Passivity-Based Position Consensus of Multiple Mechanical Integrators with Communication Delay

Transcription

1 To appear in ACC 010 Passivity-Base Position Consensus of Multiple Mechanical Integrators with Communication Delay Ke Huang an Dongjun Lee Abstract We present a consensus framework for multiple variable-rate heterogeneous mechanical integrators (i.e. multiimensional integrators with mass, amping an spring matrices) on unirecte graph with constant, yet, non-uniform communication elay. By connecting multiple mechanical integrators via (iscrete-time) spring connections over elaye links with some amping injection, our propose framework not only achieves position consensus, but also enforces close-loop iscrete-time passivity. Moreover, it allows arbitrary control gains regarless of integration steps, if there is no elay. Simulation result is also given to support the theory. I. ITRODUCTIO Multiagent consensus problem has been an active research area, ue to its implications across many application omains an isciplines: multirobot coorination 1, sensor network, flocking an swarming in nature an animation 3, istribute computing 4, human collective behavior 5, to name a few. Many strong results have been reporte for both the continuous-time an iscrete-time first-orer systems (e.g. 6, 7, 8). Relatively rare results have been reporte for the secon-orer systems (e.g. 9, 10), with even lesser results available for iscrete-time secon-orer consensus with communication elays (e.g. 11, 1, 13). In this paper, we present a new consensus framework for multiple mechanical integrators (i.e. multi-imensional secon-orer iscrete-time integrator with mass, amping an spring matrices) on unirecte graph with constant, yet, non-uniform inter-agent communication elays. One of the unique properties of our propose framework is that, by connecting the mechanical integrators (MI) via (iscretetime) spring connections over elaye communication links with some amping injection, it not only achieves asymptotic position consensus among MIs, but also enforces iscretetime passivity of the close-loop multiple MIs. For this, we first utilize our recently propose non-iterative variable-rate passive MI 14, which is iscrete-time passive unlike explicit integrators use in other works (i.e. upate epens solely on the values of previous cycle), yet, can still be simulate fast enough (particularly for haptics) ue to its non-iterativeness. We then exten the PD (proportionalerivative) scheme of 15 into the iscrete-time omain to enforce close-loop passivity an position consensus of The authors are with the Department of Mechanical, Aerospace & Biomeical Engineering, University of Tennessee, Knoxville T 37996, Phone: , {khuang1,jlee}@utk.eu. : Corresponing author. Research supporte in part by the ational Science Founation CAREER Awar IIS the multiple MIs over unirecte communication graph with non-uniform constant elays. The motivating application of this work is peer-to-peer share multiuser haptic interaction over packet-switching communication network (e.g. Internet with some ata buffering), where istribute users haptically interact with their own MI (i.e. local copy of a share virtual environment) via their haptic evice, while these MIs are being synchronize to provie consistent perception of the share virtual environment among the users. For this, the aforementione closeloop passivity is very useful, since it woul allow us to ensure interaction stability with wie-ranges of (continuous-time) heterogeneous/unmoele haptic evices or human users, as long as they behave as passive systems. See 16. To our knowlege, none of results so far on iscrete-time secon-orer consensus (e.g. 11, 1, 13) has achieve this close-loop passivity with elay. In aition to its unique passivity property, our framework here also can hanle with: non-uniform elay (unlike 11, 1); multi-imensionality (unlike 13); an variable upate rate an non-uniformity among the MIs (unlike 11, 1, 13). Our framework also allows arbitrary gains regarless of sampling rate, if there is no elay (unlike 11, 1). The rest of the paper is organize as follows. We review some relate graph theory an non-iterative variable-rate passive MI of 14, an also introuce our problem setting in Sec. II. The consensus of multiple MIs is then iscusse with a numerical example in Sec. III, an some concluing remarks are given in Sec. IV. A. Graph Theory II. BACKGROUD In this paper, we consier unirecte an simple (i.e. no self-loop/multiple-eges) graph G(V, E), where V : {v 1,...,v } is the set of vertexes; an E refers to the set of eges. If v i an v j are connecte, we say that the ege from v i to v j (enote as e i j ) belongs to E. The set of information neighbors of a vertex v i is efine as i : { j e i j E}. If G(V,E) is unirecte, e i j E implies e ji E. Suppose the total number of (irecte) eges e i j of G is. Then, there exists a bijective map F : S E s.t. F(l) (p l,q l ) where S : {1,..., } an, with a slight abuse of notation, we enote the set of {(i, j) e i j E} by the ege set E. For simplicity, we will often omit subscript l for (p l,q l ).

