Passivity-Based Position Consensus of Multiple Mechanical

To appear in ACC 2010

Passivity-Based Position Consensus of Multiple Mechanical Integrators with Communication Delay Ke Huang and Dongjun Lee∗ Abstract— We present a consensus framework for multiple variable-rate heterogeneous mechanical integrators (i.e. multidimensional integrators with mass, damping and spring matrices) on undirected graph with constant, yet, non-uniform communication delay. By connecting multiple mechanical integrators via (discrete-time) spring connections over delayed links with some damping injection, our proposed framework not only achieves position consensus, but also enforces closed-loop discrete-time passivity. Moreover, it allows arbitrary control gains regardless of integration steps, if there is no delay. Simulation result is also given to support the theory.

I. I NTRODUCTION Multiagent consensus problem has been an active research area, due to its implications across many application domains and disciplines: multirobot coordination [1], sensor network [2], flocking and swarming in nature and animation [3], distributed computing [4], human collective behavior [5], to name a few. Many strong results have been reported for both the continuous-time and discrete-time first-order systems (e.g. [6], [7], [8]). Relatively rare results have been reported for the second-order systems (e.g. [9], [10]), with even lesser results available for discrete-time second-order consensus with communication delays (e.g. [11], [12], [13]). In this paper, we present a new consensus framework for multiple mechanical integrators (i.e. multi-dimensional second-order discrete-time integrator with mass, damping and spring matrices) on undirected graph with constant, yet, non-uniform inter-agent communication delays. One of the unique properties of our proposed framework is that, by connecting the mechanical integrators (MI) via (discretetime) spring connections over delayed communication links with some damping injection, it not only achieves asymptotic position consensus among MIs, but also enforces discretetime passivity of the closed-loop multiple MIs. For this, we first utilize our recently proposed non-iterative variable-rate passive MI [14], which is discrete-time passive unlike explicit integrators used in other works (i.e. update depends solely on the values of previous cycle), yet, can still be simulated fast enough (particularly for haptics) due to its non-iterativeness. We then extend the PD (proportionalderivative) scheme of [15] into the discrete-time domain to enforce closed-loop passivity and position consensus of The authors are with the Department of Mechanical, Aerospace & Biomedical Engineering, University of Tennessee, Knoxville TN 37996, Phone: +1-865-974-5309, Email: {khuang1,djlee}@utk.edu. ∗ : Corresponding author. Research supported in part by the National Science Foundation CAREER Award IIS-0844890.

the multiple MIs over undirected communication graph with non-uniform constant delays. The motivating application of this work is peer-to-peer shared multiuser haptic interaction over packet-switching communication network (e.g. Internet with some data buffering), where distributed users haptically interact with their own MI (i.e. local copy of a shared virtual environment) via their haptic device, while these MIs are being synchronized to provide consistent perception of the shared virtual environment among the users. For this, the aforementioned closedloop passivity is very useful, since it would allow us to ensure interaction stability with wide-ranges of (continuous-time) heterogeneous/unmodeled haptic devices or human users, as long as they behave as passive systems. See [16]. To our knowledge, none of results so far on discrete-time second-order consensus (e.g. [11], [12], [13]) has achieved this closed-loop passivity with delay. In addition to its unique passivity property, our framework here also can handle with: non-uniform delay (unlike [11], [12]); multi-dimensionality (unlike [13]); and variable update rate and non-uniformity among the MIs (unlike [11], [12], [13]). Our framework also allows arbitrary gains regardless of sampling rate, if there is no delay (unlike [11], [12]). The rest of the paper is organized as follows. We review some related graph theory and non-iterative variable-rate passive MI of [14], and also introduce our problem setting in Sec. II. The consensus of multiple MIs is then discussed with a numerical example in Sec. III, and some concluding remarks are given in Sec. IV. II. BACKGROUND A. Graph Theory In this paper, we consider undirected and simple (i.e. no self-loop/multiple-edges) graph G(V, E), where V := {v1 , . . . , vN } is the set of vertexes; and E refers to the set of edges. If vi and v j are connected, we say that the edge from vi to v j (denote as ei j ) belongs to E. The set of information neighbors of a vertex vi is defined as Ni := { j|ei j ∈ E}. If G(V, E) is undirected, ei j ∈ E implies e ji ∈ E. Suppose the total number of (directed) edges ei j of G is Ne . Then, there exists a bijective map F : S → E s.t. F (l) = (pl , ql ) where S := {1, . . . , Ne } and, with a slight abuse of notation, we denote the set of {(i, j)|ei j ∈ E} by the edge set E. For simplicity, we will often omit subscript l for (pl , ql ).

