Control of Synchronization for Multi-Agent Systems in Acceleration Motion with Additional Analysis of Formation Control

Transcription

1 2 American Contro Conference on O'Farre Street San Francisco CA USA June 29 - Juy 2 Contro of Synchronization for uti-agent Systems in Acceeration otion with Additiona Anaysis of Formation Contro Haopeng Zhang Kaana Pothuvia Qing Hui Ran Yang and Jordan. Berg Abstract he synchronization contro of muti-agent systems pays a significant roe in miitary and civi appications. his paper investigates the acceeration motion of muti-agent systems unike other synchronization protocos we introduce the constant vaue into the protoco which acts as the acceeration of the agents. Furthermore we expand the acceeration protoco into the formation contro protoco by which the muti-agents can achieve the desired formations. Simuations are provided for both protocos and additionay we adopt the Amigobots as our experiment environment to verify our theoretica resuts. Compared with the atab simuations we concude that under the proposed protocos the system goes to acceeration synchronization or desired formation with acceptabe error. I. INRODUCION Synchronization contro for muti-agent systems has attracted more and more attention and research for its wide appications in miitary and civi aspects -4 incuding UAV s (Unmanned Air Vehices) and UGV s (Unmanned Ground Vehices) distributed wireess sensor networks and swarms of heterogeneous air and space vehices. he main research in this area has been focusing on synchronization protoco design. any different design approaches have been proposed to guarantee synchronization of muti-agent systems under different circumstances. Specificay in 5 an observation-based synchronization protoco was presented and a necessary and sufficient condition was provided to guarantee the system synchronization. 2 adopted the eader-foower mode to investigate the synchronization probem for muti-agent systems. he pinning contro approach for the synchronization of muti-agent systems was investigated in 3 in which a genera criterion for ensuring network synchronization has been derived. In this paper two probems are investigated. Firsty the averaging probem is discussed in arge voumes of iterature but there is sti one simpe question remaining: can we achieve the desired synchronization vaue not just the average for the muti-agent system? he second question is his work was supported by the Defense hreat Reduction Agency Basic Research Award #HDRA---9 to exas ech University. H. Zhang K. Pothuvia Q. Hui and J.. Berg are with the Department of echanica Engineering exas ech University Lubbock. X USA (haopeng.zhang@ttu.edu; qing.hui@ttu.edu; kaana.pothuvia@ttu.edu; jordan.berg@ttu.edu). R. Yang is with the Schoo of Information Science and echnoogy Sun Yat-Sen University Guangzhou Guangdong 5275 China yangran@mai.sysu.edu.cn. can we achieve the desired formation for the muti-agent system by a itte change in the synchronization protoco? With these questions in mind we propose a nove synchronization protoco to achieve the acceeration motion in synchronization for the muti-agent system characterized by doube integrators. Under our protoco the agents approach the same acceeration motion by anayzing the zero-state and zero-input effect for the system. oreover unike other synchronization 4 protocos under which the veocity of the synchronization is the average of the initia veocities of each agent we can adjust the synchronization veocity by adjusting the constant in our protoco. Furthermore based on the proposed protoco we present a formation protoco under which the system can achieve the desired formation. Not merey restricted to one dimension we expand the protocos into the x-y pane and even x-y-z space. Additionay we adopt the Amigobots as our experiment environment to test the acceeration synchronization protoco and formation