Consensus control of multi-manipulator systems based on disturbance observer under Markov switching topologies

Chang-E Ren · Quanxin Fu

Abstract In this paper, to solve the consensus control problem of multi-manipulator systems under Markov switching topologies,we propose a distributed consensus control strategy based on disturbance observer. In multi-manipulator systems, external disturbance described by heterogeneous exogenous systems is considered, and all communication topologies are directed.First, a disturbance observer is presented to suppress the inf luence of unknown external disturbance, and the equivalent compensation is introduced into the control protocol in multi-manipulator systems. Then, a novel control protocol based on neighbor information is designed, which guarantees that multi-manipulator systems reach consensus under Markov switching topologies. Finally, two simulation examples verify the validity of the theoretical result.

Keywords Multi-manipulator systems · Disturbance observer · Markov switching topologies · Consensus control

1 Introduction

The cooperative control of multi-agent systems (MASs)has become a hot issue in the f ield of control for the past few years. As a fundamental problem of cooperative control, the consensus problem has been studied from diff erent aspects, such as [ 1- 7]. The basic idea of the distributed consensus control problem is that the controller based on the information of the agent’s neighbors is designed so that the relevant states of all agents eventually tend to the same values. The work reported in [ 3, 5, 6] solved the consensus control problem of MASs by presenting the diff erent adaptive control algorithms. In [ 6], an adaptive control algorithm was presented to solve the consensus control problem of MASs, in which the internal model was constructed to solve the unknown disturbance. Two adaptive control algorithms based on fuzzy NNs were proposed for the consensus problem of MASs with constant and time-varying unknown control direction in [ 5]. In [ 3], an adaptive control algorithm was proposed for nonlinear MASs with unknown control direction, and then an iterative learning strategy was used to respond to the proposed control algorithm. The f initetime consensus control problem of MASs was addressed in [ 7] and [ 4]. Zuo and Tie [ 7] proposed two robust nonlinear control protocols to solve the f inite-time consensus control problem of MASs. Using the sliding mode control method, Ren and Chen [ 4] not only solved the asymptotic consensus problem of MASs but also solved the f inite-time consensus problem of MASs. In [ 1], a distributed controller was proposed by combining backstepping techniques and NNs-based state observer, which can ensure that MASs are uniformly ultimately bounded. Du et al. [ 2] proposed a control algorithm to reach consensus for the second-order MASs, in which each agent contained unknown nonlinear dynamic and unmeasurable velocity. Because of the rapid development of artif icial intelligence and robotics, robot manipulator has been applied in many f ields [ 8- 11], such as industrial manufacturing, medical treatment, which can simultaneously perform some tasks to improve effi ciency.Therefore, it is of practical signif icance to study the consensus control problem of multi-manipulator systems.

In some existing literature on distributed consensus control problems, it often happens that the disturbance is not considered in the modeling process of the agent, such as [ 12] and [ 13]. However, the controller designed in the above situation would cause the performance of the system to decline or even malfunction. Traditional distributed control methods, such as robust control [ 14, 15], and backstepping method [ 16], are passively anti-disturbance and cannot quickly suppress the eff ects of the disturbance. As a result, anti-disturbance control methods get more and more attention [ 17- 20]. As one of the anti-disturbance control methods, the disturbance observer is designed in many papers to estimate unknown disturbance, and the equivalent compensation is introduced into the control protocol,such as [ 21] and [ 22]. In [ 21], a disturbance observer based on fuzzy logic systems was f irst adopted to estimate the diff erences between the nonlinear dynamic of diff erent agents, and an adaptive control algorithm was proposed for estimation errors and consensus problems. To compensate for the eff ects of mismatched disturbance, Wang et al.[ 22] designed a disturbance observer and proposed two distributed controllers based on the disturbance observer for homogeneous MASs. In this paper, the diff erence is that a disturbance observer is designed for the unknown external disturbance described by the heterogeneous exosystems, which can improve the control accuracy of multimanipulator systems.

In the actual application, packet loss and node connection failure are frequent and often random in the process of information transmission, which makes the consensus control under f ixed topology no longer suitable. Therefore, it is more practical to consider the consensus control problem of MASs under random switching topologies, such as [ 23- 27]. It was required in [ 25] and [ 26] that the communication topology switched between some directed graphs, and all graphs contained a directed spanning tree. The work in [ 23] required that the communication topology G(t) across the time interval[tk,tk+1) was jointly connected, i.e., the union of graphs in the interval [tk,tk+1) was an undirected connected graph. Whereas,the study [ 24] only required that the union of graphs had a directed spanning tree. It means that the restriction of [ 24] is weaker than that of [ 23, 25, 26]. Like [ 24], it was also required in [ 27] that the union of graphs in the interval [tk,tk+1) had a directed spanning tree. Besides, the Markov process was introduced in [ 27] to drive the communication topology.

