Neural Dynamics for Distributed Collaborative Control of Manipulators With Time Delays

Long Jin,, Xin Zheng, and Xin Luo,

Abstract—Time-delay phenomena extensively exist in practical systems, e.g., multi-agent systems, bringing negative impacts on their stabilities. This work analyzes a collaborative control problem of redundant manipulators with time delays and proposes a time-delayed and distributed neural dynamics scheme.Under assumptions that the network topology is fixed and connected and the existing maximal time delay is no more than a threshold value, it is proved that all manipulators in the distributed network are able to reach a desired motion. The proposed distributed scheme with time delays considered is converted into a time-variant optimization problem, and a neural dynamics solver is designed to solve it online. Then, the proposed neural dynamics solver is proved to be stable, convergent and robust. Additionally, the allowable threshold value of time delay that ensures the proposed scheme’s stability is calculated.Illustrative examples and comparisons are provided to intuitively prove the validity of the proposed neural dynamics scheme and solver.

I. INTRODUCTION

NEURAL dynamics approaches have long been utilized as effective tools in control analysis and are increasingly employed in many kinds of systems [1]?[4]. There are plenty of outstanding outcomes on neural dynamics because of its prominent performance [5]?[8]. For instance, Jinet al.present two neural dynamics algorithms in [5] to compute the online future matrix inversion with superior computational precision.In [6], the complex-variable pseudoconvex optimization problems are analyzed by providing a neural dynamics method. It can be seen from [5], [6] that neural dynamics methods, which incorporate neural networks and dynamic systems together, possess their both merits, such as low cost and high computation power, and thus are better choices to deal with complex tasks than traditional methods [9].

Time delay is a basic phenomenon existing in various circumstances. For example, some financial or population policies that need a certain period of time to be effective can be seen as systems with time delays. In fact, practical systems possess the property of causality, which means that more or less time delays exist in all these systems. It is known that unmodeled time delays may destabilize an otherwise stable system. Besides, with the increase of the number of manipulators involved in systems, negative impacts on their stabilities brought by time delays may result in the collapse of practical systems. Therefore, taking time delays into consideration when analyzing a distributed multi-robot system is evidently necessary. In the past few decades, a mass of research on time delays in distributed systems has been carried out with fruitful outcomes [10]?[13]. As a seminal work, Olfati-Saber and Murray investigate consensus problems in a distributed undirected network with fixed topologies and time delays in [10]. Moreau [11] analyzes the stability of multi-agent consensus problems with time-variant topologies. Münzet al.present a standard structure for multiagent systems with time delays and provide the condition which can ensure their stability [12]. Recently, two stability criteria from a matrix-valued polynomial are investigated in[14] for time-delay systems. Zhanget al.provide a review of recent outcomes in [15] on some challenging issues brought by various integral vectors inherited from the Bessel-Legendre inequality to analyze the stability of linear systems with timevarying delays. However, as far as the authors know, there is no existing work considering time delay phenomena in distributed collaborative control of multiple redundant manipulators. Most of the existing results on distributed collaborative control for multi-agent systems with time delays are investigated at the particle level [16], [17], in which their theories do not involve related kinematic analyses of manipulators. Only small number of outcomes take multiple manipulators or robots into account when analyzing problems with time delays. For example, a control scheme for carrying an object cooperatively by multiple mobile manipulators with time delays is investigated [18], whose scheme is not distributed and thus not available to distributed circumstances.Therefore, existing vast results cannot be used to address our problem.

Robotics has developed rapidly in both scientific and engineering fields in recent years and been extensively utilized in many sorts of scenarios, for example, industrial robots [19], pneumatic soft ones [20], and social ones [21]. It is known that a manipulator whose degree of freedom (DOF)is more than needed is called a redundant manipulator, which can be more effective when executing complicated tasks than a non-redundant one. As a result, the research on redundant manipulators is becoming increasingly popular. For example,in [22], a sensor-based single redundant industrial robot system for automated object detection and exploration is presented. Additionally, analyses on controllers of redundant manipulators have gained numerous achievements. In [23],the moving object’s obstacle avoidance and tracking problems are described as the kinematic control of redundant manipulators. Adaptive discrete zeroing neural dynamics models are designed to analyze the tracking controller of redundant manipulators [24]. Moreover, neural dynamics methods mentioned above are utilized in the control analysis of both single and dual redundant manipulators. In [25], a neural dynamics method is able to handle single manipulator redundancy resolution by solving a constrained quadratic programming (QP) problem. In order to investigate the kinematic control of model-unknown manipulators, a datadriven scheme and its corresponding neural dynamics method are designed and applied [26]. The control of a dual-arm humanoid manipulator is exploited via a neural dynamics method as well [27].