protoco for the muti-agent system but due to the imit of the robots a one-dimensiona experiment is investigated. he corresponding atab simuations are aso provided which dispay the exact synchronization or formation for the agents. he robots move into synchronization with acceptabe error thus verifying the theoretica anaysis of the protoco. oreover the atab simuations in the x- y-z space are provided. he organization of the rest of paper is as foows: the basic graph theory is introduced in Section II and in Section II the acceeration synchronization protoco is proposed and fuy anayzed; in Section III we adjust the proposed protoco into the formation protoco. Finay the Amigobots experiment and atab simuation resuts are shown in Section IV and Section V concudes the paper. II. AHEAICAL PRELIINARIES Graph theory is a powerfu too for investigating networked systems. In this paper we use graph-reated notation to describe our network mode. ore specificay et G = (V E A ) denote an undirected graph with the set of vertices V. = {v... and E V V which represents the set of edges. he matrix A with nonnegative adjacency eements a ij serves as the weighted adjacency matrix. he node index of G is denoted as a finite index set N = { An edge of G is denoted by e ij = (V i V j ) and the adjacency //$26. 2 AACC 59

2 eements associated with the edges are positive. We assume e ij E a ij = and a ii = for a i N. he set of neighbors of the node V i is denoted by N = {V j V (v i v j ) E j = : j i. he degree matrix of a graph G is defined as where δ ij = = δ ij () { N j= a ij if i = j if i j. he Lapacian matrix of graph G is defined by L = A. (2) If there is a path from any node to any other node in the graph then we ca the graph connected. In this paper we assume that the topoogy of the muti-agent system is connected. AIN RESUL Given a connected graph according to Newtonian mechanics we adopt the same dynamic mode as 9 for each agent: ẋ i = v i m i v i = u i (3) where x i v i m i are the position veocity and mass of the node i respectivey. oreover for a system of agents we assume m = m 2 =... = m =. Definition 2.: he synchronization probem in acceeration motion for muti-agent systems is to find a contro aw u i such that v i (t) = v() + bt x i (t) = x() + v()t + 2 bt2 (4) I.e. the trajectory of each agent of the muti-agent system is under the same equation of acceeration motion. Here x() = x () () x () and v() = v () () v () and xi () v i () are the initia states of the muti-agent system = R. he foowing theorem is one of the main resuts of the paper. heorem 2.: Given a connected muti-agent system the system achieves synchronization in acceeration motion as defined in Definition 2. under the foowing synchronization protoco u i = j=j i a ij g(v j v i ) + k(x j x i ) + b (5) with k > g > a ij is the (ij)th eement of the matrix A and b serves as the acceeration. With E i = x i v i and ψi = b under the protoco (5) we obtain the vector form for the muti-agent system E i = AE i + B j=j i a ij (E j E i ) + ψ i (6) where A = B =. Furthermore define E = k g E E 2 E hen the muti-agent system dynamics become E = Φ E + ψ (7) where Φ = I A L B and ψ = where b denotes the Kronecker product. o prove our main resut we investigate the the zero-state effect and zero-input effect of the system. A. Zero-input effect of the system In this subsection we wi study the zero-input effect of the system in other words the constant vector ψ is assumed to be a zero vector. he foowing two emmas are needed. Lemma 2.: he matrix Φ in the system (7) has the foowing properties:. he matrix Φ has the eigenvaue λ = with agebraic mutipicity of two which is denoted by λ Φ = λ Φ2 =. 2. Except for λ Φ and λ Φ2 a the other eigenvaues of Φ satisfy Re(λ Φi ) < i = 3 2. Proof: Based on the fact that the eigenvaues of the matrix L can be denoted by = λ L < λ L2 < λ L3 < < λ L and there exists an orthogona matrix W such that then it foows that W LW = λ L λ L2 λ L (W I 2 )Φ(W I 2 ) = I A λ L λ L2 λ L B = A A λ L2 B A λ L B (8) Since the matrix A has two eigenvaues λ A = λ A2 = it gives that λ Φ = λ Φ2 =. For i = 2 3 det(a λ Li B) = det = λ λ Li g λ Li k Li g hus rank(φ) = rank((w I 2 )Φ(W I 2 )) = rank(a) + rank(a λ Li B) i=2 = + 2( ) = 2 (9) 5

