Tracking task-space synchronization of networked ... - Springer Link

Nonlinear Dyn (2016) 83:1673–1685 DOI 10.1007/s11071-015-2439-9

ORIGINAL PAPER

Tracking task-space synchronization of networked Lagrangian systems with switching topology Liyun Zhao · Jinchen Ji · Jun Liu · Quanjun Wu · Jin Zhou

Received: 30 April 2015 / Accepted: 28 September 2015 / Published online: 22 October 2015 © Springer Science+Business Media Dordrecht 2015

Abstract This paper investigates the tracking synchronization problem of networked Lagrangian systems with directed switching topologies in task space. A tracking synchronization protocol is developed for the systems with uncertainties in kinematic, dynamic and actuator models. The estimated parameters are updated by using three adaptive control laws to account for the uncertainties. It is found that the positions and velocities of networked Lagrangian systems can track the desired position and velocity in task space, under the condition that the graph topologies are jointly conL. Zhao · J. Liu · J. Zhou (B) Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China e-mail: [email protected] L. Zhao School of Mathematics, Physics and Biological Engineering, Inner Mongolia University of Science and Technology, Baotou 014010, China e-mail: [email protected] J. Ji Faculty of Engineering and IT, University of Technology Sydney, PO Box 123, Broadway, Sydney, NSW 2007, Australia J. Liu Department of Mathematics, Jining University, Qufu 273155, Shandong, China Q. Wu School of Mathematics and Physics, Shanghai University of Electric Power, Shanghai 200090, China

nected and balanced pointwise in time. Specifically, if the dynamic, kinematic and actuator parameters are certain, the tracking synchronization error will be exponentially convergent during a periodically intermittent interaction. Numerical examples and simulations are given to verify the theoretical analysis and demonstrate the effectiveness of the proposed control approach. Keywords Tracking synchronization · Lagrangian systems · Switching topologies · Task space

1 Introduction Recently, the synchronization or consensus of networked Lagrangian systems has received significant attention due to its broad engineering applications in industrial process control, hazardous material exploration, environmental monitoring and so forth [1–10]. The intrinsic strong nonlinearity of multiple Lagrangian systems makes their synchronization entirely different from the common second-order consensus problem, and more interesting and challenging. A large quantity of research results about the synchronization of networked Lagrangian systems have been reported in the literature in the past few years (see [1– 9,11–13] and references cited therein). For instance, Spong applied the passivity property to address the synchronization of such systems [3]. Ren investigated the consensus problem of such systems over undi-

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rected connected topology without any leader [4]. Wu et al presented an impulsive control scheme for networked Lagrangian systems and obtained some simple and important synchronization criteria [5]. Dong proposed a distributed adaptive control algorithm to study the tracking synchronization problem of such systems [7]. Mei et al focused on the coordinated tracking problem for multiple mechanical systems under the constraints that only local followers can obtain the information of a dynamic leader [11]. Sun et al studied the position synchronization of multi-axis motions by using a model-independent cross-coupled controller [12]. It should be noted that most existing studies on the synchronization of multiple Lagrangian systems have considered the exact kinematic parameters and actuator models [2–4,7,8]. This means that all physical parameters of every agent including the length of links, joint offsets and the object held by the robot are known beforehand. However, in practical engineering, some physical parameters are usually hard to be predicted in advance and sometimes changeable with time. For example, when a two-link manipulator picks up an object of uncertain length, generally, the gripping points or orientations are not known precisely [14]. The overall kinematics are changing and difficult to derive exactly. That is to say, the uncertainties of dynamic, kinematic and actuator models can usually occur. Therefore, Cheah et al. in [14] firstly presented adaptive tracking control for Lagrangian systems in which kinematic and dynamic parameters are uncertain. Later, Cheah et al. extended such an adaptive tracking scheme to the case of uncertain actuator models [15]. Apart from the parameter uncertainties, the communication structure of the networked systems may be changed during the process. This can happen in two ways. On the one hand, the failure of information transmission of some agents is inevitable, because of the possible message dropouts or congestion of the communication channels, external disturbances and limitations of sensing ranges, etc [9,16–18]. On the other hand, some new connections between the nearby agents can be created because the agents come to an effective range of the detection with respect to each other. For the network topology, this means that some edges are added or removed from the graph. That is to say, the switching phenomenon of network topology is actually a common phenomenon in practical applications.

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Thus, it is worth studying synchronization problem over switching communication topologies. Although there have been some results on this issue (see [16,19] for example), most of researches are concerned with the multi-agent systems modeled by the first- or secondorder integrator dynamics. Motivated by the above background, this paper focuses on the coordinated tracking issue of multiple Lagrangian systems in task space with uncertain kinematic, dynamic and actuator models over directed switching topologies. A tracking synchronization protocol will be developed for such systems. It will be shown that the systems can follow a desired trajectory in task space if the graph topologies are jointly connected and balanced pointwise in time. Particularly, if the kinematic, dynamic and actuator models are certain, the position and velocity errors of the systems are exponentially convergent during the periodically intermittent interaction of agents. The rest of this paper is organized as follows: Sect. 2 presents some notations, the elementary knowledge of graph theory and an extended Barbalat’s lemma. Section 3 analyzes the coordinated tracking problem of networked Lagrangian systems in the task space over directed time-variant topologies. Illustrative examples and numerical simulations are provided in Sect. 4. Finally, conclusion and future work are provided in Sect. 5.

