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Time-varying Formation Tracking with Collision Avoidance for Multi-agent Systems*
Proceedings of the 20th World The International Federation of Congress Automatic Control Proceedings of the 20th9-14, World The International Federation of Congress Automatic Control Toulouse, France, July 2017 Available online at www.sciencedirect.com The International of Automatic Control Toulouse, France,Federation July 9-14, 2017 Toulouse, France, July 9-14, 2017
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IFAC PapersOnLine 50-1 (2017) 309–314 Time-varying Formation Tracking Time-varying Formation Tracking Time-varying Formation Tracking with Collision Avoidance for with Collision Avoidance for with Collision Avoidance Multi-agent Systems for Multi-agent Systems Multi-agent Systems ∗
J. Santiaguillo-Salinas and E. Aranda-Bricaire ∗ J. Santiaguillo-Salinas and E. Aranda-Bricaire ∗∗ J. Santiaguillo-Salinas and E. Aranda-Bricaire ∗ Department of Electrical Engineering, Mechatronics Section, ∗ ∗ Department of Electrical Engineering,exico, Mechatronics D.F., M´eSection, xico. ∗ CINVESTAV AP 14-740, 07000 M´ Department of Electrical Engineering, Mechatronics CINVESTAV AP 14-740, 07000 M´ e xico, D.F., M´eSection, xico. (e-mail: [email protected], [email protected]). CINVESTAV AP 14-740, 07000 M´ exico, D.F., M´exico. (e-mail: [email protected], [email protected]). (e-mail: [email protected], [email protected]). Abstract: This paper studies the collision avoidance problem in the time-varying formation Abstract: This for paper studies the collision in the time-varying formation tracking control multi-agent systems. Theavoidance proposedproblem control strategy is decentralized, since Abstract: This for paper studies the collision avoidance problem in the time-varying formation tracking control multi-agent Theto proposed control strategy decentralized, since agents have no global knowledgesystems. of the goal achieve, knowing only theisposition and velocity tracking control for multi-agent systems. Thetoproposed control strategy isposition decentralized, since agents no global knowledge of theallows goal knowing onlyto thetrack and velocity of somehave agents. This control strategy a achieve, set of mobile agents a predetermined agents have no global knowledge of theallows goal toa achieve, knowing onlyto thetrack position and velocity of some agents. This control strategy set of mobile agents a predetermined trajectory while they achieve a time-varying formation. For the collision avoidance, we add a of some agents. This control strategy allows formation. a set of mobile agents to track a predetermined trajectory while field they of achieve a time-varying For the collision avoidance, wecontrol add a repulsive vector the unstable focus type to the time-varying formation tracking trajectory while they achieve a time-varying formation. For the collision avoidance, wecontrol add a repulsive vector field of scheme the unstable focus type to the time-varying tracking law. A leader-followers is employed, using formation graphsformation to represent interactions repulsive vector field of scheme the unstable focus type to the time-varying formation tracking control law. A leader-followers is employed, using formation graphs to represent interactions between agents. The results are presented for the front points of differential-drive mobile robots. law. A leader-followers scheme is employed, using formation to represent interactions between agents. results The results are presented for the front points ofgraphs differential-drive mobile robots. The theoretical are verified by numerical simulation. between agents. The results are presented for the front points of differential-drive mobile robots. The theoretical results are verified by numerical simulation. The theoretical results areFederation verified by numericalControl) simulation. © 2017, IFAC (International of Automatic Hosting by Elsevier Ltd. All rights reserved. Keywords: Multi-agent systems, time-varying formation, collision avoidance, differential-drive Keywords: Multi-agent time-varying formation, collision avoidance, differential-drive mobile robots, repulsivesystems, vector fields. Keywords: Multi-agent time-varying formation, collision avoidance, differential-drive mobile robots, repulsivesystems, vector fields. mobile robots, repulsive vector fields. 1. INTRODUCTION This time-varying formation allows the group of agents to 1. INTRODUCTION This time-varying formation allows thetranslated, group of agents to behave as a rigid body which can be rotated 1. INTRODUCTION This time-varying formation allows thetranslated, group of agents to behave as a rigid body which can be rotated and scaled in the plane. Multi-agent Systems are conceived as bundles of multiple behave as a rigid body which can be translated, rotated Multi-agent are conceived as bundles cooperative of multiple and scaled in the plane. autonomous Systems robots coordinated to accomplish scaledubiquitous in the plane. Another problem in all areas of motion coMulti-agent Systems are conceived as bundles cooperative of multiple and autonomous robots coordinated tomulti-agent accomplish tasks. In recent years, the study of systems has Another ubiquitous problem in allbetween areas ofagents motionwhen coordination is the posible collision autonomous robots coordinated tomulti-agent accomplish systems cooperative tasks. In recent years, the study of has Another ubiquitous problem in allbetween areas ofagents motionwhen cogained special interest, because multiple agents can solve ordination isachieve the posible collision they try to a desired position into a formation tasks. In recent years, the study of multi-agent systems has gained special cooperatively; interest, because multiple agents solve ordination isachieve the posible collision between agents when tasks working making them morecan reliable, they try to a desired position into a formation the trajectory tracking. In the literature, we gained special cooperatively; interest, because multiple agents can solve or during tasks reliable, try tothe achieve a desired position intoliterature, a formation faster working and cheaper than it is making possible them with amore single agent they or during trajectory the we can find different methodstracking. to avoidIncollisions. In Qiantasks working cooperatively; making them more reliable, faster cheaper than it is possible with a single agent or during the trajectory tracking. In the literature, we Cao etand al. (1997). can find different methods to avoid collisions. In Qianwei et al. (2003) a mechanism for collision avoidance fasteretand cheaper than it is possible with a single agent can find different methods to avoid collisions. In QianCao al. (1997). wei et al. (2003) a mechanism for collision avoidance on traffic control type is presented. In De Gennaro Cao al. (1997). The et main applications of multi-agent systems include based wei etonal.traffic (2003) a mechanism for collision control type is presented. Deavoidance Gennaro The main and applications and Jadbabaie (2006); Dimarogonas and In Kyriakopoulos of multi-agent include transport manipulation of objects, systems localization, ex- based based on traffic control type is presented. In De Gennaro The main applications of multi-agent systems include and Jadbabaie (2006); Dimarogonas and Kyriakopoulos transport and manipulation of objects, localization, exDimarogonas et al. (2006) navigation functions ploration and motion coordination Arai et al. (2002); Cao (2006); Jadbabaie (2006); Dimarogonas and Kyriakopoulos transport andmotion manipulation of objects, ex- and Dimarogonas al. (2006) functions ploration and coordination Arai etlocalization, al. (2002); Cao and artificial potentialetfunctions arenavigation used to avoid colliet al. (1997). The main idea of motion coordination is (2006); (2006); Dimarogonas et al. (2006) navigation functions ploration and motion coordination Arai et al. (2002); Cao and artificial potential functions are used avoid colliet (1997).navigation The mainofidea of motion coordination is sions between agents. These non collision to strategies are theal. strategic a group of agents. One of the and artificial potential functions are used to avoid colliet al. (1997). The main idea of motion coordination is sions between agents. These non collision strategies are the a group of coordination agents. One of the developed based on a combination of attractive potential mainstrategic areas ofnavigation research inofthe motion is the between agents. These non collision strategies are the strategic navigation ofthe a group of coordination agents. One of the sions main areas of research in motion is the developed based on a combination of attractive potential (APF) and repulsive potential functions (RPF). formation tracking problem, where the goal is to track functions developed based on a combination of attractive potential main areas of research in the motion coordination is the (APF) and repulsive potential functions (RPF). formation tracking problem,while wherethethe goal maintain is to tracka functions Works Hern´ andez-Mart´ ınez and Aranda-Bricaire (2009, a preestablished trajectory agents functions (APF) and repulsive potential functions (RPF). formation tracking problem,while where the goal maintain is to tracka Works Hern´ andez-Mart´ ınez and (2009, adesired preestablished trajectory 2013); Loizou et al. (2003); YaoAranda-Bricaire et al. (2006) address agents pattern defined by relative the position vectors. Hern´andez-Mart´ ınez and Aranda-Bricaire (2009, a preestablished trajectory while the agents maintain a Works 2013); Loizou et al. (2003); Yao et al. (2006) address desired pattern defined by relative position vectors. the formation control problem without collisions using 2013); Loizou et al. (2003); Yao et al. (2006) address desired pattern defined by relative position vectors. The time-varying formation control can be applied as the discontinuous the formationvector control problem without collisions using fields. The time-varying formation can be applied the the formationvector control problem without collisions using solution to complex motioncontrol coordination problemsas and discontinuous fields. The time-varying formation control can be applied as and the The purpose ofvector solution to complex fields.is to design a decentralized conthis paper some examples can be motion found incoordination Bri˜ n´ on Arranzproblems et al. (2014); discontinuous solution to complex problems and trol strategy purpose for of this paper is systems to design a decentralized consome be motion found incoordination Bri˜ n´ on Arranz et al. (2014); multi-agent that allows the trajecGonz´ aexamples lez-Sierracan and Aranda-Bricaire (2013); Pe˜ naloza The The purpose for of this paper is systems to design a decentralized consome examples can be found in Bri˜ n o ´ n Arranz et al. (2014); trol strategy multi-agent that allows the trajecGonz´ a lez-Sierra and Aranda-Bricaire (2013); Pe˜ n aloza tory tracking with a time-varying formation. In our control Mendoza et al. (2011); Rend´ on-Benitez et al. (2012). In our trol strategy for multi-agent systems that allows the trajecGonz´ a lez-Sierra and Aranda-Bricaire (2013); Pe˜ n aloza tory tracking with a time-varying formation. In our control Mendoza et al. (2011); formation Rend´ on-Benitez al. (2012). In our strategy the collision avoidance between agents is also concase, the time-varying allows et trajectory tracking tracking with a time-varying formation. In our control Mendoza et al. (2011); formation Rend´ on-Benitez et al. (2012). In our tory the it collision avoidance between agents is also concase, time-varying trajectory sidered and is based on previous works Flores-Resendiz with the formations oriented to the allows heading angle of tracking a leader strategy strategy the collision avoidance between agents is also concase, the time-varying formation allows trajectory tracking sidered and it is based on previous works Flores-Resendiz with formations oriented to the heading angle of a leader and Aranda-Bricaire (2014); Flores-Resendiz et al. (2015). robot, as well as changes in the physical dimensions of the sidered and it is based on previous works Flores-Resendiz with formations oriented to the heading angle of a leader Aranda-Bricaire (2014); Flores-Resendiz et al. (2015). robot, as well as changes in thethe physical dimensions We use bounded control strategies based on sigmoid funcof the and formations. More specifically, time-varying formation Aranda-Bricaire (2014); Flores-Resendiz et al. (2015). robot, as well as changes in the physical dimensions of the and bounded control strategies based on sigmoid funcformations. time-varying tionsuse adding a repulsive vector field. is composedMore of aspecifically, predefined the static formation formation which is We We use bounded control vector strategies based on sigmoid funcformations. More specifically, the time-varying formation is composed of a predefined static formation which is tions adding a repulsive field. transformed by a rotation matrix, that depends on the tions adding vector field. between agents of a thea repulsive interaction topology is composed of aa rotation predefined static formation which is To model transformed matrix, on the orientation ofby a specific leader robot, that and adepends scaling matrix, To model the interaction topology between agents of isa system we use formation graphs, where each agent transformed by a rotation matrix, that depends on the orientation of on a specific scaling to matrix, model the interaction topology between agents of isa that depends a factorleader whichrobot, varies and witharespect time. To system we use formation graphs, where each agent by a vertex and the sharing of information orientation of on a specific scaling to matrix, that depends a factorleader whichrobot, varies and witharespect time. represented system we use formation graphs, where each agent is by is a represented vertex and by thean sharing between agents edge. of information This that depends a factor which by varies with respect time. represented work wason partially supported CONACyT, M´ exico,to through represented by a vertex and the sharing of information between agents is represented by an edge. This work was partially supported by CONACyT, M´ exico, through scholarship No. 243226. between agents is represented by an edge. This workholder was partially supported by CONACyT, M´ exico, through
scholarship holder No. 243226. scholarship holder No. 243226. Copyright © 2017 IFAC 311 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2017 IFAC 311 Peer review under responsibility of International Federation of Automatic Copyright © 2017 IFAC 311 Control. 10.1016/j.ifacol.2017.08.051
Proceedings of the 20th IFAC World Congress 310 J. Santiaguillo-Salinas et al. / IFAC PapersOnLine 50-1 (2017) 309–314 Toulouse, France, July 9-14, 2017
trajectory. The n − 1 remaining agents are follower, responsible for performing a time-varying formation with respect to the leader. The leader agent does not know the position and velocities of the followers agents, only knows the desired trajectory and velocity. The followers do not know the desired trajectory and velocity, only knows the positions and velocities of others agents in the system. Definition 2. (Laplacian). Let us have a formation graph G, the Laplacian associated with G is given by (4) L(G) = ∆ − Ad where ∆ is the degree matrix defined by (5) ∆ = diag {g1 , . . . , gn } where gi = card {Ni }, i = 1, . . . , n and Ad is the adjacency matrix of G defined by 1, if (Rj , Ri ) ∈ E (6) aij = 0, otherwise.
Fig. 1. Scheme of the differential-drive mobile robot. 2. PRELIMINARIES 2.1 Differential-drive Mobile Robots Let N = {R1 , . . . , Rn } be a set of differential-drive mobile T robots moving on the plane with positions ξi = [xi , yi ] , i = 1, . . . , n. The kinematic model for each robot, according to Fig. 1, is given by x˙ i cos θi 0 vi y˙ i = sin θi 0 , i = 1, . . . , n (1) wi ˙θi 0 1
where vi is the longitudinal velocity of the middle point of wheels axis of the i-th robot, wi its angular velocity and θi the orientation with respect to the X axis. Taking as output of the system (1) the position ξi , the so called decoupling matrix becomes singular. For this reason, to avoid singularities in the control law, it is common to study the kinematics of a point αi off the wheels axis. The coordinates of point αi are given by xi + cos θi αxi = (2) αi = αyi yi + sin θi The kinematics of point αi is given by v cos θi − sin θi vi α˙ xi = = Ai (θi ) i sin θi cos θi wi wi α˙ yi
(3)
where Ai (θi ) is the decoupling matrix for each robot Ri and this is non-singular since det (Ai (θi )) = = 0.
2.2 Algebraic Graph Theory Definition 1. (Formation Graph). Let N = {R1 , . . . , Rn } be a set of mobile agents and Ni be the subset of agents which have a flow of information towards the i-th agent. A formation graph G = {V, E, C} consists of 1) A set of vertices V = {R1 , . . . , Rn } corresponding to the n agents of the system, 2) A set of edges E = {(Rj Ri ) ∈ V × V |j ∈ Ni } where each edge represents a flow of information that goes from agent j towards agent i and 3) A set of labels C = {cji = Ri − Rj } with (Rj Ri ) ∈ E, cji ∈ R2 , with cji being a vector specifying a desired relative position between of agent Rj with respect to agent Ri . In the leader-followers scheme used in this work, the agent Rn is the leader, responsible for tracking a desired 312
In the rest of this paper we make the following assumption Assumption 1. For each follower agent, there is a communication either direct or indirect, with the leader agent; i.e., for all Ri , i = 1, . . . , n − 1 there are edges Rn Rm1 , Rm1 Rm2 , . . . , Rmr Ri ∈ E. For further details about algebraic graph theory, the reader is referred to Fax and Murray (2002); Desai (2002); Lafferriere et al. (2004). 2.3 Notations Let us introduce some notation: T
• Let m(t) = [mp (t), mq (t)] be a continuously differentiable preestablished trajectory, where m(t) ˙ ≤ ηm , ∀t ≥ 0. • The desired relative position of the i-th follower within the desired time-varying formation is given by 1 αi∗ (t) = (αj (t) + Cji (t)) , i = 1, ..., n − 1, gi j∈Ni
