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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. Heckman, D. Mox, and M.A. Hsieh, Collective motion patterns of swarms with delay coupling: Theory and experiment, Physical Revie E, 93(3), 2016, 11.
[2] R. Sepulchre, D. Paley, and N. Leonard, Stabilization of planar collection motion: All-to-all communication, IEEE Transactions on Automatic Control, 52(5), 2007, 811–824.
[3] R. Sepulchre, D. Paley, and N. Leonard, Stabilization of planar collective motion with limited communication, IEEE Transactions on Automatic Control, 53(3), 2008, 706–719.
[4] A. Jain, D. Ghose, and P. Menon, Stabilization of balanced circular motion about a desired center, International Conference on Advances in Control and Optimization of Dynamical Systems, Kanpur, India, 2014.
[5] A. Jain and D. Ghose, Stabilization of collective motion in synchronized, balanced and splay phase arrangements on a desired circle, American Control Conference, Chicago, IL.
[6] G. Mallik and A. Sinha, A study of balanced circular formation under deviated cyclic pursuit strategy, IFAC-PapersOnLine, 48(5), 2015, 41–46.
[7] R. Zheng, Z. Lin, M. Fu, and D. Sun, Distributed control for uniform circumnavigation of ring-coupled unicycles, Automatica, 53, 2015, 23–29.
[8] N. Kokolakis and N. Koussoulas, Coordinated standoﬀ tracking of a ground moving target and the phase separation problem, International Conference on Unmanned Aircraft Systems, Dallas, TX, 2018.
[9] J. Guo, G. Yan, and Z. Lin, Cooperative control synthesis for moving-target-enclosing with changing topologies, International Conference on Robotics and Automation, Anchorage, AK, 2010.
[10] Z. Zhou, H. Wang, and Z. Hu, Event-based time varying formation control for multiple quadrotor UAVs with Markovian switching topologies, Complexity, 2018, 1–15.
[11] Y. Sun and L. Wang, Consensus of multi-agent systems in directed networks with nonuniform time-varying delays, IEEE Transactions on Automatic Control, 54(7), 2009, 1607–1613.
[12] J. Marshall, M. Broucke, and B. Francis, Formations of vehicles in cyclic pursuit, IEEE Transactions on Automatic Control, 49(11), 2004, 1963–1974.
[13] J. Marshall, M. Broucke, and B. Francis, Pursuit formations of unicycles, Automatica, 42, 2006, 3–12.
[14] M. Pavone and E. Frazzoli, Decentralized policies for geometric pattern formation and path coverage, ASME Journal of Dynamic Systems, Measurement, and Control, 129(5), 2007, 633–643.
[15] B. Wu, D. Wang, and E. Poh, Cyclic formation control for satellite formation using local relative measurements, Mechatronic Systems and Control, 40(1), 2012, 11–21.
[16] J. Juang, On the formation patterns under generalized cyclic pursuit, IEEE Transactions on Automatic Control, 58(9), 2013, 2401–2405, 2013.
[17] J. Ramirez, M. Pavone, E. Frazzoli, and D. Miller, Distributed control of spacecraft formations via cyclic pursuit: Theory and experiments, 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 tracking with onboard sensing, International Journal of Robotics Research, 34(13), 2015, 1660–1677.
[19] P. Jimenez, B. Shirinzadeh, D. Oetomo, and A. Nicholson, Swarm aggregation and formation control for robots with limited perception, International Journal of Robotics and Automation, 26(4), 2011, 255–263.
[20] P. Zhu and W. Ren, Multi-robot joint localization and target tracking with local sensing and communication, American Control Conference, Charlotte, NC, 2019.
[21] L. Luo, N. Chakraborty, and K. Sycara, Provably-good distributed algorithm for constrained multi-robot task assignment for grouped tasks, IEEE Transactions on Robotics, 31(1), 2015, 19–30.
[22] D. Panagou, M. Turpin, and V. Kumar, Decentralized goal assignment and safe trajectory generation in multi-robot networks via multiple Lyapunov functions, IEEE Transactions on Automatic Control, 65(8), 2020, 3365–3380.
[23] J. Ni, X. Yang, J. Chen, and S. Yang, Dynamic bioinspired neural network for multi-robot formation control in unknown environments, International Journal of Robotics and Automation, 30(3), 2015, 256–266.
[24] M. Khan and C. Silva, Autonomous and robust multi-robot cooperation using an artiﬁcial immune system, International Journal of Robotics and Automation, 27(1), 2012, 60–75.
[25] X. Yu, L. Liu, and G. Feng, Coordinated control of multiple unicycles for escorting and patrolling task based on a cyclic pursuit strategy, American Control Conference, Boston, MA, 2016.
[26] M. Zhang and H. Liu, Cooperative tracking a moving target using multiple ﬁxed-wing UAVs, Journal of Intelligent and Robotic Systems, 81(3-4), 2016, 505–529.
[27] X. Yu and L. Liu, Cooperative control for moving-target circular formation of nonholonomic vehicles, IEEE Transactions on Automatic Control, 62(7), 2017, 3448–3454.
[28] L. Brinon-Arranz, A. Seuret, and A. Pascoal, Target tracking via a circular formation of unicycles, IFAC World Congress, Toulouse, France, 2017.
[29] A. Miao, Y. Wang, and R. Fierro, Cooperative circumnavigation of a moving target with multiple nonholonomic robots using backstepping design, Systems and Control Letters, 103, 2017, 58–65.
[30] L. Ma and N. Hovakimyan, Vision-based cyclic pursuit for cooperative target tracking, Journal of Guidance, Control, and Dynamics, 36(2), 2013, 617–622.
[32], is adapted into the chain-type communication topology. Controller (7) that is expressed as a function of the relative bearing angle(s) βij(t) alone is proposed in this paper. Controller (7) regulates the separation angles between a pair of agents, instead of the separation distances between them as in (6). The diﬀerence between regulating separation distances and separation angles is not obvious when UAVs circle on the same orbit because a constant separation distance is equivalent to a constant separation angle under this circumstance. However, when agents circle on diﬀerent orbits (circles of diﬀerent radii around the target), the diﬀerence is clear, as to be demonstrated in Figs. 6 and 7. By properly specifying the desired 2D ranges and setting the UAV’s linear velocities accordingly, UAVs can now be controlled to circle on diﬀerent orbits. With this feature in place, uniform spacing formations are achieved by using the virtual agent 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 versatile concentric formation patterns, such as patterns with local geometric shapes. This is made possible utilizing existing works (tracking [38], balanced circular formation [35], [36]) and the new results presented in this paper (uniform spacing formations that regulate either separation distances or angles for agents circling on the same or diﬀerent orbits, and hierarchy). Results presented in this paper provide more complete solution to cooperative target tracking in concentric formations, by designing the control input of each UAV as a sum of several individual control components. One practical issue of avoiding inter-vehicle collision is to be addressed in Section 5. The combination of Attraction (another way of saying tracking), Alignment (formation), and Avoidance (collision avoidance) provides a solid framework to achieve formations in the context of target tracking. Another advantage of having versatile formation patterns is that it can possibly allow agents to acquire information of each other using a mixture of information exchange (over communication channels, for agents that are far away) and onboard sensing/perception (for agents that are close to each other). Refer to the two patterns in Fig. 5(a) and (c), where overall balanced circular formations have been achieved with local geometric shapes of either straight lines or triangles. The “leaders” of the three subgroups, can exchange information over the communication channels because they are relatively far away and might not be able to “see” each other. The other two members of each subgroup can possibly use onboard sensing to obtain the information; they need to achieve local formations (because they are close and can perceive each other). 5. Inter-vehicle Collision Avoidance When developing the uniform spacing formation, collision between UAVs is more likely to occur than other patterns where agents are far away from each other. A strategy 8 Figure 6. Cooperative target tracking using formation controller (17): (a) 2D trajectories; (b) 2D range; (c) formation in the end; (d) separation distance; (e) separation angle; and (f) formation control uic(t). Target undergoes a linear motion. Agents circle on two diﬀerent orbits. of preventing the inter-vehicle collision is thus much in demand. Among methods that prevent collisions, one way is to apply a force that repels the agents once they get closer. This force should also be strong enough to defend other forces pushing agents to a collision [45]–[49]. Let dmin denote the minimal distance allowed among agents. The following collision avoidance control component helps to repel agents to steer away from each other once they get too close [50]: uia(t) = −Kr j∈N (ri) dmin dij sin βij (18) where Kr is the controller gain. The repulsion term (18) adjusts each agent’s heading to the opposite direction of its neighbours in N(ri). Notice that N (ri) and Ni denote diﬀerent sets. The set Ni is the set of agents whose information can be obtained by the ith agent via communication, whereas the set N (ri) denotes the set of agents that are too close to agent i. With (18) in place, each agent’s control input becomes [50]: ˙ψi(t) = ui(t) = uit(t) + uic(t) + uia(t) (19) Totally three control components are added together to achieve the objective of simultaneous tracking, formation, and inter-vehicle collision avoidance. 6. Simulation Results The proposed control laws were simulated in Matlab to verify their performance of tracking, formation, and intervehicle collision avoidance. 6.1 Achieving Uniform Spacing Formations on Diﬀerent Orbits This example demonstrates achievement of uniform spacing formations when agents circle on diﬀerent orbits. To achieve formations under this circumstance, formation controllers (17) and (16) are used, which resort to virtual agents when needed, i.e., when the two agents are not on the same orbit. It is worth mentioning that the formation controller (16) aims at regulating the relative separation distances to a constant, while the controller (17) regulates the relative separation angles to a desired value. The 2D ranges are set to be ρd,i = {15, 15, 18, 18} (m), for i = 1, 2, 3, 4, respectively. Correspondingly, UAVs’ linear velocities are Vg,i = {30, 30, 36, 36} (m/s), satisfying the relationship in (10). To focus on the formation patterns, the target’s motion is assumed linear (the target moves on a straight line). We ﬁrst present simulation results applying the formation controller (17). The 2D trajectories of the agents and the target are plotted in Fig. 6(a). Details of the tracking are given in Fig. 6(b), where ρi(t) approach their prescribed values (that are diﬀerent). All agents’ positions at the end of the simulation are shown in Fig. 6(c), 9 Figure 7. Cooperative target tracking using formation controller (16): (a) 2D trajectories; (b) 2D range; (c) formation in the end; (d) separation distance; (e) separation angle; and (f) formation control uic(t). Target undergoes a linear motion. Agents circle on two diﬀerent orbits. where all agents scatter around the target on two diﬀerent orbits. Both the separation distances (Fig. 6(d)) and the separation angles (Fig. 6(e)) are plotted. When controller (17) is applied, we expect the separation angles to approach a constant. However, the separation distances may or may not approach to the same value because the distances between two agents also depend on the orbits that they lie on. For ds = 10 (m), the desired separation angle is either φs = 2 sin−1 (ds/30) ≈ 39◦ or φs = 2π − 2 sin−1 (ds/30) ≈ 321◦ . Because all separation angles converge to 39◦ (Fig. 6(e)), formation is obtained successfully. Figure 6(f) shows the formation component uic(t), which converges to zero upon formation. Results when applying the controller (16) are shown in Fig. 7. Figure 7(a) shows the general picture of cooperative target tracking. Figure 7(b) plots the 2D ranges between the agents and the target, demonstrating successful tracking with two diﬀerent range values. Figure 7(c) shows the formation achieved in the end, where agents circle on two diﬀerent orbits around the target. Figure 7(d) shows that the separation distances between two successive agents are now regulated to the speciﬁed value of ds = 10 (m) (same for all agents). On the contrary, the separation angles do not approach to one value. From Fig. 7(c), it can be seen that agents {1, 4} and agents {2, 3} circle on two diﬀerent orbits. The separation angles between these two pairs of consecutive agents should be smaller than that between agents {1, 2}, which circle on the same orbit. This is conﬁrmed in Fig. 7(e). Figure 7(f) shows that the formation control component uic(t) also vanishes to zero upon formation. Figures 6 and 7 together show the diﬀerence between the two formation controllers (16) and (17). The formation controller (16) regulates the relative separation distances between agents to a speciﬁed value ds, while the controller (17) regulates the separation angles to φs. 6.2 Achieving Concentric Formations with Local Geometric Shapes This section presents cooperative target tracking in concentric formations with local geometric shapes of straight lines and triangles, as those shown in Fig. 5. To focus on formation patterns, the target’s motion is assumed linear, i.e., the target moves on a straight line. Corresponding to the four patterns in Fig. 5, simulation results are presented in Fig. 8, with several snapshots showing how formation is achieved over time. Cooperative target tracking in these concentric formations are successfully obtained. 6.3 Achieving Inter-vehicle Collision Avoidance This example demonstrates cooperative target tracking with inter-vehicle collision avoidance. The two scenarios of without and with collision avoidance, i.e., before and after applying uia(t), are shown and compared in Fig. 9. We select the target’s velocity to be piecewise-constant, n = 5, ds = 10 (m), and dmin = 7 (m). The ﬁrst row of Fig. 9 is 10 Figure 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 is either straight line or triangle. for the scenario without the collision avoidance capability, i.e., before the control component uia(t) is applied. After uia(t) is applied, results are shown in the second row of Fig. 9. In each scenario, the 2D trajectories are plotted 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 component uia(t) (Fig. 9(b) versus (d)). For the second scenario with collision avoidance, the component uia(t) for each agent is shown in Fig. 9(e), where a saturation of |uia(t)| ≤ 3 has been used for all agents. Comparison between Fig. 9(b) and (d) shows that the control component uia(t) helps to keep the minimal distance to be greater than the allowed value. Otherwise, the minimal distance can be much smaller, as indicated in Fig. 9(b). The zigzag area in Fig. 9(d), corresponds to the circumstances when uia(t) takes eﬀect. Figure 9(e) shows that uia(t) only takes eﬀect when needed, i.e., when agents {3, 5} get too close to each other. 