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Multi-agent Consensus using Generalized CyclicPursuit Strategies
A Thesis
Submitted For the Degree of
Doctor of Philosophy
in the Faculty of Engineering
by
Arpita Sinha
Department of Aerospace Engineering
Indian Institute of Science
BANGALORE – 560 012
July 2007
c©Arpita Sinha
July 2007
All rights reserved
Dedicated to
My Parents
Acknowledgements
I would like to acknowledge and extend my deep, heartfelt and sincere gratitude to by
my advisor Prof. Debasish Ghose. This doctoral work has been completed successfully
due to his continuous guidance, encouragement and support. The knowledge which I
acquired from him both in academic and non-academic areas proved to be immensely
beneficial and would be a great resource in my future endeavors. I would also like to
thank him for his patience in explaining every minute things and clearing my smallest
doubts.
I would like to take this opportunity to thank Prof B.N. Raghunandan and Prof V
Mani for opportunities and facilities that I received from them throughout this period.
I am also thankful to Prof M.S. Bhat and Dr. R. Padhi for their valuable advice and
support.
I am especially thankful to my fellow lab-mate Dhananjay for clearing my doubts
by his in-depth knowledge in mathematics. I would also like to express my gratefulness
to one of my other lab-mates Sheeba for her care and companionship. I want to thank
my all other lab-mates Bulbul Mukherjee, Ashwini, Guruprasd, Prasanna, Rajanikant
and Krishnanand who have directly or indirectly helped me in my research. I have
furthermore to thank Sujit for his help during the initial phase of my research work. I
like to thank Mangal, Priya, Gowri, Titas, Madhumita, Kaushik, Sarachchandra, Joseph,
Akhil, Urmila, Praveen, Kannan, Srivardhan, Suresh, Hassan, Amit, Varun, Vaibhav,
Anup and Powly for their supports and company. I would also like to extend my sincere
thanks to all the staffs of the Aerospace Department and STC for their constant help in
i
Acknowledgements ii
all the official formalities during these days.
I am deeply indebted to my parents, Sri Somendra Prasad Sinha and Smt Arati
Sinha for their unconditional love, support and blessings. Their inspirations helped me
to reach the highest degree in education. I enjoyed my stay in the institute mostly due to
the affection and guardianship of my cousin Subimal Ghosh and my friend Aditi Datta
(whom I consider as my elder sister). I am grateful to my grandmother, aunt, my cousin
Joydeep Roy and all others in my family for their love and best wishes. I would also like
to thank Chaitali Misra, Subhankar Karmakar, Priyanko Ghosh, Sudipta Das, Vidya,
Vinoj, Gargi and all my juniors at IISc. It was a great pleasure being in their company.
I would like to give my special thanks to my friend Pramit Basu, whose presence was a
great support to me.
Above all, I am grateful to GOD, who made all these things possible.
Publications based on this Thesis
Journals:
1. A. Sinha and D. Ghose, “Generalization of linear cyclic pursuit with application
to rendezvous of multiple autonomous agents” IEEE Transactions on Automatic
Control, vol. 51, no. 11, Nov 2006, pp. 1819 - 1824.
2. A. Sinha and D. Ghose, “Control of multi-agent systems using linear cyclic pur-
suit with heterogonous controller gains” to appear in ASME, Journal of Dynamic
Systems, Measurement and Control.
3. A. Sinha and D. Ghose, “Generalization of nonlinear cyclic pursuit”, accepted in
Automatica.
Conferences:
1. A. Sinha and D. Ghose, “Line formation of a swarm of autonomous agents with
centroidal cyclic pursuit” Proceedings of the International Conference on Advances
in Control and Optimization of Dynamical Systems, ACODS, Bangalore, India,
Feb 2007, pp. 447-450.
2. A. Sinha and D. Ghose, “Control of agent swarm using generalized centroidal
cyclic pursuit laws”, Proceedings of Twentieth International Joint Conference on
Artificial Intelligence, IJCAI, Hyderabad, India, Jan 2007, pp. 1525 - 1530.
iii
Publications based on this Thesis iv
3. A. Sinha and D. Ghose, “Behavior of autonomous mobile agents using linear cyclic
pursuit laws” Proceedings of the American Control Conference, Minneapolis, MN,
June 2006, pp. 4964 - 4969.
4. A. Sinha and D. Ghose, “Generalization of nonlinear cyclic pursuit”, Proceedings
of the American Control Conference, Portland, OR, June 2005, pp. 4997 - 5002.
5. A. Sinha and D. Ghose, “Some generalization of linear cyclic pursuit” Proceedings
of IEEE India Annual Conference, INDICON 2004, Kharagpur, India, Dec 2004,
pp. 210 - 213.
Abstract
One of the main focus of research on multi-agent systems is that of coordination in a
group of agents to solve problems that are beyond the capability of a single agent. Each
agent in the multi-agent system has limited capacity and/or knowledge which makes
coordination a challenging task. Applications of multi-agent systems in space and ocean
exploration, military surveillance and rescue missions, require the agents to achieve some
consensus in their motion. The consensus have to be achieved and maintained without
a centralized controller. Multi-agent system research borrows ideas from the biological
world where such motion consensus strategies can be found in the flocking of birds,
schooling of fishes, and colony of ants. One such class of strategies are the cyclic pursuit
strategies which mimic the behavior of dogs, birds, ants, or beetles, where one agent
pursues another in a cyclic manner, and are commonly referred to as the ‘bugs’ problem.
In the literature, cyclic pursuit laws have been applied to a swarm of homogenous
agents, where there exists a predefined cyclic connection between agents and each agent
follows its predecessor. At equilibrium, the agents reach consensus in relative positions.
Equilibrium formation, convergence, rate of convergence, and stability are some of the
aspects that has been studied under cyclic pursuit.
In this thesis, the notion of cyclic pursuit has been generalized. In cyclic pursuit,
usually agents are homogenous in the sense of having identical speeds and controller gains
where an agent has an unique predecessor whom it follows. This is defined as the basic
cyclic pursuit (BCP) and the sequence of connection among the agents is defined as the
pursuit sequence (PS). We first generalize this system by assuming heterogeneous speed
v
Abstract vi
and controller gains. Then, we consider a strategy where an agent can follow a weighted
centroid of a group of other agents instead of a single agent. This is called centroidal
cyclic pursuit (CCP). In CCP, the set of weights used by the agents are assumed to be
the same. We generalize this further by considering the set of weights adopted by each
agent to be different. This defines a generalized centroidal cyclic pursuit (GCCP). The
behavior of the agents under BCP, CCP and GCCP are studied in this thesis.
We show that a group of holonomic agents, under the cyclic pursuit laws − BCP,
CCP and GCCP − can be represented as a linear system. The stability of this system
is shown to depend on the gains of the agents. A stable system leads to a rendezvous
of the agents. The point of rendezvous, also called the reachable point, is a function
of the gains. In this thesis, the conditions for stability of the heterogeneous system of
agents in cyclic pursuit are obtained. Also, the reachable point is obtained as a function
of the controller gains. The reachable set, which is a region in space where rendezvous
can occur, given the initial positions of the agents, are determined and a procedure is
proposed for calculating the gains of the agents for rendezvous to occur at any desired
point within the reachable set. Invariance properties of stability, reachable point and
reachable set, with respect to the pursuit sequence and the weights are shown to exist
for these linear cyclic pursuit laws.
When the linear system is unstable, the agents are shown to exhibit directed motion.
We obtain the conditions under which such directed motion is possible. The straight line
asymptote to which the agents converge is characterized by the gains and the pursuit
sequence of the agents. The straight lines asymptote always passes through a point,
called the asymptote point, for given initial positions and gains of the agents. This
invariance property of the asymptote point with respect to the pursuit sequence and the
weights are proved.
For non-holonomic agents, cyclic pursuit strategies give rise to a system of nonlin-
ear state equations. It is shown that the system at equilibrium converges to a rigid
Abstract vii
polygonal formation that rotates in space. The agents move in concentric circles at equi-
librium. The formation at equilibrium and the conditions for equilibrium are obtained
for heterogeneous speeds and controller gains.
The application of cyclic pursuit strategies to autonomous vehicles requires the sat-
isfaction of some realistic restrictions like maximum speed limits, maximum latax limits,
etc. The performances of the strategies with these limitations are discussed. It is also
observed that the cyclic pursuit strategies can also be used to model some behavior of
biological organisms such as schools of fishes.
Abstract viii
Contents
Acknowledgements i
Publications based on this Thesis iii
Abstract v
Notation and Abbreviations xix
1 Introduction 1
1.1 Multi-agent consensus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Cyclic pursuit as consensus protocol . . . . . . . . . . . . . . . . . . . . . 5
1.3 Generalization of cyclic pursuit . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Contributions and organization of the thesis . . . . . . . . . . . . . . . . 8
2 Rendezvous using linear Basic Cyclic Pursuit 11
2.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Rendezvous and Reachable point . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Invariance properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.5.1 Fixed pursuit sequence: Varying controller gains . . . . . . . . . . 35
2.5.2 Computation of controller gains for a rendezvous point . . . . . . 38
2.5.3 Pursuit sequence invariance properties . . . . . . . . . . . . . . . 40
2.5.4 Finite and infinite switching of pursuit sequences . . . . . . . . . 41
ix
CONTENTS x
2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3 Rendezvous using linear Centroidal Cyclic Pursuit 47
3.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3 Rendezvous and Reachable point . . . . . . . . . . . . . . . . . . . . . . 56
3.4 Invariance properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.5 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.5.1 Fixed pursuit sequence: Varying controller gains . . . . . . . . . . 63
3.5.2 Pursuit sequence invariance properties . . . . . . . . . . . . . . . 64
3.5.3 Computation of controller gains for a rendezvous point . . . . . . 67
3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4 Rendezvous using linear Generalized Centroidal Cyclic Pursuit 69
4.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3 Rendezvous and Reachable point . . . . . . . . . . . . . . . . . . . . . . 76
4.4 Invariance properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.5 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.5.1 Fixed pursuit sequence: Varying controller gains . . . . . . . . . . 85
4.5.2 Computation of controller gains for a rendezvous point . . . . . . 87
4.5.3 Pursuit sequence invariance properties . . . . . . . . . . . . . . . 88
4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5 Directed motion using linear cyclic pursuit 93
5.1 Directed motion using basic cyclic pursuit (BCP) . . . . . . . . . . . . . 93
5.2 Invariance properties under basic cyclic pursuit (BCP) . . . . . . . . . . 101
5.3 Directed motion using generalized centroidal cyclic pursuit (GCCP) . . . 102
5.4 Directed motion: An alternate approach . . . . . . . . . . . . . . . . . . 105
5.5 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.5.1 Directed motion under BCP . . . . . . . . . . . . . . . . . . . . . 107
CONTENTS xi
5.5.2 Pursuit sequence invariance . . . . . . . . . . . . . . . . . . . . . 110
5.5.3 Directed motion under GCCP . . . . . . . . . . . . . . . . . . . . 111
5.5.4 Alternate approach of directed motion . . . . . . . . . . . . . . . 113
5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6 Circular motion using nonlinear Basic Cyclic Pursuit 117
6.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.2 Analysis for possible formations . . . . . . . . . . . . . . . . . . . . . . . 119
6.3 Special case: Homogeneous system . . . . . . . . . . . . . . . . . . . . . 127
6.4 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7 Realistic cyclic pursuit 137
7.1 Autonomous vehicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.1.1 Linear cyclic pursuit with limitations on the maximum speeds ofthe agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.1.2 Linear cyclic pursuit with fixed turn rate . . . . . . . . . . . . . . 143
7.1.3 Linear cyclic pursuit with fixed turn rate and limitation on themaximum speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.1.4 Linear cyclic pursuit with fixed turn rate and limitation on themaximum speed for unstable gains . . . . . . . . . . . . . . . . . 145
7.1.5 Nonlinear cyclic pursuit with limitations on the maximum lateralacceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.2 Schooling of fishes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
8 Conclusions 151
References 155
CONTENTS xii
List of Tables
2.1 The coefficients of ρ(s) aipqr for different pursuit sequences . . . . . . . . 33
2.2 Initial positions of the agents and their gains for different cases of BCP . 35
3.1 Initial positions of the agents and their gains for different cases of CCP . 63
4.1 Initial positions of the agents and their gains for different cases of GCCP 84
5.1 Initial positions of the agents and their gains for different cases of directedmotion under BCP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.2 Eigenvalues of A for Cases I-IV . . . . . . . . . . . . . . . . . . . . . . . 108
5.3 Initial positions of the agents and their gains for different cases of directedmotion using GCCP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.4 Eigenvalues of A for Cases VI-VII . . . . . . . . . . . . . . . . . . . . . . 113
6.1 Velocities and gains of the agents for different cases of nonlinear BCP . . 128
6.2 The range of Ri . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
xiii
LIST OF TABLES xiv
List of Figures
1.1 Generalization of cyclic pursuit . . . . . . . . . . . . . . . . . . . . . . . 8
2.1 Gershgorin Discs of A when all the gains are positive . . . . . . . . . . . 15
2.2 Condition (b) of Theorem 2.2 as a function of the gain ki . . . . . . . . . 16
2.3 Gershgorin Discs of A when only one gains is negative . . . . . . . . . . . 16
2.4 Gershgorin Discs of A when α changes from 0 to 1 . . . . . . . . . . . . . 17
2.5 The convex hull and cone for a given initial position of some agents inR2 (d = 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.6 The reachable set (gray shaded region) for a group of agents in d = 2 . . 27
2.7 Bounds on xi(t) along d1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.8 Bounds on xi(t) along d2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.9 Trajectories of a team of 5 agents with all positive gains (Case I) . . . . . 36
2.10 Trajectories of a team of 5 agents with one gain zero and all other gainspositive such that Theorem 2.2 is satisfied (Case II) . . . . . . . . . . . . 37
2.11 Trajectories of a team of 5 agents with the gain of first agent is negativeand other gains positive such that the gains satisfy Theorem 2.2 (Case III) 37
2.12 Trajectories of a team of 5 agents with the gain of first agent is negativeand other gains positive, such that Theorem 2.2 is not satisfied and theagents do not converge to a point (Case IV). . . . . . . . . . . . . . . . . 38
2.13 Trajectories of a team of 5 agents with two negative gains. Theorem 2.2is not satisfied and the agents do not converge to a point (Case V). . . . 39
2.14 Trajectories of the agents converging to a desired points Zf = (0, 0) ∈Co(Z0) (Case VI) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.15 Trajectories of the agents converging to desired points Zf = (20,−5) /∈Co(Z0) (Case VII) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
xv
LIST OF FIGURES xvi
2.16 Trajectories of the agents, with all positive gains, for different pursuitsequences (Case VIII) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.17 Trajectories of the agents, with the gain of first agent negative, for differentpursuit sequences (Case IX) . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.18 Trajectories of the agents, with all positive gains, when the pursuit se-quence switches as BPS1 → BPS2 → BPS3 (Case X) . . . . . . . . . . . 43
2.19 Trajectories of the agents, with the gain of the first agent negative, whenthe pursuit sequence switches as BPS1 → BPS2 → BPS3 (Case XI) . . . 43
2.20 Trajectories of the agents, with all positive gains, when the pursuit se-quence switches infinitely at regular time intervals as BPS1 → BPS2 →BPS3 → BPS1 → BPS2 → . . . (Case XII) . . . . . . . . . . . . . . . . . 44
3.1 Centroidal cyclic pursuit . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2 Trajectories of a swarm of 12 agents when all gains positive (Case I) . . . 64
3.3 Trajectories of a swarm of 12 agents when the gain of one of the agent isnegative while the others are positive (Case II) . . . . . . . . . . . . . . . 64
3.4 Invariance property of the reachable point, Zf = (−16.47, 2.52) for a givenbasic pursuit sequence and different weights (Case III) . . . . . . . . . . 65
3.5 Invariance property of the reachable point, Zf = (−16.47, 2.52) for a givenweight and different basic pursuit sequences (Case IV) . . . . . . . . . . 66
3.6 Invariance property of the reachable point, Zf = (−16.47, 2.52) withswitching of pursuit sequence from (BPS1, w1) → (BPS1, w2) → (BPS2,w1) (Case V) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.7 Invariance property of the reachable point, Zf = (−16.47, 2.52) with infi-nite switching from (BPS1, w1) → (BPS1, w2) → (BPS2, w1) → (BPS1,w1) → (BPS1, w2) → . . . (Case VI) . . . . . . . . . . . . . . . . . . . . . 67
3.8 Trajectories of 5 agents using centroidal cyclic pursuit (CCP) convergingat Zf = (0, 0) (Case VII) . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.1 Trajectories of a swarm of 12 agents when the gains of all the agents arepositive (Case I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.2 Trajectories of a swarm of 12 agents when the gain of one of the agent isnegative and the other gains are positive such that Theorem 4.2 is satisfied(Case II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.3 Trajectories of the agents converging to Zf = (60, 60, 60) (Case III) . . . 87
4.4 Trajectories of the agents converging to Zf /∈ Co(Z0) (Case IV) . . . . . 88
LIST OF FIGURES xvii
4.5 Trajectories of the agents under centroidal cyclic pursuit (CCP) and gen-eralized centroidal cyclic (GCCP) (satisfying some properties) demon-strating the pursuit sequence invariance of the rendezvous point Zf =(64.3, 41.3, 58.7) (Case V) . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.6 Trajectories of the agents under CCP and GCCP demonstrating that therendezvous point is not pursuit sequence invariance (Case VI) . . . . . . 90
5.1 The trajectories of 5 agents when the gains of the agents satisfies Condition(i) of Theorem 5.2 (Case I) . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.2 The trajectories of 5 agents when the gains of the agents satisfies Condi-tion(ii) of Theorem 5.2 (Case II) . . . . . . . . . . . . . . . . . . . . . . . 109
5.3 The trajectories of 5 agents when the gains of the agents satisfies Condi-tion(iii) of Theorem 5.2 (Case III) . . . . . . . . . . . . . . . . . . . . . . 109
5.4 The trajectories of 5 agents when the gains of the agents satisfies none ofthe conditions of Theorem 5.2 (Case IV) . . . . . . . . . . . . . . . . . . 110
5.5 Simulation to demonstrate pursuit sequence invariance of the asymptotepoint (Case V) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.6 Simulation to demonstrate finite pursuit sequence switching invariance ofthe asymptote point (Case VI) . . . . . . . . . . . . . . . . . . . . . . . . 112
5.7 Trajectories of a swarm of agents when one gain is negative and Theorem5.5 is satisfied (Case VII) . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.8 Trajectories of a swarm of agents when three gain are negative and The-orem 5.5 is not satisfied (Case IX) . . . . . . . . . . . . . . . . . . . . . . 114
5.9 Directed motion with combination of stable and unstable gains . . . . . . 115
6.1 Basic formation geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.2 Representation of the angles with respect to a fixed reference . . . . . . . 119
6.3 Multi-vehicle formation with circular trajectory . . . . . . . . . . . . . . 120
6.4 Angle calculation for a general polygon of n sides . . . . . . . . . . . . . 122
6.5 Representation of the range of φ in polar coordinate . . . . . . . . . . . . 124
6.6 A representation of the ranges of φ and ρ for different agents . . . . . . . 124
6.7 Range of ρi for Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.8 Trajectories of n = 5 agents for Case I ( • - initial position, N - finalposition of the UAVs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.9 Range of ρi for Case II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
LIST OF FIGURES xviii
6.10 The roots of (6.19) for Cases II, III, and IV . . . . . . . . . . . . . . . . 131
6.11 Trajectories of n = 5 agents for Case II (• - initial position, N - finalposition of the UAVs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.12 Range of ρi for Case III . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.13 Trajectories of n = 5 agents for Case III (• - initial position, N - finalposition of the UAVs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.14 Range of ρi for Case IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.15 Trajectories of n = 5 agents for Case IV (stable equilibrium) (• - initialposition, N - final position of the UAVs) . . . . . . . . . . . . . . . . . . 134
6.16 Trajectories of n = 5 agents showing unstable equilibrium (a) initial equi-librium configuration, (b) intermediate configuration, (c) final stable con-figuration corresponding to q = 2 . . . . . . . . . . . . . . . . . . . . . . 134
6.17 Roots of (6.19) for Case V . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.18 Trajectories of n = 5 agents for case V for different initial conditions (• -initial position, N - final position of the UAVs) . . . . . . . . . . . . . . . 135
7.1 Trajectories of the agents with speed saturation . . . . . . . . . . . . . . 141
7.2 Trajectories of the agents with appropriate selection of gains such thatthe speed do not saturate and rendezvous occur at Zf = (0, 0) . . . . . . 143
7.3 Trajectories of the agents with fixed turn rate . . . . . . . . . . . . . . . 144
7.4 Rendezvous of the agents with fixed turn rate and speed saturation . . . 145
7.5 Directed motion of the agents with fixed turn rate and speed saturation . 146
7.6 Circular motion of the agents with latax saturation . . . . . . . . . . . . 147
7.7 Schooling of fishes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.8 Simulated schooling of fishes . . . . . . . . . . . . . . . . . . . . . . . . . 148
Notation and Abbreviations
A − Matrix representing the equation of motion of the agents.
ai − Lateral acceleration of the non-holonomic agent i.
Co(Z0) − Convex hull of a Z0.
Cp − Finitely generated cone
d − Dimension of the space.
ki − Controller gain of agent i.
ki − Lower bound on ki.
K − Gain matrix.
n − Number of agents.
P(Z0) − Reachable set.
Rn − n-dimensional space.
Rp − pth eigenvalue of A.
ui − Control of the holonomic agent i.
vi − Unit velocity vector of the holonomic agent i.
vmi − Maximum speed of holonomic agent i.
Vi − Constant speed of non-holonomic agent i.
w − Set of weights for computing the centroid in CCP and GCCP.
xi(t) − Position of the agent i at time t along a given axis.
Zi(t) − Position of the agent i at time t in d-space.
Z0 − Set of initial positions of the agents.
Zf − Rendezvous/reachable point.
xix
Notation and Abbreviations xx
Zic − The weighted centroid that agent i follows.
Zf (Z0) − Reachable set.
Γ − Set of all possible weights.
Γ − Subset of Γ.
ξij − Elements of Adj(χ)
χ − Pursuit sequence matrix.
ηi − Weight for calculating the centroid in CCP and GCCP.
φ − Deviation between the LOS and the non-holonomic agent orienntation.
ρ − Radius of the circle traversed by non-holonomic agent i (Chaper 6).
ρ(s) − Characteristic equation of A
θγδ − Direction of motion of the holonomic agent i in (γ, δ)−plane.
Adj(A) − Adjoint of A.
Rank(A) − Rank of A.
trace(A) − Trace of A.
BCP − Basic cyclic pursuit.
BPS − Basic pursuit sequence.
BPS − Set of all possible basic pursuit sequences.
CCP − Centroidal cyclic pursuit.
GCCP − Generalized centroidal cyclic pursuit.
PS − Pursuit sequence.
LOS − Line of sight.
Chapter 1
Introduction
Multi-agent systems are groups of intelligent agents, interacting with each other and
the environment, to accomplish certain tasks that are difficult to achieve by a single
agent. Multi-agent systems are characterized by limited information gathering and pro-
cessing capability of each agent, decentralized control, and asynchronous computation.
Research on multi-agent systems started in the late 1980s as a subfield of Distributed
Artificial Intelligence. Presently, multi-agent system theory is applied widely from au-
tonomous vehicles to e-commerce. The advantages of multi-agent systems are well-
established in the literature. Firstly, multi-agent systems are robust. Since each agent
is autonomous, multi-agent systems degrade gracefully with agent failure. They are
also scalable. Another advantage, mainly from the robotics research point of view, is the
“performance/cost ratio”. A single robot is much costlier than many simple robots. This
is also true in the case of sensor networks. However, multi-agent systems also introduce
many new challenges like the coordination and communication that should exist between
the agents to perform a task. The issues of information exchange and design of control
strategies for coordination of agents are current topics of research interest.
1
Chapter 1. Introduction 2
1.1 Multi-agent consensus
A major part of the researches on multi-agent systems, with reference to autonomous
agents, are related to the consensus problem. Consensus, as defined in [1], implies
reaching an agreement regarding a certain quantity of interest that depends on the
states of all the agents. The quantity of interest can be position, direction of motion, the
relative distance between the agents, or some other functions of the states. Consensus of
multi-agent systems can be found in nature, e.g., flock of flying birds, schools of fishes or
herd of land animals. Reynolds [2] and Viscek et al. [3] are pioneers in modeling these
types of behaviours, which were later used in achieving consensus of groups of agents
like robots, UAVs, or satellites.
To reach a consensus, each agent in a multi-agent system should have information
about certain state(s) of all or some of the agents. The flow of information defines
the interaction topology in multi-agent systems, which can be represented by a directed
graph G = (V,E) where V = {1, 2, · · · , n} represents the agents and E ⊆ V × V defines
the connection between the agents. Thus, the neighbors of agent i are Ni = {j ∈ V :
(i, j) ∈ E}. In continuous time consensus protocol, if xi(t) denotes the information state
of the ith agent at time t, then the dynamics of agent i is given as
xi(t) =∑j∈Ni
aij(t){xj(t)− xi(t)} (1.1)
where, aij(t) are the time varying weighting factors. For the group of n agents, the
dynamics can be written as x = −Lx where L = [lij], the graph Laplacian, is given as
lij =
−1, j ∈ Ni, i 6= j;
|Ni|, i = j.(1.2)
where, |Ni| is the cardinality of Ni. Consensus is achieved when xi = xf , ∀i, or, in other
words, ||xi − xj|| → 0,∀j 6= i as t →∞.
Chapter 1. Introduction 3
The consensus problems are solved using concepts from algebraic graph theory [4]
and matrix theory [5]. Convergence results are obtained from spectral analysis of the
graph Laplacian. It is shown in [6], [7] that for a time invariant information exchange
topology, the consensus is reached if, and only if, the topology has a spanning tree. The
Fiedler eigenvalue [8] of G gives a measure of the rate of convergence of the consensus
protocols [9].
The consensus protocol becomes more practical and challenging under dynamic in-
formation exchange topology. Jadbabaie et al. [10] show that consensus can be reached
in a switching network if the union of the information exchange graphs is connected most
of the time. This result is further extended in [6], [11], [12]. The other aspects studied in
this problem are reaching consensus under communication delays [9] and under relative
information uncertainties [13]. A comprehensive study of the consensus and cooperative
control of multi-agent system can be found in [1], [14].
The consensus algorithms are used in several applications as described below:
Flocking : Flocking of a group of mobile agents are obtained by aligning the velocities of
all the agents while maintaining a certain distance between them and avoiding collision
with each other and with obstacles. This implies a consensus in the velocity of all the
agents. Jadbabaie et al. [10] proved the convergence results for a group of agents using
distributed control laws, where the information topology changes with time. In [15],
different flocking algorithms that are scalable and has obstacle avoidance capabilities are
proposed. Other flocking algorithms and their stability analysis are addressed in [16],
[17], [18], [19], [20], [21].
Formation control : In formation control problems, the relative positions of the agents are
maintained. It is shown in [1] that the distributed formation control can be considered
as a consensus problem. The different approaches to formation control can be broadly
classified as leader-follower, behaviour-based, and rigid body type formations. These
strategies have been reviewed in [22]. The stability of the formation for leader-follower
and virtual leader strategy has been studied in [23] and [24], respectively. In [25], the
Chapter 1. Introduction 4
effect of communication topology on the stability of agent formation is studied using
the Nyquist criterion. Tabuada et al. [26] obtained the feasibility of motion of a rigid
formation given the kinematics and inter-agent constraints. A lower dimensional control
system is also obtained for a formation to move on a given feasible trajectory.
Rendezvous : Rendezvous of a group of mobile agents implies reaching a consensus in
position of all the agents. Lin et al. [27], [28] considered the agents to move towards
rendezvous through a “stop-and-go” strategy, which can be synchronous or asynchronous.
In [29], [30], [31], the concept of rendezvous of agents include the notion that all the agents
should reach the rendezvous point at the same time. A robust algorithm for rendezvous
of a group of agents under switching communication topology and communication failure
is studied in [32].
Distributed sensor fusion: In a sensor network, the measurements taken by each node
are often corrupted with noise. In [33], [34], distributed Kalman filters are designed that
allow each node of the sensor network to track the average of all sensor measurements.
This is called the consensus filter. The stability properties of this filter is studied in
[34]. In [35], data fusion in the presence of package loss in the communication channel
is discussed.
Coupled oscillators : Synchronization of the frequency of coupled nonlinear oscillators
can be considered as a nonlinear extension of a consensus problem and is analyzed by
linearizing about the equilibrium point. The classic Kuramoto model of coupled non-
linear oscillators assumes identical oscillators and all-to-all connection. In [36], [37], the
stability of the coupled oscillator is studied when the natural frequency of the oscillators
are different and the interconnection between the agents are not all-to-all. The sufficient
conditions for synchronization and desynchronization of the nonlinear coupled oscillator,
in terms of the eigenvalues of the graph Laplacian, is obtained in [38]. Papachristodoulou
et al. [39] studied the synchronization problem under variable time delays.
Chapter 1. Introduction 5
1.2 Cyclic pursuit as consensus protocol
Cyclic pursuit of a group of agents implies that there exists a predefined cyclic connection
between the agents and each agent follows its predecessor, called its leader. The problem
of cyclic pursuit originated from a mathematical study of the path traveled by 3 dogs,
placed at the three vertices of an equilateral triangle, chasing one another in a cyclic order
along the instantaneous line of sight, with constant speed. Edouard Lucas posed this
problem in 1877 [Nouvelles Correspondance Mathematique 3 (1877)] and, in 1880, Henri
Brocard showed that the dogs follow a logarithmic spiral [Nouvelles Correspondance
Mathematique 6 (1880)]. Several researchers [40], [41], [42] generalized this problem
where n bugs are considered and studied the conditions for mutual capture of the bugs.
