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COOPERATIVE TARGET TRACKING IN CONCENTRIC FORMATIONS
Lili Ma
References
[1] K. Szwaykowska, I.B. Schwartz, L.M.-T. Romero, C.R. Heck-man, D. Mox, and M.A. Hsieh, Collective motion patterns ofswarms with delay coupling: Theory and experiment, PhysicalRevie E, 93(3), 2016, 11.
[2] R. Sepulchre, D. Paley, and N. Leonard, Stabilization ofplanar collection motion: All-to-all communication, IEEETransactions on Automatic Control, 52(5), 2007, 811–824.
[3] R. Sepulchre, D. Paley, and N. Leonard, Stabilization ofplanar collective motion with limited communication, IEEETransactions on Automatic Control, 53(3), 2008, 706–719.
[4] A. Jain, D. Ghose, and P. Menon, Stabilization of balancedcircular motion about a desired center, International Confer-ence on Advances in Control and Optimization of DynamicalSystems, Kanpur, India, 2014.
[5] A. Jain and D. Ghose, Stabilization of collective motion insynchronized, balanced and splay phase arrangements on adesired circle, American Control Conference, Chicago, IL.
[6] G. Mallik and A. Sinha, A study of balanced circular formationunder deviated cyclic pursuit strategy, IFAC-PapersOnLine,48(5), 2015, 41–46.
[7] R. Zheng, Z. Lin, M. Fu, and D. Sun, Distributed control foruniform circumnavigation of ring-coupled unicycles, Automat-ica, 53, 2015, 23–29.
[8] N. Kokolakis and N. Koussoulas, Coordinated standoﬀ track-ing of a ground moving target and the phase separation prob-lem, International Conference on Unmanned Aircraft Systems,Dallas, TX, 2018.
[9] J. Guo, G. Yan, and Z. Lin, Cooperative control syn-thesis for moving-target-enclosing with changing topologies,International Conference on Robotics and Automation, An-chorage, AK, 2010.
[10] Z. Zhou, H. Wang, and Z. Hu, Event-based time varyingformation control for multiple quadrotor UAVs with Markovianswitching topologies, Complexity, 2018, 1–15.
[11] Y. Sun and L. Wang, Consensus of multi-agent systems indirected networks with nonuniform time-varying delays, IEEETransactions on Automatic Control, 54(7), 2009, 1607–1613.
[12] J. Marshall, M. Broucke, and B. Francis, Formations of vehiclesin cyclic pursuit, IEEE Transactions on Automatic Control,49(11), 2004, 1963–1974.
[13] J. Marshall, M. Broucke, and B. Francis, Pursuit formationsof unicycles, Automatica, 42, 2006, 3–12.
[14] M. Pavone and E. Frazzoli, Decentralized policies for geomet-ric pattern formation and path coverage, ASME Journal ofDynamic Systems, Measurement, and Control, 129(5), 2007,633–643.
[15] B. Wu, D. Wang, and E. Poh, Cyclic formation control for satel-lite formation using local relative measurements, MechatronicSystems and Control, 40(1), 2012, 11–21.
[16] J. Juang, On the formation patterns under generalized cyclicpursuit, IEEE Transactions on Automatic Control, 58(9), 2013,2401–2405, 2013.
[17] J. Ramirez, M. Pavone, E. Frazzoli, and D. Miller, Distributedcontrol of spacecraft formations via cyclic pursuit: Theory andexperiments, Journal of Guidance, Control, and Dynamics,33(5), 2010, 1655–1669.
[18] K. Hausmany, J. Muller, A. Hariharan, N. Ayanian, and G.Sukhatme, Cooperative multi-robot control for target track-ing with onboard sensing, International Journal of RoboticsResearch, 34(13), 2015, 1660–1677.
[19] P. Jimenez, B. Shirinzadeh, D. Oetomo, and A. Nicholson,Swarm aggregation and formation control for robots withlimited perception, International Journal of Robotics andAutomation, 26(4), 2011, 255–263.
[20] P. Zhu and W. Ren, Multi-robot joint localization and targettracking with local sensing and communication, AmericanControl Conference, Charlotte, NC, 2019.
[21] L. Luo, N. Chakraborty, and K. Sycara, Provably-good dis-tributed algorithm for constrained multi-robot task assignmentfor grouped tasks, IEEE Transactions on Robotics, 31(1), 2015,19–30.
[22] D. Panagou, M. Turpin, and V. Kumar, Decentralized goalassignment and safe trajectory generation in multi-robot net-works via multiple Lyapunov functions, IEEE Transactions onAutomatic Control, 65(8), 2020, 3365–3380.
[23] J. Ni, X. Yang, J. Chen, and S. Yang, Dynamic bioinspiredneural network for multi-robot formation control in unknownenvironments, International Journal of Robotics and Automa-tion, 30(3), 2015, 256–266.
[24] M. Khan and C. Silva, Autonomous and robust multi-robotcooperation using an artiﬁcial immune system, InternationalJournal of Robotics and Automation, 27(1), 2012, 60–75.
[25] X. Yu, L. Liu, and G. Feng, Coordinated control of multipleunicycles for escorting and patrolling task based on a cyclicpursuit strategy, American Control Conference, Boston, MA,2016.
[26] M. Zhang and H. Liu, Cooperative tracking a moving targetusing multiple ﬁxed-wing UAVs, Journal of Intelligent andRobotic Systems, 81(3-4), 2016, 505–529.
[27] X. Yu and L. Liu, Cooperative control for moving-target circularformation of nonholonomic vehicles, IEEE Transactions onAutomatic Control, 62(7), 2017, 3448–3454.
[28] L. Brinon-Arranz, A. Seuret, and A. Pascoal, Target trackingvia a circular formation of unicycles, IFAC World Congress,Toulouse, France, 2017.
[29] A. Miao, Y. Wang, and R. Fierro, Cooperative circumnavi-gation of a moving target with multiple nonholonomic robotsusing backstepping design, Systems and Control Letters, 103,2017, 58–65.
[30] L. Ma and N. Hovakimyan, Vision-based cyclic pursuit forcooperative target tracking, Journal of Guidance, Control, andDynamics, 36(2), 2013, 617–622.
[32],is adapted into the chain-type communication topology.Controller (7) that is expressed as a function of the rela-tive bearing angle(s) βij(t) alone is proposed in this paper.Controller (7) regulates the separation angles between apair of agents, instead of the separation distances betweenthem as in (6). The diﬀerence between regulating separa-tion distances and separation angles is not obvious whenUAVs circle on the same orbit because a constant separa-tion distance is equivalent to a constant separation angleunder this circumstance.However, when agents circle on diﬀerent orbits (circlesof diﬀerent radii around the target), the diﬀerence is clear,as to be demonstrated in Figs. 6 and 7. By properlyspecifying the desired 2D ranges and setting the UAV’slinear velocities accordingly, UAVs can now be controlledto circle on diﬀerent orbits. With this feature in place, uni-form spacing formations are achieved by using the virtualagent to represent an actual agent in need. Controllers (6)and (7) are modiﬁed to use relative bearing angles ˜βi,j(t)referring to the virtual agent(s), yielding new controllers(16) and (17) for the case of circling on diﬀerent orbits.The ultimate objective is to achieve more versatileconcentric formation patterns, such as patterns with localgeometric shapes. This is made possible utilizing existingworks (tracking [38], balanced circular formation [35], [36])and the new results presented in this paper (uniform spac-ing formations that regulate either separation distances orangles for agents circling on the same or diﬀerent orbits,and hierarchy). Results presented in this paper providemore complete solution to cooperative target tracking inconcentric formations, by designing the control input ofeach UAV as a sum of several individual control compo-nents. One practical issue of avoiding inter-vehicle col-lision is to be addressed in Section 5. The combinationof Attraction (another way of saying tracking), Alignment(formation), and Avoidance (collision avoidance) providesa solid framework to achieve formations in the context oftarget tracking.Another advantage of having versatile formation pat-terns is that it can possibly allow agents to acquire in-formation of each other using a mixture of informationexchange (over communication channels, for agents thatare far away) and onboard sensing/perception (for agentsthat are close to each other). Refer to the two patternsin Fig. 5(a) and (c), where overall balanced circular for-mations have been achieved with local geometric shapesof either straight lines or triangles. The “leaders” of thethree subgroups, can exchange information over the com-munication channels because they are relatively far awayand might not be able to “see” each other. The othertwo members of each subgroup can possibly use onboardsensing to obtain the information; they need to achievelocal formations (because they are close and can perceiveeach other).5. Inter-vehicle Collision AvoidanceWhen developing the uniform spacing formation, collisionbetween UAVs is more likely to occur than other patternswhere agents are far away from each other. A strategy8Figure 6. Cooperative target tracking using formation controller (17): (a) 2D trajectories; (b) 2D range; (c) formation in theend; (d) separation distance; (e) separation angle; and (f) formation control uic(t). Target undergoes a linear motion. Agentscircle on two diﬀerent orbits.of preventing the inter-vehicle collision is thus much indemand. Among methods that prevent collisions, one wayis to apply a force that repels the agents once they getcloser. This force should also be strong enough to defendother forces pushing agents to a collision [45]–[49].Let dmin denote the minimal distance allowed amongagents. The following collision avoidance control compo-nent helps to repel agents to steer away from each otheronce they get too close [50]:uia(t) = −Krj∈N (ri)dmindijsin βij (18)where Kr is the controller gain. The repulsion term (18)adjusts each agent’s heading to the opposite directionof its neighbours in N(ri). Notice that N (ri) and Nidenote diﬀerent sets. The set Ni is the set of agentswhose information can be obtained by the ith agent viacommunication, whereas the set N (ri) denotes the set ofagents that are too close to agent i.With (18) in place, each agent’s control inputbecomes [50]:˙ψi(t) = ui(t) = uit(t) + uic(t) + uia(t) (19)Totally three control components are added together toachieve the objective of simultaneous tracking, formation,and inter-vehicle collision avoidance.6. Simulation ResultsThe proposed control laws were simulated in Matlab toverify their performance of tracking, formation, and inter-vehicle collision avoidance.6.1 Achieving Uniform Spacing Formations onDiﬀerent OrbitsThis example demonstrates achievement of uniform spac-ing formations when agents circle on diﬀerent orbits. Toachieve formations under this circumstance, formation con-trollers (17) and (16) are used, which resort to virtualagents when needed, i.e., when the two agents are not onthe same orbit. It is worth mentioning that the formationcontroller (16) aims at regulating the relative separationdistances to a constant, while the controller (17) regulatesthe relative separation angles to a desired value. The2D ranges are set to be ρd,i = {15, 15, 18, 18} (m), fori = 1, 2, 3, 4, respectively. Correspondingly, UAVs’ linearvelocities are Vg,i = {30, 30, 36, 36} (m/s), satisfying therelationship in (10). To focus on the formation patterns,the target’s motion is assumed linear (the target moves ona straight line).We ﬁrst present simulation results applying the for-mation controller (17). The 2D trajectories of the agentsand the target are plotted in Fig. 6(a). Details of thetracking are given in Fig. 6(b), where ρi(t) approach theirprescribed values (that are diﬀerent). All agents’ posi-tions at the end of the simulation are shown in Fig. 6(c),9Figure 7. Cooperative target tracking using formation controller (16): (a) 2D trajectories; (b) 2D range; (c) formation in theend; (d) separation distance; (e) separation angle; and (f) formation control uic(t). Target undergoes a linear motion. Agentscircle on two diﬀerent orbits.where all agents scatter around the target on two diﬀer-ent orbits. Both the separation distances (Fig. 6(d))and the separation angles (Fig. 6(e)) are plotted. Whencontroller (17) is applied, we expect the separation an-gles to approach a constant. However, the separationdistances may or may not approach to the same valuebecause the distances between two agents also depend onthe orbits that they lie on. For ds = 10 (m), the desiredseparation angle is either φs = 2 sin−1(ds/30) ≈ 39◦orφs = 2π − 2 sin−1(ds/30) ≈ 321◦. Because all separationangles converge to 39◦(Fig. 6(e)), formation is obtainedsuccessfully. Figure 6(f) shows the formation componentuic(t), which converges to zero upon formation.Results when applying the controller (16) are shownin Fig. 7. Figure 7(a) shows the general picture ofcooperative target tracking. Figure 7(b) plots the 2Dranges between the agents and the target, demonstratingsuccessful tracking with two diﬀerent range values. Figure7(c) shows the formation achieved in the end, where agentscircle on two diﬀerent orbits around the target. Figure7(d) shows that the separation distances between twosuccessive agents are now regulated to the speciﬁed valueof ds = 10 (m) (same for all agents). On the contrary,the separation angles do not approach to one value. FromFig. 7(c), it can be seen that agents {1, 4} and agents{2, 3} circle on two diﬀerent orbits. The separation anglesbetween these two pairs of consecutive agents should besmaller than that between agents {1, 2}, which circle onthe same orbit. This is conﬁrmed in Fig. 7(e). Figure 7(f)shows that the formation control component uic(t) alsovanishes to zero upon formation.Figures 6 and 7 together show the diﬀerence betweenthe two formation controllers (16) and (17). The formationcontroller (16) regulates the relative separation distancesbetween agents to a speciﬁed value ds, while the controller(17) regulates the separation angles to φs.6.2 Achieving Concentric Formations with LocalGeometric ShapesThis section presents cooperative target tracking in con-centric formations with local geometric shapes of straightlines and triangles, as those shown in Fig. 5. To focus onformation patterns, the target’s motion is assumed linear,i.e., the target moves on a straight line. Corresponding tothe four patterns in Fig. 5, simulation results are presentedin Fig. 8, with several snapshots showing how formationis achieved over time. Cooperative target tracking in theseconcentric formations are successfully obtained.6.3 Achieving Inter-vehicle Collision AvoidanceThis example demonstrates cooperative target trackingwith inter-vehicle collision avoidance. The two scenarios ofwithout and with collision avoidance, i.e., before and afterapplying uia(t), are shown and compared in Fig. 9. Weselect the target’s velocity to be piecewise-constant, n = 5,ds = 10 (m), and dmin = 7 (m). The ﬁrst row of Fig. 9 is10Figure 8. Cooperative target tracking in local geometric shapes: (a) balanced-line; (b) uniform-line; (c) balanced-triangle;and (d) uniform-triangle. The overall formation is either balanced circular or uniform spacing. The local geometric shape iseither straight line or triangle.for the scenario without the collision avoidance capability,i.e., before the control component uia(t) is applied. Afteruia(t) is applied, results are shown in the second rowof Fig. 9. In each scenario, the 2D trajectories areplotted to show the general picture (Fig. 9(a) versus (c)).Then, the minimal distance among all agents is plotted,demonstrating the eﬀect of the added control componentuia(t) (Fig. 9(b) versus (d)). For the second scenario withcollision avoidance, the component uia(t) for each agent isshown in Fig. 9(e), where a saturation of |uia(t)| ≤ 3 hasbeen used for all agents. Comparison between Fig. 9(b)and (d) shows that the control component uia(t) helps tokeep the minimal distance to be greater than the allowedvalue. Otherwise, the minimal distance can be muchsmaller, as indicated in Fig. 9(b). The zigzag area in Fig.9(d), corresponds to the circumstances when uia(t) takeseﬀect. Figure 9(e) shows that uia(t) only takes eﬀect whenneeded, i.e., when agents {3, 5} get too close to each other.7. Comparison with Prior StudiesComparing with the distance-based coordination controllaw (6), our proposed bearing-angle-based coordinationcontrol law (7) has one potential advantage. Consider a11Figure 9. With inter-vehicle collision avoidance (before and after): (a) before 2D trajectories; (b) before minimum distance;(c) after 2D trajectories; (b) after minimum distance; and (e) after uic(t) (rad/s).scenario when information exchange among/between someagents becomes unavailable (e.g., due to communicationloss or out of communication range). Instead of computingthe relative bearing angles from the exchanged positionsas in (12), the needed information of relative bearing an-gles can be estimated/obtained from a local vision systeminstalled on each UAV. In other words, when the expectedinformation from others is unavailable (either permanentlyor temporarily), the objective of achieving and maintainingformation could still be achieved by using local measure-ments and estimates.Regarding formation control, most existing results areeither leaderless or leader-following [51], [52]. The proposedmethod of obtaining versatile formation patterns allows acombination of both. This can be seen in the “Balanced-Line” and “Balanced-Triangle” patterns (Fig. 5). Theoverall balanced circular formation can be obtained us-ing a leaderless communication topology, whereas achieve-ment of local geometric shapes can be implemented in aleader-based manner. The adopted hierarchical formationstructure allows selection of appropriate communicationtopologies on diﬀerent layers.This paper also considers a practical issue that wouldoccur, i.e., collisions among agents. This issue was tackledby adding another control component into each UAV’s con-trol input. As can be seen from (18), this added collision-avoidance control component can also be expressed as afunction of bearing angles. As mentioned earlier, a bearing-angle-based control law has the potential of still achiev-ing its control objective (formation or collision avoidance)during communication loss, by using local measurementsfrom each UAV’s onboard sensors.8. ConclusionsThis paper is to obtain more versatile concentric forma-tions in cooperative target tracking where a ﬂeet of UAVs iscommanded to circle above (and around) a moving groundtarget. On the basis of our previous results, versatile for-mation patterns are achieved with the help of three newfeatures. The ﬁrst feature is a new formation pattern,the uniform spacing formation where either the relativeseparation distances or the separation angles can be regu-lated to a desired value. Diﬀerent from a balanced circularformation where agents spread evenly over a full circle,agents can now spread evenly over a portion of a circle.Two kinds of uniform spacing formation control laws areproposed, where one regulates the separation distances be-tween two agents and the other regulates the separationangles in between. The second feature allows UAVs tocircle on diﬀerent orbits. To achieve formation under thiscircumstance, formation controllers will resort to virtualagents representing the actual agents in need. The thirdfeature is the usage of a (two-layer) hierarchical formationstructure, which allows selection of formation patterns fordiﬀerent layers. Combinations of these new features withour existing results yield more versatile concentric forma-tion patterns with diﬀerent local geometric shapes, such asstraight lines and triangles. Inter-vehicle collision avoid-ance is also addressed. Agents will be repelled to steeraway from each other once they get too close.12All UAVs are assumed to have constant linear veloci-ties. Control of each UAV is via its yaw rate. The designidea is to add three control components (three headingcontrollers) together to achieve the overall objective. Eachcontrol component has a goal. The proposed extensions tospreading agents on a portion of a circle, circling agentson orbits of diﬀerent radii, formation in local geometricshapes, and avoiding inter-vehicle collisions, provide morecomplete solution to cooperative target tracking in theconcentric manner.This paper also raises several questions for future in-vestigations. The implementation of the proposed schemeson physical robots and the extension of the developed tech-niques to 3D scenarios and cooperative tracking of multipletargets with obstacle avoidance capability [53], [54] will beof particular interest. Stability analyses in the presenceof formation pattern switching and broken communicationlinks are another research direction to look into. Also, in-vestigations of the time delay factor for obtaining stabilityconditions as well as desirable performance with reason-able computation complexity [55]–[57] are needed. Finally,Artiﬁcial Intelligence (AI) techniques have recently beendeveloped for robotic communication to enhance the com-munication capability of robotic networks for coordinatedactions. Application of the AI and/or Neural Networksto the ﬁeld of robotic networks in the context of coopera-tive target tracking is a promising research area to pursue[58]–[60].References[1] K. Szwaykowska, I.B. Schwartz, L.M.