Distributed Synchronization Control to Trajectory Tracking of Multiple Robot Manipulators

Abstract

This paper investigates the issue of designing decentralized control laws to cooperatively command a team of general fully actuated manipulators. The purpose is to synchronize their movements while tracking a common desired trajectory. Based on the well-known consensus algorithm, the control strategy consists in synchronizing the joint position and the velocity of each robot in the network with respect to neighboring robots' joints and velocities. Modeled by an undirected graph, the cooperative robot network requires just local neighbor-to-neighbor information exchange between manipulators. So, it does not assume the existence of an explicit leader in the team. Based above all on combination of Lyapunov direct method and cross-coupling strategy, the proposed decentralized control law is extended to an adaptive synchronization control taking into account parameter uncertainties. To address the time delay problems in the network communication channels, the suggested synchronization control law robustly synchronizes robots to track a given trajectory. To this end, Krasovskii functional method has been used to deal with the delay-dependent stability problem. A real-time software simulator is developed to visualize the robot manipulators coordination.

1. Introduction

There has been a great research effort for synchronization problems, in distributed cooperative control strategies where robot control laws are coupled and each robot control is updated using local rule based on its own sensors and the states of its neighbors [1, 2]. Consensus algorithms, distributed coordination, and passivity-based output synchronization for networked Euler-Lagrange systems have been studied in [3–5], respectively. In this context, one recent representative work [6] shows that we can synchronize the multicomposed system when only position measurements are available. Synchronization framework that can be directly applied to cooperative control of multiagent systems in robotic manipulation and teleoperation has been presented in [1]. Specifically, the design based on the graph theory and the Laplacian matrix produces interesting results [7–9]. Adaptive control is an effective strategy used to address the synchronization problem [10, 11].

In the literature, most of earlier works on multiagent coordination and consensus [2, 8, 9, 12] mainly deal with very simple dynamic models such as linear systems and focuses on algorithm taking the form of first-order dynamics or second-order dynamics without nonlinear inertia matrices [13–15]. Most previous works on consensus and coordination of multiagent systems using the graph theory and the Laplacian [2, 8, 9, 16, 17] have presented a synchronization to the weighted average of initial conditions but they do not consider multiagent systems where there is a desired path to follow.

The objective of this paper is to develop a synchronized trajectory-tracking control of multiple robot manipulators. The proposed controller relies principally on a consensus algorithm for systems modeled by nonlinear second-order dynamics and applies the algorithm for the synchronization control problem by choosing appropriately information states on which consensus is reached. The concept key of the new synchronizing controller is the introduction of a state vector that quantifies the coordination degree between robot manipulator positions and different positions of its neighbors, using cross-coupling technique. Robot manipulators are widely used in production processes. In tasks that can not be fulfilled by a single robot, either because of the complexity of the task or the spatial and temporal limits of the robot, the use of cooperative robots proved to be a good determination [18]. The tasks being executed by each robot may terminate in different time due to nonhomogenity of robots or even electronics that drives motors of robots’ joints. Despite the common trajectory robots have to track, it is not almost sure that the final running time of each robot will definitely be the same. So, in order to ensure this same final running time, robots have to have image of the current state of the neighboring robots as such to synchronize their movement together.

This paper presents a distributed control strategy. Through local interactions, the proposed approach can achieve more efficient performances, in particular in the presence of external disturbances. The proposed method based on a combination of Lyapunov direct method and concepts of cross-coupling, through a convenient mathematical manipulation, has converted the motion control problem of multiple systems into a stabilization problem for one single system. Compared to existing approaches, main contributions of this paper can be stated as follows. (1)The present work deals with highly nonlinear systems. While [13–15], to name a few, deal with simple dynamic models. (2)In contrast to [2, 8, 9, 16, 17] the proposed approach achieves not only global asymptotical synchronization of the configuration variables, but also global asymptotical convergence to the desired trajectory. (3)The theory is extended to adaptive control and time delay control. (4)In [10, 11, 19], the position synchronization error of each robot is defined as the differential position error between this robot and its two adjacent robots. However, the proposed design allows interconnections between all robots, such that all robots have direct influence in the combined dynamics, and consequently the synchronous behavior is the result of interactions between all robots. Moreover, it is straightforward to show that the particular choice of the tracking error surface in the proposed approach provides an exponential convergence of 𝑞_𝑖 to 𝑞_𝑑 if we have a constant tracking error surface. (5)Using the Virtual Reality Modeling Language (VRML) a virtual world was developed to simulate the robot synchronization application in 3D scenes.