7. Comparison with Prior Studies Comparing with the distance-based coordination control law (6), our proposed bearing-angle-based coordination control law (7) has one potential advantage. Consider a 11 Figure 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 some agents becomes unavailable (e.g., due to communication loss or out of communication range). Instead of computing the relative bearing angles from the exchanged positions as in (12), the needed information of relative bearing angles can be estimated/obtained from a local vision system installed on each UAV. In other words, when the expected information from others is unavailable (either permanently or temporarily), the objective of achieving and maintaining formation could still be achieved by using local measurements and estimates. Regarding formation control, most existing results are either leaderless or leader-following [51], [52]. The proposed method of obtaining versatile formation patterns allows a combination of both. This can be seen in the “BalancedLine” and “Balanced-Triangle” patterns (Fig. 5). The overall balanced circular formation can be obtained using a leaderless communication topology, whereas achievement of local geometric shapes can be implemented in a leader-based manner. The adopted hierarchical formation structure allows selection of appropriate communication topologies on diﬀerent layers. This paper also considers a practical issue that would occur, i.e., collisions among agents. This issue was tackled by adding another control component into each UAV’s control input. As can be seen from (18), this added collisionavoidance control component can also be expressed as a function of bearing angles. As mentioned earlier, a bearingangle-based control law has the potential of still achieving its control objective (formation or collision avoidance) during communication loss, by using local measurements from each UAV’s onboard sensors. 8. Conclusions This paper is to obtain more versatile concentric formations in cooperative target tracking where a ﬂeet of UAVs is commanded to circle above (and around) a moving ground target. On the basis of our previous results, versatile formation patterns are achieved with the help of three new features. The ﬁrst feature is a new formation pattern, the uniform spacing formation where either the relative separation distances or the separation angles can be regulated to a desired value. Diﬀerent from a balanced circular formation 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 are proposed, where one regulates the separation distances between two agents and the other regulates the separation angles in between. The second feature allows UAVs to circle on diﬀerent orbits. To achieve formation under this circumstance, formation controllers will resort to virtual agents representing the actual agents in need. The third feature is the usage of a (two-layer) hierarchical formation structure, which allows selection of formation patterns for diﬀerent layers. Combinations of these new features with our existing results yield more versatile concentric formation patterns with diﬀerent local geometric shapes, such as straight lines and triangles. Inter-vehicle collision avoidance is also addressed. Agents will be repelled to steer away from each other once they get too close. 12 All UAVs are assumed to have constant linear velocities. Control of each UAV is via its yaw rate. The design idea is to add three control components (three heading controllers) together to achieve the overall objective. Each control component has a goal. The proposed extensions to spreading agents on a portion of a circle, circling agents on orbits of diﬀerent radii, formation in local geometric shapes, and avoiding inter-vehicle collisions, provide more complete solution to cooperative target tracking in the concentric manner. This paper also raises several questions for future investigations. The implementation of the proposed schemes on physical robots and the extension of the developed techniques to 3D scenarios and cooperative tracking of multiple targets with obstacle avoidance capability [53], [54] will be of particular interest. Stability analyses in the presence of formation pattern switching and broken communication links are another research direction to look into. Also, investigations of the time delay factor for obtaining stability conditions as well as desirable performance with reasonable computation complexity [55]–[57] are needed. Finally, Artiﬁcial Intelligence (AI) techniques have recently been developed for robotic communication to enhance the communication capability of robotic networks for coordinated actions. Application of the AI and/or Neural Networks to the ﬁeld of robotic networks in the context of cooperative target tracking is a promising research area to pursue [58]–[60]. References [1] K. Szwaykowska, I.B. Schwartz, L.M.-T. Romero, C.R. Heckman, D. Mox, and M.A. Hsieh, Collective motion patterns of swarms with delay coupling: Theory and experiment, Physical Revie E, 93(3), 2016, 11. [2] R. Sepulchre, D. Paley, and N. Leonard, Stabilization of planar collection motion: All-to-all communication, IEEE Transactions on Automatic Control, 52(5), 2007, 811–824. [3] R. Sepulchre, D. Paley, and N. Leonard, Stabilization of planar collective motion with limited communication, IEEE Transactions on Automatic Control, 53(3), 2008, 706–719. [4] A. Jain, D. Ghose, and P. Menon, Stabilization of balanced circular motion about a desired center, International Conference on Advances in Control and Optimization of Dynamical Systems, Kanpur, India, 2014. [5] A. Jain and D. Ghose, Stabilization of collective motion in synchronized, balanced and splay phase arrangements on a desired circle, American Control Conference, Chicago, IL. [6] G. Mallik and A. Sinha, A study of balanced circular formation under deviated cyclic pursuit strategy, IFAC-PapersOnLine, 48(5), 2015, 41–46. [7] R. Zheng, Z. Lin, M. Fu, and D. Sun, Distributed control for uniform circumnavigation of ring-coupled unicycles, Automatica, 53, 2015, 23–29. [8] N. Kokolakis and N. Koussoulas, Coordinated standoﬀ tracking of a ground moving target and the phase separation problem, International Conference on Unmanned Aircraft Systems, Dallas, TX, 2018. [9] J. Guo, G. Yan, and Z. Lin, Cooperative control synthesis for moving-target-enclosing with changing topologies, International Conference on Robotics and Automation, Anchorage, AK, 2010. [10] Z. Zhou, H. Wang, and Z. Hu, Event-based time varying formation control for multiple quadrotor UAVs with Markovian switching topologies, Complexity, 2018, 1–15. [11] Y. Sun and L. Wang, Consensus of multi-agent systems in directed networks with nonuniform time-varying delays, IEEE Transactions on Automatic Control, 54(7), 2009, 1607–1613. [12] J. Marshall, M. Broucke, and B. Francis, Formations of vehicles in cyclic pursuit, IEEE Transactions on Automatic Control, 49(11), 2004, 1963–1974. [13] J. Marshall, M. Broucke, and B. Francis, Pursuit formations of unicycles, Automatica, 42, 2006, 3–12. [14] M. Pavone and E. Frazzoli, Decentralized policies for geometric pattern formation and path coverage, ASME Journal of Dynamic Systems, Measurement, and Control, 129(5), 2007, 633–643. [15] B. Wu, D. Wang, and E. Poh, Cyclic formation control for satellite formation using local relative measurements, Mechatronic Systems and Control, 40(1), 2012, 11–21. [16] J. Juang, On the formation patterns under generalized cyclic pursuit, IEEE Transactions on Automatic Control, 58(9), 2013, 2401–2405, 2013. [17] J. Ramirez, M. Pavone, E. Frazzoli, and D. Miller, Distributed control of spacecraft formations via cyclic pursuit: Theory and experiments, 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 tracking with onboard sensing, International Journal of Robotics Research, 34(13), 2015, 1660–1677. [19] P. Jimenez, B. Shirinzadeh, D. Oetomo, and A. Nicholson, Swarm aggregation and formation control for robots with limited perception, International Journal of Robotics and Automation, 26(4), 2011, 255–263. [20] P. Zhu and W. Ren, Multi-robot joint localization and target tracking with local sensing and communication, American Control Conference, Charlotte, NC, 2019. [21] L. Luo, N. Chakraborty, and K. Sycara, Provably-good distributed algorithm for constrained multi-robot task assignment for grouped tasks, IEEE Transactions on Robotics, 31(1), 2015, 19–30. [22] D. Panagou, M. Turpin, and V. Kumar, Decentralized goal assignment and safe trajectory generation in multi-robot networks via multiple Lyapunov functions, IEEE Transactions on Automatic Control, 65(8), 2020, 3365–3380. [23] J. Ni, X. Yang, J. Chen, and S. Yang, Dynamic bioinspired neural network for multi-robot formation control in unknown environments, International Journal of Robotics and Automation, 30(3), 2015, 256–266. [24] M. Khan and C. Silva, Autonomous and robust multi-robot cooperation using an artiﬁcial immune system, International Journal of Robotics and Automation, 27(1), 2012, 60–75. [25] X. Yu, L. Liu, and G. Feng, Coordinated control of multiple unicycles for escorting and patrolling task based on a cyclic pursuit strategy, American Control Conference, Boston, MA, 2016. [26] M. Zhang and H. Liu, Cooperative tracking a moving target using multiple ﬁxed-wing UAVs, Journal of Intelligent and Robotic Systems, 81(3-4), 2016, 505–529. [27] X. Yu and L. Liu, Cooperative control for moving-target circular formation of nonholonomic vehicles, IEEE Transactions on Automatic Control, 62(7), 2017, 3448–3454. [28] L. Brinon-Arranz, A. Seuret, and A. Pascoal, Target tracking via a circular formation of unicycles, IFAC World Congress, Toulouse, France, 2017. [29] A. Miao, Y. Wang, and R. Fierro, Cooperative circumnavigation of a moving target with multiple nonholonomic robots using backstepping design, Systems and Control Letters, 103, 2017, 58–65. [30] L. Ma and N. Hovakimyan, Vision-based cyclic pursuit for cooperative target tracking, Journal of Guidance, Control, and Dynamics, 36(2), 2013, 617–622. [31] N. Moshtagh, N. Michael, A. Jadbabaie, and K. Daniilidis, Vision-based, distributed control laws for motion coordination of 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, Triangular formation control using range measurements: An application to marine robotic vehicles, IFAC Proceedings, 45(5), 2012, 112–117. 13
[35]–[37] control all agents to circle on one orbit (the same orbit). For more ﬂexibility and versatility, one would want all agents to be able to circle on diﬀerent orbits. This section describes the modiﬁcations and arrangements needed to make this happen. Start by considering target tracking. When all UAVs are commanded to orbit on the same circle, the 2D range reference is the same for all UAVs and is denoted by ρd so far. Accordingly, the linear velocities Vg,i are the same for all UAVs. To circle UAVs on diﬀerent orbits, the range references will be diﬀerent. As a result, UAVs’ linear velocities, which are the same before, need to be diﬀerent. Let ρd,i and Vg,i denote the prescribed range reference and linear velocity of agent i. Using agents i and j as an example, the relationship between their linear velocities and 2D range references needs to satisfy: Vg,i Vg,j = ρd,i ρd,j (10) For coordination, our approach to coordinate all agents to circle on diﬀerent orbits is to refer to virtual agent(s). Suppose agent i needs information from agent j to achieve coordination. Instead of using agent j’s information, agent i will use information from a virtual agent that represents agent j but orbits on the same orbit as agent i. This virtual agent, denoted by the big dot in Fig. 4, lies in the intersection of agent i’s own orbit and the line connecting the target and the actual agent j. That is, the virtual agent is the projection of agent j onto agent i’s orbit w.r.t. the target. No modiﬁcation is needed to an existing communication topology. No additional information exchange among agents is required. Based on the target’s position (measured/estimated by each agent), each agent’s position (known from the agent’s onboard sensor), and the neighbouring agent’s position (obtained via information exchange), the position of the virtual agent can be computed. Let [xi(t), yi(t)] be the agent i’s Cartesian coordinate in the world frame. Suppose that the coordination control law of agent i requires information of agent j. Instead 5 Figure 4. Virtual agent representing agent j used by agent i. of using agent j’s information [xj (t) , yj(t)] directly, a virtual agent will be ﬁrst computed and then used. The virtual agent does not necessarily have a speed associated with it. It is its relative position (distance or separation angle) w.r.t. agent i that matters. The coordinates of this virtual agent, denoted by [˜xi(t), ˜y(t)] , are computed as: ˜xj(t) = ρd,i ρd,j (xj(t) − xt(t)) + xt(t), ˜yj(t) = ρd,i ρd,j (yj(t) − yt(t)) + yt(t) (11) Letting ˜βi,j(t) ∈ [−π, π] denote the relative bearing angle between agent i and the virtual agent representing agent 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 vector w.r.t. the x-axis, which can be computed as θi (t) = π 2 − ψi(t). To extend to the scenario of circling on diﬀerent orbits, we reﬁne the deﬁnition of uniform spacing to have either desired distance spacing or desired angular spacing between two agents. In the case of spacing in distance, the separation distances between two consecutive agents will be regulated to the desired value. For spacing in angular distance, the separation angles between each two consecutive agents w.r.t. the target are regulated to the desired value, regardless of the orbits they are on. Under this extension, the case of circling on the same orbit becomes a special case with all orbits reducing to one, where constant relative distance is equivalent to constant separation angle, and the virtual agent representing agent j is the agent j itself. These are no longer true for circling on diﬀerent orbits. To circle on diﬀerent orbits, our formation control laws, which were all designed as functions of relative bearing angles βi,j(t) referring to the actual agents, will be modiﬁed to use relative bearing angles ˜βi,j(t) referring to the virtual agents. To be speciﬁc, consider the balanced circular formation. Simply by replacing βi,j(t) with ˜βi,j(t), the balanced circular formation can be obtained in a straightforward manner for the case of circling on diﬀerent orbits. Particularly, the three formation control laws in [35] are modiﬁed to be: • Achieving balanced circular formation under all-to-all: uic(t) = −κ n j=1,j=i cos ˜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) − cos π n , κ > 0 (15) For uniform spacing formations, controller (6) is modiﬁed to be: uic(t) = ⎧ ⎨ ⎩ 0, i = k −kv ˜βi(i+1)(t) ln (cv−1)di(i+1)(t)+ds cvds , i = k (16) Our coordination controller (7) is a function of both βi(i+1)(t) and β0. Similarly, βi(i+1)(t) will be replaced by ˜βi(i+1)(t). For circling on diﬀerent orbits, it is not convenient to use ds to specify the desired separation angle, as in (8). Instead, the desired separation angle, denoted by 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 law might be more convenient than the other. For instance, when the formation is speciﬁed literally using relative distance(s), the coordination control laws (6) or (16) might be more convenient. If the formation is concerned with relative angular distance, the coordination control laws (7) or (17) might suit better. 4. Concentric Formations with Local Geometric Shapes So far, we have achieved two formation patterns (balanced circular in [35],