-T. Romero, C.R. Heck-man, D. Mox, and M.A. Hsieh, Collective motion patterns ofswarms with delay coupling: Theory and experiment, PhysicalRevie E, 93(3), 2016, 11.[2] R. Sepulchre, D. Paley, and N. Leonard, Stabilization ofplanar collection motion: All-to-all communication, IEEETransactions on Automatic Control, 52(5), 2007, 811–824.[3] R. Sepulchre, D. Paley, and N. Leonard, Stabilization ofplanar collective motion with limited communication, IEEETransactions on Automatic Control, 53(3), 2008, 706–719.[4] A. Jain, D. Ghose, and P. Menon, Stabilization of balancedcircular motion about a desired center, International Confer-ence on Advances in Control and Optimization of DynamicalSystems, Kanpur, India, 2014.[5] A. Jain and D. Ghose, Stabilization of collective motion insynchronized, balanced and splay phase arrangements on adesired circle, American Control Conference, Chicago, IL.[6] G. Mallik and A. Sinha, A study of balanced circular formationunder deviated cyclic pursuit strategy, IFAC-PapersOnLine,48(5), 2015, 41–46.[7] R. Zheng, Z. Lin, M. Fu, and D. Sun, Distributed control foruniform circumnavigation of ring-coupled unicycles, Automat-ica, 53, 2015, 23–29.[8] N. Kokolakis and N. Koussoulas, Coordinated standoﬀ track-ing of a ground moving target and the phase separation prob-lem, International Conference on Unmanned Aircraft Systems,Dallas, TX, 2018.[9] J. Guo, G. Yan, and Z. Lin, Cooperative control syn-thesis for moving-target-enclosing with changing topologies,International Conference on Robotics and Automation, An-chorage, AK, 2010.[10] Z. Zhou, H. Wang, and Z. Hu, Event-based time varyingformation control for multiple quadrotor UAVs with Markovianswitching topologies, Complexity, 2018, 1–15.[11] Y. Sun and L. Wang, Consensus of multi-agent systems indirected networks with nonuniform time-varying delays, IEEETransactions on Automatic Control, 54(7), 2009, 1607–1613.[12] J. Marshall, M. Broucke, and B. Francis, Formations of vehiclesin cyclic pursuit, IEEE Transactions on Automatic Control,49(11), 2004, 1963–1974.[13] J. Marshall, M. Broucke, and B. Francis, Pursuit formationsof unicycles, Automatica, 42, 2006, 3–12.[14] M. Pavone and E. Frazzoli, Decentralized policies for geomet-ric pattern formation and path coverage, ASME Journal ofDynamic Systems, Measurement, and Control, 129(5), 2007,633–643.[15] B. Wu, D. Wang, and E. Poh, Cyclic formation control for satel-lite formation using local relative measurements, MechatronicSystems and Control, 40(1), 2012, 11–21.[16] J. Juang, On the formation patterns under generalized cyclicpursuit, IEEE Transactions on Automatic Control, 58(9), 2013,2401–2405, 2013.[17] J. Ramirez, M. Pavone, E. Frazzoli, and D. Miller, Distributedcontrol of spacecraft formations via cyclic pursuit: Theory andexperiments, Journal of Guidance, Control, and Dynamics,33(5), 2010, 1655–1669.[18] K. Hausmany, J. Muller, A. Hariharan, N. Ayanian, and G.Sukhatme, Cooperative multi-robot control for target track-ing with onboard sensing, International Journal of RoboticsResearch, 34(13), 2015, 1660–1677.[19] P. Jimenez, B. Shirinzadeh, D. Oetomo, and A. Nicholson,Swarm aggregation and formation control for robots withlimited perception, International Journal of Robotics andAutomation, 26(4), 2011, 255–263.[20] P. Zhu and W. Ren, Multi-robot joint localization and targettracking with local sensing and communication, AmericanControl Conference, Charlotte, NC, 2019.[21] L. Luo, N. Chakraborty, and K. Sycara, Provably-good dis-tributed algorithm for constrained multi-robot task assignmentfor grouped tasks, IEEE Transactions on Robotics, 31(1), 2015,19–30.[22] D. Panagou, M. Turpin, and V. Kumar, Decentralized goalassignment and safe trajectory generation in multi-robot net-works via multiple Lyapunov functions, IEEE Transactions onAutomatic Control, 65(8), 2020, 3365–3380.[23] J. Ni, X. Yang, J. Chen, and S. Yang, Dynamic bioinspiredneural network for multi-robot formation control in unknownenvironments, International Journal of Robotics and Automa-tion, 30(3), 2015, 256–266.[24] M. Khan and C. Silva, Autonomous and robust multi-robotcooperation using an artiﬁcial immune system, InternationalJournal of Robotics and Automation, 27(1), 2012, 60–75.[25] X. Yu, L. Liu, and G. Feng, Coordinated control of multipleunicycles for escorting and patrolling task based on a cyclicpursuit strategy, American Control Conference, Boston, MA,2016.[26] M. Zhang and H. Liu, Cooperative tracking a moving targetusing multiple ﬁxed-wing UAVs, Journal of Intelligent andRobotic Systems, 81(3-4), 2016, 505–529.[27] X. Yu and L. Liu, Cooperative control for moving-target circularformation of nonholonomic vehicles, IEEE Transactions onAutomatic Control, 62(7), 2017, 3448–3454.[28] L. Brinon-Arranz, A. Seuret, and A. Pascoal, Target trackingvia a circular formation of unicycles, IFAC World Congress,Toulouse, France, 2017.[29] A. Miao, Y. Wang, and R. Fierro, Cooperative circumnavi-gation of a moving target with multiple nonholonomic robotsusing backstepping design, Systems and Control Letters, 103,2017, 58–65.[30] L. Ma and N. Hovakimyan, Vision-based cyclic pursuit forcooperative target tracking, Journal of Guidance, Control, andDynamics, 36(2), 2013, 617–622.[31] N. Moshtagh, N. Michael, A. Jadbabaie, and K. Daniilidis,Vision-based, distributed control laws for motion coordinationof nonholonomic robots, IEEE Transactions on Robotics, 25(4),2009, 851–860.[32] N. Ceccarelli, M. Marco, A. Garulli, and A. Giannitrapani,Collective circular motion of multi-vehicle systems, Automatica,44, 2008, 3025–3035.
[33] J. Soares, A. Aguiar, A. Pascoal, and M. Gallieri, Triangularformation control using range measurements: An applicationto marine robotic vehicles, IFAC Proceedings, 45(5), 2012,112–117.13
[35]–[37] control all agents to circle on one orbit (the sameorbit). For more ﬂexibility and versatility, one wouldwant all agents to be able to circle on diﬀerent orbits.This section describes the modiﬁcations and arrangementsneeded to make this happen.Start by considering target tracking. When all UAVsare commanded to orbit on the same circle, the 2D rangereference is the same for all UAVs and is denoted by ρdso far. Accordingly, the linear velocities Vg,i are the samefor all UAVs. To circle UAVs on diﬀerent orbits, therange references will be diﬀerent. As a result, UAVs’ linearvelocities, which are the same before, need to be diﬀerent.Let ρd,i and Vg,i denote the prescribed range reference andlinear velocity of agent i. Using agents i and j as anexample, the relationship between their linear velocitiesand 2D range references needs to satisfy:Vg,iVg,j=ρd,iρd,j(10)For coordination, our approach to coordinate all agentsto circle on diﬀerent orbits is to refer to virtual agent(s).Suppose agent i needs information from agent j to achievecoordination. Instead of using agent j’s information,agent i will use information from a virtual agent thatrepresents agent j but orbits on the same orbit as agenti. This virtual agent, denoted by the big dot in Fig. 4,lies in the intersection of agent i’s own orbit and the lineconnecting the target and the actual agent j. That is, thevirtual agent is the projection of agent j onto agent i’sorbit w.r.t. the target. No modiﬁcation is needed to an ex-isting communication topology. No additional informationexchange among agents is required. Based on the target’sposition (measured/estimated by each agent), each agent’sposition (known from the agent’s onboard sensor), andthe neighbouring agent’s position (obtained via informa-tion exchange), the position of the virtual agent can becomputed.Let [xi(t), yi(t)] be the agent i’s Cartesian coordinatein the world frame. Suppose that the coordination controllaw of agent i requires information of agent j. Instead5Figure 4. Virtual agent representing agent j used byagent i.of using agent j’s information [xj (t) , yj(t)] directly, avirtual agent will be ﬁrst computed and then used. Thevirtual agent does not necessarily have a speed associatedwith it. It is its relative position (distance or separationangle) w.r.t. agent i that matters. The coordinates of thisvirtual agent, denoted by [˜xi(t), ˜y(t)] , are computed as:˜xj(t) =ρd,iρd,j(xj(t) − xt(t)) + xt(t),˜yj(t) =ρd,iρd,j(yj(t) − yt(t)) + yt(t) (11)Letting ˜βi,j(t) ∈ [−π, π] denote the relative bearingangle between agent i and the virtual agent representingagent j, it can be computed as:˜βi,j(t) = tan−1 ˜yj(t) − yi(t)˜xj(t) − xi(t)− θi(t) (12)where θi(t) is the angle of the ith agent’s velocity vectorw.r.t. the x-axis, which can be computed as θi (t) =π2 − ψi(t).To extend to the scenario of circling on diﬀerent or-bits, we reﬁne the deﬁnition of uniform spacing to haveeither desired distance spacing or desired angular spacingbetween two agents. In the case of spacing in distance, theseparation distances between two consecutive agents willbe regulated to the desired value. For spacing in angulardistance, the separation angles between each two consecu-tive agents w.r.t. the target are regulated to the desiredvalue, regardless of the orbits they are on. Under thisextension, the case of circling on the same orbit becomes aspecial case with all orbits reducing to one, where constantrelative distance is equivalent to constant separation angle,and the virtual agent representing agent j is the agent jitself. These are no longer true for circling on diﬀerentorbits. To circle on diﬀerent orbits, our formation con-trol laws, which were all designed as functions of relativebearing angles βi,j(t) referring to the actual agents, will bemodiﬁed to use relative bearing angles ˜βi,j(t) referring tothe virtual agents.To be speciﬁc, consider the balanced circular forma-tion. Simply by replacing βi,j(t) with ˜βi,j(t), the balancedcircular formation can be obtained in a straightforwardmanner for the case of circling on diﬀerent orbits. Particu-larly, the three formation control laws in [35] are modiﬁedto be:• Achieving balanced circular formation under all-to-all:uic(t) = −κnj=1,j=icos ˜βij(t), κ > 0 (13)• Achieving balanced circular formation under ring:uic(t) = −κ cos ˜βi(i+1)(t) + cos ˜βi(i−1)(t) , κ > 0(14)• Achieving balanced circular formation under cyclic:uic(t) = −κ cos ˜βi(i+1)(t) − cosπn, κ > 0 (15)For uniform spacing formations, controller (6) is mod-iﬁed to be:uic(t) =⎧⎨⎩0, i = k−kv˜βi(i+1)(t) ln(cv−1)di(i+1)(t)+dscvds, i = k(16)Our coordination controller (7) is a function of bothβi(i+1)(t) and β0. Similarly, βi(i+1)(t) will be replacedby ˜βi(i+1)(t). For circling on diﬀerent orbits, it is notconvenient to use ds to specify the desired separation angle,as in (8). Instead, the desired separation angle, denotedby φs, will be speciﬁed directly. Our formation controller(7) is modiﬁed as shown below with β0 = φs/2:uic(t) =⎧⎨⎩0, i = k−κ(cos ˜βi(i+1)(t) − cos β0), i = k, κ > 0(17)In certain circumstance, one coordination control lawmight be more convenient than the other. For instance,when the formation is speciﬁed literally using relativedistance(s), the coordination control laws (6) or (16) mightbe more convenient. If the formation is concerned withrelative angular distance, the coordination control laws (7)or (17) might suit better.4. Concentric Formations with Local GeometricShapesSo far, we have achieved two formation patterns (balancedcircular in [35],
[37] control all agents to circle on one orbit (the sameorbit). For more ﬂexibility and versatility, one wouldwant all agents to be able to circle on diﬀerent orbits.This section describes the modiﬁcations and arrangementsneeded to make this happen.Start by considering target tracking. When all UAVsare commanded to orbit on the same circle, the 2D rangereference is the same for all UAVs and is denoted by ρdso far. Accordingly, the linear velocities Vg,i are the samefor all UAVs. To circle UAVs on diﬀerent orbits, therange references will be diﬀerent. As a result, UAVs’ linearvelocities, which are the same before, need to be diﬀerent.Let ρd,i and Vg,i denote the prescribed range reference andlinear velocity of agent i. Using agents i and j as anexample, the relationship between their linear velocitiesand 2D range references needs to satisfy:Vg,iVg,j=ρd,iρd,j(10)For coordination, our approach to coordinate all agentsto circle on diﬀerent orbits is to refer to virtual agent(s).Suppose agent i needs information from agent j to achievecoordination. Instead of using agent j’s information,agent i will use information from a virtual agent thatrepresents agent j but orbits on the same orbit as agenti. This virtual agent, denoted by the big dot in Fig. 4,lies in the intersection of agent i’s own orbit and the lineconnecting the target and the actual agent j. That is, thevirtual agent is the projection of agent j onto agent i’sorbit w.r.t. the target. No modiﬁcation is needed to an ex-isting communication topology. No additional informationexchange among agents is required. Based on the target’sposition (measured/estimated by each agent), each agent’sposition (known from the agent’s onboard sensor), andthe neighbouring agent’s position (obtained via informa-tion exchange), the position of the virtual agent can becomputed.Let [xi(t), yi(t)] be the agent i’s Cartesian coordinatein the world frame. Suppose that the coordination controllaw of agent i requires information of agent j. Instead5Figure 4. Virtual agent representing agent j used byagent i.of using agent j’s information [xj (t) , yj(t)] directly, avirtual agent will be ﬁrst computed and then used. Thevirtual agent does not necessarily have a speed associatedwith it. It is its relative position (distance or separationangle) w.r.t. agent i that matters. The coordinates of thisvirtual agent, denoted by [˜xi(t), ˜y(t)] , are computed as:˜xj(t) =ρd,iρd,j(xj(t) − xt(t)) + xt(t),˜yj(t) =ρd,iρd,j(yj(t) − yt(t)) + yt(t) (11)Letting ˜βi,j(t) ∈ [−π, π] denote the relative bearingangle between agent i and the virtual agent representingagent j, it can be computed as:˜βi,j(t) = tan−1 ˜yj(t) − yi(t)˜xj(t) − xi(t)− θi(t) (12)where θi(t) is the angle of the ith agent’s velocity vectorw.r.t. the x-axis, which can be computed as θi (t) =π2 − ψi(t).To extend to the scenario of circling on diﬀerent or-bits, we reﬁne the deﬁnition of uniform spacing to haveeither desired distance spacing or desired angular spacingbetween two agents. In the case of spacing in distance, theseparation distances between two consecutive agents willbe regulated to the desired value. For spacing in angulardistance, the separation angles between each two consecu-tive agents w.r.t. the target are regulated to the desiredvalue, regardless of the orbits they are on. Under thisextension, the case of circling on the same orbit becomes aspecial case with all orbits reducing to one, where constantrelative distance is equivalent to constant separation angle,and the virtual agent representing agent j is the agent jitself. These are no longer true for circling on diﬀerentorbits. To circle on diﬀerent orbits, our formation con-trol laws, which were all designed as functions of relativebearing angles βi,j(t) referring to the actual agents, will bemodiﬁed to use relative bearing angles ˜βi,j(t) referring tothe virtual agents.To be speciﬁc, consider the balanced circular forma-tion. Simply by replacing βi,j(t) with ˜βi,j(t), the balancedcircular formation can be obtained in a straightforwardmanner for the case of circling on diﬀerent orbits. Particu-larly, the three formation control laws in [35] are modiﬁedto be:• Achieving balanced circular formation under all-to-all:uic(t) = −κnj=1,j=icos ˜βij(t), κ > 0 (13)• Achieving balanced circular formation under ring:uic(t) = −κ cos ˜βi(i+1)(t) + cos ˜βi(i−1)(t) , κ > 0(14)• Achieving balanced circular formation under cyclic:uic(t) = −κ cos ˜βi(i+1)(t) − cosπn, κ > 0 (15)For uniform spacing formations, controller (6) is mod-iﬁed to be:uic(t) =⎧⎨⎩0, i = k−kv˜βi(i+1)(t) ln(cv−1)di(i+1)(t)+dscvds, i = k(16)Our coordination controller (7) is a function of bothβi(i+1)(t) and β0. Similarly, βi(i+1)(t) will be replacedby ˜βi(i+1)(t). For circling on diﬀerent orbits, it is notconvenient to use ds to specify the desired separation angle,as in (8). Instead, the desired separation angle, denotedby φs, will be speciﬁed directly. Our formation controller(7) is modiﬁed as shown below with β0 = φs/2:uic(t) =⎧⎨⎩0, i = k−κ(cos ˜βi(i+1)(t) − cos β0), i = k, κ > 0(17)In certain circumstance, one coordination control lawmight be more convenient than the other. For instance,when the formation is speciﬁed literally using relativedistance(s), the coordination control laws (6) or (16) mightbe more convenient. If the formation is concerned withrelative angular distance, the coordination control laws (7)or (17) might suit better.4. Concentric Formations with Local GeometricShapesSo far, we have achieved two formation patterns (balancedcircular in [35], [36] and uniform spacing in Sections 2.2and 3) and allowed agents to circle on either the same orbitor diﬀerent orbits. In the context of concentric formations,we would like to generate more versatile formations, suchas formations with local geometric shapes. Clearly, com-binations of circular and/or uniform formations are more6Figure 5. Formations in local geometric shapes (straight lines and triangles): (a) balanced-line; (b) uniform-line; (c)balanced-triangle; and (d) uniform-triangle.versatile than one ﬁxed pattern. This is done by utilizingexisting features (tracking, formation in diﬀerent patterns,circling on diﬀerent orbits), with the help of a hierarchicalformation structure [41]–[44]. The simplest hierarchicalscheme, the two-layer hierarchical structure, can be de-scribed as follows. A collection of n agents is divided inton2 subgroups, each containing n1 agents (n1 × n2 = n).The local control strategy is chosen such that the agentswithin each subgroup can be commanded to achieve cer-tain formation pattern [41]. In [44], a two-layer hybridpursuit system was described, where cyclic pursuit strat-egy was considered at the higher layer (the ﬁrst layer) andchain-like communication topology was used at the lowerlayer (the second layer).The concept of hierarchy is now applied to cooper-ative target tracking. The idea of hierarchy allows dif-ferent subgroups to select diﬀerent formation laws thatare already known to be stable. Similar to [44], a two-layer hierarchical formation structure is used. The ﬁrstlayer can be set to achieve either the balanced circu-lar or the uniform spacing formation. The second layercan be set to achieve the uniform spacing formation byspecifying either a desired separation distance or an-gle. Two examples are given below to demonstrate howversatile patterns are achieved by determining the 2D rangereferences of the agents (same or diﬀerent) and the forma-tion pattern on each layer (balanced circular or uniformspacing).The ﬁrst example achieves concentric formations withlocal geometric shapes in straight lines:(1) The agents {1, 4, 7}, {2, 5, 8}, and {3, 6, 9} areassigned to be on the inner, middle, and outer orbits,respectively, by specifying their 2D range references tobe {15, 18, 25} (m). Correspondingly, the UAVs’ linearvelocities are set to be {30, 36, 50} (m/s), satisfyingthe relationship in (10).