2. Preliminaries

The dynamic equation of a general rigid link manipulator having 𝑛 degrees of freedom in a free space can be written as 𝑀_𝑖𝑞_𝑖𝑞̈_𝑖+𝐶_𝑖𝑞_𝑖,𝑞̇_𝑖𝑞̇_𝑖+𝑔_𝑖𝑞_𝑖=𝜏_𝑖, (1) where 𝑖(1≤𝑖≤𝑝) denotes the 𝑖th robot index in the network and 𝑝 is the total number of the individual elements. In addition, 𝑞_𝑖∈ℝ^𝑛 denotes the vector of generalized displacements of the 𝑖th robot coordinates, and 𝜏_𝑖∈ℝ^𝑛 denotes the vector of generalized control input torques in robot coordinates; 𝑀_𝑖(𝑞_𝑖)∈ℝ^𝑛×𝑛 inertia matrix which is symmetric uniformly bounded and positive definite, 𝐶_𝑖(𝑞_𝑖,𝑞̇_𝑖)𝑞̇_𝑖∈ℝ^𝑛 is a vector function containing Coriolis and centrifugal forces, and 𝑔_𝑖(𝑞_𝑖)∈ℝ^𝑛 is a vector function consisting of gravitational forces. According to [20, 21], we have some fundamental properties of motion equations.

(i) The inertia matrix 𝑀_𝑖(𝑞_𝑖) is symmetric, positive definite, and uniformly bounded: 𝑀_𝑖𝑞_𝑖=𝑀^𝑇_𝑖𝑞_𝑖>0. (2)

(ii) Using a proper definition, 𝑀̇_𝑖(𝑞_𝑖)−2𝐶_𝑖(𝑞_𝑖,𝑞̇_𝑖) is a skew symmetric matrix, satisfying 𝑋^𝑇𝑀̇_𝑖𝑞_𝑖−2𝐶_𝑖𝑞,𝑞𝑋=0, (3) where 𝑋^𝑇 is the transpose of a vector 𝑋∈ℝ^𝑛.

(iii) The Euler-Lagrange equation (1) is linear with respect to the structural parameter 𝜃, hence, 𝑀_𝑖𝑞_𝑖𝑞̈_𝑖+𝐶_𝑖𝑞_𝑖,𝑞̇_𝑖𝑞̇_𝑖+𝑔_𝑖𝑞_𝑖=𝑌𝑞_𝑖,𝑞̇_𝑖,𝑞̈_𝑖𝜃_𝑖, (4) where 𝑌∈ℝ^𝑛×𝑎 is the regressor matrix composed of known functions of 𝑞, 𝑞̇ and 𝑞̈, 𝜃_𝑖∈ℝ^𝑎 is the vector of structural parameters of the manipulator, and 𝑎 is the number of unknown parameters. In the present topology, the edge represents bidirectional communication links. This consists of a group of 𝑝 manipulators interchanging information that can be viewed as an undirected graph (Figure 1).

Figure 1 

Multirobot system under mutual synchronization scheme.