The following two properties will be used with their proof omitted here, ∀εi j ∈ ℜn×n Ne

N

∑∑

i=1 j∈Ni

εi j = ∑ ε pq

(1)

l=1

and further, if G(V, E) is undirected, Ne

1

Ne

∑ ε pq = 2 ∑ (ε pq + εqp)

l=1

(2)

l=1

Fig. 1. Multiple MIs on undirected graph G with non-uniform delays Ni j .

where (p, q) = F (l). The incidence matrix D := {dil } ∈ ℜN×Ne of G is defined by   1 if vi is the initial end of the connection el −1 if vi is the terminal end of the connection el dil =  0 otherwise

while the Laplacian matrix L = {li j } ∈ ℜN×N of G is given by   deg(vi ) if i = j −1 if ei j ∈ E li j = (3)  0 otherwise where deg(vi ) refers to the degree (number of edges) of vi . We then have [17, Prop.4.8]: L = DDT

(4)

It is well-known that, if G is undirected and connected, the zero eigenvalue of L is simple with the eigenvector 1 := [1 . . . 1]T ∈ ℜN , and all the other eigenvalues have strictly positive real parts [6], [7], [17]. B. Non-Iterative Variable-Rate Passive MIs In this paper, we utilize the non-iterative variable-rate passive mechanical integrators (MIs) proposed in [14]: for a single mass-spring-damper type MI, during the integration step Tk > 0, vk+1 + vk xk+1 + xk vk+1 − vk +B +K = τk + fk M Tk 2 2 xk+1 − xk vk+1 + vk = 2 Tk where M, B, K ∈ ℜn×n are respectively mass, damper, and spring matrices (with M, K being symmetric/positive-definite, and B being positive-semidefinite); xk , vk ∈ ℜn is the position/velocity; and τk , fk ∈ ℜn are the control and human/environmental force respectively. Here, the first line is the dynamics update; while the second kinematics. This integrator is implicit (i.e. the future state does not only depend on the previous one but also on itself too), yet, still non-iterative (i.e. both xk+1 and vk+1 can be expressed as functions depending on xk and vk explicitly, so they can be solved without iterations), thus, can still be simulated fast. The main reason of using this MI over other integrators (e.g. integrators used in [11], [12], [13], [18]) is that it possesses the following open-loop discrete passivity: M¯

∑ vˆTk (τk + fk )Tk ≥ Vk+1 − Vo ≥ −Vo

k=0

∀M¯ ≥ 0

(5)

where vˆk := (vk+1 + vk )/2 and Vk := vTk Mvk /2 + xTk Kxk /2 is the total energy with Vo being the initial energy. This discrete-time passivity allows us to connect this MI with a variety of haptic devices or human users while ensuring interaction stability, provided that they behave as passive systems. In this paper, we will utilize and also aim to preserve this passivity of the MIs. See [14] for more details on this non-iterative passive integrator. C. Consensus Setting We consider N MIs on an undirected communication graph G(V, E), where each edge ei j ∈ E (i.e. from ith -MI to jth MI) is associated with constant indexing delay N ji ≥ 0 and a symmetric/positive-definite spring matrix Ki j ∈ ℜn×n . See Fig. 1. Then, following Sec. II-B, we write down the following consensus protocol for the ith -MI: during the update interval Ti (k) > 0, vi (k + 1) − vi(k) = τi (k) + fi (k) Ti (k) xi (k + 1) − xi(k) vˆi (k) = Ti (k) τi (k) := −Bi vˆi (k) − ∑ Ki j (xˆi (k) − xˆ j (k − Ni j )) Mi