[37] control all agents to circle on one orbit (the same orbit). For more ﬂexibility and versatility, one would want all agents to be able to circle on diﬀerent orbits. This section describes the modiﬁcations and arrangements needed to make this happen. Start by considering target tracking. When all UAVs are commanded to orbit on the same circle, the 2D range reference is the same for all UAVs and is denoted by ρd so far. Accordingly, the linear velocities Vg,i are the same for all UAVs. To circle UAVs on diﬀerent orbits, the range references will be diﬀerent. As a result, UAVs’ linear velocities, which are the same before, need to be diﬀerent. Let ρd,i and Vg,i denote the prescribed range reference and linear velocity of agent i. Using agents i and j as an example, the relationship between their linear velocities and 2D range references needs to satisfy: Vg,i Vg,j = ρd,i ρd,j (10) For coordination, our approach to coordinate all agents to circle on diﬀerent orbits is to refer to virtual agent(s). Suppose agent i needs information from agent j to achieve coordination. Instead of using agent j’s information, agent i will use information from a virtual agent that represents agent j but orbits on the same orbit as agent i. This virtual agent, denoted by the big dot in Fig. 4, lies in the intersection of agent i’s own orbit and the line connecting the target and the actual agent j. That is, the virtual agent is the projection of agent j onto agent i’s orbit w.r.t. the target. No modiﬁcation is needed to an existing communication topology. No additional information exchange among agents is required. Based on the target’s position (measured/estimated by each agent), each agent’s position (known from the agent’s onboard sensor), and the neighbouring agent’s position (obtained via information exchange), the position of the virtual agent can be computed. Let [xi(t), yi(t)] be the agent i’s Cartesian coordinate in the world frame. Suppose that the coordination control law of agent i requires information of agent j. Instead 5 Figure 4. Virtual agent representing agent j used by agent i. of using agent j’s information [xj (t) , yj(t)] directly, a virtual agent will be ﬁrst computed and then used. The virtual agent does not necessarily have a speed associated with it. It is its relative position (distance or separation angle) w.r.t. agent i that matters. The coordinates of this virtual agent, denoted by [˜xi(t), ˜y(t)] , are computed as: ˜xj(t) = ρd,i ρd,j (xj(t) − xt(t)) + xt(t), ˜yj(t) = ρd,i ρd,j (yj(t) − yt(t)) + yt(t) (11) Letting ˜βi,j(t) ∈ [−π, π] denote the relative bearing angle between agent i and the virtual agent representing agent 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 vector w.r.t. the x-axis, which can be computed as θi (t) = π 2 − ψi(t). To extend to the scenario of circling on diﬀerent orbits, we reﬁne the deﬁnition of uniform spacing to have either desired distance spacing or desired angular spacing between two agents. In the case of spacing in distance, the separation distances between two consecutive agents will be regulated to the desired value. For spacing in angular distance, the separation angles between each two consecutive agents w.r.t. the target are regulated to the desired value, regardless of the orbits they are on. Under this extension, the case of circling on the same orbit becomes a special case with all orbits reducing to one, where constant relative distance is equivalent to constant separation angle, and the virtual agent representing agent j is the agent j itself. These are no longer true for circling on diﬀerent orbits. To circle on diﬀerent orbits, our formation control laws, which were all designed as functions of relative bearing angles βi,j(t) referring to the actual agents, will be modiﬁed to use relative bearing angles ˜βi,j(t) referring to the virtual agents. To be speciﬁc, consider the balanced circular formation. Simply by replacing βi,j(t) with ˜βi,j(t), the balanced circular formation can be obtained in a straightforward manner for the case of circling on diﬀerent orbits. Particularly, the three formation control laws in [35] are modiﬁed to be: • Achieving balanced circular formation under all-to-all: uic(t) = −κ n j=1,j=i cos ˜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) − cos π n , κ > 0 (15) For uniform spacing formations, controller (6) is modiﬁed to be: uic(t) = ⎧ ⎨ ⎩ 0, i = k −kv ˜βi(i+1)(t) ln (cv−1)di(i+1)(t)+ds cvds , i = k (16) Our coordination controller (7) is a function of both βi(i+1)(t) and β0. Similarly, βi(i+1)(t) will be replaced by ˜βi(i+1)(t). For circling on diﬀerent orbits, it is not convenient to use ds to specify the desired separation angle, as in (8). Instead, the desired separation angle, denoted by 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 law might be more convenient than the other. For instance, when the formation is speciﬁed literally using relative distance(s), the coordination control laws (6) or (16) might be more convenient. If the formation is concerned with relative angular distance, the coordination control laws (7) or (17) might suit better. 4. Concentric Formations with Local Geometric Shapes So far, we have achieved two formation patterns (balanced circular in [35], [36] and uniform spacing in Sections 2.2 and 3) and allowed agents to circle on either the same orbit or diﬀerent orbits. In the context of concentric formations, we would like to generate more versatile formations, such as formations with local geometric shapes. Clearly, combinations of circular and/or uniform formations are more 6 Figure 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 utilizing existing features (tracking, formation in diﬀerent patterns, circling on diﬀerent orbits), with the help of a hierarchical formation structure [41]–[44]. The simplest hierarchical scheme, the two-layer hierarchical structure, can be described as follows. A collection of n agents is divided into n2 subgroups, each containing n1 agents (n1 × n2 = n). The local control strategy is chosen such that the agents within each subgroup can be commanded to achieve certain formation pattern [41]. In [44], a two-layer hybrid pursuit system was described, where cyclic pursuit strategy was considered at the higher layer (the ﬁrst layer) and chain-like communication topology was used at the lower layer (the second layer). The concept of hierarchy is now applied to cooperative target tracking. The idea of hierarchy allows different subgroups to select diﬀerent formation laws that are already known to be stable. Similar to [44], a twolayer hierarchical formation structure is used. The ﬁrst layer can be set to achieve either the balanced circular or the uniform spacing formation. The second layer can be set to achieve the uniform spacing formation by specifying either a desired separation distance or angle. Two examples are given below to demonstrate how versatile patterns are achieved by determining the 2D range references of the agents (same or diﬀerent) and the formation pattern on each layer (balanced circular or uniform spacing). The ﬁrst example achieves concentric formations with local geometric shapes in straight lines: (1) The agents {1, 4, 7}, {2, 5, 8}, and {3, 6, 9} are assigned to be on the inner, middle, and outer orbits, respectively, by specifying their 2D range references to be {15, 18, 25} (m). Correspondingly, the UAVs’ linear velocities are set to be {30, 36, 50} (m/s), satisfying the relationship in (10). (2) On the ﬁrst layer, the balanced circular formation is used for agents {1, 4, 7}, which lie on the inner circle. On the second layer, a uniform spacing formation is used inside each subgroup. There are three subgroups: {1, 2, 3}, {4, 5, 6}, and {7, 8, 9}. The agent that is involved in formation on the ﬁrst layer works as “leader” of its subgroup. As the intended geometric shape is in straight line and the radial diﬀerences between each two adjacent orbits can be diﬀerent, it is more convenient to use (7) with β0 = π/2. Actual agents are used when obtaining bearing angles βi(i+1)(t). The formation pattern is shown in Fig. 5(a). 7 (3) Simply changing the formation pattern on the ﬁrst layer from the balanced circular to the uniform spacing pattern, an overall uniform spacing formation is achieved with local geometric shapes in straight lines. The formation pattern is shown in Fig. 5(b). The second example achieves concentric formations with local geometric shapes in triangles: (1) The agents {1, 3, 4, 6, 7, 9} are assigned to circle on the inner orbit. All other agents circle on the outer orbit. 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 circular formation on the ﬁrst layer. On the second layer, a uniform spacing formation is used. To achieve local shapes in triangles, it is convenient to use (16), which regulates distances directly. Virtual agents are used. The achieved formation pattern is shown in Fig. 5(c). (3) Simply changing the formation pattern on the ﬁrst layer from the balanced circular to the uniform spacing pattern, an overall uniform spacing formation is obtained with local geometric shapes in triangles. The formation pattern is shown in Fig. 5(d). To clarify how subgroups are deﬁned, how the leader of each subgroup is chosen, and if the robots know the size of their subgroups a priori, we use Fig. 5(a) as an example. For the subgroup consisting agents {1, 2, 3}, members in this subgroup are deﬁned by specifying agent 1 not to “seek” anyone else; agent 2 “seeking” agent 1; and agent 3 “seeking” agent 2. In each subgroup, all members do 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 only know which one to “seek”. In the simulation examples, the knowledge of which agent to “seek” is assigned. In reality, these knowledges can be perceived by the agents so that formation can still be maintained with agents joining or leaving the group. At this point, we would like to highlight diﬀerences between the control laws of previous work [35], [36] and the new work in this paper, i.e., formation controllers (6), (7), (16), and (17), as well as discussing some interesting aspects of the new control laws. Three formation controllers were reported in [35], [36] each achieving a balanced circular formation under one of the following three communication topologies: (1) all-to-all, (2) ring, and (3) cyclic pursuit. Using the balanced circular formation, all agents, which circle on the same orbit around the target, spread evenly around a full circle, whose centre resides in the moving target. These previous works are focused on studying the feasibility of achieving formations for target tracking. Having successfully obtained cooperative tracking in the balanced circular formation, one would naturally wonder what other formation patterns can be obtained that also ﬁt into the tracking scenario. Instead of spreading all agents over one full circle, spreading them over a portion of a circle (i.e., an arc) is one natural variation. Two uniform spacing formation controllers (6) and (7) are presented 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 relative bearing angle(s) βij(t) alone is proposed in this paper. Controller (7) regulates the separation angles between a pair of agents, instead of the separation distances between them as in (6). The diﬀerence between regulating separation distances and separation angles is not obvious when UAVs circle on the same orbit because a constant separation distance is equivalent to a constant separation angle under this circumstance. However, when agents circle on diﬀerent orbits (circles of diﬀerent radii around the target), the diﬀerence is clear, as to be demonstrated in Figs. 6 and 7. By properly specifying the desired 2D ranges and setting the UAV’s linear velocities accordingly, UAVs can now be controlled to circle on diﬀerent orbits. With this feature in place, uniform spacing formations are achieved by using the virtual agent 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 versatile concentric formation patterns, such as patterns with local geometric shapes. This is made possible utilizing existing works (tracking
[38], balanced circular formation [35], [36]) and the new results presented in this paper (uniform spacing formations that regulate either separation distances or angles for agents circling on the same or diﬀerent orbits, and hierarchy). Results presented in this paper provide more complete solution to cooperative target tracking in concentric formations, by designing the control input of each UAV as a sum of several individual control components. One practical issue of avoiding inter-vehicle collision is to be addressed in Section 5. The combination of Attraction (another way of saying tracking), Alignment (formation), and Avoidance (collision avoidance) provides a solid framework to achieve formations in the context of target tracking. Another advantage of having versatile formation patterns is that it can possibly allow agents to acquire information of each other using a mixture of information exchange (over communication channels, for agents that are far away) and onboard sensing/perception (for agents that are close to each other). Refer to the two patterns in Fig. 5(a) and (c), where overall balanced circular formations have been achieved with local geometric shapes of either straight lines or triangles. The “leaders” of the three subgroups, can exchange information over the communication channels because they are relatively far away and might not be able to “see” each other. The other two members of each subgroup can possibly use onboard sensing to obtain the information; they need to achieve local formations (because they are close and can perceive each other). 5. Inter-vehicle Collision Avoidance When developing the uniform spacing formation, collision between UAVs is more likely to occur than other patterns where agents are far away from each other. A strategy 8 Figure 6. Cooperative target tracking using formation controller (17): (a) 2D trajectories; (b) 2D range; (c) formation in the end; (d) separation distance; (e) separation angle; and (f) formation control uic(t). Target undergoes a linear motion. Agents circle on two diﬀerent orbits. of preventing the inter-vehicle collision is thus much in demand. Among methods that prevent collisions, one way is to apply a force that repels the agents once they get closer. This force should also be strong enough to defend other forces pushing agents to a collision [45]–[49]. Let dmin denote the minimal distance allowed among agents. The following collision avoidance control component helps to repel agents to steer away from each other once they get too close [50]: uia(t) = −Kr j∈N (ri) dmin dij sin βij (18) where Kr is the controller gain. The repulsion term (18) adjusts each agent’s heading to the opposite direction of its neighbours in N(ri). Notice that N (ri) and Ni denote diﬀerent sets. The set Ni is the set of agents whose information can be obtained by the ith agent via communication, whereas the set N (ri) denotes the set of agents that are too close to agent i. With (18) in place, each agent’s control input becomes [50]: ˙ψi(t) = ui(t) = uit(t) + uic(t) + uia(t) (19) Totally three control components are added together to achieve the objective of simultaneous tracking, formation, and inter-vehicle collision avoidance. 6. Simulation Results The proposed control laws were simulated in Matlab to verify their performance of tracking, formation, and intervehicle collision avoidance. 