Mutual capture implies that the bugs reached a consensus in position. Other problems
looked at in the pursuit literature includes the study of evolution of the path traveled by
a trail of ants from one point to another [43], stability of the regular geometries of cyclic
pursuit [44], forward-time (when an agent moves towards its leader) and reverse-time
(when an agent moves away from its leader) cyclic pursuit [45], cyclic pursuit games [46]
where the evader and pursuer moves on a cyclic graph.
Bruckstein et al. [47] studied the evolution of the cyclic pursuits for ants, crickets and
frogs. Ants represent the continuous time cyclic pursuit with varying speeds, while crick-
ets and frogs represent discrete time cyclic pursuit with constant speeds. The possible
outcomes of these pursuits − collision, limit cycle, equilibrium states, periodic motion −are studied. In another paper, Bruckstein et al. [48] studied the linear and cyclic pursuit
on grids, where the ants are allowed to move from one grid point to another.
The application of cyclic pursuit to multi-agent systems was demonstrated by Mar-
shall et al. [49] where two types of agents are considered − holonomic agents that do
not have any motion constraints and nonholonomic agents, like the wheeled robots, that
have turn rate constraints. Cyclic pursuit for holonomic agents give rise to linear cyclic
pursuit and for non-holonomic agents, to nonlinear cyclic pursuit.
Chapter 1. Introduction 6
For linear cyclic pursuit, it is assumed that each agent i knows the position xi+1(t)
of its leader and the pursuit law is given as
xi(t) = κ(xi+1(t)− xi(t)) (1.3)
where, κ is the gain of all the agents. Consensus is reached when all the agents converge
to a point. Marshall et al. [49], [50] proved that for every initial condition, the centroid
of the agents remains constant and the agents exponentially converge to the centroid.
Nonlinear cyclic pursuit is also studied in [49], where each agent is homogenous,
that is, the agents have same speed and gain, and each agent knows the position and
orientation of its leader. Equilibrium is reached when the agents form a stable polygon
in space. This can be considered as a formation control problem. The stability of
the formation is obtained by linearizing the system about the equilibrium point and
evaluating the eigenvalues of the linearized system. In [50], the speeds of the agents
are assumed to be proportional to the distance between an agent and its leader and the
limits on the constant of proportionality (or the gains) are found for stable formation.
These results are experimentally verified in [51].
Lin et al. [52] used linear cyclic pursuit laws to obtain different formations of the
agents, like line formation or triangle formation and derived the conditions for collision
avoidance during rendezvous. They also studied the rendezvous of the agents under
limited field of view. In [7], the feasibility of obtaining different formations of the agents
having motion constraints are discussed. Smith et al. [53] used hierarchical cyclic pursuit
and compared the rate of convergence with the traditional cyclic pursuit scheme. Linear
cyclic pursuit concept was also applied to Euclidean curve shortening [54].
Chapter 1. Introduction 7
1.3 Generalization of cyclic pursuit
Certain generalizations of the basic cyclic pursuit, as described in the literature surveyed
in Section 1.2, is of interest and forms the subject matter of this thesis. In cyclic pursuit, a
group of n agents, ordered from 1 to n, are considered. A cyclic connection exists between
the agents with each agent following its predecessor. The sequence in which each agent
pursues another is called the Pursuit Sequence (PS) of the agents. The basic pursuit
sequence is BPS= {1, 2, . . . , n} which implies that the agents are following each other in
the sequence 1 → 2 → · · · → n → 1. Assume a pursuit sequence BPS={p1, p2, . . . , pn}where, pi ∈ {1, 2, . . . , n}, ∀i and pi 6= pj,∀i, j. This is a generalization in terms of
the pursuit sequence. An agent pi, instead of following the agent pi+1, can follow a
point which is the weighted centroid of the remaining n− 1 agents. Let, the weights be
w = [η1, η2, . . . , ηn−1] where an agent pi associates the weight ηj with the agent pi+j (mod
n) while calculating the centroid. The weight w can be same or different for different
agents. Then, the pursuit sequence of a group of agents are given by (BCP, {wi}ni=1).
With this, the following definitions follow:
Definition 1.1 (Basic cyclic pursuit) If a group of agents follow one another in a
cyclic order, they execute a basic cyclic pursuit (BCP).
For basic cyclic pursuit, the elements of w or the weights ηi, are only 0 and 1, such that
the cyclic structure is preserved. Note that any arbitrary distribution of 0 or 1 may not
preserve the cyclic structure.
Definition 1.2 (Centroidal cyclic pursuit) In a group of agents, if each agent fol-
lows a point that is the weighted centroid of the other agents and the weights used by each
of the agents are the same, then the agents are said to execute centroidal cyclic pursuit
(CCP).
Thus, the weight w is same for all the agents and the pursuit sequence is PS = (BPS, w).
Chapter 1. Introduction 8
Z2
Z1
Z4Z3
(a) Basic cyclic pursuit
x
xx
xZ2
Z1
Z4Z3
(b) Generalized centroidal cyclic pursuit
Figure 1.1: Generalization of cyclic pursuit
Definition 1.3 (Generalized centroidal cyclic pursuit) In a group of agents, if each
agent follows a point that is the weighted centroid of the other agents and the weights
used by different agents are different, then the agents execute generalized centroidal cyclic
pursuit (GCCP).
Here, we have a set of n weights that each of the agents follow and hence the pursuit
sequence is PS = (BPS, {w1, . . . , wn})
The basic cyclic pursuit and generalized centroidal cyclic pursuit is illustrated in
Figure 1.1. These generalized cyclic pursuit laws are studied in this thesis.
Another generalization that we consider in this thesis is the concept of using hetero-
geneous agents. A heterogeneous group of agents will have different speeds and controller
gains. In reality, a group of agents cannot be identical in all respects. Thus, it is log-
ical to study heterogeneous systems. Moreover, heterogeneity gives more flexibility in
controlling the behaviour of the agents.
1.4 Contributions and organization of the thesis
(i) Generalization of the concept of cyclic pursuit − BCP, CCP and GCCP
(ii) Analysis of cyclic pursuit laws for heterogenous agents.
Chapter 1. Introduction 9
(iii) Conditions for stability of different generalized linear cyclic pursuit laws.
(iv) Characterizing stable behaviour under generalized linear cyclic pursuit laws.
(v) Characterizing unstable behaviour under generalized linear cyclic pursuit laws.
(vi) Invariance properties of generalized linear cyclic pursuit laws.
(vii) Equilibrium formation of heterogenous agents under nonlinear basic cyclic pursuit.
(viii) Behaviour of cyclic pursuit laws under realistic constraints.
This thesis is organized according to the sequence of generalization of the cyclic
pursuit strategies. The analysis is carried out for heterogeneous agents, that is, agents
with different gains and speeds. The heterogeneity of the agents are utilized to obtain
different behaviours of the agents. Initially, the holonomic agents are studied, followed
by the study of non-holonomic agents.
Holonomic agents under cyclic pursuit strategies give rise to a linear system of state
equations. In Chapter 2, linear basic cyclic pursuit is analyzed for a group of heteroge-
neous agents. The agents converge to a point when the linear system is stable. The point
of convergence, called the reachable point or the rendezvous point, can be controlled by
the controller gains of the agents. Thus, we show that, with heterogeneous agents, the
rendezvous point can occur at any desired point. The stability and the rendezvous point
also exhibit some invariance properties with respect to the pursuit sequence of the agents,
which allow changing the connection between the agents while executing the same goal.
In Chapter 3, we formulate and analyze linear centroidal cyclic pursuit. The agents
under this strategy follow a group of other agents instead of only one of them. The
behaviour of the agents are similar to basic cyclic pursuit. A stable system results in
rendezvous of the agents, where the rendezvous points are functions of the controller
gains. The invariance of stability and rendezvous point for centroidal cyclic pursuit are
addressed and compared with basic cyclic pursuit.
Chapter 1. Introduction 10
The behaviour of a stable linear system under generalized centroid cyclic pursuit is
studied in Chapter 4. Generalized cyclic pursuit gives the flexibility that each agent can
select independently the group of agents it will follow. In this case, the stability and
rendezvous point depends on the pursuit sequence of the agents. Thus, the invariance
properties of the system do not hold in general except under certain conditions.
In Chapter 5, we shift our attention to the analysis of unstable linear system under
different cyclic pursuit strategies. The instability of the system is utilized to obtain
directed motion of the agents. The direction of motion changes with the pursuit sequence,
but there exists a point, called the asymptote point that remains invariant to pursuit
sequences. All the asymptotes of the directed motion passes through this point. An
alternate approach to obtain directed motion is also proposed.
Chapter 6 focuses on the non-holonomic agents that gives rise to nonlinear cyclic
pursuit. At equilibrium, the agents under basic cyclic pursuit exhibit circular motion
about a point. The radius of the circles are different for heterogeneous agents. The
equilibrium formation and the necessary conditions for equilibrium are studied.
In Chapter 7, cyclic pursuit strategies are applied to coordinate a group of au-
tonomous vehicles and to model the behaviour of the biological organisms like the fish
schools. These applications require imposition of realistic constraints to the basic cyclic
pursuit strategies. The behaviour of the autonomous vehicles like the robots and UAVs
and the schools of fishes under realistic cyclic pursuit are observed through simulation.
Chapter 8 concludes the thesis with a summary of the work done and some discussions
on the future promising directions of research.
Chapter 2
Rendezvous using linear Basic
Cyclic Pursuit
In this chapter, the behavior of a swarm of heterogenous agents in linear cyclic pursuit
is analyzed. The agents follow basic cyclic pursuit (BCP) laws as discussed in Chapter
1. The trajectories of the agents are studied as a function of the controller gains of
the agents. The conditions for rendezvous, under which the agents converge to a point,
called the reachable point, are obtained. The complete set of reachable points, called the
reachable set, is characterized. The possible points at which convergence or rendezvous
can occur are also obtained. Some interesting properties of the reachable point are
discussed.
2.1 Problem formulation
Linear cyclic pursuit between n agents indexed from 1 to n, in a d dimensional space, is
formulated as follows: The position of the agent i at any time t ≥ 0 is given by
Zi(t) = [z1i (t) z2
i (t) . . . zdi (t)]
T ∈ Rd, i = 1, 2, . . . , n. (2.1)
11
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 12
The equation of motion of agent i is
Zi = ui (2.2)
where ui is the control of agent i.
For basic cyclic pursuit, we assume the pursuit sequence (discussed in Section 1.3)
to be BPS=(1, 2, . . . , n). Then, ui is given as
ui = ki
[Zi+1(t)− Zi(t)
](2.3)
where, ki is the gain of the agent i. Let
k = {ki}ni=1 (2.4)
define the set of gains of all the agents. Thus, the equation of motion of the agent i is
given by
Zi(t) = ki
[Zi+1(t)− Zi(t)
](2.5)
From (2.5), it can be seen that, for every agent i, each coordinate zδi , δ = 1, · · · , d,
of Zi, evolves independently in time. Hence, these equations can be decoupled into d
identical linear system of equations and can be represented as
X = AX (2.6)
where
A =
−k1 k1 0 · · · 0
0 −k2 k2 · · · 0...
kn 0 0 · · · −kn
(2.7)
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 13
The characteristic polynomial of A is
ρ(s) =n∏
i=1
(s + ki)−n∏
i=1
ki (2.8)
We can expand (2.8) as
ρ(s) = sn + Bn−1sn−1 + Bn−2s
n−2 . . . + B2s2 + B1s + B0 (2.9)
where the coefficients B0 and B1, which we will need for our analysis later, can be
obtained directly from (2.8), as
B0 = 0 (2.10)
B1 =n∑
i=1
n∏
j=1,j 6=i
kj (2.11)
This shows that there is exactly one eigenvalue of A at the origin, provided not more
than one gain is zero. The stability of the linear system, given in (2.6), is analyzed in
the next section.
2.2 Stability analysis
We prove stability using the Gershgorin’s disc theorem [5] which is stated below:
Theorem 2.1 (Gershgorin’s Theorem) Let A = [aij] ∈ Mn, and let
Ri(A) ≡n∑
j=1,j 6=i
|aij|, 1 ≤ i ≤ n
denote the deleted absolute row sums of A. Then, all the eigenvalues of A are located
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 14
in the union of n discs
n⋃i=1
{z ∈ C : |z − aii| ≤ Ri(A)
}≡ G(A)
Therefore, for a n × n square matrix A, we can draw n circles with centers at the
diagonal elements of A, i.e., aii, i = 1, 2, . . . , n, and with radius equal to the sum of the
absolute values of the other elements in the same row, that is,∑
j 6=i |aij|. Such circles
are called Gershgorin’s discs. All the eigenvalues of A lie in the region formed by the
union of all the n discs.
Theorem 2.2 The linear system, given by (2.6), is stable if and only if the following
conditions hold
(a) At most one ki is negative or zero, that is, at most for one i, ki ≤ 0 and kj > 0,
∀j, j 6= i.
(b)∑n
i=1
(∏nj=1,j 6=i kj
)> 0
Proof. From (2.11), it can be seen that Condition (b) implies B1 > 0. First, we prove
the “if” part of the theorem, that is, if both the Conditions (a) and (b) hold, then the
system is stable. Consider the following cases:
Case 1: All the gains are positive.
Condition (b) is satisfied. For ki > 0,∀i, the Gershgorin’s discs of A are shown in Figure
2.1. It can be seen that A does not have any eigenvalue on the right-hand side of the
s-plane and on the imaginary axis except at the origin. At the origin, there is only
one eigenvalue. Hence, the system is stable (in the sense that the output will remain
bounded).
Case 2: One gain is zero and other gains are positive.
Condition (b) is satisfied. Let, ki = 0 and kj > 0,∀j, j 6= i. Then the Gershgorin’s discs
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 15
-k2-kn
-k1
Complex Plane
jω
σ
Figure 2.1: Gershgorin Discs of A when all the gains are positive
for the A matrix will be similar to Case I (Figure 2.1) and all the roots of A will lie either
on the left hand side of the s-plane or at the origin. From (2.11), B1 6= 0, therefore,
there is only one root of A at the origin. Hence, the system is stable.
Case 3: One gain is negative and other gains are positive.
Let ki < 0 and kj > 0,∀j, j 6= i. Then, for Condition (b) to be satisfied, ki > ki where
ki = −∏n
j=1,j 6=i kj∑nl=1,l 6=i
∏nj=1,j 6=i,l kj
(2.12)
It is to be shown that, given the gains kj > 0,∀j, j 6= i, if ki > ki, then the system is
stable. We prove this by contradiction. Using (2.11), we can rewrite B1, as a function
of ki, as
B1 = ki
(n∑
l=1,l 6=i
n∏
j=1,j 6=i,l
kj
)+
n∏
j=1,j 6=i
kj (2.13)
Let us plot B1 as a function of ki (Figure 2.2). The system should be stable in [ki,∞).
Let β ≤ α ≤ 0 and the system be unstable in [β, α] ⊆ [ki,∞) which implies that there
are some roots of A on the right-hand side of the s-plane. The Gershgorin’s discs of A is
shown in Figure 2.3. Since the root locus is continuous and the roots of A should always
remain within the Gershgorin’s disc, at ki = α, at least two roots of A should be at the
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 16
B1
kiαβ
(0,0)k_
i
Figure 2.2: Condition (b) of Theorem 2.2 as a function of the gain ki
-k2-kn -k1
Complex Plane
jω
σ-ki
Figure 2.3: Gershgorin Discs of A when only one gains is negative
origin (since, one root of A is always at the origin). This requires B1 = 0 in (2.9). But,
from (2.11), B1 6= 0 for ki = α > ki. This leads to a contradiction and hence the system
is stable.
The “only if” part is proved by contradiction. Assume the system is stable but any
one or both the conditions do not hold. We consider the following cases separately.
Case 1: Two or more gains are zero.
When two or more gains are zero and others are either positive or negative, B1 = 0 in
(2.9), which implies that more than one root is at the origin, and hence the system is
unstable.
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 17
-kn -k1
Complex Plane
jω
σ-kj
α
-ki
Figure 2.4: Gershgorin Discs of A when α changes from 0 to 1
Case 2: Two or more gains are negative.
Assume only two gains are negative. More than two negative gains can be proved simi-
larly. Let, ki < 0, kj < 0 and kl ≥ 0, ∀l, l 6= i, j. Consider a matrix
A =
−k1 αk1 0 · · · 0
0 −k2 αk2 · · · 0...
αkn 0 0 · · · −kn
(2.14)
where, α ∈ [0, 1]. The characteristic polynomial of A is
ρ(s) =n∏
i=1
(s + ki)− αn
n∏i=1
ki (2.15)
When α = 0, the eigenvalues of A are −ki,∀i. The corresponding Gershgorin’s discs are
points (circles of zero radius) at (−ki, 0), i = 1, 2, · · · , n. When α = 1, A = A, and the
characteristic polynomial is ρ(s). The Gershgorin’s discs, as α varies from 0 to 1, are
shown in Figure 2.4.
Now, as α goes from 0 to 1, by continuity of the root locus, the root locus starting
from (−ki, 0) and (−kj, 0) remain within the discs centered at (−ki, 0) and (−kj, 0) with
radius α|ki| and α|kj|. At α = 1, these roots are either still on the right hand side or
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 18
are both at the origin or one at the origin and the other at the right hand side of the s
plane. All these yield an unstable system.
Case 3 :∑n
i=1
(∏nj=1,j 6=i kj
)≤ 0
This case implies B1 ≤ 0. If B1 = 0, then two roots are at the origin and if B1 < 0,
then one root is at the origin and at least one root on the right hand side of the s plane.
Hence, the system cannot be stable. ¤
Therefore, when the system is stable, there is one and only one eigenvalue of A at
the origin. The solution of (2.6), in the frequency domain, is
X(s) = (sI − A)−1X(t0) (2.16)
Expanding the ith component of X(s)
xi(s) =1
ρ(s)
n∑q=1
biq(s)xq(t0) , i = 1, . . . , n (2.17)
where, ρ(s) is the characteristic polynomial of A and biq(s),∀i, are functions of k and can
be expressed as
biq(s) =
∏nl=1,l 6=q (s + kl), q = i;
∏q−il=1 kl
∏nl=q−i+1,l 6=q (s + kl), q > i;
∏n−q+i−1l=1,i6=q kl
∏nl=n−q+i (s + kl), q < i.
(2.18)
Let the non-zero eigenvalues of A be Rp = (σp + jωp), p = 1, · · · , n − 1, where n is the
number of distinct eigenvalues of A, with the pth eigenvalue having algebraic multiplicity
of np. Let Sr = {Rp|ωp = 0} be the set of real eigenvalues and Si = {Rp|ωp 6= 0} be the
set of complex conjugate eigenvalues. Then, taking inverse Laplace transform of (2.17),
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 19
we get
xi(t) = xf +∑p∈Sr
{n∑
q=1
(np∑
r=1
aipqrt
r−1
)xq(t0)
}eσpt
+∑p∈Si
[n∑
q=1
{np∑
r=1
aipqrt
r−1 cos(ωpt) + ai∗pqrt
r−1 sin(ωpt)
}xq(t0)
]eσpt (2.19)
where, for the pth eigenvalue, Rp
aipqr =
1
r!
dr
dsr
[{s− (σp + jωp)
}np biq(s)
ρ(s)
] ∣∣∣∣s=Rp
(2.20)
and ai∗pqr is the complex conjugate of ai
pqr, and when Rp = 0
xf =
∑nq=1(1/kq)xq∑n
q=1(1/kq)(2.21)
When the system is stable, i.e., σp < 0,∀p, as t → ∞, xi(t) = xf ,∀i. This implies that
all the the agents will converge to the point xf . In the next section, the rendezvous of
the agents is analyzed.
2.3 Rendezvous and Reachable point
For a stable system, the agents will converge to a point. We analyze the point of con-
vergence in the following theorem.
Theorem 2.3 (Reachable Point) If a system of n-agents, with equation of motion
given in (2.5), have their initial positions at Z0 = {Zi(t0)}ni=1, and gains k that satisfies
Theorem 2.2, then they converge to a point Zf given by,
Zf =n∑
i=1
{(1/ki∑n
j=1 1/kj
)Zi(t0)
}=
∑ni=1 Zi(t0)/ki∑n
i=1 1/ki
(2.22)
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 20
where Zf is called a reachable point or the rendezvous point of this system of n agents.
Proof. Summing (2.5) for all n, in the sense of mod n, we get
n∑i=1
Zi(t)
ki
=n∑
i=1
(Zi+1(t)− Zi(t)) = 0 (2.23)
⇒n∑
i=1
Zi(t)
ki
= constant (2.24)
for all time, t. Then, considering the initial position Zi(t0) and final position Zi(tf ) of
the agent i, we can write
n∑i=1
Zi(t0)
ki
=n∑
i=1
Zi(tf )
ki
(2.25)
When the system is stable, all the agents converge to a point, that is, Zi(tf ) = Zf , ∀i.Thus
n∑i=1
Zi(t0)
ki
=n∑
i=1
Zf
ki
= Zf
n∑i=1
1
ki
(2.26)
⇒ Zf =
∑ni=1 Zi(t0)/ki∑n
i=1 1/ki
(2.27)
from which we get (2.22) or the rendezvous point. ¤
We can compare (2.19) with (2.22) and observe that xf is same as one of the coordi-
nates of Zf .
Now, let us denote
Zf (Z0, k) =
∑ni=1 Zi(t0)/ki∑n
i=1 1/ki
(2.28)
as the reachable point obtained from the initial positions Z0, and gains k that satisfy
Theorem 2.2. Then, the set of reachable points (called the reachable set), at which
rendezvous can occurs, starting from the initial point Z0, be denoted as Zf (Z0) and
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 21
defined as,
Zf (Z0) =
{Zf (Z
0, k)∣∣∣ ∀k satisfying Theorem 2.2
}(2.29)
Thus for every Z ∈ Zf (Z0), there exists a k satisfying Theorem 2.2, such that Zf (Z
0, k) =
Z. Hence, the point of convergence of the n agents, given their initial positions, can be
controlled by a judicious selection of the gains k.
Next, the region in Rd, where a rendezvous of n agents is possible, is obtained. Let
Co(Z0) be the convex hull of Z0. A finitely generated cone [55] can be defined as,
Definition 2.1 (Finitely generate cone) A cone C is finitely generated by vectors a1, . . . ,
am, if C consists of all the vectors of the form
x = λ1a1 + λ2a2 + . . . + λmam (2.30)
with λ1 = 1, λi ≥ 0 for i = 2, . . . , m. This cone C has a vertex at a1.
With this, we define a cone Cp as follows:
Definition 2.2 A cone Cp is finitely generated by the vectors [Zp(t0) − Zi(t0)], i =
1, · · · , p− 1, p + 1, · · · , n if Cp consist of all vector of the form
Z = Zp(t0) +n∑
i=1,i6=p
λi(Zp(t0)− Zi(t0)) (2.31)
where λi ≥ 0, i = 1, . . . , p− 1, p + 1, . . . , n. Cp has a vertex at Zp(t0).
Theorem 2.4 Consider a system of n agents, with equation of motion given in (2.5)
and initial positions at Z0. A point Z is reachable if and only if,
Z ∈ Co(Z0)⋃ { n⋃
p=1
Cp
}= P(Z0) (say) (2.32)
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 22
that is, Zf (Z0) = P(Z0).
Proof. First, we show that Zf (Z0) ⊆ P(Z0). Let Z ∈ Zf (Z
0). Then, by definition of
Zf (Z0), there exists a k, satisfying Theorem 2.2, such that
Z =
∑ni=1 Zi(t0)/ki∑n
i=1 1/ki
(2.33)
holds. We will show that Z ∈ P(Z0). Consider the following cases.
Case I : Let ki > 0, ∀i. Then, (2.33) can be written as
Z =n∑
i=1
(1/ki)∑nj=1(1/kj)
Zi(t0) (2.34)
Thus, Z is a convex combination of Zi(t0), i = 1, ..., n. Hence, Z ∈ Co(Z0) and so
Z ∈ P(Z0).
Case II : Let one of the gains kp < 0 and the remaining gains ki > 0, ∀i, i 6= p. Then,
from (2.34),
Z =n∑
i=1,i6=p
(1/ki)∑nj=1(1/kj)
Zi(t0) +(1/kp)∑nj=1(1/kj)
Zp(t0) (2.35)
⇒ Z
n∑i=1
1
ki
=n∑
i=1,i6=p
1
ki
Zi(t0) +1
kp
Zp(t0) (2.36)
⇒ Z
n∑i=1
1
ki
− Zp(t0)n∑
i=1
1
ki
=n∑
i=1,i 6=p
1
ki
Zi(t0) +1
kp
Zp(t0)− Zp(t0)n∑
i=1
1
ki
(2.37)
⇒{
Z − Zp(t0)} n∑
i=1
1
ki
=n∑
i=1,i 6=p
1
ki
{Zi(t0)− Zp(t0)
}(2.38)
Since only kp < 0, and ki > 0,∀i, i 6= p, we have
n∏i=1
ki < 0 (2.39)
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 23
From Condition (b) of Theorem 2.2, for stable systems
n∑i=1
( n∏
j=1,j 6=i
kj
)> 0 (2.40)
Dividing the above equation by 2.39, we get
n∑i=1
1
ki
< 0 (2.41)
Let
n∑i=1
1
ki
= −1
c(2.42)
where, c > 0. Then, from (2.38),
−1
c
{Z − Zp(t0)
}=
n∑
i=1,i6=p
1
ki
{Zi(t0)− Zp(t0)
}(2.43)
⇒ Z − Zp(t0) =n∑
i=1,i 6=p
− c
ki
{Zi(t0)− Zp(t0)
}(2.44)
⇒ Z = Zp(t0) +n∑
i=1,i6=p
c
ki
{Zp(t0)− Zi(t0)
}(2.45)
Then, from (2.31), Z ∈ Cp and so Z ∈ P(Z0).
Case III : Let one of the gains kp = 0 and the remaining gains ki > 0,∀i, i 6= p. Then,
(2.33) can be written as,
Z =
∑ni=1
(∏nj=1,j 6=i kj
)Zi(t0)∑n
i=1
∏nj=1,j 6=i kj
(2.46)
Putting kp = 0 in the above equation,
Z =
∏nj=1,j 6=p kjZp(t0)∏n
j=1,j 6=p kj
= Zp(t0) (2.47)
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 24
Thus, Z ∈ Co(Z0) and so Z ∈ P(Z0).
Therefore, from Case I-III above, if Z ∈ Zf (Z0), then Z ∈ P(Z0). Therefore,
Zf (Z0) ⊆ P(Z0).
Now, to show that P(Z0) ⊆ Zf (Z0) it has to be shown that for any point Z ∈ P(Z0),
there exists k such that (2.33) holds.
We denote int{P(Z0)} as the interior of the set P(Z0) and define it as: α ∈int{P(Z0)} if there exists an ε > 0 such that, for all β satisfying de(α, β) < ε, β ∈ P(Z0),
where de(α, β) is the Euclidean distance between α and β. Then, boundary of P(Z0),
denoted by ∂{P(Z0)}, is defined as: α ∈ ∂{P(Z0)} if α does not belong to int{P(Z0)}.
Thus, P(Z0) can be partitioned as
P(Z0) = P1(Z0) ∪ P2(Z
0) ∪ P3(Z0) (2.48)
where,
P1(Z0) = int{P(Z0)} (2.49)
P2(Z0) =
{Zi(t0)
∣∣∣Zi(t0) ∈ ∂{P(Z0)}}
(2.50)
P3(Z0) = ∂{P(Z0)} \ P2(Z
0) (2.51)
Then, P1(Z0) is the interior of P(Z0), P2(Z
0) is the set of vertices of the convex set
P(Z0), and P3(Z0) is the boundary of P(Z0) without the vertices. We will consider
these sets separately.
Case I : Z ∈ P1(Z0). We have the following cases:
Case Ia: Let Z ∈ int{Co(Z0)}. Then, there exists αi, i = 1, . . . , n,∑n
i=1 αi = 1 with αi >
0,∀i such that
n∑i=1
αiZi(t0) = Z (2.52)
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 25
Let
ki =c
αi
, i = 1, 2, . . . , n (2.53)
where, c > 0 is any positive constant. Thus, ki > 0,∀i, and
n∑i=1
1
ki
=1
c(2.54)
Replacing αi by c/ki in (2.52),
Z =n∑
i=1
(c
ki
)Zi(t0) =
n∑i=1
(1/ki
1/c
)Zi(t0) =
n∑i=1
{1/ki∑n
j=1 1/kj
}Zi(t0) (2.55)
The above equation is the same as (2.33) and all the gains satisfy Theorem 2.2. Therefore,
Z ∈ Zf (Z0)
Case Ib: Let Z ∈ int{Cp} for some p. Then, there exist βi > 0, i = 1, 2, . . . , n, such that
Z can be expressed as
Z = Zp(t0) +n∑
i=1,i 6=p
βi
{Zp(t0)− Zi(t0)
}(2.56)
=[ n∑
i=1,i6=p
−βiZi(t0)]
+ (1 + β1 + · · ·+ βp−1 + βp+1 + · · ·+ βn)Zp(t0) (2.57)
Let us define a quantity βp as
βp = −(1 + β1 + · · ·+ βp−1 + βp+1 + · · ·+ βn) (2.58)
and
ki =c
βi
, i = 1, 2, . . . , n (2.59)
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 26
where, c > 0 is any positive constant. Then, ki ≥ 0, i = 1, . . . , n, i 6= p and kp < 0. Also,
n∑i=1
βi = −1 ⇒n∑
i=1
1
ki
= −1
c(2.60)
Since c > 0, using (2.41), it can be seen that the gains satisfy Theorem 2.2. Then, (2.57)
can be written as
Z =n∑
i=1
−βiZi(t0) (2.61)
Replacing βi by c/ki from (2.59), we get
Z =n∑
i=1
(− c
ki
)Zi(t0) =
n∑i=1
{1/ki
−1/c
}Zi(t0) =
n∑i=1
{1/ki∑n
j=1 1/kj
}Zi(t0) (2.62)
This is the same as (2.33). Hence, Z ∈ Zf (Z0).