where Cji , defined in Section IV, is a time-varying position vector between the agents i and j. The time derivative of Cji (t) satisfies C˙ ji (t) ≤ ηc , ∀t ≥ 0. • Given a vector z = [z1 , . . . , zp ]T , we define tanh(z) = [tanh(z1 ), . . . , tanh(zp )]T . • Given a matrix X ∈ Cn×n with λ1 , . . . , λn eigenvalues, then its spectral radius ρ(X) is defined as ρ(X) = max{| λ1 |, . . . , | λn |}. 3. PROBLEM STATEMENT
The goal of this work is to design a decentralized control law ui = (αi , Ni ), i = 1, . . . , n that achieves • Asymptotic tracking of a prescribed trajectory by the leader agent, i.e. lim (αn (t) − m(t)) = 0. t→∞
• Asymptotic time-varying formation by the follower agents, i.e. for i = 1, ..., n − 1 lim (αi (t) − αi∗ (t)) = 0. t→∞
Proceedings of the 20th IFAC World Congress J. Santiaguillo-Salinas et al. / IFAC PapersOnLine 50-1 (2017) 309–314 Toulouse, France, July 9-14, 2017
• Collision avoidance between agents; that is, all agents remain at some distance greater than or equal to a predefined minimum distance d from each other, i.e. αi (t) − αj (t) ≥ d, i, j = 1, . . . , n, i = j, ∀t ≥ 0. 4. CONTROL STRATEGY 4.1 Time-varying Position Vector In order to maintain a formation by the follower agents oriented to the direction of the leader agent and resize the formation, we use a time-varying position vector given by Cji (t) = µ(t)R(θn )cji (7) where cji is a position vector corresponding to the static desired formation, R(θn ) is a rotation matrix given by cos θn − sin θn (8) R(θn ) = sin θn cos θn and µ(t) is a scaling factor. The time-derivative of (7) is given by ˙ ˙ (9) C˙ ji (t) = µ(t)R(θ n )cji + µ(t)R(θn )cji where ˙ n ) = − sin θn − cos θn wn (10) R(θ cos θn − sin θn
0 A = (ΛL(G)) + Γ = (ΛL(G)) + ... 0
i = 1, . . . , n − 1
(12)
A−1 i (θi )
where is the inverse of the decoupling matrix, m(t) is the desired trajectory, m(t) ˙ is the desired velocity, km and kf are the tracking and formation control gains. The first main result of this paper is the following Theorem 1. Consider the system (3) and the control laws (11)-(12). Suppose that km , kf > 0. Then in the closedloop system defined by (3), (11)-(12), it follows that the leader agent Rn converge to the desired marching trajectory, i.e. limt→∞ (αn (t) − m (t)) = 0, whereas the follower agents converge to the desired formation, i.e. limt→∞ (αi (t) − αi∗ (t)) = 0, for i = 1, . . . , n − 1. Proof. The closed-loop system (3), (11)-(12) is given by α˙ = (A ⊗ I2 )−1 [−(K ⊗ I2 ) tanh((A ⊗ I2 )α − C) + M ] (13) T
where α = [α1 , . . . , αn ] , K = diag {kf , . . . , kf , km }, ⊗ denote the Kronecker product, I2 is the identity matrix,
1 1 C = Cji (t), . . . , g1 gn−1
j∈N1
1 ˙ 1 M = Cji (t), . . . , g1 gn−1 j∈N1
j∈Nn−1
j∈Nn−1
T
Cji (t), m(t)
T
˙ C˙ ji (t), m(t)
0 .. . 1
At this point, we have to show that (A ⊗ I2 ) is invertible. From the properties of the Kronecker product we have (A⊗ I2 )−1 = A−1 ⊗ I2−1 . Since I2 is the identity matrix, then I2−1 exits and we address in the matrix A = (ΛL(G)) + Γ. The matrix ΛL(G) is positive semidefinite. Hence, the matrix (ΛL(G)) + Γ is positive definite. In addition, (ΛL(G)) + Γ = (aij ) satisfies aij ≤ 0 for i = j and aii > 0, ∀i, so that from definition of M -matrix Horn and Johnson (2011); Poole (1975) and Assumption 1, (ΛL(G)) + Γ has a inverse and therefore A−1 exist. Now define the errors of the system as en = αn − m(t) ei = αi −
αi∗ , i
= 1, . . . , n − 1
(14) (15)
The system errors in matrix form are given by e = (A ⊗ I2 ) α − C
(16)
T
where e = [e1 , . . . , en ] . The dynamics of the error coordinates are given by (17) e˙ = − (K ⊗ I2 ) tanh(e) We propose a Lyapunov function candidate given by 1 (18) V = eT (K ⊗ I2 )−1 e 2 and evaluating its time derivative along the trajectories of the system we have V˙ = −eT (K ⊗ I2 )−1 (K ⊗ I2 ) tanh(e) = −eT tanh(e) < 0, ∀e with e = 0
(19)
then the errors converge asymptotically to zero. 4.3 Collision Avoidance Once the control strategy for time-varying formation tracking has been designed, we address the problem of collision avoidance between agents by designing a complementary control law based on repulsive vector fields depending on the distance among agents. The distance between any pair of agents is given by αi − αj , ∀i, j ∈ N , i = j. Then, the agents αj which are at risk of collision with the agent αi belong to the set (20) Mi = {αj ∈ N | αi − αj ≤ d} , i = 1, . . . , n where d is the minimum distance allowed between agents. In order to avoid collisions between agents, we propose repulsive vector fields given by (αxi − αxj ) − (αyi − αyj ) βi = (21) δij (αxi − αxj ) + (αyi − αyj ) j∈Mi
313
··· .. . ···
where L(G) of the formation graph G and is the Laplacian 1 1 Λ = diag g1 , . . . , gn−1 , 0 .
4.2 Time-varying Formation Tracking we propose a control law defined by vn ˙ (11) = A−1 n (θn ) (−km tanh (αn − m(t)) + m(t)) wn vi ∗ = A−1 ˙ i∗ ) , i (θi ) (−kf tanh (αi − αi ) + α wi
311
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where > 0 is a design parameter to be defined later and δij is given by 1, if αi − αj ≤ d (22) δij = 0, if αi − αj > d The vector fields are proposed so that, for agent Ri there exists an unstable counterclockwise focus, centered at the position of the other agents in risk of collision. Remark 1. It should be clear that the minimum distance allowed between agents d must be less than the minimum distance between agents in the desired formation, i. e. d < min{ Cij } Moreover we make the following assumption Assumption 2. The initial conditions of all agents satisfy αi (0) − αj (0) ≥ d, ∀i, j ∈ N . Namely, there is no risk of collision between any agent at t = 0. Finally, the control strategy for the agents are given by vn =A−1 ˙ + βn ) n (θn ) (−km tanh(αn − m(t)) + m(t) wn (23) vi ∗ ˙ i∗ + βi ) , =A−1 i (θi ) (−kf tanh (αi − αi ) + α wi i = 1, ..., n − 1 (24) To analyze the relative distance among j-th and i-th agents, we define the variables pji = αxi − αxj and qji = αyi − αyj , j, i = 1, . . . , n, j = i which correspond to the horizontal and vertical distances between agents. In the planes pji − qji , we identify the origin as the point where collision between j-th and i-th agents occurs and a circle of radius d, centered at the origin, as the influence region between the two agents. Outside the circle, only the time-varying formation tracking control law acts while inside the circle the repulsive vector fields appear. In order to present our second main result, we need to establish the following Technical Lemma. Lemma 2. Consider the system (3) and the control laws (11)-(12) along with definitions k ∗ = max(kf , km ) and η ∗ = max(ηm , ηc ). Then in the closed-loop system (3)(11)-(12) are bounded by the velocities of √ agents∗ √ ∗the −1 T −1 ηˆ = ρ ((A ) (A )) k 2n + η n .
Proof. Taking the norm of the system (13) we get
α˙ ≤ (A ⊗ I2 )−1 [−(K ⊗ I2 ) tanh((A ⊗ I2 )α − C) +M ] ≤ (A ⊗ I2 )−1 ( −(K ⊗ I2 )
tanh((A ⊗ I2 )α − C) + M ) √ √ where tanh((A ⊗ I2 )α − C) ≤ 2n, M ≤ η ∗ n, with −(K ⊗ I2 ) = ρ (K ⊗ I2 )T (K ⊗ I2 ) and (A ⊗ I2 )−1 = ρ ([(A ⊗ I2 )−1 ]T [(A ⊗ I2 )−1 ]), but since I2 is the identity matrix with two eigenvalues 1 and from the spectrum properties of the Kronecker product we have −(K ⊗ I2 ) = ρ K T K = max(kf , km ) = k ∗ and (A ⊗ I2 )−1 = ρ ((A−1 )T (A−1 )). Finally, we have √ √ α˙ ≤ ρ ((A−1 )T (A−1 )) k ∗ 2n + η ∗ n = η. (25)
314
Now, we can state our second main result. First, we consider the case when only two agents are in risk of collision. From this simplest case, we state a series of theorems leading to the general case. Theorem 3. Considere the system (3) and the control laws (23)-(24). Suppose that there exists risk of collision between only two agents at time instant t and satisfies > ηdˆ . Then, in the closed-loop system (3)-(23)-(24) the agents tend asymptotically to their desired positions and they stay at a distance greater than or equal to d, ∀t ≥ 0. Proof. We show that the r-th and s-th agents will avoid collision between them and they stay at some minimum distance from each other. Define a surface given by 2 − d2 = 0 (26) σrs = p2rs + qrs To determine the behavior under the action of the repulsive vector fields we use the positive definite function 1 2 V = σrs (27) 2 whose time derivative is given by V˙ = σrs σ˙ rs . The time derivative of (26) along the trajectories of the closed-loop system is given by p˙ rs σ˙ rs = 2 [ prs qrs ] q˙rs
= −2 [ prs qrs ] ((−ks tanh(αs − αs∗ ) + α˙ s∗ )
2 −(−kr tanh(αr − αr∗ ) + α˙ r∗ )) + 4 (p2rs + qrs )(28)
Therefore, V˙ ≤ 0 is achieved if σrs σ˙ rs ≤ 0. When there exists risk of collision, (prs , qrs ) lies in the inner region of σrs = 0, that is σrs ≤ 0, then the analysis reduces to show that σ˙ rs ≥ 0. That means the resulting vector fields inside the circle points outwards; that is, to the region free of collision. Using the definition of the cross product we have 2 2 η σ˙ rs = 4 p2rs + qrs ˆ cos θrs + 4 (p2rs + qrs ) 2 2 2 2 ≥ −4 prs + qrs ηˆ + 4 (prs + qrs ) > 0. (29) Solving for we have that, if > ηˆ/d then σ˙ rs > 0. This implies that rth and sth agents move away from each other until they reach a distance d. Since αs (0) − αr (0) ≥ d, then the agents not only avoid collision but also satisfy αs (t) − αr (t) ≥ d for all time.