3 Furthermore the characteristic poynomia of A λ Li B is given as foows s f i (s) = det λ Li g s + λ Li k Put f i (s) = then = s 2 + λ Li ks + λ Li g () s = λ Lik ± (λ Li k) 2 4λ Li g 2 Since k g > it foows that Re(λ Φi ) < for i = Lemma 2.2: et w r be the right eigenvector associated with eigenvaue zero and w be the eft eigenvector associated with eigenvaue zero. hen w r = w = () and the generaized right eigenvector and the generaized eft eigenvector of Φ denoted by v r and v respectivey are v r = k/g v = k/g Proof: Note that Φ w r = ( I L) k g ( ) = (I ) + 2 = 2 Φ v r = ( I L) k g k/g ( ) = I Simiary (2) = w r (3) w Φ = 2 (4) aso v Φ = w which competes the proof. hen we propose the state-input effect for the muti-agent system. Lemma 2.3: Consider the dynamic system (7) with ψ =. hen for any k g > im ζ(t) = im exp(φ)ζ() (5) = N x() + N v()t (6) N v() Proof: Based on Lemma 2. and 2.2 there must exist a nonsinguar matrix P such that Φ can be factored into the Jordan canonica form Φ = PJP... = P λ Φ3... P (7) λ Φ2 where P = w r v r p 3 p 2 P = v w q 3 q 2 and p 3 p 2 (q 3 q 2 ) are the right (eft) eigenvectors or generaized right eigenvectors associated with eigenvaues λ Φ3 λ Φ2 respectivey. Since λ Φ3 λ Φ2 < it foows that im E(t) which competes the proof. = im exp(φt)ζ() = k g t k g = x() + v()t (8) v() B. Zero-state effect of the system o prove heorem 2. we need to investigate the zerostate effect of the system (7) in this subsection. Lemma 2.4: Given a connected topoogy for the mutiagent system the zero-state effect of the system (7) is im t Proof: Note that t im expφ(t τ)ψ dτ = = P im = w r v r t 2 t2 t expφ(t τ) dτ exp(j(t))p v wr + p 3 p 4... p 2 λ Φ3 2 bt2 bt (9)... λ Φ2 q 3 q 4 q 2 (2) It is easy to verify that t wr v 2 t2 v r t wr = ( )( ) 2 t2 t Next we wi prove p3 p 4... p 2 λ Φ3... = L k/g + S λ Φ2 /g p 3 p 4... p 2 (2) (22) 5

4 where L = and S is a matrix satisfying SL = L and a the row sums of S and the coumn sums of S are zero. First we introduce two parameters α and β. Note that On the other hand and P α β... P λ Φ3 λ Φ2 Pα β λ Φ3...λ Φ2 P = 2 (23) P α β λ Φ3... = w r v r α β v + p 3... p 2 λ Φ3 λ Φ2 P w... λ Φ2 q 3 q 4 q 2 (24) Pα β λ Φ3...λ Φ2 P = Φ + α v w r v r w β (25) it is easy to verify that ( k/g /g L + S + w r v r α β ) v w (Φ + α v ) w r v r w β = 2 (26) hen together with (23) and (24) one can obtain (22). Furthermore q p3 p 4... p 2 λ Φ3... λ Φ2 3. q 2 = b 2 (27) t wr v 2 t2 v r t wr = b 2 bt2 bt (28) Now based on Lemma 2. and Lemma 2.2 heorem 2. is obvious. From this point we can sove the former first question that is how we can achieve the desired consensus vaue which is defined here as the achievement of the desired veocity for the muti-agent system. his desired veocity is stated by the foowing theorem. heorem 2.2: For the connected system (7) with k g > and a constant b a ij as defined before the muti-agent system achieves desired veocity v d under the foowing contro protoco. u i = C(t){ a ij g(v j v i ) + k(x j x i ) + b (29) where and j=j i C(t) { if t td ) otherwise (3) t d = v d N v()) (3) b Proof: According to heorem 2. the system goes into acceeration synchronization at time t s and the expression of the veocity of each agent is with t d > t s and at time t d v i (t) = N v()) + bt (32) v i (t d ) = v d (33) and u i (t) = for t t d so the desired consensus is achieved. III. FORAION CONROL o address the second question in regard to formation contro for muti-agent systems in this section we propose a formation contro protoco which is the transformation of our synchronization