Case II : Let Z = Zp(t0) ∈ P2(Z0). Then, we can write
Z = Zp(t0) =
∑ni=1
∏nj=1,j 6=i kjZi(t0)∑n
i=1
∏nj=1,j 6=i kj
(2.63)
where, kp = 0 and ki > 0,∀i, i 6= p. This is same as (2.33) and hence Z ∈ Zf (Z0).
Case III : When Z ∈ ∂{P2(Z0)}, Z can be expressed as in (2.52) and (2.56) where some
i, αi = 0 and βi = 0, respectively. This will result in some of the gains ki to be infinite.
If we do not restrict the gains to be finite, then following similar arguments as in Case
Ia and Case Ib, Z ∈ Zf (Z0).
Thus, Cases I, II and III together proves that if Z ∈ P(Z0), then Z ∈ Zf (Z0)
and so P(Z0) ⊆ Zf (Z0). From these we get Zf (Z
0) = P(Z0) and hence all points in
Zf (Z0) = P(Z0) are reachable. ¤
Some examples of Co(Z0) and Cp are shown in Figure 2.5 for a 2-dimensional plane.
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 27
Zi8
1Zi
ZiZiZiZiZiZiZiZi5
ZiZiZiZi7
Zi2Zi6
Zi4 Zi3
0Co(Z )
Zi8
1Zi
ZiZiZiZiZiZiZiZi5
ZiZiZiZi7
Zi2Zp
Zi4 Zi3
Cp
Figure 2.5: The convex hull and cone for a given initial position of some agents inR2 (d = 2)
(a) All agents at the ver-tices of the convex hull
(b) One agent on the edgeof the convex hull
(c) One agent inside of theconvex hull
Figure 2.6: The reachable set (gray shaded region) for a group of agents in d = 2
The set of points Zf (Z0) = P(Z0) forms the reachable set of the system of n-agents
with the given initial positions. In fact, the agents can be made to converge to any
desirable point within this reachable set by suitably selecting the gains. The gains can
be selected as given in (2.53) or (2.59), depending on where the reachable point is located
in P(Z0). It can be seen that these gains are not unique, since none of αi, βi, and c need
be unique. Some examples of P(Z0) are given in Figure 2.6 for d = 2 (that is, a 2-
dimensional case).
In the next section, some interesting properties of the reachable point and the reach-
able set are obtained.
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 28
2.4 Invariance properties
So far, the analysis is done for pursuit sequence BPS= (1, 2, . . . , n). Now, if a different
pursuit sequence is considered, certain properties of the system do not change, i.e., these
properties are pursuit sequence invariant. These are discussed below.
Consider a system of n agents, with gains k and equations of motion given in (2.5).
Assume that the initial positions of the agents are at Z0 and the pursuit sequence is
BPS= (1, 2, . . . , n). Let this system be stable according to Theorem 2.2. Suppose,
the pursuit sequence of the agents is changed, keeping all the other parameters (gains)
same. Then, the following theorem proves that the system will remain stable even with
a different pursuit sequence.
Theorem 2.5 The stability of the linear basic cyclic pursuit is pursuit sequence invari-
ant.
Proof. It can be seen from Theorem 2.2 that the stability of the system depends only
on the gains k and not on the pursuit sequence. Thus, given a set of stable gains, the
system is stable for any pursuit sequence and hence, it is pursuit sequence invariant. ¤
Next, we consider the same stable system with pursuit sequence BPS= (1, 2, . . . , n).
Let, the agents converge at the point Zf . Now, if a different pursuit sequence is consid-
ered, keeping the gains same, then the agents converge to the same point Zf as proved
in the next theorem.
Theorem 2.6 The reachable point, and thus the reachable set, of a linear basic cyclic
pursuit is pursuit sequence invariant.
Proof. Consider (2.22), which gives the expression of the reachable point. This equation
depends only on the initial positions of the agents and their gains and is independent
of the pursuit sequence of the agents. Hence, the reachable point is pursuit sequence
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 29
invariant. Since, the reachable point is pursuit sequence invariance, the reachable set is
also pursuit sequence invariant. ¤
Given n agents, there are (n − 1)! different basic pursuit sequences possible. Let,
BPS ={
BPSi
}(n−1)!
i=1be the set of (n − 1)! pursuit sequences. From Theorem 2.5 and
Theorem 2.6, for all the pursuit sequences, the stability, the reachable point and the
reachable set remain the same. Now, if the pursuit sequence changes en route, while the
agents are moving towards the rendezvous point, it is called switching of pursuit sequence.
If the switching occurs a finite number of times, then it is called finite switching. If
the pursuit sequence switches infinite number of times such that the time between any
two consecutive switches is greater than some constant ε > 0, then it is called infinite
switching. Finite switching can be considered as a special case of infinite switching where
ε will be the smallest interval between any two consecutive switches. In the following
theorems, stability and rendezvous point under finite and infinite switching is discussed.
Theorem 2.7 (Stability with finite switching) The stability of the linear basic cyclic
pursuit is invariant under finite switching of pursuit sequences.
Proof. It is proved in Theorem 2.5 that if a system of n agents are stable for a given
pursuit sequence BPS ∈ BPS, then it is stable for all the pursuit sequences in BPS. Now,
if the pursuit sequences switches, it implies that the switch occur between two stable
systems. Since the number of switches is finite, the system, after the last switch, is stable
and hence, the stability is invariant under finite switching. ¤
Theorem 2.8 (Reachability with finite switching) The reachable point of a linear
basic cyclic pursuit is invariant under finite switching of pursuit sequences.
Proof. Let the switching of pursuit sequences occur at t1, · · · , tm, m < ∞ such that
0 < t1 < · · · < tm < ∞, and the pursuit sequence during tj ≤ t < tj+1 is BPSj ∈ BPS.
The switching invariance property is proved by showing that the reachable point, given
in (2.22), remains the same after a switch. At t = tj, the connection among the agents
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 30
is BPSj and their positions are at Zi(tj),∀i. If there is no further switching, then the
reachable point, from (2.22), is
Zf =
∑ni=1(1/ki)Zi(tj)∑n
i=1 1/ki
(2.64)
Let the next switching occur at tj+1, when the position of agent i is Zi(tj+1). For t ≥ tj+1,
the pursuit sequence is BPSj+1. If there are no more switching of connections, let the
reachable point be Z ′f . Now, from (2.22) and (2.24)
Z ′f
n∑i=1
1
ki
=n∑
i=1
Zi(tj+1)
ki
=n∑
i=1
Zi(tj)
ki
= Zf
n∑i=1
1
ki
(2.65)
This shows that Z ′f = Zf . Hence, when there is one switching of connection, the ren-
dezvous point does not change. This can be extended to a finite number of switchings
to show that the reachable point remains unchanged after the final switch tm. ¤
Theorem 2.9 (Stability with infinite switching) The stability of the linear basic
cyclic pursuit is invariant under infinite switching of pursuit sequences.
Proof. As seen in Section 2.1, the system of n agents can be decoupled along each
direction and can be analyzed separately. Consider (2.19). Let us define,
xmax = max{xf , x1(t0), x2(t0), . . . , xn(t0)} (2.66)
xmin = min{xf , x1(t0), x2(t0), . . . , xn(t0)} (2.67)
Then, I0 = [xmin, xmax] is the closed interval that contains the initial positions of all the
agents. It will be shown that xi(t) ∈ I0,∀i, ∀t. Consider the case when xp(t) ∈ ∂{I0},that is, at the boundary of I0, for some p and some t. The gain kp can be positive, zero
or negative. We consider these cases separately.
Case I : xp(t) ∈ ∂{I0} and kp ≥ 0
From (2.6), xp(t) = kp(xp+1 − xp). Assume xp(t) = xmin. Then, xp+1 ≥ xp, and so,
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 31
xp(t) ≥ 0. Therefore xp(t) ∈ I0. Similarly, when xp(t) = xmax, xp+1 ≤ xp, and thus
xp(t) ≥ 0. Therefore xp(t) ∈ I0
Case II : xp(t) ∈ ∂{I0} and kp < 0
Here, we prove here that, if kp < 0, then xp /∈ ∂{I0}. From (2.21), we can write
xf
n∑i=1
1
ki
=n∑
i=1,i6=p
1
ki
xi(t) +1
kp
xp(t) (2.68)
⇒ xp(t) =
(∑ni=1 1/ki
1/kp
)xf +
n∑
i=1,i6=p
(− 1/ki
1/kp
)xi (2.69)
Let cp =(Pn
i=1 1/ki
1/kp
)and ci =
(− 1/ki
1/kp
). Then,
xp(t) = cpxf +n∑
i=1,i6=p
cixi (2.70)
Since∑n
i=1 1/ki < 0 (from (2.41)), it can be seen that ci > 0,∀i and also∑n
i=1 ci = 1.
Hence, xp(t) is a convex combination of the xf and xi,∀i, i 6= p. Therefore, xp(t) cannot
lie on the boundary of I0, that is, xp(t) /∈ ∂{I0} if kp < 0.
Therefore, xi(t) ∈ I0,∀i, ∀t.
Now, if the pursuit sequence switches, the characteristic polynomial of A, given by
(2.8), does not change. Hence, the eigenvalues of A remain the same. In (2.19), Rp will
be same but aipqr will change if the pursuit sequence changes. Let, for pursuit sequence
BPSj, the coefficients aipqr be represented as a
i,BPSjpqr . Since, there are only (n−1)! possible
pursuit sequences, let us define aimax as
aimax = max
BPS ∈ BPS
{∣∣∣ai,BPSjpqr
∣∣∣,∀p, q, r}
(2.71)
Let
xm = max{|xf |, |x1(t0)|, |x2(t0)|, . . . , |xn(t0)|
}(2.72)
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 32
Then, using (2.19), we can write, for all i
L ≤ xi(t) ≤ U (2.73)
where
L = xf − naimax
(n∑
r=1
tr−1
)xmeσmaxt (2.74)
U = xf + naimax
(n∑
r=1
tr−1
)xmeσmaxt (2.75)
and σmax is the real part of the most positive eigenvalue of A. If we consider a stable
system, then σmax < 0. Therefore, as t → ∞, both the LHS and RHS of (2.73) tends
towards xf and so xi(t) → xf as t → ∞. This implies that the system is stable with
infinite switching. ¤
Theorem 2.10 (Reachability with infinite switching) The reachable point of a lin-
ear basic cyclic pursuit is invariant under infinite switching of pursuit sequences.
Proof. It is shown in Theorem 2.9 that, in the case of infinite switching, as t → ∞,
xi(t) → xf ,∀i, where xf is given in (2.21). Hence, the reachable point xf is invariant
under infinite switching. ¤
Note that the trajectory of the agents may change due to switching but the stability
and the reachable point remain unchanged.
A. Example
Here, we will give a numerical example to illustrate the bounds on xi(t) used in (2.73).
Consider four agents (n = 4) in R2 with initial positions Z0 = {(10,−1), (7, 2), (0, 10),
(−7, 5)} and gains k = {4, 6, 8, 10}. There are (n − 1)! = 6 possible pursuit sequences.
From (2.8), the characteristic polynomial of the system is ρ(s) = s4+28s3+284s2+1232s
and the eigenvalues are 0, −7.0±j√
39, −14.0. The coefficients aipqr of (2.19) for different
pursuit sequences are given in Table 2.1.
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 33
Pursuit Sequences (1,2,3,4) (1,2,4,3) (1,3,2,4) (1,3,4,2) (1,4,2,3) (1,4,2,3)
Constant
x1(t0) 30/77 30/77 30/77 30/77 30/77 30/77
x2(t0) 20/77 20/77 20/77 20/77 20/77 20/77
x3(t0) 15/77 15/77 15/77 15/77 15/77 15/77
x4(t0) 12/77 12/77 12/77 12/77 12/77 12/77
e−7t ×cos(
√39t)
x1(t0) 5/77 5/77 5/77 5/77 5/77 5/77
x2(t0) -2/77 -2/77 -4/77 0 -5/77 0
x3(t0) -3/77 0 -1/77 -1/77 0 -5/77
x4(t0) 0 -3/77 0 -4/77 0 0
e−7t ×sin(
√39t)
x1(t0) 5√
39143
5√
39143
5√
39143
5√
39143
5√
39143
5√
39143
x2(t0) 6√
39143
6√
39143
−4√
39143
−40/3√
39143
−5/3√
39143
−40/3√
39143
x3(t0) −3√
39143
−10√
39143
7√
39143
7√
39143
−10√
39143
5/3√
39143
x4(t0) −8√
39143
−1√
39143
−8√
39143
4/3√
39143
20/3√
39143
20/3√
39143
e−14t
x1(t0) 12/77 12/77 12/77 12/77 12/77 12/77
x2(t0) -6/77 6/77 8/77 -20/77 15/77 -20/77
x3(t0) 6/77 -15/77 -8/77 -8/77 -15/77 20/77
x4(t0) -12/77 9/77 -12/77 16/77 -12/77 -12/77
Table 2.1: The coefficients of ρ(s) aipqr for different pursuit sequences
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 34
0 0.2 0.4 0.6 0.8 1−30
−15
0
15
30
U
L
x3(t)x
2(t)
x1(t)x
4(t)
Figure 2.7: Bounds on xi(t) along d1
Then, from (2.73), the bounds on xi(t),∀i is given as
L = 4.62− 4(40/3)
√39
143
(1 + t + t2 + t3
)10e−7t (2.76)
U = 4.62 + 4(40/3)
√39
143
(1 + t + t2 + t3
)10e−7t (2.77)
for one direction (say, for d1) and
L = 2.86− 4(40/3)
√39
143
(1 + t + t2 + t3
)10e−7t (2.78)
U = 2.86 + 4(40/3)
√39
143
(1 + t + t2 + t3
)10e−7t (2.79)
for the other (d2). The plot of xi(t),∀i, along with the bounds, are shown in Figures 2.7
and 2.8 for the two directions respectively.
In the next section, some simulations are given that verify the results obtained in
this chapter.
2.5 Simulation results
A group of 5 agents are considered in a plane. The initial positions of the agents and
different sets of gains, that are selected to demonstrate the results obtained in this
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 35
0 0.2 0.4 0.6 0.8 1−30
−15
0
15
30
x1(t)x
2(t)
x4(t)
x3(t)
U
L
Figure 2.8: Bounds on xi(t) along d2
Agent Initialpositions
Gains
Case I Case II Case III Case IV Case V Case VI Case VII
1 (10, -1) 4 0 -1 -3 -1 15.35 -0.46
2 (7, 2) 6 6 6 6 6 12.42 1.67
3 (0, 10) 8 8 8 8 -2 10.50 33.33
4 (-7, 5) 10 10 10 10 10 2.59 2.33
5 (4, -8) 12 12 12 12 12 2.68 10.00
Table 2.2: Initial positions of the agents and their gains for different cases of BCP
chapter, are given in Table 2.2.
2.5.1 Fixed pursuit sequence: Varying controller gains
Consider the pursuit sequence as BPS1 = (1, 2, 3, 4, 5).
Case I : We show that when the gains of all the agents are positive, they converge
within Co(Z0). The gains of the agents are shown in Table 2.2. For this set of gains,∑n
i=1
∏j=1,j 6=i kj = 16704 > 0 and therefore Theorem 2.2 is satisfied and the system
is stable. From Theorems 2.3 and 2.4, the agents will converge at Zf given by (2.22).
Satisfying the initial conditions and gains, we get Zf = (4.55, 1.6) ∈ Co(Z0). This is
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 36
−10 −5 0 5 10 15−10
−5
0
5
10
15
1
3
5
2
4Co(Z0)
Figure 2.9: Trajectories of a team of 5 agents with all positive gains (Case I)
verified in the trajectories of the agents shown in Figure 2.9.
Case II : When one gain is zero and others are all positive, we show that the agents
converge to a point. Let, the gains of the agents be as shown in the Table 2.2 Here,∑n
i=1
∏j=1,j 6=i kj = 5760 > 0 and thus the system is stable. The rendezvous point can
be calculated from (2.22) and it is equal to Zf = (10,−1). The trajectories are shown
in Figure 2.10 and we can verify the rendezvous point.
Case III : When the gain of only one agent is negative, but the system satisfies
Theorem 2.2, the agents converge outside the convex hull Co(Z0). The gains selected
in this case is shown in Table 2.2. These gains satisfy the conditions in Theorem 2.2,
since∑n
i=1
∏j=1,j 6=i kj = 3024 > 0. Thus, the system is stable. Also from (2.12),
k1 = −2.86 < k1 . From Theorems 2.3 and 2.4, the rendezvous occurs at Zf given
by (2.22). For the initial conditions and the gains, we get Zf = (17.5,−4.6) ∈ C1.
The simulation result is shown in Figure 2.11. We observe that the rendezvous point
Zf /∈ Co(Z0) and the figure illustrates that Zf ∈ C1.
Case IV : We consider the gain of one of the agent to be negative and the others
to be positive such that the system is not stable according to Theorem 2.2. Then, the
agents should not converge to a point. The gains of the agents are shown in Table 2.2.
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 37
−10 −5 0 5 10 15−10
−5
0
5
10
15
2
5
1
3
4Co(Z0)
Figure 2.10: Trajectories of a team of 5 agents with one gain zero and all other gainspositive such that Theorem 2.2 is satisfied (Case II)
−10 −5 0 5 10 15 20−10
−5
0
5
10
15
3
4
1
5
2
C1
Figure 2.11: Trajectories of a team of 5 agents with the gain of first agent is negativeand other gains positive such that the gains satisfy Theorem 2.2 (Case III)
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 38
−20 0 20 40 60−30
−10
10
20
21
5
3
4
Figure 2.12: Trajectories of a team of 5 agents with the gain of first agent is negativeand other gains positive, such that Theorem 2.2 is not satisfied and the agents do notconverge to a point (Case IV).
Theorem 2.2 is not satisfied in this case, since∑n
i=1
∏j=1,j 6=i kj = −2448 < 0. Also, from
(2.12), k1 = −2.86 > k1. The trajectories of the agents are shown in Figure 2.12 and we
observe that rendezvous does not occur.
Case V : Consider that the gains of more than one agent is negative. Then, the
system will be unstable according to Theorem 2.2. The gains of the agents are shown
in Table 2.2. These gains violate Condition (a) of Theorem 2.2. Figure 2.13 shows the
trajectories of the agents and we observe that the agents do not converge to a point.
2.5.2 Computation of controller gains for a rendezvous point
Pursuit sequence BPS1 = (1, 2, 3, 4, 5) is considered for these cases.
Case VI : Let the desired rendezvous point be Zf = (0, 0). For the given the ini-
tial positions (in Table 2.2), Zf ∈ int{Co(Z0)}. Therefore, the gains of the agents
can be calculated using (2.52) and (2.53). One of the set of αis that satisfy (2.52) is
[0.07, 0.08, 0.095, 0.39, 0.37] . Assuming c = 1, the gains of the agents are given in
Table 2.2. The trajectories of the agents are shown in Figure 2.14(a) and we observe
that the agents converge to the point Zf = (0, 0). If we assumed c = 2, the gains of the
agents will be doubled, but the trajectories remain the same as shown in Figure 2.14(b)
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 39
−40 −20 0 20 40 60−40
−20
0
20
40
12
5
4
3
Figure 2.13: Trajectories of a team of 5 agents with two negative gains. Theorem 2.2 isnot satisfied and the agents do not converge to a point (Case V).
−10 0 10 15−10
0
10
15
3
2
1
4
5
(a) c = 1−10 0 10 15
−10
0
10
15
3
2
1
4
5
(b) c = 2
Figure 2.14: Trajectories of the agents converging to a desired points Zf = (0, 0) ∈Co(Z0) (Case VI)
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 40
−10 0 10 20 25−10
0
10
15
1
2
4
5
3
Figure 2.15: Trajectories of the agents converging to desired points Zf = (20,−5) /∈Co(Z0) (Case VII)
Case VII : Let us consider the desired rendezvous point to be Zf = (20,−5). For
the given initial position of the agents, Zf /∈ Co(Z0). Since Z2(t0) ∈ int{Co(Z0)}, all
points in R2 are reachable. Note that Zf ∈ C1 also. So we can select a negative gain for
the first agent. One of the sets of βis that satisfy (2.56) is [−2.16, 0.6, 0.03, 0.43, 0.1].
Using (2.59), the gains of the agents are calculated with c = 1 and the values are given in
Table 2.2. Figure 2.15 shows that the trajectories of the agents converging at the desired
point Zf = (20,−5).
2.5.3 Pursuit sequence invariance properties
To demonstrate the invariance properties, we consider two different pursuit sequences
BPS1 = (1, 2, 3, 4, 5)
BPS2 = (1, 3, 5, 2, 4)
Case VIII : We demonstrate the invariance of stability and rendezvous point with
respect to the pursuit sequence when all the gains are positive. Consider the gains same
as in Case I. Figure 2.16 show the trajectories of the agents for the two different pursuit
sequences BPS1 and BPS2. We observe, from Theorem 2.2, that the system is stable for
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 41
−10 0 10 15−10
0
10
15
2
1
5
4
3
(a) Pursuit sequence BPS2
−10 −5 0 5 10 15−10
−5
0
5
10
15
1
3
5
2
4
(b) Pursuit sequence BPS1
Figure 2.16: Trajectories of the agents, with all positive gains, for different pursuitsequences (Case VIII)
both the pursuit sequences and the rendezvous point is the same and is given by (2.22).
However, the trajectories of the agents are different.
Case IX : Here, we demonstrate the invariance property when one gain is negative
and all the others are positive. We select the same gains as in Case III. The system
is stable and has the same rendezvous point for both the pursuit sequences, BPS1 and
BPS2 as shown in Figure 2.17. The figure also shows that the trajectories of the agents
are different for the two pursuit sequences.
2.5.4 Finite and infinite switching of pursuit sequences
We consider the following pursuit sequences
BPS1 = (1, 2, 3, 4, 5)
BPS2 = (1, 3, 5, 2, 4)
BPS3 = (1, 4, 2, 5, 3).
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 42
−10 −5 0 5 10 15 20−10
−5
0
5
10
15
2
1
4
5
3
(a) Pursuit sequence BPS2
−10 −5 0 5 10 15 20−10
−5
0
5
10
15
3
4
1
5
2
(b) Pursuit sequence BPS1
Figure 2.17: Trajectories of the agents, with the gain of first agent negative, for differentpursuit sequences (Case IX)
Case X : Invariance properties with finite switching is demonstrated when the gains of
all the agents are positive. We consider the same gains as in Case I. At t = 0, the pursuit
sequence is BPS1. At t = 0.05, it switches to BPS2 and at t = 0.15, it switches to BPS3.
The trajectories of the agents are shown in Figure 2.18(a). Comparing with Figure
2.18(b), we observe that the rendezvous point is the same and hence the rendezvous
point is invariant under finite switching.
Case XI : We demonstrate the invariance properties with respect to the finite switch-
ing when the gain of one of the agent is negative and the other gains are positive.
Consider the same gains as in Case III. The pursuit sequence switches from BPS1 to
BPS2 to BPS3 at t = 0.05 and t = 0.15, respectively. The simulation is shown in Figure
2.19 and it can seen that the agents converge to the same point without switching (Figure
2.19(b)) and with finite switching (Figure 2.19(a)).
Case XII : We demonstrate the invariance properties with respect to infinite switching
of the pursuit sequence. Infinite switching is simulated by switching between the three
pursuit sequences BPS1, BPS2 and BPS3 repeatedly with the time between consecutive
switching ∆t = 0.02. The controller gains are the same as in Case I and the trajectories
are shown in Figure 2.20. The figure verifies the stability and rendezvous point invariance
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 43
−10 −5 0 5 10 15−10
−5
0
5
10
15
4
2
5
3
1
(a) Finite switching−10 −5 0 5 10 15
−10
−5
0
5
10
15
1
3
5
2
4
(b) No switching
Figure 2.18: Trajectories of the agents, with all positive gains, when the pursuit sequenceswitches as BPS1 → BPS2 → BPS3 (Case X)
−10 −5 0 5 10 15 20−10
−5
0
5
10
15
2
1
5
3
4
(a) Finite switching−10 −5 0 5 10 15 20
−10
−5
0
5
10
15
3
4
1
5
2
(b) No switching
Figure 2.19: Trajectories of the agents, with the gain of the first agent negative, whenthe pursuit sequence switches as BPS1 → BPS2 → BPS3 (Case XI)
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 44
−10 −5 0 5 10 15−10
−5
0
5
10
15
1
2
4
5
3
(a) Infinite switching−10 −5 0 5 10 15
−10
−5
0
5
10
15
1
3
5
2
4
(b) No switching
Figure 2.20: Trajectories of the agents, with all positive gains, when the pursuit sequenceswitches infinitely at regular time intervals as BPS1 → BPS2 → BPS3 → BPS1 → BPS2
→ . . . (Case XII)
with respect to infinite switching.
2.6 Conclusions
In this chapter, the stable behaviour of a group of heterogenous agents under linear
cyclic pursuit is studied. The conditions for stability are obtained. The stable group of
agents converge to a point, called the reachable or rendezvous point, which is obtained
as a function of the gains and the initial positions of the agents. The reachable set
is determined for a given initial position of the agents. It is found that the stability,
reachable points and the reachable sets are not affected by the basic pursuit sequence
that the agents follow. These invariance properties are also proved to be valid for finite
and infinite switching of the basic pursuit sequences.
The analysis done in this chapter assumes that the agents follow a basic cyclic pursuit,
where one agent follows another in a cyclic manner. This can be generalized further
where an agent can follow a point in the convex combination of the other agents. This is
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 45
called a centroidal cyclic pursuit. In the next chapter, the behaviour of the agents under
centroidal cyclic pursuit is studied.
Chapter 2. Rendezvous using linear Basic Cyclic Pursuit 46
Chapter 3
Rendezvous using linear Centroidal
Cyclic Pursuit
In basic cyclic pursuit, addressed in Chapter 2, an agent pursues another agent according
to the basic pursuit sequence of the agents. In this chapter, we consider a generalized
pursuit strategy where an agent follows a point which is a convex combination of the
positions of the other agents. This strategy is called centroidal cyclic pursuit. Similar
to the basic cyclic pursuit, the stability of the centroidal cyclic pursuit, and the be-
haviour of the stable system of agents, are analyzed and several invariance properties are
demonstrated.
3.1 Problem formulation
Consider a group of n agents, ordered from 1 to n, in a d dimensional space, as in Chapter
2. The position of the agent i at any time t ≥ 0 is Zi(t) = [z1i (t) z2
i (t) . . . zdi (t)]
T ∈ Rd.
For centroidal cyclic pursuit, the agent i follows a point, Zic, which is the weighted cen-
troid of the other agents’ position (Figure 3.1). Let the weights be w = (η1, η2, . . . , ηn−1)
with∑n
i=1 ηi = 1 and ηi ≥ 0,∀i. Then, assuming a basic pursuit sequence BPS=(1, 2, . . . ,
47
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 48
x
x
x x
x
xx
Z3
Z1
Z2Z1
Z2Z4
Z3
Z3cZ3c
Z2c Z4c
Z1c
Z2cZ1c
Figure 3.1: Centroidal cyclic pursuit
n), agent i will follow agent i + j (mod n) with weights ηj. For a general basic pursuit
sequence BPS= (p1, p2, . . . , pn), where pi ∈ {1, 2, . . . , n}, ∀i, and pi 6= pj, agent pi will
follow agent pi+j with weights ηj.
Thus, the equation of motion of agent i can be written as
Zi(t) = ui(t) = ki[Zic(t)− Zi(t)] (3.1)
where, assuming BPS=(1, 2, . . . , n), Zic is given by
Zic = η1Zi+1 + . . . + ηn−iZn + ηn−i+1Z1 + . . . + ηn−1Zi−1 =n−1∑j=1
ηjZi+j (3.2)
where the summation is mod n. Let,
Γ =
{w = (η1, η2, . . . , ηn−1)
∣∣∣n∑
i=1
ηi = 1, 0 ≤ ηi ≤ 1
}(3.3)
Then, Γ defines the set of all possible weights that can be used with each basic pursuit
sequence to obtain the centroidal cyclic pursuit. Now, if ηi = 1 for some i, then either
it will correspond to one of the basic pursuit sequences, or it will destroy the cyclic
structure. For example, when n > 2 and n is even, η2 = 1 will give rise to two distinct
cycles instead of one. Hence, in this chapter, the weights are assumed to belong to the
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 49
set
Γ =
{w = (η1, η2, . . . , ηn−1)
∣∣∣n∑
i=1
ηi = 1, 0 < ηi < 1
}(3.4)
Note that, Γ ⊆ Γ. Here, we assume that ηi 6= 0,∀i. The cases when one of the ηi = 1
and the cyclic structure is maintained, is already discussed in Chapter 2.