(2) On the ﬁrst layer, the balanced circular formation isused for agents {1, 4, 7}, which lie on the inner circle.On the second layer, a uniform spacing formation isused inside each subgroup. There are three subgroups:{1, 2, 3}, {4, 5, 6}, and {7, 8, 9}. The agent that is in-volved in formation on the ﬁrst layer works as “leader”of its subgroup. As the intended geometric shapeis in straight line and the radial diﬀerences betweeneach two adjacent orbits can be diﬀerent, it is moreconvenient to use (7) with β0 = π/2. Actual agentsare used when obtaining bearing angles βi(i+1)(t). Theformation pattern is shown in Fig. 5(a).7(3) Simply changing the formation pattern on the ﬁrstlayer from the balanced circular to the uniform spac-ing pattern, an overall uniform spacing formation isachieved with local geometric shapes in straight lines.The formation pattern is shown in Fig. 5(b).The second example achieves concentric formationswith local geometric shapes in triangles:(1) The agents {1, 3, 4, 6, 7, 9} are assigned to circle onthe inner orbit. All other agents circle on the outerorbit. The 2D range references ρd,i are {15, 19} (m),which requires UAVs’ linear velocities to be {30, 38}(m/s).(2) Agents {1, 4, 7} are used to form a balanced circularformation on the ﬁrst layer. On the second layer, auniform spacing formation is used. To achieve localshapes in triangles, it is convenient to use (16), whichregulates distances directly. Virtual agents are used.The achieved formation pattern is shown in Fig. 5(c).(3) Simply changing the formation pattern on the ﬁrstlayer from the balanced circular to the uniform spac-ing pattern, an overall uniform spacing formation isobtained with local geometric shapes in triangles. Theformation pattern is shown in Fig. 5(d).To clarify how subgroups are deﬁned, how the leaderof each subgroup is chosen, and if the robots know thesize of their subgroups a priori, we use Fig. 5(a) as anexample. For the subgroup consisting agents {1, 2, 3},members in this subgroup are deﬁned by specifying agent1 not to “seek” anyone else; agent 2 “seeking” agent 1; andagent 3 “seeking” agent 2. In each subgroup, all membersdo not know the total number of that subgroup a priori.The “leader” only knows that it does not need to “seek”anyone else in its subgroup. The rest of the members onlyknow which one to “seek”. In the simulation examples, theknowledge of which agent to “seek” is assigned. In reality,these knowledges can be perceived by the agents so thatformation can still be maintained with agents joining orleaving the group.At this point, we would like to highlight diﬀerencesbetween the control laws of previous work [35], [36] andthe new work in this paper, i.e., formation controllers (6),(7), (16), and (17), as well as discussing some interestingaspects of the new control laws. Three formation con-trollers were reported in [35], [36] each achieving a bal-anced circular formation under one of the following threecommunication topologies: (1) all-to-all, (2) ring, and (3)cyclic pursuit. Using the balanced circular formation, allagents, which circle on the same orbit around the target,spread evenly around a full circle, whose centre resides inthe moving target. These previous works are focused onstudying the feasibility of achieving formations for targettracking. Having successfully obtained cooperative track-ing in the balanced circular formation, one would naturallywonder what other formation patterns can be obtainedthat also ﬁt into the tracking scenario. Instead of spread-ing all agents over one full circle, spreading them over aportion of a circle (i.e., an arc) is one natural variation.Two uniform spacing formation controllers (6) and (7) arepresented in this paper. Controller (6), adopted from [32],is adapted into the chain-type communication topology.Controller (7) that is expressed as a function of the rela-tive bearing angle(s) βij(t) alone is proposed in this paper.Controller (7) regulates the separation angles between apair of agents, instead of the separation distances betweenthem as in (6). The diﬀerence between regulating separa-tion distances and separation angles is not obvious whenUAVs circle on the same orbit because a constant separa-tion distance is equivalent to a constant separation angleunder this circumstance.However, when agents circle on diﬀerent orbits (circlesof diﬀerent radii around the target), the diﬀerence is clear,as to be demonstrated in Figs. 6 and 7. By properlyspecifying the desired 2D ranges and setting the UAV’slinear velocities accordingly, UAVs can now be controlledto circle on diﬀerent orbits. With this feature in place, uni-form spacing formations are achieved by using the virtualagent to represent an actual agent in need. Controllers (6)and (7) are modiﬁed to use relative bearing angles ˜βi,j(t)referring to the virtual agent(s), yielding new controllers(16) and (17) for the case of circling on diﬀerent orbits.The ultimate objective is to achieve more versatileconcentric formation patterns, such as patterns with localgeometric shapes. This is made possible utilizing existingworks (tracking
[38], balanced circular formation [35], [36])and the new results presented in this paper (uniform spac-ing formations that regulate either separation distances orangles for agents circling on the same or diﬀerent orbits,and hierarchy). Results presented in this paper providemore complete solution to cooperative target tracking inconcentric formations, by designing the control input ofeach UAV as a sum of several individual control compo-nents. One practical issue of avoiding inter-vehicle col-lision is to be addressed in Section 5. The combinationof Attraction (another way of saying tracking), Alignment(formation), and Avoidance (collision avoidance) providesa solid framework to achieve formations in the context oftarget tracking.Another advantage of having versatile formation pat-terns is that it can possibly allow agents to acquire in-formation of each other using a mixture of informationexchange (over communication channels, for agents thatare far away) and onboard sensing/perception (for agentsthat are close to each other). Refer to the two patternsin Fig. 5(a) and (c), where overall balanced circular for-mations have been achieved with local geometric shapesof either straight lines or triangles. The “leaders” of thethree subgroups, can exchange information over the com-munication channels because they are relatively far awayand might not be able to “see” each other. The othertwo members of each subgroup can possibly use onboardsensing to obtain the information; they need to achievelocal formations (because they are close and can perceiveeach other).5. Inter-vehicle Collision AvoidanceWhen developing the uniform spacing formation, collisionbetween UAVs is more likely to occur than other patternswhere agents are far away from each other. A strategy8Figure 6. Cooperative target tracking using formation controller (17): (a) 2D trajectories; (b) 2D range; (c) formation in theend; (d) separation distance; (e) separation angle; and (f) formation control uic(t). Target undergoes a linear motion. Agentscircle on two diﬀerent orbits.of preventing the inter-vehicle collision is thus much indemand. Among methods that prevent collisions, one wayis to apply a force that repels the agents once they getcloser. This force should also be strong enough to defendother forces pushing agents to a collision [45]–[49].Let dmin denote the minimal distance allowed amongagents. The following collision avoidance control compo-nent helps to repel agents to steer away from each otheronce they get too close [50]:uia(t) = −Krj∈N (ri)dmindijsin βij (18)where Kr is the controller gain. The repulsion term (18)adjusts each agent’s heading to the opposite directionof its neighbours in N(ri). Notice that N (ri) and Nidenote diﬀerent sets. The set Ni is the set of agentswhose information can be obtained by the ith agent viacommunication, whereas the set N (ri) denotes the set ofagents that are too close to agent i.With (18) in place, each agent’s control inputbecomes [50]:˙ψi(t) = ui(t) = uit(t) + uic(t) + uia(t) (19)Totally three control components are added together toachieve the objective of simultaneous tracking, formation,and inter-vehicle collision avoidance.6. Simulation ResultsThe proposed control laws were simulated in Matlab toverify their performance of tracking, formation, and inter-vehicle collision avoidance.6.1 Achieving Uniform Spacing Formations onDiﬀerent OrbitsThis example demonstrates achievement of uniform spac-ing formations when agents circle on diﬀerent orbits. Toachieve formations under this circumstance, formation con-trollers (17) and (16) are used, which resort to virtualagents when needed, i.e., when the two agents are not onthe same orbit. It is worth mentioning that the formationcontroller (16) aims at regulating the relative separationdistances to a constant, while the controller (17) regulatesthe relative separation angles to a desired value. The2D ranges are set to be ρd,i = {15, 15, 18, 18} (m), fori = 1, 2, 3, 4, respectively. Correspondingly, UAVs’ linearvelocities are Vg,i = {30, 30, 36, 36} (m/s), satisfying therelationship in (10). To focus on the formation patterns,the target’s motion is assumed linear (the target moves ona straight line).We ﬁrst present simulation results applying the for-mation controller (17). The 2D trajectories of the agentsand the target are plotted in Fig. 6(a). Details of thetracking are given in Fig. 6(b), where ρi(t) approach theirprescribed values (that are diﬀerent). All agents’ posi-tions at the end of the simulation are shown in Fig. 6(c),9Figure 7. Cooperative target tracking using formation controller (16): (a) 2D trajectories; (b) 2D range; (c) formation in theend; (d) separation distance; (e) separation angle; and (f) formation control uic(t). Target undergoes a linear motion. Agentscircle on two diﬀerent orbits.where all agents scatter around the target on two diﬀer-ent orbits. Both the separation distances (Fig. 6(d))and the separation angles (Fig. 6(e)) are plotted. Whencontroller (17) is applied, we expect the separation an-gles to approach a constant. However, the separationdistances may or may not approach to the same valuebecause the distances between two agents also depend onthe orbits that they lie on. For ds = 10 (m), the desiredseparation angle is either φs = 2 sin−1(ds/30) ≈ 39◦orφs = 2π − 2 sin−1(ds/30) ≈ 321◦. Because all separationangles converge to 39◦(Fig. 6(e)), formation is obtainedsuccessfully. Figure 6(f) shows the formation componentuic(t), which converges to zero upon formation.Results when applying the controller (16) are shownin Fig. 7. Figure 7(a) shows the general picture ofcooperative target tracking. Figure 7(b) plots the 2Dranges between the agents and the target, demonstratingsuccessful tracking with two diﬀerent range values. Figure7(c) shows the formation achieved in the end, where agentscircle on two diﬀerent orbits around the target. Figure7(d) shows that the separation distances between twosuccessive agents are now regulated to the speciﬁed valueof ds = 10 (m) (same for all agents). On the contrary,the separation angles do not approach to one value. FromFig. 7(c), it can be seen that agents {1, 4} and agents{2, 3} circle on two diﬀerent orbits. The separation anglesbetween these two pairs of consecutive agents should besmaller than that between agents {1, 2}, which circle onthe same orbit. This is conﬁrmed in Fig. 7(e). Figure 7(f)shows that the formation control component uic(t) alsovanishes to zero upon formation.Figures 6 and 7 together show the diﬀerence betweenthe two formation controllers (16) and (17). The formationcontroller (16) regulates the relative separation distancesbetween agents to a speciﬁed value ds, while the controller(17) regulates the separation angles to φs.6.2 Achieving Concentric Formations with LocalGeometric ShapesThis section presents cooperative target tracking in con-centric formations with local geometric shapes of straightlines and triangles, as those shown in Fig. 5. To focus onformation patterns, the target’s motion is assumed linear,i.e., the target moves on a straight line. Corresponding tothe four patterns in Fig. 5, simulation results are presentedin Fig. 8, with several snapshots showing how formationis achieved over time. Cooperative target tracking in theseconcentric formations are successfully obtained.6.3 Achieving Inter-vehicle Collision AvoidanceThis example demonstrates cooperative target trackingwith inter-vehicle collision avoidance. The two scenarios ofwithout and with collision avoidance, i.e., before and afterapplying uia(t), are shown and compared in Fig. 9. Weselect the target’s velocity to be piecewise-constant, n = 5,ds = 10 (m), and dmin = 7 (m). The ﬁrst row of Fig. 9 is10Figure 8. Cooperative target tracking in local geometric shapes: (a) balanced-line; (b) uniform-line; (c) balanced-triangle;and (d) uniform-triangle. The overall formation is either balanced circular or uniform spacing. The local geometric shape iseither straight line or triangle.for the scenario without the collision avoidance capability,i.e., before the control component uia(t) is applied. Afteruia(t) is applied, results are shown in the second rowof Fig. 9. In each scenario, the 2D trajectories areplotted to show the general picture (Fig. 9(a) versus (c)).Then, the minimal distance among all agents is plotted,demonstrating the eﬀect of the added control componentuia(t) (Fig. 9(b) versus (d)). For the second scenario withcollision avoidance, the component uia(t) for each agent isshown in Fig. 9(e), where a saturation of |uia(t)| ≤ 3 hasbeen used for all agents. Comparison between Fig. 9(b)and (d) shows that the control component uia(t) helps tokeep the minimal distance to be greater than the allowedvalue. Otherwise, the minimal distance can be muchsmaller, as indicated in Fig. 9(b). The zigzag area in Fig.9(d), corresponds to the circumstances when uia(t) takeseﬀect. Figure 9(e) shows that uia(t) only takes eﬀect whenneeded, i.e., when agents {3, 5} get too close to each other.7. Comparison with Prior StudiesComparing with the distance-based coordination controllaw (6), our proposed bearing-angle-based coordinationcontrol law (7) has one potential advantage. Consider a11Figure 9. With inter-vehicle collision avoidance (before and after): (a) before 2D trajectories; (b) before minimum distance;(c) after 2D trajectories; (b) after minimum distance; and (e) after uic(t) (rad/s).scenario when information exchange among/between someagents becomes unavailable (e.g., due to communicationloss or out of communication range). Instead of computingthe relative bearing angles from the exchanged positionsas in (12), the needed information of relative bearing an-gles can be estimated/obtained from a local vision systeminstalled on each UAV. In other words, when the expectedinformation from others is unavailable (either permanentlyor temporarily), the objective of achieving and maintainingformation could still be achieved by using local measure-ments and estimates.Regarding formation control, most existing results areeither leaderless or leader-following [51], [52]. The proposedmethod of obtaining versatile formation patterns allows acombination of both. This can be seen in the “Balanced-Line” and “Balanced-Triangle” patterns (Fig. 5). Theoverall balanced circular formation can be obtained us-ing a leaderless communication topology, whereas achieve-ment of local geometric shapes can be implemented in aleader-based manner. The adopted hierarchical formationstructure allows selection of appropriate communicationtopologies on diﬀerent layers.This paper also considers a practical issue that wouldoccur, i.e., collisions among agents. This issue was tackledby adding another control component into each UAV’s con-trol input. As can be seen from (18), this added collision-avoidance control component can also be expressed as afunction of bearing angles. As mentioned earlier, a bearing-angle-based control law has the potential of still achiev-ing its control objective (formation or collision avoidance)during communication loss, by using local measurementsfrom each UAV’s onboard sensors.8. ConclusionsThis paper is to obtain more versatile concentric forma-tions in cooperative target tracking where a ﬂeet of UAVs iscommanded to circle above (and around) a moving groundtarget. On the basis of our previous results, versatile for-mation patterns are achieved with the help of three newfeatures. The ﬁrst feature is a new formation pattern,the uniform spacing formation where either the relativeseparation distances or the separation angles can be regu-lated to a desired value. Diﬀerent from a balanced circularformation where agents spread evenly over a full circle,agents can now spread evenly over a portion of a circle.Two kinds of uniform spacing formation control laws areproposed, where one regulates the separation distances be-tween two agents and the other regulates the separationangles in between. The second feature allows UAVs tocircle on diﬀerent orbits. To achieve formation under thiscircumstance, formation controllers will resort to virtualagents representing the actual agents in need. The thirdfeature is the usage of a (two-layer) hierarchical formationstructure, which allows selection of formation patterns fordiﬀerent layers. Combinations of these new features withour existing results yield more versatile concentric forma-tion patterns with diﬀerent local geometric shapes, such asstraight lines and triangles. Inter-vehicle collision avoid-ance is also addressed. Agents will be repelled to steeraway from each other once they get too close.12All UAVs are assumed to have constant linear veloci-ties. Control of each UAV is via its yaw rate. The designidea is to add three control components (three headingcontrollers) together to achieve the overall objective. Eachcontrol component has a goal. The proposed extensions tospreading agents on a portion of a circle, circling agentson orbits of diﬀerent radii, formation in local geometricshapes, and avoiding inter-vehicle collisions, provide morecomplete solution to cooperative target tracking in theconcentric manner.This paper also raises several questions for future in-vestigations. The implementation of the proposed schemeson physical robots and the extension of the developed tech-niques to 3D scenarios and cooperative tracking of multipletargets with obstacle avoidance capability [53], [54] will beof particular interest. Stability analyses in the presenceof formation pattern switching and broken communicationlinks are another research direction to look into. Also, in-vestigations of the time delay factor for obtaining stabilityconditions as well as desirable performance with reason-able computation complexity [55]–[57] are needed. Finally,Artiﬁcial Intelligence (AI) techniques have recently beendeveloped for robotic communication to enhance the com-munication capability of robotic networks for coordinatedactions. Application of the AI and/or Neural Networksto the ﬁeld of robotic networks in the context of coopera-tive target tracking is a promising research area to pursue[58]–[60].References[1] K. Szwaykowska, I.B. Schwartz, L.M.-T. Romero, C.R. Heck-man, D. Mox, and M.A. Hsieh, Collective motion patterns ofswarms with delay coupling: Theory and experiment, PhysicalRevie E, 93(3), 2016, 11.[2] R. Sepulchre, D. Paley, and N. Leonard, Stabilization ofplanar collection motion: All-to-all communication, IEEETransactions on Automatic Control, 52(5), 2007, 811–824.[3] R. Sepulchre, D. Paley, and N. Leonard, Stabilization ofplanar collective motion with limited communication, IEEETransactions on Automatic Control, 53(3), 2008, 706–719.[4] A. Jain, D. Ghose, and P. Menon, Stabilization of balancedcircular motion about a desired center, International Confer-ence on Advances in Control and Optimization of DynamicalSystems, Kanpur, India, 2014.[5] A. Jain and D. Ghose, Stabilization of collective motion insynchronized, balanced and splay phase arrangements on adesired circle, American Control Conference, Chicago, IL.[6] G. Mallik and A. Sinha, A study of balanced circular formationunder deviated cyclic pursuit strategy, IFAC-PapersOnLine,48(5), 2015, 41–46.[7] R. Zheng, Z. Lin, M. Fu, and D. Sun, Distributed control foruniform circumnavigation of ring-coupled unicycles, Automat-ica, 53, 2015, 23–29.[8] N. Kokolakis and N. Koussoulas, Coordinated standoﬀ track-ing of a ground moving target and the phase separation prob-lem, International Conference on Unmanned Aircraft Systems,Dallas, TX, 2018.[9] J. Guo, G. Yan, and Z. Lin, Cooperative control syn-thesis for moving-target-enclosing with changing topologies,International Conference on Robotics and Automation, An-chorage, AK, 2010.