3. Controller Design

We design decentralized control laws for 𝑝 robot manipulators such that all joint positions mutually synchronize and track a common desired trajectory. The control objective of the proposed synchronization controller scheme is to synchronize the 𝑖th-joint position and velocity 𝑞_𝑖, 𝑞̇_𝑖 to the state of any manipulator 𝑞_𝑗, 𝑞̇_𝑗. Besides the controller is required to regulate the joint position 𝑞_𝑖 to track a desired trajectory 𝑞_𝑑. Specifically, the control torque for the 𝑖th-robot is to control the tracking error to converge to zero and at the same time to synchronize its motion with respect to motions of the 𝑝−1 robots in the network, so that the synchronization error converges to zero. To this end, we define the tracking error surface of the 𝑖th manipulator as 𝜀_1𝑖(𝑡)=𝑞_𝑖(𝑡)−𝑞_𝑑(𝑡)+^𝑡_𝑡0Λ_𝑖𝑞_𝑖(𝜆)−𝑞_𝑑(𝜆)𝑑𝜆, (5) where Λ_𝑖 is a diagonal positive-definite matrix. Information on the vector 𝜀_1𝑖 will give insight on the convergence of the joint positions to the desired trajectory. However, it is straightforward to show that this particular choice of this tracking error surface provides an exponential convergence of 𝑞_𝑖 to 𝑞_𝑑 if we have a constant tracking error surface. But the defined sliding error only guarantees the trajectory tracking. Nevertheless, it is required to know the performance of the controller, that is, to know how the trajectory of each robot manipulator converges with respect to each other. There are various ways to choose the synchronization error. For example in [6], authors include the error information of all systems involved in the synchronization. In [10], authors use the cross-coupling concept to solve the synchronization problem. Our approach will make use of the cross-coupling to propose a feasible and efficient synchronization error, which consists on a measure of the synchronization for robot manipulators as defined as follows: 𝜀_2𝑖(𝑡)=^𝑝_𝑗≠𝑖𝐾_𝑖𝑗𝑞_𝑖−𝑞_𝑑−𝑞_𝑗−𝑞_𝑑=^𝑝_𝑗≠𝑖𝐾_𝑖𝑗𝑞_𝑖−𝑞_𝑗, (6) where 𝐾_𝑖𝑗 is a symmetric positive-definite matrix that gives insight on the communication quality between the 𝑖th and 𝑗th robot manipulators. We note that 𝐾_𝑖𝑗≠0 if and only if there is information exchange between robot manipulators 𝑖 and 𝑗. Consequently, each robot is not necessarily aware of all other robots. Therefore, we define the synchronizing tracking error surface which encompasses both synchronization error and trajectory tracking error for manipulator 𝑖 as: 𝑒_𝑖=𝜀_1𝑖+^𝑡_𝑡0𝜀_2𝑖(𝜆)𝑑𝜆. (7) The objective is to design a control law such that coupling errors, that is, position errors, velocity errors, and synchronization errors, all converge to zero. For each manipulator, the control law 𝜏_𝑖 is defined as follows: 𝜏_𝑖=𝐶_𝑖𝑞_𝑖,𝑞̇_𝑖𝑞̇_𝑖+𝑔_𝑖𝑞_𝑖+𝑀_𝑖𝑞_𝑖𝑞̈_𝑑−𝐾_𝑝𝑖𝑒_𝑖−𝐾_𝑑𝑖𝑒_𝑖̇−Λ_𝑖𝑞̇_𝑖−𝑞̇_𝑑+𝑀_𝑖𝑞_{𝑖𝑗≠𝑖}𝐾_𝑖𝑗𝜀_2𝑖−𝜀_2𝑗+Λ_𝑖𝜀_2𝑖, (8) where 𝑞_𝑑 is a common trajectory reference to be tracked, which is a smooth time-varying trajectory and for which the first and the second derivative exist for all 𝑡≥0. 𝐾_𝑑𝑖 and 𝐾_𝑝𝑖 are symmetric positive-definite matrices.