Now, we consider the case when three agents are in risk of collision, that is, agent r is in risk of collision against agents s1 and s2. Theorem 4. Consider the system (3) and the control laws (23)-(24). Suppose that there exists risk of collision between three agents and satisfies > 2(ˆ η /d). Then, in the closed-loop system (3)-(23)-(24) the agents converge asymptotically to their desired positions and they stay at a distance greater than or equal to d, ∀t ≥ 0. Proof. We define a surface composed of 2 components given by 2 2 σrs1 prs1 + qrs1 − d2 σ= = = 0. (30) 2 σrs2 p2rs2 + qrs2 − d2 We use the positive definite function
Proceedings of the 20th IFAC World Congress J. Santiaguillo-Salinas et al. / IFAC PapersOnLine 50-1 (2017) 309–314 Toulouse, France, July 9-14, 2017
(31)
V˙ = σ T σ˙ = σrs1 σ˙ rs1 + σrs2 σ˙ rs2 ≤ σ ∗ (σ˙ rs1 + σ˙ rs2 )(32) where σ ∗ = max{σrs1 , σrs2 }. Evaluating V˙ considering that the trajectories lie in the inner region of σ = 0, that is, σrs1 , σrs2 < 0 then the analysis reduces to show that σ˙ rs1 + σ˙ rs2 > 0. Hence ∗ ∗ ) + α˙ s1 ) σ˙ rs1 + σ˙ rs2 = 2 [ prs1 qrs1 ] ((−ks1 tanh(αs1 − αs1 2 ) −(−kr tanh(αr − αr∗ )α˙ r∗ )) + 4(p2rs1 + qrs1 ∗ ∗ + 2 [ prs2 qrs2 ] ((−ks2 tanh(αs2 − αs2 ) + α˙ s2 ) ∗ ∗ 2 2 −(−kr tanh(αr − αr ) + α˙ r )) + 4(prs2 + qrs2 ) prs2 + 4 [ prs1 qrs1 ] qrs2 2 2 η ≥ −4 p2rs1 + qrs1 ˆ + 4(p2rs1 + qrs1 ) 2 2 η − 4 p2rs2 + qrs2 ˆ + 4(p2rs2 + qrs2 ) 2 2 cos θ + 4 p2rs1 + qrs1 p2rs2 + qrs2 rs1,rs2 > 0. (33) In this scenario, agents s1 and s2 can be positioned at any point of the circumference of radius d around the agent r, considering that, from Theorem 3, they must remain at a distance greater than or equal to d between them. The worst case occurs when the agents s1 and s2 are uniformly distributed over the circumference of radius d. Thus, cos θrs1,rs2 = −1 and solving for we have that, if > 2( η /d) then σ˙ rs1 + σ˙ rs2 > 0. This implies that agents s1, s2 and r avoid collision between them. Geometrically, the most general case occurs when the r-th agent is surrounded by six agents, i.e. seven agents are in danger of collision. Theorem 5. Consider the system (3) and the control laws (23)-(24). Suppose that there exists risk of collision between n ≥ 3 agents and satisfies > 2(ˆ η /d). Then, in the closed-loop system (3)-(23)-(24) the agents converge asymptotically to their desired positions and they stay at a distance greater than or equal to d, ∀t ≥ 0. Proof. We follow a similar procedure to that presented in the proof of Theorem 4, considering a surface with n − 1 components and showing that, if σ˙ rs1 + . . . + σ˙ r(n−1) > 0, then V˙ < 0, taking into account that the worst case is presented when the n − 1 agents are uniformly distributed over the circumference of radio d around the r-th agent, so the agents avoid collision between them. 5. NUMERICAL SIMULATION The results of a numerical simulation using the control strategy given by (23)-(24) are shown below. For the simulation, we considered 5 differential-drive mobile robots, where the point αi to control is located 0.15m ahead the midpoint of the wheel axis. The formation graph employed in the simulation is shown in Fig. 2. The control gains used in the simulation are km = 2 and kf = 3. The desired trajectory is a quadrifolium curve 315
Fig. 2. Formation graph for the simulation.
Initial Pos.
End Pos.
Agents
Desired Trajectory
3
2
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1 V = σT σ 2 whose time derivative is given by
313
1
0
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-1
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1
2
3
4
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Fig. 3. Trajectories of the agents in the plane. given by m(t) = [4 sin (2ωt) cos (ωt) , 4 sin (2ωt) sin (ωt)], where ω = 2π T with a period of T = 80 s. The static position given by c12 = [0, 0.6], c21 = [0, −0.6], vectors are π π c32 = −0.6 cos( 10 ), −0.6 sin( 10 ) , c34 = [0, −0.97], c43 = 3π 3π [0, 0.97] and c54 = −0.6 cos( 10 ), −0.6 sin( 10 ) . The scaling factor is given by µ(t) = 1 + 0.2 sin(ωt). The minimum allowed distance between agents is d = 0.3 m and was set to = 4(2(ˆ η /d)), to ensure the minimum distance condition will not be violated. Fig. 3 shows the motion of the agents in the plane. The initial position of the agents are indicated with an “x” and positions in different time instants are represented with a circle “o”. Is observed how the leader follows the desired trajectory while the followers achieve a timevarying formation. Furthermore, the minimum distance requirement is satisfied. Fig. 4 show the errors of the agents, such errors converge to zero. Fig. 5 depicts all the posible distances between agents. The distances between any pair of agents is always greater than or equal to the predefined distance d = 0.3. 6. CONCLUSIONS AND OUTLOOKS This paper presents a time-varying formation tracking control with collision avoidance for multi-agent systems. We propose a bounded control law complemented with a repulsive vector field of the unstable focus type, so that trajectory tracking with a time-varying formation and collision avoidance between agents are guaranteed. As shown in the numerical simulation, the goals are achieved. As future work, it is proposed to control the midpoint of wheel axis of the differential-drive mobile robots and include a strategy for obstacle avoidance. It is also intended to validate the theoretical results obtained through realtime experiments.
Proceedings of the 20th IFAC World Congress 314 J. Santiaguillo-Salinas et al. / IFAC PapersOnLine 50-1 (2017) 309–314 Toulouse, France, July 9-14, 2017
4.5
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Fig. 4. Errors of the agents. 2 1.8 1.6
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Fig. 5. Distances among agents of the system. REFERENCES Arai, T., Pagello, E., and Parker, L.E. (2002). Guest editorial advances in multirobot systems. IEEE Transactions on Robotics and Automation, 18, 655–661. Bri˜ n´ on Arranz, L., Seuret, A., and Canudas-de Wit, C. (2014). Cooperative control design for time-varying formations of multi-agent systems. IEEE Transactions on Automatic Control, 59(8), 2283–2288. Cao, Y.U., Fukunaga, A.S., and Kahng, A.B. (1997). Cooperative mobile robotics: antecedents and directions. Autonomous Robots, 4, 226–234. De Gennaro, M.C. and Jadbabaie, A. (2006). Formation control for a cooperative multi-agent system using decentralized navigation functions. American Control Conference, 1346–1351. Desai, J.P. (2002). A graph theoretic approach for modeling mobile robot team formations. Journal of Robotic Systems, 19(11), 511–525. Dimarogonas, D.V. and Kyriakopoulos, K.J. (2006). Distributed cooperative control and collision avoidance for multiple kinematic agents. In Proc. of the 45th IEEE Conference on Decision and Control, 721–726. doi: 10.1109/CDC.2006.376884. Dimarogonas, D.V., Loizou, S.G., Kyriakopoulos, K.J., and Zavlanos, M.M. (2006). A feedback stabilization 316
and collision avoidance scheme for multiple independent non-point agents. Automatica, 42, 229–243. Fax, J.A. and Murray, R.M. (2002). Graph laplacians and stabilization of vehicle formations. In Proc. of the 15th IFAC World Congress, 15. Flores-Resendiz, J.F. and Aranda-Bricaire, E. (2014). Cyclic pursuit formation control without collisions in multi-agent systems using discontinuous vector fields. In Proc. of the XVI Congreso Latinoamericano de Control Autom´ atico, 941–946. Flores-Resendiz, J.F., Aranda-Bricaire, E., Gonzlez-Sierra, J., and Santiaguillo-Salinas, J. (2015). Finite-time formation control without collisions for multiagent systems with communication graphs composed of cyclic paths. Mathematical Problems in Engineering, 17. doi: 10.1155/2015/948086. Gonz´alez-Sierra, J. and Aranda-Bricaire, E. (2013). Design of a virtual mechanism for trajectory tracking of convoys of mobile robots. 10th Int. Conf. on Electrical Engineering, Computing Science and Automatic Control, 364– 368. Hern´andez-Mart´ınez, E. and Aranda-Bricaire, E. (2009). Multi-agent formation control with collision avoidance based on discontinuous vector fields. In Proc. of the 35th Annual Conference of IEEE Industrial Electronics, 2283–2288. doi:10.1109/IECON.2009.5415217. Hern´andez-Mart´ınez, E. and Aranda-Bricaire, E. (2013). Collision avoidance in formation control using discontinuous vector fields. In Proc. of the 9th IFAC Symposium on Nonlinear Control Systems, 797–802. Horn, R.A. and Johnson, C.R. (2011). Topics in matrix analysis. Cambridge University Press. Lafferriere, G., Caughman, J., and Williams, A. (2004). Graph theoretic methods in the stability of vehicle formations. American Control Conference, 4, 3729– 3734. Loizou, S.G., Tannert, H.G., Kumar, V., and kyriakopoulos, K.J. (2003). Closed loop navigation for mobile agents in dynamic environments. In Proc. of the 2003 IEEE/RSJ Int. Conf. on Intelligent Robots and Systems, 4, 3769–3774. doi:10.1109/IROS.2003.1249741. Pe˜ naloza Mendoza, G.R., Hern´andez-Mendoza, D.E., and Aranda-Bricaire, E. (2011). Time-varying formation control for multi-agent systems applied to n-trailer configuration. 8th Int. Conf. on Electrical Engineering, Computing Science and Automatic Control, 1–6. Poole, G. (1975). Generalized m-matrices and applications. Mathematics of Computation, 29(131), 903–910. Qianwei, H., Hongxu, M., and Hui, Z. (2003). Collisionavoidance mechanism of multi agent system. In Proc. of the 2003 IEEE Int. Conf. on Robotics, Intelligent Systems and Signal Processing, 2, 1036–1040. doi: 10.1109/RISSP.2003.1285732. Rend´on-Benitez, F., Santiaguillo-Salinas, J., Gonz´ alezSierra, J., and Aranda-Bricaire, E. (2012). Marching control of multi-agent systems with orientation to the marching angle of the leader (in spanish). In Proc. of the XV Congreso Latinoamericano de Control Autom´ atico. Yao, J., Ordoez, R., and Gazi, V. (2006). Swarm tracking using artificial potentials and sliding mode control. In Proc. of the 45th IEEE Conference on Decision and Control, 4670–4675. doi:10.1109/CDC.2006.377565.