Now, (3.1) can be decoupled into d identical linear system of equations, as in Section
(2.6), and we can write
X = AX (3.5)
where
A =
−k1 η1k1 η2k1 · · · ηn−1k1
ηn−1k2 −k2 η1k2 · · · ηn−2k2
...
η1kn η2kn η3kn · · · −kn
(3.6)
The above expression can be written as A = Kχ, where
K =
k1 0 0 · · · 0
0 k2 0 · · · 0...
0 0 0 · · · kn
(3.7)
where K is the gain matrix and
χ =
−1 η1 η2 · · · ηn−1
ηn−1 −1 η1 · · · ηn−2
...
η1 η2 η3 · · · −1
(3.8)
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 50
is the weight matrix of the centroidal cyclic pursuit, called the pursuit sequence matrix.
The characteristic polynomial of A can be written as
ρ(s) = sn + Bn−1sn−1 + Bn−2s
n−2 . . . + B2s2 + B1s + B0 (3.9)
This is similar to (2.9) in Chapter 2, but the values of the coefficients are different.
Below, we evaluate B0 and B1 which are required for later analysis.
Since the columns of A are linearly dependent, A is singular and B0 = 0. Further,
Rank(A) = Rank(χ) as Rank(K) = n. Consider a matrix χii formed by removing the
ith row and ith column of χ. The Gershgorin’s discs of χii will have center at (−1, 0)
and radius less than one, since ηi > 0,∀i. Thus, χii does not have any eigenvalue at the
origin and so is of full rank. Therefore, Rank(A) = Rank(χ) = n− 1 and B1 6= 0.
The expression for B1 can be obtained using the L’Hospital rule as
B1 = lims→0
det(sI − A)
s= lim
s→0
[ d
ds
{det(sI − A)
}](3.10)
The derivative of the determinant of any n × n matrix P (y) with respect to y, is
ddy{det(P )} and is the sum of the n determinants obtained by replacing in all possi-
ble ways the elements of one row (column) of P by their derivatives with respect to y
[56]. If mij represents the minor of the (i, j) − th element of A, then from (3.10), as
s → 0
B1 = (−1)n−1
n∑i=1
mii
= (−1)n−1trace(Adj(A))
= (−1)n−1trace {Adj(χ) Adj(K)} (3.11)
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 51
Let the adjoint of χ be denoted by,
Adj(χ) =
ξ11 ξ12 ξ13 · · · ξ1n
ξ21 ξ22 ξ23 · · · ξ1n
...
ξn1 ξn2 ξn3 · · · ξnn
(3.12)
Theorem 3.1 The adjoint of the pursuit sequence matrix χ for centroidal cyclic pursuit
is given by
Adj(χ) = (−1)n+1|ξ|1n×n (3.13)
for some ξ, that is in (3.12), ξij = ξ, with the sign of ξ determined by n.
Proof. It is to be shown here that all the elements of Adj(χ) are identical, that is,
ξij = ξ, ∀i, j.
First, we show that ξil = ξjl,∀i, j, and for a given l, where ξil is the cofactor of the
(l, i)-th element of χ, that is, ξil = (−1)i+l|χli|. Here, χij is the matrix obtained by
removing the ith row and jth column of χ.
Consider two matrices, χli and χl(i+1), i = 1, . . . , n − 1. For both of these matrices,
only the ith column is different and the other columns are the same. We can perform
an elementary column transformation on χli, such that the ith column is replaced by the
sum of all the columns of χli (including the ith column). Let the new matrix be χli. Note
that χli will have the same determinant as χli. Now, since the sum of the elements of a
row of χ is zero, the ith column of χli will be equal to the ith column of χl(i+1) with a
negative sign. Then, χli has the same elements as χl(i+1) except that the sign of all the
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 52
elements of the ith column of these two matrices are opposite. Thus,
|χli| = |χli| = −|χl(i+1)| (3.14)
⇒ (−1)l+i|χli| = (−1)l+i+1|χl(i+1)| (3.15)
⇒ ξil = ξ(i+1)l (3.16)
Since the above equation is true for all i, ξil = ξjl,∀i, j and for a given l.
Since the rows of χ also sum to zero, it can be similarly shown that ξli = ξlj, ∀i, j,for a given l. Hence, ξij = ξ, ∀i, j.
Now, the determinant of a matrix is equal to the product of its eigenvalues, so ξii
is the product of the eigenvalues of χii. From the Gershgorin’s disc theorem, all the
eigenvalues of χii have negative real parts. Therefore, if n is odd, ξii > 0 and if n is even,
ξii < 0.1 Since ξ = ξii, ξ > 0, if n is odd and ξ < 0, if n is even. Thus, each element of
Adj(χ) in (3.12) can be written as (−1)n+1|ξ| and hence we get (3.13). ¤
Below we give an example to illustrate Theorem 3.1.
Illustrative example: Consider a matrix
χ =
−1 η1 η2 η3
η3 −1 η1 η2
η2 η3 −1 η1
η1 η2 η3 −1
(3.17)
Then,
χ11 =
−1 η1 η2
η3 −1 η1
η2 η3 −1
, χ12 =
η3 η1 η2
η2 −1 η1
η1 η3 −1
(3.18)
1This is true in general because, even if the eigenvalues are imaginary, the imaginary roots come incomplex conjugate pairs and will contribute a positive value.
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 53
Here, the second and third columns of χ11 and χ12 are the same. Consider the matrix
χ11 = χ11e(I) =
−1 η1 η2
η3 −1 η1
η2 η3 −1
1 0 0
1 1 0
1 0 1
=
−η3 η1 η2
−η2 −1 η1
−η1 η3 −1
(3.19)
since∑3
i=1 ηi = 1. As |e(I)| = 1, |χ11| = |χ11| = −|χ12|. This shows that ξ11 = ξ21.
Then, from (3.11), we can write
B1 = (−1)n−1
n∑i=1
{(−1)n+1|ξ|
n∏
j=1,j 6=i
kj
}
= |ξ|n∑
i=1
{n∏
j=1,j 6=i
kj
}(3.20)
Using the above, the stability of the system is analyzed in the next section.
3.2 Stability analysis
We state and prove a theorem identical to Theorem 2.2 in Chapter 2
Theorem 3.2 The linear system, given by (3.5), is stable if and only if the following
conditions hold
(a) At most one ki is negative or zero, that is, at most for one i, ki ≤ 0 and kj > 0,
∀j, j 6= i.
(b)∑n
i=1
( ∏nj=1,j 6=i kj
)> 0
Proof. From (3.20), Condition (b) implies B1 > 0 as |ξ| > 0. Since ηi > 0, ∀i and∑n
i=1 ηi = 1, the Gershgorin’s discs of A (for centroidal cyclic pursuit, CCP given in
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 54
(3.6)), are the same as that of A (for basic cyclic pursuit, BCP given in (2.7)). Thus,
this proof follows in the similar lines as Theorem 2.2.
First, we assume that Conditions (a) and (b) hold. We have to show that the system
is stable. Consider the following cases:
Case 1: All the gains are positive.
When all the gains are positive, Condition (b) is satisfied and all the Gershgorin’s discs
lie on the left hand side of the s plane. Therefore, none of the eigenvalues of A can have
positive real parts and, since B1 6= 0, there is only one eigenvalue of A at the origin.
Hence, the system is stable.
Case 2: One gain is zero and other gains are positive.
In this case, Condition (b) is satisfied and the Gershgorin’s disc remains the same as in
Case 1. Also, B1 6= 0 and thus, the system is stable.
Case 3: One gain is negative and other gains are positive.
For Condition (b) to be satisfied, we can find the lower bound on the gain, ki, given by
ki = −∏n
j=1,j 6=i kj∑nl=1,l 6=i
∏nj=1,j 6=i,l kj
(3.21)
Note that, this equation is the same as (2.12) in Chapter 2. It has been proved in
Theorem 2.2 that, given the gains kj > 0,∀j, j 6= i, if ki > ki, then the system is stable.
We omit the details of the proof here.
To prove the “only if” part, assume that the system is stable but any one or both
the conditions do not hold. Consider the different possible cases:
Case 1: More than one gain is zero.
From (3.20), B1 = 0 and therefore ρ(s), given in (3.9), will have more than one root at
the origin. Hence the system is unstable.
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 55
Case 2: More than one gain is negative
Consider a matrix, similar to (2.14)
A =
−k1 αη1k1 αη2k1 · · · αη(n−1)k1
αη(n−1)k2 −k2 αη1k2 · · · αη(n−2)k2
...
αη1kn αη2kn αη3kn · · · −kn
(3.22)
The Gershgorin’s discs of A when α = 0 are points at (−ki, 0),∀i. When α = 1, A = A.
From the continuity of the root locus and the necessity of the roots to be within the
Gershgorin’s disc, it can be argued, similar to Theorem 2.2, that as α goes from 0 to 1,
more than one root will always remain on the right hand side of the s plane. Therefore,
at α = 1, there will be more than one roots on the right hand side or at the origin.
Hence, the system is unstable.
Case 3 :∑n
i=1
( ∏nj=1,j 6=j kj
)≤ 0.
This case implies B1 ≤ 0 and hence, the system can not be stable ¤
Thus, the conditions for stability of centroidal cyclic pursuit (CCP) are same as that
for basic cyclic pursuit (BCP). When the system is stable, there is one and only one
eigenvalue of A at the origin. As in (2.19), the solution of (3.5) can be written as
xi(t) = xif +
∑p∈Sr
{ n∑q=1
( np∑r=1
aipqrt
r−1)xq(t0)
}eσpt
+∑p∈Si
{ n∑q=1
( np∑r=1
aipqrt
r−1 cos(ωpt) + ai∗pqrt
r−1 sin(ωpt))xq(t0)
}eσpt (3.23)
where, xif corresponds to the zero eigenvalue, and ai
pqr and ai∗pqr are functions of χ and
K, and are complex conjugates. This equation is the same as (2.19), except that the
values of the coefficients aiprq and ai∗
prq are different as they now depend on the weights
w. When the system is stable, i.e., σp < 0,∀p, as t →∞, xi(t) = xif ,∀i.
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 56
The eigenvector corresponding to the zero eigenvalues is v = 11×n. Since the Rank(A)
= n − 1, the dimension of the null space N (A) = 1 and it is spanned by v. Therefore,
at equilibrium, X = 0, which implies AX = 0 and the nontrivial solution of this is
X = cv = c11×n for some c. Hence, at equilibrium x1 = x2 = . . . = xn, or in other word,
the agents will converge to a point. Therefore, xif = xf ,∀i. In the next section, the point
of rendezvous is analyzed.
3.3 Rendezvous and Reachable point
When the system is stable, the agents converge to a point at equilibrium. We state and
prove the rendezvous point theorem that is identical to Theorem 2.3.
Theorem 3.3 (Reachable Point) If a system of n-agents, with equation of motion
given in (3.1), have their initial positions at Z0 = {Zi(t0)}ni=1 and gains matrix K, that
satisfies Theorem 3.2, then they converge to a point Zf given by,
Zf =n∑
i=1
{( 1/ki∑nj=1 1/kj
)Zi(t0)
}=
∑ni=1 Zi(t0)/ki∑n
i=1 1/ki
(3.24)
where Zf is called a reachable point or the rendezvous point of this system of n agents.
Proof. This theorem is identical to Theorem 2.3 and the proof follows in the same line.
Summing (3.1) for all n, in the sense of mod n, we get
n∑i=1
Zi(t)
ki
=n∑
i=1
(Zic(t)− Zi(t))
=n∑
i=1
[n−1∑j=1
ηjZi+j(t)− Zi(t)
](3.25)
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 57
Changing the order of the summations and assuming ηn = −1,
n∑i=1
Zi(t)
ki
=n∑
j=1
[n∑
i=1
ηn−j+iZj(t)
](3.26)
=n∑
j=1
[Zj(t)
n∑i=1
ηn−j+i
](3.27)
Since∑n
i=1 ηn−j+i = 0,
n∑i=1
Zi(t)
ki
= 0 (3.28)
⇒n∑
i=1
Zi(t)
ki
= constant (3.29)
Then, considering the initial position Zi(t0) and final position Zi(tf ) of the agent i,
n∑i=1
Zi(t0)
ki
=n∑
i=1
Zi(tf )
ki
(3.30)
When the system is stable, all the agents converge to a point, Zi(tf ) = Zf , ∀i. Thus,
n∑i=1
Zi(t0)
ki
=n∑
i=1
Zf
ki
= Zf
n∑i=1
1
ki
(3.31)
⇒ Zf =
∑ni=1 Zi(t0)/ki∑n
i=1 1/ki
(3.32)
Hence, we get (3.24) ¤
From (3.23), the final value of xif = xf ,∀i. Therefore, using (3.24), we can write
xf =
∑nq=1(1/kq)xq(t0)∑n
q=1(1/kq)(3.33)
Comparing (3.24) with (2.22), it can be seen that, given the initial positions of
the agents and the gains, the reachable point is the same for both basic cyclic pursuit
(BCP) and centroidal cyclic pursuit (CCP). The weights w ∈ Γ do not play any role in
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 58
determining the rendezvous point. Thus, the reachable set remains the same for BCP
and CCP. To make the agents converge to a desired point within the reachable set, the
gains can be selected as discussed in Theorem 2.4. Any weight w ∈ Γ can be chosen.
The difference in the two strategies, BCP and CCP, are reflected in the trajectory of
the agents, which changes with the weight w. This leads automatically to the invariance
properties of the reachable point and will be discussed in the next section.
3.4 Invariance properties
Similar to Section 2.4, we study the invariance properties of the system (3.1) with respect
to the pursuit sequence of the agents. For centroidal cyclic pursuit, the pursuit sequence
can change in two ways − (i) by changing the weights while keeping the basic pursuit
sequence same or (ii) by changing the basic pursuit sequence while keeping the weights
same. We show that the stability and rendezvous point do not change with
(a) Different sets of weights for a given basic pursuit sequence.
(b) Different basic pursuit sequence for a given set of weights.
(c) Different sets of weights and basic pursuit sequence.
Theorem 3.4 The stability of the linear centroidal cyclic pursuit is pursuit sequence
invariant.
Proof. It can be seen from Theorem 3.2 that the stability of the system depends only
on the gains K. Thus, given a set of stable gains, the system is stable for any basic
pursuit sequence and any weights. Therefore, under Conditions (a), (b) and (c) stated
above, the stability of the system remains unchanged and hence, it is pursuit sequence
invariant. ¤
Theorem 3.5 The reachable point and thus the reachable set of a linear centroidal cyclic
pursuit is pursuit sequence invariant.
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 59
Proof. Consider (3.24), which gives the expression for the reachable or the rendezvous
point. This equation depends only on the initial positions of the agents and their gains
and is independent of the basic pursuit sequence and the weights of the agents. Hence,
the reachable point is pursuit sequence invariant. Similarly, the reachable set is also
pursuit sequence invariant. ¤
Now, let us consider switching of pursuit sequence. Again, we can have three different
types of pursuit sequence switching
a) Switching of the weights while the same basic pursuit sequence is followed.
b) Switching of the basic pursuit sequence keeping the weights same.
c) Switching both the weights and the basic pursuit sequence.
The definition of finite and infinite switching of pursuit sequence is the same as given in
Section 2.4.
Theorem 3.6 (Stability with finite switching) The stability of the linear centroidal
cyclic pursuit is invariant under finite switching of pursuit sequences.
Proof. It is proved in Theorem 3.4 that if a system of n agents is stable for a given basic
pursuit sequence and weights, then it is stable for all the basic pursuit sequences and
weights. Therefore, if the pursuit sequence switches, it implies that the switch occurs
between two stable systems. Since the number of switches are finite, the system, after
the last switch, is stable and hence the stability is invariant under finite switching of
pursuit sequences. ¤
Theorem 3.7 (Reachability with finite switching) The reachable point of linear cen-
troidal cyclic pursuit is invariant under finite switching of pursuit sequences.
Proof. Let the switching of pursuit sequences occur at t1, · · · , tm, m < ∞ such that
0 < t1 < · · · < tm < ∞. During tj ≤ t < tj+1, let the pursuit sequence be (BPSj, wj).
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 60
The switching invariance property is proved by showing that the reachable point, given
in (3.24), remains the same after a switch of either the basic cyclic pursuit or the weights
or both. At t = tj, the positions of the agents are Zi(tj),∀i. If there are no further
switchings, then the reachable point, from (3.24), is
Zf =
∑ni=1 Zi(tj)/ki∑n
i=1 1/ki
(3.34)
Let, at tj+1, the weight wj switch to wj+1 while BPSj = BPSj+1. The position of agent
i at tj+1 is Zi(tj+1). For t ≥ tj+1, the pursuit sequence is (BPSj+1, wj+1). If there are
no more switching of connections, let the reachable point be Z ′f . Now, from (3.24) and
(3.29), we have
Z ′f
n∑i=1
1
ki
=n∑
i=1
Zi(tj+1)
ki
=n∑
i=1
Zi(tj)
ki
= Zf
n∑i=1
1
ki
(3.35)
This shows that Z ′f = Zf . It can be similarly shown that if at tj+1, BPSj switches to
BPSj+1 while wj = wj+1, Zf will remain as the rendezvous point. This is also true if
both basic pursuit sequence and weights changes. Hence, when there is one switching
of connection, the rendezvous point does not change. This can be extended to a finite
number of switchings to show that the reachable point remains unchanged after the final
switch tm. ¤
Theorem 3.8 (Stability with infinite switching) The stability of the linear centroidal
cyclic pursuit is invariant under infinite switching of pursuit sequences.
Proof. The stability is proved similar to Theorem 2.9. We analyze the system along
one direction, since from Section 3.1, we have seen that the system can be decoupled
and analyzed separately along each direction. Consider (3.23). We define
xmax = max{
xf , x1(t0), x2(t0), . . . , xn(t0)}
(3.36)
xmin = min{
xf , x1(t0), x2(t0), . . . , xn(t0)}
(3.37)
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 61
which are the same as in (2.66) and (2.67). We prove that xi(t) ∈ I0 = [xmin, xmax],∀i, ∀t.Now, as long as xi(t) ∈ int{I0}, that is, interior of I0, xi(t) ∈ I0. We consider the case
when xi(t) ∈ ∂{I0}, that is, on the boundary of I0, for some i and some t. Consider the
following cases:
Case I : xp(t) ∈ ∂{I0} and kp ≥ 0.
Assume xp(t) = xmin. From (3.5), xp(t) = kp(xpc−xp). Since, xpc is the weighted centroid
of the remaining n−1 agents, xpc ≥ xp, and thus xp(t) ≥ 0. Hence, xp(t) ∈ I0. Similarly,
when, xp(t) = xmax, xpc ≤ xp, and thus xp(t) ≤ 0. Hence, again xp(t) ∈ I0.
Case II : xp(t) ∈ ∂{I0} and kp < 0.
It has been shown in the proof of Theorem 2.9 that, when kp < 0, xp can be expressed
as a convex combinations of xf and xi,∀i, i 6= p. This is true even for CCP. Hence,
xp /∈ ∂{I0}. Therefore, xi(t) ∈ I0,∀i, ∀t.
Now, if either the basic pursuit sequence or the weights change, the characteristics
equation will be different and so will be the eigenvalues of A. But, from the continuity of
the root locus, given ε > 0, ∃ δ > 0 such that if 0 < ε ≤ ηi ≤ 1− ε < 1,∀i, then the real
part of all the eigenvalues σp < −δ,∀p, except for the one at the origin. Let, for basic
pursuit sequence BPSj and weights wl, the coefficients aipqr be represented as a
i,BPSj ,wlpqr .
Now, let us define
aimax = sup
BPS∈BPS, w∈Γ
{|ai,BPSj ,wl
pqr |,∀p, q, r}∀j,l
(3.38)
Let
xm = max{|xf |, |x1(t0)|, |x2(t0)|, . . . , |xn(t0)|
}(3.39)
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 62
Then, using (3.23), we can write the bounds L and U similar to (2.74) and (2.75), as
L = xf − naimax
(n∑
r=1
tr−1
)xmeσmaxt (3.40)
U = xf + naimax
(n∑
r=1
tr−1
)xmeσmaxt (3.41)
where, we assume σmax = 1− ε and
L ≤ xi(t) ≤ U (3.42)
Therefore, as t →∞, both the LHS and RHS of (3.42) tends towards xf and so xi(t) → xf
as t →∞. This implies that the system is stable with infinite switching. ¤
Theorem 3.9 (Reachability with infinite switching) The reachable point of linear
centroidal cyclic pursuit is invariant under infinite switching of pursuit sequences.
Proof. It is shown in Theorem 3.8 that at t → ∞, xi(t) = xf , ∀i, even with infinite
switching of pursuit sequences. Hence, the reachable point xf is invariant under infinite
switching. ¤
In the next section, simulations are carried out to verify the results obtained in this
chapter.
3.5 Simulation results
A swarm of 12 agents is considered in R2. The initial positions of the agents are shown
in Table 3.1. Different sets of gains are selected (as shown in Table 3.1) to demonstrate
the results obtained in this chapter.
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 63
Agent Initial PositionGains ki
Case I Case II
1 ( 4 , -8 ) 2 2
2 ( 3 , 9 ) 3 3
3 ( 0 , 2 ) 9 9
4 ( -7 , -1 ) 7 7
5 ( 7 , -3 ) 10 10
6 ( 14 , 5 ) 5 5
7 ( -3 , -6 ) 5 5
8 (-12 , 2 ) 6 -0.1
9 ( 11 , 5 ) 4 4
10 ( 2 , 4 ) 8 8
11 ( 1 , 7 ) 9 9
12 ( 4 , -6 ) 8 8
Table 3.1: Initial positions of the agents and their gains for different cases of CCP
3.5.1 Fixed pursuit sequence: Varying controller gains
Consider the basic pursuit sequence and weight of the agents as
BPS1 = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12)
w1 = (0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.9)
Case I : We demonstrate that when the gains of all the agents are positive, the system
is stable and rendezvous occurs. The gains of the agents are shown in Table 3.1. For this
set of gains,∑n
i=1
∏j=1,j 6=i kj = 3.1 × 109 > 0 and thus, Theorem 3.2 is satisfied. From
(3.24), the reachable point Zf = (2.75, 0.29) ∈ Co(Z0). The simulation, given in Figure
3.2, validates the reachable point.
Case II : We show that when only one gain is negative, but Theorem 3.2 is satisfied,
the agents converge to a point outside Co(Z0). The set of gains are shown in Table 3.1,
where agent 8 has negative gains, while the others have positive gains.∑n
i=1
∏j=1,j 6=i kj =
1.7 × 108 > 0 and hence Theorem 3.2 is satisfied and the system is stable. The lower
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 64
−15 −10 −5 0 5 10 15−10
−5
0
5
10
Figure 3.2: Trajectories of a swarm of 12 agents when all gains positive (Case I)
−20 −10 0 10 20−15
−10
0
10
15
Figure 3.3: Trajectories of a swarm of 12 agents when the gain of one of the agent isnegative while the others are positive (Case II)
bound of k8 = −0.45 < k8. From Theorem 3.3, Zf = (−16.47, 2.52) /∈ Co(Z0). The
trajectories of the agents are shown in Figure 3.3, and it confirms the reachable point.
3.5.2 Pursuit sequence invariance properties
We consider the basic pursuit sequences
BPS1 = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12)
BPS2 = (1, 3, 4, 5, 6, 12, 11, 10, 9, 8, 7, 2)
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 65
−20 −10 0 10 20−15
−10
0
10
15
(a) Pursuit sequence (BPS1, w1)−20 −10 0 10 20
−15
−10
0
10
15
(b) Pursuit sequence (BPS1, w2)
Figure 3.4: Invariance property of the reachable point, Zf = (−16.47, 2.52) for a givenbasic pursuit sequence and different weights (Case III)
and the weights
w1 = (0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.9)
w2 = (0.05, 0.05, 0.05, 0.01, , 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.015)
Case III : We demonstrate the stability and rendezvous point invariance with respect
to the weights used by the agents for a given basic pursuit sequence. We consider pursuit
sequence BPS1 and two different sets of the weights w1 and w2. The gains are same as
in Case II. The agents converge to the point Zf = (−16.47, 2.52) as seen in Figure 3.4
Case IV : The invariance properties for a given weight but different pursuit sequences
are demonstrated. Consider the basic pursuit sequences BPS1 and BPS2 and weight
w1. Assuming the gains are same as in Case II, the agents converge to the point Zf =
(−16.47, 2.52) for both the pursuit sequence as seen in Figure 3.5.
Case V : We demonstrate that the stability and the rendezvous point is invariant
under finite switching of the pursuit sequences and weights. The gains are taken to be
the same as in Case II. At t = 0, the pursuit sequence is (BPS1, w1). The pursuit sequence
switches to (BPS1, w2) at t = 0.05, and to (BPS2, w1) at t = 0.2. The trajectories are
shown in Figure 3.6(a). We observe that the reachable point Zf = (−16.47, 2.52) remains
the same as in the case of no switching (Figure 3.6(b)).
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 66
−20 −10 0 10 20−15
−10
0
10
15
(a) Pursuit sequence (BPS1, w1)−20 −10 0 10 20
−15
−10
0
10
15
(b) Pursuit sequence (BPS2, w1)
Figure 3.5: Invariance property of the reachable point, Zf = (−16.47, 2.52) for a givenweight and different basic pursuit sequences (Case IV)
−20 −10 0 10 20−15
−10
0
10
15
(a) Finite switching−20 −10 0 10 20
−15
−10
0
10
15
(b) No switching
Figure 3.6: Invariance property of the reachable point, Zf = (−16.47, 2.52) with switch-ing of pursuit sequence from (BPS1, w1) → (BPS1, w2) → (BPS2, w1) (Case V)
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 67
−20 −10 0 10 20−15
−10
0
10
15
(a) Infinite switching−20 −10 0 10 20
−15
−10
0
10
15
(b) No switching
Figure 3.7: Invariance property of the reachable point, Zf = (−16.47, 2.52) with infiniteswitching from (BPS1, w1) → (BPS1, w2) → (BPS2, w1) → (BPS1, w1) → (BPS1, w2)→ . . . (Case VI)
Case VI : We demonstrate that the stability and the rendezvous point is invariant
under infinite switching of the pursuit sequence. The gains of the agents are same as in
Case II. The pursuit sequences are switched from (BPS1, w1) → (BPS1, w2) → (BPS2,
w1) → (BPS1, w1) → (BPS1, w2) → . . . and the time between each switch ∆t = 0.02.
The trajectories are shown in Figure 3.7 and we observe the invariance property.
3.5.3 Computation of controller gains for a rendezvous point
Here, we demonstrate that to make the agents converge to a desired point, we can select
the gains as illustrated in Theorem 2.4.
Case VII : In this case, we consider 5 agents with initial positions and gains as in
Case VI of Section 2.5. We want the agents to converge at Zf = (0, 0). Assume the
pursuit sequence as BPS=(1, 2, 3, 4, 5) and weights as w = (0.5, 0.5, 0, 0). Considering
the same gains as in Case VI of Section 2.5, the trajectories are shown in Figure 3.8.
We observe the trajectories are different but the rendezvous occurs at Zf = (0, 0). This
shows that the computations give in Chapter 2 are sufficient to determine the required
controller gain for rendezvous at the specified point.
Chapter 3. Rendezvous using linear Centroidal Cyclic Pursuit 68
−10 0 10 15−10
0
10
15
2
1
3
5
4
Figure 3.8: Trajectories of 5 agents using centroidal cyclic pursuit (CCP) converging atZf = (0, 0) (Case VII)
3.6 Conclusions
In this chapter, the stability, rendezvous and invariance properties of the centroidal
cyclic pursuit (CCP) are studied. The stability, reachable point and reachable sets are
the same for both the basic and centroidal cyclic pursuit case. We observe the invariance
properties of stability and rendezvous point with respect to pursuit sequence. In the
next chapter, we generalize centroidal cyclic pursuit by relaxing the requirement that
the weights w used by each agent is the same and consider different weights.
Chapter 4
Rendezvous using linear Generalized
Centroidal Cyclic Pursuit
This chapter generalizes the cyclic pursuit strategies discussed in the previous two chap-
ters. Here, the agents follow a centroidal cyclic pursuit but the weights used by each
agent to compute the centroid are different. The stability, reachable/rendezvous point,
reachable set and the invariance properties are studied under this generalized centroidal
cyclic pursuit (GCCP) strategy.
4.1 Problem formulation
A group of n agents are considered in a d dimensional space as in Chapters 2 and 3.
They are ordered from 1 to n. At any time t, the agents are at positions Zi(t) =
[z1i (t) z2
i (t) . . . zdi (t)]
T ∈ Rd. Each agent i follows a point Zic which is the weighted
centroid of the position of the remaining n − 1 agents, as discussed in Chapter 3. The
agent i uses a weight wi ∈ Γ where Γ is the set defined in (3.4) and is given by
Γ =
{w = (η1, η2, . . . , ηn−1)
∣∣∣n∑
i=1
ηi = 1, 0 < ηi < 1
}(4.1)
69
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit70
Assuming the basic pursuit sequence BPS=(1, 2, . . . , n), the state equation of the agent
i can be written as
Zi = ui = ki(Zic − Zi) (4.2)
where
Zic = ηi1Zi+1 + . . . + ηi
n−iZn + ηin−i+1Z1 + . . . + ηi
n−1Zi+1 =n−1∑j=1
ηijZi−j (4.3)
Here, the summation is mod n and the weights wi = (ηi1, η
i2, . . . , η
in−1) ∈ Γ. Thus, the
set of weights wi that the agent i uses may be different from the set of weights wj used
by agent j. This is a generalization of the concept of centroidal cyclic pursuit and hence
is called generalized centroidal cyclic pursuit (GCCP).