2 The following two properties will be use with their proof omitte here, ε i j R n n j i ε i j an further, if G(V, E) is unirecte, ε pq (1) ε pq 1 (ε pq + ε qp ) () where (p,q) F(l). The incience matrix D : { il } R of G is efine by 1 if v i is the initial en of the connection e l il 1 if v i is the terminal en of the connection e l while the Laplacian matrix L {l i j } R of G is given by eg(v i ) if i j l i j 1 if e i j E (3) where eg(v i ) refers to the egree (number of eges) of v i. We then have 17, Prop.4.8: L DD T (4) It is well-known that, if G is unirecte an connecte, the zero eigenvalue of L is simple with the eigenvector 1 : T R, an all the other eigenvalues have strictly positive real parts 6, 7, 17. B. on-iterative Variable-Rate Passive MIs In this paper, we utilize the non-iterative variable-rate passive mechanical integrators (MIs) propose in 14: for a single mass-spring-amper type MI, uring the integration step T k > 0, M v k+1 v k + B v k+1 + v k + K x k+1 + x k τ k + f k T k v k+1 + v k x k+1 x k T k where M,B,K R n n are respectively mass, amper, an spring matrices (with M, K being symmetric/positive-efinite, an B being positive-semiefinite); x k,v k R n is the position/velocity; an τ k, f k R n are the control an human/environmental force respectively. Here, the first line is the ynamics upate; while the secon kinematics. This integrator is implicit (i.e. the future state oes not only epen on the previous one but also on itself too), yet, still non-iterative (i.e. both x k+1 an v k+1 can be expresse as functions epening on x k an v k explicitly, so they can be solve without iterations), thus, can still be simulate fast. The main reason of using this MI over other integrators (e.g. integrators use in 11, 1, 13, 18) is that it possesses the following open-loop iscrete passivity: ˆv T k (τ k + f k )T k V k+1 V o V o 0 (5) Fig. 1. Multiple MIs on unirecte graph G with non-uniform elays i j. where ˆv k : (v k+1 + v k )/ an V k : v T k Mv k/ + x T k Kx k/ is the total energy with V o being the initial energy. This iscrete-time passivity allows us to connect this MI with a variety of haptic evices or human users while ensuring interaction stability, provie that they behave as passive systems. In this paper, we will utilize an also aim to preserve this passivity of the MIs. See 14 for more etails on this non-iterative passive integrator. C. Consensus Setting We consier MIs on an unirecte communication graph G(V,E), where each ege e i j E (i.e. from i th -MI to j th - MI) is associate with constant inexing elay ji 0 an a symmetric/positive-efinite spring matrix K i j R n n. See Fig. 1. Then, following Sec. II-B, we write own the following consensus protocol for the i th -MI: uring the upate interval > 0, M i v i (k+1) v i (k) τ i (k)+ f i (k) (6) ˆv i (k) x i(k+1) x i (k) (7) τ i (k) : B i ˆv i (k) K i j ( ˆx i (k) ˆx j (k i j )) j i (8) where ˆv i (k) : (v i (k + 1) + v i (k))/, ˆx i (k) : (x i (k + 1) + x i (k))/; i is the information neighbors of the i th -MI; B i R n n is the amping matrix of i th MI; K i j R n n is the spring matrix associate with e i j ; i j 0 is the constant inexing elay from j th -MI to i th -MI; an f i (k) is the force acting on i th -MI (e.g. virtual coupling/contact with an virtual