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IFAC PapersOnLine 50-1 (2017) 309–314 Time-varying Formation Tracking Time-varying Formation Tracking Time-varying Formation Tracking with Collision Avoidance for with Collision Avoidance for with Collision Avoidance Multi-agent Systems for Multi-agent Systems Multi-agent Systems ∗
J. Santiaguillo-Salinas and E. Aranda-Bricaire ∗ J. Santiaguillo-Salinas and E. Aranda-Bricaire ∗∗ J. Santiaguillo-Salinas and E. Aranda-Bricaire ∗ Department of Electrical Engineering, Mechatronics Section, ∗ ∗ Department of Electrical Engineering,exico, Mechatronics D.F., M´eSection, xico. ∗ CINVESTAV AP 14-740, 07000 M´ Department of Electrical Engineering, Mechatronics CINVESTAV AP 14-740, 07000 M´ e xico, D.F., M´eSection, xico. (e-mail: [email protected], [email protected]). CINVESTAV AP 14-740, 07000 M´ exico, D.F., M´exico. (e-mail: [email protected], [email protected]). (e-mail: [email protected], [email protected]). Abstract: This paper studies the collision avoidance problem in the time-varying formation Abstract: This for paper studies the collision in the time-varying formation tracking control multi-agent systems. Theavoidance proposedproblem control strategy is decentralized, since Abstract: This for paper studies the collision avoidance problem in the time-varying formation tracking control multi-agent Theto proposed control strategy decentralized, since agents have no global knowledgesystems. of the goal achieve, knowing only theisposition and velocity tracking control for multi-agent systems. Thetoproposed control strategy isposition decentralized, since agents no global knowledge of theallows goal knowing onlyto thetrack and velocity of somehave agents. This control strategy a achieve, set of mobile agents a predetermined agents have no global knowledge of theallows goal toa achieve, knowing onlyto thetrack position and velocity of some agents. This control strategy set of mobile agents a predetermined trajectory while they achieve a time-varying formation. For the collision avoidance, we add a of some agents. This control strategy allows formation. a set of mobile agents to track a predetermined trajectory while field they of achieve a time-varying For the collision avoidance, wecontrol add a repulsive vector the unstable focus type to the time-varying formation tracking trajectory while they achieve a time-varying formation. For the collision avoidance, wecontrol add a repulsive vector field of scheme the unstable focus type to the time-varying tracking law. A leader-followers is employed, using formation graphsformation to represent interactions repulsive vector field of scheme the unstable focus type to the time-varying formation tracking control law. A leader-followers is employed, using formation graphs to represent interactions between agents. The results are presented for the front points of differential-drive mobile robots. law. A leader-followers scheme is employed, using formation to represent interactions between agents. results The results are presented for the front points ofgraphs differential-drive mobile robots. The theoretical are verified by numerical simulation. between agents. The results are presented for the front points of differential-drive mobile robots. The theoretical results are verified by numerical simulation. The theoretical results areFederation verified by numericalControl) simulation. © 2017, IFAC (International of Automatic Hosting by Elsevier Ltd. All rights reserved. Keywords: Multi-agent systems, time-varying formation, collision avoidance, differential-drive Keywords: Multi-agent time-varying formation, collision avoidance, differential-drive mobile robots, repulsivesystems, vector fields. Keywords: Multi-agent time-varying formation, collision avoidance, differential-drive mobile robots, repulsivesystems, vector fields. mobile robots, repulsive vector fields. 1. INTRODUCTION This time-varying formation allows the group of agents to 1. INTRODUCTION This time-varying formation allows thetranslated, group of agents to behave as a rigid body which can be rotated 1. INTRODUCTION This time-varying formation allows thetranslated, group of agents to behave as a rigid body which can be rotated and scaled in the plane. Multi-agent Systems are conceived as bundles of multiple behave as a rigid body which can be translated, rotated Multi-agent are conceived as bundles cooperative of multiple and scaled in the plane. autonomous Systems robots coordinated to accomplish scaledubiquitous in the plane. Another problem in all areas of motion coMulti-agent Systems are conceived as bundles cooperative of multiple and autonomous robots coordinated tomulti-agent accomplish tasks. In recent years, the study of systems has Another ubiquitous problem in allbetween areas ofagents motionwhen coordination is the posible collision autonomous robots coordinated tomulti-agent accomplish systems cooperative tasks. In recent years, the study of has Another ubiquitous problem in allbetween areas ofagents motionwhen cogained special interest, because multiple agents can solve ordination isachieve the posible collision they try to a desired position into a formation tasks. In recent years, the study of multi-agent systems has gained special cooperatively; interest, because multiple agents solve ordination isachieve the posible collision between agents when tasks working making them morecan reliable, they try to a desired position into a formation the trajectory tracking. In the literature, we gained special cooperatively; interest, because multiple agents can solve or during tasks reliable, try tothe achieve a desired position intoliterature, a formation faster working and cheaper than it is making possible them with amore single agent they or during trajectory the we can find different methodstracking. to avoidIncollisions. In Qiantasks working cooperatively; making them more reliable, faster cheaper than it is possible with a single agent or during the trajectory tracking. In the literature, we Cao etand al. (1997). can find different methods to avoid collisions. In Qianwei et al. (2003) a mechanism for collision avoidance fasteretand cheaper than it is possible with a single agent can find different methods to avoid collisions. In QianCao al. (1997). wei et al. (2003) a mechanism for collision avoidance on traffic control type is presented. In De Gennaro Cao al. (1997). The et main applications of multi-agent systems include based wei etonal.traffic (2003) a mechanism for collision control type is presented. Deavoidance Gennaro The main and applications and Jadbabaie (2006); Dimarogonas and In Kyriakopoulos of multi-agent include transport manipulation of objects, systems localization, ex- based based on traffic control type is presented. In De Gennaro The main applications of multi-agent systems include and Jadbabaie (2006); Dimarogonas and Kyriakopoulos transport and manipulation of objects, localization, exDimarogonas et al. (2006) navigation functions ploration and motion coordination Arai et al. (2002); Cao (2006); Jadbabaie (2006); Dimarogonas and Kyriakopoulos transport andmotion manipulation of objects, ex- and Dimarogonas al. (2006) functions ploration and coordination Arai etlocalization, al. (2002); Cao and artificial potentialetfunctions arenavigation used to avoid colliet al. (1997). The main idea of motion coordination is (2006); (2006); Dimarogonas et al. (2006) navigation functions ploration and motion coordination Arai et al. (2002); Cao and artificial potential functions are used avoid colliet (1997).navigation The mainofidea of motion coordination is sions between agents. These non collision to strategies are theal. strategic a group of agents. One of the and artificial potential functions are used to avoid colliet al. (1997). The main idea of motion coordination is sions between agents. These non collision strategies are the a group of coordination agents. One of the developed based on a combination of attractive potential mainstrategic areas ofnavigation research inofthe motion is the between agents. These non collision strategies are the strategic navigation ofthe