Now, for each agent i, (4.2) can be decoupled along each coordinate and as in (2.6)
and (3.5), we will have, for all the agents, d identical linear systems of equations, given
by
X = AX (4.4)
where,
A =
−k1 η11k1 η1
2k1 · · · η1n−1k1
η2n−1k2 −k2 η2
1k2 · · · η2n−2k2
...
ηn1 kn ηn
2 kn ηn3 kn · · · −kn
(4.5)
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit71
Equation (4.4) can be analyzed similar to (3.5). We can write A = Kχ where
K =
k1 0 0 · · · 0
0 k2 0 · · · 0...
0 0 0 · · · kn
(4.6)
is the gain matrix and
χ =
−1 η11 η1
2 · · · η1n−1
η2n−1 −1 η2
1 · · · η2n−2
...
ηn1 ηn
2 ηn3 · · · −1
(4.7)
is the weight matrix or the pursuit sequence matrix of genralized centroidal cyclic pursuit
(GCCP).
We can write the characteristic polynomial of A as
ρ(s) = sn + Bn−1sn−1 + Bn−2s
n−2 . . . + B2s2 + B1s + B0 (4.8)
This has the same form as (2.9) and (3.9). Similar to Section 3.1, we will evaluate B0
and B1 for further analysis.
The sum of the elements of the rows of χ is zero. Therefore, using Gershgorin’s disc
theorem, we have Rank(χ) = n − 1, as shown in Section 3.1. Thus, Rank(A) = n − 1
and B0 = 0, B1 6= 0. The expression for B1 can be obtained using the L’Hospital rule
B1 = lims→0
det(sI − A)
s= lim
s→0
[ d
ds
{det(sI − A)
}](4.9)
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit72
Let, mij represents the minor of the (i, j)-th element of A, then as s → 0
B1 = (−1)n−1
n∑i=1
mii = (−1)n−1trace(Adj(A))
= (−1)n−1trace{
Adj(χ) Adj(K)}
(4.10)
Let, the adjoint of χ be
Adj(χ) =
ξ11 ξ12 ξ13 · · · ξ1n
ξ21 ξ22 ξ23 · · · ξ1n
...
ξn1 ξn2 ξn3 · · · ξnn
(4.11)
Theorem 4.1 The adjoint of the pursuit sequence matrix χ, for generalized centroidal
cyclic pursuit, is given by
Adj(χ) = (−1)n+11n×1
[|ξ1| |ξ2| . . . |ξn|
](4.12)
for some ξj that is in (3.12), ξij = ξj,∀i, j, with the sign of each ξj determined by n.
Proof. Since the sum of the columns of χ = 0, as shown in Section 3.1, the elements of
a column of Adj(χ) are identical, that is, ξij = ξj,∀i, j. However, since the sum of the
elements of a column of χ need not necessary be zero, the elements of a row of Adj(χ)
are not necessarily identical.
Again, from Gershgorin’s Disc Theorem, it can be proved as in Section 3.1 that, if n
is odd, ξi > 0 and if n is even, ξi < 0. Thus, ξi,∀i have the same sign and each row of
Adj(χ) can be written as (−1)n+11n×1
[|ξ1| |ξ2| . . . |ξn|
]. Hence, we get (4.12). ¤
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit73
Then, from (4.10), we can write
B1 = (−1)n+1
n∑i=1
{(−1)n+1|ξi|
n∏
j=1,j 6=i
kj
}
=n∑
i=1
|ξi|{ n∏
j=1,j 6=i
kj
}(4.13)
B1 has similar form as in (3.20). Now, since χ is singular,
Adj(χ).χ = 0 (4.14)
Using (4.4) and (4.14), let us evaluate
(−1)n+1[|ξ1| |ξ2| . . . |ξn|
]K−1X
= (−1)n+1[|ξ1| |ξ2| . . . |ξn|
]K−1(Kχ)X
= (−1)n+1[|ξ1| |ξ2| . . . |ξn|
]χ X
= 01×nX = 0 (4.15)
Simplifying (4.15),
n∑i=1
|ξi|xi
ki
= 0 (4.16)
⇒n∑
i=1
|ξi|xi
ki
= constant (4.17)
In the next section, the conditions for stability of generalized centroidal cyclic pursuit
(GCCP) is analyzed.
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit74
4.2 Stability analysis
The stability of the system, under generalized centroidal cyclic pursuit, is analyzed sim-
ilar to the analysis done in Sections 2.2 and 3.2.
Theorem 4.2 The linear system, given by (4.4), is stable if and only if the following
conditions hold
(a) At most one ki is negative or zero, that is, at most for one i, ki ≤ 0 and kj > 0,
∀j, j 6= i.
(b)∑n
i=1 |ξi|( ∏n
j=1,j 6=i kj
)> 0
Proof. This proof is similar to Theorem 2.2 and 3.2. From (4.13), Condition (b) implies
B1 > 0. The Gershgorin’s discs of A are similar to that in Theorem 3.2, since the sum
of the elements of the rows of χ are zero for both the strategies.
First, we prove the ’if’ part, that is, if Conditions (a) and (b) are satisfied, then the
system is stable. Three different cases are considered.
Case 1: All the gains are positive.
When ki > 0,∀i, Condition (b) is automatically satisfied. The Gershgorin’s discs of A
are centered at (−ki, 0) with radius −ki, i = 1, . . . , n. Therefore, all the eigenvalues of
A has negative real part except only one at the origin, since B0 = 0 and B1 6= 0. Hence
the system is stable.
Case 2: One gain is zero and other gains are positive.
This case is similar to Case 1. All the Gershgorin’s discs lie on left half of the s plane,
and B0 = 0 and B1 6= 0. Hence, the system is stable.
Case 3: One gain is negative and other gains are positive.
Let ki < 0 and kj > 0, ∀j, j 6= i. Then, for Condition (b) to be satisfied, ki > ki where
ki = − |ξi|∏n
j=1,j 6=i kj∑nl=1,l 6=i |ξl|
∏nj=1,j 6=i,l kj
(4.18)
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit75
This equation is similar to (2.12) and (3.21). It can be shown that, given the gains
kj > 0,∀j, j 6= i, if ki > ki, then the system is stable as discussed in the proof of
Theorem 2.2. We omit the details of the proof here.
The “only if” part is proved by contradiction. Assume the system is stable but any
one or both the conditions do not hold. We consider the following cases separately.
Case 1: More than one gain is zero
When two or more gains are zero, B1 = 0, which implies more than one root at the
origin, and hence the system is unstable.
Case 1: More than one gain is negative.
Consider a matrix
A =
−k1 αη11k1 αη1
2k1 · · · αη1(n−1)k1
αη2(n−1)k2 −k2 αη2
1k2 · · · αη2(n−2)k2
...
αηn1 kn αηn
2 kn αηn3 kn · · · −kn
(4.19)
where α varies from 0 to 1. This matrix is similar to (3.22). At α = 0, A has more than
one eigenvalues on the right hand side. From the continuity of the root locus and the
Gershgorin’s disc theorem, as α varies from 0 to 1, there will be more than one root of
A on the right hand side or the origin of the s plane, and hence the system is not stable.
This proof is illustrated in Theorem 2.2.
Case 3 :∑n
i=1 |ξi|( ∏n
j=1,j 6=j kj
)≤ 0
This case implies B1 ≤ 0 and hence, the system is unstable. ¤
Therefore, as in basic cyclic pursuit (BCP) and centroidal cyclic pursuit (CCP), when
the system under generalized centroidal cyclic pursuit (GCCP) is stable, there is one and
only one eigenvalue of A at the origin and the others are all on the left hand side of the
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit76
s plane. Thus, we can write the solution of (4.4), similar to (2.19) and (3.23), as
xi(t) = xif +
∑p∈Sr
{ n∑q=1
( np∑r=1
aipqrt
r−1)xq(0)
}eσpt
+∑p∈Si
{ n∑q=1
( np∑r=1
aipqrt
r−1 cos(ωpt) + ai∗pqrt
r−1 sin(ωpt))xq(0)
}eσpt (4.20)
where again xif corresponds to the zero eigenvalue and ai
pqr and ai∗pqr are complex conjugate
and are functions of χ and K. When the system is stable, i.e., σp < 0,∀p, as t → ∞,
xi(t) = xif ,∀i. Now, the eigenvector corresponding to the zero eigenvalue is v = 11×n
and it spans the null space of A, as Nullity(A) = 1. Thus, at equilibrium, the solution
of X = AX = 0 is X = c11×n for some c. Therefore, all the agents will converge to a
point and xif = xf , ∀i. In the next section, the point of rendezvous is analyzed.
4.3 Rendezvous and Reachable point
We analyze the rendezvous of a stable system similar to Sections 2.3 and 3.3
Theorem 4.3 (Reachable Point) If a system of n-agents, with equations of motion
given in (4.2), have their initial positions at Z0 = {Zi(t0)}ni=1, gain matrix K and pursuit
sequence matrix χ, that satisfies Theorem 4.2, then they converge to a point Zf given by,
Zf =n∑
i=1
{( |ξi|/ki∑nj=1 |ξj|/kj
)Zi(t0)
}=
∑ni=1(|ξi|/ki)Zi(t0)∑n
i=1 |ξi|/ki
(4.21)
where Zf is called a reachable point or the rendezvous point of this system of n agents.
Proof. Equation (4.17) holds for all the directions. Thus, in general, we can write, ∀t
n∑i=1
|ξi|ki
Zi(t) = constant (4.22)
⇒n∑
i=1
|ξi|ki
Zi(t0) =n∑
i=1
|ξi|ki
Zi(tf ) (4.23)
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit77
where, Zi(t0) and Zi(tf ) are the initial and final position of the ith agent, respectively.
When the system is stable, all the agents converge to a point, and Zi(tf ) = Zf , ∀i. Thus,
n∑i=1
|ξi|ki
Zi(t0) =n∑
i=1
|ξi|ki
Zf = Zf
n∑i=1
|ξi|ki
(4.24)
⇒ Zf =
∑ni=1(|ξi|/ki)Zi(t0)∑n
i=1 |ξi|/ki
(4.25)
Hence, we get (4.21). ¤
The reachable point Zf , given in (4.21), is different from (2.22) and (3.24). Here,
the gains ki are multiplied by a factor 1/|ξi|. However, since 1/|ξi| > 0, the reachable
set remains the same as for basic cyclic pursuit and centroidal cyclic pursuit. This is
discussed in the next theorem.
Given the initial positions of the agents Z0 and the weight matrix χ, let us define the
reachable set as
Zf (Z0) =
{Zf (Z
0, k, χ)∣∣∣ ∀k satisfying Theorem 4.2
}(4.26)
Here, Co(Z0) and Cp has the same definition as in Section 2.3.
Theorem 4.4 Consider a system of n agents with equation of motion given in (4.2) and
initial positions at Z0. A point Z is reachable if and only if,
Z ∈ Co(Z0)⋃ { n⋃
p=1
Cp
}= P(Z0) (4.27)
that is, Zf (Z0) = P(Z0).
Proof. The proof follows similar to Theorem 2.4. First, we show that Zf (Z0) ⊆ P(Z0).
Let Z ∈ Zf (Z0). Then, by definition of Zf (Z
0), there exists a gain matrix K, satisfying
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit78
Theorem 4.2, such that,
Z =
∑ni=1(|ξi|/ki)Zi(t0)∑n
i=1 |ξi|/ki
(4.28)
holds. We will show that Z ∈ P(Z0). Consider the following cases.
Case I : Let ki > 0, ∀i. Then, (4.28) can be written as
Z =n∑
i=1
(|ξi|/ki)∑nj=1(|ξj|/kj)
Zi(t0) (4.29)
Thus, Z is a convex combination of Zi(t0), i = 1, ..., n. Hence, Z ∈ Co(Z0) and so
Z ∈ P(Z0).
Case II : Let one of the gains kp < 0 and the remaining ki > 0, ∀i, i 6= p. Then, from
(4.29),
Z =n∑
i=1,i6=p
(|ξi|/ki)∑nj=1(|ξj|/kj)
Zi(t0) +(|ξp|/kp)∑nj=1(|ξj|/kj)
Zp(t0) (4.30)
⇒ Z
n∑i=1
|ξi|ki
=∑
i=1,i 6=p
|ξi|ki
Zi(t0) +|ξp|kp
Zp(t0) (4.31)
⇒ Z
n∑i=1
|ξi|ki
− Zp(t0)n∑
i=1
|ξi|ki
=∑
i=1,i6=p
|ξi|ki
Zi(t0) +|ξp|kp
Zp(t0)− Zp(t0)n∑
i=1
|ξi|ki
(4.32)
⇒{
Z − Zp(t0)} n∑
i=1
|ξi|ki
=n∑
i=1,i6=p
|ξi|ki
{Zi(t0)− Zp(t0)
}(4.33)
Since kp < 0 and ki > 0, ∀i, i 6= p, we have
n∏i=1
ki < 0 (4.34)
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit79
From Condition (b) of Theorem 4.2, a stable system will have
n∑i=1
|ξi|( n∏
j=1,j 6=i
kj
)> 0 (4.35)
Dividing the above equation by∏n
i=1 ki < 0, we get
n∑i=1
|ξi|ki
< 0 (4.36)
Let
n∑i=1
|ξi|ki
= −1
c(4.37)
where c > 0. Then, from (4.33)
−1
c
{Z − Zp(t0)
}=
n∑
i=1,i 6=p
|ξi|ki
(Zi(t0)− Zp(t0)) (4.38)
⇒ Z − Zp(t0) =n∑
i=1,i6=p
−c|ξi|ki
(Zi(t0)− Zp(t0)) (4.39)
⇒ Z = Zp(t0) +n∑
i=1,i 6=p
c|ξi|ki
(Zp(t0)− Zi(t0)) (4.40)
Then, from (2.31), Z ∈ Cp and so Z ∈ P(Z0).
Case III : Let one of the gains kp = 0 and the remaining ki > 0, ∀i, i 6= p. We can write
(4.28) as
Z =
∑ni=1 |ξi|(
∏nj=1,j 6=i kj)Zi(t0)∑n
i=1 |ξi|(∏n
j=1,j 6=i kj)(4.41)
Putting kp = 0 in the above equation
Z =|ξp|(
∏nj=1,j 6=p kj)Zp(t0)
|ξp|(∏n
j=1,j 6=p kj)= Zp(t0) (4.42)
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit80
Thus, Z ∈ Co(Z0) and so Z ∈ P(Z0).
Therefore, from Cases I−III, if Z ∈ Zf (Z0), then Z ∈ P(Z0) or Zf (Z
0) ⊆ P(Z0).
Now, to prove P(Z0) ⊆ Zf (Z0), we will show that for any point Z ∈ P(Z0), there
exists K such that (4.28) holds. We can partition P(Z0), similar to (2.48), as
P(Z0) = P1(Z0) ∪ P2(Z
0) ∪ P3(Z0) (4.43)
where,
P1(Z0) = int{P(Z0)} (4.44)
P2(Z0) =
{Zi(t0)
∣∣∣Zi(t0) ∈ ∂{P(Z0)}}
(4.45)
P3(Z0) = ∂{P(Z0)} \ P2(Z
0) (4.46)
We will consider these sets separately.
Case I : Z ∈ P1(Z0). We have the following cases:
Case Ia: Let Z ∈ int{Co(Z0)}. Then, there exists αi, i = 1, . . . , n,∑n
i=1 αi = 1 with αi >
0,∀i such that
n∑i=1
αiZi(t0) = Z (4.47)
Let
ki =c|ξi|αi
, i = 1, 2, . . . , n (4.48)
where, c > 0 is any positive constant. Thus, ki > 0, ∀i, and
n∑i=1
|ξi|ki
=1
c(4.49)
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit81
Replacing αi by (c|ξi|)/ki in (4.47),
Z =n∑
i=1
(c|ξi|ki
)Zi(t0) (4.50)
=n∑
i=1
( |ξi|/ki
1/c
)Zi(t0) (4.51)
=n∑
i=1
{|ξi|/ki∑n
j=1 |ξj|/kj
}Zi(t0) (4.52)
The above equation is the same as (4.28) and all the gains satisfy Theorem 4.2. Therefore,
Z ∈ Zf (Z0)
Case Ib: Let Z ∈ int{Cp} for some p. Then, there exist βi > 0, i = 1, 2, . . . , n, such that
Z can be expressed as
Z = Zp(t0) +n∑
i=1,i6=p
βi {Zp(t0)− Zi(t0)} (4.53)
=( n∑
i=1,i6=p
−βiZi(t0))
+ (1 + β1 + · · ·+ βp−1 + βp+1 + · · ·+ βn)Zp(t0) (4.54)
=n∑
i=1
−βiZi(t0) (4.55)
where
βp = −(1 + β1 + · · ·+ βp−1 + βp+1 + · · ·+ βn) (4.56)
Now, let
ki =c|ξi|βi
, i = 1, 2, . . . , n (4.57)
where, c > 0 is any positive constant. Then, ki ≥ 0, i = 1, . . . , n, i 6= p and kp < 0. Also
n∑i=1
βi = −1 ⇒n∑
i=1
|ξi|ki
= −1
c(4.58)
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit82
Since c > 0, using (4.36), it can be seen that the gains satisfy Theorem 4.2. Now,
replacing βi by (c|ξi|)/ki from (4.57) into (4.55), we get
Z =n∑
i=1
{−c|ξi|
ki
}Zi(t0) (4.59)
=n∑
i=1
{ |ξi|/ki
−1/c
}Zi(t0) (4.60)
=n∑
i=1
{|ξi|/ki∑n
j=1 |ξi|/kj
}Zi(t0) (4.61)
This is the same as (4.28). Hence, Z ∈ Zf (Z0).
Case II : Let Z = Zp(t0) ∈ P2(Z0). Then, we can write
Z = Zp(t0) =
∑ni=1 |ξi|
(∏nj=1,j 6=i kj
)Zi(t0)
∑ni=1 |ξi|
(∏nj=1,j 6=i kj
) (4.62)
where, kp = 0 and ki > 0,∀i, i 6= p. This is same as (4.28) and hence Z ∈ Zf (Z0).
Case III : When Z ∈ ∂{P2(Z0)}, Z can be expressed as in (4.47) and (4.53) where for
some i, αi = 0 and βi = 0 respectively. This will result in some of the gains ki to be
infinite. If we do not restrict the gains to be finite, then following similar arguments as
in Case Ia and Case Ib, Z ∈ Zf (Z0).
Thus, Cases I, II and III together proved that if Z ∈ P(Z0), then Z ∈ Zf (Z0) and so
P(Z0) ⊆ Zf (Z0). Therefore, Zf (Z
0) = P(Z0) and hence all points in Zf (Z0) = P(Z0)
are reachable. ¤
In the next section, the invariance properties of generalized centroidal cyclic pursuit
are discussed.
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit83
4.4 Invariance properties
From Theorems 4.2 and 4.3, we find that the stability and the rendezvous point depends
on ξi, ∀i, which implies that these properties are not independent of the pursuit sequence
matrix χ, or in other words, they are not pursuit sequence invariant.
However, if the values of ηji are such that the sum of the elements of the rows of χ is
zero, then ξi = ξ, ∀i as proved in Section 3.1. This condition is automatically satisfied for
centroidal cyclic pursuit (CCP). Under this condition, Theorem 4.2 is same as Theorem
3.2 and the reachable point, given by
Zf =n∑
i=1
1/ki∑nj=1 1/kj
Zi(t0) (4.63)
is also the same. Thus, the invariance properties of stability and reachable point with
respect to the pursuit sequence and finite and infinite switching of pursuit sequence will
hold under the special condition of sum of the elements of the rows of χ being zero.
Theorem 4.5 For linear generalized centroidal cyclic pursuit, where the elements of the
column of the pursuit sequence matrix sum up to zero, the stability and rendezvous point
is pursuit sequence invariant and also invariant to finite and infinite switching of pursuit
sequences.
The proof of the theorem are similar to those in Section 3.4 and are omitted here. In
the next section, simulations are carried out to verify the results obtained in this chapter.
4.5 Simulation results
Unlike Chapters 2 and 3, we will do 3 -D simulations. Consider a swarm of 12 agents.
The initial positions of the agents are shown in Table 4.1 and the basic pursuit sequences
is
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit84
Agent Initial positionGain
Case I Case II Case III Case IV
1 (48 , 83 , 37) 6 6 4.6159 7.2817
2 (47 , 97 , 14) 8 8 2.6689 3.6511
3 (71 , 30 , 1) 7 7 2.7034 1.0543
4 (24 ,100 , 64) 10 10 1.2246 5.1982
5 (72 , 44 , 74) 1 1 1.0289 1.8322
6 (87 , 1 , 3) 10 10 4.0504 1.1620
7 (41 , 29 , 10) 3 3 20.2304 1.6995
8 (42 , 69 , 89) 10 10 1.3097 5.0688
9 (95 , 49 , 76) 7 7 0.9346 2.1936
10 (88 , 8 , 58) 5 5 1.9492 1.6415
11 (70 , 20 , 69) 1 -0.25 2.6679 2.6159
12 (31 , 98 , 97) 5 5 3.4745 -0.2245
Table 4.1: Initial positions of the agents and their gains for different cases of GCCP
χ =
−1 0.05 0.05 0.05 0.05 0.10 0.10 0.10 0.10 0.10 0.10 0.20
0.02 −1 0.01 0.10 0.10 0.20 0.25 0.05 0.05 0.05 0.05 0.12
0.01 0.01 −1 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.90
0.01 0.84 0.02 −1 0.02 0.02 0.01 0.01 0.01 0.02 0.02 0.02
0.02 0.05 0.05 0.05 −1 0.06 0.04 0.04 0.04 0.07 0.03 0.55
0.10 0.09 0.09 0.09 0.09 −1 0.09 0.09 0.09 0.09 0.09 0.09
0.10 0.10 0.10 0.10 0.10 0.10 −1 0.05 0.05 0.05 0.05 0.20
0.20 0.10 0.30 0.01 0.01 0.05 0.05 −1 0.06 0.16 0.05 0.01
0.40 0.40 0.01 0.01 0.01 0.01 0.01 0.01 −1 0.02 0.02 0.10
0.90 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 −1 0.01 0.01
0.30 0.03 0.03 0.03 0.03 0.05 0.11 0.06 0.04 0.06 −1 0.26
0.24 0.04 0.04 0.04 0.04 0.06 0.06 0.07 0.08 0.08 0.25 −1
(4.64)
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit85
Adj(χ) = −0.01
43 25 13 11 11 16 18 13 14 16 20 41
43 25 13 11 11 16 18 13 14 16 20 41
43 25 13 11 11 16 18 13 14 16 20 41
43 25 13 11 11 16 18 13 14 16 20 41
43 25 13 11 11 16 18 13 14 16 20 41
43 25 13 11 11 16 18 13 14 16 20 41
43 25 13 11 11 16 18 13 14 16 20 41
43 25 13 11 11 16 18 13 14 16 20 41
43 25 13 11 11 16 18 13 14 16 20 41
43 25 13 11 11 16 18 13 14 16 20 41
43 25 13 11 11 16 18 13 14 16 20 41
43 25 13 11 11 16 18 13 14 16 20 41
(4.65)
BPS=(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12)
The weight matrix χ is given in (4.64). Different sets of gains are selected (as shown in
Table 4.1) to demonstrate the results obtained in this chapter.
4.5.1 Fixed pursuit sequence: Varying controller gains
Case I : We demonstrate that when the gains of all the agents are positive, rendezvous will
occur within Co(Z0). Consider the gains given in Table 4.1. Based on χ given in (4.64),
we compute the adjoint matrix given in (4.65). Thus, we get∑n
i=1 |ξi|( ∏n
j=1,j 6=i kj
)=
1.17×108 > 0. Therefore, Theorem 4.2 is satisfied. From Theorem 4.3 and 4.4, the agents
converge to the point Zf given in (4.21). For the given initial conditions, pursuit sequence
matrix and gains, we compute Zf = (60.12, 12.47, 58.40) ∈ Co(Z0). The trajectories of
the agents are shown in Figure 4.1, which verifies the rendezvous point.
Case II : Here, we consider one gain to be negative and all others gains to be positive
such that Theorem 4.2 is satisfied. We show that the agents converge to a point. The
gains of the agents are given in Table 4.1, where the agent 11 has a negative gain. For
this set of gains,∑n
i=1 |ξi|( ∏n
j=1,j 6=i kj
)= 1.47× 107 > 0 and hence the system is stable
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit86
0
50
100
0
50
1000
50
100
Figure 4.1: Trajectories of a swarm of 12 agents when the gains of all the agents arepositive (Case I)
0
50
100
−50
0
50
1000
50
100
Figure 4.2: Trajectories of a swarm of 12 agents when the gain of one of the agent isnegative and the other gains are positive such that Theorem 4.2 is satisfied (Case II)
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit87
0
50
100
0
50
1000
50
100
Figure 4.3: Trajectories of the agents converging to Zf = (60, 60, 60) (Case III)
as the gains satisfy Theorem 4.2. The lower bound k11 = −0.43 < k11. From Theorems
4.3 and 4.4, Zf = (89.64,−34.05, 90.09) /∈ Co(Z0). The simulation, given in Figure 4.2,
confirms the reachable point.
4.5.2 Computation of controller gains for a rendezvous point
Case III : Let the desired rendezvous point be Zf = (60, 60, 60) ∈ Co(Z0). To satisfy
(4.47) we get αi, ∀i as
α = 10−2[9.24 9.26 4.77 9.19 10.63 4.06 0.90 9.89 14.56 8.14 7.49 11.88]
Assuming c = 1, the gains of the agents are given in Table 4.1. The trajectories of the
agents with these gains are shown in Figure 4.3 and we observe that the agents converge
to (60, 60, 60).
Case IV : To make the agents converge at Zf = (0, 150, 150) /∈ Co(Z0), the gain of
one of the agents has to be negative. Assume that the gain of the agent 12 is negative.
From (4.53), we get
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit88
0
100
0
100
2000
100
200
Figure 4.4: Trajectories of the agents converging to Zf /∈ Co(Z0) (Case IV)
β = 10−2[5.85 6.77 12.22 2.17 5.97 14.16 10.72 2.55 6.20 9.66 7.63 − 183.91]
Assuming c = 1, the gains of the agents are given in Table 4.1. The trajectories of the
agents are shown in Figure 4.4 and we observe that the agents converge to the desired
point.
4.5.3 Pursuit sequence invariance properties
For GCCP, the stability and rendezvous point invariance with respect to the pursuit
sequence hold only when the elements of the columns of χ sum up to zero. Consider the
pursuit sequence matrix χ, given in (4.66), which has the property that the sum of the
elements columns sum χ is zero.
We compare the invariance property of GCCP with CCP. For comparison, we simulate
the CCP in 3-D with the initial positions given in Table 4.1 and consider the basic pursuit
sequence as (BPS, w), where
w = (0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.9)
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit89
0
50
100
0
50
1000
50
100
(a) CCP with pursuit sequence (BPS, w)
0
50
100
0
50
1000
50
100
(b) GCCP with pursuit sequence (BPS, χ)
Figure 4.5: Trajectories of the agents under centroidal cyclic pursuit (CCP) and gener-alized centroidal cyclic (GCCP) (satisfying some properties) demonstrating the pursuitsequence invariance of the rendezvous point Zf = (64.3, 41.3, 58.7) (Case V)
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit90
0
50
100
0
50
1000
50
100
(a) CCP Zf = (64.3, 41.3, 58.7)
0
50
100
0
50
1000
50
100
(b) GCCP Zf = (60.12, 12.47, 58.40)
Figure 4.6: Trajectories of the agents under CCP and GCCP demonstrating that therendezvous point is not pursuit sequence invariance (Case VI)
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit91
χ =
−1 0.01 0.01 0.01 0.01 0.01 0.31 0.01 0.01 0.01 0.60 0.01
0.01 −1 0.70 0.21 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01
0.01 0.01 −1 0.01 0.01 0.01 0.01 0.10 0.01 0.81 0.01 0.01
0.25 0.01 0.01 −1 0.66 0.01 0.01 0.01 0.01 0.01 0.01 0.01
0.01 0.01 0.01 0.01 −1 0.01 0.01 0.01 0.01 0.01 0.01 0.90
0.01 0.01 0.01 0.01 0.01 −1 0.60 0.01 0.01 0.01 0.31 0.01
0.01 0.01 0.01 0.01 0.01 0.90 −1 0.01 0.01 0.01 0.01 0.01
0.01 0.90 0.01 0.01 0.01 0.01 0.01 −1 0.01 0.01 0.01 0.01
0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.81 −1 0.10 0.01 0.01
0.01 0.01 0.21 0.70 0.01 0.01 0.01 0.01 0.01 −1 0.01 0.01
0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.90 0.01 −1 0.01
0.66 0.01 0.01 0.01 0.25 0.01 0.01 0.01 0.01 0.01 0.01 −1
(4.66)
Case V : We demonstrate the invariance property of the GCCP when the sum of the
elements of the rows of the pursuit sequence matrix is zero. We consider the gains same
as in Case I and the basic pursuit sequence as BPS. For GCCP, the pursuit sequence
matrix χ is given in (4.66) and for the CCP, the weight considered are w. The trajectories
of the agents under CCP and GCCP are shown in Figure 4.5. The rendezvous point
Zf = (64.3, 41.3, 58.7) is the same in both the cases and demonstrates the invariance of
the rendezvous point.