protoco presented before. he formation contro protoco for the muti-agent system is: u i = a ij g(v j v i ) + k(x j x i ij ) (34) j=j i where ij is the idea distance between i and j. Here we try to contro the formation for the muti-agent system via the contro of the distance between each agent. We can obtain the compact vector form of the muti-agent system E = Φ E + ψ (35) where Φ = I A L B AB is as defined before and ψ = L g 2 3. Define d ij = x j x i and we have the foowing theorem. heorem 3.: Given a connected muti-agent system (35) under the formation contro protoco (34) the system goes to v i (t) = vi ()d ij (t) = ij. Proof: For the zero-state effect for the system because of (2) and (22) we have t wr v 2 t2 v r t wr ψ = ( )( ) L t 2 t2 t = t g 2 t2 t = g 2 (36) 52

5 Furthermore p3 p 4...p 2 λ Φ3... q3 q 4 q 2 ψ ( k g = L ) + S g ψ ( ) ( ) = S g L g λ Φ2 2 3 = L 2 3 = j=2 j (37) hen together with emma 2.3 the theorem can foow immediatey. Furthermore one can extend our synchronization protoco and formation contro protoco into the x-y pane; even x-y-z space. Without osing generaity the X-Y mode is deveoped here. X-Y pane mode: ẋ i ẏ i v x i v y i = v x i = v y i = u x i = u y i where x i y i is the position vi x and vy i is the x-axis veocity and y-axis veocity respectivey. u x i and u y i are the contro inputs of the agent i respectivey. Furthermore we propose the x-y pane synchronization protoco for the muti-agent system u x i u y i = j=j i a ijg x (v x j vx i ) + kx (x j x i ) + b x = j=j i a ijg y (v y j vy i ) + ky (y j y i ) + b y (38) where a ij is defined as before and g x g y k x k y are positive constants. Based on the heorem (2.) we can obtain the foowing coroary. Coroary 3.: Given a connected muti-agent system (7) the muti-agent system achieves acceeration synchronization under the contro protoco (38). Furthermore we propose the foowing x-y pane formation contro protoco for the muti-agent system: u x i = u y i = j=j i j=j i a ij g x (v x j vx i ) + kx (x j x i x ij ) a ij g y (v y j vy i ) + ky (y j y i y ij ) (39) Since the goa is to achieve a particuar formation for the muti-agent system it is necessary for the muti-agent system veocity(m/s) Seting point (a) Simuation of Veocity vs ime v veocity(m/s) (b) Experiment of Veocity vs ime Fig.. Comparison of the Simuation and Experimenta Resuts for System s Veocity Seting point (a) Simuation of Position vs ime x (b) Experiment of Veocity vs ime Fig. 2. Comparison of the Simuation and Experimenta Resut for System s Position to achieve the same veocity so here we assume v x () = v y (). Based on heorem (3.) we can arrive at the foowing coroary. Coroary 3.2: Given a connected muti-agent system (7) the system achieves the desired formation under the contro protoco (39). IV. SIULAION AND EXPERIENS Given a connected topoogy there are four agents in the system and the topoogy is connected. Under our protoco (5) with k = g = 2 and b = 2 Fig. and Fig. 2 are the atab simuation resuts i.e. the veocities and positions vs time t respectivey. We can concude that the system goes into acceeration synchronization with the same equation of motion. oreover we appy the protoco to achieve the desired consensus veocity. As Fig. shows the (a) Initia Condition of the (b) Fina Condition of the uti-agent System s synchronization uti-agent System s synchronization Fig. 3. Amigobots Experiment Set-up v x 53