2 Preliminaries 2.1 Notations Throughout this paper, the following notations and definitions will be used. N = {0, 1, 2, · · · } is used to express the natural number set. R n represents an n dimension column vector set, for the vector u ∈ R n , u T stands for its transpose, and u˙ is the derivative of the differentiable vector u. Let R n×N be the set of n × N real matrices. λmin (•) and λmax (•) represent the minimum and maximum eigenvalues of the matrix “•”. 0 is a vector with all zeros. In ∈ R n×n is the n-order identity matrix, and diag(γ1 , γ2 , · · · , γn ) ∈ R n×n is the diagonal matrix with diagonal entries γi (i = 1, 2, · · · , n). For the vector u = (u 1 , . . . , u n )T ∈ R n , diag(u) denotes the diagonal matrix with the ith diagonal entry u i . The symbol ⊗ is used to indicate the Kronecker product of two matrices [20].

Tracking task-space synchronization of networked Lagrangian systems...

2.2 Graph theory Assume n-order weighted graph is represented by G = (V, E, A), where V = {v1 , v2 , . . . , vn }, E ⊆ V × V, A = (ai j )n×n are the set of nodes, the set of edges and the weighted adjacency matrix, respectively. A directed edge in a directed graph is denoted by (vi , v j ). (vi , v j ) ∈ V means the node v j can receive information from the node vi , where vi is called the parent node of the v j , while v j is called the child node of vi . The in-degree of node vi is the number of edges that have this node as a child node. Similarly, the out-degree of node vi is the number of edges that have node vi as the parent node. A directed path is the sequence of edges of the form (vi1 , vi2 ), (vi2 , vi3 ), · · · , (vi j−1 , vi j ), where (vik , vik+1 )(k = 1, 2, . . . , j − 1) is an edge of graph G. Directed graph G is said to be strongly connected if there exists a directed path between any two different nodes [3,21]. The element ai j of the weighted adjacency matrix A is positive if and only if there is a directed edge (v j , vi ) in G; otherwise, ai j = 0. All of the graphs are assumed to have no self-loop. The elements of the Laplacian matrix n×n L = (li j ) ∈ R n associated with graph G are defined as: lii = i j and li j = −ai j , where j=1, j=i a  i = j. If nj=1, j=i ai j = nj=1, j=i a ji for each of i = 1, . . . , n, then the graph G is said to be balanced. The jointly connected topologies will be introduced below. Consider that time sequence {tk }k≥0 satisfies 0 = t0 < t1 < · · · < tk−1 < tk < · · · (k ∈ N) with limk→∞ tk = +∞ and tk − tk−1 ≤  for a certain constant number  > 0. The time interval [tk−1 , tk ) n k i−1 i [tk−1 , tk−1 ) such can be written as [tk−1 , tk ) = i=1 i−1 i that tk−1 − tk−1 ≥ 1 > 0 [22]. The network topoli−1 i ogy Gσ (t) is fixed with t ∈ [tk−1 , tk−1 ), where the map σ : [0, +∞) →  = {1, 2, · · · , P} is a right continuous piecewise and constant stochastic switching signal with P being the total number of all possible interaction graphs. Every Gσ (t) is permitted not to be strongly connected. The directed graphs Gσ (t) are said to be jointly connected  across the interval [t, t + ) if the directed graph s∈[t,t+) Gσ (s) is strongly connected [3,9,23,24]. 2.3 An extended Barbalat’s lemma A generalized version of Barbalat’s lemma which is applicable to piecewise continuous functions will be

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given in order to address the tracking synchronization problem of networked Lagrangian systems in this paper. Lemma 1 [9,17,25] Assume that the sequence {tk }k∈N with t0 = 0 is strictly increasing along with k, and Q(t) : [0, +∞) → R + satisfies 1. 2. 3.

limt→∞ Q(t) exists; Q(t) is twice differentiable for t ∈ [tk , tk+1 ); ¨ Q(t) is bounded in interval [0, +∞), i.e., there exists a constant positive number K¯ such that ¨ < K¯ . suptk−1 ≤t 0, x˙ri is the reference velocity and X˜ i = xi − xd is the tracking error vector.

123

(7)

where θˆki is an estimated kinematic parameter vector and Ji (qi , θˆki ) is an approximate Jacobian matrix. Taking time derivative of Eq. (7) yields the following jointspace reference acceleration q¨ri = Ji−1 (qi , θˆki )x¨ri + J˙i−1 (qi , θˆki )x˙ri .

(8)

By using Eq. (7), the joint-space sliding variable can be written as si = q˙i − q˙ri = q˙i − Ji−1 (qi , θˆki )x˙ri .