object). For the spring connection K i j, we assume K i j K ji (9) i.e. symmetric (yet, still can be non-uniform) spring connection, although their elays may not be so (i.e. i j ji ). ote that the upate rate can be varying. With this variable, the constant inexing elay i j oes not necessarily imply constant time elay. III. COSESUS OF MULTIPLE MECHAICAL ITEGRATORS O DELAYED UDIRECTED GRAPH As mentione in Sec. II-C, the spring connections in the system are multi-imensional an non-uniform (i.e. the spring matrix K i j can vary from connection to connection). These features cannot be capture by the graph Laplacian L ue to its efinition (3). Therefore, we will nee to use the

3 following stiffness matrix, instea of the graph Laplacian L, as the tool to show the position consensus among MIs connecte via K i j with each other. The stiffness matrix P R n n with its i j th block matrices P i j R n n is given as j i K i j i j P i j K i j i j an e i j E (10) where K i j R n n is the spring matrix associate with e i j. The next proposition shows that the stiffness matrix has close relation with incience matrix on an unirecte graph. Proposition 1: Consier MIs on an unirecte communication graph G(V, E). The stiffness matrix as efine in (10) can be represente as, P 1 (D I n)k (D I n ) T (11) where D is the incience matrix of G(V, E) (efine in Sec. II), refers to the Kronecker prouct, an K : iag(k 1,K,...,K e ) with K l the spring matrix associate with e l,l S {1,..., }. Proof: Let us efine ˆD : (D I n )K 1 R n n. Then, its il-th block matrix ˆ il R n n is given by K 1 l if v i is the initial en of the ege e l ˆ il K 1 l if v i is the terminal en of the ege e l ±K 1 l where i represents the inex of MI; while K l is the spring matrix associate with the ege e l,l S. Let us enote the eges connecting k th MI by Ñ k : {r if e r is incoming or outgoing ege of v k }. Then, ˆ kl 0 n (n n zero matrix), if l Ñk; an ˆ kl 0 n otherwise. ow, we are going to show that ˆD ˆD T P. First, note that, the k th n n iagonal block of ˆD ˆD T is given by ˆ kl K l l Ñ k j k K k j where the last equality is because G(V,E) is unirecte an Ñ k inclues both the incoming an outgoing eges of v k. Also, for the off-iagonal block matrices of ˆD ˆD T, we have, for i j ˆ il ˆ jl l Ñ i Ñ j ˆ il ˆ jl (1) since, for ˆ il ˆ jl 0 n, it requires l Ñi Ñ j. Suppose e a,e b are the 1-tuple equivalence of e i j,e ji E respectively (i.e. F(a) (i, j),f(b) ( j,i)). Then, we have Ñi Ñ j {a,b}. An (1) will become (K a + K b ) (K i j + K ji ) K i j from the efinition of ˆ il above an (9). By comparing these iagonal an off-iagonal block matrices of ˆD ˆD T with P in (11), we can then see that P 1 ˆD ˆD T One of the strong properties of the graph Laplacian L is: for a connecte unirecte graph G(V,E) with noes, rank(l) 1. The following lemma extens this property to stiffness matrix. Lemma 1: For a simple unirecte graph G(V,E), we have rank(p) n rank(l) where n is imension of MI; L is graph Laplacian of G(V, E); an P is the stiffness matrix in (10). Moreover, if G(V,E) is connecte, rank(p) n( 1) Proof: By Proposition 1, the properties of rank an Kronecker prouct, an (4), we have, rank(p) rank( (D I n )K 1 (D In )K 1 T) rank(k T (D T D I n )K 1 ) rank(i n )rank(d T D) n rank(l) since K is nonsingular. Also, if