6 veocity(m/s) Veocity vs ime (a) Formation simuation of Veocity vs ime v veocity(m/s) Veocity vs ime (b) Formation experiment of Veocity vs ime Fig. 4. Compare of the Simuation and Experiment Resut for System s formation Position vs ime (a) Formation simuation of Position vs ime x Position vs ime (b) Formation Experiment of Position vs ime Fig. 5. Compare of the Simuation and Experiment Resut for System s formation agents achieve the desired veocity v d =.225m/s using our protoco (29). he next goa is for the system to reach the desired formation which is achieved using proposed formation contro protoco (34). he desired formation is one in which the distances between the position of each agent from agent to agent 4 are the same. Fig. 4 dispays the agent speeds approaching the average of the initia speed vaues and Fig. 5 dispays the desired formation. Furthermore the simuation of the 3D formation is provided in Fig. 7. In this paper together with the simuation resuts we adopt the Amigobot mobie robots as our experiment environment to verify our theoretica anaysis. he same muti-agent (a) Initia Condition of the (b) Fina Condition of the uti-agent System s formation uti-agent System s formation Y axis veocity X axis veocity 4 6 Fig. 6. (a) uti-agent System s 3D formation contro ime Amigobots Experiment Set-up Y axis position X axis position 2 4 (b) uti-agent System s 3D formation contro v x ime graph topoogy is investigated in the experiment. Under synchronization protoco (29) and formation protoco (34) in Fig. Fig. 2 Fig. 4 and Fig. 5 we compare the experimenta resuts with the atab simuations. Fig. 3 and Fig. 6 are photographs of the experiment set-ups having impemented the synchronization and formation protocos respectivey. V. CONCLUSION his paper investigates the synchronization probem as it reates to acceeration motion as we as the formation contro probem for muti-agent systems. Regarding our proposed synchronization protoco the agents of the system move with the same equation of motion. Aso we provide an appication of this protoco the resut being that we can obtain the desired consensus veocity instead of the average of the initia veocities of the agents. Regarding our proposed formation protoco the desired formation can be achieved. Both contro protocos can be extended from a singe axis to 3D space yet no discrepancies are encountered. he Amigobots experiments and the atab simuations aso verify our theoretica anaysis. REFERENCES R. Ofati-Saber J. A. Fax and R.. urray Consensus and cooperation in networked muti-agent systems Proc. IEEE vo pp R. Ofati-Saber and R.. urray Consensus probems in networks of agents with switching topoogy and time-deays IEEE rans. Autom. Contro vo pp L. Schenato and G. Gamba A distributed consensus protoco for cock synchronization in wireess sensor network in Proc. 46th IEEE Conf. Decision Contro New Oreans LA 27 pp A. L. Fradkov B. Andrievsky and R. J. Evans Synchronization of passifiabe Lurie systems via imited-capacity communication channe IEEE rans. Circuit Syst. Part I: Reg. Papers vo pp Z. Li Z. Duan G. Chen and L. Huang Consensus of mutiagent systems and synchronization of comp ex networks: a unified viewpoint IEEE rans. on Circuits Syst. Part I: Reg. Papers vo pp R. W. Beard. W. clain. A. Goddrich and E. P. Anderson Coordinated target assignment and intercept for unmanned air vehices IEEE rans. Robot. Autom. vo pp J. G. Bender An overview of systems studies of automated highway systems IEEE rans. Vehic. echno. vo pp J. R. Carpenter Decentraized contro of sateite formations Int. J. Robust Nonin. Contro vo pp G. Xie and L. Wang Consensus contro for a cass of networks of dynamic agents Int. J. Robust Nonin. Contro vo pp D. Zheng Linear Systems heory Beijing China: singhua Univ. Press 22. W. Ren and R. W. Beard Distributed Consensus in uti-vehice Cooperative Contro heory and Appications Springer-Verag London U.K A. K. Bondhus K. Y. Pettersen and J.. Gravdah Leader foower synchronization of sateite attitude without anguar veocity measurements in Proc. 44th IEEE Conf. Decision Contro Sevie Spain 25 pp W. Yu G. Chen and J. Liu On pinning synchronization of undirected and directed compex dynamica networks Automatica vo pp W. Yu G. Chen and. Cao Some necessary and sufficient conditions for second-order consensus in muti-agent dynamica systems Automatica vo. 46 pp Fig. 7. 3D formation contro for the muti-agent system 54