Case VI : We demonstrate that GCCP does not have the invariance properties with
respect to the pursuit sequences when the sum of the rows of pursuit sequence matrix is
not zero. We consider the same gains as in Case I. The basic pursuit sequence is BPS.
For GCCP, the pursuit sequence matrix χ is given in (4.64) and for CCP, the weights
are w. Figure 4.6 shows the trajectories of the agents under GCCP and CCP. It can be
seen from the figure that the agents converge at different points Zf = (64.3, 41.3, 58.7)
for CCP (Figure 4.6(a)) and Zf = (60.12, 12.47, 58.40) for GCCP (Figure 4.6(b)) and
hence, the rendezvous point is not pursuit sequence invariant.
Chapter 4. Rendezvous using linear Generalized Centroidal Cyclic Pursuit92
4.6 Conclusions
In this chapter, stable behaviour of a swarm of agents under generalized centroidal cyclic
pursuit (GCCP) is studied. The stability, reachable point and the reachable sets are
obtained. In Chapter 2-4, we considered gains to be such that the system is stable. In
the next chapter, we will study the behaviour of the system for unstable gains, and show
that instability can be exploited to obtain directed motion of the swarm of agents.
Chapter 5
Directed motion using linear cyclic
pursuit
Under linear cyclic pursuit, a system of n agents, with stable gains, will converge to a
point. In this chapter we show that, when the gains are unstable, the agents will converge
to a directed motion, if certain conditions are satisfied. These conditions are explored
under different cyclic pursuit strategies and the directed motion is characterized. An
alternate approach for directed motion is also discussed.
5.1 Directed motion using basic cyclic pursuit (BCP)
Basic cyclic pursuit has been formulated in Section 2.1. Given a group of n agents with
BPS = (1, 2, . . . , n), the equation of motion of the agents is given as
Zi(t) = ki(Zi+1(t)− Zi(t)) (5.1)
The system of agents can be analyzed using (2.6), which is reproduced here as
X = AX (5.2)
93
Chapter 5. Directed motion using linear cyclic pursuit 94
The solution of (5.2) is given in (2.19). We rewrite (2.19) as
xi(t) = xf +n−1∑p=1
{ n∑q=1
( np∑r=1
aipqrt
r−1)xq(0)
}eRpt (5.3)
where, we consider aipqr to be a complex number. The other variables are as defined in
(2.19). Unlike Chapter 2, we analyze the unstable system in this chapter. When the
system is unstable, one or more eigenvalues of A will be on the RHS of the s-plane. The
behaviour of the agents depends on the most positive eigenvalue which is defined as
Definition 5.1 The most positive eigenvalue of a linear system is defined as the eigen-
value with the largest real part.
Note that, this is different from the notion of dominant eigenvalue [57] which is the
eigenvalue with the highest absolute value. With this definition, we state the following
theorem.
Theorem 5.1 Consider a system of n-agents with equation of motion given in (5.1).
The trajectory of all the agents converge to a straight line as t → ∞ if and only if the
most positive eigenvalue of A is real and positive.
Proof. If the most positive eigenvalue is positive, then (5.2) is unstable. Let the unit
vector along the velocity vector of agent i at time t be −→vi , where
−→vi (t) =1
vi(t)
[v1
i (t) v2i (t) . . . vd
j (t)]T
(5.4)
and vδi (t) = zδ
i (t),∀δ, vi(t) =√{z1
i (t)}2 + · · ·+ {zdi (t)}2. If all the agents have to
converge to a straight line as t →∞, then
limt→∞
vδi (t)
vi(t)= lim
t→∞vδ
j (t)
vj(t), ∀i, j, δ (5.5)
Chapter 5. Directed motion using linear cyclic pursuit 95
Equivalently, for all i, j ∈ {1, · · · , n} and δ, γ ∈ {1, · · · , d},
limt→∞
vδi (t)
vγi (t)
= limt→∞
vδj (t)
vγj (t)
= θγδ (5.6)
where, θγδ is a constant independent of time and agent identity. To prove (5.6), consider
any one of the d-dimensions, represented by xi(t). Differentiating (5.3), we get
xi(t) =n−1∑p=1
[n∑
q=1
{ np∑r=1
aipqr
((r − 1)tr−2 + Rpt
r−1)}
xq(0)
]eRpt (5.7)
Let V = X, then
V = X = AX = AV , V (0) = AX(0) (5.8)
Thus, V has the same dynamics as (5.2) and vi(t) can be obtained, similar to (5.3), as
vi(t) =n−1∑p=1
{ n∑q=1
( np∑r=1
aipqrt
r−1)Axq(0)
}eRpt (5.9)
Here, vf = 0 (corresponding to xf in (5.3)). Comparing (5.7) and (5.9), for r = np, we
get
aipqnp
Rp = kq−1aip(q−1)np
− kqaipqnp
(5.10)
Then, from (5.10),
aipqnp
aip1np
=
q∏
l=2
kl−1
(Rp + kl)= Mpq (5.11)
for q > 1 and Mp1 = 1. It can be seen that Mpq is independent of the agent identity i.
Chapter 5. Directed motion using linear cyclic pursuit 96
Now, the instantaneous slope of the trajectory of the agent i in the (γ, δ) plane is
given by
vδi (t)
vγi (t)
=zδ
i (t)
zγi (t)
=
∑n−1p=1
[ ∑nq=1
{ ∑np
r=1 aipqr
((r − 1)tr−2 + Rpt
r−1)}
zδq(0)
]eRpt
∑n−1p=1
[ ∑nq=1
{∑np
r=1 aipqr
((r − 1)tr−2 + Rptr−1
)}zγ
q (0)
]eRpt
(5.12)
The trajectory (5.12) is evaluated as t → ∞. Let σm > σp, ∀p, p 6= m, then σm is the
real part of the most positive eigenvalue of A. Now, if ωm = 0, then the most positive
eigenvalue is real. Dividing the numerator and denominator of (5.12) by eσm and equating
all the terms containing e(Rp−σm)t to zero as t →∞, we get,
limt→∞
vδi (t)
vγi (t)
= limt→∞
∑nq=1
{∑nm
r=1 aimqr
((r − 1)tr−2 + Rmtr−1
)}zδ
q(0)
∑nq=1
{ ∑nm
r=1 aimqr
((r − 1)tr−2 + Rmtr−1
)}zγ
q (0)
(5.13)
Similarly, dividing the numerator and denominator of (5.13) by tnm−1
limt→∞
vδi (t)
vγi (t)
= limt→∞
∑nq=1 ai
mqnmRmzδ
q(0)∑nq=1 ai
mqrRmzγq (0)
= limt→∞
∑nq=1 ai
mqnmzδ
q(0)∑nq=1 ai
mqrzγq (0)
(5.14)
Using (5.11), Eqn. (5.14) can be written as
limt→∞
vδi (t)
vγi (t)
=
∑nq=1 Mmqz
δq(0)∑n
q=1 Mmqzγq (0)
= θγδ (5.15)
where, θγδ is independent of time and the agent identity i. It is a constant and a function
of k, η, σm and Zi0, ∀i. Let,
Zf =n∑
i=1
( 1/ki∑nj=1 1/kj
)Zi(0) (5.16)
Chapter 5. Directed motion using linear cyclic pursuit 97
which is the reachable point in case of stable system. For the unstable system, we define
this point as the asymptote point. Following a similar procedure,
limt→∞
zδi − zδ
f
zγi − zγ
f
=
∑nq=1 Mmqz
δq(0)∑n
q=1 Mmqzγq (0)
(5.17)
Therefore, as t →∞, from (5.15) and (5.17), we can show that
zδi
zγi
=zδ
i − zδf
zγi − zγ
f
(5.18)
Hence, (zγf , zδ
f ) is on the straight line along which the agents converge as t →∞.
Now, to prove the converse, let ωm 6= 0, then
limt→∞
vδi
vγi
= limt→∞
∑nq=1
(ai
mqnmRmejωmt + ai∗
mqnmR∗
me−jωmt)zδ
q(0)
∑nq=1
(ai
mqnmRmejωmt + ai∗
mqnmR∗
me−jωmt)zγ
q (0)(5.19)
where R∗m is the conjugate of Rm and
aimqnp
=(s−Rm
)nm biq(s)
ρ(s)
∣∣∣∣s=Rm
, ai∗mqnp
=(s−R∗
m
)nm biq(s)
ρ(s)
∣∣∣∣s=R∗m
(5.20)
where ρ(s) is the characteristic equation of A as given in (2.8). Therefore, (5.19) can be
written as
limt→∞
vδi
vγi
= limt→∞
∑nq=1 rq cos(φq + ωmt)zγ
q (0)∑nq=1 rq cos(φq + ωmt)zγ
q (0)(5.21)
where, aimqnm
Rm = rqejφq . From the above, it is seen that if ωm 6= 0, the agents will not
converge to a straight line. ¤
Remark 5.1: The straight line asymptote of the trajectories (after sufficiently large
time) passes through Zf = [z1f z2
f . . . zdf ]
T ∈ Rd. We call this point as the asymptote
point as stated before.
Chapter 5. Directed motion using linear cyclic pursuit 98
Remark 5.2: Even though the agents converge to a straight line, the direction of motion
of all the agents need not be the same. In fact, if the gain of only one agent is negative,
all the agents move in the same direction, otherwise they may move in two opposite
directions along the straight line.
Remark 5.3: When ωm 6= 0, the agents do not converge to a straight line. However,
the direction in which agent i moves, after sufficiently large t, can be calculated from
(5.21).
The condition in Theorem 5.1 can be further simplified. Instead of finding the eigen-
values of A, the condition on the gains of the agents can be found, under which the
agents will converge to a straight line. To obtain the conditions, the following lemma is
required.
Lemma 5.1 If α± jβ be a complex conjugate root of
f(s) =n∏
i=1
(s + ki)− κ (5.22)
where κ ∈ R, then f(α) > 0 if κ < 0 and f(α) < 0 if κ > 0
Proof. Since α± jβ is a complex conjugate pair of roots of f(s),
f(α± jβ) =n∏
i=1
(α + ki ± jβ)− κ = 0 ⇒ 1
κ
[ n∏i=1
{(α + ki)
2 + β2}] 1
2
= 1 (5.23)
If β 6= 0, then 1κ2
∏ni=1(α + ki)
2 < 1. Hence, if κ > 0,
−κ <
n∏i=1
(α + ki) < κ ⇒ f(α) =n∏
i=1
(α + ki)− κ < 0 (5.24)
For κ < 0, it can be similarly shown that f(α) > 0 ¤
In (5.22), if κ =∏n
i=1 ki, then f(s) = ρ(s), where ρ(s) is the characteristic equation
of A as given in (2.8). Let f(s) = q(s) when κ = 0. Now, the gains are arranged in an
Chapter 5. Directed motion using linear cyclic pursuit 99
increasing sequence as k1 ≤ k2 ≤ · · · ≤ kn to prove the following result.
Theorem 5.2 The n agents, with equation of motion given in (5.1), will converge to a
straight line asymptotically if and only if the gains are unstable and any of the following
conditions are satisfied
1)∏n
i=1 ki =∏n
i=1 ki > 0
2) k1 < 0, and ki > 0, i = 2, . . . , n
3) ∃ ξm ∈ (−k2,−k1) such that ξm is a root of dq(s)ds
and satisfies q(ξm) <∏n
i=1 ki < 0
Proof. First, it is proved that if system of n agents is unstable and any one of the three
conditions is satisfied, the agents converge to a straight line, that is, from Theorem 5.1,
the most positive eigenvalue of ρ(s) is real and positive. Since, the system is unstable,
at least the most positive eigenvalue has a positive real part. It remains to be shown
that the most positive eigenvalue is also real.
Case 1 : When∏n
i=1 ki > 0, from Lemma 5.1, if α ± jβ (β 6= 0) is a root of ρ(s), then
ρ(α) < 0. Since ρ(s) → +∞ as Re(s) →∞, the most positive eigenvalue of ρ(s) has to
be real.
Case 2 : This can be proved as a special case of Case 3 and will be proved after Case 3.
Case 3 : Consider the root locus of (5.22) parameterized by κ where κ varies from 0 to
−∞. The breakaway points of f(s) is the solution of
dq(s)
ds=
d
ds
( n∏i=1
(s + ki))
= 0 (5.25)
Let ξ be a solution of (5.25), and let ξ is real. Now, if the gain of f(s) at s = ξ is more
than∏n
i=1 ki, i.e.,∏n
i=1(ξ + ki) = q(ξ) >∏n
i=1 ki, the roots of f(s) that are approaching
the point ξ will remain real when κ =∏n
i=1 ki, that is, when f(s) = ρ(s).
Chapter 5. Directed motion using linear cyclic pursuit 100
The root locus exists between (−k2,−k1). Let the breakaway point in the region
(−k2,−k1) be s = ξm. If q(ξm) <∏n
i−1 ki, ρ(s) has a real root, say ξr, where ξr > ξm
and ξr is the most positive eigenvalue of ρ(s).
We need to show that none of the complex conjugate roots of ρ(s) has a real part
more than ξr. We use Lemma 5.1 to prove it. Let ξ1 and ξ2 be the two real roots of f(s)
approaching the breakaway point ξm. From the structure of f(s), we see that f(s) < 0
for s ∈ (ξ1, ξ2). Thus, from Lemma 5.1, f(s) cannot have any complex conjugate roots
whose real part lies in (ξ1, ξ2). Hence, the locus of complex conjugate roots (which is
continuous) cannot cross ξm for 0 < κ <∏n
i=1 ki. Therefore, ξr remains the most positive
root of ρ(s).
Case 2 : Here, k1 < 0 and k2 > 0. Considering (5.22), the root locus will exist in the
region (−k2,−k1). We know that ρ(s) has a root at the origin and the origin lies in
(−k2,−k1). Therefore, when κ =∏n
i=1 ki, f(s) will have two real roots in the region
(−k2,−k1), one at the origin and the other positive (say ξr > 0), since the system is
unstable. This is the only real positive root of f(s) when κ =∏n
i=1 ki. As proved in
Case 3, no other complex conjugate roots will have a real part more than ξr. Thus, the
most positive eigenvalue of ρ(s) is real.
The converse is proved by contradiction. Let the agents converge to a straight line
as t → ∞, i.e., the most positive eigenvalue of ρ(s) is real and positive but either the
system is not unstable or none of the three conditions holds. This means that one of the
following two condition is true:
(a) The system is stable
(b)∏n
j=1 kj < q(ξm) < 0 and more than one gain is negative.
If the system is stable, then the agents converge to a point. Therefore, they cannot
converge to a straight line. This contradicts our assumption.
Chapter 5. Directed motion using linear cyclic pursuit 101
If more than one gain is negative and∏n
j=1 kj < q(ξm) < 0, then following the
arguments in Case 3 above, we can conclude that the most positive eigenvalue cannot
be real. Hence, the agents cannot converge to a straight line. ¤
Therefore, under basic cyclic pursuit, whether the agents will converge to straight
line can be determined from the value of the gains. In the next section, the invariance
properties of the directed motion are discussed.
5.2 Invariance properties under basic cyclic pursuit
(BCP)
Similar to stability and rendezvous for the stable system, the unstable system leading to
directed motion of the agents has some invariance properties. Here, the same definition
is used for finite and infinite pursuit sequence switching as used in the case of stability
and rendezvous.
Theorem 5.3 Asymptote point of a linear basic cyclic pursuit, that satisfied Theorem
5.1, is pursuit sequence invariant.
Proof. For the agent i, as t →∞, the unit velocity vector can be written, using (5.15),
as υi(t) = (v1i /vi)[1, θ12, θ13, . . . , θ1d]
T . Since θγδ,∀γ, δ; γ 6= δ depends on the pursuit
sequence of the agents, the unit velocity vector varies as the pursuit sequence is changed.
Thus, for different pursuit sequence, the asymptote along which the agents converge
is different. However, from (5.17), all the asymptotes pass through the point Zf (the
asymptote point), which is independent of the pursuit sequence. ¤
Thus, the asymptote point does not change with the pursuit sequence. However, the
slope of the asymptote depends on the pursuit sequence, as (5.15) has Mmq, which is
dependent on the pursuit sequence of the agents. Hence, the slope of the asymptote is
not pursuit sequence invariant.
Chapter 5. Directed motion using linear cyclic pursuit 102
Theorem 5.4 Asymptote point of a linear basic cyclic pursuit, that satisfies Theorem
5.1 is invariant under finite switching.
Proof. From Theorem 2.8 (reachability with switching), we see that the point Zf
remains invariant even after a finite number of switching of connection among the agents.
Theorem 5.3 shows that irrespective of the connection between the agents, the asymptote,
along which the agents converge, passes through the point Zf . Hence, even after a finite
number of switching of connections, the agents will converge to a line that will pass
though Zf . ¤
Note that the trajectory of the agents may change due to switching but the asymp-
tote point remains unchanged. The direction of motion of the agents change with each
switching. As t →∞, the agents move along the direction corresponding to the pursuit
sequence after the last switch. The asymptote the agents follow after the last switch
passes through the asymptote point as proved in the above theorem.
For infinite switching, we can find the asymptote after each switch, assuming that
there will be no more switching of the pursuit sequence. If all these asymptotes pass
through the asymptote point, we can call the asymptote point to be invariant under
infinite switching of pursuit sequence. Then, for linear basic cyclic pursuit, satisfying
Theorem 5.1, the asymptote point is invariant under infinite switching of pursuit se-
quence. This is just an extension of the Theorem 5.4.
In the next section, we discuss the directed motion of the system under generalized
centroidal cyclic pursuit.
5.3 Directed motion using generalized centroidal cyclic
pursuit (GCCP)
Under centroidal cyclic pursuit (CCP) and generalized centroidal cyclic pursuit (GCCP),
the agents can execute directed motion under the conditions stated in Theorem 5.1. We
Chapter 5. Directed motion using linear cyclic pursuit 103
prove the theorem for GCCP only, as CCP can be considered as a special case of GCCP.
The equation of motion of agent i under GCCP is given in (4.2) and is reproduced here
Zi(t) = ki(Zic − Zi) (5.26)
Theorem 5.5 Consider a system of n-agents with equation of given in (5.26). The
trajectory of all the agents converge to a straight line as t → ∞ if and only if the most
positive eigenvalue of A is real and positive.
Proof. The proof is similar to that of Theorem 5.1. For the agents to converge to a
straight line (5.6) has to be satisfied. For GCCP, (5.7) and (5.9) hold and we rewrite
them for further analysis
xi(t) =n−1∑p=1
[n∑
q=1
{ np∑r=1
aipqr
((r − 1)tr−2 + Rpt
r−1)}
xq(0)
]eRpt (5.27)
vi(t) =n−1∑p=1
{ n∑q=1
( np∑r=1
aipqrt
r−1)Axq(0)
}eRpt (5.28)
Comparing (5.27) and (5.28), for r = np
n∑q=1
Rpaipqnp
=n∑
q=1
aipqnp
A (5.29)
Let aip = [ai
p1np· · · ai
pnnp]T , then (5.29) can be written in the matrix form as
aip(RpI − A) = 0 (5.30)
Since Rp is an eigenvalue of A, aip is the right eigenvector of A. Then, each element of
aip can be expressed, independent of the agent identity i, as
aipqnp
= Mpq(k, η, Rp) (5.31)
Chapter 5. Directed motion using linear cyclic pursuit 104
Then, the instantaneous slope of the trajectory of agent i in the (γ, δ) plane can be
obtained, similar to (5.15), as
limt→∞
vδi (t)
vγi (t)
=
∑nq=1 Mmqz
δq(0)∑n
q=1 Mmqzγq (0)
= θγδ (5.32)
where Mmq is as obtained in (5.31). Here, θγδ is independent of time and the agent
identity i and gives the slope of the straight line asymptote.
Now, let Zf =∑n
i=1
(|ξi|/kiPn
j=1 |ξj |/kj
)Zi(0), which is the reachable point of GCCP, when
the system is stable. Following a similar procedure,
limt→∞
zδi − zδ
f
zγi − zγ
f
=
∑nq=1 Mmqz
δq(0)∑n
q=1 Mmqzγq (0)
(5.33)
Therefore, as t →∞,
zδi
zγi
=zδ
i − zδf
zγi − zγ
f
(5.34)
and hence, (zγf , zδ
f ) is on the straight line asymptote.
The ‘only if’ part, that is, if the most positive eigenvalue is real, then the agents
will execute a directed motion, can be proved similar to Theorem 5.1 and the details are
omitted here. ¤
Remark 5.4: For GCCP, we do not have any results similar to Theorem 5.2, that
simplifies Theorem 5.5. This is because, it is difficult to locate the eigenvalues of (5.2)
in the s plane, for GCCP.
Remark 5.5: The asymptote point of the unstable GCCP is the same as the rendezvous
point of the stable GCCP. Hence, the invariance property of the asymptote point for
GCCP will not hold in general as seen in the case of rendezvous point. However, if the
pursuit sequence matrix of A is such that the sum of the elements of all the column of
the pursuit sequence matrix is zero (as discussed in Section 4.4), then the asymptote
point will be invariant with respect to the pursuit sequence as proved in Section 5.2.
Chapter 5. Directed motion using linear cyclic pursuit 105
The next section combines the ideas of stable and unstable system to derive an
alternate method for directed motion of the agents.
5.4 Directed motion: An alternate approach
So far, it is seen that if the system is stable, rendezvous occurs and if unstable, then
under certain conditions, the agents asymptotically converge to a straight line. Also, the
equation of motion along each coordinate axis is evolves independently. This inspires
us to use different sets of gains along different coordinate axes such that the system is
stable along some axis and unstable long the other.
Assume d = 2, that is, the agents are in a plane. A stable set of gains can be assigned
along one axis while along the other, the set of gains can be unstable. Then, the agents
will converge to a point along the stable axis while converge to a straight line along the
other. When these two motions are combined, a directed motion is obtained. The agents
can converge along any direction by appropriate rotation of the axis.
Suppose, the agents have to move in a line formation along a direction given by θ.
The coordinate system can be rotated such that in the new coordinate system (y1, y2),
the axis y1 is aligned along θ. Therefore,
[y1
y2
]=
[cos θ sin θ
− sin θ cos θ
][z1
z2
](5.35)
The state equation of the ith agents in the new coordinate system are
y1i = k1
i (y1ic − y1
i ) (5.36)
y2i = k2
i (y2ic − y2
i ) (5.37)
where, yjic is the weighted centroid that the ith agent follows in the direction j, k1
i and
Chapter 5. Directed motion using linear cyclic pursuit 106
k2i are the gains along the y1 and y2 directions, respectively. Equation (5.36) and (5.37)
can be expressed in the previous coordinate system as
z1i = (k1
i cos2 θ + k2i sin2 θ)(z1
ic − z1i ) + (k1
i − k2i ) sin θ cos θ(z2
ic − z2i ) (5.38)
z2i = (k2
i cos2 θ + k1i sin2 θ)(z2
ic − z2i ) + (k1
i − k2i ) sin θ cos θ(z1
ic − z1i ) (5.39)
Thus, we get the control law as
uji = (kj
i cos2 θ + kj+1i sin2 θ)(zj
ic − zji ) + (k1
i − k2i ) sin θ cos θ(zj+1
ic − zj+1i ) (5.40)
where j = 1, 2. Now, let the gains k1i , ∀i, satisfy Theorem 5.5. Then the agents will
asymptotically converge to a straight line parallel to the y1 axis. If the gains k2i , ∀i, are
stable, then the agents will converge to the point (say, y2f ) on the y2 axis given by
y2f =
n∑i=1
|ξi|/ki∑nj=1 |ξj|/kj
{cos θz2
i (t0)− sin θz1i (t0)
}(5.41)
Thus, with this control law, the agents will move in a line formation along the straight
line which is inclined at θ and passes through the point (z1f , z
2f ) given as
z1f =
n∑i=1
|ξi|/ki∑nj=1 |ξj|/kj
{sin2 θz1
i (t0)− sin θ cos θz2i (t0)
}(5.42)
z2f =
n∑i=1
|ξi|/ki∑nj=1 |ξj|/kj
{cos2 θz2
i (t0)− sin θ cos θz1i (t0)
}(5.43)
This law can be extended to a general d dimensional space.
In the next section, simulations are carried out to verify the results obtained in this
chapter.
Chapter 5. Directed motion using linear cyclic pursuit 107
Agents Initial Position
Gains
Case I Case II Case III Case IVCase X
π/4 axis 3π/4 axis
1 (10,-1) -3 -3 -3 -3 -0.5 1
2 (7,2) 6 6 -6 -6 1 1
3 (0,10) 8 8 8 -8 1 1
4 (-7,5) -10 10 -15 -10 1 1
5 (4,-8) 12 12 12 -12 1 1
Table 5.1: Initial positions of the agents and their gains for different cases of directedmotion under BCP
5.5 Simulation results
5.5.1 Directed motion under BCP
A system of 5 agents is considered in R2. The initial positions of the agents are given in
Table 5.1. We assume the pursuit sequence of the agents are
BPS0 = (1, 2, 3, 4, 5)
Case I : Consider the gains given in Table 5.1. This set of gains satisfies Condition (i)
of Theorem 5.2. The agents converge to a straight line (Figure 5.1). The eigenvalues
of this system are given in Table 5.2, from which it can be seen that the most positive
eigenvalue is real. The slope of the asymptote, calculated from (5.15), is −33.46◦ which
matches with the simulation result. Moreover, it can be seen that some agents move in
one direction while the others in the opposite direction along the asymptote as t →∞.
Case II : Consider the gains given in Table 5.1, which satisfies Condition (ii) of Theorem
5.2. The agents asymptotically converge to a straight line (Figure 5.2). The eigenvalues
of this system are shown in Table 5.2, from which it can obtained that the most positive
eigenvalue is real. The slope of the asymptote, calculated from (5.15), is −34.9238◦ which
Chapter 5. Directed motion using linear cyclic pursuit 108
−50 −25 0 25 50−50
−25
0
25
50
12
3
5
4
Figure 5.1: The trajectories of 5 agents when the gains of the agents satisfies Condition(i) of Theorem 5.2 (Case I)
Case I Case II Case III Case IV
-11.3 + j2.3 -15.0 -12.7 0
-11.3 - j2.3 -9.6 + j5.5 -6.1 5.5 + j6.05
-0.7 -9.6 - j5.5 0 5.5 - j6.05
0 0 8.2 14.0 + j3.8
10.4 1.3 14.6 14.0 - j3.8
Table 5.2: Eigenvalues of A for Cases I-IV
Chapter 5. Directed motion using linear cyclic pursuit 109
−20 0 20 40 60−30
−10
10
20
21
5
3
4
Figure 5.2: The trajectories of 5 agents when the gains of the agents satisfies Condition(ii)of Theorem 5.2 (Case II)
−50 −25 0 25 50−50
−25
0
25
50
1
2
3
4
5
Figure 5.3: The trajectories of 5 agents when the gains of the agents satisfies Condi-tion(iii) of Theorem 5.2 (Case III)
matches with the simulation.
Case III : Consider the gains given in Table 5.1. This set of gains does not satisfy
Conditions (i) and (ii) of Theorem 5.2. Since q(ξm) = −7.8×104 <∏n
i=1 ki = −2.5×104,
Condition(iii) of Theorem 5.2 is satisfied and the agents converge asymptotically to
a straight line (Figure 5.3). The eigenvalues are shown in Table 5.2 and the slope,
calculated from (5.15), is −36.08◦ which matches with the simulation.
Case IV : Consider the gains given in Table 5.1. Here, none of the conditions of Theorem
5.2 are satisfied as, q(ξm) = −132.15 >∏n
i=1 ki = 17280. Hence, the agents do not
Chapter 5. Directed motion using linear cyclic pursuit 110
−100 −50 0 50 100−100
−50
0
50
100
1
2
5
3
4
Figure 5.4: The trajectories of 5 agents when the gains of the agents satisfies none of theconditions of Theorem 5.2 (Case IV)
converge asymptotically to a straight line (Figure 5.4). The eigenvalues are shown in
Table 5.2 The most positive eigenvalue is not real and hence, the system also violates
the condition of Theorem 5.1.
5.5.2 Pursuit sequence invariance
We demonstrate the invariance of the asymptote point with respect to the pursuit se-
quence and also with respect to the finite switching of the pursuit sequences. Let us
consider the following basic pursuit sequence
BPS0 = (1, 2, 3, 4, 5)
BPS1 = (1, 5, 4, 3, 2)
BPS2 = (1, 3, 5, 2, 4)
BPS3 = (1, 4, 2, 5, 3)
Case V : Consider the same gains as in Case II and two pursuit sequences BPS0 and BPS1
(Figure 5.5). The slopes of the asymptotes are −34.9238◦ and −21.74◦. However, in both
the cases, the asymptotes pass through the same asymptote point Zf = (−17.9, 12.3).
Chapter 5. Directed motion using linear cyclic pursuit 111
−40 0 40 80 120−80
−40
0
40
2
5
3
14
Asymptote Point
(a) Pursuit sequence BPS1
−40 0 40 80 120−80
−40
0
40
5
21
3
4
Asymptote Point
(b) Pursuit sequence BPS0
Figure 5.5: Simulation to demonstrate pursuit sequence invariance of the asymptotepoint (Case V)
Here, the direction of motion varies with the pursuit sequence but the asymptote point
remains the same.
