# Decentralized Hybrid Formation Control of Unmanned Aerial Vehicles

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2 (a) Fig. 1. (a) Vertices of the element R i,j. Edges of the element R i,j. velocity V rel. Problem 1: Given the dynamics of the follower UAVs as (1) and their velocity in the form of (2), design the formation controller to generate the relative velocity of the followers, V relk, such that starting from any initial state inside the control horizon, the follower UAVs eventually reach their desired positions, while avoiding the collision with other follower UAVs. Moreover, after reaching the formation, the follower UAVs should remain at the desired positions. III. DISCRETE MODEL OF THE UAV MOTION DYNAMICS OVER THE POLAR PARTITIONED SPACE To address this problem, for each UAV consider a circle with the radius of R m that is centered at its desired position. With the aid of the partitioning curves {r i = Rm n (i r 1 1), i = 1,..., n r } and {θ j = 2π n θ 1 (j 1), j = 1,..., n θ}, this circle can be partitioned into (n r 1)(n θ 1) partitioning elements. In this partitioned space, an element R i,j = {p = (r, θ) r i r r i+1, θ j θ θ j+1 }, has four vertices, v 0, v 1, v 2, v 3 (Fig. 1(a)), four edges, E r +, Er, E + θ, E θ (Fig. 1). The set V ( ) stands for the vertices that belong to ( can be an edge, or a region R i,j ). As shown in [6], for a system with a multi-affine dynamics ẋ = h(x, u(x)) defined over this polar partitioned space, two control features can be designed. First, the region R i,j can be invariant, i.e., the trajectories of the system remain inside the region forever. The other control feature is the exit edge. It is possible to design a controller to drive the system s trajectory to exit from the edge Eq, s q {r, θ} and s {+, }, by choosing the control values u(v m ) at the vertices. According to the properties of multi-affine systems, the control value at any point inside the region can be achieved based on the control values at the vertices as u(x) = Σ 3 m=0λ m (x)u(v m ), which λ m (x) is a coefficient that determines the weight of u(v m ) in the control value u(x). We denote the controller + for having a region invariant by C 0k. Also, C r k, C r k, C θ + k, and C θ k are respectively the controllers for having the edges F + r, F r, F + θ, and F θ, as exit edges. Further details on how to design these controllers are provided in [6]. Now, this model of the UAV motion dynamics over the partitioned space can be abstracted to a finite state machine and can be presented by a discrete automaton. An automaton can be formally defined as follows: Definition 1: (Automaton)[11]. A deterministic automaton is a tuple A := (Q, q 0, E, δ, Q m ) consisting of a set of states Q; an initial state q 0 Q; a set of events E that causes transitions between the states, and a transition relation δ Q E Q (with a partial map δ : Q E Q), such that (q, e, q ) δ if and only if state q is transited to state q by event e, denoted by q e q (or δ(q, e) = q ). Q m Q represents the marked states to assign a meaning of accomplishment to some states. For supervisor automaton whose all states are marked, Q m is omitted from the tuple. For this automaton, the sequence of these events forms a string. We use ε to denote an empty string, and Σ to denote the set of all possible strings over the set Σ including ε. The language of the automaton G, denoted by L(G), is the set of all strings that can be generated by G, starting from the initial states. The marked language, L m (G), is the set of strings that belong to L(G) and end with the marked states. For UAV 1, the discrete model of the system over the partitioned space can be described by the automaton A 1 = (Q 1, q 01, E 1, δ 1, Q m1 ) whose set of discrete states is Q 1 = {R 1, O 1 }, and its event set is E 1 = + + C 1 {C 01 } D 1 Ex, where C 1 = {C r 1, C r1, C θ 1, C θ1 } and D 1 = {d i,j 1 1 i n r 1, 1 j n θ 1}. When UAV 1 is in one of the regions R i,j, in the abstract model it is considered to be in the discrete state R 1. Then, one of the actuation commands belong to C 1 drives the UAV to one of its adjacent regions. In this case, right after issuing the actuation commands, the system transits to the detection state O 1 and waits until the UAV enters a new region. Crossing boundaries of the new region, a detection event belonging to D 1 will be generated which shows the UAV has entered the new region R i,j. The command C 01, keeps the UAV in the current region and does not change the discrete state of the system. We use the notation D M 1 = {d i,j 1 1 i = 1, 1 j n θ 1} to denote the detection events, which show entering a region in the first circle, and d 1 = D 1 D M 1 = {d i,j 1 1 < i n r 1, 1 j n θ 1} D 1 for the rest of detection events. Here, Ex = {Ca 12F, Ca 12N, Ca 21F, Ca 21N, Stop 1, Stop 2, R 21, R 12 } is the set of external events which are required for the collision avoidance and do not change the state of the system. The events belong to CA = {Ca 12F, Ca 12N, Ca 21F, Ca 21N } show the collision alarms, in which the events in CA 1 = {Ca 12F, Ca 12N } show that UAV 2 enters the alarm zone of UAV 1 and accordingly, the events in CA 2 = {Ca 21F, Ca 21N } show that UAV 1 enters the alarm zone of UAV 2. The details will be discussed in Section IV-B. The events Stop 1 and Stop 2 are the commands that request UAV 1 and UAV 2 to stop at their current position in the relative frame and the command R 12 and R 21 release them, respectively. Similar definitions can be given for the DES model of UAV 2. The graph representation of the discrete models of UAV 1 and UAV 2 are shown in Fig. 2. In these graphs, the arrows starting from one state and ending to another state represent