(9)

By incorporating the definition of joint-space sliding variable (9) and Property 4 into Eq. (6), the task-space sliding variable is given by ri = Ji (qi , θki )q˙i − Ji (qi , θˆki )q˙ri   = Ji (qi , θˆki )q˙i − Ji (qi , θˆki )q˙ri   + Ji (qi , θki )q˙i − Ji (qi , θˆki )q˙i = Ji (qi , θˆki )si + Yki (qi , q˙i )θ˜ki ,

(10)

where θ˜ki = θki −θˆki is the estimated error of kinematic parameter. For the sake of convenience, simple notations of Jˆi = Ji (qi , θˆki ), Ydi = Ydi (qi , q˙i , q˙ri , q¨ri ), Yki = Yki (qi , q˙i ) will be used. The directed switching topologies composed of n agents are assumed to be jointly connected and balanced information graph pointwise in time. According to Lemma 1 in [3], these switching topologies only have two types of nodes, i.e., isolated nodes and strongly connected nodes. For notations, Nς (t) and Nc (t) are used to denote the set of isolated node and the set of strongly connected nodes of the graph formed by n agents at moment t. Then, the adaptive control laws proposed for system (1) are assumed to consist of the following parts: (i) The control input ⎧ − Kˆ i−1 JˆiT i X˜ i + Kˆ i−1 Ydi θˆdi + Kˆ i−1 Yai (τoi )θˆai , i ∈ Nς (t), ⎪ ⎪ ⎪ ⎪ ⎪ n

⎪  ⎪ ⎪ − Kˆ −1 Jˆ T b r + ⎪ ai j (t)(ri − r j ) − Kˆ i−1 JˆiT i X˜ i i i ⎪ i i ⎪ ⎨

  j=1 τi = feedback   ⎪ ⎪ coordination ⎪ ⎪ ⎪ ⎪ −1 ⎪ ˆ ⎪ θˆ + K Y + Kˆ i−1 Yai (τoi )θˆai , i ∈ Nc (t). d ⎪ i i di ⎪ ⎪

 

  ⎩ dynamic compensation

uncertainty compensation

(11)

Tracking task-space synchronization of networked Lagrangian systems...

(ii) The adaptive law of dynamic parameter θ˙ˆdi = − d−1 YdT si .

(12)

i

i

(iii) The adaptive law of kinematic parameter ⎧ ⎪

−1 Y T X˜ , i ∈ Nς (t), ⎪ ⎨ ki ki i i ˙θˆ =

n  ki  −1 T  ⎪ ai j (t)(ri −r j ) + i X˜ i , i ∈ Nc (t). ⎪ ⎩ k Yk bi ri + i

i

j=1

(13)

i

i

⎧ Mi (qi )˙si + Ci (qi , q˙i )si = − JˆiT i X˜ i − Ydi θ˜di ⎪ ⎪ ⎪ ⎪ ⎪ ˆ −1 ˆ −1 ˆ ⎪ i ∈ Nς (t), ⎪ ⎨ −(K i K i − In )τoi + K i K i Yai (τoi )θai ,

n  ⎪ ai j (t)(ri −r j ) − JˆiT i X˜ i ⎪ Mi (qi )˙si +Ci (qi , q˙i )si = − JˆiT bi ri + ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎩ −Ydi θ˜di − (K i Kˆ i−1 − In )τoi + K i Kˆ i−1 Yai (τoi )θˆai , i ∈ Nc (t),

(16) with θ˜di = θdi − θˆdi being the estimated error of dynamic parameter. Due to (K i Kˆ i−1 − In )τoi − K i Kˆ i−1 Yai (τoi )θˆai

(iv) The adaptive law for actuator θ˙ˆai = − a−1 YaT (τoi )si ,

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(14)

where the diagonal transmission matrix Kˆ i is the estimator of the transmission matrix K i , the matrices i , di , ki are positive definite, ai is the positive definite diagonal matrix, Yai (τoi ) = diag{−τoi } with τoi being defined as ⎧ T Jˆi i X˜ i − Ydi θˆdi , i ∈ Nς (t), ⎪ ⎪ ⎪ ⎪ ⎨

n  T ai j (t)(ri − r j ) τoi = Jˆi bi ri + (15) ⎪ j=1 ⎪ ⎪ ⎪ ⎩ + JˆiT i X˜ i − Ydi θˆdi , i ∈ Nc (t), where ai j (t) is the entry of the adjacency matrix corresponding to n agents at time instant t and bi is a feedback gain for the ith strongly connected agent (bi > 0 if the feedback gain is applied to the ith strongly connected agent and bi = 0 otherwise).

