# Decentralized coordination of multiple autonomous vehicles

vii Contents Page Abstract.........................................................iii Acknowledgments..................................................v List of Figures.....................................................x 1 Introduction...................................................1 1.1 Problem Statement..................................2 1.2 Overview of Related Work..............................4 1.2.1 General Coordination Approaches......................4 1.2.2 Coordination Without any Group Reference State..............5 1.2.3 Coordination With a Group Reference State.................7 1.2.4 Coordination With Multiple Group Reference States............8 1.3 Contributions of Dissertation.............................9 1.4 Dissertation Outline..................................10 2 Decentralized Coordination Algorithms Without a Group Reference State........12 2.1 Continuous-time Coordination Algorithms for Double-integrator Dynamics....12 2.2 Sampled-data Coordination Algorithms for Double-integrator Dynamics......13 2.3 Fixed Interaction Case................................14 2.3.1 Convergence Analysis of Sampled-data Coordination Algorithm with Ab- solute Damping................................14 2.3.2 Convergence Analysis of Sampled-data Coordination Algorithmwith Rela- tive Damping.................................23 2.3.3 Simulation..................................29 2.4 Dynamic Interaction Case...............................30 2.4.1 Convergence Analysis of Sampled-data Algorithm with Absolute Damping under Dynamic Directed Interaction.....................31 2.4.2 Convergence Analysis of Sampled-data Algorithm with Relative Damping under Dynamic Directed Interaction.....................37 2.4.3 Simulation..................................41 3 Decentralized Coordination Algorithms with a Group Reference State...........45 3.1 Continuous-time Setting...............................45 3.1.1 Decentralized Coordinated Tracking for Single-integrator Kinematics...46 3.1.2 Decentralized Coordinated Tracking for Second-order Dynamics......52 3.2 PD-like Discrete-time Consensus Tracking Algorithms with a Reference State...77 3.2.1 Existing PD-like Continuous-time Consensus Algorithm..........77 3.2.2 PD-like Discrete-time Consensus Algorithm.................81

viii 3.2.3 Convergence Analysis of the PD-like Discrete-time Consensus Algorithm with a Time-varying Reference State.....................82 3.2.4 Comparison Between P-like and PD-like Discrete-time Consensus Algo- rithms with a Time-varying Reference State.................88 3.2.5 Simulation..................................90 4 Decentralized Containment Control with Multiple Group Reference States........94 4.1 Deﬁnitions and Notations...............................94 4.2 Single-integrator Kinematics.............................95 4.2.1 Stability Analysis with Multiple Stationary Leaders.............95 4.2.2 Stability Analysis with Multiple Dynamic Leaders.............101 4.3 Double-integrator Dynamics.............................108 4.3.1 Stability Analysis with Multiple Stationary Leaders.............108 4.3.2 Stability Analysis with Multiple Dynamic Leaders.............111 4.3.3 Simulation..................................118 4.3.4 Experimental Validation...........................121 5 LQR-based Optimal Consensus Algorithms.............................125 5.1 Deﬁnitions.......................................125 5.2 Global Cost Functions................................126 5.2.1 Continuous-time Case............................126 5.2.2 Discrete-time Case..............................127 5.3 LQR-based Optimal Linear Consensus Algorithms in a Continuous-time Setting..128 5.3.1 Optimal Laplacian Matrix Using Interaction-free Cost Function......129 5.3.2 Optimal Scaling Factor Using Interaction-related Cost Function......133 5.3.3 Illustrative Examples.............................136 5.4 LQR-based Optimal Linear Consensus Algorithms in a Discrete-time Setting...137 5.4.1 Optimal Laplacian Matrix Using Interaction-free Cost Function......137 5.4.2 Optimal Scaling Factor Using Interaction-related Cost Function......147 5.4.3 Illustrative Examples.............................148 6 Decentralized Coordination Algorithms of Networked Fractional-order Systems....150 6.1 Fractional Calculus..................................150 6.2 Mathematical Model.................................151 6.3 Coordination Algorithms for Fractional-order Systems Without Damping Terms..152 6.3.1 Convergence Analysis............................152 6.3.2 Comparison Between Coordination for Fractional-order Systems and Integer- order Systems.................................161 6.3.3 Simulation Illustrations and Discussions...................165 6.4 Convergence Analysis of Fractional-order Coordination Algorithms with Absolute/ Relative Damping...................................167 6.4.1 Absolute Damping..............................168 6.4.2 Relative Damping..............................171 6.4.3 Simulation..................................175

