming perspective, and have overcome lack of integrability by adopting
a numerical approach. The methodologies developed in this chapter has been used
later in Chapter 5 to reconstruct flight trajectories for bat foraging events. We
have also proposed an ordinary cross validation approach, based on leaving-one-
out strategy, to select an optimal amount of regularization, which plays a crucial
role in maintaining a proper balance between goodness of fit and smoothness of the
trajectory.
However, as leaving-one-out requires the reconstruction to be carried out mul-
tiple times (same as the data-set size) by dropping out a single observation every
time, implementation of ordinary cross validation is quite demanding from a com-
putational perspective. It would be interesting to investigate if a computationally
efficient alternative can replace this algorithmic approach for optimal amount of
regularization. Unbiassed risk estimators (Li [1985]; Solo [1996]) based on Charles
Stein’s work [Stein, 1981] on mean estimation for multivariate normal distribution,
appears very relevant in our context.
204
In Chapter 3, we have introduced a linear generative model (basically a triple
integrator), and imposed regularization by trading total fit error against high values
of jerk (i.e. third derivative of position) path integral. Then, by exploiting integra-
bility of the generative model and quadratic nature of the cost functional, we have
derived a closed form solution. Moreover, we have shown that the reconstructed
position, velocity and acceleration can be expressed a linear combination of the
observed position data. By choosing appropriate linear generative model, one can
easily show that smoothing splines can be posed as a special case in this framework.
The trajectory reconstruction algorithm developed in this chapter has later been
used in Chapter 6 to reconstruct flight trajectories of starling flocks.
Our numerical example in this chapter (Section 3.5) attempted to reconstruct
a curve on a sphere from a set of discrete and noisy observations. Although this
approach yields a satisfactory performance in terms of enforcing the reconstructed
trajectory to lie on the sphere, it would be exciting to construct data smoothing
algorithms to reconstruct trajectories which lie on a lower dimensional algebraic
manifold in the ambient space. Another potential extension of this work lies in the
area of reconstructing periodic curves.
In Chapter 4, we have exploited the theory of Pontryagin’s maximum princi-
ple to solve data smoothing posed as a continuous time optimal control problem.
The proposed results are capable of dealing with data smoothing problems in both
Euclidean (Rn) and matrix Lie group (G) settings, and they yield result in a semi-
analytic way. Example problem demonstrates that this theory enables us to turn an
optimal control problem over an infinite dimensional function space into a two-point
205
boundary value problem, which can be tackled via an appropriate multiple-shooting
method [Morrison et al., 1962]. It is worth mentioning here that this regularized
inversion problem can also be viewed from the perspective of waypoint tracking. If
someone attempts to achieve reduction in some path cost, by sacrificing exactness
in its traversal of way-points, that problem can easily be cast in this framework.
However, there are some issues which requires attention before we attempt to
broaden the scope of this framework. One obvious direction along this line is to
consider penalty functionals involving derivatives of control inputs. Although this
issue can be addressed by augmenting new states to the system, some care should
be taken to set up the associated symplectic framework in a proper way. Another
pressing concern lies in the singularity issues of the Hamiltonian. The following data
smoothing problem exemplifies this issue in an efficient way.
Let us consider the generative model, governed by the natural Frenet frame
equations, for evolution of a trajectory in a three-dimensional space. As discussed
in Section 1.3.1, we can pack the position vector and the associated frame vectors,
inside a 4ˆ 4 matrix gptq defined as
gptq “
»
——–
xptq yptq zptq rptq
0 0 0 1
fi
ffiffifl P SEp3q.