(6) (7) (8)

j∈Ni

where vˆi (k) := (vi (k + 1) + vi (k))/2, xˆi (k) := (xi (k + 1) + xi (k))/2; Ni is the information neighbors of the ith -MI; Bi ∈ ℜn×n is the damping matrix of ith MI; Ki j ∈ ℜn×n is the spring matrix associated with ei j ; Ni j ≥ 0 is the constant indexing delay from jth -MI to ith -MI; and fi (k) is the force acting on ith -MI (e.g. virtual coupling/contact with an virtual object). For the spring connection Ki j , we assume Ki j = K ji

(9)

i.e. symmetric (yet, still can be non-uniform) spring connection, although their delays may not be so (i.e. Ni j 6= N ji ). Note that the update rate Ti (k) can be varying. With this variable Ti (k), the constant indexing delay Ni j does not necessarily imply constant time delay. III. C ONSENSUS OF M ULTIPLE M ECHANICAL I NTEGRATORS ON D ELAYED U NDIRECTED G RAPH As mentioned in Sec. II-C, the spring connections in the system are multi-dimensional and non-uniform (i.e. the spring matrix Ki j can vary from connection to connection). These features cannot be captured by the graph Laplacian L due to its definition (3). Therefore, we will need to use the

following stiffness matrix, instead of the graph Laplacian L, as the tool to show the position consensus among N MIs connected via Ki j with each other. The stiffness matrix P ∈ ℜnN×nN with its i jth block matrices Pi j ∈ ℜn×n is given as   ∑ j∈Ni Ki j i = j −Ki j i 6= j and ei j ∈ E (10) Pi j =  0 otherwise

where Ki j ∈ ℜn×n is the spring matrix associated with ei j . The next proposition shows that the stiffness matrix has close relation with incidence matrix on an undirected graph. Proposition 1: Consider N MIs on an undirected communication graph G(V, E). The stiffness matrix as defined in (10) can be represented as, 1 (11) P = (D ⊗ In)Kd (D ⊗ In )T 2 where D is the incidence matrix of G(V, E) (defined in Sec. II), ⊗ refers to the Kronecker product, and Kd := diag(K1 , K2 , . . . , KNe ) with Kl the spring matrix associated with el , l ∈ S = {1, ..., Ne }. 1 Proof: Let us define Dˆ := (D⊗ In )Kd2 ∈ ℜnN×nNe . Then, its il-th block matrix dˆil ∈ ℜn×n is given by  1  if vi is the initial end of the edge el  Kl2 1 dˆil = 2 −Kl if vi is the terminal end of the edge el   0 otherwise

where i represents the index of MI; while Kl is the spring matrix associated with the edge el , l ∈ S. Let ˜ k := us denote the edges connecting kth MI by N {r|if er is incoming or outgoing edge of vk }. Then, dˆkl = 1 ˜ k ; and dˆkl = 0n oth±Kl2 6= 0n (n × n zero matrix), if l ∈ N erwise. Now, we are going to show that Dˆ Dˆ T = 2P. First, note that, the kth n × n diagonal block of Dˆ Dˆ T is given by Ne

∑ dˆkl2 = ∑

l=1

Kl = 2

˜k l∈N

∑

Kk j

where the last equality is because G(V, E) is undirected and ˜ k includes both the incoming and outgoing edges of vk . N Also, for the off-diagonal block matrices of Dˆ Dˆ T , we have, for i 6= j Ne

l=1

rank(P) = n rank(L) where n is dimension of MI; L is graph Laplacian of G(V, E); and P is the stiffness matrix in (10). Moreover, if G(V, E) is connected, rank(P) = n(N − 1) Proof: By Proposition 1, the properties of rank and Kronecker product, and (4), we have, 1 1   T rank(P) = rank( (D ⊗ In)Kd2 (D ⊗ In )Kd2 ) T