ix 7 Conclusion and Future Research.....................................177 7.1 Summary of Contributions..............................177 7.2 Ongoing and Future Research............................178 References.......................................................179 Appendices.......................................................188 Appendix A Graph Theory Notions...........................189 Appendix B Caputo Fractional Operator.........................190 Vita............................................................192

x List of Figures Figure Page 1.1 Flock of birds.Alarge population of birds ﬂy in a regular formation.Photo courtesy of Prof.A.Menges,A.Ziliken.............................1 2.1 Directed graph G for four vehicles.An arrow from j to i denotes that vehicle i can receive information fromvehicle j...........................30 2.2 Convergence results using (2.6) and (2.7) with different α and T values.Note that coordination is reached in (a) and (c) but not in (b) and (d) depending on different choices of α and T...................................30 2.3 Interaction graphs for four vehicles.An arrow fromj to i denotes that vehicle i can receive information fromvehicle j..........................42 2.4 Convergence results using (2.6) with a switching interaction.............43 2.5 Interaction graphs for four vehicles.An arrow fromj to i denotes that vehicle i can receive information fromvehicle j...........................43 2.6 Convergence results using (2.7) with α = 1,T = 0.1 sec,and the interaction graph switches from a set {G (4) ,G (5) ,G (6) }.........................44 2.7 Convergence results using (2.7) with α = 1,T = 0.1 sec,and the interaction graph switches from a set {G (1) ,G (2) ,G (3) }.........................44 3.1 Potential functions V 1 ij and V 2 ij with R = 2.5 and d ij = 2...............68 3.2 Network topology for a group of six followers with a virtual leader.Here L denotes the virtual leader while F i ,i = 1, ,6,denote the followers............68 3.3 Trajectories of the followers and the virtual leader using (3.2) in 2D.The circle de- notes the starting position of the virtual leader while the squares denote the starting positions of the follwers................................69 3.4 Position tracking errors using (3.2) in 2D.......................69 3.5 Decentralized swarmtracking for 48 followers using (3.11) in 2Din the presence of a virtual leader.The circles denote the positions of the followers while the square denotes the position of the virtual leader.An undirected edge connecting two fol- lowers means that the two followers are neighbors of each other while a directed edge fromthe virtual leader to a follower means that the virtual leader is a neighbor of the follower.....................................71

xi 3.6 Trajectories of the followers and the virtual leader using (3.15) in 2D.The circle de- notes the starting position of the virtual leader while the squares denote the starting positions of the followers...............................72 3.7 Position and velocity tracking errors using (3.15) in 2D................72 3.8 Decentralized swarmtracking for 49 followers using (3.31) in 2Din the presence of a virtual leader.The circles denote the positions of the followers while the square denotes the position of the virtual leader.An undirected edge connecting two fol- lowers means that the two followers are neighbors of each other while a directed edge fromthe virtual leader to a follower means that the virtual leader is a neighbor of the follower.....................................73 3.9 Decentralized swarmtracking for 50 followers using (3.34) in 2Din the presence of a virtual leader.The circles denote the positions of the followers while the square denotes the position of the virtual leader.An undirected edge connecting two fol- lowers means that the two followers are neighbors of each other while a directed edge fromthe virtual leader to a follower means that the virtual leader is a neighbor of the follower.....................................74 3.10 Trajectories of the followers and the virtual leader using (3.26) in 2D with connec- tivity maintenance mechanism.The circle denotes the starting position of the virtual leader while the squares denote the starting positions of the followers........76 3.11 Position tracking errors using (3.26) in 2D in the presence of connectivity mainte- nance mechanism....................................76 3.12 Trajectories of the followers and the virtual leader using (3.23) in 2D with connec- tivity maintenance mechanism.The circle denotes the starting position of the virtual leader while the squares denote the starting positions of the followers........76 3.13 Position tracking errors using (3.23) in 2D in the presence of connectivity mainte- nance mechanism....................................77 3.14 Decentralized swarm tracking for 50 followers with a virtual leader using (3.11) in 2D in the presence of the connectivity maintenance mechanism in Remark 3.1.19. The circles denote the positions of the followers while the square denotes the posi- tion of the virtual leader.An undirected edge connecting two followers means that the two followers are neighbors of each other while a directed edge fromthe virtual leader to a follower means that the virtual leader is a neighbor of the follower....78 3.15 Decentralized swarm tracking for 50 followers with a virtual leader using (3.31) in 2D in the presence of the connectivity maintenance mechanism in Remark 3.1.19. The circles denote the positions of the followers while the square denotes the posi- tion of the virtual leader.An undirected edge connecting two followers means that the two followers are neighbors of each other while a directed edge fromthe virtual leader to a follower means that the virtual leader is a neighbor of the follower....79