G(V,E) is connecte, we have rank(l) 1. Thus, rank(p) n( 1) For the MIs system with the upate law (6)-(7), we efine the close-loop iscrete-time passivity s.t.: 0 ˆv i (k) T f i (k) c (13) where c R is a boune constant. Following 15, 19, 0, we also efine the controller passivity s.t.: 0 ˆv i (k) T τ i (k) (14) where R is a boune constant. The following lemma shows the connection between close-loop iscrete-time passivity an controller passivity. Lemma : For the MIs system with the upate law (6)- (7), controller passivity (14) implies the close-loop iscretetime passivity (13). Proof: By (6), (7), an (14), ˆv i (k) T f i (k) 1 ˆv i (k) T M i v i (k+1) v i (k) τ i (k) v i (0) M i : c (15) where 1 v i(0) M i represents the initial kinetic energy insie the system, with the notation A : T A (A R n n is require to be symmetric an positive-efinite),

4 As mentione in Sec. I, close-loop iscrete-time passivity an position consensus among MIs ensure the interaction stability an consistent perception of the share virtual environment among the users. The following theorem shows that, with enough amping injection, our consensus protocol (6)-(8) achieves both the close-loop iscrete-time passivity an position consensus. Theorem 1: Consier MIs on an unirecte connecte graph G(V, E) with the consensus protocol (6)-(8). Suppose that we set the gains B i,k i j s.t. i j + ji B i T j K i j, i 1,..., (16) j i where T j max k (T j (k)), an v(k) 0, k 0. Then, 1) the system achieves controller passivity (14), thus, also close-loop passivity (13); an ) if there is aitional positive-efinite amping B e i R n n on each MI, an f i (k) 0, the position consensus is achieve s.t. x i (k) x j (k) an v i (k) 0, i, j 1,..., Proof: The controller passivity conition (14) means that the controller only generates boune energy the whole time. Hence we can prove the first item by showing this energy bouneness. Denote the energy generate by the controller uring k th integration step as s E (k) : ˆv i (k) T τ i (k) j i D i (k) ˆv i (k) T K i j ( ˆxi (k) ˆx j (k i j ) ) D i (k) ˆv T p (k)k pq ˆxp (k) ˆx q (k pq ) T p (k) where D i (k) : ˆv i (k) B i, an (p,q) F(l). The last term can then be rewritten as ˆv T p(k)k pq ˆxq (k) ˆx q (k pq ) T p (k) + ˆv T p(k)k pq ˆxp (k) ˆx q (k) T p (k) ˆv T p(k)k pq ˆxq (k) ˆx q (k pq ) T p (k) (17) + 1 ( ˆvp (k)t p (k) ˆv q (k)t q (k)) T K pq ( ˆx p (k) ˆx q (k)) where we use (). Denote x i j (k) : x i (k) x j (k) an ϕ i j : 1 4 x i j(k) K i j. Then, the last terms of (17) can be simplifie s.t. ( ˆv p (k)t p (k) ˆv q (k)t q (k)) T K pq ( ˆx p (k) ˆx q (k)) 1 ( x pq(k+1) x pq (k)) T K pq ( x pq (k+1)+ x pq (k)) (ϕ pq (k+1) ϕ pq (k)) Then, we can rewrite s E (k) s.t. s E (k) Let us efine ϕpq (k) ϕ pq (k+1) D i (k) ˆv T p (k)k pq ˆxq (k) ˆx q (k pq ) T p (k) (18) Θ pq (k) : ˆv T p (k)k pq ˆxq (k) ˆx q (k pq ) T p (k) By inserting intermeiate terms k 1 jk+1 pq ˆxq ( j)+ ˆx q ( j) between ˆx q (k) an ˆx q (k pq ), we can then rewrite Θ pq (k) as Θ pq (k) T p (k) ˆv T p (k)k pq T p (k) ˆv T p (k)k pq k 1 ( ˆxq ( j + 1) ˆx q ( j) ) jk pq k 1 1 ˆvq ( j + 1)T q ( j + 1)+ ˆv q ( j)t q ( j) jk pq T p (k) ˆv T p(k)k pq k 1 jk+1 pq ˆv q ( j)t q ( j) + 1 ( ˆvq (k)t q (k)+ ˆv q (k pq )T q (k pq ) ) where the secon line is ue to (7). Since K pq is