3 (a) Fig. 2. (a) DES model of UAV 1. DES model of UAV 2. the transitions, labeled by the events belong to E i. The entering arrows stand for the initial states. Marked states are shown by double circles. In the DES model of UAV k, k = 1, 2, the event set E k + consists of the controllable event set E ck = {C 0k, C r C r k, C θ + k, C θ k, Stop 1, Stop 2, R 21, R 12 } and the uncontrollable event set E uck = {Ca 12F, Ca 12N, Ca 21F, Ca 21N } D 1. The uncontrollable events are those that cannot be affected by the supervisor. A language K is controllable with respect to the language L(A) and the event set E uc if and only if s K and σ E uc, if sσ L(A), then sσ K. Indeed, the controllability is the existence condition of a supervisor for the control goal described by the specification K [11]. IV. DESIGNING A DECENTRALIZED MODULAR SUPERVISOR FOR THE FORMATION CONTROL OF THE UAVS Given the discrete model of follower UAVs over the partitioned space, it is possible to design the supervisor to achieve a desired order of events to accomplish the formation. Indeed, the supervisor, S, observes the executed strings of the plant A and disables the undesirable controllable events. Here, we assume that all of the events are observable. The generated language and marked language of the closedloop system, L(S/A) and L m (S/A), can be constructed as follows: (1) ε L(S/A) (2) [(s L(S/A)) and (sσ L(A)) and (σ L(S))] (sσ L(S/A)) (3) L m (S/A) = L(S/A) L m (A) where s is the string that has been generated so far by the plant A, and σ is an event, which the supervisor S should decide whether keep it active or not in the supervised system S/A. Within this framework one can use parallel composition to facilitate the control synthesis. Parallel composition is a binary operation between two automata which can be defined as follows: Definition 2: (Parallel Composition [11]) Let A i = ( Qi, q 0 i, E i, δ i, Q mi ), i = 1, 2, be automata. The parallel composition (synchronous composition) of A 1 and A 2 is the automaton A 1 A 2 = (Q = Q 1 Q 2, q 0 = (q 0 1, q 0 2), E = E 1 E 2, δ, Q m = Q m1 Q m2 ), with δ defined as (q 1, q 2 ) Q, e E : δ((q 1, q 2 ), e) = k, (δ 1 (q 1, e), δ 2 (q 2, e)), if δ 1 (q 1, e)!, δ 2 (q 2, e)!, e E 1 E 2 ; (δ 1 (q 1, e), q 2 ), if δ 1 (q 1, e)!, e E 1 \E 2 ; (q 1, δ 2 (q 2, e)), if δ 2 (q 2, e)!, e E 2 \E 1 ; undefined, otherwise. Here, the parallel composition is used to combine the plant s discrete model and the supervisor as follows: Lemma 1: [12] Let A = (Q, q 0, E, α, Q m ), be the plant automaton and K E be the desired marked language. There exists a nonblocking supervisor S such that L m (S/A) = L m (S A) = K if = K = K L m (A) and K is controllable. In this case, S could be any automaton with L(S) = L m (S) = K. Now, using the above lemma, it is possible to design the supervisor for the formation problem described in Problem 1, which includes two modules: 1- Reaching and keeping the formation and 2- Avoiding collision. Next lemma describes how to design the supervisors in a modular way. Lemma 2: [12] Let A = (Q, q 0, E, α, Q m ) be the plant automaton and the prefix-closed controllable languages K 1, K 2 E be the desired marked specifications. Suppose there exist nonblocking supervisors S 1 and S 2 such that L m (S 1 /A) = L m (S 1 A) = K 1 and L m (S 2 /A) = L m (S 2 A) = K 2, then S = S 1 S 2 is a nonblocking supervisor with L m (S A) = K 1 K2. A. Designing the supervisor for reaching and keeping the formation For reaching the formation, it is sufficient to directly drive each of the follower UAVs towards one of the regions R 1,j, 1 j n θ 1, located in the first circle in their corresponding partitioned motion space. After reaching R 1,j, the UAVs should remain inside it, to keep the formation. The specifications K F 1 and K F 2 for reaching and keeping the specification for UAV 1 and UAV 2 are realized in Fig. 3. When the k th follower UAV is not in the first circle, the command C r k will be generated to push the UAV towards the origin. Entering a new region, one of the events from d k = {d i,j k 1 < i n r 1, 1 j n θ 1} will appear. This will continue until one of the events from D M k = {d i,j k i = 1, 1 j n θ 1} be generated, which shows that the formation is reached. In this case, the event C 0k is activated, which keeps the system trajectory inside the first region. If a collision alarm happens to UAV k, the formation supervisor does not change the generable language after the events belonging to CA, and lets the collision avoidance supervisor handle it until the collision be avoided and the UAV be released to resume the formation task. It can be seen that K F k, k = 1, 2 are controllable with respect to the plant language L(A k ) and the event set E uck, as they do not disable any uncontrollable event. Therefore, based on Lemma 1, there exist supervisors that can control the plants A 1 and A 2 to achieve these specifications. The supervisors are the realization of the above specifications in which all states are marked. Marking all states of the supervisors allows the closed-loop marked states to be solely determined by the plants marked states. The supervisor for