1

= rank(Kd2 (DT D ⊗ In)Kd2 ) = rank(In )rank(DT D) = n rank(L) since Kd is nonsingular. Also, if G(V, E) is connected, we have rank(L) = N − 1. Thus, rank(P) = n(N − 1) For the N MIs system with the update law (6)-(7), we define the close-loop discrete-time passivity s.t.: ∀M¯ ≥ 0 N

M¯

∑ ∑ vˆi (k)T fi (k)Ti (k) ≥ −c2

(13)

i=1 k=0

where c ∈ ℜ is a bounded constant. Following [15], [19], [20], we also define the controller passivity s.t.: ∀M¯ ≥ 0 N

M¯

∑ ∑ vˆi (k)T τi (k)Ti (k) ≤ d 2

(14)

i=1 k=0

j∈Nk

∑ dˆil dˆjl = ∑

One of the strong properties of the graph Laplacian L is: for a connected undirected graph G(V, E) with N nodes, rank(L) = N − 1. The following lemma extends this property to stiffness matrix. Lemma 1: For a simple undirected graph G(V, E), we have

dˆil dˆjl

(12)

˜ i ∩N ˜j l∈N

˜ i ∩N ˜ j . Suppose ea , eb since, for dˆil dˆjl 6= 0n , it requires l ∈ N are the 1-tuple equivalence of ei j , e ji ∈ E respectively (i.e. ˜i ∩N ˜j = F (a) = (i, j), F (b) = ( j, i)). Then, we have N {a, b}. And (12) will become −(Ka + Kb ) = −(Ki j + K ji ) = −2Ki j from the definition of dˆil above and (9). By comparing these diagonal and off-diagonal block matrices of Dˆ Dˆ T with P in (11), we can then see that 1 P = Dˆ Dˆ T 2

where d ∈ ℜ is a bounded constant. The following lemma shows the connection between closed-loop discrete-time passivity and controller passivity. Lemma 2: For the N MIs system with the update law (6)(7), controller passivity (14) implies the closed-loop discretetime passivity (13). Proof: By (6), (7), and (14), N

M¯

∑ ∑ vˆi (k)T fi (k)Ti (k)

i=1 k=0

N M¯  vi (k + 1) − vi(k)  = ∑ ∑ vˆi (k)T Mi − τi (k) Ti (k) Ti (k) i=1 k=0

≥−

1 N ∑ ||vi (0)||2Mi − d 2 =: −c2 2 i=1

(15)

where 12 ∑Ni=1 ||vi (0)||2Mi represents the initial kinetic energy inside the system, with the notation ||?||2A := ?T A? (A ∈ ℜn×n is required to be symmetric and positive-definite),

As mentioned in Sec. I, closed-loop discrete-time passivity and position consensus among MIs ensure the interaction stability and consistent perception of the shared virtual environment among the users. The following theorem shows that, with enough damping injection, our consensus protocol (6)-(8) achieves both the closed-loop discrete-time passivity and position consensus. Theorem 1: Consider N MIs on an undirected connected graph G(V, E) with the consensus protocol (6)-(8). Suppose that we set the gains Bi , Ki j s.t. Bi ≥

Ni j + N ji ¯ T j Ki j , 2 j∈Ni

∑

i = 1, . . . , N

(16)

where T¯j ≥ maxk (T j (k)), and v(k) = 0, ∀k ≤ 0. Then, 1) the system achieves controller passivity (14), thus, also closed-loop passivity (13); and 2) if there is additional positive-definite damping Bei ∈ ℜn×n on each MI, and fi (k) = 0, the position consensus is achieved s.t. xi (k) → x j (k) and vi (k) → 0,

N

sE (k) := ∑ vˆi (k)T τi (k)Ti (k) i=1

=∑

i=1

h

∑

T

−Di (k) − vˆi (k) Ki j

j∈Ni

N

Ne

= − ∑ Di (k) − ∑ i=1

l=1

vˆTp (k)K pq

i
xˆi (k) − xˆ j (k − Ni j ) Ti (k)

Ne

h   = ∑ vˆTp (k)K pq xˆq (k) − xˆq (k − N pq) Tp (k)