6.1 Achieving Uniform Spacing Formations on Diﬀerent Orbits This example demonstrates achievement of uniform spacing formations when agents circle on diﬀerent orbits. To achieve formations under this circumstance, formation controllers (17) and (16) are used, which resort to virtual agents when needed, i.e., when the two agents are not on the same orbit. It is worth mentioning that the formation controller (16) aims at regulating the relative separation distances to a constant, while the controller (17) regulates the relative separation angles to a desired value. The 2D ranges are set to be ρd,i = {15, 15, 18, 18} (m), for i = 1, 2, 3, 4, respectively. Correspondingly, UAVs’ linear velocities are Vg,i = {30, 30, 36, 36} (m/s), satisfying the relationship in (10). To focus on the formation patterns, the target’s motion is assumed linear (the target moves on a straight line). We ﬁrst present simulation results applying the formation controller (17). The 2D trajectories of the agents and the target are plotted in Fig. 6(a). Details of the tracking are given in Fig. 6(b), where ρi(t) approach their prescribed values (that are diﬀerent). All agents’ positions at the end of the simulation are shown in Fig. 6(c), 9 Figure 7. Cooperative target tracking using formation controller (16): (a) 2D trajectories; (b) 2D range; (c) formation in the end; (d) separation distance; (e) separation angle; and (f) formation control uic(t). Target undergoes a linear motion. Agents circle on two diﬀerent orbits. where all agents scatter around the target on two diﬀerent orbits. Both the separation distances (Fig. 6(d)) and the separation angles (Fig. 6(e)) are plotted. When controller (17) is applied, we expect the separation angles to approach a constant. However, the separation distances may or may not approach to the same value because the distances between two agents also depend on the orbits that they lie on. For ds = 10 (m), the desired separation angle is either φs = 2 sin−1 (ds/30) ≈ 39◦ or φs = 2π − 2 sin−1 (ds/30) ≈ 321◦ . Because all separation angles converge to 39◦ (Fig. 6(e)), formation is obtained successfully. Figure 6(f) shows the formation component uic(t), which converges to zero upon formation. Results when applying the controller (16) are shown in Fig. 7. Figure 7(a) shows the general picture of cooperative target tracking. Figure 7(b) plots the 2D ranges between the agents and the target, demonstrating successful tracking with two diﬀerent range values. Figure 7(c) shows the formation achieved in the end, where agents circle on two diﬀerent orbits around the target. Figure 7(d) shows that the separation distances between two successive agents are now regulated to the speciﬁed value of ds = 10 (m) (same for all agents). On the contrary, the separation angles do not approach to one value. From Fig. 7(c), it can be seen that agents {1, 4} and agents {2, 3} circle on two diﬀerent orbits. The separation angles between these two pairs of consecutive agents should be smaller than that between agents {1, 2}, which circle on the same orbit. This is conﬁrmed in Fig. 7(e). Figure 7(f) shows that the formation control component uic(t) also vanishes to zero upon formation. Figures 6 and 7 together show the diﬀerence between the two formation controllers (16) and (17). The formation controller (16) regulates the relative separation distances between agents to a speciﬁed value ds, while the controller (17) regulates the separation angles to φs. 6.2 Achieving Concentric Formations with Local Geometric Shapes This section presents cooperative target tracking in concentric formations with local geometric shapes of straight lines and triangles, as those shown in Fig. 5. To focus on formation patterns, the target’s motion is assumed linear, i.e., the target moves on a straight line. Corresponding to the four patterns in Fig. 5, simulation results are presented in Fig. 8, with several snapshots showing how formation is achieved over time. Cooperative target tracking in these concentric formations are successfully obtained. 6.3 Achieving Inter-vehicle Collision Avoidance This example demonstrates cooperative target tracking with inter-vehicle collision avoidance. The two scenarios of without and with collision avoidance, i.e., before and after applying uia(t), are shown and compared in Fig. 9. We select the target’s velocity to be piecewise-constant, n = 5, ds = 10 (m), and dmin = 7 (m). The ﬁrst row of Fig. 9 is 10 Figure 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 is either straight line or triangle. for the scenario without the collision avoidance capability, i.e., before the control component uia(t) is applied. After uia(t) is applied, results are shown in the second row of Fig. 9. In each scenario, the 2D trajectories are plotted 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 component uia(t) (Fig. 9(b) versus (d)). For the second scenario with collision avoidance, the component uia(t) for each agent is shown in Fig. 9(e), where a saturation of |uia(t)| ≤ 3 has been used for all agents. Comparison between Fig. 9(b) and (d) shows that the control component uia(t) helps to keep the minimal distance to be greater than the allowed value. Otherwise, the minimal distance can be much smaller, as indicated in Fig. 9(b). The zigzag area in Fig. 9(d), corresponds to the circumstances when uia(t) takes eﬀect. Figure 9(e) shows that uia(t) only takes eﬀect when needed, i.e., when agents {3, 5} get too close to each other. 7. Comparison with Prior Studies Comparing with the distance-based coordination control law (6), our proposed bearing-angle-based coordination control law (7) has one potential advantage. Consider a 11 Figure 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 some agents becomes unavailable (e.g., due to communication loss or out of communication range). Instead of computing the relative bearing angles from the exchanged positions as in (12), the needed information of relative bearing angles can be estimated/obtained from a local vision system installed on each UAV. In other words, when the expected information from others is unavailable (either permanently or temporarily), the objective of achieving and maintaining formation could still be achieved by using local measurements and estimates. Regarding formation control, most existing results are either leaderless or leader-following [51], [52]. The proposed method of obtaining versatile formation patterns allows a combination of both. This can be seen in the “BalancedLine” and “Balanced-Triangle” patterns (Fig. 5). The overall balanced circular formation can be obtained using a leaderless communication topology, whereas achievement of local geometric shapes can be implemented in a leader-based manner. The adopted hierarchical formation structure allows selection of appropriate communication topologies on diﬀerent layers. This paper also considers a practical issue that would occur, i.e., collisions among agents. This issue was tackled by adding another control component into each UAV’s control input. As can be seen from (18), this added collisionavoidance control component can also be expressed as a function of bearing angles. As mentioned earlier, a bearingangle-based control law has the potential of still achieving its control objective (formation or collision avoidance) during communication loss, by using local measurements from each UAV’s onboard sensors. 8. Conclusions This paper is to obtain more versatile concentric formations in cooperative target tracking where a ﬂeet of UAVs is commanded to circle above (and around) a moving ground target. On the basis of our previous results, versatile formation patterns are achieved with the help of three new features. The ﬁrst feature is a new formation pattern, the uniform spacing formation where either the relative separation distances or the separation angles can be regulated to a desired value. Diﬀerent from a balanced circular formation 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 are proposed, where one regulates the separation distances between two agents and the other regulates the separation angles in between. The second feature allows UAVs to circle on diﬀerent orbits. To achieve formation under this circumstance, formation controllers will resort to virtual agents representing the actual agents in need. The third feature is the usage of a (two-layer) hierarchical formation structure, which allows selection of formation patterns for diﬀerent layers. Combinations of these new features with our existing results yield more versatile concentric formation patterns with diﬀerent local geometric shapes, such as straight lines and triangles. Inter-vehicle collision avoidance is also addressed. Agents will be repelled to steer away from each other once they get too close. 12 All UAVs are assumed to have constant linear velocities. Control of each UAV is via its yaw rate. The design idea is to add three control components (three heading controllers) together to achieve the overall objective. Each control component has a goal. The proposed extensions to spreading agents on a portion of a circle, circling agents on orbits of diﬀerent radii, formation in local geometric shapes, and avoiding inter-vehicle collisions, provide more complete solution to cooperative target tracking in the concentric manner. This paper also raises several questions for future investigations. The implementation of the proposed schemes on physical robots and the extension of the developed techniques to 3D scenarios and cooperative tracking of multiple targets with obstacle avoidance capability [53], [54] will be of particular interest. Stability analyses in the presence of formation pattern switching and broken communication links are another research direction to look into. Also, investigations of the time delay factor for obtaining stability conditions as well as desirable performance with reasonable computation complexity [55]–[57] are needed. Finally, Artiﬁcial Intelligence (AI) techniques have recently been developed for robotic communication to enhance the communication capability of robotic networks for coordinated actions. Application of the AI and/or Neural Networks to the ﬁeld of robotic networks in the context of cooperative target tracking is a promising research area to pursue [58]–[60]. References [1] K. Szwaykowska, I.B. Schwartz, L.M.-T. Romero, C.R. Heckman, D. Mox, and M.A. Hsieh, Collective motion patterns of swarms with delay coupling: Theory and experiment, Physical Revie E, 93(3), 2016, 11. [2] R. Sepulchre, D. Paley, and N. Leonard, Stabilization of planar collection motion: All-to-all communication, IEEE Transactions on Automatic Control, 52(5), 2007, 811–824. [3] R. Sepulchre, D. Paley, and N. Leonard, Stabilization of planar collective motion with limited communication, IEEE Transactions on Automatic Control, 53(3), 2008, 706–719. [4] A. Jain, D. Ghose, and P. Menon, Stabilization of balanced circular motion about a desired center, International Conference on Advances in Control and Optimization of Dynamical Systems, Kanpur, India, 2014. [5] A. Jain and D. Ghose, Stabilization of collective motion in synchronized, balanced and splay phase arrangements on a desired circle, American Control Conference, Chicago, IL. [6] G. Mallik and A. Sinha, A study of balanced circular formation under deviated cyclic pursuit strategy, IFAC-PapersOnLine, 48(5), 2015, 41–46. [7] R. Zheng, Z. Lin, M. Fu, and D. Sun, Distributed control for uniform circumnavigation of ring-coupled unicycles, Automatica, 53, 2015, 23–29. [8] N. Kokolakis and N. Koussoulas, Coordinated standoﬀ tracking of a ground moving target and the phase separation problem, International Conference on Unmanned Aircraft Systems, Dallas, TX, 2018. [9] J. Guo, G. Yan, and Z. Lin, Cooperative control synthesis for moving-target-enclosing with changing topologies, International Conference on Robotics and Automation, Anchorage, AK, 2010. [10] Z. Zhou, H. Wang, and Z. Hu, Event-based time varying formation control for multiple quadrotor UAVs with Markovian switching topologies, Complexity, 2018, 1–15. [11] Y. Sun and L. Wang, Consensus of multi-agent systems in directed networks with nonuniform time-varying delays, IEEE Transactions on Automatic Control, 54(7), 2009, 1607–1613. [12] J. Marshall, M. Broucke, and B. Francis, Formations of vehicles in cyclic pursuit, IEEE Transactions on Automatic Control, 49(11), 2004, 1963–1974. [13] J. Marshall, M. Broucke, and B. Francis, Pursuit formations of unicycles, Automatica, 42, 2006, 3–12. [14] M. Pavone and E. Frazzoli, Decentralized policies for geometric pattern formation and path coverage, ASME Journal of Dynamic Systems, Measurement, and Control, 129(5), 2007, 633–643. [15] B. Wu, D. Wang, and E. Poh, Cyclic formation control for satellite formation using local relative measurements, Mechatronic Systems and Control, 40(1), 2012, 11–21. [16] J. Juang, On the formation patterns under generalized cyclic pursuit, IEEE Transactions on Automatic Control, 58(9), 2013, 2401–2405, 2013. [17] J. Ramirez, M. Pavone, E. Frazzoli, and D. Miller, Distributed control of spacecraft formations via cyclic pursuit: Theory and experiments, 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 tracking with onboard sensing, International Journal of Robotics Research, 34(13), 2015, 1660–1677. [19] P. Jimenez, B. Shirinzadeh, D. Oetomo, and A. Nicholson, Swarm aggregation and formation control for robots with limited perception, International Journal of Robotics and Automation, 26(4), 2011, 255–263. [20] P. Zhu and W. Ren, Multi-robot joint localization and target tracking with local sensing and communication, American Control Conference, Charlotte, NC, 2019. [21] L. Luo, N. Chakraborty, and K. Sycara, Provably-good distributed algorithm for constrained multi-robot task assignment for grouped tasks, IEEE Transactions on Robotics, 31(1), 2015, 19–30. [22] D. Panagou, M. Turpin, and V. Kumar, Decentralized goal assignment and safe trajectory generation in multi-robot networks via multiple Lyapunov functions, IEEE Transactions on Automatic Control, 65(8), 2020, 3365–3380. [23] J. Ni, X. Yang, J. Chen, and S. Yang, Dynamic bioinspired neural network for multi-robot formation control in unknown environments, International Journal of Robotics and Automation, 30(3), 2015, 256–266. [24] M. Khan and C. Silva, Autonomous and robust multi-robot cooperation using an artiﬁcial immune system, International Journal of Robotics and Automation, 27(1), 2012, 60–75. [25] X. Yu, L. Liu, and G. Feng, Coordinated control of multiple unicycles for escorting and patrolling task based on a cyclic pursuit strategy, American Control Conference, Boston, MA, 2016. [26] M. Zhang and H. Liu, Cooperative tracking a moving target using multiple ﬁxed-wing UAVs, Journal of Intelligent and Robotic Systems, 81(3-4), 2016, 505–529. [27] X. Yu and L. Liu, Cooperative control for moving-target circular formation of nonholonomic vehicles, IEEE Transactions on Automatic Control, 62(7), 2017, 3448–3454. [28] L. Brinon-Arranz, A. Seuret, and A. Pascoal, Target tracking via a circular formation of unicycles, IFAC World Congress, Toulouse, France, 2017. [29] A. Miao, Y. Wang, and R. Fierro, Cooperative circumnavigation of a moving target with multiple nonholonomic robots using backstepping design, Systems and Control Letters, 103, 2017, 58–65. [30] L. Ma and N. Hovakimyan, Vision-based cyclic pursuit for cooperative target tracking, Journal of Guidance, Control, and Dynamics, 36(2), 2013, 617–622. [31] N. Moshtagh, N. Michael, A. Jadbabaie, and K. Daniilidis, Vision-based, distributed control laws for motion coordination of 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, Triangular formation control using range measurements: An application to marine robotic vehicles, IFAC Proceedings, 45(5), 2012, 112–117. 13 [34] Z. He and J. Xu, Moving target tracking by UAVs in an urban area, 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 in balanced circular formation: Multiple UAVs tracking a ground vehicle, American Control Conference, Washington, DC, USA, 2013, 5386–5391. [36] L. Ma, Cooperative target tracking with time-varying formation radius, European Control Conference, Linz, Austria, 2015. [37] L. Ma, Cooperative target tracking in balanced circular formation with time-varying radius, International Journal of Robotics and 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 guaranteed performance bounds, Journal of Guidance, Control, and Dynamics, 3, 2010, 33.
[39] V. Cichella, I. Kaminer, V. Dobrokhodov, and N. Hovakimyan, Coordinated vision-based tracking for multiple UAVs, American Control Conference, Hamburg, Germany, 2015.