Theorem 1. If 𝐾_𝑑𝑖>2∑_𝑗≠𝑖𝐾_𝑖𝑗, the proposed synchronization tracking controller (8) guarantees asymptotic convergence to zero of position errors, velocity errors, and synchronization errors, that is, 𝑒_𝑖→0 as 𝑡→∞.

Remark 1. Note that the synchronization controller only requires local information of the joint position 𝑞_𝑖, the joint velocity and the information about the desired trajectory. Also in order to ensure synchronization with other robots it needs information on the joint position from neighbors. It has to be noticed that the joint position of the neighboring robots are lumped into the error variable 𝜀_2𝑗.

Proof. Notice that the controller law (8) contains PD controller terms. However, these terms are premultiplied by the inertia matrix 𝑀_𝑖(𝑞_𝑖). Therefore it is clear that this is not a linear controller as the PD control law, since the position and velocity gains are not constant but they depend explicitly on the position error.
Substituting (8) in (1) yields: 𝑀_𝑖𝑞_𝑖𝑞̈_𝑖=𝑀_𝑖𝑞_𝑖𝑞̈_𝑑−𝐾_𝑝𝑖𝑒_𝑖−𝐾_𝑑𝑖𝑒_𝑖̇−Λ_𝑖𝑞̇_𝑖−𝑞̇_𝑑+𝑀_𝑖𝑞_{𝑖𝑗≠𝑖}𝐾_𝑖𝑗𝜀_2𝑖−𝜀_2𝑗+Λ_𝑖𝜀_2𝑖. (9) Multiplying by 𝑀_𝑖⁻¹ in both sides yields 𝑞̈_𝑖=𝑞̈_𝑑−𝐾_𝑝𝑖𝑒_𝑖−𝐾_𝑑𝑖𝑒_𝑖̇−Λ_𝑖𝑞̇_𝑖−𝑞̇_𝑑+_𝑗≠𝑖𝐾_𝑖𝑗𝜀_2𝑖−𝜀_2𝑗+Λ_𝑖𝜀_2𝑖. (10) Adding 𝜀̇_2𝑖 in both sides yields 𝑞̈_𝑖−𝑞̈_𝑑+Λ_𝑖𝑞̇_𝑖−𝑞̇_𝑑+𝜀̇_2𝑖=−𝐾_𝑝𝑖𝑒_𝑖−𝐾_𝑑𝑖𝑒_𝑖̇+_𝑗≠𝑖𝐾_𝑖𝑗𝜀_2𝑖−𝜀_2𝑗+Λ_𝑖𝜀_2𝑖+𝜀̇_2𝑖. (11) This results in 𝑒_𝑖̈=−𝐾_𝑝𝑖𝑒_𝑖−𝐾_𝑑𝑖𝑒_𝑖̇+_𝑗≠𝑖𝐾_𝑖𝑗𝜀_2𝑖−𝜀_2𝑗+Λ_𝑖𝜀_2𝑖+𝜀̇_2𝑖. (12) Using the expression of the synchronization error 𝜀_2𝑖 and its first derivative gives 𝑒_𝑖̈=−𝐾_𝑝𝑖𝑒_𝑖−𝐾_𝑑𝑖𝑒_𝑖̇+_𝑗≠𝑖𝐾_𝑖𝑗𝜀_2𝑖−𝜀_2𝑗+_𝑗≠𝑖𝐾_𝑖𝑗𝑞̇_𝑖−𝑞̇_𝑑−𝑞̇_𝑗−𝑞̇_𝑑+_𝑗≠𝑖Λ_𝑖𝐾_𝑖𝑗𝑞_𝑖−𝑞_𝑑−𝑞_𝑗−𝑞_𝑑. (13) Further calculation will result in 