(17)

l=1

i 1 (vˆ p (k)Tp (k) − vˆq (k)Tq (k))T K pq (xˆ p (k) − xˆq (k)) 2

where we use (2). Denote ∆xi j (k) := xi (k) − x j (k) and ϕi j := 1 2 4 ||∆xi j (k)||Ki j . Then, the last terms of (17) can be simplified s.t. (vˆ p (k)Tp (k) − vˆq (k)Tq (k))T K pq (xˆ p (k) − xˆq (k)) 1 = (∆x pq(k + 1) − ∆x pq(k))T K pq (∆x pq (k + 1) + ∆x pq(k)) 2 = 2 (ϕ pq (k + 1) − ϕ pq(k))

∑

l=1



 N ϕ pq (k) − ϕ pq (k + 1) − ∑ Di (k) i=1

Ne

  − ∑ vˆTp (k)K pq xˆq (k) − xˆq (k − N pq ) Tp (k) (18) l=1

Let us define   Θ pq(k) := vˆTp (k)K pq xˆq (k) − xˆq (k − N pq ) Tp (k)  By inserting intermediate terms ∑k−1 j=k+1−N pq − xˆq ( j)+ xˆq ( j)] between xˆq (k) and xˆq (k − N pq ), we can then rewrite Θ pq(k) as k−1

Θ pq (k) = Tp (k)vˆTp (k)K pq

∑

j=k−N pq


xˆq ( j + 1) − xˆq( j)

k−1

= Tp (k)vˆTp (k)K pq

 1 vˆq ( j + 1)Tq ( j + 1) + vˆq( j)Tq ( j) j=k−N pq 2

= Tp (k)vˆTp (k)K pq

h

∑

k−1

∑

vˆq ( j)Tq ( j)

j=k+1−N pq


i 1 vˆq (k)Tq (k) + vˆq (k − N pq)Tq (k − N pq ) 2 where the second line is due to (7). Since K pq is symmetric and positive-definite, we have the following fact s.t.  1 |aT K pq b| ≤ ||a||2Kpq + ||b||2Kpq , ∀a, b ∈ ℜn 2 Using this, we can then show that +

k−1   1 Tq ( j) ||vˆ p (k)||2Kpq + ||vˆq ( j)||2Kpq |Θ pq (k)| ≤ Tp (k) ∑ 2 j=k+1−N pq

  xˆ p (k) − xˆq (k − N pq) Tp (k)

where Di (k) := ||vˆi (k)||2Bi Ti (k), and (p, q) = F (l). The last term can then be rewritten as Ne h   ∑ vˆTp (k)Kpq xˆq(k) − xˆq(k − Npq) Tp(k) l=1 i   + vˆTp (k)K pq xˆ p (k) − xˆq (k) Tp (k)

+

Ne

sE (k) =

∀i, j = 1, . . . , N

Proof: The controller passivity condition (14) means that the controller only generates bounded energy the whole time. Hence we can prove the first item by showing this energy boundedness. Denote the energy generated by the controller during kth integration step as

N

Then, we can rewrite sE (k) s.t.

=

  1 + Tp (k)Tq (k) ||vˆ p (k)||2Kpq + ||vˆq (k)||2Kpq 4   1 + Tp (k)Tq (k − N pq ) ||vˆ p (k)||2Kpq + ||vˆq (k − N pq )||2Kpq 4 k−1   1 α pq (k) Tq (k) + Tq (k − N pq) + 2 Tq ( j) ∑ 4 j=k+1−N pq k−1   1 + Tp (k) αqp (k) + αqp (k − N pq ) + 2 ∑ αqp( j) 4 j=k+1−N pq

where α pq (k) := Tp (k)||vˆ p (k)||2Kpq ≥ 0. Since G(V, E) is undirected, for each |Θ pq (k)|, we will also have the corresponding |Θqp (k)|. By summing them up and collecting the terms containing α pq and αqp respectively, we can show that |Θ pq(k)| + |Θqp(k)| ≤ Ω pq (k) + Ωqp (k) where h1 i k−1 1 α pq ( j) Ω pq (k) := Tq (k) α pq (k − Nqp) + ∑ 2 2 j=k+1−Nqp i h k−1 1 1 Tq ( j) + α pq (k) Tq (k) + Tq (k − N pq) + ∑ 2 2 j=k+1−N pq