xii 3.16 Decentralized swarm tracking for 50 followers with a virtual leader using (3.34) in 2D in the presence of the connectivity maintenance mechanism in Remark 3.1.19. The circles denote the positions of the followers while the square denotes the posi- tion of the virtual leader.An undirected edge connecting two followers means that the two followers are neighbors of each other while a directed edge fromthe virtual leader to a follower means that the virtual leader is a neighbor of the follower....80 3.17 Directed graph for four vehicles.A solid arrow from j to i denotes that vehicle i can receive information from vehicle j.A dashed arrow from ξ r to l denotes that vehicle l can receive information from the virtual leader...............91 3.18 Consensus tracking with a time-varying reference state using PD-like discrete-time consensus algorithm (3.42) under different T and γ..................92 3.19 Consensus tracking with a time-varying reference state using P-like discrete-time consensus algorithm (3.57)...............................93 4.1 Containment control under different coordinate frames in the two-dimensional space. The squares denote the positions of the four leaders.The blue and red rectan- gles represent the smallest rectangles containing the leaders under,respectively,the (X 1 ,Y 1 ) coordinate frame and the (X 2 ,Y 2 ) coordinate frame............99 4.2 A special network topology when a subgroup of agents can be viewed as a leader..101 4.3 Switching directed network topologies for a group of agents with four leaders and one follower.Here L i ,i = 1, ,4,denote the leaders while F denotes the follower.106 4.4 A counterexample to illustrate that the follower cannot converge to the dynamic convex hull in the two-dimensional space.The red square represents the position of the follower and the blue circles represent the positions of the four leaders.....107 4.5 A counterexample to illustrate that the follower cannot converge to the dynamic convex hull in the two-dimensional space when sgn() is deﬁned by (4.12).The red square represents the position of the follower and the blue circles represent the positions of the four leaders..............................108 4.6 Network topology for a group of vehicles with multiple leaders.L i ,i = 1, ,4, denote the leaders.F i ,i = 1, ,6,denote the followers...............119 4.7 Trajectories of the agents using (4.14) under a ﬁxed and a switching directed net- work topology in the two-dimensional space.Circles denote the starting positions of the stationary leaders while the red and black squares denote,respectively,the starting and ending positions of the followers.....................119 4.8 Trajectories of the agents using (4.17) under a ﬁxed directed network topology in the two-dimensional space.Circles denote the positions of the dynamic leaders while the squares denote the positions of the followers.Two snapshots at t = 25s and t = 50 s show that all followers remain in the dynamic convex hull formed by the dynamic leaders....................................120

xiii 4.9 Trajectories of the vehicles using (4.19) under a ﬁxed directed network topology in the two-dimensional space.Circles denote the positions of the dynamic leaders while the squares denote the positions of the followers.Two snapshots at t = 25 s and t = 50 s show that all followers remain in the dynamic convex hull formed by the dynamic leaders..................................120 4.10 Trajectories of the vehicles using (4.23) under a ﬁxed directed network topology in the two-dimensional space.Circles denote the positions of the dynamic lead- ers while the squares denote the positions of the followers.Two snapshots at t = 21.47 s and t = 45.17 s show that all followers remain in the dynamic convex hull formed by the dynamic leaders.............................122 4.11 Multi-vehicle experimental platform at Utah State University.............122 4.12 Network topology for ﬁve mobile robots.L i ,i = 1, ,3,denote the leaders. F i ,i = 1, ,2,denote the followers.........................123 4.13 Trajectories of the ﬁve mobile robots using (4.17)..................124 4.14 Trajectories of the ﬁve mobile robots using (4.23)..................124 5.1 Evolution of cost function J r as a function of β....................137 5.2 Evolution of cost function J r as a function of β....................149 6.1 Mittag-Lefﬂer functions and the derivatives......................164 6.2 Interaction graph for twelve agents.An arrow from j to i denotes that agent i can receive information fromagent j...........................165 6.3 Simulation results using (6.4) with different orders..................166 6.4 Simulation result using (6.4) with varying orders.(|r i (t) − r j (t)| < 0.1 for any t > 21.73 s.)......................................168 6.5 Directed network topology for four systems.An arrow from j to i denotes that system i can receive information from system j...................175 6.6 States of the four systems using (6.22) with α = 1.6 and β = 1 with the directed ﬁxed network topology given by Fig.6.5.......................176 6.7 States of the four systems using (6.28) with α = 1.2 and γ = 1 with the directed ﬁxed network topology given by Fig.6.5.......................176