206
Now, by letting
X1 “
»
——————————–
0 ´1 0 0
1 0 0 0
0 0 0 0
0 0 0 0
fi
ffiffiffiffiffiffiffiffiffiffifl
, X2 “
»
——————————–
0 0 1 0
0 0 0 0
´1 0 0 0
0 0 0 0
fi
ffiffiffiffiffiffiffiffiffiffifl
, X3 “
»
——————————–
0 0 0 0
0 0 ´1 0
0 1 0 0
0 0 0 0
fi
ffiffiffiffiffiffiffiffiffiffifl
,
X4 “
»
——————————–
0 0 0 1
0 0 0 0
0 0 0 0
0 0 0 0
fi
ffiffiffiffiffiffiffiffiffiffifl
, X5 “
»
——————————–
0 0 0 0
0 0 0 1
0 0 0 0
0 0 0 0
fi
ffiffiffiffiffiffiffiffiffiffifl
, X6 “
»
——————————–
0 0 0 0
0 0 0 0
0 0 0 1
0 0 0 0
fi
ffiffiffiffiffiffiffiffiffiffifl
,
denote a basis for the associated Lie algebra sep3q, the underlying generative model
can be expressed as the following left-invariant dynamical system on SEp3q,
9g “ gξu “ g
`
νpX4 ` uX1 ´ vX2q
˘
(8.1)
where u, v are natural curvatures (steering control) and ν is the speed of the trajec-
tory. Then, by letting triuNi“0 denote the set of observed positions, and by imposing
regularization via trading total fit error against high values of the curvatures and
speed path integral, one can formulate the trajectory reconstruction problem as the
207
following optimal control problem
Minimize
gpt0q,u,v,ν
Nÿ
i“0
}rptiq ´ ri}2 ` λ
tNż
t0
`
u2 ` v2 ` ν2
˘
dt
subject to gpt0q P SEp3q, u, v P U , ν P U`
9g “ g
`
νpX4 ` uX1 ´ vX2q
˘
“ TeLg ¨
`
νpX4 ` uX1 ´ vX2q
˘
,
(8.2)
where U(U`) is the space of real(positive) valued functions on rt0, tN s and λ ą 0 acts
as a regularization parameter for the inverse problem. By comparing this optimal
control problem (8.2) with the one mentioned in the statement of maximum principle
(4.45) we have
Lpuq “ λpu2 ` v2 ` ν2q
fpgptiq, riq “ }Agptiqe4 ´ ri}2
where A “ re1 e2 e3sT and teiu4i“1 denotes a standard basis vector in R4. By
following an approach similar to the one adopted for the previous example problems,
we define the pre-Hamiltonian as
Hpg, p, uq “ xp, TeLg ¨ ξuy ´ Lpuq (8.3)
where p P T ˚g SEp3q and TeLg represents the tangent map of the left translation
by g on SEp3q. Now we introduce µ P se˚p3q defined as µ “ TeL˚g ¨ p. By letting
208
X5i , i “ 1, ¨ ¨ ¨ , 6 denote a dual basis for se˚p3q, µ can be represented as
µ “
6ÿ
i“1
µiX5i .
Then, by leveraging the left-invariance of the dynamics (8.1), the pre-hamiltonian
can be expressed as
Hpg, p, uq “ xTeL˚g ¨ p, ξuy ´ Lpuq
“ x
6ÿ
i“1
µiX5i , νpX4 ` uX1 ´ vX2qy ´ Lpuq
“ νpµ4 ` uµ1 ´ vµ2q ´ λpu2 ` v2 ` ν2q. (8.4)
As both ξu and Lpuq are differentiable with respect to the control inputs, the optimal
control can be derived by solving
BH
Bu
ˇˇ
ˇˇ
pu˚,v˚,ν˚q
“ ν˚µ1 ´ 2λu˚ “ 0
BH
Bv
ˇˇ
ˇˇ
pu˚,v˚,ν˚q
“ ´ν˚µ2 ´ 2λv˚ “ 0 (8.5)
BH
Bν
ˇˇ
ˇˇ
pu˚,v˚,ν˚q
“ µ4 ` u˚µ1 ´ v˚µ2 ´ 2λν˚ “ 0.
Hence the optimal control inputs can be expressed by
¨
˚˚
˚˚
˚˚
˝
u˚
v˚
ν˚
˛
‹‹‹‹‹‹‚
“ 14λ2 ´ pµ21 ` µ22q
¨
˚˚
˚˚
˚˚
˝
µ1µ4
´µ2µ4
2λµ4
˛
‹‹‹‹‹‹‚
, (8.6)
209
and by substituting the optimal controls into the pre-hamiltonian, (4.87) yields an
SEp2q-invariant reduced hamiltonian, given by
h “ λµ
2
4
4λ2 ´ pµ21 ` µ22q
. (8.7)
Now, we can derive the reduced dynamics on se˚p3q by computing ∇µh and the
associated structure constants. However from (8.7) one can notice that the reduced
hamiltonian becomes singular whenever µ21 ` µ22 “ 4λ2. With this perspective, it
would be interesting to seek answer for such questions as:
• Is the submanifold Msing “ tpµ1, ¨ ¨ ¨ , µ6q P R6|µ21`µ22 “ 4λ2u invariant under
the reduced dynamics?
• Does this submanifold (Msing) attract any trajectory originating outside the
submanifold? Under what conditions one can avoid the singularity?