[10] Z. Zhou, H. Wang, and Z. Hu, Event-based time varyingformation control for multiple quadrotor UAVs with Markovianswitching topologies, Complexity, 2018, 1–15.[11] Y. Sun and L. Wang, Consensus of multi-agent systems indirected networks with nonuniform time-varying delays, IEEETransactions on Automatic Control, 54(7), 2009, 1607–1613.[12] J. Marshall, M. Broucke, and B. Francis, Formations of vehiclesin cyclic pursuit, IEEE Transactions on Automatic Control,49(11), 2004, 1963–1974.[13] J. Marshall, M. Broucke, and B. Francis, Pursuit formationsof unicycles, Automatica, 42, 2006, 3–12.[14] M. Pavone and E. Frazzoli, Decentralized policies for geomet-ric pattern formation and path coverage, ASME Journal ofDynamic Systems, Measurement, and Control, 129(5), 2007,633–643.[15] B. Wu, D. Wang, and E. Poh, Cyclic formation control for satel-lite formation using local relative measurements, MechatronicSystems and Control, 40(1), 2012, 11–21.[16] J. Juang, On the formation patterns under generalized cyclicpursuit, IEEE Transactions on Automatic Control, 58(9), 2013,2401–2405, 2013.[17] J. Ramirez, M. Pavone, E. Frazzoli, and D. Miller, Distributedcontrol of spacecraft formations via cyclic pursuit: Theory andexperiments, Journal of Guidance, Control, and Dynamics,33(5), 2010, 1655–1669.[18] K. Hausmany, J. Muller, A. Hariharan, N. Ayanian, and G.Sukhatme, Cooperative multi-robot control for target track-ing with onboard sensing, International Journal of RoboticsResearch, 34(13), 2015, 1660–1677.[19] P. Jimenez, B. Shirinzadeh, D. Oetomo, and A. Nicholson,Swarm aggregation and formation control for robots withlimited perception, International Journal of Robotics andAutomation, 26(4), 2011, 255–263.[20] P. Zhu and W. Ren, Multi-robot joint localization and targettracking with local sensing and communication, AmericanControl Conference, Charlotte, NC, 2019.[21] L. Luo, N. Chakraborty, and K. Sycara, Provably-good dis-tributed algorithm for constrained multi-robot task assignmentfor grouped tasks, IEEE Transactions on Robotics, 31(1), 2015,19–30.[22] D. Panagou, M. Turpin, and V. Kumar, Decentralized goalassignment and safe trajectory generation in multi-robot net-works via multiple Lyapunov functions, IEEE Transactions onAutomatic Control, 65(8), 2020, 3365–3380.[23] J. Ni, X. Yang, J. Chen, and S. Yang, Dynamic bioinspiredneural network for multi-robot formation control in unknownenvironments, International Journal of Robotics and Automa-tion, 30(3), 2015, 256–266.[24] M. Khan and C. Silva, Autonomous and robust multi-robotcooperation using an artiﬁcial immune system, InternationalJournal of Robotics and Automation, 27(1), 2012, 60–75.[25] X. Yu, L. Liu, and G. Feng, Coordinated control of multipleunicycles for escorting and patrolling task based on a cyclicpursuit strategy, American Control Conference, Boston, MA,2016.[26] M. Zhang and H. Liu, Cooperative tracking a moving targetusing multiple ﬁxed-wing UAVs, Journal of Intelligent andRobotic Systems, 81(3-4), 2016, 505–529.[27] X. Yu and L. Liu, Cooperative control for moving-target circularformation of nonholonomic vehicles, IEEE Transactions onAutomatic Control, 62(7), 2017, 3448–3454.[28] L. Brinon-Arranz, A. Seuret, and A. Pascoal, Target trackingvia a circular formation of unicycles, IFAC World Congress,Toulouse, France, 2017.[29] A. Miao, Y. Wang, and R. Fierro, Cooperative circumnavi-gation of a moving target with multiple nonholonomic robotsusing backstepping design, Systems and Control Letters, 103,2017, 58–65.[30] L. Ma and N. Hovakimyan, Vision-based cyclic pursuit forcooperative target tracking, Journal of Guidance, Control, andDynamics, 36(2), 2013, 617–622.[31] N. Moshtagh, N. Michael, A. Jadbabaie, and K. Daniilidis,Vision-based, distributed control laws for motion coordinationof nonholonomic robots, IEEE Transactions on Robotics, 25(4),2009, 851–860.[32] N. Ceccarelli, M. Marco, A. Garulli, and A. Giannitrapani,Collective circular motion of multi-vehicle systems, Automatica,44, 2008, 3025–3035.[33] J. Soares, A. Aguiar, A. Pascoal, and M. Gallieri, Triangularformation control using range measurements: An applicationto marine robotic vehicles, IFAC Proceedings, 45(5), 2012,112–117.13[34] Z. He and J. Xu, Moving target tracking by UAVs in an urbanarea, Mechatronic Systems and Control, 42(2), 2014. DOI:10.2316/Journal.201.2014.2.201-2572.[35] L. Ma and N. Hovakimyan, Cooperative target tracking inbalanced circular formation: Multiple UAVs tracking a groundvehicle, American Control Conference, Washington, DC, USA,2013, 5386–5391.[36] L. Ma, Cooperative target tracking with time-varying formationradius, European Control Conference, Linz, Austria, 2015.[37] L. Ma, Cooperative target tracking in balanced circular forma-tion with time-varying radius, International Journal of Roboticsand Automation, 35(4), 2020. DOI: 10.2316/J.2020.206-0086.[38] L. Ma, C. Cao, N. Hovakimyan, V. Dobrokhodov, and I.Kaminer, Adaptive vision-based guidance law with guaran-teed performance bounds, Journal of Guidance, Control, andDynamics, 3, 2010, 33.
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[41]–[44]. The simplest hierarchicalscheme, the two-layer hierarchical structure, can be de-scribed as follows. A collection of n agents is divided inton2 subgroups, each containing n1 agents (n1 × n2 = n).The local control strategy is chosen such that the agentswithin each subgroup can be commanded to achieve cer-tain formation pattern [41]. In [44], a two-layer hybridpursuit system was described, where cyclic pursuit strat-egy was considered at the higher layer (the ﬁrst layer) andchain-like communication topology was used at the lowerlayer (the second layer).The concept of hierarchy is now applied to cooper-ative target tracking. The idea of hierarchy allows dif-ferent subgroups to select diﬀerent formation laws thatare already known to be stable. Similar to [44], a two-layer hierarchical formation structure is used. The ﬁrstlayer can be set to achieve either the balanced circu-lar or the uniform spacing formation. The second layercan be set to achieve the uniform spacing formation byspecifying either a desired separation distance or an-gle. Two examples are given below to demonstrate howversatile patterns are achieved by determining the 2D rangereferences of the agents (same or diﬀerent) and the forma-tion pattern on each layer (balanced circular or uniformspacing).The ﬁrst example achieves concentric formations withlocal geometric shapes in straight lines:(1) The agents {1, 4, 7}, {2, 5, 8}, and {3, 6, 9} areassigned to be on the inner, middle, and outer orbits,respectively, by specifying their 2D range references tobe {15, 18, 25} (m). Correspondingly, the UAVs’ linearvelocities are set to be {30, 36, 50} (m/s), satisfyingthe relationship in (10).(2) On the ﬁrst layer, the balanced circular formation isused for agents {1, 4, 7}, which lie on the inner circle.On the second layer, a uniform spacing formation isused inside each subgroup. There are three subgroups:{1, 2, 3}, {4, 5, 6}, and {7, 8, 9}. The agent that is in-volved in formation on the ﬁrst layer works as “leader”of its subgroup. As the intended geometric shapeis in straight line and the radial diﬀerences betweeneach two adjacent orbits can be diﬀerent, it is moreconvenient to use (7) with β0 = π/2. Actual agentsare used when obtaining bearing angles βi(i+1)(t). Theformation pattern is shown in Fig. 5(a).7(3) Simply changing the formation pattern on the ﬁrstlayer from the balanced circular to the uniform spac-ing pattern, an overall uniform spacing formation isachieved with local geometric shapes in straight lines.The formation pattern is shown in Fig. 5(b).The second example achieves concentric formationswith local geometric shapes in triangles:(1) The agents {1, 3, 4, 6, 7, 9} are assigned to circle onthe inner orbit. All other agents circle on the outerorbit. The 2D range references ρd,i are {15, 19} (m),which requires UAVs’ linear velocities to be {30, 38}(m/s).(2) Agents {1, 4, 7} are used to form a balanced circularformation on the ﬁrst layer. On the second layer, auniform spacing formation is used. To achieve localshapes in triangles, it is convenient to use (16), whichregulates distances directly. Virtual agents are used.The achieved formation pattern is shown in Fig. 5(c).(3) Simply changing the formation pattern on the ﬁrstlayer from the balanced circular to the uniform spac-ing pattern, an overall uniform spacing formation isobtained with local geometric shapes in triangles. Theformation pattern is shown in Fig. 5(d).To clarify how subgroups are deﬁned, how the leaderof each subgroup is chosen, and if the robots know thesize of their subgroups a priori, we use Fig. 5(a) as anexample. For the subgroup consisting agents {1, 2, 3},members in this subgroup are deﬁned by specifying agent1 not to “seek” anyone else; agent 2 “seeking” agent 1; andagent 3 “seeking” agent 2. In each subgroup, all membersdo not know the total number of that subgroup a priori.The “leader” only knows that it does not need to “seek”anyone else in its subgroup. The rest of the members onlyknow which one to “seek”. In the simulation examples, theknowledge of which agent to “seek” is assigned. In reality,these knowledges can be perceived by the agents so thatformation can still be maintained with agents joining orleaving the group.At this point, we would like to highlight diﬀerencesbetween the control laws of previous work [35], [36] andthe new work in this paper, i.e., formation controllers (6),(7), (16), and (17), as well as discussing some interestingaspects of the new control laws. Three formation con-trollers were reported in [35], [36] each achieving a bal-anced circular formation under one of the following threecommunication topologies: (1) all-to-all, (2) ring, and (3)cyclic pursuit. Using the balanced circular formation, allagents, which circle on the same orbit around the target,spread evenly around a full circle, whose centre resides inthe moving target. These previous works are focused onstudying the feasibility of achieving formations for targettracking. Having successfully obtained cooperative track-ing in the balanced circular formation, one would naturallywonder what other formation patterns can be obtainedthat also ﬁt into the tracking scenario. Instead of spread-ing all agents over one full circle, spreading them over aportion of a circle (i.e., an arc) is one natural variation.Two uniform spacing formation controllers (6) and (7) arepresented in this paper. Controller (6), adopted from [32],is adapted into the chain-type communication topology.Controller (7) that is expressed as a function of the rela-tive bearing angle(s) βij(t) alone is proposed in this paper.Controller (7) regulates the separation angles between apair of agents, instead of the separation distances betweenthem as in (6). The diﬀerence between regulating separa-tion distances and separation angles is not obvious whenUAVs circle on the same orbit because a constant separa-tion distance is equivalent to a constant separation angleunder this circumstance.However, when agents circle on diﬀerent orbits (circlesof diﬀerent radii around the target), the diﬀerence is clear,as to be demonstrated in Figs. 6 and 7. By properlyspecifying the desired 2D ranges and setting the UAV’slinear velocities accordingly, UAVs can now be controlledto circle on diﬀerent orbits. With this feature in place, uni-form spacing formations are achieved by using the virtualagent to represent an actual agent in need. Controllers (6)and (7) are modiﬁed to use relative bearing angles ˜βi,j(t)referring to the virtual agent(s), yielding new controllers(16) and (17) for the case of circling on diﬀerent orbits.The ultimate objective is to achieve more versatileconcentric formation patterns, such as patterns with localgeometric shapes. This is made possible utilizing existingworks (tracking [38], balanced circular formation [35], [36])and the new results presented in this paper (uniform spac-ing formations that regulate either separation distances orangles for agents circling on the same or diﬀerent orbits,and hierarchy). Results presented in this paper providemore complete solution to cooperative target tracking inconcentric formations, by designing the control input ofeach UAV as a sum of several individual control compo-nents. One practical issue of avoiding inter-vehicle col-lision is to be addressed in Section 5. The combinationof Attraction (another way of saying tracking), Alignment(formation), and Avoidance (collision avoidance) providesa solid framework to achieve formations in the context oftarget tracking.Another advantage of having versatile formation pat-terns is that it can possibly allow agents to acquire in-formation of each other using a mixture of informationexchange (over communication channels, for agents thatare far away) and onboard sensing/perception (for agentsthat are close to each other). Refer to the two patternsin Fig. 5(a) and (c), where overall balanced circular for-mations have been achieved with local geometric shapesof either straight lines or triangles. The “leaders” of thethree subgroups, can exchange information over the com-munication channels because they are relatively far awayand might not be able to “see” each other. The othertwo members of each subgroup can possibly use onboardsensing to obtain the information; they need to achievelocal formations (because they are close and can perceiveeach other).5. Inter-vehicle Collision AvoidanceWhen developing the uniform spacing formation, collisionbetween UAVs is more likely to occur than other patternswhere agents are far away from each other. A strategy8Figure 6. Cooperative target tracking using formation controller (17): (a) 2D trajectories; (b) 2D range; (c) formation in theend; (d) separation distance; (e) separation angle; and (f) formation control uic(t). Target undergoes a linear motion. Agentscircle on two diﬀerent orbits.of preventing the inter-vehicle collision is thus much indemand. Among methods that prevent collisions, one wayis to apply a force that repels the agents once they getcloser. This force should also be strong enough to defendother forces pushing agents to a collision [45]–[49].Let dmin denote the minimal distance allowed amongagents. The following collision avoidance control compo-nent helps to repel agents to steer away from each otheronce they get too close [50]:uia(t) = −Krj∈N (ri)dmindijsin βij (18)where Kr is the controller gain. The repulsion term (18)adjusts each agent’s heading to the opposite directionof its neighbours in N(ri). Notice that N (ri) and Nidenote diﬀerent sets. The set Ni is the set of agentswhose information can be obtained by the ith agent viacommunication, whereas the set N (ri) denotes the set ofagents that are too close to agent i.With (18) in place, each agent’s control inputbecomes [50]:˙ψi(t) = ui(t) = uit(t) + uic(t) + uia(t) (19)Totally three control components are added together toachieve the objective of simultaneous tracking, formation,and inter-vehicle collision avoidance.6. Simulation ResultsThe proposed control laws were simulated in Matlab toverify their performance of tracking, formation, and inter-vehicle collision avoidance.6.1 Achieving Uniform Spacing Formations onDiﬀerent OrbitsThis example demonstrates achievement of uniform spac-ing formations when agents circle on diﬀerent orbits. Toachieve formations under this circumstance, formation con-trollers (17) and (16) are used, which resort to virtualagents when needed, i.e., when the two agents are not onthe same orbit. It is worth mentioning that the formationcontroller (16) aims at regulating the relative separationdistances to a constant, while the controller (17) regulatesthe relative separation angles to a desired value. The2D ranges are set to be ρd,i = {15, 15, 18, 18} (m), fori = 1, 2, 3, 4, respectively. Correspondingly, UAVs’ linearvelocities are Vg,i = {30, 30, 36, 36} (m/s), satisfying therelationship in (10). To focus on the formation patterns,the target’s motion is assumed linear (the target moves ona straight line).We ﬁrst present simulation results applying the for-mation controller (17). The 2D trajectories of the agentsand the target are plotted in Fig. 6(a). Details of thetracking are given in Fig. 6(b), where ρi(t) approach theirprescribed values (that are diﬀerent). All agents’ posi-tions at the end of the simulation are shown in Fig. 6(c),9Figure 7. Cooperative target tracking using formation controller (16): (a) 2D trajectories; (b) 2D range; (c) formation in theend; (d) separation distance; (e) separation angle; and (f) formation control uic(t). Target undergoes a linear motion. Agentscircle on two diﬀerent orbits.where all agents scatter around the target on two diﬀer-ent orbits. Both the separation distances (Fig. 6(d))and the separation angles (Fig. 6(e)) are plotted. Whencontroller (17) is applied, we expect the separation an-gles to approach a constant. However, the separationdistances may or may not approach to the same valuebecause the distances between two agents also depend onthe orbits that they lie on. For ds = 10 (m), the desiredseparation angle is either φs = 2 sin−1(ds/30) ≈ 39◦orφs = 2π − 2 sin−1(ds/30) ≈ 321◦. Because all separationangles converge to 39◦(Fig. 6(e)), formation is obtainedsuccessfully. Figure 6(f) shows the formation componentuic(t), which converges to zero upon formation.Results when applying the controller (16) are shownin Fig. 7. Figure 7(a) shows the general picture ofcooperative target tracking. Figure 7(b) plots the 2Dranges between the agents and the target, demonstratingsuccessful tracking with two diﬀerent range values. Figure7(c) shows the formation achieved in the end, where agentscircle on two diﬀerent orbits around the target. Figure7(d) shows that the separation distances between twosuccessive agents are now regulated to the speciﬁed valueof ds = 10 (m) (same for all agents). On the contrary,the separation angles do not approach to one value. FromFig. 