a group of coordination agents. One of the sions main areas of research in motion is the developed based on a combination of attractive potential (APF) and repulsive potential functions (RPF). formation tracking problem, where the goal is to track functions developed based on a combination of attractive potential main areas of research in the motion coordination is the (APF) and repulsive potential functions (RPF). formation tracking problem,while wherethethe goal maintain is to tracka functions Works Hern´ andez-Mart´ ınez and Aranda-Bricaire (2009, a preestablished trajectory agents functions (APF) and repulsive potential functions (RPF). formation tracking problem,while where the goal maintain is to tracka Works Hern´ andez-Mart´ ınez and (2009, adesired preestablished trajectory 2013); Loizou et al. (2003); YaoAranda-Bricaire et al. (2006) address agents pattern defined by relative the position vectors. Hern´andez-Mart´ ınez and Aranda-Bricaire (2009, a preestablished trajectory while the agents maintain a Works 2013); Loizou et al. (2003); Yao et al. (2006) address desired pattern defined by relative position vectors. the formation control problem without collisions using 2013); Loizou et al. (2003); Yao et al. (2006) address desired pattern defined by relative position vectors. The time-varying formation control can be applied as the discontinuous the formationvector control problem without collisions using fields. The time-varying formation can be applied the the formationvector control problem without collisions using solution to complex motioncontrol coordination problemsas and discontinuous fields. The time-varying formation control can be applied as and the The purpose ofvector solution to complex fields.is to design a decentralized conthis paper some examples can be motion found incoordination Bri˜ n´ on Arranzproblems et al. (2014); discontinuous solution to complex problems and trol strategy purpose for of this paper is systems to design a decentralized consome be motion found incoordination Bri˜ n´ on Arranz et al. (2014); multi-agent that allows the trajecGonz´ aexamples lez-Sierracan and Aranda-Bricaire (2013); Pe˜ naloza The The purpose for of this paper is systems to design a decentralized consome examples can be found in Bri˜ n o ´ n Arranz et al. (2014); trol strategy multi-agent that allows the trajecGonz´ a lez-Sierra and Aranda-Bricaire (2013); Pe˜ n aloza tory tracking with a time-varying formation. In our control Mendoza et al. (2011); Rend´ on-Benitez et al. (2012). In our trol strategy for multi-agent systems that allows the trajecGonz´ a lez-Sierra and Aranda-Bricaire (2013); Pe˜ n aloza tory tracking with a time-varying formation. In our control Mendoza et al. (2011); formation Rend´ on-Benitez al. (2012). In our strategy the collision avoidance between agents is also concase, the time-varying allows et trajectory tracking tracking with a time-varying formation. In our control Mendoza et al. (2011); formation Rend´ on-Benitez et al. (2012). In our tory the it collision avoidance between agents is also concase, time-varying trajectory sidered and is based on previous works Flores-Resendiz with the formations oriented to the allows heading angle of tracking a leader strategy strategy the collision avoidance between agents is also concase, the time-varying formation allows trajectory tracking sidered and it is based on previous works Flores-Resendiz with formations oriented to the heading angle of a leader and Aranda-Bricaire (2014); Flores-Resendiz et al. (2015). robot, as well as changes in the physical dimensions of the sidered and it is based on previous works Flores-Resendiz with formations oriented to the heading angle of a leader Aranda-Bricaire (2014); Flores-Resendiz et al. (2015). robot, as well as changes in thethe physical dimensions We use bounded control strategies based on sigmoid funcof the and formations. More specifically, time-varying formation Aranda-Bricaire (2014); Flores-Resendiz et al. (2015). robot, as well as changes in the physical dimensions of the and bounded control strategies based on sigmoid funcformations. time-varying tionsuse adding a repulsive vector field. is composedMore of aspecifically, predefined the static formation formation which is We We use bounded control vector strategies based on sigmoid funcformations. More specifically, the time-varying formation is composed of a predefined static formation which is tions adding a repulsive field. transformed by a rotation matrix, that depends on the tions adding vector field. between agents of a thea repulsive interaction topology is composed of aa rotation predefined static formation which is To model transformed matrix, on the orientation ofby a specific leader robot, that and adepends scaling matrix, To model the interaction topology between agents of isa system we use formation graphs, where each agent transformed by a rotation matrix, that depends on the orientation of on a specific scaling to matrix, model the interaction topology between agents of isa that depends a factorleader whichrobot, varies and witharespect time. To system we use formation graphs, where each agent by a vertex and the sharing of information orientation of on a specific scaling to matrix, that depends a factorleader whichrobot, varies and witharespect time. represented system we use formation graphs, where each agent is by is a represented vertex and by thean sharing between agents edge. of information This that depends a factor which by varies with respect time. represented work wason partially supported CONACyT, M´ exico,to through represented by a vertex and the sharing of information between agents is represented by an edge. This work was partially supported by CONACyT, M´ exico, through scholarship No. 243226. between agents is represented by an edge. This workholder was partially supported by CONACyT, M´ exico, through
scholarship holder No. 243226. scholarship holder No. 243226. Copyright © 2017 IFAC 311 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2017 IFAC 311 Peer review under responsibility of International Federation of Automatic Copyright © 2017 IFAC 311 Control. 10.1016/j.ifacol.2017.08.051
Proceedings of the 20th IFAC World Congress 310 J. Santiaguillo-Salinas et al. / IFAC PapersOnLine 50-1 (2017) 309–314 Toulouse, France, July 9-14, 2017
trajectory. The n − 1 remaining agents are follower, responsible for performing a time-varying formation with respect to the leader. The leader agent does not know the position and velocities of the followers agents, only knows the desired trajectory and velocity. The followers do not know the desired trajectory and velocity, only knows the positions and velocities of others agents in the system. Definition 2. (Laplacian). Let us have a formation graph G, the Laplacian associated with G is given by (4) L(G) = ∆ − Ad where ∆ is the degree matrix defined by (5) ∆ = diag {g1 , . . . , gn } where gi = card {Ni }, i = 1, . . . , n and Ad is the adjacency matrix of G defined by 1, if (Rj , Ri ) ∈ E (6) aij = 0, otherwise.
Fig. 1. Scheme of the differential-drive mobile robot. 2. PRELIMINARIES 2.1 Differential-drive Mobile Robots Let N = {R1 , . . . , Rn } be a set of differential-drive mobile T robots moving on the plane with positions ξi = [xi , yi ] , i = 1, . . . , n. The kinematic model for each robot, according to Fig. 1, is given by x˙ i cos θi 0 vi y˙ i = sin θi 0 , i = 1, . . . , n (1) wi ˙θi 0 1
where vi is the longitudinal velocity of the middle point of wheels axis of the i-th robot, wi its angular velocity and θi the orientation with respect to the X axis. Taking as output of the system (1) the position ξi , the so called decoupling matrix becomes singular. For this reason, to avoid singularities in the control law, it is common to study the kinematics of a point αi off the wheels axis. The coordinates of point αi are given by xi + cos θi αxi = (2) αi = αyi yi + sin θi The kinematics of point αi is given by v cos θi − sin θi vi α˙ xi = = Ai (θi ) i sin θi cos θi wi wi α˙ yi
(3)
where Ai (θi ) is the decoupling matrix for each robot Ri and this is non-singular since det (Ai (θi )) = = 0.