[41]–[44]. The simplest hierarchical scheme, the two-layer hierarchical structure, can be described as follows. A collection of n agents is divided into n2 subgroups, each containing n1 agents (n1 × n2 = n). The local control strategy is chosen such that the agents within each subgroup can be commanded to achieve certain formation pattern [41]. In [44], a two-layer hybrid pursuit system was described, where cyclic pursuit strategy was considered at the higher layer (the ﬁrst layer) and chain-like communication topology was used at the lower layer (the second layer). The concept of hierarchy is now applied to cooperative target tracking. The idea of hierarchy allows different subgroups to select diﬀerent formation laws that are already known to be stable. Similar to [44], a twolayer hierarchical formation structure is used. The ﬁrst layer can be set to achieve either the balanced circular or the uniform spacing formation. The second layer can be set to achieve the uniform spacing formation by specifying either a desired separation distance or angle. Two examples are given below to demonstrate how versatile patterns are achieved by determining the 2D range references of the agents (same or diﬀerent) and the formation pattern on each layer (balanced circular or uniform spacing). The ﬁrst example achieves concentric formations with local geometric shapes in straight lines: (1) The agents {1, 4, 7}, {2, 5, 8}, and {3, 6, 9} are assigned to be on the inner, middle, and outer orbits, respectively, by specifying their 2D range references to be {15, 18, 25} (m). Correspondingly, the UAVs’ linear velocities are set to be {30, 36, 50} (m/s), satisfying the relationship in (10). (2) On the ﬁrst layer, the balanced circular formation is used for agents {1, 4, 7}, which lie on the inner circle. On the second layer, a uniform spacing formation is used inside each subgroup. There are three subgroups: {1, 2, 3}, {4, 5, 6}, and {7, 8, 9}. The agent that is involved in formation on the ﬁrst layer works as “leader” of its subgroup. As the intended geometric shape is in straight line and the radial diﬀerences between each two adjacent orbits can be diﬀerent, it is more convenient to use (7) with β0 = π/2. Actual agents are used when obtaining bearing angles βi(i+1)(t). The formation pattern is shown in Fig. 5(a). 7 (3) Simply changing the formation pattern on the ﬁrst layer from the balanced circular to the uniform spacing pattern, an overall uniform spacing formation is achieved with local geometric shapes in straight lines. The formation pattern is shown in Fig. 5(b). The second example achieves concentric formations with local geometric shapes in triangles: (1) The agents {1, 3, 4, 6, 7, 9} are assigned to circle on the inner orbit. All other agents circle on the outer orbit. 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 circular formation on the ﬁrst layer. On the second layer, a uniform spacing formation is used. To achieve local shapes in triangles, it is convenient to use (16), which regulates distances directly. Virtual agents are used. The achieved formation pattern is shown in Fig. 5(c). (3) Simply changing the formation pattern on the ﬁrst layer from the balanced circular to the uniform spacing pattern, an overall uniform spacing formation is obtained with local geometric shapes in triangles. The formation pattern is shown in Fig. 5(d). To clarify how subgroups are deﬁned, how the leader of each subgroup is chosen, and if the robots know the size of their subgroups a priori, we use Fig. 5(a) as an example. For the subgroup consisting agents {1, 2, 3}, members in this subgroup are deﬁned by specifying agent 1 not to “seek” anyone else; agent 2 “seeking” agent 1; and agent 3 “seeking” agent 2. In each subgroup, all members do 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 only know which one to “seek”. In the simulation examples, the knowledge of which agent to “seek” is assigned. In reality, these knowledges can be perceived by the agents so that formation can still be maintained with agents joining or leaving the group. At this point, we would like to highlight diﬀerences between the control laws of previous work [35], [36] and the new work in this paper, i.e., formation controllers (6), (7), (16), and (17), as well as discussing some interesting aspects of the new control laws. Three formation controllers were reported in [35], [36] each achieving a balanced circular formation under one of the following three communication topologies: (1) all-to-all, (2) ring, and (3) cyclic pursuit. Using the balanced circular formation, all agents, which circle on the same orbit around the target, spread evenly around a full circle, whose centre resides in the moving target. These previous works are focused on studying the feasibility of achieving formations for target tracking. Having successfully obtained cooperative tracking in the balanced circular formation, one would naturally wonder what other formation patterns can be obtained that also ﬁt into the tracking scenario. Instead of spreading all agents over one full circle, spreading them over a portion of a circle (i.e., an arc) is one natural variation. Two uniform spacing formation controllers (6) and (7) are presented 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 relative bearing angle(s) βij(t) alone is proposed in this paper. Controller (7) regulates the separation angles between a pair of agents, instead of the separation distances between them as in (6). The diﬀerence between regulating separation distances and separation angles is not obvious when UAVs circle on the same orbit because a constant separation distance is equivalent to a constant separation angle under this circumstance. However, when agents circle on diﬀerent orbits (circles of diﬀerent radii around the target), the diﬀerence is clear, as to be demonstrated in Figs. 6 and 7. By properly specifying the desired 2D ranges and setting the UAV’s linear velocities accordingly, UAVs can now be controlled to circle on diﬀerent orbits. With this feature in place, uniform spacing formations are achieved by using the virtual agent 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 versatile concentric formation patterns, such as patterns with local geometric shapes. This is made possible utilizing existing works (tracking [38], balanced circular formation [35], [36]) and the new results presented in this paper (uniform spacing formations that regulate either separation distances or angles for agents circling on the same or diﬀerent orbits, and hierarchy). Results presented in this paper provide more complete solution to cooperative target tracking in concentric formations, by designing the control input of each UAV as a sum of several individual control components. One practical issue of avoiding inter-vehicle collision is to be addressed in Section 5. The combination of Attraction (another way of saying tracking), Alignment (formation), and Avoidance (collision avoidance) provides a solid framework to achieve formations in the context of target tracking. Another advantage of having versatile formation patterns is that it can possibly allow agents to acquire information of each other using a mixture of information exchange (over communication channels, for agents that are far away) and onboard sensing/perception (for agents that are close to each other). Refer to the two patterns in Fig. 5(a) and (c), where overall balanced circular formations have been achieved with local geometric shapes of either straight lines or triangles. The “leaders” of the three subgroups, can exchange information over the communication channels because they are relatively far away and might not be able to “see” each other. The other two members of each subgroup can possibly use onboard sensing to obtain the information; they need to achieve local formations (because they are close and can perceive each other). 5. Inter-vehicle Collision Avoidance When developing the uniform spacing formation, collision between UAVs is more likely to occur than other patterns where agents are far away from each other. A strategy 8 Figure 6. Cooperative target tracking using formation controller (17): (a) 2D trajectories; (b) 2D range; (c) formation in the end; (d) separation distance; (e) separation angle; and (f) formation control uic(t). Target undergoes a linear motion. Agents circle on two diﬀerent orbits. of preventing the inter-vehicle collision is thus much in demand. Among methods that prevent collisions, one way is to apply a force that repels the agents once they get closer. This force should also be strong enough to defend other forces pushing agents to a collision [45]–[49]. Let dmin denote the minimal distance allowed among agents. The following collision avoidance control component helps to repel agents to steer away from each other once they get too close [50]: uia(t) = −Kr j∈N (ri) dmin dij sin βij (18) where Kr is the controller gain. The repulsion term (18) adjusts each agent’s heading to the opposite direction of its neighbours in N(ri). Notice that N (ri) and Ni denote diﬀerent sets. The set Ni is the set of agents whose information can be obtained by the ith agent via communication, whereas the set N (ri) denotes the set of agents that are too close to agent i. With (18) in place, each agent’s control input becomes [50]: ˙ψi(t) = ui(t) = uit(t) + uic(t) + uia(t) (19) Totally three control components are added together to achieve the objective of simultaneous tracking, formation, and inter-vehicle collision avoidance. 6. Simulation Results The proposed control laws were simulated in Matlab to verify their performance of tracking, formation, and intervehicle collision avoidance. 6.1 Achieving Uniform Spacing Formations on Diﬀerent Orbits This example demonstrates achievement of uniform spacing formations when agents circle on diﬀerent orbits. To achieve formations under this circumstance, formation controllers (17) and (16) are used, which resort to virtual agents when needed, i.e., when the two agents are not on the same orbit. It is worth mentioning that the formation controller (16) aims at regulating the relative separation distances to a constant, while the controller (17) regulates the relative separation angles to a desired value. The 2D ranges are set to be ρd,i = {15, 15, 18, 18} (m), for i = 1, 2, 3, 4, respectively. Correspondingly, UAVs’ linear velocities are Vg,i = {30, 30, 36, 36} (m/s), satisfying the relationship in (10). To focus on the formation patterns, the target’s motion is assumed linear (the target moves on a straight line). We ﬁrst present simulation results applying the formation controller (17). The 2D trajectories of the agents and the target are plotted in Fig. 6(a). Details of the tracking are given in Fig. 6(b), where ρi(t) approach their prescribed values (that are diﬀerent). All agents’ positions at the end of the simulation are shown in Fig. 6(c), 9 Figure 7. Cooperative target tracking using formation controller (16): (a) 2D trajectories; (b) 2D range; (c) formation in the end; (d) separation distance; (e) separation angle; and (f) formation control uic(t). Target undergoes a linear motion. Agents circle on two diﬀerent orbits. where all agents scatter around the target on two diﬀerent orbits. Both the separation distances (Fig. 6(d)) and the separation angles (Fig. 6(e)) are plotted. When controller (17) is applied, we expect the separation angles to approach a constant. However, the separation distances may or may not approach to the same value because the distances between two agents also depend on the orbits that they lie on. For ds = 10 (m), the desired separation angle is either φs = 2 sin−1 (ds/30) ≈ 39◦ or φs = 2π − 2 sin−1 (ds/30) ≈ 321◦ . Because all separation angles converge to 39◦ (Fig. 6(e)), formation is obtained successfully. Figure 6(f) shows the formation component uic(t), which converges to zero upon formation. Results when applying the controller (16) are shown in Fig. 7. Figure 7(a) shows the general picture of cooperative target tracking. Figure 7(b) plots the 2D ranges between the agents and the target, demonstrating successful tracking with two diﬀerent range values. Figure 7(c) shows the formation achieved in the end, where agents circle on two diﬀerent orbits around the target. Figure 7(d) shows that the separation distances between two successive agents are now regulated to the speciﬁed value of ds = 10 (m) (same for all agents). On the contrary, the separation angles do not approach to one value. From Fig. 7(c), it can be seen that agents {1, 4} and agents {2, 3} circle on two diﬀerent orbits. The separation angles between these two pairs of consecutive agents should be smaller than that between agents {1, 2}, which circle on the same orbit. This is conﬁrmed in Fig. 7(e). Figure 7(f) shows that the formation control component uic(t) also vanishes to zero upon formation. Figures 6 and 7 together show the diﬀerence between the two formation controllers (16) and (17). The formation controller (16) regulates the relative separation distances between agents to a speciﬁed value ds, while the controller (17) regulates the separation angles to φs. 6.2 Achieving Concentric Formations with Local Geometric Shapes This section presents cooperative target tracking in concentric formations with local geometric shapes of straight lines and triangles, as those shown in Fig. 5. To focus on formation patterns, the target’s motion is assumed linear, i.e., the target moves on a straight line. Corresponding to the four patterns in Fig. 5, simulation results are presented in Fig. 8, with several snapshots showing how formation is achieved over time. Cooperative target tracking in these concentric formations are successfully obtained. 6.3 Achieving Inter-vehicle Collision Avoidance This example demonstrates cooperative target tracking with inter-vehicle collision avoidance. The two scenarios of without and with collision avoidance, i.e., before and after applying uia(t), are shown and compared in Fig. 9. We select the target’s velocity to be piecewise-constant, n = 5, ds = 10 (m), and dmin = 7 (m). The ﬁrst row of Fig. 9 is 10 Figure 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 is either straight line or triangle. for the scenario without the collision avoidance capability, i.e., before the control component uia(t) is applied. After uia(t) is applied, results are shown in the second row of Fig. 9. In each scenario, the 2D trajectories are plotted 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 component uia(t) (Fig. 9(b) versus (d)). For the second scenario with collision avoidance, the component uia(t) for each agent is shown in Fig. 9(e), where a saturation of |uia(t)| ≤ 3 has been used for all agents. Comparison between Fig. 9(b) and (d) shows that the control component uia(t) helps to keep the minimal distance to be greater than the allowed value. Otherwise, the minimal distance can be much smaller, as indicated in Fig. 9(b). The zigzag area in Fig. 9(d), corresponds to the circumstances when uia(t) takes eﬀect. Figure 9(e) shows that uia(t) only takes eﬀect when needed, i.e., when agents {3, 5} get too close to each other. 7. Comparison with Prior Studies Comparing with the distance-based coordination control law (6), our proposed bearing-angle-based coordination control law (7) has one potential advantage. Consider a 11 Figure 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 some agents becomes unavailable (e.g., due to communication loss or out of communication range). Instead of computing the relative bearing angles from the exchanged positions as in (12), the needed information of relative bearing angles can be estimated/obtained from a local vision system installed on each UAV. In other words, when the expected information from others is unavailable (either permanently or temporarily), the objective of achieving and maintaining formation could still be achieved by using local measurements and estimates. Regarding formation control, most existing results are either leaderless or leader-following [51], [52]. The proposed method of obtaining versatile formation patterns allows a combination of both. This can be seen in the “BalancedLine” and “Balanced-Triangle” patterns (Fig. 5). The overall balanced circular formation can be obtained using a leaderless communication topology, whereas achievement of local geometric shapes can be implemented in a leader-based manner. The adopted hierarchical formation structure allows selection of appropriate communication topologies on diﬀerent layers. This paper also considers a practical issue that would occur, i.e., collisions among agents. This issue was tackled by adding another control component into each UAV’s control input. As can be seen from (18), this added collisionavoidance control component can also be expressed as a function of bearing angles. As mentioned earlier, a bearingangle-based control law has the potential of still achieving its control objective (formation or collision avoidance) during communication loss, by using local measurements from each UAV’s onboard sensors. 