7(c), it can be seen that agents {1, 4} and agents{2, 3} circle on two diﬀerent orbits. The separation anglesbetween these two pairs of consecutive agents should besmaller than that between agents {1, 2}, which circle onthe same orbit. This is conﬁrmed in Fig. 7(e). Figure 7(f)shows that the formation control component uic(t) alsovanishes to zero upon formation.Figures 6 and 7 together show the diﬀerence betweenthe two formation controllers (16) and (17). The formationcontroller (16) regulates the relative separation distancesbetween agents to a speciﬁed value ds, while the controller(17) regulates the separation angles to φs.6.2 Achieving Concentric Formations with LocalGeometric ShapesThis section presents cooperative target tracking in con-centric formations with local geometric shapes of straightlines and triangles, as those shown in Fig. 5. To focus onformation patterns, the target’s motion is assumed linear,i.e., the target moves on a straight line. Corresponding tothe four patterns in Fig. 5, simulation results are presentedin Fig. 8, with several snapshots showing how formationis achieved over time. Cooperative target tracking in theseconcentric formations are successfully obtained.6.3 Achieving Inter-vehicle Collision AvoidanceThis example demonstrates cooperative target trackingwith inter-vehicle collision avoidance. The two scenarios ofwithout and with collision avoidance, i.e., before and afterapplying uia(t), are shown and compared in Fig. 9. Weselect the target’s velocity to be piecewise-constant, n = 5,ds = 10 (m), and dmin = 7 (m). The ﬁrst row of Fig. 9 is10Figure 8. Cooperative target tracking in local geometric shapes: (a) balanced-line; (b) uniform-line; (c) balanced-triangle;and (d) uniform-triangle. The overall formation is either balanced circular or uniform spacing. The local geometric shape iseither straight line or triangle.for the scenario without the collision avoidance capability,i.e., before the control component uia(t) is applied. Afteruia(t) is applied, results are shown in the second rowof Fig. 9. In each scenario, the 2D trajectories areplotted to show the general picture (Fig. 9(a) versus (c)).Then, the minimal distance among all agents is plotted,demonstrating the eﬀect of the added control componentuia(t) (Fig. 9(b) versus (d)). For the second scenario withcollision avoidance, the component uia(t) for each agent isshown in Fig. 9(e), where a saturation of |uia(t)| ≤ 3 hasbeen used for all agents. Comparison between Fig. 9(b)and (d) shows that the control component uia(t) helps tokeep the minimal distance to be greater than the allowedvalue. Otherwise, the minimal distance can be muchsmaller, as indicated in Fig. 9(b). The zigzag area in Fig.9(d), corresponds to the circumstances when uia(t) takeseﬀect. Figure 9(e) shows that uia(t) only takes eﬀect whenneeded, i.e., when agents {3, 5} get too close to each other.7. Comparison with Prior StudiesComparing with the distance-based coordination controllaw (6), our proposed bearing-angle-based coordinationcontrol law (7) has one potential advantage. Consider a11Figure 9. With inter-vehicle collision avoidance (before and after): (a) before 2D trajectories; (b) before minimum distance;(c) after 2D trajectories; (b) after minimum distance; and (e) after uic(t) (rad/s).scenario when information exchange among/between someagents becomes unavailable (e.g., due to communicationloss or out of communication range). Instead of computingthe relative bearing angles from the exchanged positionsas in (12), the needed information of relative bearing an-gles can be estimated/obtained from a local vision systeminstalled on each UAV. In other words, when the expectedinformation from others is unavailable (either permanentlyor temporarily), the objective of achieving and maintainingformation could still be achieved by using local measure-ments and estimates.Regarding formation control, most existing results areeither leaderless or leader-following [51], [52]. The proposedmethod of obtaining versatile formation patterns allows acombination of both. This can be seen in the “Balanced-Line” and “Balanced-Triangle” patterns (Fig. 5). Theoverall balanced circular formation can be obtained us-ing a leaderless communication topology, whereas achieve-ment of local geometric shapes can be implemented in aleader-based manner. The adopted hierarchical formationstructure allows selection of appropriate communicationtopologies on diﬀerent layers.This paper also considers a practical issue that wouldoccur, i.e., collisions among agents. This issue was tackledby adding another control component into each UAV’s con-trol input. As can be seen from (18), this added collision-avoidance control component can also be expressed as afunction of bearing angles. As mentioned earlier, a bearing-angle-based control law has the potential of still achiev-ing its control objective (formation or collision avoidance)during communication loss, by using local measurementsfrom each UAV’s onboard sensors.8. ConclusionsThis paper is to obtain more versatile concentric forma-tions in cooperative target tracking where a ﬂeet of UAVs iscommanded to circle above (and around) a moving groundtarget. On the basis of our previous results, versatile for-mation patterns are achieved with the help of three newfeatures. The ﬁrst feature is a new formation pattern,the uniform spacing formation where either the relativeseparation distances or the separation angles can be regu-lated to a desired value. Diﬀerent from a balanced circularformation where agents spread evenly over a full circle,agents can now spread evenly over a portion of a circle.Two kinds of uniform spacing formation control laws areproposed, where one regulates the separation distances be-tween two agents and the other regulates the separationangles in between. The second feature allows UAVs tocircle on diﬀerent orbits. To achieve formation under thiscircumstance, formation controllers will resort to virtualagents representing the actual agents in need. The thirdfeature is the usage of a (two-layer) hierarchical formationstructure, which allows selection of formation patterns fordiﬀerent layers. Combinations of these new features withour existing results yield more versatile concentric forma-tion patterns with diﬀerent local geometric shapes, such asstraight lines and triangles. Inter-vehicle collision avoid-ance is also addressed. Agents will be repelled to steeraway from each other once they get too close.12All UAVs are assumed to have constant linear veloci-ties. Control of each UAV is via its yaw rate. The designidea is to add three control components (three headingcontrollers) together to achieve the overall objective. Eachcontrol component has a goal. The proposed extensions tospreading agents on a portion of a circle, circling agentson orbits of diﬀerent radii, formation in local geometricshapes, and avoiding inter-vehicle collisions, provide morecomplete solution to cooperative target tracking in theconcentric manner.This paper also raises several questions for future in-vestigations. The implementation of the proposed schemeson physical robots and the extension of the developed tech-niques to 3D scenarios and cooperative tracking of multipletargets with obstacle avoidance capability [53], [54] will beof particular interest. Stability analyses in the presenceof formation pattern switching and broken communicationlinks are another research direction to look into. Also, in-vestigations of the time delay factor for obtaining stabilityconditions as well as desirable performance with reason-able computation complexity [55]–[57] are needed. Finally,Artiﬁcial Intelligence (AI) techniques have recently beendeveloped for robotic communication to enhance the com-munication capability of robotic networks for coordinatedactions. Application of the AI and/or Neural Networksto the ﬁeld of robotic networks in the context of coopera-tive target tracking is a promising research area to pursue[58]–[60].References[1] K. Szwaykowska, I.B. Schwartz, L.M.-T. Romero, C.R. Heck-man, D. Mox, and M.A. Hsieh, Collective motion patterns ofswarms with delay coupling: Theory and experiment, PhysicalRevie E, 93(3), 2016, 11.[2] R. Sepulchre, D. Paley, and N. Leonard, Stabilization ofplanar collection motion: All-to-all communication, IEEETransactions on Automatic Control, 52(5), 2007, 811–824.[3] R. Sepulchre, D. Paley, and N. Leonard, Stabilization ofplanar collective motion with limited communication, IEEETransactions on Automatic Control, 53(3), 2008, 706–719.[4] A. Jain, D. Ghose, and P. Menon, Stabilization of balancedcircular motion about a desired center, International Confer-ence on Advances in Control and Optimization of DynamicalSystems, Kanpur, India, 2014.[5] A. Jain and D. Ghose, Stabilization of collective motion insynchronized, balanced and splay phase arrangements on adesired circle, American Control Conference, Chicago, IL.[6] G. Mallik and A. Sinha, A study of balanced circular formationunder deviated cyclic pursuit strategy, IFAC-PapersOnLine,48(5), 2015, 41–46.[7] R. Zheng, Z. Lin, M. Fu, and D. Sun, Distributed control foruniform circumnavigation of ring-coupled unicycles, Automat-ica, 53, 2015, 23–29.[8] N. Kokolakis and N. Koussoulas, Coordinated standoﬀ track-ing of a ground moving target and the phase separation prob-lem, International Conference on Unmanned Aircraft Systems,Dallas, TX, 2018.[9] J. Guo, G. Yan, and Z. Lin, Cooperative control syn-thesis for moving-target-enclosing with changing topologies,International Conference on Robotics and Automation, An-chorage, AK, 2010.[10] Z. Zhou, H. Wang, and Z. Hu, Event-based time varyingformation control for multiple quadrotor UAVs with Markovianswitching topologies, Complexity, 2018, 1–15.[11] Y. Sun and L. Wang, Consensus of multi-agent systems indirected networks with nonuniform time-varying delays, IEEETransactions on Automatic Control, 54(7), 2009, 1607–1613.[12] J. Marshall, M. Broucke, and B. Francis, Formations of vehiclesin cyclic pursuit, IEEE Transactions on Automatic Control,49(11), 2004, 1963–1974.[13] J. Marshall, M. Broucke, and B. Francis, Pursuit formationsof unicycles, Automatica, 42, 2006, 3–12.[14] M. Pavone and E. Frazzoli, Decentralized policies for geomet-ric pattern formation and path coverage, ASME Journal ofDynamic Systems, Measurement, and Control, 129(5), 2007,633–643.[15] B. Wu, D. Wang, and E. Poh, Cyclic formation control for satel-lite formation using local relative measurements, MechatronicSystems and Control, 40(1), 2012, 11–21.[16] J. Juang, On the formation patterns under generalized cyclicpursuit, IEEE Transactions on Automatic Control, 58(9), 2013,2401–2405, 2013.[17] J. Ramirez, M. Pavone, E. Frazzoli, and D. Miller, Distributedcontrol of spacecraft formations via cyclic pursuit: Theory andexperiments, Journal of Guidance, Control, and Dynamics,33(5), 2010, 1655–1669.[18] K. Hausmany, J. Muller, A. Hariharan, N. Ayanian, and G.Sukhatme, Cooperative multi-robot control for target track-ing with onboard sensing, International Journal of RoboticsResearch, 34(13), 2015, 1660–1677.[19] P. Jimenez, B. Shirinzadeh, D. Oetomo, and A. Nicholson,Swarm aggregation and formation control for robots withlimited perception, International Journal of Robotics andAutomation, 26(4), 2011, 255–263.[20] P. Zhu and W. Ren, Multi-robot joint localization and targettracking with local sensing and communication, AmericanControl Conference, Charlotte, NC, 2019.[21] L. Luo, N. Chakraborty, and K. Sycara, Provably-good dis-tributed algorithm for constrained multi-robot task assignmentfor grouped tasks, IEEE Transactions on Robotics, 31(1), 2015,19–30.[22] D. Panagou, M. Turpin, and V. Kumar, Decentralized goalassignment and safe trajectory generation in multi-robot net-works via multiple Lyapunov functions, IEEE Transactions onAutomatic Control, 65(8), 2020, 3365–3380.[23] J. Ni, X. Yang, J. Chen, and S. Yang, Dynamic bioinspiredneural network for multi-robot formation control in unknownenvironments, International Journal of Robotics and Automa-tion, 30(3), 2015, 256–266.[24] M. Khan and C. Silva, Autonomous and robust multi-robotcooperation using an artiﬁcial immune system, InternationalJournal of Robotics and Automation, 27(1), 2012, 60–75.[25] X. Yu, L. Liu, and G. Feng, Coordinated control of multipleunicycles for escorting and patrolling task based on a cyclicpursuit strategy, American Control Conference, Boston, MA,2016.[26] M. Zhang and H. Liu, Cooperative tracking a moving targetusing multiple ﬁxed-wing UAVs, Journal of Intelligent andRobotic Systems, 81(3-4), 2016, 505–529.[27] X. Yu and L. Liu, Cooperative control for moving-target circularformation of nonholonomic vehicles, IEEE Transactions onAutomatic Control, 62(7), 2017, 3448–3454.[28] L. Brinon-Arranz, A. Seuret, and A. Pascoal, Target trackingvia a circular formation of unicycles, IFAC World Congress,Toulouse, France, 2017.[29] A. Miao, Y. Wang, and R. 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DOI:10.2316/Journal.201.2014.2.201-2572.[35] L. Ma and N. Hovakimyan, Cooperative target tracking inbalanced circular formation: Multiple UAVs tracking a groundvehicle, American Control Conference, Washington, DC, USA,2013, 5386–5391.[36] L. Ma, Cooperative target tracking with time-varying formationradius, European Control Conference, Linz, Austria, 2015.[37] L. Ma, Cooperative target tracking in balanced circular forma-tion with time-varying radius, International Journal of Roboticsand Automation, 35(4), 2020. DOI: 10.2316/J.2020.206-0086.[38] L. Ma, C. Cao, N. Hovakimyan, V. Dobrokhodov, and I.Kaminer, Adaptive vision-based guidance law with guaran-teed performance bounds, Journal of Guidance, Control, andDynamics, 3, 2010, 33.[39] V. Cichella, I. Kaminer, V. Dobrokhodov, and N. Hovakimyan,Coordinated vision-based tracking for multiple UAVs, Ameri-can Control Conference, Hamburg, Germany, 2015.[40] Q. Han, S. Sun, and H. 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[42] L. Consolini, F. Morbidi, D. Prattichizzo, and M. Tosques,Steering hierarchical formations of unicycle robots, IEEE Con-ference on Decision and Control, New Orleans, LA, 2007.
[44]. The simplest hierarchicalscheme, the two-layer hierarchical structure, can be de-scribed as follows. A collection of n agents is divided inton2 subgroups, each containing n1 agents (n1 × n2 = n).The local control strategy is chosen such that the agentswithin each subgroup can be commanded to achieve cer-tain formation pattern [41]. In [44], a two-layer hybridpursuit system was described, where cyclic pursuit strat-egy was considered at the higher layer (the ﬁrst layer) andchain-like communication topology was used at the lowerlayer (the second layer).The concept of hierarchy is now applied to cooper-ative target tracking. The idea of hierarchy allows dif-ferent subgroups to select diﬀerent formation laws thatare already known to be stable. Similar to [44], a two-layer hierarchical formation structure is used. The ﬁrstlayer can be set to achieve either the balanced circu-lar or the uniform spacing formation. The second layercan be set to achieve the uniform spacing formation byspecifying either a desired separation distance or an-gle. Two examples are given below to demonstrate howversatile patterns are achieved by determining the 2D rangereferences of the agents (same or diﬀerent) and the forma-tion pattern on each layer (balanced circular or uniformspacing).The ﬁrst example achieves concentric formations withlocal geometric shapes in straight lines:(1) The agents {1, 4, 7}, {2, 5, 8}, and {3, 6, 9} areassigned to be on the inner, middle, and outer orbits,respectively, by specifying their 2D range references tobe {15, 18, 25} (m). Correspondingly, the UAVs’ linearvelocities are set to be {30, 36, 50} (m/s), satisfyingthe relationship in (10).(2) On the ﬁrst layer, the balanced circular formation isused for agents {1, 4, 7}, which lie on the inner circle.On the second layer, a uniform spacing formation isused inside each subgroup. There are three subgroups:{1, 2, 3}, {4, 5, 6}, and {7, 8, 9}. The agent that is in-volved in formation on the ﬁrst layer works as “leader”of its subgroup. As the intended geometric shapeis in straight line and the radial diﬀerences betweeneach two adjacent orbits can be diﬀerent, it is moreconvenient to use (7) with β0 = π/2. Actual agentsare used when obtaining bearing angles βi(i+1)(t). Theformation pattern is shown in Fig. 5(a).7(3) Simply changing the formation pattern on the ﬁrstlayer from the balanced circular to the uniform spac-ing pattern, an overall uniform spacing formation isachieved with local geometric shapes in straight lines.The formation pattern is shown in Fig. 5(b).The second example achieves concentric formationswith local geometric shapes in triangles:(1) The agents {1, 3, 4, 6, 7, 9} are assigned to circle onthe inner orbit. All other agents circle on the outerorbit. The 2D range references ρd,i are {15, 19} (m),which requires UAVs’ linear velocities to be {30, 38}(m/s).(2) Agents {1, 4, 7} are used to form a balanced circularformation on the ﬁrst layer. On the second layer, auniform spacing formation is used. To achieve localshapes in triangles, it is convenient to use (16), whichregulates distances directly. Virtual agents are used.The achieved formation pattern is shown in Fig. 5(c).(3) Simply changing the formation pattern on the ﬁrstlayer from the balanced circular to the uniform spac-ing pattern, an overall uniform spacing formation isobtained with local geometric shapes in triangles. Theformation pattern is shown in Fig. 5(d).To clarify how subgroups are deﬁned, how the leaderof each subgroup is chosen, and if the robots know thesize of their subgroups a priori, we use Fig. 5(a) as anexample. For the subgroup consisting agents {1, 2, 3},members in this subgroup are deﬁned by specifying agent1 not to “seek” anyone else; agent 2 “seeking” agent 1; andagent 3 “seeking” agent 2. In each subgroup, all membersdo not know the total number of that subgroup a priori.The “leader” only knows that it does not need to “seek”anyone else in its subgroup. The rest of the members onlyknow which one to “seek”. In the simulation examples, theknowledge of which agent to “seek” is assigned. In reality,these knowledges can be perceived by the agents so thatformation can still be maintained with agents joining orleaving the group.At this point, we would like to highlight diﬀerencesbetween the control laws of previous work [35], [36] andthe new work in this paper, i.e., formation controllers (6),(7), (16), and (17), as well as discussing some interestingaspects of the new control laws. Three formation con-trollers were reported in [35], [36] each achieving a bal-anced circular formation under one of the following threecommunication topologies: (1) all-to-all, (2) ring, and (3)cyclic pursuit. Using the balanced circular formation, allagents, which circle on the same orbit around the target,spread evenly around a full circle, whose centre resides inthe moving target. These previous works are focused onstudying the feasibility of achieving formations for targettracking. Having successfully obtained cooperative track-ing in the balanced circular formation, one would naturallywonder what other formation patterns can be obtainedthat also ﬁt into the tracking scenario. Instead of spread-ing all agents over one full circle, spreading them over aportion of a circle (i.e., an arc) is one natural variation.Two uniform spacing formation controllers (6) and (7) arepresented in this paper. Controller (6), adopted from [32],is adapted into the chain-type communication topology.Controller (7) that is expressed as a function of the rela-tive bearing angle(s) βij(t) alone is proposed in this paper.Controller (7) regulates the separation angles between apair of agents, instead of the separation distances betweenthem as in (6). The diﬀerence between regulating separa-tion distances and separation angles is not obvious whenUAVs circle on the same orbit because a constant separa-tion distance is equivalent to a constant separation angleunder this circumstance.However, when agents circle on diﬀerent orbits (circlesof diﬀerent radii around the target), the diﬀerence is clear,as to be demonstrated in Figs. 6 and 7. By properlyspecifying the desired 2D ranges and setting the UAV’slinear velocities accordingly, UAVs can now be controlledto circle on diﬀerent orbits. With this feature in place, uni-form spacing formations are achieved by using the virtualagent to represent an actual agent in need. Controllers (6)and (7) are modiﬁed to use relative bearing angles ˜βi,j(t)referring to the virtual agent(s), yielding new controllers(16) and (17) for the case of circling on diﬀerent orbits.The ultimate objective is to achieve more versatileconcentric formation patterns, such as patterns with localgeometric shapes. This is made possible utilizing existingworks (tracking [38], balanced circular formation [35], [36])and the new results presented in this paper (uniform spac-ing formations that regulate either separation distances orangles for agents circling on the same or diﬀerent orbits,and hierarchy). Results presented in this paper providemore complete solution to cooperative target tracking inconcentric formations, by designing the control input ofeach UAV as a sum of several individual control compo-nents. One practical issue of avoiding inter-vehicle col-lision is to be addressed in Section 5. The combinationof Attraction (another way of saying tracking), Alignment(formation), and Avoidance (collision avoidance) providesa solid framework to achieve formations in the context oftarget tracking.Another advantage of having versatile formation pat-terns is that it can possibly allow agents to acquire in-formation of each other using a mixture of informationexchange (over communication channels, for agents thatare far away) and onboard sensing/perception (for agentsthat are close to each other). Refer to the two patternsin Fig. 5(a) and (c), where overall balanced circular for-mations have been achieved with local geometric shapesof either straight lines or triangles. The “leaders” of thethree subgroups, can exchange information over the com-munication channels because they are relatively far awayand might not be able to “see” each other. The othertwo members of each subgroup can possibly use onboardsensing to obtain the information; they need to achievelocal formations (because they are close and can perceiveeach other).5. Inter-vehicle Collision AvoidanceWhen developing the uniform spacing formation, collisionbetween UAVs is more likely to occur than other patternswhere agents are far away from each other. A strategy8Figure 6. Cooperative target tracking using formation controller (17): (a) 2D trajectories; (b) 2D range; (c) formation in theend; (d) separation distance; (e) separation angle; and (f) formation control uic(t). Target undergoes a linear motion. Agentscircle on two diﬀerent orbits.of preventing the inter-vehicle collision is thus much indemand. Among methods that prevent collisions, one wayis to apply a force that repels the agents once they getcloser. This force should also be strong enough to defendother forces pushing agents to a collision