= Yai (τoi )θ˜ai ,

(17)

where θ˜ai = θai − K i Kˆ i−1 θˆai with θai = (ki1 /kˆi1 − j j 1, ki2 /kˆi2 − 1, . . . , kiN /kˆiN − 1)T and ki , kˆi are, respectively, the jth diagonal entries of matrices K i , Kˆ i . It is noted that Eq. (17) is similar to Eq. (22) in [15]. Equation (16) can be rewritten as ⎧ Mi (qi )˙si + Ci (qi , q˙i )si ⎪ ⎪ ⎪ ⎪ ⎪ T ⎪ i ∈ Nς (t), ⎪ = − Jˆi i X˜ i − Ydi θ˜di − Yai (τoi )θ˜ai , ⎨

n  T ⎪ ai j (t)(ri − r j ) ⎪ Mi (qi )˙si + Ci (qi , q˙i )si = − Jˆi bi ri + ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎩ T ˆ ˜ ˜ ˜ − Ji i X i − Ydi θdi − Yai (τoi )θai , i ∈ Nc (t).

(18) For n networked systems (18), the Lyapunov-like function candidate is chosen as n    V (t) = Vi (t) = Vi (t) + Vi (t), i∈Nς (t)

i=1

Remark 1 For strongly connected nodes, the first term of the control protocol (11) acts as coordinating taskspace coordination; the second term is feedback of position error; the third term is dynamic compensation term; and the fourth term compensates for the uncertainty introduced by actuator. Therefore, it can be seen that the main function of the control input (11) is to make the strongly connected agents synchronize each other and the isolated agents track the desired trajectory, and thus, all agents achieve the desired tracking synchronization. For the uncertainty in dynamic, kinematic and actuator models, the estimated parameters are updated by using the respective adaptive control laws (ii), (iii) and (iv). Substituting the input (11) into systems (1), and using the definition of the joint-space sliding variable si and Eq. (15), yield

i∈Nc (t)

where 1 T 1 1 si Mi (qi )si + θ˜dT di θ˜di + θ˜kT ki θ˜ki i 2 2 2 i 1 T 1 + θ˜a ai K i Kˆ i−1 θ˜ai + X˜ iT i X˜ i . 2 i 2 It is clear that Vi (t) ≥ 0. By using the definition of θ˜di , θ˜ki , θ˜ai , the derivative of Vi (t) can be written as Vi (t) =

1 V˙i (t) = siT Mi (qi )˙si + siT M˙ i (qi )si + θ˜dT di θ˙˜di i 2 ˙ ˙ T T T + θ˜k k θ˜k + θ˜a a θ˜a + X˜ i i X˙˜ i i

i

i

= siT Mi (qi )˙si − θ˜kT ki θ˙ˆki i

i

i

i

1 + siT M˙ i (qi )si − θ˜dT di θ˙ˆdi i 2 ˙ − θ˜aT a θˆa + X˜ iT i X˙˜ i . i

i

i

When i ∈ Nς (t), based on the first equation of systems (18), the parameter adaptive law (12) and the first

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equation in (13) and (14), the derivative of Vi (t) can be further expressed as

V˙i (t) = siT − Ci (qi , q˙i )si − JˆiT i X˜ i − Yd θ˜d i



i

1 − Yai (τoi )θ˜ai + siT M˙ i (qi )si + θ˜dT YdT si i i 2 T T T T T ˜ ˜ ˜ ˜ − θk Yk i X i + θa Ya (τoi )si + X i i X˙˜ i i

i

i

−

i

= −siT JˆiT i X˜ i − θ˜kT YkT i X˜ i + X˜ iT i X˙˜ i . i

Here, the second equality in (19) is obtained by using Property 2, the equations siT Ydi θ˜di = θ˜dT YdT si i i and siT Yai (τoi )θ˜ai = θ˜aT YaT (τoi )si . Substituting (10) i i into (19) yields the following equality: V˙i (t) = −siT JˆiT i X˜ i − θ˜kT YkT i X˜ i + X˜ iT i X˙˜ i i

= −(ri −Yki θ˜ki )T i X˜ i − θ˜kT YkT i X˜ i + X˜ iT i X˙˜ i i

i

= −( X˙˜ i + α X˜ i − Yki θ˜ki )T i X˜ i − θ˜kT YkT i X˜ i i

i

+ X˜ iT i X˙˜ i = −α X˜ iT i X˜ i .

(20)

For i ∈ Nc (t), from systems (18), the parameter adaptive law (12) and the second equation in (13) and (14), the corresponding derivative of Vi (t) can be obtained as n

   ai j (t)(ri − r j ) = siT − JˆiT bi ri + j=1

− Ci (qi , q˙i )si − JˆiT i X˜ i − Ydi θ˜di − Yai (τoi )θ˜ai



1 + siT M˙ i (qi )si + θ˜dT YdT si i i 2 n    − θ˜kT YkT bi ri + ai j (t)(ri − r j ) + i X˜ i i

j=1

+ θ˜aT YaT (τoi )si + X˜ iT i X˙˜ i i

i

n    = − (siT JˆiT + θ˜kT YkT ) bi ri + ai j (t)(ri − r j ) i

i

j=1

− siT JˆiT i X˜ i

− θ˜kT YkT i X˜ i i i

+ X˜ iT i X˙˜ i

n

 = − riT bi ri + ai j (t)(ri − r j ) − α X˜ iT i X˜ i . j=1

(21)

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j=1

α X˜ iT i X˜ i .