Motivated by the above analyses, the objective of this paper is to design a consensus algorithm for multi-manipulator systems with the unknown external disturbance under Markov switching topologies. A disturbance observer is constructed to suppress the eff ects of the unknown disturbance described by a heterogeneous exogenous system.Moreover, the actual value of the disturbance is replaced by the estimated value of the disturbance observer in the control algorithm. It shows that the proposed control algorithm can well control multi-manipulator systems. The main contributions are stated as follows:

(1) In contrast to the literature that does not consider the disturbance, such as [ 12, 28], multi-manipulator systems considered in this paper contain the unknown external disturbance, which indicates multi-manipulator systems model is more general. Furthermore, the unknown external disturbance considered in this paper is generated by an exogenous system, rather than being assumed to be bounded as in literature [ 15]. Moreover, a disturbance observer is presented to suppress the inf luence of the unknown external disturbance, which improves the control accuracy and robustness of multi-manipulator systems.

(2) Compared with the existing literature on the consensus problem of multi-manipulator systems in fixed topology [ 11] and [ 29], this paper studies the consensus control problem under Markov switching topologies. Furthermore, it is only required that the union of graphs contains a directed spanning tree.Hence, the restriction for communication topology in this paper is much weaker than [ 25] and [ 26] in which each communication topology has a directed spanning tree.

The proposed control scheme consists of the following parts. First, a disturbance observer will be used to compensate for the eff ects of unknown disturbance. Then, to realize the consensus of multi-manipulator systems under Markov switching topologies, a disturbance observer-based control algorithm is designed and proved by the Lyapunov stability theory. Finally, two simulations are given to verify the validity of the control algorithm.

2 Preliminaries and problem statement

2.1 Graph theory

where k＞0,and P∈?n×n is a positive def inite matrix.

2.2 Problem statement

Theith manipulator can be modeled as follows:

whereqi,˙qiand¨qi∈?2represent the link’s position, velocity, and acceleration of theith follower manipulator respectively.Ti∈?2is the input torque vector, andωi∈?2is the external disturbance torque of theith follower manipulator.Gi(qi,˙qi) is aff ected by Coriolis, gravitational, and centripetal torque.Ji(qi) is a positive def inite inertia matrix,then ( 2) is rewritten as the following equations:

Diff erent from [ 29], the external disturbance of the multimanipulator systems in this paper is generated by the following heterogeneous linear exogenous systems:

Remark 1Systems ( 4) can be used to describe many types of disturbances, such as harmonic disturbance with unknown frequency, amplitude and initial phase.

Define a probability space (Ω,F,P) , whereΩis the sample space, F is the event f ield, and P is a probability function def ined on F . Let the transition rates matrix of the homogeneous continuous-time Markov process be denoted byQ=(qmn) , which is given by

Assumption 2 [ 32] The transition rates matrixQis ergodic.

Remark 3From Assumption 1, it can draw a conclusion that all eigenvalues ofunare positive. According to Assumption 2, we can obtain that the Markov process has a unique invariant distributionι=[ι1,ι2,…,ιs] withι=min{ιm|m=1,2,…,s} , and all states of the Markov process can be mutually reachable in the state space. Thus,ιm＞0 , for allm.

Def inition 1 For any initial stateξi(0) , under the designed control protocol, the following equation is satisf ied:

where E[·] is the expectation, then, multi-manipulator systems ( 6) reach mean square consensus.

3 Main results

This section f irstly designs a disturbance observer and then proposes a control protocol for multi-manipulator systems( 6) under Markov switching topologies.

On the basis of [ 34], the following disturbance observer is proposed:

where＾ζiand＾ωidenote the estimations ofζiandωirespectively.ziis the disturbance observer’s state.pi(vi) andLi(vi)are the nonlinear function and the nonlinear observer gain to be designed, respectively, and satisfy

By substituting ( 11) and ( 12) into ( 13), it follows that

The nonlinear observer gainLi(vi) is chosen as

whereΘiis a matrix to be designed. Then, the estimation error dynamic of the disturbance observer can be rewritten as follows:

Since (Ei,Fi) is observable, we can f ind a matrixΘisuch that˙ei=(Ei-ΘiFi)eiis asymptotic stable. Thereby, the estimated value of the disturbance observer ( 11) can asymptotically track the actual value of disturbance ( 4).

whereJi(qi) is the inertia matrix,Gi(qi,vi) is torque def ined in ( 2),Kis a feedback matrix to be designed, and＾ωiis the disturbance estimation.