2.2 Algebraic Graph Theory Definition 1. (Formation Graph). Let N = {R1 , . . . , Rn } be a set of mobile agents and Ni be the subset of agents which have a flow of information towards the i-th agent. A formation graph G = {V, E, C} consists of 1) A set of vertices V = {R1 , . . . , Rn } corresponding to the n agents of the system, 2) A set of edges E = {(Rj Ri ) ∈ V × V |j ∈ Ni } where each edge represents a flow of information that goes from agent j towards agent i and 3) A set of labels C = {cji = Ri − Rj } with (Rj Ri ) ∈ E, cji ∈ R2 , with cji being a vector specifying a desired relative position between of agent Rj with respect to agent Ri . In the leader-followers scheme used in this work, the agent Rn is the leader, responsible for tracking a desired 312
In the rest of this paper we make the following assumption Assumption 1. For each follower agent, there is a communication either direct or indirect, with the leader agent; i.e., for all Ri , i = 1, . . . , n − 1 there are edges Rn Rm1 , Rm1 Rm2 , . . . , Rmr Ri ∈ E. For further details about algebraic graph theory, the reader is referred to Fax and Murray (2002); Desai (2002); Lafferriere et al. (2004). 2.3 Notations Let us introduce some notation: T
• Let m(t) = [mp (t), mq (t)] be a continuously differentiable preestablished trajectory, where m(t) ˙ ≤ ηm , ∀t ≥ 0. • The desired relative position of the i-th follower within the desired time-varying formation is given by 1 αi∗ (t) = (αj (t) + Cji (t)) , i = 1, ..., n − 1, gi j∈Ni
where Cji , defined in Section IV, is a time-varying position vector between the agents i and j. The time derivative of Cji (t) satisfies C˙ ji (t) ≤ ηc , ∀t ≥ 0. • Given a vector z = [z1 , . . . , zp ]T , we define tanh(z) = [tanh(z1 ), . . . , tanh(zp )]T . • Given a matrix X ∈ Cn×n with λ1 , . . . , λn eigenvalues, then its spectral radius ρ(X) is defined as ρ(X) = max{| λ1 |, . . . , | λn |}. 3. PROBLEM STATEMENT
The goal of this work is to design a decentralized control law ui = (αi , Ni ), i = 1, . . . , n that achieves • Asymptotic tracking of a prescribed trajectory by the leader agent, i.e. lim (αn (t) − m(t)) = 0. t→∞
• Asymptotic time-varying formation by the follower agents, i.e. for i = 1, ..., n − 1 lim (αi (t) − αi∗ (t)) = 0. t→∞
Proceedings of the 20th IFAC World Congress J. Santiaguillo-Salinas et al. / IFAC PapersOnLine 50-1 (2017) 309–314 Toulouse, France, July 9-14, 2017
• Collision avoidance between agents; that is, all agents remain at some distance greater than or equal to a predefined minimum distance d from each other, i.e. αi (t) − αj (t) ≥ d, i, j = 1, . . . , n, i = j, ∀t ≥ 0. 4. CONTROL STRATEGY 4.1 Time-varying Position Vector In order to maintain a formation by the follower agents oriented to the direction of the leader agent and resize the formation, we use a time-varying position vector given by Cji (t) = µ(t)R(θn )cji (7) where cji is a position vector corresponding to the static desired formation, R(θn ) is a rotation matrix given by cos θn − sin θn (8) R(θn ) = sin θn cos θn and µ(t) is a scaling factor. The time-derivative of (7) is given by ˙ ˙ (9) C˙ ji (t) = µ(t)R(θ n )cji + µ(t)R(θn )cji where ˙ n ) = − sin θn − cos θn wn (10) R(θ cos θn − sin θn
0 A = (ΛL(G)) + Γ = (ΛL(G)) + ... 0
i = 1, . . . , n − 1
(12)
A−1 i (θi )
where is the inverse of the decoupling matrix, m(t) is the desired trajectory, m(t) ˙ is the desired velocity, km and kf are the tracking and formation control gains. The first main result of this paper is the following Theorem 1. Consider the system (3) and the control laws (11)-(12). Suppose that km , kf > 0. Then in the closedloop system defined by (3), (11)-(12), it follows that the leader agent Rn converge to the desired marching trajectory, i.e. limt→∞ (αn (t) − m (t)) = 0, whereas the follower agents converge to the desired formation, i.e. limt→∞ (αi (t) − αi∗ (t)) = 0, for i = 1, . . . , n − 1. Proof. The closed-loop system (3), (11)-(12) is given by α˙ = (A ⊗ I2 )−1 [−(K ⊗ I2 ) tanh((A ⊗ I2 )α − C) + M ] (13) T
where α = [α1 , . . . , αn ] , K = diag {kf , . . . , kf , km }, ⊗ denote the Kronecker product, I2 is the identity matrix,
1 1 C = Cji (t), . . . , g1 gn−1
j∈N1
1 ˙ 1 M = Cji (t), . . . , g1 gn−1 j∈N1
j∈Nn−1
j∈Nn−1
T
Cji (t), m(t)
T
˙ C˙ ji (t), m(t)
0 .. . 1
At this point, we have to show that (A ⊗ I2 ) is invertible. From the properties of the Kronecker product we have (A⊗ I2 )−1 = A−1 ⊗ I2−1 . Since I2 is the identity matrix, then I2−1 exits and we address in the matrix A = (ΛL(G)) + Γ. The matrix ΛL(G) is positive semidefinite. Hence, the matrix (ΛL(G)) + Γ is positive definite. In addition, (ΛL(G)) + Γ = (aij ) satisfies aij ≤ 0 for i = j and aii > 0, ∀i, so that from definition of M -matrix Horn and Johnson (2011); Poole (1975) and Assumption 1, (ΛL(G)) + Γ has a inverse and therefore A−1 exist. Now define the errors of the system as en = αn − m(t) ei = αi −
αi∗ , i
= 1, . . . , n − 1
(14) (15)
The system errors in matrix form are given by e = (A ⊗ I2 ) α − C
(16)
T
where e = [e1 , . . . , en ] . The dynamics of the error coordinates are given by (17) e˙ = − (K ⊗ I2 ) tanh(e) We propose a Lyapunov function candidate given by 1 (18) V = eT (K ⊗ I2 )−1 e 2 and evaluating its time derivative along the trajectories of the system we have V˙ = −eT (K ⊗ I2 )−1 (K ⊗ I2 ) tanh(e) = −eT tanh(e) < 0, ∀e with e = 0
(19)
then the errors converge asymptotically to zero. 4.3 Collision Avoidance Once the control strategy for time-varying formation tracking has been designed, we address the problem of collision avoidance between agents by designing a complementary control law based on repulsive vector fields depending on the distance among agents. The distance between any pair of agents is given by αi − αj , ∀i, j ∈ N , i = j. Then, the agents αj which are at risk of collision with the agent αi belong to the set (20) Mi = {αj ∈ N | αi − αj ≤ d} , i = 1, . . . , n where d is the minimum distance allowed between agents. In order to avoid collisions between agents, we propose repulsive vector fields given by (αxi − αxj ) − (αyi − αyj ) βi = (21) δij (αxi − αxj ) + (αyi − αyj ) j∈Mi
313
··· .. . ···
where L(G) of the formation graph G and is the Laplacian 1 1 Λ = diag g1 , . . . , gn−1 , 0 .
4.2 Time-varying Formation Tracking we propose a control law defined by vn ˙ (11) = A−1 n (θn ) (−km tanh (αn − m(t)) + m(t)) wn vi ∗ = A−1 ˙ i∗ ) , i (θi ) (−kf tanh (αi − αi ) + α wi
311
Proceedings of the 20th IFAC World Congress 312 J. Santiaguillo-Salinas et al. / IFAC PapersOnLine 50-1 (2017) 309–314 Toulouse, France, July 9-14, 2017
where > 0 is a design parameter to be defined later and δij is given by 1, if αi − αj ≤ d (22) δij = 0, if αi − αj > d The vector fields are proposed so that, for agent Ri there exists an unstable counterclockwise focus, centered at the position of the other agents in risk of collision. Remark 1. It should be clear that the minimum distance allowed between agents d must be less than the minimum distance between agents in the desired formation, i. e. d < min{ Cij } Moreover we make the following assumption Assumption 2. The initial conditions of all agents satisfy αi (0) − αj (0) ≥ d, ∀i, j ∈ N . Namely, there is no risk of collision between any agent at t = 0. Finally, the control strategy for the agents are given by vn =A−1 ˙ + βn ) n (θn ) (−km tanh(αn − m(t)) + m(t) wn (23) vi ∗ ˙ i∗ + βi ) , =A−1 i (θi ) (−kf tanh (αi − αi ) + α wi i = 1, ..., n − 1 (24) To analyze the relative distance among j-th and i-th agents, we define the variables pji = αxi − αxj and qji = αyi − αyj , j, i = 1, . . . , n, j = i which correspond to the horizontal and vertical distances between agents. In the planes pji − qji , we identify the origin as the point where collision between j-th and i-th agents occurs and a circle of radius d, centered at the origin, as the influence region between the two agents. Outside the circle, only the time-varying formation tracking control law acts while inside the circle the repulsive vector fields appear. In order to present our second main result, we need to establish the following Technical Lemma. Lemma 2. Consider the system (3) and the control laws (11)-(12) along with definitions k ∗ = max(kf , km ) and η ∗ = max(ηm , ηc ). Then in the closed-loop system (3)(11)-(12) are bounded by the velocities of √ agents∗ √ ∗the −1 T −1 ηˆ = ρ ((A ) (A )) k 2n + η n .