8. Conclusions This paper is to obtain more versatile concentric formations in cooperative target tracking where a ﬂeet of UAVs is commanded to circle above (and around) a moving ground target. On the basis of our previous results, versatile formation patterns are achieved with the help of three new features. The ﬁrst feature is a new formation pattern, the uniform spacing formation where either the relative separation distances or the separation angles can be regulated to a desired value. Diﬀerent from a balanced circular formation 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 are proposed, where one regulates the separation distances between two agents and the other regulates the separation angles in between. The second feature allows UAVs to circle on diﬀerent orbits. To achieve formation under this circumstance, formation controllers will resort to virtual agents representing the actual agents in need. The third feature is the usage of a (two-layer) hierarchical formation structure, which allows selection of formation patterns for diﬀerent layers. Combinations of these new features with our existing results yield more versatile concentric formation patterns with diﬀerent local geometric shapes, such as straight lines and triangles. Inter-vehicle collision avoidance is also addressed. Agents will be repelled to steer away from each other once they get too close. 12 All UAVs are assumed to have constant linear velocities. Control of each UAV is via its yaw rate. The design idea is to add three control components (three heading controllers) together to achieve the overall objective. Each control component has a goal. The proposed extensions to spreading agents on a portion of a circle, circling agents on orbits of diﬀerent radii, formation in local geometric shapes, and avoiding inter-vehicle collisions, provide more complete solution to cooperative target tracking in the concentric manner. This paper also raises several questions for future investigations. The implementation of the proposed schemes on physical robots and the extension of the developed techniques to 3D scenarios and cooperative tracking of multiple targets with obstacle avoidance capability [53], [54] will be of particular interest. Stability analyses in the presence of formation pattern switching and broken communication links are another research direction to look into. Also, investigations of the time delay factor for obtaining stability conditions as well as desirable performance with reasonable computation complexity [55]–[57] are needed. Finally, Artiﬁcial Intelligence (AI) techniques have recently been developed for robotic communication to enhance the communication capability of robotic networks for coordinated actions. Application of the AI and/or Neural Networks to the ﬁeld of robotic networks in the context of cooperative target tracking is a promising research area to pursue [58]–[60]. References [1] K. Szwaykowska, I.B. Schwartz, L.M.-T. Romero, C.R. Heckman, D. Mox, and M.A. Hsieh, Collective motion patterns of swarms with delay coupling: Theory and experiment, Physical Revie E, 93(3), 2016, 11. [2] R. Sepulchre, D. Paley, and N. Leonard, Stabilization of planar collection motion: All-to-all communication, IEEE Transactions on Automatic Control, 52(5), 2007, 811–824. [3] R. Sepulchre, D. Paley, and N. Leonard, Stabilization of planar collective motion with limited communication, IEEE Transactions on Automatic Control, 53(3), 2008, 706–719. [4] A. Jain, D. Ghose, and P. Menon, Stabilization of balanced circular motion about a desired center, International Conference on Advances in Control and Optimization of Dynamical Systems, Kanpur, India, 2014. [5] A. Jain and D. Ghose, Stabilization of collective motion in synchronized, balanced and splay phase arrangements on a desired circle, American Control Conference, Chicago, IL. [6] G. Mallik and A. Sinha, A study of balanced circular formation under deviated cyclic pursuit strategy, IFAC-PapersOnLine, 48(5), 2015, 41–46. [7] R. Zheng, Z. Lin, M. Fu, and D. Sun, Distributed control for uniform circumnavigation of ring-coupled unicycles, Automatica, 53, 2015, 23–29. [8] N. Kokolakis and N. Koussoulas, Coordinated standoﬀ tracking of a ground moving target and the phase separation problem, International Conference on Unmanned Aircraft Systems, Dallas, TX, 2018. [9] J. Guo, G. Yan, and Z. Lin, Cooperative control synthesis for moving-target-enclosing with changing topologies, International Conference on Robotics and Automation, Anchorage, AK, 2010. [10] Z. Zhou, H. Wang, and Z. Hu, Event-based time varying formation control for multiple quadrotor UAVs with Markovian switching topologies, Complexity, 2018, 1–15. [11] Y. Sun and L. Wang, Consensus of multi-agent systems in directed networks with nonuniform time-varying delays, IEEE Transactions on Automatic Control, 54(7), 2009, 1607–1613. [12] J. Marshall, M. Broucke, and B. Francis, Formations of vehicles in cyclic pursuit, IEEE Transactions on Automatic Control, 49(11), 2004, 1963–1974. [13] J. Marshall, M. Broucke, and B. Francis, Pursuit formations of unicycles, Automatica, 42, 2006, 3–12. [14] M. Pavone and E. Frazzoli, Decentralized policies for geometric pattern formation and path coverage, ASME Journal of Dynamic Systems, Measurement, and Control, 129(5), 2007, 633–643. [15] B. Wu, D. Wang, and E. Poh, Cyclic formation control for satellite formation using local relative measurements, Mechatronic Systems and Control, 40(1), 2012, 11–21. [16] J. Juang, On the formation patterns under generalized cyclic pursuit, IEEE Transactions on Automatic Control, 58(9), 2013, 2401–2405, 2013. [17] J. Ramirez, M. Pavone, E. Frazzoli, and D. Miller, Distributed control of spacecraft formations via cyclic pursuit: Theory and experiments, 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 tracking with onboard sensing, International Journal of Robotics Research, 34(13), 2015, 1660–1677. [19] P. Jimenez, B. Shirinzadeh, D. Oetomo, and A. Nicholson, Swarm aggregation and formation control for robots with limited perception, International Journal of Robotics and Automation, 26(4), 2011, 255–263. [20] P. Zhu and W. Ren, Multi-robot joint localization and target tracking with local sensing and communication, American Control Conference, Charlotte, NC, 2019. [21] L. Luo, N. Chakraborty, and K. Sycara, Provably-good distributed algorithm for constrained multi-robot task assignment for grouped tasks, IEEE Transactions on Robotics, 31(1), 2015, 19–30. [22] D. Panagou, M. Turpin, and V. Kumar, Decentralized goal assignment and safe trajectory generation in multi-robot networks via multiple Lyapunov functions, IEEE Transactions on Automatic Control, 65(8), 2020, 3365–3380. [23] J. Ni, X. Yang, J. Chen, and S. Yang, Dynamic bioinspired neural network for multi-robot formation control in unknown environments, International Journal of Robotics and Automation, 30(3), 2015, 256–266. [24] M. Khan and C. Silva, Autonomous and robust multi-robot cooperation using an artiﬁcial immune system, International Journal of Robotics and Automation, 27(1), 2012, 60–75. [25] X. Yu, L. Liu, and G. Feng, Coordinated control of multiple unicycles for escorting and patrolling task based on a cyclic pursuit strategy, American Control Conference, Boston, MA, 2016. [26] M. Zhang and H. Liu, Cooperative tracking a moving target using multiple ﬁxed-wing UAVs, Journal of Intelligent and Robotic Systems, 81(3-4), 2016, 505–529. [27] X. Yu and L. Liu, Cooperative control for moving-target circular formation of nonholonomic vehicles, IEEE Transactions on Automatic Control, 62(7), 2017, 3448–3454. [28] L. Brinon-Arranz, A. Seuret, and A. Pascoal, Target tracking via a circular formation of unicycles, IFAC World Congress, Toulouse, France, 2017. [29] A. Miao, Y. Wang, and R. Fierro, Cooperative circumnavigation of a moving target with multiple nonholonomic robots using backstepping design, Systems and Control Letters, 103, 2017, 58–65. [30] L. Ma and N. Hovakimyan, Vision-based cyclic pursuit for cooperative target tracking, Journal of Guidance, Control, and Dynamics, 36(2), 2013, 617–622. [31] N. Moshtagh, N. Michael, A. Jadbabaie, and K. Daniilidis, Vision-based, distributed control laws for motion coordination of 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, Triangular formation control using range measurements: An application to marine robotic vehicles, IFAC Proceedings, 45(5), 2012, 112–117. 13 [34] Z. He and J. Xu, Moving target tracking by UAVs in an urban area, 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 in balanced circular formation: Multiple UAVs tracking a ground vehicle, American Control Conference, Washington, DC, USA, 2013, 5386–5391. [36] L. Ma, Cooperative target tracking with time-varying formation radius, European Control Conference, Linz, Austria, 2015. [37] L. Ma, Cooperative target tracking in balanced circular formation with time-varying radius, International Journal of Robotics and 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 guaranteed performance bounds, Journal of Guidance, Control, and Dynamics, 3, 2010, 33. [39] V. Cichella, I. Kaminer, V. Dobrokhodov, and N. Hovakimyan, Coordinated vision-based tracking for multiple UAVs, American Control Conference, Hamburg, Germany, 2015. [40] Q. Han, S. Sun, and H. Lang, Leader-follower formation control 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. 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[44]. The simplest hierarchical scheme, the two-layer hierarchical structure, can be described as follows. A collection of n agents is divided into n2 subgroups, each containing n1 agents (n1 × n2 = n). The local control strategy is chosen such that the agents within each subgroup can be commanded to achieve certain formation pattern [41]. In [44], a two-layer hybrid pursuit system was described, where cyclic pursuit strategy was considered at the higher layer (the ﬁrst layer) and chain-like communication topology was used at the lower layer (the second layer). The concept of hierarchy is now applied to cooperative target tracking. The idea of hierarchy allows different subgroups to select diﬀerent formation laws that are already known to be stable. Similar to [44], a twolayer hierarchical formation structure is used. The ﬁrst layer can be set to achieve either the balanced circular or the uniform spacing formation. The second layer can be set to achieve the uniform spacing formation by specifying either a desired separation distance or angle. Two examples are given below to demonstrate how versatile patterns are achieved by determining the 2D range references of the agents (same or diﬀerent) and the formation pattern on each layer (balanced circular or uniform spacing). The ﬁrst example achieves concentric formations with local geometric shapes in straight lines: (1) The agents {1, 4, 7}, {2, 5, 8}, and {3, 6, 9} are assigned to be on the inner, middle, and outer orbits, respectively, by specifying their 2D range references to be {15, 18, 25} (m). Correspondingly, the UAVs’ linear velocities are set to be {30, 36, 50} (m/s), satisfying the relationship in (10). (2) On the ﬁrst layer, the balanced circular formation is used for agents {1, 4, 7}, which lie on the inner circle. On the second layer, a uniform spacing formation is used inside each subgroup. There are three subgroups: {1, 2, 3}, {4, 5, 6}, and {7, 8, 9}. The agent that is involved in formation on the ﬁrst layer works as “leader” of its subgroup. As the intended geometric shape is in straight line and the radial diﬀerences between each two adjacent orbits can be diﬀerent, it is more convenient to use (7) with β0 = π/2. Actual agents are used when obtaining bearing angles βi(i+1)(t). The formation pattern is shown in Fig. 5(a). 7 (3) Simply changing the formation pattern on the ﬁrst layer from the balanced circular to the uniform spacing pattern, an overall uniform spacing formation is achieved with local geometric shapes in straight lines. The formation pattern is shown in Fig. 5(b). The second example achieves concentric formations with local geometric shapes in triangles: (1) The agents {1, 3, 4, 6, 7, 9} are assigned to circle on the inner orbit. All other agents circle on the outer orbit. 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 circular formation on the ﬁrst layer. On the second layer, a uniform spacing formation is used. To achieve local shapes in triangles, it is convenient to use (16), which regulates distances directly. Virtual agents are used. The achieved formation pattern is shown in Fig. 5(c). (3) Simply changing the formation pattern on the ﬁrst layer from the balanced circular to the uniform spacing pattern, an overall uniform spacing formation is obtained with local geometric shapes in triangles. The formation pattern is shown in Fig. 5(d). To clarify how subgroups are deﬁned, how the leader of each subgroup is chosen, and if the robots know the size of their subgroups a priori, we use Fig. 5(a) as an example. For the subgroup consisting agents {1, 2, 3}, members in this subgroup are deﬁned by specifying agent 1 not to “seek” anyone else; agent 2 “seeking” agent 1; and agent 3 “seeking” agent 2. In each subgroup, all members do 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 only know which one to “seek”. In the simulation examples, the knowledge of which agent to “seek” is assigned. In reality, these knowledges can be perceived by the agents so that formation can still be maintained with agents joining or leaving the group. At this point, we would like to highlight diﬀerences between the control laws of previous work [35], [36] and the new work in this paper, i.e., formation controllers (6), (7), (16), and (17), as well as discussing some interesting aspects of the new control laws. Three formation controllers were reported in [35], [36] each achieving a balanced circular formation under one of the following three communication topologies: (1) all-to-all, (2) ring, and (3) cyclic pursuit. Using the balanced circular formation, all agents, which circle on the same orbit around the target, spread evenly around a full circle, whose centre resides in the moving target. These previous works are focused on studying the feasibility of achieving formations for target tracking. Having successfully obtained cooperative tracking in the balanced circular formation, one would naturally wonder what other formation patterns can be obtained that also ﬁt into the tracking scenario. Instead of spreading all agents over one full circle, spreading them over a portion of a circle (i.e., an arc) is one natural variation. Two uniform spacing formation controllers (6) and (7) are presented 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 relative bearing angle(s) βij(t) alone is proposed in this paper. Controller (7) regulates the separation angles between a pair of agents, instead of the separation distances between them as in (6). The diﬀerence between regulating separation distances and separation angles is not obvious when UAVs circle on the same orbit because a constant separation distance is equivalent to a constant separation angle under this circumstance. However, when agents circle on diﬀerent orbits (circles of diﬀerent radii around the target), the diﬀerence is clear, as to be demonstrated in Figs. 6 and 7. By properly specifying the desired 2D ranges and setting the UAV’s linear velocities accordingly, UAVs can now be controlled to circle on diﬀerent orbits. With this feature in place, uniform spacing formations are achieved by using the virtual agent 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 versatile concentric formation patterns, such as patterns with local geometric shapes. This is made possible utilizing existing works (tracking [38], balanced circular formation [35], [36]) and the new results presented in this paper (uniform spacing formations that regulate either separation distances or angles for agents circling on the same or diﬀerent orbits, and hierarchy). Results presented in this paper provide more complete solution to cooperative target tracking in concentric formations, by designing the control input of each UAV as a sum of several individual control components. One practical issue of avoiding inter-vehicle collision is to be addressed in Section 5. The combination of Attraction (another way of saying tracking), Alignment (formation), and Avoidance (collision avoidance) provides a solid framework to achieve formations in the context of target tracking. Another advantage of having versatile formation patterns is that it can possibly allow agents to acquire information of each other using a mixture of information exchange (over communication channels, for agents that are far away) and onboard sensing/perception (for agents that are close to each other). Refer to the two patterns in Fig. 5(a) and (c), where overall balanced circular formations have been achieved with local geometric shapes of either straight lines or triangles. The “leaders” of the three subgroups, can exchange information over the communication channels because they are relatively far away and might not be able to “see” each other. The other two members of each subgroup can possibly use onboard sensing to obtain the information; they need to achieve local formations (because they are close and can perceive each other). 5. Inter-vehicle Collision Avoidance When developing the uniform spacing formation, collision between UAVs is more likely to occur than other patterns where agents are far away from each other. A strategy 8 Figure 6. Cooperative target tracking using formation controller (17): (a) 2D trajectories; (b) 2D range; (c) formation in the end; (d) separation distance; (e) separation angle; and (f) formation control uic(t). Target undergoes a linear motion. Agents circle on two diﬀerent orbits. of preventing the inter-vehicle collision is thus much in demand. Among methods that prevent collisions, one way is to apply a force that repels the agents once they get closer. This force should also be strong enough to defend other forces pushing agents to a collision