symmetric an positive-efinite, we have the following fact s.t. a T K pq b 1 ) ( a K pq + b K pq, a,b R n Using this, we can then show that Θ pq (k) 1 k 1 T p(k) T q ( j) ˆv p (k) K pq + ˆv q ( j) K pq jk+1 pq T p(k)t q (k) ˆv p (k) K pq + ˆv q (k) K pq T p(k)t q (k pq ) ˆv p (k) K pq + ˆv q (k pq ) K pq 1 4 α pq(k) k 1 T q (k)+t q (k pq )+ T q ( j) jk+1 pq T p(k) k 1 α qp (k)+α qp (k pq )+ α qp ( j) jk+1 pq where α pq (k) : T p (k) ˆv p (k) K pq 0. Since G(V,E) is unirecte, for each Θ pq (k), we will also have the corresponing Θ qp (k). By summing them up an collecting the terms containing α pq an α qp respectively, we can show that Θ pq (k) + Θ qp (k) Ω pq (k)+ω qp (k) where Ω pq (k) : 1 1 k 1 q(k) T α pq(k qp )+ α pq ( j) jk+1 qp + 1 α pq(k) T q (k)+ 1 k 1 T q(k pq )+ T q ( j) jk+1 pq

5 Then, by summing Ω pq (k) over time, we have 1 Ω pq (k) α pq (k) T q (k)+ 1 k 1 T q(k pq )+ T q ( j) jk+1 pq k pq (k qp )T q (k)+ 4 α 1 T q (k) α pq ( j) jk+1 qp 1 α pq (k) T q (k)+ 1 k 1 T q(k pq )+ T q ( j) jk+1 pq α pq (k)t q (k+ qp ) α pq (k)t q (k+ qp ) k +1 qp + 1 k 1+ qp α pq (k) jk+1 qp T q ( j) α pq ( k) k 1+ qp T q ( j) j +1 ( ) 1 k 1+ qp 4 α pq(k) T q (k pq )+T q (k+ qp )+ T q ( j) jk+1 pq ( ) where the last term in (**) comes from the 1 st,3 r an 6 th terms in (*). From (**), with α pq (k) 0 an T q T q (k), we have Ω pq (k) pq + qp Then, we can rewrite (18) s.t.: 0, s E (k) ϕ pq (0) ϕ pq (0) ˆv τ i(k) p1 p1 D p (k)+ D p (k)+ T q α pq (k) p1 p1 q p Θ pq (k) q p Ω pq (k) ϕ pq (0) : (19) where is a boune constant, an we use the fact that: uner the gain-setting conition (16), q p Ω pq (k) pq + qp q p D p (k) T q α pq (k) D p (k) ˆv(k) T pq + qp T q K pq B p ˆv(k)T p (k) 0 q p where the thir line is ue to (16). This proves the controller passivity (14). Thus, by Lemma, close-loop passivity (13) is achieve. For the secon item, from the close-loop passivity (13), with the aitional B e i an f i (k) 0, we have: > 0 c ˆv i (k) B e i (0) which implies that, with c being boune, ˆv i (k) 0. With this, (6) at k th an k+1 th integration steps are reuce to M i v i (k+1) v i (k) j i K i j ( ˆx i (k) ˆx j (k i j )) (1) v i (k+) v i (k+1) M i T i (k+1) K i j ( ˆx i (k+1) ˆx j (k+1 i j )) () j i Also, from (7) an ˆv i (k 1) 0, x i (k) x i (k 1) an x i (k) ˆx i (k 1) ˆx i (k ) ˆx i (k i j ). Therefore, the spring forces of the two successive time steps converge to each other, i.e. j i K i j ( ˆx i (k) ˆx j (k i j )) j i K i j ( ˆx i (k+1) ˆx j (k+1 i j )) thus, combining this with (1)-(), we have (v i (k+1) v i (k))t i (k+1) (v i (k+) v i (k+1)) where, from ˆv i (k) 0, v i (k+) v i (k). Therefore, (v i (k+1) v i (k))t i (k+1) (v i (k+1) v i (k)) implying that, with > 0, v i (k + 1) v i (k). Moreover, with ˆv i (k) 0, we have v i (k) 0. Substituting this into (6) with f i (k) 0, we have j i K i j (x i (k) x j (k)) 0, i 1,... which can be written by Px(k) 0 (3) where P is the stiffness matrix efine in (10), an x(k) : x 1 (k) T...x (k) T T. This (3) then implies x(k) 1 T... T T because: 1) by Lemma 1, the imension of ker(p) is n. ) P(1 ) 0, R n (i.e. ker(p) {1 R n }). Thus, by setting the control gains accoring to the conition (16), the consensus protocol (6)-(8) enforces the -port iscrete-time passivity for the close-loop system. If there is no elay, the gain setting conition (16) vanishes an in contrast to 11,1, we can choose arbitrary control gains, while enforcing passivity. On top of the passivity, with the conition (16), f i (k) 0, an extra amping B e i, the protocol (6)-(8) also achieves the position consensus among all MIs. This is very similar to the results of continuous-time elaye