4 (a) Fig. 3. (a) The specification for reaching and keeping the formation for UAV1. The specification for reaching and keeping the formation for UAV2. reaching the formation and keeping the formation of UAV k is denoted by A F k. B. Designing the supervisor for collision avoidance Fig. 4. UAV 2 enters the alarm zone of UAV 2. When UAV 1 is going to reach its desired position, in some situations, the other follower, UAV 2, may enter the alarm zone of UAV 1 (Fig. 4), which requires these UAVs to cooperatively avoid the collision. For this purpose, first, UAV 1 asks UAV 2 to stop in the relative frame and then, UAV 1 finds a path to safely get away from UAV 2. After avoiding the collision, UAV 1 releases UAV 2 and both UAVs resume their normal operation for reaching the formation. Similar strategy is taken when UAV 1 enters the alarm zone of UAV 2. This specification, K C, is shown in Fig. 5 whose left side shows that after appearing one of the events ca 12F or ca 12N, UAV 1 realizes that UAV 2 has entered its alarm zone. Therefore, by event Stop 2, UAV 1 requests UAV 2 to stop for a while to safely manage the situation. The event ca 12F shows that UAV 2 is in front of the path of UAV 1 towards its destination and hence, to avoid the collision it is sufficient that UAV 1 turns anticlockwise to change its azimuth angle, θ, by activating the command C + θ. This will continue until removing the collision alarm. Then, UAV 1 releases UAV 2, and reaching the formation can be resumed by the reaching formation supervisor which was explained in the previous section. Meanwhile, if UAV 1 enters one of the regions in the first circle, one of the events belong to D M 1 appears which means that UAV 1 has reached its desired formation and should remain there for the rest of mission. Similarly, the right side of Fig. 5, shows the collision avoidance mechanism when UAV 1 enters the alarm zone of UAV 2. If neither of collision avoidance alarms from the set CA happens, then UAV 1 and UAV 2 can do their normal operations by independent enabling of events C 1 and C 2 followed by the detection signals D 1 and D 2 in any order as shown on the top of Fig. 5. The other module, the reaching formation supervisor, will manage this situation. It can be verified that K C is controllable with respect to the language L(A 1 A 2 ) and the event set E uc1 E uc2. Therefore, based on Lemma 1, there exists a supervisor A c that can control the plants A 1 and A 2 to achieve this joint specification. The supervisor is the realization of the specification K C in which all states are marked. The collision avoidance supervisor, A C, is a centralized supervisor which manages both UAV 1 and UAV 2. To make this supervisor decentralized and to achieve local supervisors, we will utilize our proposed decomposition scheme introduced in [13]. Here, local supervisors can be achieved by the projection of the global supervisor to each agent s local event set. The projection of the global supervisor A C to the event set of UAV i, E i, is denoted by P Ei (A C ), and can be obtained by replacing the events that belong to E\E i by ε-moves, and then, merging the ε-related states. Once the local supervisor automata are derived through the natural projection, the decentralized supervisor is then obtained using the parallel composition of local supervisor automata. Parallel composition captures the logical behavior of concurrent distributed systems by allowing each subsystem to evolve individually on its private events, while synchronize with its neighbors on shared events for cooperative tasks. The obtained decentralized supervisor is then compared with the original global supervisor automaton using the bisimulation relation. Definition 3: Consider two automata A i = (Q i, qi 0, E, δ i ), i = 1, 2. The automaton A 1 is said to be similar to A 2 (or A 2 simulates A 1 ), denoted by A 1 A 2, if there exists a relation R from A 1 to A 2 over Q 1, Q 2 and with respect to E, such that (1) (q1, 0 q2) 0 R, and (2) (q 1, q 2 ) R, q 1 δ 1 (q 1, e), then q 2 Q 2 such that q 2 δ 2 (q 2, e), (q 1, q 2) R. Automata A 1 and A 2 are said to be bisimilar (bisimulate each other), denoted by A 1 = A2 if A 1 A 2 with a simulation relation R 1, A 2 A 1 with a simulation relation R 2 and R1 1 = R 2, where R1 1 = {(y, x) Q 2 Q 1 (x, y) R 1 }.