[45]–[49]. Let dmin denote the minimal distance allowed among agents. The following collision avoidance control component helps to repel agents to steer away from each other once they get too close [50]: uia(t) = −Kr j∈N (ri) dmin dij sin βij (18) where Kr is the controller gain. The repulsion term (18) adjusts each agent’s heading to the opposite direction of its neighbours in N(ri). Notice that N (ri) and Ni denote diﬀerent sets. The set Ni is the set of agents whose information can be obtained by the ith agent via communication, whereas the set N (ri) denotes the set of agents that are too close to agent i. With (18) in place, each agent’s control input becomes [50]: ˙ψi(t) = ui(t) = uit(t) + uic(t) + uia(t) (19) Totally three control components are added together to achieve the objective of simultaneous tracking, formation, and inter-vehicle collision avoidance. 6. Simulation Results The proposed control laws were simulated in Matlab to verify their performance of tracking, formation, and intervehicle collision avoidance. 6.1 Achieving Uniform Spacing Formations on Diﬀerent Orbits This example demonstrates achievement of uniform spacing formations when agents circle on diﬀerent orbits. To achieve formations under this circumstance, formation controllers (17) and (16) are used, which resort to virtual agents when needed, i.e., when the two agents are not on the same orbit. It is worth mentioning that the formation controller (16) aims at regulating the relative separation distances to a constant, while the controller (17) regulates the relative separation angles to a desired value. The 2D ranges are set to be ρd,i = {15, 15, 18, 18} (m), for i = 1, 2, 3, 4, respectively. Correspondingly, UAVs’ linear velocities are Vg,i = {30, 30, 36, 36} (m/s), satisfying the relationship in (10). To focus on the formation patterns, the target’s motion is assumed linear (the target moves on a straight line). We ﬁrst present simulation results applying the formation controller (17). The 2D trajectories of the agents and the target are plotted in Fig. 6(a). Details of the tracking are given in Fig. 6(b), where ρi(t) approach their prescribed values (that are diﬀerent). All agents’ positions at the end of the simulation are shown in Fig. 6(c), 9 Figure 7. Cooperative target tracking using formation controller (16): (a) 2D trajectories; (b) 2D range; (c) formation in the end; (d) separation distance; (e) separation angle; and (f) formation control uic(t). Target undergoes a linear motion. Agents circle on two diﬀerent orbits. where all agents scatter around the target on two diﬀerent orbits. Both the separation distances (Fig. 6(d)) and the separation angles (Fig. 6(e)) are plotted. When controller (17) is applied, we expect the separation angles to approach a constant. However, the separation distances may or may not approach to the same value because the distances between two agents also depend on the orbits that they lie on. For ds = 10 (m), the desired separation angle is either φs = 2 sin−1 (ds/30) ≈ 39◦ or φs = 2π − 2 sin−1 (ds/30) ≈ 321◦ . Because all separation angles converge to 39◦ (Fig. 6(e)), formation is obtained successfully. Figure 6(f) shows the formation component uic(t), which converges to zero upon formation. Results when applying the controller (16) are shown in Fig. 7. Figure 7(a) shows the general picture of cooperative target tracking. Figure 7(b) plots the 2D ranges between the agents and the target, demonstrating successful tracking with two diﬀerent range values. Figure 7(c) shows the formation achieved in the end, where agents circle on two diﬀerent orbits around the target. Figure 7(d) shows that the separation distances between two successive agents are now regulated to the speciﬁed value of ds = 10 (m) (same for all agents). On the contrary, the separation angles do not approach to one value. From Fig. 7(c), it can be seen that agents {1, 4} and agents {2, 3} circle on two diﬀerent orbits. The separation angles between these two pairs of consecutive agents should be smaller than that between agents {1, 2}, which circle on the same orbit. This is conﬁrmed in Fig. 7(e). Figure 7(f) shows that the formation control component uic(t) also vanishes to zero upon formation. Figures 6 and 7 together show the diﬀerence between the two formation controllers (16) and (17). The formation controller (16) regulates the relative separation distances between agents to a speciﬁed value ds, while the controller (17) regulates the separation angles to φs. 6.2 Achieving Concentric Formations with Local Geometric Shapes This section presents cooperative target tracking in concentric formations with local geometric shapes of straight lines and triangles, as those shown in Fig. 5. To focus on formation patterns, the target’s motion is assumed linear, i.e., the target moves on a straight line. Corresponding to the four patterns in Fig. 5, simulation results are presented in Fig. 8, with several snapshots showing how formation is achieved over time. Cooperative target tracking in these concentric formations are successfully obtained. 6.3 Achieving Inter-vehicle Collision Avoidance This example demonstrates cooperative target tracking with inter-vehicle collision avoidance. The two scenarios of without and with collision avoidance, i.e., before and after applying uia(t), are shown and compared in Fig. 9. We select the target’s velocity to be piecewise-constant, n = 5, ds = 10 (m), and dmin = 7 (m). The ﬁrst row of Fig. 9 is 10 Figure 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 is either straight line or triangle. for the scenario without the collision avoidance capability, i.e., before the control component uia(t) is applied. After uia(t) is applied, results are shown in the second row of Fig. 9. In each scenario, the 2D trajectories are plotted 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 component uia(t) (Fig. 9(b) versus (d)). For the second scenario with collision avoidance, the component uia(t) for each agent is shown in Fig. 9(e), where a saturation of |uia(t)| ≤ 3 has been used for all agents. Comparison between Fig. 9(b) and (d) shows that the control component uia(t) helps to keep the minimal distance to be greater than the allowed value. Otherwise, the minimal distance can be much smaller, as indicated in Fig. 9(b). The zigzag area in Fig. 9(d), corresponds to the circumstances when uia(t) takes eﬀect. Figure 9(e) shows that uia(t) only takes eﬀect when needed, i.e., when agents {3, 5} get too close to each other. 7. Comparison with Prior Studies Comparing with the distance-based coordination control law (6), our proposed bearing-angle-based coordination control law (7) has one potential advantage. Consider a 11 Figure 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 some agents becomes unavailable (e.g., due to communication loss or out of communication range). Instead of computing the relative bearing angles from the exchanged positions as in (12), the needed information of relative bearing angles can be estimated/obtained from a local vision system installed on each UAV. In other words, when the expected information from others is unavailable (either permanently or temporarily), the objective of achieving and maintaining formation could still be achieved by using local measurements and estimates. Regarding formation control, most existing results are either leaderless or leader-following [51], [52]. The proposed method of obtaining versatile formation patterns allows a combination of both. This can be seen in the “BalancedLine” and “Balanced-Triangle” patterns (Fig. 5). The overall balanced circular formation can be obtained using a leaderless communication topology, whereas achievement of local geometric shapes can be implemented in a leader-based manner. The adopted hierarchical formation structure allows selection of appropriate communication topologies on diﬀerent layers. This paper also considers a practical issue that would occur, i.e., collisions among agents. This issue was tackled by adding another control component into each UAV’s control input. As can be seen from (18), this added collisionavoidance control component can also be expressed as a function of bearing angles. As mentioned earlier, a bearingangle-based control law has the potential of still achieving its control objective (formation or collision avoidance) during communication loss, by using local measurements from each UAV’s onboard sensors. 8. Conclusions This paper is to obtain more versatile concentric formations in cooperative target tracking where a ﬂeet of UAVs is commanded to circle above (and around) a moving ground target. On the basis of our previous results, versatile formation patterns are achieved with the help of three new features. The ﬁrst feature is a new formation pattern, the uniform spacing formation where either the relative separation distances or the separation angles can be regulated to a desired value. Diﬀerent from a balanced circular formation 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 are proposed, where one regulates the separation distances between two agents and the other regulates the separation angles in between. The second feature allows UAVs to circle on diﬀerent orbits. To achieve formation under this circumstance, formation controllers will resort to virtual agents representing the actual agents in need. The third feature is the usage of a (two-layer) hierarchical formation structure, which allows selection of formation patterns for diﬀerent layers. Combinations of these new features with our existing results yield more versatile concentric formation patterns with diﬀerent local geometric shapes, such as straight lines and triangles. Inter-vehicle collision avoidance is also addressed. Agents will be repelled to steer away from each other once they get too close. 12 All UAVs are assumed to have constant linear velocities. Control of each UAV is via its yaw rate. The design idea is to add three control components (three heading controllers) together to achieve the overall objective. Each control component has a goal. The proposed extensions to spreading agents on a portion of a circle, circling agents on orbits of diﬀerent radii, formation in local geometric shapes, and avoiding inter-vehicle collisions, provide more complete solution to cooperative target tracking in the concentric manner. This paper also raises several questions for future investigations. The implementation of the proposed schemes on physical robots and the extension of the developed techniques to 3D scenarios and cooperative tracking of multiple targets with obstacle avoidance capability [53], [54] will be of particular interest. Stability analyses in the presence of formation pattern switching and broken communication links are another research direction to look into. Also, investigations of the time delay factor for obtaining stability conditions as well as desirable performance with reasonable computation complexity [55]–[57] are needed. 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Feng, Coordinated control of multiple unicycles for escorting and patrolling task based on a cyclic pursuit strategy, American Control Conference, Boston, MA, 2016. [26] M. Zhang and H. Liu, Cooperative tracking a moving target using multiple ﬁxed-wing UAVs, Journal of Intelligent and Robotic Systems, 81(3-4), 2016, 505–529. [27] X. Yu and L. Liu, Cooperative control for moving-target circular formation of nonholonomic vehicles, IEEE Transactions on Automatic Control, 62(7), 2017, 3448–3454. [28] L. Brinon-Arranz, A. Seuret, and A. Pascoal, Target tracking via a circular formation of unicycles, IFAC World Congress, Toulouse, France, 2017. [29] A. Miao, Y. Wang, and R. Fierro, Cooperative circumnavigation of a moving target with multiple nonholonomic robots using backstepping design, Systems and Control Letters, 103, 2017, 58–65. [30] L. Ma and N. Hovakimyan, Vision-based cyclic pursuit for cooperative target tracking, Journal of Guidance, Control, and Dynamics, 36(2), 2013, 617–622. [31] N. Moshtagh, N. Michael, A. Jadbabaie, and K. Daniilidis, Vision-based, distributed control laws for motion coordination of 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, Triangular formation control using range measurements: An application to marine robotic vehicles, IFAC Proceedings, 45(5), 2012, 112–117. 13 [34] Z. He and J. Xu, Moving target tracking by UAVs in an urban area, 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 in balanced circular formation: Multiple UAVs tracking a ground vehicle, American Control Conference, Washington, DC, USA, 2013, 5386–5391. [36] L. Ma, Cooperative target tracking with time-varying formation radius, European Control Conference, Linz, Austria, 2015. [37] L. Ma, Cooperative target tracking in balanced circular formation with time-varying radius, International Journal of Robotics and 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 guaranteed performance bounds, Journal of Guidance, Control, and Dynamics, 3, 2010, 33. [39] V. Cichella, I. Kaminer, V. Dobrokhodov, and N. Hovakimyan, Coordinated vision-based tracking for multiple UAVs, American Control Conference, Hamburg, Germany, 2015. [40] Q. Han, S. Sun, and H. Lang, Leader-follower formation control 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 cyclic pursuit 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 Conference on Decision and Control, New Orleans, LA, 2007. [43] D. Mukherjee and D. Ghose, Generalized hierarchical cyclic pursuit, Automatica, 71, 2016, 318–323. [44] W. Ding, G. Yan, and Z. Lin, Formations on two-layer pursuit systems, IEEE International Conference on Robotics and Automation, Japan, 2009. [45] A. Satici, H. Poonawala, H. Eckert, and M. Spong, Connectivity preserving formation control with collision avoidance for nonholonomic wheeled mobile robots, IEEE International Conference on Intelligent Robots and Systems, Tokyo, Japan, 2013.
[46] J. Santiaguillo-Salinas and E. Arando-bricaire, Containment problem with time-varying formation and collision avoidance for multiagent systems, International Journal of Advanced Robotic Systems, 13, 2017, 1–13.
[47] J. Flores-Resendiz, E. Aranda-Bricaire, J. González-Sierra, and J. Santiaguillo-Salinas, Finite-time formation control without collisions for multiagent systems with communication graphs composed of cyclic paths, Mathematical Problems in Engineering, 1, 2015, 1–17.