𝑒_𝑖̈=−𝐾_𝑝𝑖𝑒_𝑖−𝐾_𝑑𝑖𝑒_𝑖̇+_𝑗≠𝑖𝐾_𝑖𝑗𝑒_𝑖̇−𝑒_𝑗̇. (14) Equation (14) represents the closed-loop synchronized system for the 𝑖th manipulator. In the sequel, we proceed to analyze the stability properties of the proposed synchronized control scheme and ultimately show that the control goals: the position error, velocity error, and synchronization error, all converge to zero. To prove stability of the overall synchronized system, we define 𝑒^𝑇=[𝑒^𝑇₁,𝑒^𝑇₂,…,𝑒^𝑇_𝑛]^𝑇.
Using (14) we obtain the synchronized error dynamics: 𝑒̈=−𝐾_𝑝𝑒−𝐾_𝑑−𝐾_𝑐𝑒,̇ (15) where 𝐾_𝑝=diag(𝐾_𝑝𝑖),𝐾_𝑑=diag(𝐾_𝑑𝑖), and 𝐾_𝑐 is given by _𝑗≠1𝐾_1𝑗⋯−𝐾_1𝑗⋯−𝐾_1𝑛⋯⋯⋯⋯⋯−𝐾_𝑖1⋯_𝑗≠𝑖𝐾_𝑖𝑗⋯−𝐾_𝑖𝑛⋯⋯⋯⋯⋯−𝐾_𝑛1⋯−𝐾_𝑛𝑗⋯_𝑗≠𝑛𝐾_𝑛𝑗. (16) Note that 𝐾_𝑐 is symmetric and positive semidefinite matrix.
The synchronized error dynamics (15) is a linear time invariant system described by a second-order linear differential equation. A sufficient condition for the error dynamics to be exponentially stable is that the matrices 𝐾_𝑝 and 𝐾_𝑑−𝐾_𝑐 are positive definite. In particular, matrices 𝐾_𝑑𝑖 can be diagonal satisfying 𝐾_𝑑𝑖>2∑_𝑖≠𝑗𝐾_𝑖𝑗.
To analyze the stability properties of the closed-loop error dynamics (15), we take the following definite and radially unbounded Lyapunov function candidate: 𝑉=𝑒^𝑇̇𝑒̇+𝑒^𝑇𝐾_𝑝𝑒. (17) Its derivative with respect to time can be expressed as 𝑉̇=−2𝑒^𝑇̇𝐾_𝑑−𝐾_𝑐𝑒̇≤0. (18) It follows by direct application of LaSalle's invariance principle that the origin (𝑒,𝑒)̇=(0,0) is globally asymptotically stable and lim𝑒̇→0 for 𝑡→∞.