Then, by summing Ω pq (k) over time, we have

where the third line is due to (16). This proves the controller passivity (14). Thus, by Lemma 2, closed-loop passivity (13) is achieved. For the second item, from the closed-loop passivity (13), with the additional Bei and fi (k) = 0, we have: ∀M¯ > 0

M¯

∑ Ω pq(k) =

k=0

¯ i h k−1 1 M 1 α pq (k) Tq (k) + Tq (k − N pq) + Tq ( j) ∑ ∑ 2 k=0 2 j=k+1−N pq ¯

c2 ≥

¯

k−1 1 M 1 M + ∑ α pq (k − Nqp)Tq (k) + ∑ Tq (k) ∑ α pq( j) 4 k=0 2 k=0 j=k+1−Nqp

=

¯ i h k−1 1 M 1 α pq (k) Tq (k) + Tq (k − N pq) + Tq ( j) ∑ ∑ 2 k=0 2 j=k+1−N pq

+

¯ i M¯ 1h M (k)T (k + N ) − (k)T (k + N ) α α pq q qp pq q qp ∑ ∑ 4 k=0 ¯ k=M+1−N

+

1h 2

k=0

Mi Mi

j=k+1

−

∑

α pq (M¯ − k)

¯ M−k−1+N qp

∑

¯ j=M+1

k=0

i Tq ( j)

(∗)

k−1+Nqp h i 1 (k) T (k − N ) + T (k + N ) + 2 α q pq q qp ∑ Tq ( j) ∑ 4 pq j=k+1−N pq k=0 M¯

≤

vi (k + 1) − vi(k) → − ∑ Ki j (xˆi (k) − xˆ j (k − Ni j )) (21) Ti (k) j∈Ni vi (k + 2) − vi(k + 1) → Ti (k + 1) − ∑ Ki j (xˆi (k + 1) − xˆ j (k + 1 − Ni j )) (22) j∈Ni

Tq ( j)

Nqp −2

(20)

which implies that, with c being bounded, vˆi (k) → 0. With this, (6) at kth and k + 1th integration steps are reduced to

k−1+Nqp

∑ α pq (k) ∑

N

k=0 i=1

qp

M¯

M¯

∑ ∑ Ti (k)||vˆi (k)||2Bei

Also, from (7) and vˆi (k − 1) → 0, xi (k) → xi (k − 1) and xi (k) → xˆi (k − 1) → xˆi (k − 2) → · · · → xˆi (k − Ni j ). Therefore, the spring forces of the two successive time steps converge to each other, i.e.

∑

Ki j (xˆi (k) − xˆ j (k − Ni j )) →

j∈Ni

∑

(∗∗)

Ki j (xˆi (k + 1) − xˆ j (k + 1 − Ni j ))

j∈Ni

where the last term in (**) comes from the 1st , 3rd and 6th terms in (*). From (**), with α pq (k) ≥ 0 and T¯q ≥ Tq (k), we have M¯

∑ Ω pq(k) ≤

k=0

¯

∑ sE (k) =

k=0

M¯

(vi (k + 1) − vi(k))Ti (k + 1) → −(vi (k + 1) − vi(k))Ti (k)

≤

N

∑

k=0 i=1

M¯

M¯

N

N

∑ ϕ pq(0) − ∑ ∑ D p(k) + ∑ ∑ ∑ k=0 p=1

l=1

M¯

Ne

≤

implying that, with Ti (k) > 0, vi (k + 1) → vi (k). Moreover, with vˆi (k) → 0, we have vi (k) → 0. Substituting this into (6) with fi (k) = 0, we have

∑ ∑ vˆTi (k)τi (k)Ti (k)

Ne

N

∑ ϕ pq(0) − ∑ ∑ D p(k) + ∑ ∑ ∑ k=0 p=1

l=1

|Θ pq(k)|

k=0 p=1 q∈N p M¯

N

k=0 p=1 q∈N p

∑ ϕ pq(0) =: d 2

(19)

l=1

where d is a bounded constant, and we use the fact that: under the gain-setting condition (16), M¯