1 Chapter 1 Introduction Agent-based system has received more and more research attention because many real-world systems,such as ﬂocks of birds,honey bee swarms,and even human society,can be considered examples of agent-based systems.Agent-based system is studied extensively in biology science, where the behavior of animals is shown to be closely related to the group in which they are involved. A prominent phenomenon in agent-based system is that each agent’s behavior is based on its local (time-varying) neighbors.For example,in Fig.1.1,ﬂock of birds ﬂy in a regular formation. Here each bird can be considered an agent.For a large population of birds,it is impossible for them to have a leader which has the capability to control the formation of the whole group by determining the movement of each individual bird.Instead,each bird determines its movement via a local mechanism.That is,each individual bird has to act based on its local neighbors. Fig.1.1:Flock of birds.A large population of birds ﬂy in a regular formation.Photo courtesy of Prof.A.Menges,A.Ziliken.

2 Recently,the collective motions of a group of autonomous vehicles have been investigated by researchers and engineers from various perspectives,where the autonomous vehicles can be con- sidered agents.An emerging topic in the study of collective motions is decentralized coordination, which can be roughly categorized as formation control,rendezvous,ﬂocking,and sensor networks based on the applications.In addition,numerous experiments were also conducted to either validate the proposed coordination schemes or apply the coordination schemes into different scenarios. In this dissertation,we mainly focus on the mathematical study of coordination algorithms under none,one,and multiple group reference states.We also investigate the optimization problem and extend the study of integer-order dynamics to fractional-order dynamics.The main framework of this dissertation is to ﬁrst propose the coordination algorithm,then analyze the stability condition, at last present simulation and/or experimental validations. 1.1 ProblemStatement Decentralized coordination among multiple autonomous vehicles,including unmanned aerial vehicles (UAVs),unmanned ground vehicles (UGVs),and unmanned underwater vehicles (UUVs) has received signiﬁcant research attention in the systems and controls community.Although indi- vidual vehicles can be employed to ﬁnish various tasks,great beneﬁts,including high adaptability, easy maintenance,and low complexity,can be achieved by having a group of vehicles work co- operatively.The cooperative behavior of a group of autonomous vehicles is called coordination. Coordination of multiple autonomous vehicles has numerous potential applications.Examples in- clude rendezvous [1–3],ﬂocking [4–6],formation control [7,8],and sensor networks [9–11]. There are mainly two approaches used to achieve coordination of multiple autonomous ve- hicles:centralized and decentralized approaches.In the centralized approach,it is assumed that there exists a central vehicle which can send and receive the information from all other vehicles. Therefore,the coordination of all vehicles can be achieved if the central vehicle has the capability to process the information and inform each individual vehicle the desired localization or command frequently enough.Although the complexity of the centralized approach is essentially the same as the traditional leader-follower approach,the stringent requirement of the stable communication