Proof. Taking the norm of the system (13) we get
α˙ ≤ (A ⊗ I2 )−1 [−(K ⊗ I2 ) tanh((A ⊗ I2 )α − C) +M ] ≤ (A ⊗ I2 )−1 ( −(K ⊗ I2 )
tanh((A ⊗ I2 )α − C) + M ) √ √ where tanh((A ⊗ I2 )α − C) ≤ 2n, M ≤ η ∗ n, with −(K ⊗ I2 ) = ρ (K ⊗ I2 )T (K ⊗ I2 ) and (A ⊗ I2 )−1 = ρ ([(A ⊗ I2 )−1 ]T [(A ⊗ I2 )−1 ]), but since I2 is the identity matrix with two eigenvalues 1 and from the spectrum properties of the Kronecker product we have −(K ⊗ I2 ) = ρ K T K = max(kf , km ) = k ∗ and (A ⊗ I2 )−1 = ρ ((A−1 )T (A−1 )). Finally, we have √ √ α˙ ≤ ρ ((A−1 )T (A−1 )) k ∗ 2n + η ∗ n = η. (25)
314
Now, we can state our second main result. First, we consider the case when only two agents are in risk of collision. From this simplest case, we state a series of theorems leading to the general case. Theorem 3. Considere the system (3) and the control laws (23)-(24). Suppose that there exists risk of collision between only two agents at time instant t and satisfies > ηdˆ . Then, in the closed-loop system (3)-(23)-(24) the agents tend asymptotically to their desired positions and they stay at a distance greater than or equal to d, ∀t ≥ 0. Proof. We show that the r-th and s-th agents will avoid collision between them and they stay at some minimum distance from each other. Define a surface given by 2 − d2 = 0 (26) σrs = p2rs + qrs To determine the behavior under the action of the repulsive vector fields we use the positive definite function 1 2 V = σrs (27) 2 whose time derivative is given by V˙ = σrs σ˙ rs . The time derivative of (26) along the trajectories of the closed-loop system is given by p˙ rs σ˙ rs = 2 [ prs qrs ] q˙rs
= −2 [ prs qrs ] ((−ks tanh(αs − αs∗ ) + α˙ s∗ )
2 −(−kr tanh(αr − αr∗ ) + α˙ r∗ )) + 4 (p2rs + qrs )(28)
Therefore, V˙ ≤ 0 is achieved if σrs σ˙ rs ≤ 0. When there exists risk of collision, (prs , qrs ) lies in the inner region of σrs = 0, that is σrs ≤ 0, then the analysis reduces to show that σ˙ rs ≥ 0. That means the resulting vector fields inside the circle points outwards; that is, to the region free of collision. Using the definition of the cross product we have 2 2 η σ˙ rs = 4 p2rs + qrs ˆ cos θrs + 4 (p2rs + qrs ) 2 2 2 2 ≥ −4 prs + qrs ηˆ + 4 (prs + qrs ) > 0. (29) Solving for we have that, if > ηˆ/d then σ˙ rs > 0. This implies that rth and sth agents move away from each other until they reach a distance d. Since αs (0) − αr (0) ≥ d, then the agents not only avoid collision but also satisfy αs (t) − αr (t) ≥ d for all time.
Now, we consider the case when three agents are in risk of collision, that is, agent r is in risk of collision against agents s1 and s2. Theorem 4. Consider the system (3) and the control laws (23)-(24). Suppose that there exists risk of collision between three agents and satisfies > 2(ˆ η /d). Then, in the closed-loop system (3)-(23)-(24) the agents converge asymptotically to their desired positions and they stay at a distance greater than or equal to d, ∀t ≥ 0. Proof. We define a surface composed of 2 components given by 2 2 σrs1 prs1 + qrs1 − d2 σ= = = 0. (30) 2 σrs2 p2rs2 + qrs2 − d2 We use the positive definite function
Proceedings of the 20th IFAC World Congress J. Santiaguillo-Salinas et al. / IFAC PapersOnLine 50-1 (2017) 309–314 Toulouse, France, July 9-14, 2017
(31)
V˙ = σ T σ˙ = σrs1 σ˙ rs1 + σrs2 σ˙ rs2 ≤ σ ∗ (σ˙ rs1 + σ˙ rs2 )(32) where σ ∗ = max{σrs1 , σrs2 }. Evaluating V˙ considering that the trajectories lie in the inner region of σ = 0, that is, σrs1 , σrs2 < 0 then the analysis reduces to show that σ˙ rs1 + σ˙ rs2 > 0. Hence ∗ ∗ ) + α˙ s1 ) σ˙ rs1 + σ˙ rs2 = 2 [ prs1 qrs1 ] ((−ks1 tanh(αs1 − αs1 2 ) −(−kr tanh(αr − αr∗ )α˙ r∗ )) + 4(p2rs1 + qrs1 ∗ ∗ + 2 [ prs2 qrs2 ] ((−ks2 tanh(αs2 − αs2 ) + α˙ s2 ) ∗ ∗ 2 2 −(−kr tanh(αr − αr ) + α˙ r )) + 4(prs2 + qrs2 ) prs2 + 4 [ prs1 qrs1 ] qrs2 2 2 η ≥ −4 p2rs1 + qrs1 ˆ + 4(p2rs1 + qrs1 ) 2 2 η − 4 p2rs2 + qrs2 ˆ + 4(p2rs2 + qrs2 ) 2 2 cos θ + 4 p2rs1 + qrs1 p2rs2 + qrs2 rs1,rs2 > 0. (33) In this scenario, agents s1 and s2 can be positioned at any point of the circumference of radius d around the agent r, considering that, from Theorem 3, they must remain at a distance greater than or equal to d between them. The worst case occurs when the agents s1 and s2 are uniformly distributed over the circumference of radius d. Thus, cos θrs1,rs2 = −1 and solving for we have that, if > 2( η /d) then σ˙ rs1 + σ˙ rs2 > 0. This implies that agents s1, s2 and r avoid collision between them. Geometrically, the most general case occurs when the r-th agent is surrounded by six agents, i.e. seven agents are in danger of collision. Theorem 5. Consider the system (3) and the control laws (23)-(24). Suppose that there exists risk of collision between n ≥ 3 agents and satisfies > 2(ˆ η /d). Then, in the closed-loop system (3)-(23)-(24) the agents converge asymptotically to their desired positions and they stay at a distance greater than or equal to d, ∀t ≥ 0. Proof. We follow a similar procedure to that presented in the proof of Theorem 4, considering a surface with n − 1 components and showing that, if σ˙ rs1 + . . . + σ˙ r(n−1) > 0, then V˙ < 0, taking into account that the worst case is presented when the n − 1 agents are uniformly distributed over the circumference of radio d around the r-th agent, so the agents avoid collision between them. 5. NUMERICAL SIMULATION The results of a numerical simulation using the control strategy given by (23)-(24) are shown below. For the simulation, we considered 5 differential-drive mobile robots, where the point αi to control is located 0.15m ahead the midpoint of the wheel axis. The formation graph employed in the simulation is shown in Fig. 2. The control gains used in the simulation are km = 2 and kf = 3. The desired trajectory is a quadrifolium curve 315
Fig. 2. Formation graph for the simulation.
Initial Pos.
End Pos.
Agents
Desired Trajectory
3
2
Distance Y (m)
1 V = σT σ 2 whose time derivative is given by
313
1
0
-1
-2
-3
-4
-3
-2
-1
0
1
2
3
4
5
6
Distance X (m)
Fig. 3. Trajectories of the agents in the plane. given by m(t) = [4 sin (2ωt) cos (ωt) , 4 sin (2ωt) sin (ωt)], where ω = 2π T with a period of T = 80 s. The static position given by c12 = [0, 0.6], c21 = [0, −0.6], vectors are π π c32 = −0.6 cos( 10 ), −0.6 sin( 10 ) , c34 = [0, −0.97], c43 = 3π 3π [0, 0.97] and c54 = −0.6 cos( 10 ), −0.6 sin( 10 ) . The scaling factor is given by µ(t) = 1 + 0.2 sin(ωt). The minimum allowed distance between agents is d = 0.3 m and was set to = 4(2(ˆ η /d)), to ensure the minimum distance condition will not be violated. Fig. 3 shows the motion of the agents in the plane. The initial position of the agents are indicated with an “x” and positions in different time instants are represented with a circle “o”. Is observed how the leader follows the desired trajectory while the followers achieve a timevarying formation. Furthermore, the minimum distance requirement is satisfied. Fig. 4 show the errors of the agents, such errors converge to zero. Fig. 5 depicts all the posible distances between agents. The distances between any pair of agents is always greater than or equal to the predefined distance d = 0.3. 6. CONCLUSIONS AND OUTLOOKS This paper presents a time-varying formation tracking control with collision avoidance for multi-agent systems. We propose a bounded control law complemented with a repulsive vector field of the unstable focus type, so that trajectory tracking with a time-varying formation and collision avoidance between agents are guaranteed. As shown in the numerical simulation, the goals are achieved. As future work, it is proposed to control the midpoint of wheel axis of the differential-drive mobile robots and include a strategy for obstacle avoidance. It is also intended to validate the theoretical results obtained through realtime experiments.
Proceedings of the 20th IFAC World Congress 314 J. Santiaguillo-Salinas et al. / IFAC PapersOnLine 50-1 (2017) 309–314 Toulouse, France, July 9-14, 2017
4.5
e1 e2
4
e3 e4
3.5
e5
Error (m)
3 2.5 2 1.5 1 0.5
0
1
2
3
4
5
6
4
5
6
Time (s)
Fig. 4. Errors of the agents. 2 1.8 1.6
||, i - , j ||
1.4 1.2 1 0.8 0.6 0.4 0.2 0
0
1
2
3
Time (s)
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