The following theorem can be given based on the above discussion. Theorem 1 Suppose n agents form a jointly connected and balanced information graph pointwise in time, and at least one bi > 0, then the positions and velocities in the task space for the mechanical systems (1) with the control law (11) and the parameter update laws (12), (13), (14) can, respectively, track to xd , x˙d . Proof Lemma 1 is used to prove this theorem. Since n agents form balanced information graph pointwise in time, the derivative of the Lyapunov-like function can be obtained as  bi riT ri V˙ (t) = − i∈Nc (t)

−

n 1   ai j (t)(ri − r j )T (ri − r j ) 2

i∈Nc (t) j=1 n  − α X˜ iT i X˜ i . i=1

V˙i (t)

i

i∈Nc (t) n  i=1

i

(19)

i

Here the last equality in (21) is obtained by using (10) and (20). Combining equations (20) and (21) gives the following expression n

  riT bi ri + ai j (t)(ri − r j ) V˙ (t) = −

Firstly, it can be seen from V˙ (t) ≤ 0 that V (t) is t monotonous decreasing and V (t) − V (0) = 0 V˙ (s) ds ≤ 0, hence, limt→∞ V (t) exists. Secondly, it is evident that V (t) is twice differentiable in every switching interval. Thirdly, it is easy to notice that si , θ˜di , θ˜ki , θ˜ai , X˜ i are bounded from the boundedness of V (t); therefore, x˙ri = x˙d − α X˜ i is bounded if x˙d is bounded. Further, it is followed from (7) that q˙ri is bounded as Ji−1 (qi , θˆki ) is the trigonometric function of qi . From q˙i = si − q˙ri , it is noted that q˙i is bounded, which implies ri is bounded by (10). Consequently, X˙˜ i is bounded by (6). Since J˙i−1 (qi , θˆki ) is the function of q , q˙ , θˆ , θ˙ˆ and q˙ , θˆ , θ˙ˆ are bounded, J˙−1 (q , θˆ ) i

i

ki

ki

i

ki

ki

i

i

ki

is bounded, which means q¨ri is bounded by (3), and thus Ydi is bounded. By the closed-loop systems (18), we can get s˙i is bounded. If x¨d is bounded, then q¨i is bounded by the derivation of (9). By differentiating

Tracking task-space synchronization of networked Lagrangian systems...

both sides of (10), it can be obtained that r˙i is bounded; therefore, V¨ (t) is bounded in the interval [0, +∞). By Lemma 1, V˙ (t) → 0. Similar to the argument of Theorem 4 in Ref. [3], there is limt→∞ ri − r j = 0 for any two agents i, j, i.e., X˙˜ i → X˙˜ j , X˜ i → X˜ j based on Lemma 1 in [9]. On the other hand, since at least one bi = 0, then at least one ri → 0 for i ∈ Nc (t), that is to say, X˙˜ i → 0, X˜ i → 0 for i ∈ Nc (t) by (6). In conclusion, X˙˜ j → X˙˜ i → 0, X˜ j → X˜ i → 0 for any agents i, j. This completes the proof of Theorem 1.  Remark 2 The cooperative control (11) and the parameter adaptation (12)–(14) may seem somewhat conservative. This is because, according to the argument of Theorem 1, if the feedback term bi ri in the control input (11) and in the adaptive laws (12)–(14) is replaced by bi X˙˜ i , then only local velocity error feedback can make the agents’ positions and velocities track to xd , x˙d . In addition, the feedback term Kˆ i−1 JˆiT bi ri in the control input (11) can be transferred to the isolated agents. Remark 3 If each agent is always independent for all the time t ∈ [0, +∞), i.e., ai j (t) ≡ 0 for arbitrary i, j = 1, . . . , n, the realization of the control objectives xi → xd , x˙i → x˙d in the task space requires that all bi are nonzero. This implies the control input must contain the feedback terms of both position and velocity errors. This observation is in line with [15], whereas the proposed controlled scheme, due to the presence of information exchange among the agents, does not require that all of bi = 0. In other words, the control input (11) for a part of agents only needs the feedback term of position error. This is one of the advantages of the proposed networked synchronization scheme.

3.3 Intermittent synchronization protocol If the kinematic, dynamic and actuator models are certain and the information interaction among the agents is periodically intermittent   coupled [30], there is a partition [kω, kω + h) [kω + h, (k + 1)ω) k∈N of time interval [0, +∞), such that there exists no information interaction when t ∈ [kω, kω + h), yet the information topology is strongly connected when t ∈ [kω + h, (k + 1)ω)). In this case, the following control input is chosen:

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⎧ Mi (qi )q¨ri + Ci (qi , q˙i )q˙ri + gi (qi ), ⎪ ⎪ ⎪ ⎪ ⎨ M (q )q¨ + C (q , q˙ )q˙ + g (q ) i i ri i i i ri i i τi =

