Cooperative Conveyance by Vehicle Swarms with Dynamic Network Topology

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1 SICE Annual Conference 2 September 3-8, 2, Waseda University, Tokyo, Japan Cooperative Conveyance by Vehicle Swarms with Dynamic Network Topology Koji Ishimura and Toru Namerikawa Department of System Design Engineering, Keio University, 3-4- Hiyoshi, Kohoku-ku, Yokohama, , Japan ( namerikawa@sdkeioacjp) Abstract: This paper deals with cooperative conveyance control by swarms with dynamic network topology First, we introduce a dynamic network topology that depends on relative distance between the s Secondly, we propose a novel conveyance strategy based on consensus seeking with dynamic network topology The proposed strategy needs at least one which can acquire the information of the target-ect and network topology among s is time-varying but always connected Furthermore the strategy is composed of Surround and Transport mode To analyze the convergence of conveyance with dynamic network topology, algebraic graph theory and matrix theory are utilized Finally, numerical simulation results demonstrate the effectiveness of the proposed method Keywords: Multi-Vehicle System, Cooperative Conveyance, Dynamic Network Topology, Surround, Transport INTRODUCTION In recent years, there have been increasing research interests in the distributed cooperative control of networked multi- systems Several research groups developed the coordination control strategies that achieve a capturing formation around a target-ect and conveyance by multiple mobile s using neighbor information []-[6] kobayashi et al [] proposed a decentralized control of multiple mobile robots for the capturing task of a target ect based on the gradient descent The proposed strategy was applied to the case with convex but unknown shape of oject So local approximation of the ect shape was introduced in [] An approach to multi-robot manipulation: Dynamic Object Closure, a closure condition that guarantees to trap a moving ect in a predefined future timing, was addressed in [2] The target-capturing strategy for networked multiple s with dynamic network topology is proposed in [3] The strategy controlled to converge to the formation while they are tracking the target-ect moving in 3 dimensional space Decentralized control policies for a group of robots to move toward a goal position while maintaining a condition of ect closure is presented in [4] A strategy which using simple vector fields to enable a team of s approaching an ect is proposed in [5] Vehicles surround ect to cage it, and transport it to a destination, while avoiding inter-robot collisions V Kumar et al[6] presented multi-robot manipulation of non-circular ects, cooperative manipulation in environments with obstacles Furthermore, their strategy enables s avoiding collision and obstacle by sensing the positions/velocities of relative neighbors In our work, we build on [3], [6] and present multirobot manipulation, cooperative conveyance with dynamic network topology, and via the composition of surrounding behaviors with feedback controllers Furthermore, our strategy enables s to avoid terminating network connection from masking by sliding the surrounding positions of s for target-ect While we assume that s are holonomic, it is possible to extend our methodology to include non-holonomic s using feedback linearization techniques Finally, the effectiveness of the proposal method is verified by the numerical simulations This paper is organized as follows Section 2 introduces the multi- systems and target-ect, network topology that depends on relative distance between the s and control ectives Section 3 describes the proposed method which enables cooperative conveyance in various initial positions and relaxing assumption for number of s which recognizing targetect Section 4 describes the results by numerical simulations Finally, we summarize the obtained results in Section 5 2 PROBLEM STATEMENT 2 Multi- Systems We consider the following N mobile s in Cartesian coordinates ṙ i = u i, i =,,N () where r i =[x i y i ] T R 2 is the position of i th and u i R 2 is the control input of i th + Target-Object th Fig Vehicles and an Object PR// SICE