3 among the vehicles is vulnerable because of inevitable disturbances,limited bandwidth,and unre- liable communication channels.In addition,the centralized approach is not scalable since a more powerful central station is required with the increasing number of vehicles in the group. Considering the aforementioned disadvantages of the centralized approach,decentralized ap- proach has been proposed and studied in the past decades.An important problem in decentralized coordination is to study the effect of communication patterns on the system stability.Recently, decentralized coordination of multi-vehicle systems has been investigated under different commu- nication patterns,including undirected/directed ﬁxed,switching,and stochastic networks.Along this direction,we try to solve the following several decentralized coordination problems. First,we study the decentralized coordination algorithms when there exists no group reference state.We mainly focus on the study of sampled-data coordination algorithms where a group of vehicles with double-integrator dynamics reach a desired geometric formation via local interaction. The main problem involved is to ﬁnd the conditions on the network topology as well as the control gains such that coordination can be achieved. Second,we study the decentralized coordination algorithms when there exists one group ref- erence state.We consider two different scenarios:consensus tracking and swarm tracking.In the continuous-time setting,the objective of consensus tracking is to propose control algorithms and study the corresponding conditions such that all followers ultimately track the leaders accurately.In the discrete-time setting,the objective of consensus tracking is to propose control algorithms,show the boundedness of the tracking errors between the followers and the leader using the proposed al- gorithms,and quantitatively characterize the bound.For swarm tracking problem,the objective is to propose control algorithms and study the corresponding conditions such that the followers move cohesively with the leaders while avoiding collision. Third,we study decentralized coordination algorithms when there exist multiple group refer- ence states.In this case,the control objective is to guarantee that the vehicles stay within the convex hull,i.e.,the minimum geometric space,formed by the leaders.Note that this problem is much more challenging because the desired state is not a unique point,but a set. Lastly,two other important problems are considered,which are the optimal linear consensus

4 problem in the presence of global cost functions and the extension from the study of integer-order dynamics to that of fractional-order dynamics.The objective of the optimal linear consensus prob- lemis to either ﬁnd the optimal Laplacian matrix or the optimal coupling factor under certain global cost functions.The objective of fractional-order coordination algorithms is to guarantee coordina- tion for multiple fractional-order systems. 1.2 Overview of Related Work Due to the abundance of existing literature on decentralized coordination,we provide here an overview that is incomplete.We summarize the related work according to the following logic. As the ﬁrst step,we brieﬂy introduce the general approaches used to achieve coordination.Then we focus on introducing those papers which are closely related to the dissertation:coordination without any group reference state,coordination with one group reference state,and coordination with multiple group reference states. 1.2.1 General Coordination Approaches The main objective of group coordination is to guarantee that a group of autonomous vehicles maintain a geometric conﬁguration.The main application of group coordination is formation control (see [12,13] and references therein).The objective of formation control is to guarantee that a group of autonomous vehicles can form certain desired (possibly dynamic) geometric behavior.In the absence of any external reference state,the objective of formation control is to design controllers such that certain desired geometric formation can be achieved for a group of autonomous vehicles. Differently,when there exists an external reference state,the objective of formation control is to design controllers for the vehicles such that they can form certain geometric formation and track the external reference state as a group.The approaches used to solve formation control can be roughly categorized as leader-follower [14–18],behavior [7,19,20],potential function [4,5,21–23], virtual leader/virtual structure [12,24–30],graph rigidity [31–35],and consensus [36–41].In the leader-follower approach,the vehicles who are designated as leaders can be designed to track the desired trajectory while the vehicles who are designated as the followers can be designed to track certain state determined by their local neighbors.Note that the leader-follower approach can be

5 considered a two-level control approach where the top level is responsible for the leaders while the low level is responsible for the followers.In the behavioral approach,the behavior of the vehicles can be categorized into several types,such as obstacle avoidance,formation maintenance,and target tracking.Accordingly,the control input for each vehicle at certain time is determined by the desired behavior of the vehicle at this time.In the potential function approach,the different behaviors used in behavioral approach are implemented via some potential function.In particular,the potential function for each vehicle is deﬁned based on its state and the states of its local neighbors.The virtual leader/virtual structure approach is quite similar to the leader-follower approach except that the leader in the virtual leader/virtual structure approach which is used to represent the desired trajectory does not exist.In the rigidity approach,the formation (or shape) of a group of vehicles is determined by the edges.By changing the edges properly,the desired geometric formation can be guaranteed.In the consensus approach,the group geometric formation is achieved by properly choosing the information states on which consensus is reached. In addition to the aforementioned approaches,consensus approach was also applied in for- mation control problems from different perspectives [4–6,11,29,42–49].We will overview the approach in detail in the following several subsection. 1.2.2 Coordination Without any Group Reference State When there exists no group reference state,the control objective is to guarantee that the ve- hicles reach desired inter-vehicle deviation,i.e.,formation stabilization.A fundamental approach used in formation stabilization is consensus (also called rendezvous or synchronization in different settings),which means that the vehicles will reach agreement on their ﬁnal states.Accordingly, group coordination can be easily obtained by introducing the state deviations into the consensus algorithms.Consensus has been investigated extensively fromdifferent perspectives.In the follow- ing,we will review the existing consensus algorithms. Consensus has an old history [50–52].In the literature,consensus means agreement of a group faced with decision making situations.As for a group behavior,sharing information with each other,or consulting more than one expert makes the decision makers more conﬁdent [50].Inspired