[45]–[49].Let dmin denote the minimal distance allowed amongagents. The following collision avoidance control compo-nent helps to repel agents to steer away from each otheronce they get too close [50]:uia(t) = −Krj∈N (ri)dmindijsin βij (18)where Kr is the controller gain. The repulsion term (18)adjusts each agent’s heading to the opposite directionof its neighbours in N(ri). Notice that N (ri) and Nidenote diﬀerent sets. The set Ni is the set of agentswhose information can be obtained by the ith agent viacommunication, whereas the set N (ri) denotes the set ofagents that are too close to agent i.With (18) in place, each agent’s control inputbecomes [50]:˙ψi(t) = ui(t) = uit(t) + uic(t) + uia(t) (19)Totally three control components are added together toachieve the objective of simultaneous tracking, formation,and inter-vehicle collision avoidance.6. Simulation ResultsThe proposed control laws were simulated in Matlab toverify their performance of tracking, formation, and inter-vehicle collision avoidance.6.1 Achieving Uniform Spacing Formations onDiﬀerent OrbitsThis example demonstrates achievement of uniform spac-ing formations when agents circle on diﬀerent orbits. Toachieve formations under this circumstance, formation con-trollers (17) and (16) are used, which resort to virtualagents when needed, i.e., when the two agents are not onthe same orbit. It is worth mentioning that the formationcontroller (16) aims at regulating the relative separationdistances to a constant, while the controller (17) regulatesthe relative separation angles to a desired value. The2D ranges are set to be ρd,i = {15, 15, 18, 18} (m), fori = 1, 2, 3, 4, respectively. Correspondingly, UAVs’ linearvelocities are Vg,i = {30, 30, 36, 36} (m/s), satisfying therelationship in (10). To focus on the formation patterns,the target’s motion is assumed linear (the target moves ona straight line).We ﬁrst present simulation results applying the for-mation controller (17). The 2D trajectories of the agentsand the target are plotted in Fig. 6(a). Details of thetracking are given in Fig. 6(b), where ρi(t) approach theirprescribed values (that are diﬀerent). All agents’ posi-tions at the end of the simulation are shown in Fig. 6(c),9Figure 7. Cooperative target tracking using formation controller (16): (a) 2D trajectories; (b) 2D range; (c) formation in theend; (d) separation distance; (e) separation angle; and (f) formation control uic(t). Target undergoes a linear motion. Agentscircle on two diﬀerent orbits.where all agents scatter around the target on two diﬀer-ent orbits. Both the separation distances (Fig. 6(d))and the separation angles (Fig. 6(e)) are plotted. Whencontroller (17) is applied, we expect the separation an-gles to approach a constant. However, the separationdistances may or may not approach to the same valuebecause the distances between two agents also depend onthe orbits that they lie on. For ds = 10 (m), the desiredseparation angle is either φs = 2 sin−1(ds/30) ≈ 39◦orφs = 2π − 2 sin−1(ds/30) ≈ 321◦. Because all separationangles converge to 39◦(Fig. 6(e)), formation is obtainedsuccessfully. Figure 6(f) shows the formation componentuic(t), which converges to zero upon formation.Results when applying the controller (16) are shownin Fig. 7. Figure 7(a) shows the general picture ofcooperative target tracking. Figure 7(b) plots the 2Dranges between the agents and the target, demonstratingsuccessful tracking with two diﬀerent range values. Figure7(c) shows the formation achieved in the end, where agentscircle on two diﬀerent orbits around the target. Figure7(d) shows that the separation distances between twosuccessive agents are now regulated to the speciﬁed valueof ds = 10 (m) (same for all agents). On the contrary,the separation angles do not approach to one value. FromFig. 7(c), it can be seen that agents {1, 4} and agents{2, 3} circle on two diﬀerent orbits. The separation anglesbetween these two pairs of consecutive agents should besmaller than that between agents {1, 2}, which circle onthe same orbit. This is conﬁrmed in Fig. 7(e). Figure 7(f)shows that the formation control component uic(t) alsovanishes to zero upon formation.Figures 6 and 7 together show the diﬀerence betweenthe two formation controllers (16) and (17). The formationcontroller (16) regulates the relative separation distancesbetween agents to a speciﬁed value ds, while the controller(17) regulates the separation angles to φs.6.2 Achieving Concentric Formations with LocalGeometric ShapesThis section presents cooperative target tracking in con-centric formations with local geometric shapes of straightlines and triangles, as those shown in Fig. 5. To focus onformation patterns, the target’s motion is assumed linear,i.e., the target moves on a straight line. Corresponding tothe four patterns in Fig. 5, simulation results are presentedin Fig. 8, with several snapshots showing how formationis achieved over time. Cooperative target tracking in theseconcentric formations are successfully obtained.6.3 Achieving Inter-vehicle Collision AvoidanceThis example demonstrates cooperative target trackingwith inter-vehicle collision avoidance. The two scenarios ofwithout and with collision avoidance, i.e., before and afterapplying uia(t), are shown and compared in Fig. 9. Weselect the target’s velocity to be piecewise-constant, n = 5,ds = 10 (m), and dmin = 7 (m). The ﬁrst row of Fig. 9 is10Figure 8. Cooperative target tracking in local geometric shapes: (a) balanced-line; (b) uniform-line; (c) balanced-triangle;and (d) uniform-triangle. The overall formation is either balanced circular or uniform spacing. The local geometric shape iseither straight line or triangle.for the scenario without the collision avoidance capability,i.e., before the control component uia(t) is applied. Afteruia(t) is applied, results are shown in the second rowof Fig. 9. In each scenario, the 2D trajectories areplotted to show the general picture (Fig. 9(a) versus (c)).Then, the minimal distance among all agents is plotted,demonstrating the eﬀect of the added control componentuia(t) (Fig. 9(b) versus (d)). For the second scenario withcollision avoidance, the component uia(t) for each agent isshown in Fig. 9(e), where a saturation of |uia(t)| ≤ 3 hasbeen used for all agents. Comparison between Fig. 9(b)and (d) shows that the control component uia(t) helps tokeep the minimal distance to be greater than the allowedvalue. Otherwise, the minimal distance can be muchsmaller, as indicated in Fig. 9(b). The zigzag area in Fig.9(d), corresponds to the circumstances when uia(t) takeseﬀect. Figure 9(e) shows that uia(t) only takes eﬀect whenneeded, i.e., when agents {3, 5} get too close to each other.7. Comparison with Prior StudiesComparing with the distance-based coordination controllaw (6), our proposed bearing-angle-based coordinationcontrol law (7) has one potential advantage. Consider a11Figure 9. With inter-vehicle collision avoidance (before and after): (a) before 2D trajectories; (b) before minimum distance;(c) after 2D trajectories; (b) after minimum distance; and (e) after uic(t) (rad/s).scenario when information exchange among/between someagents becomes unavailable (e.g., due to communicationloss or out of communication range). Instead of computingthe relative bearing angles from the exchanged positionsas in (12), the needed information of relative bearing an-gles can be estimated/obtained from a local vision systeminstalled on each UAV. In other words, when the expectedinformation from others is unavailable (either permanentlyor temporarily), the objective of achieving and maintainingformation could still be achieved by using local measure-ments and estimates.Regarding formation control, most existing results areeither leaderless or leader-following [51], [52]. The proposedmethod of obtaining versatile formation patterns allows acombination of both. This can be seen in the “Balanced-Line” and “Balanced-Triangle” patterns (Fig. 5). Theoverall balanced circular formation can be obtained us-ing a leaderless communication topology, whereas achieve-ment of local geometric shapes can be implemented in aleader-based manner. The adopted hierarchical formationstructure allows selection of appropriate communicationtopologies on diﬀerent layers.This paper also considers a practical issue that wouldoccur, i.e., collisions among agents. This issue was tackledby adding another control component into each UAV’s con-trol input. As can be seen from (18), this added collision-avoidance control component can also be expressed as afunction of bearing angles. As mentioned earlier, a bearing-angle-based control law has the potential of still achiev-ing its control objective (formation or collision avoidance)during communication loss, by using local measurementsfrom each UAV’s onboard sensors.8. ConclusionsThis paper is to obtain more versatile concentric forma-tions in cooperative target tracking where a ﬂeet of UAVs iscommanded to circle above (and around) a moving groundtarget. On the basis of our previous results, versatile for-mation patterns are achieved with the help of three newfeatures. The ﬁrst feature is a new formation pattern,the uniform spacing formation where either the relativeseparation distances or the separation angles can be regu-lated to a desired value. Diﬀerent from a balanced circularformation where agents spread evenly over a full circle,agents can now spread evenly over a portion of a circle.Two kinds of uniform spacing formation control laws areproposed, where one regulates the separation distances be-tween two agents and the other regulates the separationangles in between. The second feature allows UAVs tocircle on diﬀerent orbits. To achieve formation under thiscircumstance, formation controllers will resort to virtualagents representing the actual agents in need. The thirdfeature is the usage of a (two-layer) hierarchical formationstructure, which allows selection of formation patterns fordiﬀerent layers. Combinations of these new features withour existing results yield more versatile concentric forma-tion patterns with diﬀerent local geometric shapes, such asstraight lines and triangles. Inter-vehicle collision avoid-ance is also addressed. Agents will be repelled to steeraway from each other once they get too close.12All UAVs are assumed to have constant linear veloci-ties. Control of each UAV is via its yaw rate. The designidea is to add three control components (three headingcontrollers) together to achieve the overall objective. Eachcontrol component has a goal. The proposed extensions tospreading agents on a portion of a circle, circling agentson orbits of diﬀerent radii, formation in local geometricshapes, and avoiding inter-vehicle collisions, provide morecomplete solution to cooperative target tracking in theconcentric manner.This paper also raises several questions for future in-vestigations. The implementation of the proposed schemeson physical robots and the extension of the developed tech-niques to 3D scenarios and cooperative tracking of multipletargets with obstacle avoidance capability [53], [54] will beof particular interest. Stability analyses in the presenceof formation pattern switching and broken communicationlinks are another research direction to look into. Also, in-vestigations of the time delay factor for obtaining stabilityconditions as well as desirable performance with reason-able computation complexity [55]–[57] are needed. Finally,Artiﬁcial Intelligence (AI) techniques have recently beendeveloped for robotic communication to enhance the com-munication capability of robotic networks for coordinatedactions. Application of the AI and/or Neural Networksto the ﬁeld of robotic networks in the context of coopera-tive target tracking is a promising research area to pursue[58]–[60].References[1] K. Szwaykowska, I.B. Schwartz, L.M.-T. Romero, C.R. Heck-man, D. Mox, and M.A. Hsieh, Collective motion patterns ofswarms with delay coupling: Theory and experiment, PhysicalRevie E, 93(3), 2016, 11.[2] R. Sepulchre, D. Paley, and N. Leonard, Stabilization ofplanar collection motion: All-to-all communication, IEEETransactions on Automatic Control, 52(5), 2007, 811–824.[3] R. Sepulchre, D. Paley, and N. Leonard, Stabilization ofplanar collective motion with limited communication, IEEETransactions on Automatic Control, 53(3), 2008, 706–719.[4] A. Jain, D. Ghose, and P. Menon, Stabilization of balancedcircular motion about a desired center, International Confer-ence on Advances in Control and Optimization of DynamicalSystems, Kanpur, India, 2014.[5] A. Jain and D. Ghose, Stabilization of collective motion insynchronized, balanced and splay phase arrangements on adesired circle, American Control Conference, Chicago, IL.[6] G. Mallik and A. Sinha, A study of balanced circular formationunder deviated cyclic pursuit strategy, IFAC-PapersOnLine,48(5), 2015, 41–46.[7] R. Zheng, Z. Lin, M. Fu, and D. Sun, Distributed control foruniform circumnavigation of ring-coupled unicycles, Automat-ica, 53, 2015, 23–29.[8] N. Kokolakis and N. Koussoulas, Coordinated standoﬀ track-ing of a ground moving target and the phase separation prob-lem, International Conference on Unmanned Aircraft Systems,Dallas, TX, 2018.[9] J. Guo, G. Yan, and Z. Lin, Cooperative control syn-thesis for moving-target-enclosing with changing topologies,International Conference on Robotics and Automation, An-chorage, AK, 2010.[10] Z. Zhou, H. Wang, and Z. Hu, Event-based time varyingformation control for multiple quadrotor UAVs with Markovianswitching topologies, Complexity, 2018, 1–15.[11] Y. Sun and L. Wang, Consensus of multi-agent systems indirected networks with nonuniform time-varying delays, IEEETransactions on Automatic Control, 54(7), 2009, 1607–1613.[12] J. Marshall, M. Broucke, and B. Francis, Formations of vehiclesin cyclic pursuit, IEEE Transactions on Automatic Control,49(11), 2004, 1963–1974.[13] J. Marshall, M. Broucke, and B. Francis, Pursuit formationsof unicycles, Automatica, 42, 2006, 3–12.[14] M. Pavone and E. Frazzoli, Decentralized policies for geomet-ric pattern formation and path coverage, ASME Journal ofDynamic Systems, Measurement, and Control, 129(5), 2007,633–643.[15] B. Wu, D. Wang, and E. Poh, Cyclic formation control for satel-lite formation using local relative measurements, MechatronicSystems and Control, 40(1), 2012, 11–21.[16] J. Juang, On the formation patterns under generalized cyclicpursuit, IEEE Transactions on Automatic Control, 58(9), 2013,2401–2405, 2013.[17] J. Ramirez, M. Pavone, E. Frazzoli, and D. Miller, Distributedcontrol of spacecraft formations via cyclic pursuit: Theory andexperiments, Journal of Guidance, Control, and Dynamics,33(5), 2010, 1655–1669.[18] K. Hausmany, J. Muller, A. Hariharan, N. Ayanian, and G.Sukhatme, Cooperative multi-robot control for target track-ing with onboard sensing, International Journal of RoboticsResearch, 34(13), 2015, 1660–1677.[19] P. Jimenez, B. Shirinzadeh, D. Oetomo, and A. Nicholson,Swarm aggregation and formation control for robots withlimited perception, International Journal of Robotics andAutomation, 26(4), 2011, 255–263.[20] P. Zhu and W. Ren, Multi-robot joint localization and targettracking with local sensing and communication, AmericanControl Conference, Charlotte, NC, 2019.[21] L. Luo, N. Chakraborty, and K. Sycara, Provably-good dis-tributed algorithm for constrained multi-robot task assignmentfor grouped tasks, IEEE Transactions on Robotics, 31(1), 2015,19–30.[22] D. Panagou, M. Turpin, and V. Kumar, Decentralized goalassignment and safe trajectory generation in multi-robot net-works via multiple Lyapunov functions, IEEE Transactions onAutomatic Control, 65(8), 2020, 3365–3380.[23] J. Ni, X. Yang, J. Chen, and S. Yang, Dynamic bioinspiredneural network for multi-robot formation control in unknownenvironments, International Journal of Robotics and Automa-tion, 30(3), 2015, 256–266.[24] M. Khan and C. Silva, Autonomous and robust multi-robotcooperation using an artiﬁcial immune system, InternationalJournal of Robotics and Automation, 27(1), 2012, 60–75.[25] X. Yu, L. Liu, and G. Feng, Coordinated control of multipleunicycles for escorting and patrolling task based on a cyclicpursuit strategy, American Control Conference, Boston, MA,2016.[26] M. Zhang and H. Liu, Cooperative tracking a moving targetusing multiple ﬁxed-wing UAVs, Journal of Intelligent andRobotic Systems, 81(3-4), 2016, 505–529.[27] X. Yu and L. Liu, Cooperative control for moving-target circularformation of nonholonomic vehicles, IEEE Transactions onAutomatic Control, 62(7), 2017, 3448–3454.[28] L. Brinon-Arranz, A. Seuret, and A. Pascoal, Target trackingvia a circular formation of unicycles, IFAC World Congress,Toulouse, France, 2017.[29] A. Miao, Y. Wang, and R. Fierro, Cooperative circumnavi-gation of a moving target with multiple nonholonomic robotsusing backstepping design, Systems and Control Letters, 103,2017, 58–65.[30] L. Ma and N. Hovakimyan, Vision-based cyclic pursuit forcooperative target tracking, Journal of Guidance, Control, andDynamics, 36(2), 2013, 617–622.[31] N. Moshtagh, N. Michael, A. Jadbabaie, and K. Daniilidis,Vision-based, distributed control laws for motion coordinationof nonholonomic robots, IEEE Transactions on Robotics, 25(4),2009, 851–860.[32] N. Ceccarelli, M. Marco, A. Garulli, and A. Giannitrapani,Collective circular motion of multi-vehicle systems, Automatica,44, 2008, 3025–3035.[33] J. Soares, A. Aguiar, A. Pascoal, and M. Gallieri, Triangularformation control using range measurements: An applicationto marine robotic vehicles, IFAC Proceedings, 45(5), 2012,112–117.13[34] Z. He and J. Xu, Moving target tracking by UAVs in an urbanarea, Mechatronic Systems and Control, 42(2), 2014. DOI:10.2316/Journal.201.2014.2.201-2572.[35] L. Ma and N. Hovakimyan, Cooperative target tracking inbalanced circular formation: Multiple UAVs tracking a groundvehicle, American Control Conference, Washington, DC, USA,2013, 5386–5391.[36] L. Ma, Cooperative target tracking with time-varying formationradius, European Control Conference, Linz, Austria, 2015.[37] L. Ma, Cooperative target tracking in balanced circular forma-tion with time-varying radius, International Journal of Roboticsand Automation, 35(4), 2020. DOI: 10.2316/J.2020.206-0086.[38] L. Ma, C. Cao, N. Hovakimyan, V. Dobrokhodov, and I.Kaminer, Adaptive vision-based guidance law with guaran-teed performance bounds, Journal of Guidance, Control, andDynamics, 3, 2010, 33.[39] V. Cichella, I. Kaminer, V. Dobrokhodov, and N. Hovakimyan,Coordinated vision-based tracking for multiple UAVs, Ameri-can Control Conference, Hamburg, Germany, 2015.[40] Q. Han, S. Sun, and H. Lang, Leader-follower formationcontrol of multi-robots based on bearing-only observations,International Journal of Robotics and Automation, 34(2), 2019.DOI: 10.2316/J.2019.206-4831.[41] S. Smith, M. Broucke, and B. Francis, A hierarchical cyclicpursuit scheme for vehicle networks, Automatica, 41, 2005,1045–1053.[42] L. Consolini, F. Morbidi, D. Prattichizzo, and M. Tosques,Steering hierarchical formations of unicycle robots, IEEE Con-ference on Decision and Control, New Orleans, LA, 2007.[43] D. Mukherjee and D. Ghose, Generalized hierarchical cyclicpursuit, Automatica, 71, 2016, 318–323.[44] W. Ding, G. Yan, and Z. Lin, Formations on two-layer pur-suit systems, IEEE International Conference on Robotics andAutomation, Japan, 2009.[45] A. Satici, H. Poonawala, H. Eckert, and M. Spong, Connec-tivity preserving formation control with collision avoidancefor nonholonomic wheeled mobile robots, IEEE InternationalConference on Intelligent Robots and Systems, Tokyo, Japan,2013.