⎪ n  ⎪ −1 ⎪ ⎪ ai j (ri − r j ) , bi ri + ⎩ −Ji

t ∈ [kω, kω + h),

t ∈ [kω + h, (k + 1)ω),

j=1

(22) where q˙ri = Ji−1 x˙ri = Ji−1 (x˙d − α X˜ i ) with Ji = Ji (qi , θki ) and ri = Ji si , ai j is the entry of adjacency matrix and identical in every interval [kω + h, (k + 1)ω). In addition, h and ω − h indicate the duration of “decoupling time” and “coupling time”, respectively. Substituting the control law (22) into system (1) yields the following closed-loop dynamics: ⎧ M (q )˙s + Ci (qi , q˙i )si = 0, t ∈ [kω, kω + h), ⎪ ⎪ ⎪ i i i ⎪

⎨ n  Mi (qi )˙si + Ci (qi , q˙i )si = −Ji−1 bi ri + ai j (ri − r j ) , ⎪ j=1 ⎪ ⎪ ⎪ ⎩ t ∈ [kω + h, (k + 1)ω).

(23) The following theorem is given for discussing the tracking issue of networked Lagrangian systems. Theorem 2 For systems (23), if n agents form a strongly connected information graph in time interval [kω + h, (k + 1)ω) and at least one bi > 0, then the agents’ positions and velocities can track to xd , x˙d , respectively. Proof Denote S = (s1T , s2T , . . . , snT )T ∈ R n N , H = 1 T 2 (L P + P L) + B, where L is Laplacian matrix corresponding to n agents and B = diag{ p1 b1 , p2 b2 , . . . pn bn } with ( p1 , p2 , . . . , pn )T being the left eigenvector of L with respect to eigenvalue 0. Since n agents form a strongly connected information graph, the matrix L is irreducible. Accordingly, pi > 0 for i = 1, 2, . . . , n [6]. Define a positive definite function V as n n  1 Vi = pi siT Mi (qi )si . V = 2 i=1

i=1

When t ∈ [kω, kω + h) for an arbitrarily given k ∈ N, taking the time derivative of V along the trajectories of (23) gives V˙ = −

n  i=1

  pi siT Mi (qi )˙si + 21 siT M˙ i (qi )si = 0.

The above equality is obtained by Property 2. Then following V˙ = 0, for any t ∈ [kω, kω + h) gives rise to V (t) = V (kω).

(24)

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For t ∈ [kω +h, (k +1)ω), the corresponding derivative of V is given by V˙ = − =−

n  i=1 n 

  1 pi siT Mi (qi )˙si + siT M˙ i (qi )si 2

i=1

+Ji−1

j=1

=−

n 



1 ai j (ri − r j ) + siT M˙ i (qi )si ] 2

n    pi siT bi Ji−1 ri + Ji−1 ai j (ri − r j )

i=1

=− =− − −

n 

j=1

pi siT

i=1 n 

1 2

n 

ai j (si − s j ) + bi si



j=1

pi ai j siT (si − s j )

i=1

n 1

2

i=1 n 

p j a ji siT (si − s j )

pi bi siT si −

i=1

1 T T S (L P ⊗ I N )S 2

1 = − S T (PL ⊗ I N )S − S T (B ⊗ I N )S 2 = − S T (H ⊗ I N )S.

V˙ ≤ −λmin (H ⊗ I N )S T S. On the other hand, V ≤ 21 M S T S with M = max {M i }. So V˙ ≤ −βV with β = 2λmin (H ⊗I N ) > 0. M

In other words, for any t ∈ [kω + h, (k + 1)ω), V (t) ≤ e−β(t−(kω+h)) V (kω + h).

(25)

For an arbitrary t > 0, when t ∈ [kω + h, (k + 1)ω), Equation (25) becomes by considering Equation (24) and the continuity of function V (t) V (t) ≤ e−β(t−(kω+h)) V (kω) ≤ e−β(t−(kω+h)) e−β(ω−h)k V (0),

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Remark 4 When t ∈ [kω + h, (k + 1)ω), V (t) is exponentially decreasing according to (25). Therefore, if the kinematic, dynamic and actuator models are certain, under periodically intermittent interaction, the convergence speed of the agents’ states in system (1) with the control input (22) is fast. Specifically, the longer the “coupling time” ω − h is, the faster the agents’ convergence speed becomes. Notably, if there exists no “decoupling time” h, i.e., h = 0, the position and velocity of every agent can, respectively, exponentially converge to xd , x˙d . The “decoupling time” h can be adjusted to improve the tracking performance of such networked Lagrangian systems. Better tracking performance can be achieved by finding a good balance between the “decoupling time” h and “coupling time” ω − h. As such, the convergence speed of the agents’ states can be as fast as possible, and the “coupling time” ω − h is acceptable in practical applications. 4 Numerical simulations

Since the information exchange graph is strongly connected, 21 (L T P + P L) has a simple zero eigenvalue and the other eigenvalues are positive [6,31]. This implies all eigenvalues of H are positive due to at least one pi bi > 0. Then the derivative of V can be further written as

1≤i≤n

V (t) ≤ V (kω) ≤ e−β(ω−h)k V (0).