6 by Vicsek et al.[52],it is shown that consensus can be achieved if the undirected communication graph is jointly connected [36].Consensus is further studied when the communication graph may be unidirectional/directed [37–39,53].In particular,average consensus is shown to be achieved if the communication graph is strongly connected and balanced at each time [37],while consensus can be achieved if the communication graph has a directed spanning tree jointly by using the prop- erties of inﬁnity products of stochastic matrices [39].By using the set-valued Lyapunov function, Moreau [38] provided a similar condition on the communication graph as that in Ren and Beard [39] to guarantee consensus. Given the aforementioned literature on the study of consensus problem,several directions have also been discussed recently.The ﬁrst direction is the study of consensus problems over stochastic networks.The motivation here is the unstable communication among the vehicles.Consensus over stochastic networks was ﬁrst studied where the communication topology is assumed to be undirected and modeled in a probabilistic setting and consensus is shown to be achieved in probability [54]. Consensus was further studied over directed stochastic networks [55–57].In particular,necessary and sufﬁcient conditions on the stochastic network topology were presented such that consensus can be achieved in probability [57]. The second direction is the study of asynchronous consensus algorithms,which is motivated by the fact that the agents may update their states asynchronously because the embedded clocks are not necessarily synchronized.Asynchronous consensus was studied from different perspectives using different approaches [41,58,59].In particular,Cao et al.[41] used the properties of “compositions” of directed graphs and the concept of “analytic synchronization.” Xiao and Wang [58] used the properties of inﬁnite products of stochastic matrices.Differently,Fang and Antsaklis [59] used the paracontracting theorem.Note that the approaches used in Cao et al.[41] and Xiao and Wang [58] are generally used for linear systems while the approach used in Fang and Antsaklis [59] can be used for nonlinear systems. The third direction is to study consensus for general systems,including systems with double- integrator dynamics,fractional-order dynamics,etc.For systems with double-integrator dynamics, two consensus algorithms were proposed which can guarantee the convergence of the states with,re-

7 spectively,(generally) nonzero ﬁnal velocity and zero ﬁnal velocity [60,61].Then the sampled-data case of the consensus algorithms was also studied [62–64].In particular,Hayakawa et al.[62] fo- cused on the undirected network topology case while Cao and Ren [63,64] focused on,respectively, the ﬁxed and switching directed network topology case.Necessary and sufﬁcient conditions on the network topology and the control gains were presented to guarantee consensus [63].However,only sufﬁcient conditions on the network topology and the control gains were presented to guarantee consensus because the switching topology case is much more complicated than the ﬁxed topology case [64].Considering the fact that the system dynamics in reality may be fractional (nonintegral), the existing study of integer-order consensus algorithms was extended to fractional-order consensus algorithms [65].Two survey papers provide more detailed information [46,66]. 1.2.3 Coordination With a Group Reference State Coordination with a group reference state is also called coordinated tracking.Here,coordi- nated tracking refers to both consensus tracking and swarm tracking.The objective of consensus tracking is that a group of followers tracks the group reference state with local interaction.The unique group reference state is also called “leader.” A consensus tracking algorithm was pro- posed and analyzed under a variable undirected network topology [67,68].In particular,the al- gorithm requires the availability of the leader’s acceleration input to all followers and/or the design of distributed observers.A proportional-and-derivative-like consensus tracking algorithm under a directed network topology was proposed and studied in both continuous-time and discrete-time set- tings [69–71].In particular,the algorithmrequires either the availability of the leader’s velocity and the followers’ velocities or their estimates,or a small sampling period.A leader-follower consen- sus tracking problem was further studied in the presence of time-varying delays [72].In particular, the algorithm requires the velocity measurements of the followers and an estimator to estimate the leader’s velocity. In addition to the consensus tracking algorithms,various ﬂocking and swarm tracking algo- rithms were also studied when there exists a leader.The objective of ﬂocking or swarm tracking with a leader is that a group of followers tracks the leader while the followers and the leader main-