To analyze the consensus of multi-manipulator systems,the global error of theith follower manipulator is def ined as～ξi=ξi-ξ0. Therefore, in accordance with Eq. ( 6), we obtain

ProofConsider the following Lyapunov function:

Remark 4The control protocol designed in this paper can solve the consensus problem studied in many literature,such as [ 35, 36, 29]. This is to say, the control algorithms designed in [ 35, 36, 29] are extended in this paper. The communication topology of [ 29] and [ 36] are, respectively,f ixed and jointly connected, which is a special case of the switching topologies considered in this paper. In [ 35], it was assumed that communication topology switches among a set of graphs and all graphs are connected. It is worth mentioning that there is no disturbance in [ 35, 36]. Moreover, the simulation experiment that the control protocol in which the disturbance estimation is set to zero in ( 18) applies to [ 35]is given in Sect. 4.2.

4 Simulation

4.1 Simulation I

This section verif ies the correctness of the above theoretical results through a simulation. In this simulation, multi-manipulator systems consist of 6 agents, one of which is the leader with label 0 and the rest are the followers. Communication topologies are switched between graph1and2shown in Fig. 1. It is obvious that1and2are balanced,uncontains a directed spanning tree in which root node is leader. Then,Assumptions 1 and 2 are satisf ied.

The dynamics of follower manipulator ( 2) are as follows and the parameters are chosen as the same as [ 37]

Fig. 1 Communication topologies. a 1 . b 2 . cun=1∪2

The disturbance ( 4) generated by exogenous systems is as follows:

Fig. 2 The estimation errors of the disturbance observer. a The disturbance estimation error ωi1-＾ωi1 . b The disturbance estimation error ωi2-＾ωi2

In Fig. 2, curves F1, F2, F3, F4 and F5 denote the trajectories of disturbance estimation errors of 5 follower manipulators, respectively. It is observed from Fig. 2 that the estimation error of the disturbance observer asymptotically converges to zero.

Fig. 3 The global errors of velocity. a The global error vi1-v01 . b The global error vi2-v02

In Figs. 3 and 4, curves F1, F2, F3, F4 and F5 denote the trajectories of velocity and position tacking errors of 5 follower manipulators, respectively. From Figs. 3 and 4,it can get that the errors of velocity and position of theith follower manipulator tend to zero, i.e.,vi1→v01,vi2→v02andqi1→q01,qi2→q02, ast→∞ . Therefore, under consensus control protocol ( 18), multiple robotic manipulators can reach consensus.

4.2 Simulation II

In this section, a simulation is given to show that the controller designed in this paper is also feasible in [ 35]. Similar to [ 35], multi-agent systems consist of f ive agents, one of which is the leader with label 0 and the rest are the followers.All possible communication topologies are shown in Fig. 5,which can satisfy Assumption 1. Figure 6 shows the value of switching signal of the communication topology.

The dynamic of theith agent is as follows:wherexiandviare states of theith agent.uiis the control input to designed. Moreover, the leader is described by

Fig. 4 The global errors of position. a The global error qi1-q01 . b The global error qi2-q02

Fig. 5 All possible communication topologies. a 1 . b 2 . c3 . d 4.

Fig. 6 The switching signal

Fig. 7 The global errors of states. a The global error vi- v0 . b The global error xi- x0.

According to Fig. 7, it can be seen that the errors of velocity and position eventually tend to zero. In other words, the consensus problem of [ 35] can also be well solved under the control protocol designed in this paper.

5 Conclusions

This paper solves the consensus control problem of multimanipulator systems under Markov switching topologies,and all communication topologies are directed. A disturbance observer is introduced for the external disturbance described by the heterogeneous exosystem. Based on the designed disturbance observer, a distributed controller of multi-manipulator systems is proposed under Markov switching topologies, which can ensure that every follower manipulator’s velocity and position reach consensus with leader’s respectively. Two simulation examples verify the validity of the theoretical result.

AcknowledgementsThis work was supported in part by the National Natural Science Foundation of China (No. 61803276), the Beijing Municipal Education Commission Science Plan (General Research Project, No. KM201910028004), the Beijing Natural Science Foundation (No. 4202011), and Key Research Grant of Academy for Multidisciplinary Studies of CNU (No. JCKXYJY2019018).