M¯

∑ ∑

k=0 q∈N p

Ω pq (k) − ∑ D p (k) k=0

¯

¯

≤

M N pq + Nqp ¯ M Tq ∑ α pq (k) − ∑ D p (k) ∑ 2 q∈N p k=0 k=0

=

ˆ T ∑ ∑ v(k)

M¯

k=0

h

Ki j (xi (k) − x j (k)) → 0, i = 1, . . . N

j∈Ni

which can be written by Px(k) → 0

Ω pq (k)

Ne

≤

(vi (k + 1) − vi(k))Ti (k + 1) → (vi (k + 2) − vi(k + 1))Ti (k) where, from vˆi (k) → 0, vi (k + 2) → vi (k). Therefore,

N pq + Nqp ¯ M Tq ∑ α pq (k) 2 k=0

Then, we can rewrite (18) s.t.: ∀M¯ ≥ 0, M¯

thus, combining this with (21)-(22), we have

i N pq + Nqp ¯ ˆ Tq K pq − B p v(k)T p (k) ≤ 0 2 q∈N p

(23)

where P is the stiffness matrix defined in (10), and x(k) := [x1 (k)T . . . xN (k)T ]T . This (23) then implies  T x(k) → 1 ⊗ d = d T . . . d T because: 1) by Lemma 1, the dimension of ker(P) is n. 2) P(1 ⊗ d) = 0, ∀d ∈ ℜn (i.e. ker(P) = {1 ⊗ d|d ∈ ℜn }). Thus, by setting the control gains according to the condition (16), the consensus protocol (6)-(8) enforces the N-port discrete-time passivity for the closed-loop system. If there is no delay, the gain setting condition (16) vanishes and in contrast to [11,12], we can choose arbitrary control gains, while enforcing passivity. On top of the passivity, with the condition (16), fi (k) = 0, and extra damping Bei , the protocol (6)-(8) also achieves the position consensus among all MIs. This is very similar to the results of continuous-time delayed

Position Consensus in X Direction position in x direction [m]

10 x1 x2 x3

5

x4 x

5

0

−5

−10

0

0.2

0.4

0.6

0.8

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PD-scheme of [15], and, in fact, extends the results of [15] to the discrete-time domain, for which the (open-loop) passivity of the MIs of Sec. II-B is indispensable (as that of the continuous-time robot is for the results of [15]). The closed-loop passivity of our framework would be very useful for multiuser peer-to-peer haptics over the Internet (with some data-buffering for constant delays), as it allows any passive humans/devices to be connected while enforcing the closed-loop passivity; as well as the design of virtual world and device servo-loop to be separated from each other. See [16]. We perform a simulation for five 2-dim. MIs, with nonuniform spring matrices Ki j = ki j I with ki j in the range of 25 to 100 N/m; and with non-uniform constant delays in the range of 40 to 1000 ms. We set Ti (k) to be the same (10ms) for simplicity, although varying Ti (k) can be incorporated. See Fig. 2 for the simulation results. Due to the delay among the agents, it takes around 1 second to achieve position consensus both in X and Y directions. The consensus time can be improved by optimizing network topology, Ki j and Bi , and it will be investigated in our future work. See [16] for a preliminary result in this direction. IV. C ONCLUSION

AND

F UTURE W ORKS

We present a novel passivity-enforcing framework for multiple second-order mechanical integrators on undirected graph with constant non-uniform communication delay. By connecting multiple non-iterative passive MIs [16] through discrete-time springs over delayed links with some damping injection, the position consensus and the discrete-time closed-loop passivity are achieved. The control gains can be arbitrarily chosen regardless of the integration steps if there is no delay. The proposed consensus framework is particularly promising for multiuser distributed haptic collaboration over the