In the last several years, investigations on the control of multiple agents were plentiful [28]?[30]. Reference [31]presents an introduction to some progress in finite-time cooperative control of multiple agents. A dynamic eventtriggered mechanism for distributed formation control of systems with multiple agents is designed [32]. An advanced structure of consensus control with fault-estimation-in-theloop is put forward to deal with the fault existing in a multiagent system [33]. Dinget al. introduce a survey of outcomes in [34] for the estimation of secure state and control of cyberphysical systems. A finite-horizon H∞containment control scheme for discrete time-varying multi-agent systems is presented [35]. In order to ensure that the final state of each agent is consistent with the command center’s output with bounded residual errors, a distributed controller is presented in[36]. In [37], a game-theoretic perspective distributed scheme is designed, and initial states of manipulators are restricted to the desired location. Reference [38] investigates a distributed neural dynamic scheme and analyzes the collaborative controller of redundant manipulators. Yet, up to now, no existing work takes time delay into consideration for distributed collaborative control of multiple redundant manipulators.

This work moves forward along this direction by taking time delays into consideration for the collaborative control of redundant manipulators. Such a scheme can synchronously complete a required task with time delays occurring among the manipulators and optimize a performance index: repetitive motion planning (RMP). After that, this scheme can be reconstructed to an optimization problem, which is solved online by building a neural dynamics solver with strict convergence proof. The allowable threshold value of time delays to guarantee the stability of the system is derived.Moreover, the validity of the proposed scheme is intuitively verified via illustrative examples and comparisons.Specifically, there are two essential challenges about the time delay in this paper. The first one is how to integrate the time delay to the control scheme of collaborative control of multiple redundant manipulators, and the second one is how to analyze the stability, convergence and robustness of the proposed scheme rigorously with redundant robots considered. These challenges are solved in Sections II and III.Furthermore, this work aims to make the following novel contributions to the field of distributed control of redundant manipulators:

1) It considers time delay for the first time when investigating the distributed collaborative control of redundant manipulators and analyzing their kinematic properties.

2) It establishes allowable upper bound of time delay based on theoretical analyses and verifies the stability, convergence,and robustness of the designed distributed collaborative controller of redundant manipulators rigorously.

3) It provides illustrative examples on CoppeliaSim and comparisons to show the validity of the proposed neural dynamics scheme.

The rest of this paper is arranged into four parts. Section II formulates the problem of the collaborative control of redundant manipulators with time delays and computes the allowable threshold value of time delay. The corresponding neural dynamic solver and theoretical analyses are given in Section III. Section IV provides illustrative examples and comparisons to verify the advantages of the proposed timedelayed and distributed scheme. Finally, Section V draws a conclusion.

II. PRELIMINARY AND SYSTEM MODELLING

This section first provides the basic knowledge of redundant robots. Then, the dynamics of redundant manipulators with time delays is modelled, and a corresponding distributed neural dynamics scheme is proposed.

A. Basic Knowledge

B. Problem Modelling and Time Delay Analysis

Prior to modelling the problem, the meanings of mathematical symbols in this paper are listed in Table I.

TABLE I MEANINGS OF MATHEMATICAL SYMBOLS

This work considers a distributed topology, which means that, in a system with multiple manipulators, each manipulator can only receive information from its adjacent manipulators and the signal from the command center can only transfer to its adjacency as well. Therefore, this work is not dependent on the global information, and the scalability of this developed method is thus guaranteed. Besides, in order to simulate a realistic scenario, we introduce the time delay by assuming that there exists a time delayτwhen signals transfer from a manipulators adjacent to the command center to its neighboring manipulators. Therefore, as a realization of the above distributed behaviors, a corresponding formula for theith manipulator is constructed as follows:

According to the stability analysis of a control system, we can analyze the root of the following equation:

C. Distributed Scheme Design

Due to the redundancy characteristics of redundant manipulators, we may obtain multiple solutions that satisfy(6), which requires us to take some performance indices into account. In this paper, we consider RMP as a performance index

A. Neural Dynamics Solver

The Lagrange function can be formulated as

where

B. Theoretical Analyses

This part contains two theorems about the distributed neural dynamics solver with time delays (29). Theorem 2 theoretically analyzes its stability and convergence, while Theorem 3 analyzes its robustness. With the time delayτno more than the allowable threshold value, we offer the following definitions: 1) The controller’s stability means that it can guide the system to complete the given task successfully and is stable in the Lyapunov sense; 2) The controller’s convergence means that the system can reach an equilibrium point under its guidance; and 3) The controller’s robustness means that the negative impacts brought by disturbances to the system can be avoided when guided by this controller.