Case VI : We demonstrate the invariance of asymptote point with respect to finite switch-
ing of the pursuit sequences. Consider the gains case as in Case II. The pursuit sequence
switches from BPS0 to BPS2 to BPS3 at t = 0.02 and t = 0.15, respectively. Figure
5.6 shows that the asymptote point, Zf = (−17.9, 12.3) remains the same even with
switching, but the slope changes.
5.5.3 Directed motion under GCCP
Unlike the previous cases, we will do 3−D simulations. Consider a swarm of 12 agents in
R3. The initial positions of the agents are shown in Table 5.3 and the pursuit sequences
of the agents are given in (4.64). Different sets of gains, shown in Table 5.3, are selected
to demonstrate the results under GCCP.
Case VII : A negative gain is selected for Agent 11 while the other gains are positive
(Table 5.3).∑n
i=1 |ξi|( ∏n
j=1,j 6=i kj
)= −4.68 × 107 < 0 and the system is unstable.
The eigenvalues of this system is shown in Table 5.4 and we observe that Theorem
5.5 is satisfied. The asymptote point is computed as Zf = (45.25, 88.11, 42.42). The
Chapter 5. Directed motion using linear cyclic pursuit 112
−40 0 40 80 120−80
−40
0
40
12
5
3
4
Asymptote Point
(a) Finite switching−40 0 40 80 120
−80
−40
0
40
5
21
3
4
Asymptote Point
(b) No switching
Figure 5.6: Simulation to demonstrate finite pursuit sequence switching invariance of theasymptote point (Case VI)
Agent Initial positionGains
Case VII Case VIII
1 (48 , 83 , 37) 6 -6
2 (47 , 97 , 14) 8 -8
3 (71 , 30 , 1) 7 -7
4 (24 ,100 , 64) 10 10
5 (72 , 44 , 74) 1 1
6 (87 , 1 , 3) 10 10
7 (41 , 29 , 10) 3 3
8 (42 , 69 , 89) 10 10
9 (95 , 49 , 76) 7 7
10 (88 , 8 , 58) 5 5
11 (70 , 20 , 69) -1 1
12 (31 , 98 , 97) 5 5
Table 5.3: Initial positions of the agents and their gains for different cases of directedmotion using GCCP
Chapter 5. Directed motion using linear cyclic pursuit 113
Case VI Case VII
5.7944 1.3862 + j4.3397
1.8749 1.3862 - j4.3397
-1.9292 + j1.2494 0.3184 + j1.8799
-1.9292 - j1.2494 0.3184 - j1.8799
-1.5682 0.8675
-0.7616 + j0.9499 -1.6111
-0.7616 - j0.9499 -1.3931
-0.5254 -0.6058 + j0.0596
-0.0229 + j0.2341 -0.6058 - j0.0596
-0.0229 - j0.2341 0.0752
-0.1182 -0.0321
-0.0299 -0.1041
Table 5.4: Eigenvalues of A for Cases VI-VII
simulation, given in Figure 5.7, shows that all the agents move in the same direction as
expected.
Case IX : We demonstrate that if the most positive eigenvalue is not real, we do not
have directed motion. Consider the gains given in Table 5.3. The gains of the first three
agents are negative and the others are positive. The eigenvalues of the system is shown
in Table 5.4. These gains do not satisfy Theorem 5.5 and the trajectories do not converge
to any straight line as expected (shown in Figure 5.8).
5.5.4 Alternate approach of directed motion
Case X : Directed motion with different sets of gains in different coordinate axis is demon-
strated in Figure 5.9. Here, we consider 5 agents with initial positions given in Table
5.3. The goal is to align the agents along π/4. The set of gains along π/4 is unstable
while the gains are stable along the other axis (3π/4 axis) (Table 5.3). The simulation
Chapter 5. Directed motion using linear cyclic pursuit 114
0
300
600
−300
0
300−100
0
300
Asymptote Point
Figure 5.7: Trajectories of a swarm of agents when one gain is negative and Theorem5.5 is satisfied (Case VII)
−400
0
400
−400
0
400−400
0
400
Figure 5.8: Trajectories of a swarm of agents when three gain are negative and Theorem5.5 is not satisfied (Case IX)
Chapter 5. Directed motion using linear cyclic pursuit 115
−20 0 50 100−20
0
50
100
2
14
5
3
Figure 5.9: Directed motion with combination of stable and unstable gains
shows that the agents move in a line formation along π/4.
5.6 Conclusions
The behaviour of the unstable system of agents are studied in this chapter. It is seen
that under some conditions, the agents perform a directed motion. An alternate method
for directional motion is demonstrated. Certain invariance properties with respect to
pursuit sequence and switching of pursuit sequence is observed. In Chapter 2-5, we
considered linear cyclic pursuit where we studied the stable and unstable behaviour of
the holonomic agents. In the next chapter, we will consider nonlinear cyclic pursuit
where the non-holonomic agents are studied.
Chapter 5. Directed motion using linear cyclic pursuit 116
Chapter 6
Circular motion using nonlinear
Basic Cyclic Pursuit
In the previous chapters, we considered holonomic agents in cyclic pursuit that gave
rise to linear kinematics. However, mobile agents, such as wheeled vehicles or UAVs
(which are non-holonomic), cannot change their direction instantaneously, and under
cyclic pursuit give rise to nonlinear kinematics. The behaviour of the resulting het-
erogeneous agents under nonlinear basic heterogenous cyclic pursuit is analyzed in this
chapter. This is a generalization of the work done for homogeneous agents, that is,
agents that have equal speeds and gains. Here, the formation of the group of heteroge-
neous agents at equilibrium is analyzed. Limited results are also obtained regarding the
conditions for equilibrium to exist.
6.1 Problem formulation
Nonlinear cyclic pursuit is formulated using n agents in R2, ordered from 1 to n, where
agent i follows agent i + 1, that is, pursuit sequence is BPS = (1, 2, . . . , n2). Each agent
has a constant velocity Vi, and orientation αi (variable), with respect to a fixed reference
117
Chapter 6. Circular motion using nonlinear Basic Cyclic Pursuit 118
Ref
Vi
Vi+1
iθ
i+1P
Pi
iα
i+1α
Ref
ri
Figure 6.1: Basic formation geometry
(Figure 6.1). The distance between the ith and i + 1th agent is ri and the angle from the
reference to the line of sight (LOS) from agents i to i + 1 is given by θi. The control
input to the ith agent is the lateral acceleration ai which is given as
ai = kiφi (6.1)
where (with reference to Figure 6.2),
αi + φi − θi =
0, if 0 ≤ αi ≤ θi;
2π, otherwise.(6.2)
Here, ki is the controller gain. The lateral acceleration, so defined, guarantees that all
the agents move in the counter-clockwise direction. Thus, the equation of motion of the
ith agent is given as follows:
ri = Vi+1 cos(αi+1 − θi)− Vi cos(αi − θi) (6.3)
riθi = Vi+1 sin(αi+1 − θi)− Vi sin(αi − θi) (6.4)
αi =ai
Vi
=kiφi
Vi
(6.5)
Chapter 6. Circular motion using nonlinear Basic Cyclic Pursuit 119
Pi
Pi+1
Vi
φi
αiθi
Ref
Pi
Pi+1
Viφ
i
αi
θi
Ref
Figure 6.2: Representation of the angles with respect to a fixed reference
When ki = k and Vi = V, ∀i, the agents are all homogeneous, and this system have been
analyzed in [49].
6.2 Analysis for possible formations
For a system of n agents, with equation of motion given in (6.3) - (6.5), equilibrium is
said to have been achieved if the relative position of the agents do not change with time.
Therefore, at equilibrium,
ri = 0 (6.6)
θi = θi+1 (6.7)
∀i. In addition, we assume that at equilibrium
φi = 0 (6.8)
which is similar to the equilibrium condition considered in [49]. For a given system,
there can be more than one equilibrium point depending on the initial conditions. In this
paper, we present and analyze a necessary condition for equilibrium in the heterogenous
gain and speed case. Assuming that the n agents eventually converge to a equilibrium
formation, we have the following theorem.
Chapter 6. Circular motion using nonlinear Basic Cyclic Pursuit 120
φi
βi+1
Vi
Vi+1
Ri+1
Riπ/2−φ
i
π/2−βi+1
Pi
Pi+1
O
φi+1
ri
A
Figure 6.3: Multi-vehicle formation with circular trajectory
Theorem 6.1 At equilibrium, a system of n agents, with equation of motion given in
(6.3)-(6.5), move in concentric circles with equal angular velocities.
Proof. φi = 0 ⇒ φi = constant ⇒ ai = constant. An agent i, having constant speed
Vi and constant lateral acceleration ai, will move in a circular trajectory. Considering
Figure 6.3, let the position of the ith agent be Pi and the center of the circle traversed by
it be O. Since, φi = constant implies ∠OPiA = (90 − φi) is a constant. Therefore, the
position of i + 1th agent should be on the line PiA or its extension. The point at which
i + 1th agent lies on PiA is determined from ri = 0 which implies that ri =constant.
Therefore, 4OPiPi+1 forms a rigid triangle. Thus, agent i + 1 also has the center of its
circular trajectory at O. Again, since the configuration of the 4OPiPi+1 remains rigid
at equilibrium, we have ωi = ωi+1, where ωi is the rate of rotation of OPi. Hence, all
agents move in concentric circles with equal angular velocity . ¤
As pointed out before, there could be zero, one or several equilibrium points for a
given system. The necessary and sufficient condition for equilibrium point to exist or
for stability of equilibrium points are bound to be complicated for the general case. In
the following, we will derive only a necessary condition for the existence of equilibrium
Chapter 6. Circular motion using nonlinear Basic Cyclic Pursuit 121
based upon the geometry of the final equilibrium configuration.
Let the radius of the circle traversed by the first agent at equilibrium be
R1 = ρ (6.9)
Since the angular velocity is the same for all the agents, we have
ωi = ωi+1 ⇒ Vi
Ri
=Vi+1
Ri+1
(6.10)
⇒ Ri =Vi
ρ/V1
(6.11)
Now,
ai =V 2
i
Ri
= kiφi (6.12)
⇒ φi =ViV1
kiρ(6.13)
Again, from 4OPiPi+1,
Ri+1
Ri
=sin(π/2− φi)
sin(π/2− βi+1)(6.14)
⇒ βi+1 = cos−1
[Vi
Vi+1
cos
{V1Vi
kiρ
}](6.15)
Let the velocity ratio
Vi+1
Vi
= γi+1 (6.16)
Therefore,
n∑i=1
(φi + βi) =n∑
i=1
[ViV1
kiρ+ cos−1
{1
γi
cosVi−1V1
ki−1ρ
}]
(6.17)
Chapter 6. Circular motion using nonlinear Basic Cyclic Pursuit 122
i-1V
ri
ri-1
iV
i+1V
Ref
Ref
Ref
φi+1
Pi+1
Pi-1
Pi
φi
φi-1
βi
βi+1
Figure 6.4: Angle calculation for a general polygon of n sides
where the subscript indices are modulo n.
Now, consider any n sided polygon (not necessary regular), a part of which is shown
in Figure 6.4. Each node i represents the position of the agent i and the vector from
that node represents the velocity of the ith agent. Thus the angle (φi + βi) is measured
counter-clockwise from the extension of the line Pi−1Pi to the line PiPi+1, as shown in
Figure 6.4. Therefore, considering all possible polygonal topologies for a given n, we
have
∑(φi + βi) = 2qπ (6.18)
where, q = 1, 2, · · · , (n − 1). The q considered here is similar to the variable d used in
Definition 2 in [49]. Thus,
n∑i=1
[ViV1
kiρ+ cos−1
{1
γi
cosVi−1V1
ki−1ρ
}]= 2qπ (6.19)
Hence, if a system of n vehicles with arbitrary speeds and controller gains ki attains
equilibrium, (6.19) must be satisfied for some ρ and for some q = 1, 2, · · · , n − 1. We
will refine this condition by further analyzing (6.19).
Chapter 6. Circular motion using nonlinear Basic Cyclic Pursuit 123
For a given q, (6.19) is a function of ρ only. Therefore, the values of ρ that satisfies
(6.19) gives the radius of the circle of the first agent. The radius for the other agents
can be obtained from (6.11). Let us denote the left hand side (LHS) of (6.19) by f(ρ).
Theorem 6.2 (Necessary condition for equilibrium) Consider a system of n agent
system with equation of motion given in (6.3)-(6.5). A necessary condition for equilib-
rium is
maxiε{j:Vj>Vj+1}
ai ≤ miniε{j:Vj>Vj+1}
bi (6.20)
where,
ai =[mπ + cos−1(γi+1)
] ki
ViV1
, m = 0, 1 (6.21)
bi =[(m + 1)π − cos−1(γi+1)
] ki
ViV1
, m = 0, 1 (6.22)
Proof. A solution of (6.19) will exist if and only if the argument of the cos−1 term in
(6.19) lies in [−1, 1], i.e.,
∣∣∣∣cos
{ViV1
kiρ
}∣∣∣∣ ≤ γi+1, ∀ i (6.23)
Let us define the sets
X1 = {i : Vi < Vi+1} (6.24)
X2 = {i : Vi ≥ Vi+1} (6.25)
Note that both X1 and X2 are nonempty sets if not all the Vi’s are same. For all i ∈ X2,
(6.23) is always satisfied irrespective of the values of ρ. For i ∈ X1, (6.23) can be written
as
ai ≤ V1Vi
kiρi
≤ bi (6.26)
Chapter 6. Circular motion using nonlinear Basic Cyclic Pursuit 124
m=0
m=1
m=0
m=1
Vi+1
Vi
LOS
LOSπ2_
π2_
cos (γ )i+1
-1
cos (γ )-1
i+2
cos (γ )i+1
-1
cos (γ )-1
i+2
Pi
Pi+1
Pi+2
Figure 6.5: Representation of the range of φ in polar coordinate
π_2
π2
3__
.... .......... ......
aj
^
ai
^
aj^
ai
^
bj
^bj^
bi
^bi
^
φ
π_2
k
1j
j
V V___
......
aj a
ibj
bi
ρ
V Vi
i 1
k___ π_
2
Figure 6.6: A representation of the ranges of φ and ρ for different agents
Chapter 6. Circular motion using nonlinear Basic Cyclic Pursuit 125
where,
ai =[mπ + cos−1(γi+1)
]
bi =[(m + 1)π − cos−1(γi+1)
]
where m = 0,±1,±2, · · · . Using (6.13), we can see that (6.26) is a bound on φi. From
Figure 6.5, it is evident that the possible values of m are 0 or 1. Thus, for a given iεX1,
the range of values that ρ can take is given by Ri,
Ri = {ρ : ai ≤ 1
ρ≤ bi} (6.27)
where, ai and bi are given by (6.21) and (6.22) and m = 0, 1.
Note that Ri has a center at
ki
ViV1
(2m + 1
2π
)(6.28)
and a spread of
π − 2ki
ViV1
cos−1(γi+1) (6.29)
The range of φi and Ri, as given in (6.26) and (6.27), are illustrated in Figure 6.6. Since
both the center and spread of Ri are functions of Vi, Vi+1 and ki, they are different for
different i’s. Similarly, the distance between the centers also vary with i.
Now, there will exist some value of ρ that will belong to Ri given in (6.27), for all
i ∈ X1, if
⋂iεX1
Ri 6= ∅ (6.30)
Chapter 6. Circular motion using nonlinear Basic Cyclic Pursuit 126
This implies that for equilibrium to exist, (6.20) should be satisfied. Also, we can find a
ρ that will satisfy
max ai ≤ 1
ρ≤ min bi (6.31)
for all i ∈ X1. ¤
It is interesting to speculate what happens when
max ai > min bi (6.32)
From Figure 6.5, at equilibrium each velocity vector Vi should be within the shaded cone
shown in the figure. Now, from (6.10) and (6.21), we can write
φi =k1Vi
kiV1
φ1 (6.33)
Hence, all the cones can be translated to the point P1 where each cone represents the
bound within which V1 should lie for ri = 0,∀i. When max ai > min bi, the intersection
of the cones is empty and thus, there does not exist a feasible direction for the velocity
V1 at equilibrium to satisfy ri = 0, ∀i. In other words, for some i, there will be no φi that
can meet the requirement of the LOS speed components to match. Hence, there will be
no equilibrium.
The set of all values of ρ obtained from (6.31) need not satisfy (6.19). Only that ρ
which satisfies (6.19), for a given value of q, gives the radius of the circle at equilibrium.
An equilibrium may or may not exist for a given q. Also, equilibrium may occur
for more than one value of q. Hence, the system can attain any one of the equilibrium
states. However, we do not discuss the stability of these equilibrium points here.
Chapter 6. Circular motion using nonlinear Basic Cyclic Pursuit 127
6.3 Special case: Homogeneous system
We may consider a special case in which Vi = V and ki = k, for all i, which is the
assumption made in [49]. Then (6.19) reduces to
2V 2n
kρ= 2qπ (6.34)
⇒ ρ =V 2n
qkπ(6.35)
Thus, all the agents will move in a fixed circle of radius (V 2n)/(qπk) at equilibrium.
This is the same result as that obtained in [49]. The difference in the exact expression
is due to the choice of the controller gain which, in [49], is defined as kV . Again, from
(6.13),
φ =qπ
n(6.36)
which is the same as in [49]. Note that in the homogeneous speed case, the bounds on
1/ρ, given in Theorem 2, do not exist.
6.4 Simulation results
We considered 5 agents, that is, n = 5. The velocities and the gains of the agents are
given in Table 6.1. The pursuit sequence is BPS = (1, 2, 3, 4, 5). We demonstrate the
equilibrium formation of the agents for these sets of speed and gains.
Case I : For the speeds and gains of the 5 agents are given in Table 6.1, we have
X1 = {1 2 3 4} (6.37)
X2 = {5} (6.38)
The range of Ri are shown in in Figure 6.7 and also in Table 6.2. For both m = 0 and 1,
Chapter 6. Circular motion using nonlinear Basic Cyclic Pursuit 128
Agent GainSpeeds for Cases
I II III IV V
1 5 25 25 25 25 20
2 5 20 20 20 20 18
3 5 15 15 15 15 16
4 5 10 10 10 10 14
5 5 5 6 7 22 12
Table 6.1: Velocities and gains of the agents for different cases of nonlinear BCP
ρ1
ρ4
ρ3
ρ2
m = 0
50
41
33
24
194
138
89
48
ρ1
ρ4
ρ3
ρ2
m = 1
3322
2618
1914
129.6
ρ
50 100 150 2000
Figure 6.7: Range of ρi for Case 1
Chapter 6. Circular motion using nonlinear Basic Cyclic Pursuit 129
−60 −40 −20 0 20 40 60 80 100
−40
−20
0
20
40
60
80
Figure 6.8: Trajectories of n = 5 agents for Case I ( • - initial position, N - final positionof the UAVs)
⋂iεX1
Ri = ∅. The simulation result (Figure 6.8) also shows that the system does not
have an equilibrium.
Case II : Consider 5 agents with speeds and gains as in Table 6.1. Here, X1 and X2 are
same as in Case I. But now, for m = 0, R1, . . . ,R4 have non-empty intersection. The
ranges of ρ for which f(ρ) exist is
50.0 ≤ ρ ≤ 53.9 (6.39)
as shown in Table 6.2 and in Figure 6.9. We plot f(ρ) for this range of ρ in Figure
6.10. There is no value of ρ and q that satisfies (6.19). The simulation result given in
Figure 6.11 shows that this case indeed does not have an equilibrium. This shows that
the existence of ρ that satisfies (6.31) is not sufficient for (6.19) to have a solution.
Case III : Consider 5 agents with gains as in Table 6.1. Again, X1 and X2 are the same
as in Case I and II. The overlapping region of ρ for m = 0 is
50.0 ≤ ρ ≤ 62.9 (6.40)
Chapter 6. Circular motion using nonlinear Basic Cyclic Pursuit 130
Case I Case II Case III Case IV Case V
m=0
R1 [50.0 194.2] [50.0 194.2] [50.0 194.2] [50.0 194.2] [29.7 177.4]
R2 [41.3 138.4] [41.3 138.4] [41.3 138.4] [41.3 138.4] [27.0 151.3]
R3 [32.6 89.2] [32.6 89.2] [32.6 89.2] [32.6 89.2] [24.3 126.6]
R4 [23.9 47.7] [22.6 53.9] [21.3 62.9] - [21.5 103.5]
m=1
R1 [22.2 33.0] [22.2 33.0] [22.2 33.0] [22.2 33.0] [13.7 22.3]
R2 [17.9 25.9] [17.9 25.9] [17.9 25.9] [17.9 25.9] [12.4 19.9]
R3 [13.8 18.8] [13.8 18.8] [13.8 18.8] [13.8 18.8] [11.1 17.5]
R4 [ 9.5 11.9] [ 9.3 12.3] [ 9.1 12.7] - [ 9.8 15.2]
Table 6.2: The range of Ri
ρ1
ρ3
ρ2
m = 0
50
41
33
194
138
89
22.5 54
m = 1
3322
2618
1914
9.3 12.3
ρ
50 100 150 2000
ρ3
ρ2
ρ1
ρ4
ρ4
Figure 6.9: Range of ρi for Case II
Chapter 6. Circular motion using nonlinear Basic Cyclic Pursuit 131
40 50 60 70 80 90 1000
5
10
15
20
25
ρ
f(ρ) Case 2
Case 3 Case 4
6π
4π
2π
m = 0
Figure 6.10: The roots of (6.19) for Cases II, III, and IV
−60 −40 −20 0 20 40 60 80 100
−40
−20
0
20
40
60
80
Figure 6.11: Trajectories of n = 5 agents for Case II (• - initial position, N - final positionof the UAVs)
Chapter 6. Circular motion using nonlinear Basic Cyclic Pursuit 132
m = 0
50
41
33
194
138
89
21 63
ρ1
ρ4
ρ3
ρ2
m = 1
ρ
3322
2618
1914
9.112.7
ρ
50 100 150 2000
ρ4
ρ3
ρ2
ρ1
Figure 6.12: Range of ρi for Case III
The range of ρ is shown in Table 6.2 and Figures 6.12 and f(ρ) is shown in 6.10, respec-
tively. At ρ = 61.7, (6.19) is satisfied. This gives the radius of the other agents (from
(6.11)) as
ρ = [61.7 49.3 37.0 27.7 17.3] (6.41)
The simulation, shown in Figure 6.13, confirms the radius of the circles evaluated ana-
lytically.
Case IV : When the necessary condition (6.20) is satisfied, (6.19) can have more than
one solution depending on the value of q. This is demonstrated for 5 agents with the
gains and speeds of the agents as given in Table 6.1. Then,
X1 = {1 2 3} (6.42)
X2 = {4, 5} (6.43)
The range of values of ρ is shown in Table 6.2 and Figure 6.14 and f(ρ) is plotted in
Figure 6.10. From the figure, it can be seen that more than one solution (ρ = 50.9
and 74.0) exists for this particular problem. The simulation is shown in Figure 6.15
Chapter 6. Circular motion using nonlinear Basic Cyclic Pursuit 133
−60 −40 −20 0 20 40 60 80 100
−40
−20
0
20
40
60
80
Figure 6.13: Trajectories of n = 5 agents for Case III (• - initial position, N - finalposition of the UAVs)
and we observe that the system converges to the formation corresponding to ρ = 70.4.
When ρ = 50.9, the system is unstable. Simulation shows that, even if we start from
a equilibrium point corresponding to ρ = 50.9, due to numerical inaccuracies, it slowly
migrates to the other equilibrium point at ρ = 74. This is shown in Figure 6.16.
m = 0
50
41
33
194
138
89
ρ1
ρ3
ρ2
m = 1
3322
2618
1914
ρ
ρ
50 100 150 2000
ρ2
ρ1
ρ3
Figure 6.14: Range of ρi for Case IV
Case V : However, if more than one equilibrium point is stable, the system converges
to one of the equilibrium formations depending on the initial configuration. Consider 5
Chapter 6. Circular motion using nonlinear Basic Cyclic Pursuit 134
−100 −80 −60 −40 −20 0 20 40 60 80 100
−60
−40
−20
0
20
40
60
80
100
Figure 6.15: Trajectories of n = 5 agents for Case IV (stable equilibrium) (• - initialposition, N - final position of the UAVs)
−100 0 100
0
50
100
150t = 0
(a)−100 0 100
0
50
100
150t = 240 sec
(b)−100 0 100
0
50
100
150t = 475 sec
(c)
Figure 6.16: Trajectories of n = 5 agents showing unstable equilibrium (a) initial equi-librium configuration, (b) intermediate configuration, (c) final stable configuration cor-responding to q = 2
Chapter 6. Circular motion using nonlinear Basic Cyclic Pursuit 135
0 20 40 60 80 100 1200
5
10
15
20
25
ρ
f(ρ)
6π
4π
2π
Figure 6.17: Roots of (6.19) for Case V
−200 −150 −100 −50 0 50 100
0
50
100
150
200
−40 −20 0 20 40 60 80
−20
0
20
40
60
80
Figure 6.18: Trajectories of n = 5 agents for case V for different initial conditions (• -initial position, N - final position of the UAVs)
agents with speeds and gains as given in Table 6.1. For this case the range of ρ and f(ρ)
are shown in Table 6.2 and Figure 6.17, respectively. The final formation for two different
initial conditions is shown in Figure 6.18. The dependence of the final configuration on
initial configuration has also been observed in [49].
We conjecture that at equilibrium, all agents move in a counter-clockwise direction,
therefore, the condition⋂
iεX1Ri 6= ∅, from (6.30), occurs only at m = 0. This simplifies
the effort required to find the overlapping region.
Chapter 6. Circular motion using nonlinear Basic Cyclic Pursuit 136
6.5 Conclusions
In this chapter, the behaviour of a swarm of heterogeneous non-holonomic agents with
motion constraints are studied. The agents form a stable polygon while circling around
a point. The necessary conditions for this formation is obtained.
Generalization similar to the linear cyclic pursuit (CCP and GCCP) are more com-
plicated and remains an open problem. However, the results give here forms the basis for
explaining some realistic behaviour of biological organisms. This will be demonstrated
in Chapter 7.
Chapter 7
Realistic cyclic pursuit
The previous chapters proved certain analytical results on rendezvous, directed motion
and circular motion of a swarm of autonomous agents under cyclic pursuit laws. Certain
assumptions were made to obtain these results, which may not be applicable for realisti-
cally modeled robots, UAVs or other autonomous vehicles. In this chapter, some realistic
constraints are imposed and the behaviour of the group of vehicles is studied through
simulation. We will also show that cyclic pursuit strategies can be used to model the
behaviour of fish schools, which exhibit certain generic behaviour very frequently. We
will demonstrate that approximate adoption of cyclic pursuit laws are most probably the
basis for such behaviour.
7.1 Autonomous vehicles
In the analysis of linear cyclic pursuit strategies, we have assumed that the agents are
point masses, they can have unbounded speeds, and unbounded lateral accelerations.
However, autonomous vehicles, be it aerial, ground or underwater, are characterized by
one or more of the following constraints, among several others.
(i) Limitations on the maximum (and/or minimum) speed.
137
Chapter 7. Realistic cyclic pursuit 138
(ii) Turn rate constraints.
(iii) Limitation on the maximum lateral acceleration.
Since linear cyclic pursuit strategies assume infinite turn rates and no speed limits, we
impose the first two constraints. For nonlinear cyclic pursuit, the speeds of the agents are
constant and the agents are non-holonomic. Hence, only the third constraint is applied.
Under these constraints, the agents are expected to behave differently than in the ideal
case. We observe and analyze some of these behaviours.
7.1.1 Linear cyclic pursuit with limitations on the maximum
speeds of the agents
In linear cyclic pursuit, the speed of an agent is proportional to the distance between
itself and the point it is following. If the distance is large, the speed will also be large.
We can limit the maximum speed to a certain value, say vmi, in two ways
(i) Saturate the speed when it goes above the speed limit.
(ii) Control the gains to keep the speed within the maximum speed limit.
We discuss these methods below.
Holonomic vehicles with speed saturation
Here, we decouple the equation of motion of the agents along each direction and analyze
for any one direction. With limitations in the maximum speed of each vehicle, the
equation of motion of the ith agent can be written as
xi =
ki(xic − xi), if ki(xic − xi) < vmi;
ki(xic−xi)|ki||xic−xi|vmi, otherwise.
(7.1)
Chapter 7. Realistic cyclic pursuit 139
where, vmi is the maximum speed of agent i and xic is the weighted centroid that the ith
agent follows.
We prove the stability of this system when all the gains are positive. We refer to
some equations given in Chapter 3 in the proof of the following theorem.
Theorem 7.1 The system of n agents, given in (7.1) with ki > 0 for all i, is stable
under centroidal cyclic pursuit if all the gains are positive.