2 +th e th + = 2 Target-Object Assumption 2 At least one can get the information of target-ect 23 Behaviors Our approach to caging and manipulation of ects can be summarized by the behavior architecture in Fig3 The architecture relies on two behaviors Fig 2 Surrounding where ξ R is the capturing radius Moreover, the ect is assumed to be always geostationary The which are disk-shaped with radius ables to sense the position of their teammates 22 Network Topology Information exchange between s or between and target-ect can be represented as a graph We give here some basic some terminology and definitions from graph theory Let G =(V, E ) denoted a graph with the set of vertices V = {, 2,, N} and the set of edges E V V The adjacency matrix A(G )=[a ij ] R N N is defined as a ij = and a ij = if (j, i) E where i=j The adjacency matrix of a undirected graph is defined accordingly except that a ij = a ji, i = j, since (j, i) E implies (i, j) E The degree of vertex i is the number of its neighbors and is denoted by deg(i) The degree matrix of graph G is diagonal matrix defined as D(G )=[d ij ] R N N where deg(i) = N a ij,i= j d ij =,j=i,i=j graph Laplacian of the graph G is defined by (2) L(G )=D(G ) A(G )=[l ij ] (3) It is well known that the graph Laplacian has some fundamental properties as follows Property All the row sums of L are zero and thus = [ ] T R N is eigenvector of L associated with the eigenvalue λ(l) = Property 2 For a connected graph, the graph Laplacian L is symmetric positive semi-definite and the eigenvalue λ(l) =is unique And we assume that the following Assumptions for s is satisfied Assumption Network topology among the s is time-varying but always connected Fig 3 Behaviors Architecture Surround: The N s are spaced out around the target-ect at intervals of the assigned angles and maintain these angles and each approaches to the target-ect and finally keeps the distance ξ Transport: The moves toward the goal configuration or tracks a reference trajectory derived from the ect s reference trajectory As shown in Fig3, transitions between behaviors are based on simple conditions derived from information among s When Number and Caging flags are both set, the enters Transport mode and starts transporting the ect Thus, reseting the flags can cause the s to regress into a different mode 3 PROPOSAL TECHNIQUE 3 Setting of mode The conveyance of the target-ect is composed of Surround and Transport mode We introduce the control law proposed by [3] in Surround modes It allows s to cooperative capturing target-ect with dynamic network topology In Transport mode, s continue to follow and surround virtual target-ect in order to transport a target-ect to goal [Control Objectives] The control ectives for the Surround mode can be formulated as follows C) lim r i (t) r (t) = ξ, t C2) lim ṙ i (t) ṙ (t) =, t C3) lim φ i+ (t) φ i (t) = 2π [rad], i=, 2,,N t N Let φ i = tan (y i /x i ) denotes the counterclockwise angle of ith and the center is the target-ect In control ective C3), if i = N, then N += In the next section, the target-capturing strategy which achieves the control ectives C)-C3) is developed

3 32 Surrounding of ect The surrounding of the ect utilizes the control law given in (4) (9) [] u i = κ i [a i { k(ˆr i r )+(Ṙi Ṙ)+ṙ } + a ij = ] a ij { k(ˆr i ˆr j )+(Ṙi Ṙj)+ ˆr j } {, (r i r j ρ j ), (r i r j >ρ j ) (4) (j =,,N,(N + )) (5) σ - (ω) t[s] Fig 5 σ + (ω) with t = t 2 =2 κ i = (6) a ij N+ R i = σ (ω) goal + σ + (ω) goal 2 (7) σ ± (ω) = ω = e ±t(ω+t2) (8) ˆr i ˆr p (9) where k> R is constant gain, R i R 2 is surrounding position for each, r ij = r i r j, ˆr i = r i R i, ρ j R, α i [, 2π) are the desired capturing angles a ij is a variable that represents whether s can recognize the target-ect Actually, ρ j is a sensor range goal, goal 2 is initial and final capturing position t,t 2 > defined as constant scalars p is a positive even number Fig4 and 5 show the graph of σ ± (ω) which enable sliding surround position and Fig6 shows that the surround positions depend on distance of s and target Fig 6 Surround position change The ect closure is used as a surrounding method We introduce N min and N max for a given ξ and D min (), minimum radius of s R N min and N max allow agents to ensure ect closure which are given as N min = 2πξ 2 + D min (), N max = πξ () Theorem : Consider the system of N s () and We apply the capturing control laws (4)-(9) to the system If the system satisfies k> and [Assumptions -2], then the system asymptotically achieves the control ective C)-C3) ProofSubstituting Eqs (4)-(9) into () 8 σ - (ω) ṙ i = κ i [a i { k(ˆr i r )+(Ṙi Ṙ)+ṙ } ] + Σ N a ij { k(ˆr i ˆr j )+(Ṙi Ṙj)+ ˆr j } () t[s] Fig 4 σ (ω) with t = t 2 =2 We assume the target-ect as N+th (r = ˆr N+,a i = a in+ ) Here, Ṙ i, as t, so we can get the following closed loop system