[49]. Let dmin denote the minimal distance allowed among agents. The following collision avoidance control component helps to repel agents to steer away from each other once they get too close
[50]: uia(t) = −Kr j∈N (ri) dmin dij sin βij (18) where Kr is the controller gain. The repulsion term (18) adjusts each agent’s heading to the opposite direction of its neighbours in N(ri). Notice that N (ri) and Ni denote diﬀerent sets. The set Ni is the set of agents whose information can be obtained by the ith agent via communication, whereas the set N (ri) denotes the set of agents that are too close to agent i. With (18) in place, each agent’s control input becomes [50]: ˙ψi(t) = ui(t) = uit(t) + uic(t) + uia(t) (19) Totally three control components are added together to achieve the objective of simultaneous tracking, formation, and inter-vehicle collision avoidance. 6. Simulation Results The proposed control laws were simulated in Matlab to verify their performance of tracking, formation, and intervehicle collision avoidance. 6.1 Achieving Uniform Spacing Formations on Diﬀerent Orbits This example demonstrates achievement of uniform spacing formations when agents circle on diﬀerent orbits. To achieve formations under this circumstance, formation controllers (17) and (16) are used, which resort to virtual agents when needed, i.e., when the two agents are not on the same orbit. It is worth mentioning that the formation controller (16) aims at regulating the relative separation distances to a constant, while the controller (17) regulates the relative separation angles to a desired value. The 2D ranges are set to be ρd,i = {15, 15, 18, 18} (m), for i = 1, 2, 3, 4, respectively. Correspondingly, UAVs’ linear velocities are Vg,i = {30, 30, 36, 36} (m/s), satisfying the relationship in (10). To focus on the formation patterns, the target’s motion is assumed linear (the target moves on a straight line). We ﬁrst present simulation results applying the formation controller (17). The 2D trajectories of the agents and the target are plotted in Fig. 6(a). Details of the tracking are given in Fig. 6(b), where ρi(t) approach their prescribed values (that are diﬀerent). All agents’ positions at the end of the simulation are shown in Fig. 6(c), 9 Figure 7. Cooperative target tracking using formation controller (16): (a) 2D trajectories; (b) 2D range; (c) formation in the end; (d) separation distance; (e) separation angle; and (f) formation control uic(t). Target undergoes a linear motion. Agents circle on two diﬀerent orbits. where all agents scatter around the target on two diﬀerent orbits. Both the separation distances (Fig. 6(d)) and the separation angles (Fig. 6(e)) are plotted. When controller (17) is applied, we expect the separation angles to approach a constant. However, the separation distances may or may not approach to the same value because the distances between two agents also depend on the orbits that they lie on. For ds = 10 (m), the desired separation angle is either φs = 2 sin−1 (ds/30) ≈ 39◦ or φs = 2π − 2 sin−1 (ds/30) ≈ 321◦ . Because all separation angles converge to 39◦ (Fig. 6(e)), formation is obtained successfully. Figure 6(f) shows the formation component uic(t), which converges to zero upon formation. Results when applying the controller (16) are shown in Fig. 7. Figure 7(a) shows the general picture of cooperative target tracking. Figure 7(b) plots the 2D ranges between the agents and the target, demonstrating successful tracking with two diﬀerent range values. Figure 7(c) shows the formation achieved in the end, where agents circle on two diﬀerent orbits around the target. Figure 7(d) shows that the separation distances between two successive agents are now regulated to the speciﬁed value of ds = 10 (m) (same for all agents). On the contrary, the separation angles do not approach to one value. From Fig. 7(c), it can be seen that agents {1, 4} and agents {2, 3} circle on two diﬀerent orbits. The separation angles between these two pairs of consecutive agents should be smaller than that between agents {1, 2}, which circle on the same orbit. This is conﬁrmed in Fig. 7(e). Figure 7(f) shows that the formation control component uic(t) also vanishes to zero upon formation. Figures 6 and 7 together show the diﬀerence between the two formation controllers (16) and (17). The formation controller (16) regulates the relative separation distances between agents to a speciﬁed value ds, while the controller (17) regulates the separation angles to φs. 6.2 Achieving Concentric Formations with Local Geometric Shapes This section presents cooperative target tracking in concentric formations with local geometric shapes of straight lines and triangles, as those shown in Fig. 5. To focus on formation patterns, the target’s motion is assumed linear, i.e., the target moves on a straight line. Corresponding to the four patterns in Fig. 5, simulation results are presented in Fig. 8, with several snapshots showing how formation is achieved over time. Cooperative target tracking in these concentric formations are successfully obtained. 6.3 Achieving Inter-vehicle Collision Avoidance This example demonstrates cooperative target tracking with inter-vehicle collision avoidance. The two scenarios of without and with collision avoidance, i.e., before and after applying uia(t), are shown and compared in Fig. 9. We select the target’s velocity to be piecewise-constant, n = 5, ds = 10 (m), and dmin = 7 (m). The ﬁrst row of Fig. 9 is 10 Figure 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 is either straight line or triangle. for the scenario without the collision avoidance capability, i.e., before the control component uia(t) is applied. After uia(t) is applied, results are shown in the second row of Fig. 9. In each scenario, the 2D trajectories are plotted 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 component uia(t) (Fig. 9(b) versus (d)). For the second scenario with collision avoidance, the component uia(t) for each agent is shown in Fig. 9(e), where a saturation of |uia(t)| ≤ 3 has been used for all agents. Comparison between Fig. 9(b) and (d) shows that the control component uia(t) helps to keep the minimal distance to be greater than the allowed value. Otherwise, the minimal distance can be much smaller, as indicated in Fig. 9(b). The zigzag area in Fig. 9(d), corresponds to the circumstances when uia(t) takes eﬀect. Figure 9(e) shows that uia(t) only takes eﬀect when needed, i.e., when agents {3, 5} get too close to each other. 7. Comparison with Prior Studies Comparing with the distance-based coordination control law (6), our proposed bearing-angle-based coordination control law (7) has one potential advantage. Consider a 11 Figure 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 some agents becomes unavailable (e.g., due to communication loss or out of communication range). Instead of computing the relative bearing angles from the exchanged positions as in (12), the needed information of relative bearing angles can be estimated/obtained from a local vision system installed on each UAV. In other words, when the expected information from others is unavailable (either permanently or temporarily), the objective of achieving and maintaining formation could still be achieved by using local measurements and estimates. Regarding formation control, most existing results are either leaderless or leader-following
[51],
[52]. The proposed method of obtaining versatile formation patterns allows a combination of both. This can be seen in the “BalancedLine” and “Balanced-Triangle” patterns (Fig. 5). The overall balanced circular formation can be obtained using a leaderless communication topology, whereas achievement of local geometric shapes can be implemented in a leader-based manner. The adopted hierarchical formation structure allows selection of appropriate communication topologies on diﬀerent layers. This paper also considers a practical issue that would occur, i.e., collisions among agents. This issue was tackled by adding another control component into each UAV’s control input. As can be seen from (18), this added collisionavoidance control component can also be expressed as a function of bearing angles. As mentioned earlier, a bearingangle-based control law has the potential of still achieving its control objective (formation or collision avoidance) during communication loss, by using local measurements from each UAV’s onboard sensors. 8. Conclusions This paper is to obtain more versatile concentric formations in cooperative target tracking where a ﬂeet of UAVs is commanded to circle above (and around) a moving ground target. On the basis of our previous results, versatile formation patterns are achieved with the help of three new features. The ﬁrst feature is a new formation pattern, the uniform spacing formation where either the relative separation distances or the separation angles can be regulated to a desired value. Diﬀerent from a balanced circular formation 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 are proposed, where one regulates the separation distances between two agents and the other regulates the separation angles in between. The second feature allows UAVs to circle on diﬀerent orbits. To achieve formation under this circumstance, formation controllers will resort to virtual agents representing the actual agents in need. The third feature is the usage of a (two-layer) hierarchical formation structure, which allows selection of formation patterns for diﬀerent layers. Combinations of these new features with our existing results yield more versatile concentric formation patterns with diﬀerent local geometric shapes, such as straight lines and triangles. Inter-vehicle collision avoidance is also addressed. Agents will be repelled to steer away from each other once they get too close. 12 All UAVs are assumed to have constant linear velocities. Control of each UAV is via its yaw rate. The design idea is to add three control components (three heading controllers) together to achieve the overall objective. Each control component has a goal. The proposed extensions to spreading agents on a portion of a circle, circling agents on orbits of diﬀerent radii, formation in local geometric shapes, and avoiding inter-vehicle collisions, provide more complete solution to cooperative target tracking in the concentric manner. This paper also raises several questions for future investigations. The implementation of the proposed schemes on physical robots and the extension of the developed techniques to 3D scenarios and cooperative tracking of multiple targets with obstacle avoidance capability
[53],
[54] will be of particular interest. Stability analyses in the presence of formation pattern switching and broken communication links are another research direction to look into. Also, investigations of the time delay factor for obtaining stability conditions as well as desirable performance with reasonable computation complexity
[55]–[57] are needed. Finally, Artiﬁcial Intelligence (AI) techniques have recently been developed for robotic communication to enhance the communication capability of robotic networks for coordinated actions. Application of the AI and/or Neural Networks to the ﬁeld of robotic networks in the context of cooperative target tracking is a promising research area to pursue [58]–[60]. References [1] K. Szwaykowska, I.B. Schwartz, L.M.-T. Romero, C.R. Heckman, D. Mox, and M.A. Hsieh, Collective motion patterns of swarms with delay coupling: Theory and experiment, Physical Revie E, 93(3), 2016, 11. [2] R. Sepulchre, D. Paley, and N. Leonard, Stabilization of planar collection motion: All-to-all communication, IEEE Transactions on Automatic Control, 52(5), 2007, 811–824. [3] R. Sepulchre, D. Paley, and N. Leonard, Stabilization of planar collective motion with limited communication, IEEE Transactions on Automatic Control, 53(3), 2008, 706–719. [4] A. Jain, D. Ghose, and P. Menon, Stabilization of balanced circular motion about a desired center, International Conference on Advances in Control and Optimization of Dynamical Systems, Kanpur, India, 2014. [5] A. Jain and D. Ghose, Stabilization of collective motion in synchronized, balanced and splay phase arrangements on a desired circle, American Control Conference, Chicago, IL. [6] G. Mallik and A. Sinha, A study of balanced circular formation under deviated cyclic pursuit strategy, IFAC-PapersOnLine, 48(5), 2015, 41–46. [7] R. Zheng, Z. Lin, M. Fu, and D. Sun, Distributed control for uniform circumnavigation of ring-coupled unicycles, Automatica, 53, 2015, 23–29. [8] N. Kokolakis and N. Koussoulas, Coordinated standoﬀ tracking of a ground moving target and the phase separation problem, International Conference on Unmanned Aircraft Systems, Dallas, TX, 2018. [9] J. Guo, G. Yan, and Z. 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Ren, Multi-robot joint localization and target tracking with local sensing and communication, American Control Conference, Charlotte, NC, 2019. [21] L. Luo, N. Chakraborty, and K. Sycara, Provably-good distributed algorithm for constrained multi-robot task assignment for grouped tasks, IEEE Transactions on Robotics, 31(1), 2015, 19–30. [22] D. Panagou, M. Turpin, and V. Kumar, Decentralized goal assignment and safe trajectory generation in multi-robot networks via multiple Lyapunov functions, IEEE Transactions on Automatic Control, 65(8), 2020, 3365–3380. [23] J. Ni, X. Yang, J. Chen, and S. Yang, Dynamic bioinspired neural network for multi-robot formation control in unknown environments, International Journal of Robotics and Automation, 30(3), 2015, 256–266. [24] M. Khan and C. Silva, Autonomous and robust multi-robot cooperation using an artiﬁcial immune system, International Journal of Robotics and Automation, 27(1), 2012, 60–75. [25] X. Yu, L. Liu, and G. Feng, Coordinated control of multiple unicycles for escorting and patrolling task based on a cyclic pursuit strategy, American Control Conference, Boston, MA, 2016. [26] M. Zhang and H. Liu, Cooperative tracking a moving target using multiple ﬁxed-wing UAVs, Journal of Intelligent and Robotic Systems, 81(3-4), 2016, 505–529. [27] X. Yu and L. Liu, Cooperative control for moving-target circular formation of nonholonomic vehicles, IEEE Transactions on Automatic Control, 62(7), 2017, 3448–3454. [28] L. Brinon-Arranz, A. Seuret, and A. Pascoal, Target tracking via a circular formation of unicycles, IFAC World Congress, Toulouse, France, 2017. [29] A. Miao, Y. Wang, and R. Fierro, Cooperative circumnavigation of a moving target with multiple nonholonomic robots using backstepping design, Systems and Control Letters, 103, 2017, 58–65. [30] L. Ma and N. Hovakimyan, Vision-based cyclic pursuit for cooperative target tracking, Journal of Guidance, Control, and Dynamics, 36(2), 2013, 617–622. 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[57] are needed. Finally, Artiﬁcial Intelligence (AI) techniques have recently been developed for robotic communication to enhance the communication capability of robotic networks for coordinated actions. Application of the AI and/or Neural Networks to the ﬁeld of robotic networks in the context of cooperative target tracking is a promising research area to pursue
[58]–[60]. References [1] K. Szwaykowska, I.B. Schwartz, L.M.-T. Romero, C.R. Heckman, D. Mox, and M.A. Hsieh, Collective motion patterns of swarms with delay coupling: Theory and experiment, Physical Revie E, 93(3), 2016, 11. [2] R. Sepulchre, D. Paley, and N. Leonard, Stabilization of planar collection motion: All-to-all communication, IEEE Transactions on Automatic Control, 52(5), 2007, 811–824. [3] R. Sepulchre, D. Paley, and N. Leonard, Stabilization of planar collective motion with limited communication, IEEE Transactions on Automatic Control, 53(3), 2008, 706–719. [4] A. Jain, D. Ghose, and P. Menon, Stabilization of balanced circular motion about a desired center, International Conference on Advances in Control and Optimization of Dynamical Systems, Kanpur, India, 2014. [5] A. Jain and D. Ghose, Stabilization of collective motion in synchronized, balanced and splay phase arrangements on a desired circle, American Control Conference, Chicago, IL. [6] G. Mallik and A. Sinha, A study of balanced circular formation under deviated cyclic pursuit strategy, IFAC-PapersOnLine, 48(5), 2015, 41–46. [7] R. Zheng, Z. Lin, M. Fu, and D. Sun, Distributed control for uniform circumnavigation of ring-coupled unicycles, Automatica, 53, 2015, 23–29. [8] N. Kokolakis and N. Koussoulas, Coordinated standoﬀ tracking of a ground moving target and the phase separation problem, International Conference on Unmanned Aircraft Systems, Dallas, TX, 2018. [9] J. Guo, G. Yan, and Z. Lin, Cooperative control synthesis for moving-target-enclosing with changing topologies, International Conference on Robotics and Automation, Anchorage, AK, 2010. [10] Z. Zhou, H. Wang, and Z. Hu, Event-based time varying formation control for multiple quadrotor UAVs with Markovian switching topologies, Complexity, 2018, 1–15. [11] Y. Sun and L. Wang, Consensus of multi-agent systems in directed networks with nonuniform time-varying delays, IEEE Transactions on Automatic Control, 54(7), 2009, 1607–1613. [12] J. Marshall, M. Broucke, and B. Francis, Formations of vehicles in cyclic pursuit, IEEE Transactions on Automatic Control, 49(11), 2004, 1963–1974. [13] J. Marshall, M. Broucke, and B. Francis, Pursuit formations of unicycles, Automatica, 42, 2006, 3–12. [14] M. Pavone and E. Frazzoli, Decentralized policies for geometric pattern formation and path coverage, ASME Journal of Dynamic Systems, Measurement, and Control, 129(5), 2007, 633–643. [15] B. Wu, D. Wang, and E. Poh, Cyclic formation control for satellite formation using local relative measurements, Mechatronic Systems and Control, 40(1), 2012, 11–21. [16] J. Juang, On the formation patterns under generalized cyclic pursuit, IEEE Transactions on Automatic Control, 58(9), 2013, 2401–2405, 2013. [17] J. Ramirez, M. Pavone, E. Frazzoli, and D. 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