It then follows from (26), (27) that V → 0, which means si converges to zero as k → ∞. Therefore, X˜ i , X˙˜ i converge to zeros because X˙˜ i + α X˜ i = Ji si . This completes the proof of Theorem 2. 

 pi [siT − Ci (qi , q˙i )si + bi Ji−1 ri n 

when t ∈ [kω, kω + h),

(26)

In this section, three two-link planar manipulators (whose mechanical structure is shown in [6]) are employed to demonstrate the effectiveness of the obtained theoretical results. The control objective is to track the desired position xd = 5 + cos(0.5t), 4.8 + T 0.2 sin t . The jointly connected, balanced pointwise in time and stochastic switching topologies G1 , G2 , G3 with 0 − 1 adjacency elements are shown in Fig. 1. The topologies G1 , G2 , G3 are balancedand none of them 3 Gi is strongly is strongly connected. However, i=1 connected; therefore, the conditions of Theorem 1 are satisfied. In general, the dynamic model of the i twolink planar manipulator is described by    q¨i1 Mi11 (qi ) Mi12 (qi ) q¨i2 Mi21 (qi ) Mi22 (qi )      q˙i1 Ci11 (qi , q˙i ) Ci12 (qi , q˙i ) gi1 + + Ci21 (qi , q˙i ) 0 q˙i2 gi2 = K i τi ,

Tracking task-space synchronization of networked Lagrangian systems... Fig. 1 Switching topologies G1 , G2 , G3

3

i1

x (t), (i=1,2,3,4)

10 8 6 4 0

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Fig. 2 Tracking process of positions in task space for three agents with the random switching sequences G1 → G2 → G3 → G1 → G2 → G3 . . ..

3

2

3

2

where Mi11 (qi ) = θi1 + θi2 + 2θi3 cos qi2 , Mi12 (qi ) = Mi21 (qi ) = θi2 + θi3 cos qi2 , Mi22 (qi ) = θi2 , Ci11 (qi , q˙i ) = −θi3 sin qi2 q˙i2 , Ci12 (qi , q˙i ) = −θi3 sin qi2 (q˙i1 + q˙i2 ), Ci21 (qi , q˙i ) = θi3 sin qi2 q˙i1 , gi1 = θi4 cos qi1 + θi5 cos(qi1 + qi2 ), gi2 = θi5 cos(qi1 + qi2 ) with 2 + m l2 + I , θ = m l2 + I , θ = θi1 = m i1 lci i2 i1 i1 i2 i 2 ci 2 i2 i3 1 m i2 li1 lci2 , θi4 = (m i1 lci1 + m i2 li1 )g, θi5 = m i2 glci2 . The parameter values are given by m i1 = 0.32+0.001i, m i2 = 0.4 + 0.03i, li1 = 0.5 + 0.06i, li2 = 0.42 + 2 /3, 0.06i, lci1 = li1 /2, lci2 = li2 /2, Ii1 = m i1 lci 1 2 Ii2 = m i2 lci2 /3. The actuator models are estimated as Kˆ 1 = Kˆ 2 = Kˆ 3 = diag(1.35, 2.19). The matrices are given by d1 = d2 = d3 = 0.01I5 , k1 =

k2 = k3 = 0.3I2 , a1 = a2 = a3 = 0.01I2 . The feedback gains b2 = 1, b1 = b3 = 0.

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4 2 0

−2 −4 0

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t

2 0

i2

e (t), (i=1,2,3)

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Fig. 4 The change process of velocity error ei (i = 1, 2, 3) in task space

The nonlinear mapping h i from the joint configuration variable qi to the generalized end-effector position xi is given by

xi = li1 cos(qi1 ) + li2 cos(qi1 + qi2 ), T li1 sin(qi1 ) + li2 sin(qi1 + qi2 ) .

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Fig. 5 The complete trajectories of three agents and the desired trajectory with the random switching sequences G3 → G2 → G1 → G3 → G2 → G1 . . .

x (t)

Figure 2 displays the tracking process of positions in task space for three agents with the control input (11) and the parameter adaptive laws (12), (13), (14) with the random switching sequences G1 → G2 → G3 → G1 → G2 → G3 . . .. The initial configurations

