Multi-agent Consensus using Generalized Cyclic …...Multi-agent Consensus using Generalized Cyclic...

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









3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4 Rendezvous using linear Generalized Centroidal Cyclic Pursuit 69









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


5.5.1 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


6.2 Analysis for possible formations . . . . . . . . . . . . . . . . . . . . . . . 119

6.3 Special case: Homogeneous system . . . . . . . . . . . . . . . . . . . . . 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 →∞.


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


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.


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.


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].


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).


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.


(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.


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)


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


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


-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


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.


-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


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),


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)


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 =


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) =


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


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)


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 =


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)


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)


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)


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)


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.


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.


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


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


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,


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)


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.


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


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


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


−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.


−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)


−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)


−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)


−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


−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).


−10 −5 0 5 10 15 20−10

−5

0

5

10

15

2

1

4

5

3


−10 −5 0 5 10 15 20−10

−5

0

5

10

15

3

4

1

5

2


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


−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)


−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


called a centroidal cyclic pursuit. In the next chapter, the behaviour of the agents under

centroidal cyclic pursuit is studied.

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.


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


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)


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)


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


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.


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.


We state and prove a theorem identical to Theorem 2.2 in Chapter 2


conditions hold


∀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


(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:


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.


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.


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.


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 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.


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.


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)

}=


i=1 1/ki

(3.24)


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)


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 =


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


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.


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.


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).


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)


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)


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.


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.


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


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


−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.


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)


−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)).


−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)


−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.


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.


−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.


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)


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)


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). ¤


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.



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.


conditions hold


∀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.


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.


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.


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)


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


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 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.


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)


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)


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 =


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


Theorem 4.2, such that,

Z =


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)


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)


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)


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)


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.



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.


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


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)


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.


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


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)


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.


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


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.


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)


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)


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)


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.


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)


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.


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)


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.


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


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).


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.


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.


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


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)


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.


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


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


{sin2 θz1

i (t0)− sin θ cos θz2i (t0)

}(5.42)

z2f =

n∑i=1


{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.


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


5.5.1 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


−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


−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


−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).


−40 0 40 80 120−80

−40

0

40

2

5

3

14

Asymptote Point


−40 0 40 80 120−80

−40

0

40

5

21

3

4

Asymptote Point


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


−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


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


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)


−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 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.


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)


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.


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


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)


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).


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)


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


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)


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.


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.


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,


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


−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)


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


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)


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


−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


−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


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.


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)


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)


−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.


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


−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


−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.


−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


−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


−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.


(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


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 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


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


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.

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