According to the neural dynamics design formula, we have

which can be further rewritten as

Fig. 1. Computer experiments of ten UR5 robots to reach a specified configuration on account of the time-delayed and distributed collaborative control scheme (25) and the distributed neural dynamics solver with time delays (29). (a) Trajectories of end-effectors with the target location on a rectangle;(b) Tridimensional graph of ten UR5 robot manipulators in the simulation period; (c) Position errors of end-effectors.

IV. ILLUSTRATIVE EXAMPLES AND COMPARISONS

This section provides several illustrative examples and comparisons to intuitively prove the validity of the proposed time-delayed and distributed collaborative control scheme(25) as well as the distributed neural dynamics solver with time delays (29).

A. Reaching a Specified Configuration

As for the proposed time-delayed and distributed collaborative control scheme (25), we assume that the initial state ψ(～u(0)) is on the desired curve in order to execute the given task well. Therefore, it is important to verify that the end-effectors of all manipulators are able to reach a specified configuration. According to the property of the time-delayed and distributed collaborative control scheme (25), the initial angles ～u(0) can be replaced by the desired angles ～udin this simulation.

In this example, we set the values of parameters defined above asc1=10, ? =100, the number of manipulatorsp=10,durationT=1 s, time delay τ=0.01 s, and the initial joint angles are generated randomly. The desired angles of each manipulator ～udare set as[2.32,0.02,?1.68,?2.14,1.11,0]Trad and the end-effectors all locate on a rectangle.Additionally,Wij= 1 for |i?j|≤1, otherwise,Wij=0;κ1=1 and κi=0,i=2,...,10. Then, the rest parameters are set as zero. The results are shown in Fig. 1.

It can be seen in Fig. 1 that end-effectors of all ten UR5 manipulators reach the blue rectangle with only the first manipulator accessed to the command center. The whole process of the motion of manipulators are illustrated in Fig. 1(b),which intuitively demonstrates the validity of the proposed time-delayed and distributed collaborative control scheme(25). The coloured lines (not the magenta lines) in Fig. 1(b)stand for every simple tridimensional posture of manipulators in the whole simulation period. Besides, from Fig. 1(c), we can get that the position errorseX,Y,Zof end-effectors converge to zero rapidly such that the convergence and the effectiveness of (25) are well-verified.

B. Collaborative Control With Time Delays

The proposed scheme (25) can be utilized as a common strategy for the collaborative control of multiple redundant manipulators, which means that numerous applications on collaborative control can be handled with this scheme.Examples of applications such as coordinate welding [43] and remote surgery with multiple manipulators [44], whose goals are collaboratively controlling multiple manipulators to execute a given curve simultaneously, may use our proposed scheme. Besides, the coordination control of multiple multijoint fish-like robots in the water presented in [45] where each robot can be seen as a redundant manipulator can be realized by using our proposed scheme. Additionally, the gait control of robots with several legs is able to be viewed as an example of collaborative control as well when each leg of the robot and robot body are regarded as a redundant manipulator and a payload, respectively, thus allowing the scheme to be used[46]. The attitude control of telescope array [47] for the clear cosmic observation can also be managed with it.

In this part, ten UR5 manipulators are investigated for the collaborative control with time delays and are required to track a path of a tricuspid valve line. In addition, we stipulate that manipulators 1 and 5 can access to the command center,and the topology of manipulators is the same. The values of parameters are set to ?=100, the simulation timeT=2π/1=6.28s, and the other parameters do not change.The values of initial states that can be observed in the figures are omitted here, and the rest parameters are kept the same as those in the previous simulation. Figs. 2 and 3 present the simulation results.

Fig. 2. Tridimensional graph of repetitive motion planning combined by the time-delayed and distributed collaborative control scheme (25) and the distributed neural dynamic solver with time delays (29).

Besides, as mentioned earlier in this paper, a too-large time delay will affect the system’s stability and performance. Then we make a further experiment by setting time delay τ =0 s,τ=0.001 s , and τ=0.016 s, respectively, and the end-effectors position errors are shown in Fig. 4.