are chosen as q1 (0) = (−0.6121, −0.3415)T , q˙1 (0) = (−1.0800, −0.4315)T , q2 (0) = (1.001, −0.6107)T , q˙2 (0) = (0.1112, −1.9010)T , q3 (0) = (2.5001, 1.1205)T , q˙3 (0) = (−0.2, −0.1)T . The initial dynamic parameters are given by θd1 = (2.5110, 2.7002, 0.8934, 2.7235, 0.2867)T , θd2 = (0.1121, −1, 2850, 2.4561, 1.7234, 0.8812)T , θd3 = (1.3412, 3.001, 1.2340, 0.8121, −1.1067)T . The initial kinematic and actuator parameters are selected as θk1 = (3.2310, 2.1231)T , θk2 = (0.2319, 2.7009)T , θk3 = (0.5444, 4.1340)T , and θa1 = (0.0211, 4.3246), θa2 = (2.4781, 3.1623)T , θa3 = (3.56329, 0.4310)T . The estimates of the first dynamics parameter θˆdi1 , kinematic parameter θˆki1 and actuator model parameter θˆai1 for i = 1, 2, 3, respectively, are depicted in Fig. 3. These parameters seem not to converge to their actual values due to the difficulty of fulfilling the persistent excitation (see [6] and the references therein). The velocity error in task space is defined as ei = (ei1 , ei2 )T = x˙i − x˙d (i = 1, 2, 3). Figure 4 shows the variation process of velocity errors ei1 and ei2 , respectively, where the initial configurations are q1 (0) = (3.5567, 5.1212)T , q˙1 (0) = (2.2410, 0.3241)T , q2 (0) = (5.7870, 6.9010)T , q˙2 (0) = (−3.3412, −6.6790)T , q3 (0) = (7.1003, 4.5601)T , q˙3 (0) = (−1.2389, −0.8912)T . The dynamic parameters are chosen as θd1 = (4.5612, −2.0121, 0.0012, 4.7891, 1.2104)T , θd2 = (0.1290, 7.1230, 8.1080, −2.0010, 1.2001)T , θd3 = (−0.7800,

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Tracking task-space synchronization of networked Lagrangian systems...

Furthermore, simulation results (see Figs. 2, 4) indicate that the tracking synchronization of such systems does not depend on the selection of initial values, which is consistent with common global synchronization [11, 30,32,33]. Moreover, as can be seen from Figs. 2 and 5, different random switching sequences have no effect on the synchronization of such systems.

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Fig. 6 Communication topologies G5 , G6 20

x11

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Fig. 7 Responses of the states xi1 (t) and xi2 (t) for i = 1, 2, 3, d

2.4510, −0.8870, 5.001, 2.3451)T . The initial kinematic and actuator parameters are θk1 = (0.7801, 4.0100)T , θk2 = (3.4510, 4.1002)T , θk3 = (1.2111, 1.7701)T , and θa1 = (2.5444, 0.5512), θa2 = (2.4561, 0.8901)T , θa3 = (2.3451, 0.4512)T . The complete trajectories of three agents and the desired trajectory are shown in Fig. 5 with the random switching sequences G3 → G2 → G1 → G3 → G2 → G1 . . . and the same initial values as ones in Fig. 2. In addition, in order to demonstrate the effectiveness of the controller (22), it is assumed that the “decoupling time” h = 0.2, “coupling time”  ω− h = 0.2 and the desired position is xd = 1.5 + 0.5 sin(2t), −2 + 4 cos(t) . The simulation result is presented in Fig. 7 with randomly selected initial conditions within [−10, 10] and the random switching topologies G5 , G6 as shown in Fig. 6. Obviously, it is in excellent agreement with the theoretical results. It can be observed from Figs 2, 4, 5 and 7 that the end effector of three agents can track the desired position approximately. Therefore, the proposed algorithm (11) and parameter adaptive laws (12), (13), (14) and control input (22) are effective to deal with the tracking issue for networked Lagrangian systems in the task space.

This paper has investigated the tracking synchronization problem of networked Lagrangian systems in task space over directed switching topologies. Some simple yet generic conditions have been presented to stabilize networked Lagrangian systems to a desired synchronization state. The obtained results show that the positions and velocities of networked Lagrangian systems can track to the desired position and velocity if the graph topologies are jointly connected and balanced pointwise in time. Compared with the existing works, this paper made the following contributions to the field of research: (i) Different from tracking problem for a single robot examined in [15], the tracking synchronization motion of multiple robots was studied in this paper. (ii) Compared with joint-space synchronization over undirected switching topologies given in [9], the task-space tracking synchronization of networked Lagrangian systems over directed switching topologies was investigated by introducing a nonlinear mapping from joint-space position to task-space position. (iii) In contrast to time-invariant topology and certain actuator models employed in [6], networked Lagrangian systems over directed switching topologies with uncertain actuator models were considered and their tracking synchronization was studied by applying a generalized version of Barbalat’s lemma. It is worth noting that the adaptive tracking synchronization of uncertain networked Lagrangian systems with switching topology is very interesting and challenging. This is because the adaptive switching laws to be designed should take into account the uncertainties in kinematic, dynamic and actuator models and thus inevitably lead to the challenge of stability analysis for such complex switched dynamical systems. For the adaptive tracking issue of networked Lagrangian systems, a natural extension to more complex communication topology with various kinds of physical con-

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straints, such as packet losses, communication delays and fading measurements [34,35], will be quite possible, which will be a challenging topic for future research in this direction. Acknowledgments This work is supported by the National Science Foundation of China (Grant Nos. 11272191 and 61203006), the Natural Science Foundation of Inner Mongolia Autonomous Region of China (Grant No. 2015MS0122), the Science and Technology Project of High Schools of Shandong Province (Grant No. J15LJ07), the Inner Mongolia University of Science and Technology Innovation Fund (Grant No. 2014QDL011) and the Key University Science Research Project of Anhui Province (Grant No. KJ2015A196).

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