4 N+ a j N+ a 2j N+ a j a 2 a (N+) N+ = k a 2 a 2j a 2(N+) a 2 a (N+) N+ a 2 a 2j a 2(N+) + [ Here I 2 means Eq (2) can be rewritten as ] is shown ˆr ˆr 2 ˆr N+ ˆr ˆr 2 ˆr N+ ˆr ˆr 2 ˆr N+ (2) (L σ I 2 ) ˆr e = k(l σ I 2 )ˆr e (3) where L σ, D σ and A are graph Laplacian, degree matrix and adjacency matrix, is Kronecker product and ˆr R 2(N+) is ˆr = [ˆr T ˆr T 2 ˆr T N+ (= rt )]T Next, we introduce error variables ˆr ei =ˆr i ˆr N+ (r ) Thus the following error system is obtained N+ a j a 2 a N N+ a 2 a 2j a 2N N+ a N a N(N ) a j N+ a j a 2 a N N+ a 2 a 2j a 2N = k N+ a N a N(N ) Eq (4) can be rewritten as a j ˆr ˆr 2 ˆr N+ ˆr e ˆr e2 ˆr en (4) (M σ I 2 ) ˆr e = k(m σ I 2 )ˆr e (5) where ˆr e = [(ˆr ˆr N+ ) T (ˆr N ˆr N+ ) T ] T R 2N andm σ R N N is new matrix to represent the network topology Here, Eq (3) and Eq (5) are equivalent equations If Assumptions, 2 is satisfied, then (M σ I 2 )ˆr e = has an evident solution ˆr e = and M σ is nonsingular matrix From the following relationship, we can verify whether M σ is nonsingular matrix M σ is positive definite = M σ is nonsingular (6) Here, we introduce a nonzero vector x = [x x 2 x n ] R N to prove it If the network of system is connected graph (a ij = a ji, i,j(i, j=n + )), then x T M σ x can be expressed as follows x T M σ x = a ln+ x 2 l + a ij (x i x j ) 2 (7) l= And furthermore, if at least one can get the information of target-ect (a in+ =) and x=α, then we can get the following condition a ln+ x 2 l > (8) l= where α R is any scalar constant From property of graph Laplacian, the second term of Eq (7) is represented as a ij (x i x j ) 2 = x T L σ x (9) From property 2 of graph Laplacian, when the eigenvector of graph Laplacian and = [ ] T are only linearly dependent, the eigenvalue of graph Laplacian is zero If the second term of the right side of Eq (7) is zero, then the first term is strictly positive Conversely, the first term of the right side of Eq (7) is zero, then the second term is always positive Here, we can get the following condition x T M σ x = a ln+ x 2 l + a ij (x i x j ) 2 > (2) l= Accordingly, if assumption 2 is satisfied, M σ is strictly positive definite and nonsingular Multiply the both sides of Eq (5) by (M σ I 2 ) Thus, we can get the following equations(m σ I 2 ) is put from the left side on both sides of the expression ˆr e = k(m σ I 2 ) (M σ I 2 )ˆr e ˆr e = kˆr e (2)

5 This closed loop system does not depend on the network topology Therefore if the network topology is the time-varying, then the closed loop system representation does not change Convergence speed of s is decided only by controller gain k and choice of the gain is easy From the control gain is positive (k >), we can get: i = argmax k ˆΓ i ϕ(r) = γ 2 [γ 2 + β ] (r k r i ) T i ϕ i r k r i (32) (33) ˆr e, ˆre as t, (22) ˆr ˆr N+, ṙ ˆr N+ as t (23) Hence, the states of the system converge to the targetect ˆr i r, ṙ i ṙ as t,i V, (24) ˆr i ˆr j, ṙ i ṙ j as t, i, j V (25) In other words, the control ectives C)-C3) are achieved 33 Transport In Transport mode, s continue to follow and sirround virtual target-ect so as to transport real targetect to goal Then, the control input is given by the following control law (26)-(28) Given the workspace W and a goal configuration r goal, β = s 2 W s 2 W(x,y) (26) Φ(r) = r r goal [ r r goal 2κ +β ] /κ (27) u = Φ(r ) (28) where κ> R 34 Achievement of surrounding and conversion We propose an algorithm for local estimation of closure Number To locally define quorum, we introduce the concept of a forward and backward neighborhood with respect to the manipulated ect and the Surround mode For i, the Surround