Proof. Let h be a positive definite function defined as
h =n∑
i=1
|ξi||xic − xi| (7.2)
Then,
h =n∑
i=1
|ξi| xic − xi
|xic − xi|(xic − xi) (7.3)
Using (3.2), we can write
xic =n−1∑j=1
ηjxi+j =n∑
j=1,j 6=i
η(n−i+j)xj (7.4)
Then,
xic − xi =
(n∑
j=1,j 6=i
η(n−i+j)xj
)− xi =
n∑j=1
η(n−i+j)xj (7.5)
assuming ηn = −1. Differentiating and replacing the above expression in (7.3)
h =n∑
i=1
n∑j=1
|ξi| xic − xi
|xic − xi|η(n−i+j)xj (7.6)
=n∑
j=1
{n∑
i=1,i6=j
|ξi|η(n−i+j)xic − xi
|xic − xi| − |ξj| xjc − xj
|xjc − xj|
}xj (7.7)
Chapter 7. Realistic cyclic pursuit 140
Since χ is singular,
Adj(χ)χ = 0 (7.8)
Then, the jth row of the above equation can be written as
n∑
i=1,i6=j
|ξi|η(n−i+j) − |ξj| = 0 ⇒n∑
i=1,i6=j
|ξi|η(n−i+j) = |ξj| (7.9)
Hence, (7.7) will be
h =n∑
j=1
n∑
i=1,i 6=j
|ξi|η(n−i+j)
(xic − xi
|xic − xi| −xjc − xj
|xjc − xj|)
xj (7.10)
Let S1 be the set of agents for which the speeds are saturated and S2 be the sets of
agents for which the speeds are not saturated. Then,
h =∑j∈S1
n∑
i=1,i 6=j
|ξi|η(n−i+j)
(xic − xi
|xic − xi| −xjc − xj
|xjc − xj|)
xjc − xj
|xjc − xj|vmj
+∑j∈S2
n∑
i=1,i6=j
|ξ1|η(n−i+j)
(xic − xi
|xic − xi| −xjc − xj
|xjc − xj|)
kj(xjc − xj)
=∑j∈S1
n∑
i=1,i 6=j
|ξi|η(n−i+j)
(xic − xi
|xic − xi| .xjc − xj
|xjc − xj| − 1
)vmj
+∑j∈S2
n∑
i=1,i6=j
kj|ξi|η(n−i+j)
(xic − xi
|xic − xi| .xjc − xj
|xjc − xj| − 1
)|xjc − xj| (7.11)
Since,(
xic−xi
|xic−xi| .xjc−xj
|xjc−xj | − 1)≤ 0,∀i, j and all the other terms are positive, h ≤ 0. Thus,
h satisfies the properties of a Lyapunov function and hence, the system is stable. ¤
Thus, if all the gains are positive, the agents will converge to a point. However, when
one of the gains is negative, but the system is stable when there is no speed saturation,
the stability of (7.1), with speed saturation, cannot be guaranteed. In this case, the
stability and the rendezvous point would be functions of the maximum speed, initial
positions and the gains.
Chapter 7. Realistic cyclic pursuit 141
−10 −5 0 5 10 15−10
−5
0
5
10
15
Figure 7.1: Trajectories of the agents with speed saturation
A simulation with speed saturation is shown in Figure 7.1. We considered 5 agents
with the same initial positions and gains as in Case VI of Section 2.5 and assumed
the maximum speed of the agents to be 5 units. It can be seen from the figure that
rendezvous occurs at (2, 1.6), and not at the desired point (0, 0) as in Case V. Hence,
speed saturation changes the rendezvous point, although rendezvous is guaranteed.
Next, we solve the problem of limiting the maximum speed by selecting the controller
gains.
Selection of controller gains for holonomic vehicles
In linear cyclic pursuit, the speed of the agents are given by
vi(t) = Zi(t) = ki [Zic(t)− Zi(t)] (7.12)
We assume ki > 0,∀i. Now, if ki is sufficiently small, vi will not exceed the maximum
speed limit. Thus, we can select the gains ki, depending on the initial position of the
agents, such that vi ≤ vmi.
Chapter 7. Realistic cyclic pursuit 142
We find the maximum gain kmi,∀i such that, if ki ≤ kmi, then vi(t) ≤ vmi, ∀t. In the
proof of Theorem 3.8, we have shown that, along a given direction d, the position of the
agents are always within the interval Id0 = I0 calculated along that direction. Now, we
define
sm = maxd
{length(Id
0 )}
(7.13)
where length(Id0 ) is the length of the interval Id
0 ⊆ R. This is the maximum distance
between any agent and its leader. Hence, given the maximum speed of an agent, we can
calculate the maximum gains of the agent as
kmi =vmi
sm
(7.14)
Then, selecting a gain ki < kmi,∀i, we can ensure that the speed of all the agents will
never cross the maximum limit.
For selecting the gains of the agents for rendezvous at a desired point, the gains can
be selected according to (4.48) and (4.57), where c can take any positive value. We can
select c such that ki < kmi, ∀i. The advantage of selecting the gains in this way is that
the rendezvous occurs at the desired point while the speed of agents do not saturate.
Consider Case VI in Section 2.5. We assume the maximum speed of the agents as
5 units. For the initial positions, and the gains of the agents, sm = 18 and so kmi =
5/18 = 0.28. Since the set of αi is [0.07, 0.08, 0.095, 0.39, 0.37], c < 0.07× 0.28 = 0.02.
Assuming c = 0.01, the trajectories are shown in Figure 7.2 and we find that the agents
converge to (0, 0). Hence, even with limitation on the maximum speed the desired
rendezvous point can be reached by appropriately selecting c.
However, the maximum gain kmi calculated above is an underestimation of the max-
imum gain limit since the distance between an agents and its leader may never be sm.
Also, the lower values of gains can make the system of agents sluggish. Thus, we can
have an adaptive gain selection technique, by which the gains will change with time such
Chapter 7. Realistic cyclic pursuit 143
−10 0 10 15−10
0
10
15
3
2
1
4
5
Figure 7.2: Trajectories of the agents with appropriate selection of gains such that thespeed do not saturate and rendezvous occur at Zf = (0, 0)
that the response of the system does not slow down.
7.1.2 Linear cyclic pursuit with fixed turn rate
In linear cyclic pursuit, we assumed holonomic agents. However, the autonomous vehicles
like robots and UAVs, have turn rate constraints. We incorporate this constraint while
the agents are under linear cyclic pursuit strategy. Instead of instantaneously turning
towards its leader, an agent turns at a fixed rate to orient itself towards its leader. The
speed of the agents are still proportional to the distance between the agent and its leader.
We observe the effect of fixed turn rate on the rendezvous point. Let, the desired
rendezvous point be Zf = (0, 0). Considering the same initial positions and gains of 5
agents, as in Case VI of Section 2.5, we assume the fixed turn rate to be 8 units. The
agents are initially oriented along the line of sight to its leader. The trajectories are
shown in Figure 7.3. We observe that the rendezvous point is Zf = (−0.2,−0.3). Note
that, since the turn rate is fixed, the turn radius is larger when the agents are far apart
as the speeds are higher. It has been observed that rendezvous is not guaranteed, if the
Chapter 7. Realistic cyclic pursuit 144
−10 −5 0 5 10 15−10
−5
0
5
10
15
Figure 7.3: Trajectories of the agents with fixed turn rate
fixed turn rate is less than a certain value. The initial orientation plays an important
role in determining this lower bound. If the fixed turn rate is lower than the bound, the
agents spiral out and rendezvous does not occur.
7.1.3 Linear cyclic pursuit with fixed turn rate and limitation
on the maximum speed
In a realistic implementation, the agents will have both the constraints − finite (or fixed)
turn rate and maximum speed limitation. We impose both the constraints and observe
the behaviour of the agents. Consider Case VI of Section 2.5. Let the fixed rate of turn
be 8 units and the maximum speed of each agent be 5 units. We assume that the speeds
are saturated above the maximum speed. The trajectories of the agents are shown in
Figure 7.4. Rendezvous occurs at Zf = (−1,−0.2). Thus, the rendezvous does occur
but not at the desired point.
Chapter 7. Realistic cyclic pursuit 145
−10 −5 0 5 10 15−10
−5
0
5
10
15
Figure 7.4: Rendezvous of the agents with fixed turn rate and speed saturation
7.1.4 Linear cyclic pursuit with fixed turn rate and limitation
on the maximum speed for unstable gains
When the turn rate constraint and the limitation on the maximum speed are imposed
on unstable system, the direction of motion changes. Consider Case II of Section 5.5.
Applying fixed turn rate of 8 units and maximum speed of 5 units, the trajectories are
shown in Figure 7.5. The dotted line shows the trajectories without the constraints. The
agents still align themselves to a directed motion but the direction is shifted from when
the constraints are not imposed.
7.1.5 Nonlinear cyclic pursuit with limitations on the maxi-
mum lateral acceleration
In nonlinear cyclic pursuit, each agent has constant speed and uses a lateral acceleration
proportional to the angular deviation between the vehicle orientation and the line of
sight angle. If the angular deviation is large, lateral acceleration (latax) requirement will
be high. Usually UAVs, robots and other autonomous vehicles have a maximum latax
Chapter 7. Realistic cyclic pursuit 146
−20 0 20 40 60−30
−20
0
20
Figure 7.5: Directed motion of the agents with fixed turn rate and speed saturation
capability. To incorporate this constraint, we assume that if the latax required is more
than the capacity of the agents, it will use the maximum latax it can pull, that is, the
latax will saturate above a certain limit.
We consider 5 agents with speed and gains same as in Case IV of Section 6.4. When
latax saturation is not applied, the maximum latax an agent applies at equilibrium is
8.44 units. Assume that the latax will saturate above 8 units. The trajectories of the
agents, with constraint on the maximum latax, are shown in Figure 7.6. We observe
that the path followed by the agents are different from that in Case IV of Section 6.4.
However, their trajectories converge to concentric circles, the radius of the corresponding
circles being the same.
7.2 Schooling of fishes
Schooling of fishes, as also flocking of birds and herding of animals, has been of interest
to many researchers. Reynolds [2] first attempted to create a computer animation of
these behaviours found in nature. A fish school is depicted in Figure 7.7 where we see
that the fishes move approximately in a circular path. This motion can be generated
using the nonlinear cyclic pursuit. However, a fish, in the group of fishes, does not follow
Chapter 7. Realistic cyclic pursuit 147
−100 0 100 200
−100
0
100
Figure 7.6: Circular motion of the agents with latax saturation
any particular fish, but instead follows some point within the group of other fishes. Thus,
the notion of generalized centroidal cyclic pursuit may be used to depict their behaviour.
Generalized centroidal nonlinear cyclic pursuit can be formulated as follows:
Let there be n agents in R2. Each agent i has constant speed vi and position of the
agent i at any time t is given by Zi(t) = [z1i (t), z
2i (t)]. Then, the state equation of the
agent i, in the cartesian coordinates, are given as
z1i = vi cos αi (7.15)
z2i = vi sin αi (7.16)
αi =ki(θi − φi)
vi
(7.17)
where
θi = tan−1
(z2
ic − z2i
z1ic − z1
i
)(7.18)
and vi is the speed of the agent i and Zic = [z1ic, z
2ic] is the position of the weighted
centroid that the agent i follows. Note that, this formulation appears different from that
in Section 6.1, since here we consider cartesian coordinates, as against polar coordinates
used earlier.
Chapter 7. Realistic cyclic pursuit 148
(courtesy www.fotosearch.com)
Figure 7.7: Schooling of fishes
−100 −50 0 50 100 150 200−100
−50
0
50
100
150
200
Figure 7.8: Simulated schooling of fishes
Chapter 7. Realistic cyclic pursuit 149
We apply the generalized centroidal cyclic pursuit strategy to simulate the movement
of a school of fishes. However, in this strategy, to make the model more realistic, we
assume that the agents have a finite sensor radius and restricted field of view angle.
Thus, each agent will follow the weighted centroid of only those agents that it can sense
within its field of view.
Fish schooling is simulated with 100 agents in a space of 300×300 units. We assumed
that each agent has a field of view angle of ±90◦, and sensor radius of 200 units. The
agents follow the generalized centroidal nonlinear cyclic pursuit strategy as described
above. The path traced by the agents after 300 secs are shown in Figure 7.8. We observe
that the agents behave like the swarm of fishes shown in Figure 7.7.
7.3 Conclusions
In this chapter, we applied the cyclic pursuit strategies to the agents with realistic
constraints. It is observed that the behaviour of the agents matches in some aspects as
predicted in the analysis for the ideal cases, considered in the previous chapters.
Chapter 7. Realistic cyclic pursuit 150
Chapter 8
Conclusions
This thesis generalizes cyclic pursuit strategies where agents pursue each other in a cycle
and that have earlier been addressed in the literature in its basic homogeneous form.
Initially, generalization in terms of heterogeneous agents, that have different gains and
speeds, is proposed. Next, the concept of pursuit sequence is extended to include the
flexibility of the agents to select their leader or group of leaders.
The basic homogeneous form that has been analyzed by other researchers is based
upon the assumption of identical controller gains and speeds of the agents and a fixed
pursuit sequence. Extensions to heterogeneous agents under linear and nonlinear basic
cyclic pursuit (BCP), linear centroidal cyclic pursuit (CCP) and linear generalized cen-
troidal cyclic pursuit (GCCP) are proposed and analyzed in this thesis. GCCP is the
most generalized among the three cyclic pursuit strategies. Conceptually, both BCP and
CCP are special cases of GCCP. In GCCP each agent has the freedom to select different
weights for the centroidal point they follow, while in CCP these weights are identical for
each agent. In BCP, these weights are such that each agent follows just one other agent.
The agents under generalized cyclic pursuit strategies reach a consensus either in po-
sition (rendezvous), direction of motion (directed motion) or relative position (polygonal
formation) of the agents. Each of these consensus problems are studied in this thesis.
151
Chapter 8. Conclusions 152
In Chapter 2, rendezvous of the agents under linear basic cyclic pursuit (BCP) is
analyzed. The conditions for stability of the system are obtained as functions of the
gains of the agents. It is shown that when the system is stable, rendezvous occurs. The
rendezvous point is determined as a function of the gains. It is shown that, unlike in the
homogeneous agents’ case, by varying the gains, the agents can be made to converge at
desired points. The set of all points where rendezvous can occur, called the reachable
set, is obtained. We prove that the stability, the rendezvous point, and the reachable set
do not change with different pursuit sequences or even with finite and infinite switching
of pursuit sequences.
In Chapter 3, the centroidal cyclic pursuit (CCP) strategies for linear/holonomic
agents are studied. We obtained the conditions for stability of the agents under CCP.
The rendezvous point and the reachable set are determined as a function of the controller
gains. It is observed that the stability and the rendezvous point are the same for both
BCP and CCP. We also proved that the stability, rendezvous point and reachable set are
pursuit sequence invariant and are invariant to switching of the pursuit sequences.
Systems under generalized cyclic pursuit are studied in Chapter 4. The analyses are
carried out similar to BCP and CCP. The conditions for stability, the rendezvous point,
and the reachable set are obtained. It is observed that these properties are no longer
invariant with respect to the pursuit sequence of the agents, except for special cases
when the pursuit sequence matrix satisfies certain conditions.
In Chapter 5, we study the behaviour of this system under linear cyclic pursuit
strategies when the controller gains are selected to make the system unstable. It is
shown that proper selection of gains can lead to directed motion of agents. We obtain the
conditions under which directed motion can occur and also characterize the straight line
asymptote along which the agent trajectories converge. The asymptote, which defines
the direction of motion, is obtained as a function of the gains and the pursuit sequence of
the agents. It is also observed that this asymptote always passes through an asymptote
point which is proved to be invariant under different pursuit sequences and switching
Chapter 8. Conclusions 153
of pursuit sequences, although the asymptote direction itself varies with the pursuit
sequence. We also show that the invariance property of the asymptote point does not
hold for GCCP except under certain special conditions. In this chapter we also propose
an alternative approach to obtain directed motion using a combination of different sets
of stable and unstable gains along different axes.
In Chapter 6, non-holonomic agents are studied under BCP. At equilibrium, the
agents converge to a polygonal formation that rotates in space about a point. It is
observed that each agent moves in concentric circles. Results available in the literature
show that when the agents are homogeneous, the radiuses of all the agents are same.
We consider heterogeneous agents and show that under certain conditions the agents
do converge to circular motion but with different radiuses. We propose a method to
compute the radius of all the circles and obtain some necessary conditions for equilibrium
formation.
In Chapters 2-6, we analyzed the cyclic pursuit strategies under ideal conditions.
However, keeping in mind the goal of using cyclic pursuit strategies to design coordinated
control of UAV swarms, in Chapter 7, some realistic constraints are imposed on the agents
such as limitation on the speed, turn rate, and latax of the agents. The behaviour of
the agents under realistic constraints is studied through simulation experiments. These
experiments reveal that even under realistic constraints, some of the basic properties
(such as rendezvous and stability and convergence to directed and circular motion) can
be preserved in some form, and hence cyclic pursuit strategies hold considerable promise
as a basic conceptual tool to devise useful coordination strategies for UAV swarms. In
Chapter 7, we additionally show that certain frequently observed behaviour of schools
of fishes can be modeled and shown to emerge from a realistic implementation of cyclic
pursuit strategies.
The studies carried out in this thesis on generalized cyclic pursuit strategies open up
several avenues for further research in this area. We will discuss some of them below:
In all the studies in this thesis we have assumed the availability of perfect information
Chapter 8. Conclusions 154
to each agent. However, in a real system information is often corrupted with noise and
is subjected to delays. The analysis of the behaviour of the agents under cyclic pursuit
strategies in the presence of noise and delays in information is an important direction
for further research.
Control of UAV swarms has to account for fault tolerance in the event of some agents
failing. The control strategies should be able to adapt to this kind of events. The
centroidal pursuit strategy, where each agent depends on more than one leader, has the
potential to adapt itself to failures. However, this aspect needs to be studied to obtain
truly adaptive cyclic strategies. Dynamically changing pursuit sequences and number of
agents are interesting topics of future research.
We have studied the behaviour of non-holonomic agents under BCP. Analysis of the
behaviour of non-holonomic agents under CCP and GCCP are still open and are expected
to yield very useful results from the realistic implementation point of view as these are
the closest to the real UAV or robotic systems.
In conclusion, the thesis makes several contributions in generalizing the basic cyclic
pursuit strategies available in the literature and proving many fundamental properties
of this class of strategies. The performances of the strategies under realistic constraints
indicate possible applications of these strategies to real systems. It is hoped that these
results will open up further avenues of research in, and practical applications of, multi-
agent systems.
References
[1] R. Olfati-Saber, J.A. Fax, and R.M. Murray. Consensus and cooperation in net-
worked multi-agent systems. Proceedings of the IEEE, 95(1):215–233, January 2007.
[2] C.W. Reynolds. Flocks, herds, and schools: A distributed behavioral model. Com-
puter Graphics, 21(4), July 1987, pp. 25-34., 21(4):25–34, July 1987.
[3] T. Vicsek, A. Czirok, E. Ben-Jacob, I. Cohen, and O. Shochet. Novel type of phase
transition in a system of self-driven particles. Physical Review Letters, 75(6):1226–
1229, August 1995.
[4] G. Royle and C. Gadsil. Algebric Graph Theory. Number 207. New York: Springer
Graduate Text in Mathematics, 2001.
[5] R.A. Horn and C.R. Johnson. Matrix Analysis. Cambridge University Press, 1985.
[6] W. Ren and R.W. Beard. Consensus seeking in multi-agent systems using dynam-
ically changing interaction topologies. IEEE Transactions on Automatic Control,
50(5):655–661, May 2005.
[7] Z. Lin, B. Francis, and M. Maggiore. Necessary and sufficient graphical condi-
tions for formation control of unicycles. IEEE Transactions on Automatic Control,
50(1):121–127, 2005.
[8] M. Fiedler. Algebraic connectivity of graphs. Czechoslovak Mathematical Journal,
23(98):298–305, 1973.
155
REFERENCES 156
[9] R. Olfati-Saber and R.M. Murray. Consensus problems in networks of agents with
switching topology and time-delays. IEEE Transactions on Automatic Control,
49(9):1520–1533, September 2004.
[10] A. Jadbabaie, J. Lin, and A.S. Morse. Coordination of groups of mobile autonomous
agents using nearest neighbor rules. IEEE Transactions on Automatic Control,
48(6):988–1001, June 2003.
[11] L. Moreau. Stability of multiagent systems with time-dependent communication
links. IEEE Transactions on Automatic Control, 50(2):169–182, February 2005.
[12] A.V. Savkin. Coordinated collective motion of groups of autonomous mobile robots:
Analysis of vicseks model. IEEE Transactions on Automatic Control, 49(6):981–989,
June 2004.
[13] W. Ren, R.W. Beard, and D.B. Kingston. Multi-agent kalman consensus with
relative uncertainty. In Proceedings of the 2005 American Control Conference, pages
1865–1870, Portland, OR, USA, June 2005.
[14] W. Ren, R.W. Beard, and E. Atkins. A survey of consensus problems in multi-
agent coordination. In Proceedings of the 2005 American Control Conference, pages
1859–1864, Portland, OR, USA, June 2005.
[15] R. Olfati-Saber. Flocking for multi-agent dynamic systems: Algorithms and theory.
IEEE Transactions on Automatic Control, 51(3):401–420, March 2006.
[16] N. Moshtagh, A. Jadbabaie, and K. Daniilidis. Distributed geodesic control laws
for flocking of nonholonomic agents. In Proceedings of the 44th IEEE Conference
on Decision and Control, and the European Control Conference, pages 2835–2840,
Seville, Spain, December 2005.
[17] H.G. Tanner, A. Jadbabaie, and G.J. Pappas. Stable flocking of mobile agents,
Part I: Fixed topology. In Proceedings ofthe 42nd IEEE Conference on Decision
and Control, pages 2010–2015, Maui, Hawaii USA, December 2003.
REFERENCES 157
[18] H.G. Tanner, A. Jadbabaie, and G.J. Pappas. Stable flocking of mobile agents Part
II: Dynamic topology. In Proceedings ofthe 42nd IEEE Conference on Decision and
Control, pages 2010–2015, Maui, Hawaii USA, December 2003.
[19] R. Winder and J.A. Reggia. Using distributed partial memories to improve self-
organizing collective movements. IEEE Transactions on Systems, Man and Cyber-
netics Part B: Cybernetics, 34(4):1697–1707, August 2004.
[20] J.S. Baras and X. Tan. Control of autonomous swarms using gibbs sampling. In
Proceedings of the 43rd IEEE Conference on Decision and Control, pages 4752–4757,
Atlantis, Paradise Island, Bahamas, December 2004.
[21] V. Gazi and K.M. Passino. Stability analysis of swarms. IEEE Transactions on
Automatic Control, 48(4):692–697, April 2003.
[22] Y.Q. Chen and Z. Wang. Formation control: A review and a new consideration.
In Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and
Systems, (IROS 2005), pages 3181–3186, Edmonton, Alberta, Canada, August 2005.
[23] H.G. Tanner, G.J. Pappas, and V. Kumar. Leader-to-formation stability. IEEE
Transactions on Robotics and Automations, 20(3):443–455, June 2004.
[24] M. Egerstedt and X. Hu. Formation constrained multi-agent control. IEEE Trans-
actions on Robotics and Automation, 17(6):947–951, December 2001.
[25] J.A. Fax and R.M. Murray. Information flow and cooperative control of vehicle
formations. IEEE Transactions on Automatic Control, 49(9):1465–1476, September
2004.
[26] P. Tabuada, G.J. Pappas, and P. Lima. Motion feasibility of multi-agent formations.
IEEE Transcation on Robotics, 3:387–392, June 2005.
[27] J. Lin, A.S. Morse, and B.D.O. Anderson. The multi-agent rendezvous problem. In
Proceedings of the 42rd IEEE Conference on Decision and Control, pages 1508–1513,
Maui, Hawaii, USA, December 2003.
REFERENCES 158
[28] J. Lin, A.S. Morse, and B.D.O. Anderson. The multi-agent rendezvous problem -
the asynchronous case. In Proceedings of the 43rd IEEE Conference on Decision
and Control, pages 1926–1931, Atlantis, Bahamas, USA, December 2004.
[29] A. Tiwari, J. Fung, J.M. Carson, R. Bhattacharya, and R.M. Murray. A framework
for Lyapunov certificates for multi-vehicle rendezvous problems. In Proceeding of
the 2004 American Control Conference, pages 5582–5587, Boston. Massachusetts,
USA, June 2004.
[30] A. Tiwari, J. Fung, R. Bhattacharya, and R.M. Murray. Polyhedral cone invariance
applied to rendezvous of multiple agents. In Proceedings of the 43rd IEEE Confer-
ence on Decision and Control, pages 165–170, Atlantis, Bahamas, USA, December
2004.
[31] R. Bhattacharya, J. Fung, A. Tiwari, and R.M. Murray. Ellipsoidal cone and ren-
dezvous of multiple agents. In Proceedings of the 43rd IEEE Conference on Decision
and Control, pages 165–170, Atlantis, Paradise Island, Bahamas, USA, December
2004.
[32] J. Corts, S. Martnez, and F. Bullo. Robust rendezvous for mobile autonomous agents
via proximity graphs in arbitrary dimensions. IEEE Transaction on Automatic
Control, 51(8):1289–1298, August 2006.
[33] R. Olfati-Saber. Distributed kalman filter with embedded consensus filters. In
Proceedings of the 44th IEEE Conference on Decision and Control, and the European
Control Conference, pages 8179–8184, Seville, Spain, December 2005.
[34] R. Olfati-Saber and J.S. Shamma. Consensus filters for sensor networks and dis-
tributed sensor fusion. In Proceedings of the 44th IEEE Conference on Decision and
Control, and the European Control Conference, pages 6698–6703, Seville, Spain,
December 2005.
[35] V. Gupta, B. Hassibi, and R.M. Murray. On sensor fusion in the presence of packet-
dropping communication channels. In Proceedings of the 44th IEEE Conference
REFERENCES 159
on Decision and Control, and the European Control Conference, pages 3547–3552,
Seville, Spain, December 2005.
[36] A. Jadbabaie, N. Motee, and M. Barahona. On the stability of the kuramoto model
of coupled nonlinear oscillators. In Proceeding of the 2004 American Control Con-
ference, pages 4296–4301, Boston, Massachusetts, USA, June 2004.
[37] N. Chopra and M.W. Spong. On synchronization of kuramoto oscillators. In Pro-
ceedings of the 44th IEEE Conference on Decision and Control, and the European
Control Conference, pages 3916–3922, Seville, Spain, December 2005.
[38] V.M. Preciado and G.C. Verghese. Synchronization in generalized Erdos-Renyi net-
works of nonlinear oscillators. In Proceedings of the 44th IEEE Conference on Deci-
sion and Control, and the European Control Conference, pages 4628–4633, Seville,
Spain, December 2005.
[39] A. Papachristodoulou and A. Jadbabaie. Synchronization in oscillator networks:
switching topologies and non-homogeneous delays. In Proceedings of the 44th IEEE
Conference on Decision and Control and the European Control Conference, pages
2835–2840, Seville, Spain, December.
[40] M.S. Klamkin and D.J. Newman. Cyclic pursuit or “the three bugs problems”. The
American Mathematical Monthly, 78(6):631–639, June-July 1971.
[41] F. Behroozi and R. Gagnon. Cyclic pursuit in a plane. Journal of Mathematical
Physics, 20(11):2212–2216, November 1979.
[42] T.J. Richardson. Non-mutual captures in cyclic pursuit. Annals of Mathematics
and Artificial Intelligence, 31:127–146, 2001.
[43] A.M. Bruckstein. Why the ant trails look so straight and nice. The Mathematical
Intelligencer, 15(2):58–62, 1993.
[44] T.J. Richardson. Stable polygons of cyclic pursuit. Annals of Mathematics and
Artificial Intelligence, 31:147–172, 2001.
REFERENCES 160
[45] F. Behroozi and R. Gagnon. A computer-assisted study of pursuit in a plane. The
American Mathematical Monthly, 82(8):804–812, October 1975.
[46] W.H. Ruckle. Pursuit on a cyclic graph - the symmetric stochastic case. Interna-
tional Journal of Game Theory, 10(2):91– 99, 1981.
[47] A.M. Bruckstein, N. Cohen, and A. Efrat. Ants, crickets and frogs in cyclic pursuit.
CIS Report 9105, Technion, IIT, Haifa, July 1991.
[48] A.M. Bruckstein, C.L. Mallows, and I. Wagner. Probabilitic pursuit on the grid.
American Mathematical Monthly, 104(4):323–343, 1997.
[49] J.A. Marshall, M.E. Broucke, and B.A. Francis. Formations of vehicles in cyclic
pursuit. IEEE Transactions on Automatic Control, 49(11):1963–1974, November
2004.
[50] J.A. Marshall, M.E. Broucke, and B.A. Francis. Pursuit formations of unicycles.
Automatica, 42(1):3–12, January 2006.
[51] J.A. Marshall, T. Fung, M.E. Broucke, G.M.T. D’Eleuterio, and B.A. Francis.
Experiments in multivehicle coordination. Robotics and Autonomous Systems,
54(3):265–275, March 2006.
[52] Z. Lin, M. Broucke, and B. Francis. Local control strategies for groups of mobile
autonomous agents. IEEE Transactions on Automatic Control, 49(4):622–629, 2004.
[53] S.L. Smith, M.E. Broucke, and B.A. Francis. A hierarchical cyclic pursuit scheme
for vehicle networks. Automatica, 41(6):1045–1053, June 2005.
[54] S.L. Smith, M.E. Broucke, and B.A. Francis. Curve shortening and its application to
multi-agent systems. In Proceedings of the Joint 44th IEEE Conference on Decision
and Control and European Control Conference, pages 2817–2822, Seville, Spain,
December 2005.
[55] R.T. Rockafellar. Convex Analysis. Princeton University Press, 1972.
REFERENCES 161
[56] F. Ayres. Theory and Problems of Matrices. Schaum’s Outline Series. McGraw-Hill
Book Company, Singapore, Asian Student Edition, 1986.
[57] D. Poole. Linear Algebra: A Modern Introduction. Thomson Brooks/Cole, January
2005.