[46] J. Santiaguillo-Salinas and E. Arando-bricaire, Containmentproblem with time-varying formation and collision avoidancefor multiagent systems, International Journal of AdvancedRobotic Systems, 13, 2017, 1–13.
[47] J. Flores-Resendiz, E. Aranda-Bricaire, J. Gonz´alez-Sierra, andJ. Santiaguillo-Salinas, Finite-time formation control withoutcollisions for multiagent systems with communication graphscomposed of cyclic paths, Mathematical Problems in Engineer-ing, 1, 2015, 1–17.
[49].Let dmin denote the minimal distance allowed amongagents. The following collision avoidance control compo-nent helps to repel agents to steer away from each otheronce they get too close
[50]:uia(t) = −Krj∈N (ri)dmindijsin βij (18)where Kr is the controller gain. The repulsion term (18)adjusts each agent’s heading to the opposite directionof its neighbours in N(ri). Notice that N (ri) and Nidenote diﬀerent sets. The set Ni is the set of agentswhose information can be obtained by the ith agent viacommunication, whereas the set N (ri) denotes the set ofagents that are too close to agent i.With (18) in place, each agent’s control inputbecomes [50]:˙ψi(t) = ui(t) = uit(t) + uic(t) + uia(t) (19)Totally three control components are added together toachieve the objective of simultaneous tracking, formation,and inter-vehicle collision avoidance.6. Simulation ResultsThe proposed control laws were simulated in Matlab toverify their performance of tracking, formation, and inter-vehicle collision avoidance.6.1 Achieving Uniform Spacing Formations onDiﬀerent OrbitsThis example demonstrates achievement of uniform spac-ing formations when agents circle on diﬀerent orbits. Toachieve formations under this circumstance, formation con-trollers (17) and (16) are used, which resort to virtualagents when needed, i.e., when the two agents are not onthe same orbit. It is worth mentioning that the formationcontroller (16) aims at regulating the relative separationdistances to a constant, while the controller (17) regulatesthe relative separation angles to a desired value. The2D ranges are set to be ρd,i = {15, 15, 18, 18} (m), fori = 1, 2, 3, 4, respectively. Correspondingly, UAVs’ linearvelocities are Vg,i = {30, 30, 36, 36} (m/s), satisfying therelationship in (10). To focus on the formation patterns,the target’s motion is assumed linear (the target moves ona straight line).We ﬁrst present simulation results applying the for-mation controller (17). The 2D trajectories of the agentsand the target are plotted in Fig. 6(a). Details of thetracking are given in Fig. 6(b), where ρi(t) approach theirprescribed values (that are diﬀerent). All agents’ posi-tions at the end of the simulation are shown in Fig. 6(c),9Figure 7. Cooperative target tracking using formation controller (16): (a) 2D trajectories; (b) 2D range; (c) formation in theend; (d) separation distance; (e) separation angle; and (f) formation control uic(t). Target undergoes a linear motion. Agentscircle on two diﬀerent orbits.where all agents scatter around the target on two diﬀer-ent orbits. Both the separation distances (Fig. 6(d))and the separation angles (Fig. 6(e)) are plotted. Whencontroller (17) is applied, we expect the separation an-gles to approach a constant. However, the separationdistances may or may not approach to the same valuebecause the distances between two agents also depend onthe orbits that they lie on. For ds = 10 (m), the desiredseparation angle is either φs = 2 sin−1(ds/30) ≈ 39◦orφs = 2π − 2 sin−1(ds/30) ≈ 321◦. Because all separationangles converge to 39◦(Fig. 6(e)), formation is obtainedsuccessfully. Figure 6(f) shows the formation componentuic(t), which converges to zero upon formation.Results when applying the controller (16) are shownin Fig. 7. Figure 7(a) shows the general picture ofcooperative target tracking. Figure 7(b) plots the 2Dranges between the agents and the target, demonstratingsuccessful tracking with two diﬀerent range values. Figure7(c) shows the formation achieved in the end, where agentscircle on two diﬀerent orbits around the target. Figure7(d) shows that the separation distances between twosuccessive agents are now regulated to the speciﬁed valueof ds = 10 (m) (same for all agents). On the contrary,the separation angles do not approach to one value. FromFig. 7(c), it can be seen that agents {1, 4} and agents{2, 3} circle on two diﬀerent orbits. The separation anglesbetween these two pairs of consecutive agents should besmaller than that between agents {1, 2}, which circle onthe same orbit. This is conﬁrmed in Fig. 7(e). Figure 7(f)shows that the formation control component uic(t) alsovanishes to zero upon formation.Figures 6 and 7 together show the diﬀerence betweenthe two formation controllers (16) and (17). The formationcontroller (16) regulates the relative separation distancesbetween agents to a speciﬁed value ds, while the controller(17) regulates the separation angles to φs.6.2 Achieving Concentric Formations with LocalGeometric ShapesThis section presents cooperative target tracking in con-centric formations with local geometric shapes of straightlines and triangles, as those shown in Fig. 5. To focus onformation patterns, the target’s motion is assumed linear,i.e., the target moves on a straight line. Corresponding tothe four patterns in Fig. 5, simulation results are presentedin Fig. 8, with several snapshots showing how formationis achieved over time. Cooperative target tracking in theseconcentric formations are successfully obtained.6.3 Achieving Inter-vehicle Collision AvoidanceThis example demonstrates cooperative target trackingwith inter-vehicle collision avoidance. The two scenarios ofwithout and with collision avoidance, i.e., before and afterapplying uia(t), are shown and compared in Fig. 9. Weselect the target’s velocity to be piecewise-constant, n = 5,ds = 10 (m), and dmin = 7 (m). The ﬁrst row of Fig. 9 is10Figure 8. Cooperative target tracking in local geometric shapes: (a) balanced-line; (b) uniform-line; (c) balanced-triangle;and (d) uniform-triangle. The overall formation is either balanced circular or uniform spacing. The local geometric shape iseither straight line or triangle.for the scenario without the collision avoidance capability,i.e., before the control component uia(t) is applied. Afteruia(t) is applied, results are shown in the second rowof Fig. 9. In each scenario, the 2D trajectories areplotted to show the general picture (Fig. 9(a) versus (c)).Then, the minimal distance among all agents is plotted,demonstrating the eﬀect of the added control componentuia(t) (Fig. 9(b) versus (d)). For the second scenario withcollision avoidance, the component uia(t) for each agent isshown in Fig. 9(e), where a saturation of |uia(t)| ≤ 3 hasbeen used for all agents. Comparison between Fig. 9(b)and (d) shows that the control component uia(t) helps tokeep the minimal distance to be greater than the allowedvalue. Otherwise, the minimal distance can be muchsmaller, as indicated in Fig. 9(b). The zigzag area in Fig.9(d), corresponds to the circumstances when uia(t) takeseﬀect. Figure 9(e) shows that uia(t) only takes eﬀect whenneeded, i.e., when agents {3, 5} get too close to each other.7. Comparison with Prior StudiesComparing with the distance-based coordination controllaw (6), our proposed bearing-angle-based coordinationcontrol law (7) has one potential advantage. Consider a11Figure 9. With inter-vehicle collision avoidance (before and after): (a) before 2D trajectories; (b) before minimum distance;(c) after 2D trajectories; (b) after minimum distance; and (e) after uic(t) (rad/s).scenario when information exchange among/between someagents becomes unavailable (e.g., due to communicationloss or out of communication range). Instead of computingthe relative bearing angles from the exchanged positionsas in (12), the needed information of relative bearing an-gles can be estimated/obtained from a local vision systeminstalled on each UAV. In other words, when the expectedinformation from others is unavailable (either permanentlyor temporarily), the objective of achieving and maintainingformation could still be achieved by using local measure-ments and estimates.Regarding formation control, most existing results areeither leaderless or leader-following
[51],
[52]. The proposedmethod of obtaining versatile formation patterns allows acombination of both. This can be seen in the “Balanced-Line” and “Balanced-Triangle” patterns (Fig. 5). Theoverall balanced circular formation can be obtained us-ing a leaderless communication topology, whereas achieve-ment of local geometric shapes can be implemented in aleader-based manner. The adopted hierarchical formationstructure allows selection of appropriate communicationtopologies on diﬀerent layers.This paper also considers a practical issue that wouldoccur, i.e., collisions among agents. This issue was tackledby adding another control component into each UAV’s con-trol input. As can be seen from (18), this added collision-avoidance control component can also be expressed as afunction of bearing angles. As mentioned earlier, a bearing-angle-based control law has the potential of still achiev-ing its control objective (formation or collision avoidance)during communication loss, by using local measurementsfrom each UAV’s onboard sensors.8. ConclusionsThis paper is to obtain more versatile concentric forma-tions in cooperative target tracking where a ﬂeet of UAVs iscommanded to circle above (and around) a moving groundtarget. On the basis of our previous results, versatile for-mation patterns are achieved with the help of three newfeatures. The ﬁrst feature is a new formation pattern,the uniform spacing formation where either the relativeseparation distances or the separation angles can be regu-lated to a desired value. Diﬀerent from a balanced circularformation where agents spread evenly over a full circle,agents can now spread evenly over a portion of a circle.Two kinds of uniform spacing formation control laws areproposed, where one regulates the separation distances be-tween two agents and the other regulates the separationangles in between. The second feature allows UAVs tocircle on diﬀerent orbits. To achieve formation under thiscircumstance, formation controllers will resort to virtualagents representing the actual agents in need. The thirdfeature is the usage of a (two-layer) hierarchical formationstructure, which allows selection of formation patterns fordiﬀerent layers. Combinations of these new features withour existing results yield more versatile concentric forma-tion patterns with diﬀerent local geometric shapes, such asstraight lines and triangles. Inter-vehicle collision avoid-ance is also addressed. Agents will be repelled to steeraway from each other once they get too close.12All UAVs are assumed to have constant linear veloci-ties. Control of each UAV is via its yaw rate. The designidea is to add three control components (three headingcontrollers) together to achieve the overall objective. Eachcontrol component has a goal. The proposed extensions tospreading agents on a portion of a circle, circling agentson orbits of diﬀerent radii, formation in local geometricshapes, and avoiding inter-vehicle collisions, provide morecomplete solution to cooperative target tracking in theconcentric manner.This paper also raises several questions for future in-vestigations. The implementation of the proposed schemeson physical robots and the extension of the developed tech-niques to 3D scenarios and cooperative tracking of multipletargets with obstacle avoidance capability
[53],
[54] will beof particular interest. Stability analyses in the presenceof formation pattern switching and broken communicationlinks are another research direction to look into. Also, in-vestigations of the time delay factor for obtaining stabilityconditions as well as desirable performance with reason-able computation complexity
[55]–[57] are needed. Finally,Artiﬁcial Intelligence (AI) techniques have recently beendeveloped for robotic communication to enhance the com-munication capability of robotic networks for coordinatedactions. Application of the AI and/or Neural Networksto the ﬁeld of robotic networks in the context of coopera-tive target tracking is a promising research area to pursue[58]–[60].References[1] K. Szwaykowska, I.B. Schwartz, L.M.-T. Romero, C.R. Heck-man, D. Mox, and M.A. Hsieh, Collective motion patterns ofswarms with delay coupling: Theory and experiment, PhysicalRevie E, 93(3), 2016, 11.[2] R. Sepulchre, D. Paley, and N. Leonard, Stabilization ofplanar collection motion: All-to-all communication, IEEETransactions on Automatic Control, 52(5), 2007, 811–824.[3] R. Sepulchre, D. Paley, and N. Leonard, Stabilization ofplanar collective motion with limited communication, IEEETransactions on Automatic Control, 53(3), 2008, 706–719.[4] A. Jain, D. Ghose, and P. 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Huang, Improvedinequality-based functions approach for stability analysis oftime delay system, Automatica, 108, 2019, 108416.