mode introduces an approach component, ie i ϕ i and rotation component, ie i ψ i, so that we can define some set of s to be in front of agent i and another sets behinds If a neighborhood ˆΓ i represents the s within a distance D min (), then ˆΓ ± i = {j ˆΓ i < ±(r j r i ) T ( i ψ i )} (29) ψ = [,, γ γ2 + β ] T (3) where γ = s(x, y), s(x, y) = is shape boundary for surrounding target-ect In addition, the of ˆΓ + i can be defined as follows i + = argmax k ˆΓ+ i (r k r i ) T i ϕ i r k r i (3) We consider the following update rule for number i if (ˆΓ + i = ) (Γ i = ), number i = N min if f(i +,i ) >N min, (34) f(i +,i ) otherwize with f(j, k) = min(number j, number k )+ We use number i and number i ± to determine whether ect closure is achieved 2 Caging We define local target-capturing as follows caging i = (number i N min ) (number i = number i +) (number i = number i ) (35) When an agent estimates that local closure has been attained, it switches to the Transport mode and begin manipulation of the ect If closure was lost during manipulation, each agent in the system returns to the Surround mode to achieve ect closure Target Fig 7 Vehicle i s neighborhoods ˆΓ ± i with i ± 4 NUMERICAL SIMULATIONS The simulation results are shown in Fig8 - Figures illustrate the trajectories of four s and the targetect in Surround and Transport mode In each figure, Red circle shows s with radius, Blue circle shows target-ect with radius, Blue circle on green circle shows surround position for each Small blue and red circles show center position of s and target-position In initial position, only one recognizes the target-ect, and network topology among the s is always connected Fig8 shows initial position of s and targetect In Fig9, s achieve surrounding the targetect Making a comparison between Fig8 and Fig9, surround positions slide from goal to goal2 Fig shows s transporting the target-ect In Fig, s achieve transporting the target-ect to the goal

6 5 CONCLUSION Fig 8 Initial position of s and target-ect Fig 9 Achieving surround the target-ect Fig Transport the target-ect Fig Achieving transport the target-ect to goal In this paper, we proposed multi-robot manipulation, cooperative conveyance with dynamic network topology, and via the composition of surrounding behaviors with feedback controllers Furthermore, their strategy enables s avoiding collision and obstacle by sensing the positions/velocities of relative neighbors The effectiveness of the proposal method was verified by the numerical simulations REFERENCES [] Yuichi Kobayashi, Kyouji Otsubo and Shigeyuki Hosoe Design of Decentralized Capturing Behavior by Multiple Mobile Robots, International Transactions on Systems Science and Applications, vol 3, no 3, pp23-2, 27 [2] ZhiDong Wang, Hidenori Matsumoto, Yasuhisa Hirata and Kazuhiro Kosuge, A Path Planning Method for Dynamic ect closure by Using Random caging Formation Testing IEEE/RSJ International Conference on Intelligent Robots and Systems, St Louis, October -5, [3] Hiroki Kawakami and Toru Namerikawa Cooperative Target-capturing Strategy for Multi- Systems with ynamic Network Topology, Proc of American Control Conference pp , 29 [4] G A S Pereira, V Kumar, and M F Campos,Decentralized algorithms for multi-robot manipulation via caging, The Int Journal of Robotics Research, vol 23, no 7/8, pp , 24 [5] J Fink, N Michael, and V Kumar, Composition of vector fields for multi-robot manipulation via caging, in Proc 27 Robotics: Science and Systems III, Atlanta, GA, 27 [6] Jonathan Fink, M Ani Hsieh, and Vijay Kumar Multi-Robot Manipulation via caging in Environments with Obstacles, Robotics and Automation, pp47-476, 28 [7] Daniel E Koditschek, Elon Rimon Robot Navigation Functions on Manifolds with Boundary Advanced Appl Math, vol, pp42-442, 99 [8] E Rimon and D E Koditschek Exact robot navigation using arti cial potential functions, IEEE Transactions on Robotics and Automation, vol8, no5, pp5-58, October 992 [9] V de Silva and R Ghrist,Homological sensor networks, Notices of the American Mathematical Society, vol 54, no, pp -7, 27 [] A Muhammad and A Jadbabaie, Decentralized computation of homology groups in networks by gossip, in Proc of the American Control Conf, New York, July