5 Fig. 5. The specification for cooperative collision avoidance. Based on these definitions we can formally describe the decomposability conditions with respect to two local event sets. Lemma 3: (Theorem 4 in [13]) A deterministic automaton A = (Q, q 0, E = E 1 E 2, δ) is decomposable with respect to parallel composition and natural projections P i, i = 1, 2, such that A = P 1 (A) P 2 (A) if and only if A satisfies the following decomposability conditions (DC): e 1 E 1 \E 2, e 2 E 2 \E 1, q Q, s E, DC1: [δ(q, e 1 )! δ(q, e 2 )!] [δ(q, e 1 e 2 )! δ(q, e 2 e 1 )!]; DC2: δ(q, e 1 e 2 s)! δ(q, e 2 e 1 s)!; DC3: s, s E, s s, p E1 E 2 (s), p E1 E 2 (s ) start with the same common event a E 1 E 2, q Q: δ(q, s)! δ(q, s )! δ(q, p 1 (s) p 2 (s ))! δ(q, p 1 (s ) p 2 (s))!; DC4: i {1, 2}, x, x 1, x 2 Q i, x 1 x 2, e E i, t E i, x 1 δ i (x, e), x 2 δ i (x, e): δ i (x 1, t)! δ i (x 2, t)!. where, K = {s Σ ( t Σ )st K} is the prefix closure of the language K. The decomposability conditions DC1 and DC2 respectively guarantee that any decision on the selection or order of two transitions can be done by the team of agent, while conditions DC3 and DC4 respectively ensure that the interaction of local automata P 1 (A) and P 2 (A) neither allows an illegal string that is not in A, nor stops a legal string of A. Now assume that given the global task and local plants, a global supervisor is designed and decomposed into local supervisors such that each closed loop system (the supervised local plant with the corresponding local controller) satisfies the global task. In this decentralized cooperative control architecture we are then interested to check whether the entire system satisfied the global task. Problem 2: (Decentralized cooperative control problem) Consider a plant, represented by a parallel distributed system A P := 2 A Pi, with local event sets E i, i = 1, 2, and let the global specification is given by a deterministic task automaton A S over E = 2 E i. Furthermore, suppose that there exist a decomposable deterministic global controller automaton A C = 2 P i (A C ), so that A P A C = AS. Then, whether the local controllers can lead the team to satisfy the global specification in a decentralized architecture, 2 (A Pi P i (A C )) = A S. Following result considers a team of two local plants and introduces the supervisor decomposability and satisfaction of the global task by each local supervised plant as a sufficient condition for the satisfaction of global task by the team. Theorem 1: (Decentralized cooperative control using supervisor decomposition) Consider a plant, represented by a parallel distributed system A P1 A P2, with local event sets E i, i = 1, 2, and let the global specification is given by a task automaton A S over E = E 1 E 2. Furthermore, suppose that there exist a deterministic global controller automaton A C = P1 (A C ) P 2 (A C ), so that A C A p = AS. Then, the entire closed loop system satisfies the global specification, in the sense of bisimilarity, i.e., 2 (A Pi P i (A C )) = A S, provided the decomposability conditions DC1, DC2, DC3 and DC4 for A C. The significance of this result is the decentralized implementation of the global supervisor, A C, given in Fig. 5,