As shown in Fig. 2, the tridimensional graph of repetitive motion planning combined by (25) and (29) for collaboratively tracking a time-varying tricuspid valve line is presented. Note that all the manipulators complete the motion collaboratively with different poses. Fig. 3(a) shows the profiles of the end-effector position errors, from which we can see that the position errors are all of the order 1 0?5m. Besides,from Fig. 3(b) we can see that the initial states of UR5 robots～u(0) are different and the final ～uroughly equals ～u(0).Moreover, joint velocities of manipulators in a simulation period are illustrated in Fig. 3(c). It can be seen from Fig. 4 that, with the increase of the time delay, the position errors of end-effectors become larger and larger. Furthermore, when τ=0.016s, the position errors fail to converge to zero but diverge. These simulations and their observable figures demonstrate the validity of (25) and (29) as well.

C. Application to UR5 Robots Control on CoppeliaSim

For the sake of intuitively visualizing the feasibility of (29),ten UR5 robots are provided to track the tricuspid valve line on CoppeliaSim. CoppeliaSim is a software platform providing a variety of physical models of real robots. The parameters are all kept the same as above. In Fig. 5, snapshots of ten UR5 robots tracking the tricuspid valve line are presented, the initial positions of all ten UR5 robots are obviously different, and we can see that the experiment completes successfully. This simulation indicates that (29)suits well on a UR5 robot system, and therefore, the effectiveness of the proposed solver is demonstrated again.

D. Application to Sawyer Robots Control on CoppeliaSim

Equation (29) can be utilized for robots with seven DOFs.In this part, six Sawyer robots, each of which has seven DOFs,are simulated in CoppeliaSim to track the camel curve. In Fig. 6,four snapshots of the tracking process are provided. It can be seen that the initial positions of all six Sawyer robots are different, and the experiment is completed satisfactorily. This simulation demonstrates the validity of the proposed solver(29) when controlling seven DOFs redundant robots collaboratively.

E. Comparisons

In this part, we make comparisons on several detailed properties between (25) and previous schemes reported in[26]?[28], [37], [38]. Reference [26] designs and applies a data-driven cyclic-motion generation scheme and the corresponding neural dynamics method to analyze the kinematic control of model-unknown manipulators. Besides,the research in [27] investigates the control scheme for dualarm robot manipulators solved by the neural dynamics models with different focuses. References [28] and [37] present distributed schemes for analyzing the collaborative controllers of multiple redundant manipulators with the topologies neighbor-to-neighbor both, but the method in [28] offers the guaranteed bound of position error. A distributed neural dynamics scheme is designed with rigorous theoretical theorems to analyze the collaborative control of redundant manipulators [38]. In this paper, time delays are taken into consideration for the collaborative control of multiple redundant manipulators solved by neural dynamics schemes in a distributed manner for the first time, in which theoretical proofs of convergence and robustness are also given, and the developed result can degenerate into the case without time delays by setting the time delay τ=0. All the comparisons mentioned above are summarized in Table II.

V. CONCLUSIONS

Fig. 3. Computer experiments integrated by the time-delayed and distributed collaborative control scheme (25) for collaborative control of ten UR5 robot manipulators. (a) Position errors of end-effectors; (b) Joint angles; (c) Joint velocities.

Fig. 4. Position errors of end-effectors with different time delays. (a) τ =0 s; (b) τ =0.001 s; (c) τ =0.016 s.

Fig. 5. Snapshots of ten UR5 robots using the proposed distributed neural dynamic solver with time delays (29) to track the time-varying tricuspid valve line.

In this paper, a time-delayed and distributed neural dynamics scheme for the collaborative control of multiple redundant manipulators has been proposed. This scheme incorporates time delays, which is the first time that time delays are considered into the collaborative control of multiple redundant manipulators with kinematic analysis involved.Besides, the distributed solver avoids the disadvantages of centralized schemes, thus making it more suitable for handling industrial systems consisting of multiple redundant manipulators. Then, the upper bound of time delays has been calculated to ensure that the scheme is stable and convergent.Several theorems have also been proved to substantiate the feasibility of the neural dynamics solver. Additionally,illustrative examples and comparisons have been presented to demonstrate the validity of the proposed distributed scheme and solver. Moreover, research on the collaborative control of redundant manipulators with weight-unbalanced directed networks and time-varying delays represent promising research directions in the future.

Fig. 6. Snapshots of six Sawyer robots using the proposed distributed neural dynamic solver with time delays (29) to track the camel curve.

TABLE II REDUNDANT ROBOTS CONTROL SCHEMES IN DIFFERENT PAPERS