[57] are needed. Finally,Artiﬁcial Intelligence (AI) techniques have recently beendeveloped for robotic communication to enhance the com-munication capability of robotic networks for coordinatedactions. Application of the AI and/or Neural Networksto the ﬁeld of robotic networks in the context of coopera-tive target tracking is a promising research area to pursue
[58]–[60].References[1] K. Szwaykowska, I.B. Schwartz, L.M.-T. Romero, C.R. Heck-man, D. Mox, and M.A. Hsieh, Collective motion patterns ofswarms with delay coupling: Theory and experiment, PhysicalRevie E, 93(3), 2016, 11.[2] R. Sepulchre, D. Paley, and N. Leonard, Stabilization ofplanar collection motion: All-to-all communication, IEEETransactions on Automatic Control, 52(5), 2007, 811–824.[3] R. Sepulchre, D. Paley, and N. Leonard, Stabilization ofplanar collective motion with limited communication, IEEETransactions on Automatic Control, 53(3), 2008, 706–719.[4] A. Jain, D. Ghose, and P. Menon, Stabilization of balancedcircular motion about a desired center, International Confer-ence on Advances in Control and Optimization of DynamicalSystems, Kanpur, India, 2014.[5] A. Jain and D. Ghose, Stabilization of collective motion insynchronized, balanced and splay phase arrangements on adesired circle, American Control Conference, Chicago, IL.[6] G. Mallik and A. Sinha, A study of balanced circular formationunder deviated cyclic pursuit strategy, IFAC-PapersOnLine,48(5), 2015, 41–46.[7] R. Zheng, Z. Lin, M. Fu, and D. Sun, Distributed control foruniform circumnavigation of ring-coupled unicycles, Automat-ica, 53, 2015, 23–29.[8] N. Kokolakis and N. Koussoulas, Coordinated standoﬀ track-ing of a ground moving target and the phase separation prob-lem, International Conference on Unmanned Aircraft Systems,Dallas, TX, 2018.[9] J. Guo, G. Yan, and Z. Lin, Cooperative control syn-thesis for moving-target-enclosing with changing topologies,International Conference on Robotics and Automation, An-chorage, AK, 2010.[10] Z. Zhou, H. Wang, and Z. Hu, Event-based time varyingformation control for multiple quadrotor UAVs with Markovianswitching topologies, Complexity, 2018, 1–15.[11] Y. Sun and L. Wang, Consensus of multi-agent systems indirected networks with nonuniform time-varying delays, IEEETransactions on Automatic Control, 54(7), 2009, 1607–1613.[12] J. Marshall, M. Broucke, and B. Francis, Formations of vehiclesin cyclic pursuit, IEEE Transactions on Automatic Control,49(11), 2004, 1963–1974.[13] J. Marshall, M. Broucke, and B. Francis, Pursuit formationsof unicycles, Automatica, 42, 2006, 3–12.[14] M. Pavone and E. Frazzoli, Decentralized policies for geomet-ric pattern formation and path coverage, ASME Journal ofDynamic Systems, Measurement, and Control, 129(5), 2007,633–643.[15] B. Wu, D. Wang, and E. Poh, Cyclic formation control for satel-lite formation using local relative measurements, MechatronicSystems and Control, 40(1), 2012, 11–21.[16] J. Juang, On the formation patterns under generalized cyclicpursuit, IEEE Transactions on Automatic Control, 58(9), 2013,2401–2405, 2013.[17] J. Ramirez, M. Pavone, E. Frazzoli, and D. Miller, Distributedcontrol of spacecraft formations via cyclic pursuit: Theory andexperiments, Journal of Guidance, Control, and Dynamics,33(5), 2010, 1655–1669.[18] K. Hausmany, J. Muller, A. Hariharan, N. Ayanian, and G.Sukhatme, Cooperative multi-robot control for target track-ing with onboard sensing, International Journal of RoboticsResearch, 34(13), 2015, 1660–1677.[19] P. Jimenez, B. Shirinzadeh, D. Oetomo, and A. Nicholson,Swarm aggregation and formation control for robots withlimited perception, International Journal of Robotics andAutomation, 26(4), 2011, 255–263.[20] P. Zhu and W. Ren, Multi-robot joint localization and targettracking with local sensing and communication, AmericanControl Conference, Charlotte, NC, 2019.[21] L. Luo, N. Chakraborty, and K. Sycara, Provably-good dis-tributed algorithm for constrained multi-robot task assignmentfor grouped tasks, IEEE Transactions on Robotics, 31(1), 2015,19–30.[22] D. Panagou, M. Turpin, and V. Kumar, Decentralized goalassignment and safe trajectory generation in multi-robot net-works via multiple Lyapunov functions, IEEE Transactions onAutomatic Control, 65(8), 2020, 3365–3380.[23] J. Ni, X. Yang, J. Chen, and S. Yang, Dynamic bioinspiredneural network for multi-robot formation control in unknownenvironments, International Journal of Robotics and Automa-tion, 30(3), 2015, 256–266.[24] M. Khan and C. Silva, Autonomous and robust multi-robotcooperation using an artiﬁcial immune system, InternationalJournal of Robotics and Automation, 27(1), 2012, 60–75.[25] X. Yu, L. Liu, and G. Feng, Coordinated control of multipleunicycles for escorting and patrolling task based on a cyclicpursuit strategy, American Control Conference, Boston, MA,2016.[26] M. Zhang and H. Liu, Cooperative tracking a moving targetusing multiple ﬁxed-wing UAVs, Journal of Intelligent andRobotic Systems, 81(3-4), 2016, 505–529.[27] X. Yu and L. Liu, Cooperative control for moving-target circularformation of nonholonomic vehicles, IEEE Transactions onAutomatic Control, 62(7), 2017, 3448–3454.[28] L. Brinon-Arranz, A. Seuret, and A. Pascoal, Target trackingvia a circular formation of unicycles, IFAC World Congress,Toulouse, France, 2017.[29] A. Miao, Y. Wang, and R. Fierro, Cooperative circumnavi-gation of a moving target with multiple nonholonomic robotsusing backstepping design, Systems and Control Letters, 103,2017, 58–65.[30] L. Ma and N. Hovakimyan, Vision-based cyclic pursuit forcooperative target tracking, Journal of Guidance, Control, andDynamics, 36(2), 2013, 617–622.[31] N. Moshtagh, N. Michael, A. Jadbabaie, and K. Daniilidis,Vision-based, distributed control laws for motion coordinationof nonholonomic robots, IEEE Transactions on Robotics, 25(4),2009, 851–860.[32] N. Ceccarelli, M. Marco, A. Garulli, and A. Giannitrapani,Collective circular motion of multi-vehicle systems, Automatica,44, 2008, 3025–3035.[33] J. Soares, A. Aguiar, A. Pascoal, and M. Gallieri, Triangularformation control using range measurements: An applicationto marine robotic vehicles, IFAC Proceedings, 45(5), 2012,112–117.13[34] Z. He and J. Xu, Moving target tracking by UAVs in an urbanarea, Mechatronic Systems and Control, 42(2), 2014. DOI:10.2316/Journal.201.2014.2.201-2572.[35] L. Ma and N. Hovakimyan, Cooperative target tracking inbalanced circular formation: Multiple UAVs tracking a groundvehicle, American Control Conference, Washington, DC, USA,2013, 5386–5391.[36] L. Ma, Cooperative target tracking with time-varying formationradius, European Control Conference, Linz, Austria, 2015.[37] L. Ma, Cooperative target tracking in balanced circular forma-tion with time-varying radius, International Journal of Roboticsand Automation, 35(4), 2020. DOI: 10.2316/J.2020.206-0086.[38] L. Ma, C. Cao, N. Hovakimyan, V. Dobrokhodov, and I.Kaminer, Adaptive vision-based guidance law with guaran-teed performance bounds, Journal of Guidance, Control, andDynamics, 3, 2010, 33.[39] V. Cichella, I. Kaminer, V. Dobrokhodov, and N. Hovakimyan,Coordinated vision-based tracking for multiple UAVs, Ameri-can Control Conference, Hamburg, Germany, 2015.[40] Q. Han, S. Sun, and H. Lang, Leader-follower formationcontrol of multi-robots based on bearing-only observations,International Journal of Robotics and Automation, 34(2), 2019.DOI: 10.2316/J.2019.206-4831.[41] S. Smith, M. Broucke, and B. Francis, A hierarchical cyclicpursuit scheme for vehicle networks, Automatica, 41, 2005,1045–1053.[42] L. Consolini, F. Morbidi, D. Prattichizzo, and M. Tosques,Steering hierarchical formations of unicycle robots, IEEE Con-ference on Decision and Control, New Orleans, LA, 2007.[43] D. Mukherjee and D. Ghose, Generalized hierarchical cyclicpursuit, Automatica, 71, 2016, 318–323.[44] W. Ding, G. Yan, and Z. Lin, Formations on two-layer pur-suit systems, IEEE International Conference on Robotics andAutomation, Japan, 2009.[45] A. Satici, H. Poonawala, H. Eckert, and M. Spong, Connec-tivity preserving formation control with collision avoidancefor nonholonomic wheeled mobile robots, IEEE InternationalConference on Intelligent Robots and Systems, Tokyo, Japan,2013.[46] J. Santiaguillo-Salinas and E. Arando-bricaire, Containmentproblem with time-varying formation and collision avoidancefor multiagent systems, International Journal of AdvancedRobotic Systems, 13, 2017, 1–13.[47] J. Flores-Resendiz, E. Aranda-Bricaire, J. Gonz´alez-Sierra, andJ. Santiaguillo-Salinas, Finite-time formation control withoutcollisions for multiagent systems with communication graphscomposed of cyclic paths, Mathematical Problems in Engineer-ing, 1, 2015, 1–17.[48] V. Freitas and E. Macau, Control strategy for symmetriccircular formations of mobile agents with collision avoidance,FHYSCON, Florence, Italy, 2017.[49] A. Burohman, E. Joelianto, and A. Widyotriatmo, Analysis ofpotential ﬁelds for formation of multi-agent robots with colli-sion avoidance, International Conference on Instrumentation,Control, and Automation, Yogyakarta, Indonesia, 2017.[50] L. Ma, Cooperative target tracking using a ﬂeet of UAVs withcollision and obstacle avoidance, International Conference onSystem Theory, Control and Computing, Sinaia, Romania,2018.[51] X. Ma, W. Dong, and B. Li, Comprehensive fault-tolerantcontrol of leader-follower unmanned aerial vehicles (UAVs)formation, International Journal of Robotics and Automation,34(6), 2019. DOI: 10.2316/J.2019.206-0301.[52] D. Atta and B. Subudhi, Decentralized formation controlof multiple autonomous underwater vehicles, InternationalJournal of Robotics and Automation, 28(4), 2013. DOI:10.2316/Journal.206.2013.4.206-3613.[53] L. Deng, X. Ma, J. Gu, Y. Li, Z. Xu, and Y. Wang, Artiﬁcial im-mune network-based multi-robot formation path planning withobstacle avoidance, International Journal of Robotics and Au-tomation, 31(3), 2016. DOI: 10.2316/Journal.206.2016.3.206-4746.[54] A. Abbaspour, A. Moosavian, and K. Alipour, Formationcontrol and obstacle avoidance of cooperative wheeled mobilerobots, International Journal of Robotics and Automation,30(5), 2015. DOI: 10.2316/Journal.206.2015.5.206-4109.[55] Z. Li, H. Yan, H. Zhang, X. Zhan, and C. Huang, Improvedinequality-based functions approach for stability analysis oftime delay system, Automatica, 108, 2019, 108416.[56] Z. Li, H. Yan, H. Zhang, Y. Peng, J. Park, and Y. He,Stability analysis of linear systems with time-varying delayvia intermediate polynomial-based functions, Automatica, 113,2019, 108756.[57] Z. Li, C. Huang, and H. Yan, Stability analysis for systems withtime delays via new integral inequalities, IEEE Transactionson Systems, Man, and Cybernetics: Systems, 48(12), 2018,2495–2501.[58] Z. Li, Y. Bai, C. Huang, H. Yan, and S. Mu, Improved stabilityanalysis for delayed neural networks, IEEE Transactions onNeural Networks and Learning Systems, 29(9), 2017, 4535–4541.
[60].References[1] K. Szwaykowska, I.B. Schwartz, L.M.-T. Romero, C.R. Heck-man, D. Mox, and M.A. Hsieh, Collective motion patterns ofswarms with delay coupling: Theory and experiment, PhysicalRevie E, 93(3), 2016, 11.[2] R. Sepulchre, D. Paley, and N. Leonard, Stabilization ofplanar collection motion: All-to-all communication, IEEETransactions on Automatic Control, 52(5), 2007, 811–824.[3] R. Sepulchre, D. Paley, and N. Leonard, Stabilization ofplanar collective motion with limited communication, IEEETransactions on Automatic Control, 53(3), 2008, 706–719.[4] A. Jain, D. Ghose, and P. Menon, Stabilization of balancedcircular motion about a desired center, International Confer-ence on Advances in Control and Optimization of DynamicalSystems, Kanpur, India, 2014.[5] A. Jain and D. Ghose, Stabilization of collective motion insynchronized, balanced and splay phase arrangements on adesired circle, American Control Conference, Chicago, IL.[6] G. Mallik and A. Sinha, A study of balanced circular formationunder deviated cyclic pursuit strategy, IFAC-PapersOnLine,48(5), 2015, 41–46.[7] R. Zheng, Z. Lin, M. Fu, and D. Sun, Distributed control foruniform circumnavigation of ring-coupled unicycles, Automat-ica, 53, 2015, 23–29.[8] N. Kokolakis and N. Koussoulas, Coordinated standoﬀ track-ing of a ground moving target and the phase separation prob-lem, International Conference on Unmanned Aircraft Systems,Dallas, TX, 2018.[9] J. Guo, G. Yan, and Z. Lin, Cooperative control syn-thesis for moving-target-enclosing with changing topologies,International Conference on Robotics and Automation, An-chorage, AK, 2010.[10] Z. Zhou, H. Wang, and Z. Hu, Event-based time varyingformation control for multiple quadrotor UAVs with Markovianswitching topologies, Complexity, 2018, 1–15.[11] Y. Sun and L. Wang, Consensus of multi-agent systems indirected networks with nonuniform time-varying delays, IEEETransactions on Automatic Control, 54(7), 2009, 1607–1613.[12] J. Marshall, M. Broucke, and B. Francis, Formations of vehiclesin cyclic pursuit, IEEE Transactions on Automatic Control,49(11), 2004, 1963–1974.[13] J. Marshall, M. Broucke, and B. Francis, Pursuit formationsof unicycles, Automatica, 42, 2006, 3–12.[14] M. Pavone and E. Frazzoli, Decentralized policies for geomet-ric pattern formation and path coverage, ASME Journal ofDynamic Systems, Measurement, and Control, 129(5), 2007,633–643.[15] B. Wu, D. Wang, and E. Poh, Cyclic formation control for satel-lite formation using local relative measurements, MechatronicSystems and Control, 40(1), 2012, 11–21.[16] J. Juang, On the formation patterns under generalized cyclicpursuit, IEEE Transactions on Automatic Control, 58(9), 2013,2401–2405, 2013.[17] J. Ramirez, M. Pavone, E. Frazzoli, and D. Miller, Distributedcontrol of spacecraft formations via cyclic pursuit: Theory andexperiments, Journal of Guidance, Control, and Dynamics,33(5), 2010, 1655–1669.[18] K. Hausmany, J. Muller, A. Hariharan, N. Ayanian, and G.Sukhatme, Cooperative multi-robot control for target track-ing with onboard sensing, International Journal of RoboticsResearch, 34(13), 2015, 1660–1677.[19] P. Jimenez, B. Shirinzadeh, D. Oetomo, and A. Nicholson,Swarm aggregation and formation control for robots withlimited perception, International Journal of Robotics andAutomation, 26(4), 2011, 255–263.[20] P. Zhu and W. Ren, Multi-robot joint localization and targettracking with local sensing and communication, AmericanControl Conference, Charlotte, NC, 2019.[21] L. Luo, N. Chakraborty, and K. Sycara, Provably-good dis-tributed algorithm for constrained multi-robot task assignmentfor grouped tasks, IEEE Transactions on Robotics, 31(1), 2015,19–30.[22] D. Panagou, M. Turpin, and V. Kumar, Decentralized goalassignment and safe trajectory generation in multi-robot net-works via multiple Lyapunov functions, IEEE Transactions onAutomatic Control, 65(8), 2020, 3365–3380.[23] J. Ni, X. Yang, J. Chen, and S. Yang, Dynamic bioinspiredneural network for multi-robot formation control in unknownenvironments, International Journal of Robotics and Automa-tion, 30(3), 2015, 256–266.[24] M. Khan and C. Silva, Autonomous and robust multi-robotcooperation using an artiﬁcial immune system, InternationalJournal of Robotics and Automation, 27(1), 2012, 60–75.[25] X. Yu, L. Liu, and G. Feng, Coordinated control of multipleunicycles for escorting and patrolling task based on a cyclicpursuit strategy, American Control Conference, Boston, MA,2016.[26] M. Zhang and H. Liu, Cooperative tracking a moving targetusing multiple ﬁxed-wing UAVs, Journal of Intelligent andRobotic Systems, 81(3-4), 2016, 505–529.[27] X. Yu and L. Liu, Cooperative control for moving-target circularformation of nonholonomic vehicles, IEEE Transactions onAutomatic Control, 62(7), 2017, 3448–3454.[28] L. Brinon-Arranz, A. Seuret, and A. Pascoal, Target trackingvia a circular formation of unicycles, IFAC World Congress,Toulouse, France, 2017.[29] A. Miao, Y. Wang, and R. Fierro, Cooperative circumnavi-gation of a moving target with multiple nonholonomic robotsusing backstepping design, Systems and Control Letters, 103,2017, 58–65.[30] L. Ma and N. Hovakimyan, Vision-based cyclic pursuit forcooperative target tracking, Journal of Guidance, Control, andDynamics, 36(2), 2013, 617–622.[31] N. Moshtagh, N. Michael, A. Jadbabaie, and K. Daniilidis,Vision-based, distributed control laws for motion coordinationof nonholonomic robots, IEEE Transactions on Robotics, 25(4),2009, 851–860.[32] N. Ceccarelli, M. Marco, A. Garulli, and A. Giannitrapani,Collective circular motion of multi-vehicle systems, Automatica,44, 2008, 3025–3035.[33] J. Soares, A. Aguiar, A. Pascoal, and M. Gallieri, Triangularformation control using range measurements: An applicationto marine robotic vehicles, IFAC Proceedings, 45(5), 2012,112–117.13[34] Z. He and J. Xu, Moving target tracking by UAVs in an urbanarea, Mechatronic Systems and Control, 42(2), 2014. DOI:10.2316/Journal.201.2014.2.201-2572.[35] L. Ma and N. Hovakimyan, Cooperative target tracking inbalanced circular formation: Multiple UAVs tracking a groundvehicle, American Control Conference, Washington, DC, USA,2013, 5386–5391.[36] L. Ma, Cooperative target tracking with time-varying formationradius, European Control Conference, Linz, Austria, 2015.[37] L. Ma, Cooperative target tracking in balanced circular forma-tion with time-varying radius, International Journal of Roboticsand Automation, 35(4), 2020. DOI: 10.2316/J.2020.206-0086.[38] L. Ma, C. Cao, N. Hovakimyan, V. Dobrokhodov, and I.Kaminer, Adaptive vision-based guidance law with guaran-teed performance bounds, Journal of Guidance, Control, andDynamics, 3, 2010, 33.[39] V. Cichella, I. Kaminer, V. Dobrokhodov, and N. Hovakimyan,Coordinated vision-based tracking for multiple UAVs, Ameri-can Control Conference, Hamburg, Germany, 2015.[40] Q. Han, S. Sun, and H. Lang, Leader-follower formationcontrol of multi-robots based on bearing-only observations,International Journal of Robotics and Automation, 34(2), 2019.DOI: 10.2316/J.2019.206-4831.[41] S. Smith, M. Broucke, and B. Francis, A hierarchical cyclicpursuit scheme for vehicle networks, Automatica, 41, 2005,1045–1053.[42] L. Consolini, F. Morbidi, D. Prattichizzo, and M. Tosques,Steering hierarchical formations of unicycle robots, IEEE Con-ference on Decision and Control, New Orleans, LA, 2007.[43] D. Mukherjee and D. Ghose, Generalized hierarchical cyclicpursuit, Automatica, 71, 2016, 318–323.[44] W. Ding, G. Yan, and Z. Lin, Formations on two-layer pur-suit systems, IEEE International Conference on Robotics andAutomation, Japan, 2009.[45] A. Satici, H. Poonawala, H. Eckert, and M. Spong, Connec-tivity preserving formation control with collision avoidancefor nonholonomic wheeled mobile robots, IEEE InternationalConference on Intelligent Robots and Systems, Tokyo, Japan,2013.[46] J. Santiaguillo-Salinas and E. Arando-bricaire, Containmentproblem with time-varying formation and collision avoidancefor multiagent systems, International Journal of AdvancedRobotic Systems, 13, 2017, 1–13.[47] J. Flores-Resendiz, E. Aranda-Bricaire, J. Gonz´alez-Sierra, andJ. Santiaguillo-Salinas, Finite-time formation control withoutcollisions for multiagent systems with communication graphscomposed of cyclic paths, Mathematical Problems in Engineer-ing, 1, 2015, 1–17.[48] V. Freitas and E. Macau, Control strategy for symmetriccircular formations of mobile agents with collision avoidance,FHYSCON, Florence, Italy, 2017.[49] A. Burohman, E. Joelianto, and A. Widyotriatmo, Analysis ofpotential ﬁelds for formation of multi-agent robots with colli-sion avoidance, International Conference on Instrumentation,Control, and Automation, Yogyakarta, Indonesia, 2017.[50] L. Ma, Cooperative target tracking using a ﬂeet of UAVs withcollision and obstacle avoidance, International Conference onSystem Theory, Control and Computing, Sinaia, Romania,2018.[51] X. Ma, W. Dong, and B. Li, Comprehensive fault-tolerantcontrol of leader-follower unmanned aerial vehicles (UAVs)formation, International Journal of Robotics and Automation,34(6), 2019. DOI: 10.2316/J.2019.206-0301.[52] D. Atta and B. Subudhi, Decentralized formation controlof multiple autonomous underwater vehicles, InternationalJournal of Robotics and Automation, 28(4), 2013. DOI:10.2316/Journal.206.2013.4.206-3613.[53] L. Deng, X. Ma, J. Gu, Y. Li, Z. Xu, and Y. Wang, Artiﬁcial im-mune network-based multi-robot formation path planning withobstacle avoidance, International Journal of Robotics and Au-tomation, 31(3), 2016. DOI: 10.2316/Journal.206.2016.3.206-4746.[54] A. Abbaspour, A. Moosavian, and K. Alipour, Formationcontrol and obstacle avoidance of cooperative wheeled mobilerobots, International Journal of Robotics and Automation,30(5), 2015. DOI: 10.2316/Journal.206.2015.5.206-4109.[55] Z. Li, H. Yan, H. Zhang, X. Zhan, and C. Huang, Improvedinequality-based functions approach for stability analysis oftime delay system, Automatica, 108, 2019, 108416.[56] Z. Li, H. Yan, H. Zhang, Y. Peng, J. Park, and Y. He,Stability analysis of linear systems with time-varying delayvia intermediate polynomial-based functions, Automatica, 113,2019, 108756.[57] Z. Li, C. Huang, and H. Yan, Stability analysis for systems withtime delays via new integral inequalities, IEEE Transactionson Systems, Man, and Cybernetics: Systems, 48(12), 2018,2495–2501.[58] Z. Li, Y. Bai, C. Huang, H. Yan, and S. Mu, Improved stabilityanalysis for delayed neural networks, IEEE Transactions onNeural Networks and Learning Systems, 29(9), 2017, 4535–4541.[59] Z. Li, H. Yan, H. Zhang, X. Zhan, and C. Huang, Stabilityanalysis for delayed neural networks via improved auxiliarypolynomial-based functions, IEEE Transactions on NeuralNetworks and Learning Systems, 30(8), 2019, 2562–2568.[60] Z. Li, H. Yan, H. Zhang, J. Sun, and H. Lam, Stability andstabilization with additive freedom for delayed Takagi–Sugenofuzzy systems by intermediary-polynomial-based functions,IEEE Transactions on Fuzzy Systems, 28(4), 2020, 692–705.
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