6 by decomposing A C, into local supervisors. As it can be seen in A C, the successive and adjacent events from pairs of private event sets (from different local event sets) (C 01, C 2 ), (C 01, D 2 ), (C 02, C 1 ), (C 02, D 1 ), (C 1, D 2 ), (C 2, D 2 ), appear in both orders in the global supervisor automaton therefore DC1 and DC2 are satisfied. Moreover, among common events R 12, R 21, CA 1 = {ca 12F, ca 12N }, CA 2 = {ca 21F, ca 21N }, Stop 1, and Stop 2, the events R 12, R 21, Stop 1, and Stop 2 are not shared between different strings. Strings just share the events CA 1, CA 2, where the corresponding local strings do not interleave on these events because of predecessor common events before CA 1, CA 2. Therefore DC3 also is fulfilled. Finally, DC4 is satisfied because of the determinism of local automata P 1 (A C ) and P 2 (A C ), and hence, the supervisor automaton A C is decomposable into A C 1 = P 1 (A C ) and A C 2 = P 2 (A C ), shown in Fig. 6, so that A C 1 A C 2 = AC. V. CONCLUSION In this paper, a collision free formation control algorithm was proposed using hybrid supervisory control techniques. The proposed supervisor has a modular structure and can accomplish three main tasks: reaching the formation, keeping the formation, and collision avoidance. This control structure was implemented decentralizedly so that local (decomposed) supervisors can treat the distributed agents to achieve a globally safe and collision free environment. Simulation results and further details are available in the extended version [14]. (a) ACKNOWLEDGMENT The financial supports from NSF-CNS and NSF- EECS for this work are greatly acknowledged. REFERENCES [1] K. P. Valavanis, K. P. Valavanis, Advances in unmanned aerial vehicles: state of the art and the road to autonomy, Springer Publishing Company, Incorporated, [2] S. A. Bortoff, The university of toronto rc helicopter: a test bed for nonlinear control, in: Control Applications, Proceedings of the 1999 IEEE International Conference on, Vol. 1, IEEE, 1999, pp [3] R. C. Michelson, S. Reece, Update on flapping wing micro air vehicle research-ongoing work to develop a flapping wing, crawling entomopter, in: 13th Bristol International RPV/UAV Systems Conference Proceedings, Bristol England, Vol. 30, 1998, pp [4] A. R. Partovi, H. Lin, G. Cai, B. Chen, A. Kevin, Development of a cross style quadrotor, in: AIAA Guidance, Navigation, and Control Conference, [5] P. J. Antsaklis, J. A. Stiver, M. Lemmon, Hybrid system modeling and autonomous control systems, in: Hybrid Systems, Springer, 1993, pp [6] A. Karimoddini, H. Lin, B. M. Chen, T. H. Lee, Hybrid formation control of the unmanned aerial vehicles, Mechatronics 21 (5) (2011) [7] C. Belta, A. Bicchi, M. Egerstedt, E. Frazzoli, E. Klavins, G. J. Pappas, Symbolic planning and control of robot motion [grand challenges of robotics], Robotics & Automation Magazine, IEEE 14 (1) (2007) [8] A. Karimoddini, H. Lin, B. M. Chen, T. H. Lee, Hybrid threedimensional formation control for unmanned helicopters, Automatica 49 (2) (2013) [9] A. Karimoddini, G. Cai, B. M. Chen, H. Lin, T. H. Lee, Hierarchical Control Design of a UAV Helicopter, in Advances in Flight Control Systems, INTECH, Vienna, Austria, Fig. 6. (a) The local supervisor for collision avoidance for UAV 1. The local supervisor for collision avoidance for UAV 2. [10] A. Karimoddini, G. Cai, B. M. Chen, H. Lin, T. H. Lee, Multi-layer flight control synthesis and analysis of a small-scale uav helicopter, in: IEEE Conference on Robotics Automation and Mechatronics, 2010, pp [11] C. G. Cassandras, S. Lafortune, Introduction to discrete event systems, Springer, [12] R. Kumar, V. K. Garg, Modeling and Control of Logical Discrete Event Systems, Vol. 300 of The Springer International Series in Engineering and Computer Science, Springer, [13] M. Karimadini, H. Lin, Guaranteed global performance through local coordinations, Automatica 47 (5) (2011) [14] A. Karimoddini, M. Karimadini, H. Lin, Decentralized Hybrid Formation Control of Unmanned Aerial Vehicles, Technical Report: NCAT-ACCESS , North Carolina Agricultural and Technical State University, Autonomous Cooperative Control of Emergent Systems of Systems (ACCESS) Lab, [Online]. Available at:

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