6 position in x irection m position in y irection m Position Consensus in X Direction Position Consensus in Y Direction time sec Fig.. Position consensus of 5 two-imensional MIs with non-uniform communication elay. PD-scheme of 15, an, in fact, extens the results of 15 to the iscrete-time omain, for which the (open-loop) passivity of the MIs of Sec. II-B is inispensable (as that of the continuous-time robot is for the results of 15). The close-loop passivity of our framework woul be very useful for multiuser peer-to-peer haptics over the Internet (with some ata-buffering for constant elays), as it allows any passive humans/evices to be connecte while enforcing the close-loop passivity; as well as the esign of virtual worl an evice servo-loop to be separate from each other. See 16. We perform a simulation for five -im. MIs, with nonuniform spring matrices K i j k i j I with k i j in the range of 5 to 100 /m; an with non-uniform constant elays in the range of 40 to 1000 ms. We set to be the same (10ms) for simplicity, although varying can be incorporate. See Fig. for the simulation results. Due to the elay among the agents, it takes aroun 1 secon to achieve position consensus both in X an Y irections. The consensus time can be improve by optimizing network topology, K i j an B i, an it will be investigate in our future work. See 16 for a preliminary result in this irection. IV. COCLUSIO AD FUTURE WORKS We present a novel passivity-enforcing framework for multiple secon-orer mechanical integrators on unirecte graph with constant non-uniform communication elay. By connecting multiple non-iterative passive MIs 16 through iscrete-time springs over elaye links with some amping injection, the position consensus an the iscrete-time close-loop passivity are achieve. The control gains can be arbitrarily chosen regarless of the integration steps if there is no elay. The propose consensus framework is particularly promising for multiuser istribute haptic collaboration over the x 1 x x 3 x 4 x 5 y 1 y y 3 y 4 y 5 Internet as eluciate in 16. There are also several possible irections for future research. In this paper, the inexes among all MIs are synchronize (i.e. same k share). Yet, we believe this assumption can be roppe (i.e. k replace by k i for each MI). Extening our result to switching network topologies woul be an interesting irection. etwork topology optimization with limite network resource is also important for improving the performance. It is also interesting to exten our results here to general irecte graphs. REFERECES 1 J. Lin, A. S. Morse, an B. D. O. Anerson. The multi-agent renezvous problem. In Proceeings of the IEEE Conference on Decision an Control, pages , 003. J. Cortes, S. Martinez, T. Karatas, an F. Bullo. Coverage control for mobile sensing networks. IEEE Transactions on Robotics an Automation, 0():43 55, E. Bonabeau, M. 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