8.1.2 Analysis of Collective Behavior in Nature
Following the main flow of this dissertation, we set out to explore the underly-
ing mechanism behind pursuit and collective motion in natural settings in “part II”
of this dissertation, and performed analysis on the flight trajectory data of echolocat-
ing bats (demonstrating pursuit) and European starlings (demonstrating flocking).
Our study adopts a two-pronged approach to investigate the underlying flight strate-
gies and feedback mechanism for steering control - first we study the statistics of
appropriate contrast functions, and then we compare the empirical values of steer-
210
ing control to the values obtained from theoretically plausible feedback laws. This
approach is also capable of providing an estimate for the associated sensorimotor
delay.
In Chapter 5, we have performed flight data analysis for big brown bats
(Eptesicus fuscus) in two different foraging contexts. This analysis has shown
quantitative evidence in favor of a context-specific switch in flight strategy. Ac-
cording to our findings, bats apply constant absolute target direction (CATD), also
known as motion camouflage (MC) in the context of visual insects, while chasing
a free flying insect (praying mantis in our study). But if the scenario is modified
into a competitive setting with another bat foraging for the same stationary food
source (meal-worm in this set of experiments), flight data show evidence that the
trailing bat resort to classical pursuit (CP) to follow the other bat. Moreover, by
comparing empirically observed curvature values with the ones obtained from the-
oretically plausible feedback laws, this study sheds light upon the steering control
mechanisms and the associated sensorimotor delay.
Recent developments by Galloway & Dey [2015] in decentralized control have
analyzed cyclic constant bearing (CB) pursuit [Galloway et al., 2013] strategy in
a multi-agent system wherein each agent pays attention to a neighbor (moving)
and a beacon (fixed). As classical pursuit can be interpreted as a special case of
the CB pursuit strategy, it would be worthwhile to study this interaction between
the stationary prey (tethered meal-worm) and the flying bats from this perspective
(each bat having its own priority level for the stationary food source). It is worth
mentioning here that our analysis does not assume any internal model of target
211
motion, and the steering control mechanism is assumed to be based on pure reaction
to the target motion. However a recent study by Mischiati et al. [2014] has shown
that, internal model and reactive pursuit strategy, both play an important role in
generating the prey interception trajectories by dragonflies (P lathemis lydia). So
it would be interesting to design appropriate experiments and investigate if such
internal models are involved in bat foraging too.
Chapter 6 describes our ongoing work on the flight strategy analysis of Eu-
ropean starling (Sturnus vulgaris) flocks. This study has revealed that the flock-
averaged coherence (the average cosine of the angle between the velocities of a focal
bird and its neighborhood center of mass, averaged over the entire flock) gets max-
imized by considering 5-7 nearest neighbors. In addition to reconfirming a previous
result highlighting the importance of topological notion of distance in starling flocks,
this study has also provided some insight about the steering control actions adopted
by the individual starlings.
However, our current approach for constructing an interaction graph (based on
a set of nearest neighbors) is oblivious to sensory perceptions by the focal individual
and spatial distribution of the neighbors. Future works will attempt to explore other
possibilities for the interaction graph based on visual cues [Strandburg-Peshkin et al.,
2013] and statistical causality (Granger causality [Granger, 1969], directed mutual
information [Massey, 1990]).
212
8.1.3 Synthesis of Collective Motion
Statistical analysis of flight behavior in European starling flocks has revealed
that the individuals in a flock tend to fly parallel to each other. This observa-
tion led us towards proposing a decentralized algorithm (called topological velocity
alignment) to make the individuals move in the same direction without colliding into
each other (in Chapter 7), and the global behavior emerges through local interac-
tion between neighbors. The fact, that the proposed feedback law does not assume
any uniformity in the individual speeds and the attention graph is directed, makes
our approach distinct from the existing models of flocking. Moreover, this proposed
control strategy has wave-like aspects, conforming to the criterion of linear growth
of information observed in starling flocks [Attanasi et al., 2014].
Numerical simulations and implementations have shown that whenever a sub-
group of agents in the flock are provided additional instruction (e.g. to follow
predefined paths, avoid obstacles, etc.) for steering control, the effect spreads across
the whole flock (without any change in the way the uninformed agents interact
with its neighbors). These observations demand further analysis, and future work
in this area will explore how these covert leaders (agents with extra information)
in a collaborative swarm affect the group behavior. This framework can also be
exploited to design a collaboration of heterogeneous agents wherein a small group
of agents are capable of sensing the environmental cue and others just interact with
their neighbors.
213
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