Internet as elucidated in [16]. There are also several possible directions for future research. In this paper, the indexes among all MIs are synchronized (i.e. same k shared). Yet, we believe this assumption can be dropped (i.e. k replaced by ki for each MI). Extending our result to switching network topologies would be an interesting direction. Network topology optimization with limited network resource is also important for improving the performance. It is also interesting to extend our results here to general directed graphs. R EFERENCES [1] J. Lin, A. S. Morse, and B. D. O. Anderson. The multi-agent rendezvous problem. In Proceedings of the IEEE Conference on Decision and Control, pages 1508–1513, 2003. [2] J. Cortes, S. Martinez, T. Karatas, and F. Bullo. Coverage control for mobile sensing networks. IEEE Transactions on Robotics and Automation, 20(2):243–255, 2004. [3] E. Bonabeau, M. Dorigo, and G. Theraulaz. Swarm intelligence: from natural to artificial systems. Oxford University Press, USA 1999. [4] V. D. Blondel, J. M. Hendrickx, A. Olshevsky, and J. N. Tsitsiklis. Convergence in multiagent coordination, consensus, and flocking. In Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, pages 2996–3000, 2005. [5] D. J. Low. Following the crowd. Nature, 407(Sep. 28):465–466, 2000. [6] R. Olfati-Saber and R. M. Murray. Consensus problems in networks of agents with switching topology and time-delays. IEEE Transactions on Automatic Control, 49(9):1520–1533, 2004. [7] W. Ren and R. Beard. Consensus seeking in multiagent systems under dynamically changing interaction topologies. IEEE Transactions on Automatic Control, 50(5):655–661, 2005. [8] F. Xiao and L. Wang. Asynchronous consensus in continuous-time multi-agent systems with switching topology and time-varying delays. IEEE Transactions on Automatic Control, 53(8):1804–1816, 2008. [9] H. G. Tanner, A. Jadbabaie, and G. J. Pappas. Flocking in fixed and switching networks. IEEE Transactions on Automatic Control, 52(5):863–868, 2007. [10] D. J. Lee and M. W. Spong. Stable flocking of multiple inertial agents on balanced graphs. IEEE Transactions on Automatic Control, 52(8):1469–1475, 2007. [11] T. Hayakawa, T. Matsuzawa, and S. Hara. Formation control of multi-agent systems with sampled information. In Proc. of the IEEE Conference on Decision & Control, pages 4333–4338, 2006. [12] W. Ren and Y. Cao. Convergence of sampled-date consensus algorithms for double-integrator dynamics. In Proc. of the IEEE Conference on Decision & Control, pages 3965–3970, 2007. [13] P. Lin and Y. Jia. Consensus of second-order discrete-time multiagent systems with nonuniform time-delays and dynamically changing topologies. Automatica, 45:2154–2158, 2009. [14] D. J. Lee and K. Huang. On passive non-iterative variable-step numerical integration of mechanical systems for haptic rendering. In ASME Dynamic Systems & Control Conference, 2008. [15] D. J. Lee and M. W. Spong. Passive bilateral teleoperation with constant time delay. IEEE Transactions on Robotics, 22(2):269–281, 2006. [16] D. J. Lee and K. Huang. Peer-to-peer control architecture for multiuser haptics collaboration over undirected packet-switching communication network. In Proc. of IEEE International Conference on Robotics & Automation, 2010. To appear. [17] N. Biggs. Algebraic Graph Theory Second Edition. Cambridge University Press, UK 1993. [18] J. M. Brown and J. E. Colgate. Minimum mass for haptic display simulations. In Proceedings of ASME International Mechanical Engineering Congress and Exposition, pages 249–256, 1998. [19] D. J. Lee Passive Decomposition and Control of Interactive Mechanical Systems under Coordination Requirements. Doctoral Dissertation, University of Minnesota, 2004. [20] D. J. Lee and P. Y. Li Toward robust passivity: a passive control implementation structure for mechanical teleoperators. In Proceedings of IEEE VR Symposium on Haptic Interface for Virtual Reality and Teleoperator System, pages 132-139, 2003. [21] B. Hannaford and J.H. Ryu. Time domain passivity control of haptic interfaces. IEEE Transactions on Robotics and Automation, 18(1):1, 2002.