Referring to the expression of the global error (7) 𝑒_𝑖=𝑞_𝑖−𝑞_𝑑+^𝑡_𝑡0Λ𝑞_𝑖(𝜆)−𝑞_𝑑(𝜆)𝑑𝜆+^𝑡_{𝑡0𝑗≠𝑖}𝐾_𝑖𝑗𝑞_𝑖−𝑞_𝑗 (19) as 𝑒̇=0 we have 𝑞̇_𝑖−𝑞̇_𝑑=−Λ_𝑖𝑞_𝑖−𝑞_𝑑−_𝑗≠𝑖𝐾_𝑖𝑗𝑞_𝑖−𝑞_𝑗. (20) We set 𝜀_𝑖=𝑞_𝑖−𝑞_𝑑. Then (20) can be written as 𝜀_𝑖̇=−Λ_𝑖𝜀_𝑖−_𝑗≠𝑖𝐾_𝑖𝑗𝜀_𝑖−𝜀_𝑗. (21) Our objective is to show that lim𝜀_𝑖 converges to zero as 𝑡↦∞. To this end, we define 𝜀=[𝜀^𝑇₁⋯𝜀^𝑇_𝑖⋯𝜀^𝑇_𝑛]^𝑇 and Λ=diag(Λ₁⋯Λ_𝑖⋯Λ_𝑛). Equation (21) can be written as 𝜀̇=𝐴⋅𝜀, (22) where matrix 𝐴 is given by −Λ₁−_𝑗≠1𝐾_1𝑗⋯𝐾_1𝑗⋯𝐾_1𝑛𝐾……………_𝑖1⋯−Λ_𝑖−_𝑗≠𝑖𝐾_𝑖𝑗⋯𝐾_𝑖𝑛𝐾⋯⋯⋯⋯⋯_𝑛1⋯𝐾_𝑛𝑗⋯−Λ_𝑛−_𝑗≠𝑛𝐾_𝑛𝑗. (23) Consider the following Lyapunov function: 𝑣(𝑡)=𝜀^𝑇𝜀. (24) Differentiating 𝑣(𝑡) with respect to time yields 𝑣̇=2^𝑛_𝑖=1𝜀^𝑇_𝑖𝜀_𝑖̇=2^𝑛_𝑖=1𝜀^𝑇_𝑖−Λ_𝑖𝜀_𝑖−_𝑗≠𝑖𝐾_𝑖𝑗𝜀_𝑖−𝜀_𝑗=−2^𝑛_𝑖=1𝜀^𝑇_𝑖Λ_𝑖𝜀_𝑖−2^𝑛_{𝑖=1𝑗≠𝑖}𝜀^𝑇_𝑖𝐾_𝑖𝑗𝜀_𝑖−𝜀_𝑗=−2^𝑛_𝑖=1𝜀^𝑇_𝑖Λ_𝑖𝜀_𝑖−2^𝑛_{𝑖=1𝑗≠𝑖}𝜀^𝑇_𝑖𝐾_𝑖𝑗𝜀_𝑖+2^𝑛_{𝑖=1𝑗≠𝑖}𝜀^𝑇_𝑖𝐾_𝑖𝑗𝜀_𝑗. (25) Knowing that ^𝑛_{𝑖=1𝑗≠𝑖}𝜀^𝑇_𝑖𝐾_𝑖𝑗𝜀_𝑖=^𝑛_{𝑗=1𝑖≠𝑗}𝜀^𝑇_𝑗𝐾_𝑗𝑖𝜀_𝑗, (26) consequently, 𝑣̇=−2^𝑛_𝑖=1𝜀^𝑇_𝑖Λ_𝑖𝜀_𝑖−^𝑛_{𝑖=1𝑗≠𝑖}𝜀_𝑖−𝜀_𝑗^𝑇𝐾_𝑖𝑗𝜀_𝑖−𝜀_𝑗≤0. (27) It follows by direct application of LaSalle's invariance that the origin is globally asymptotically stable. Consequently we obtain lim𝜀_𝑖(𝑡)→0 for 𝑡→∞. Then 𝑞_𝑖→𝑞_𝑑 and 𝑞̇_𝑖→𝑞̇_𝑑 for 𝑡→∞.
Referring to (19), we show that 𝑞_𝑖→𝑞_𝑗 for 𝑡→∞.

4. Adaptive Synchronization

In this section, we consider an uncertainty in the model parameters. We propose to extend the previous decentralized control law to an adaptive version. For that, we consider the following adaptive control law, which has the similar local coupling structure as the proposed control law in (8): 𝜏_𝑖=𝐶_𝑖𝑞_𝑖,𝑞̇_𝑖+𝑔_𝑖𝑞_𝑖+𝑀_𝑖𝑞_𝑖×𝑞̈_𝑑−𝐾_𝑝𝑖𝑒_𝑖−𝐾_𝑑𝑖𝑒_𝑖̇−Λ_𝑖𝑞̇_𝑖−𝑞̇_𝑑+𝑀_𝑖𝑞_{𝑖𝑗≠𝑖}𝐾_𝑖𝑗𝜀_2𝑖−𝜀_2𝑗+Λ_𝑖𝜀_2𝑖. (28) We recall that a similar relation to (1) holds when the estimate parameter vector 𝜃 is used to replace the exact parameter vector 𝜃: 𝑀_𝑖𝑞_𝑖𝑞̈_𝑖+𝐶_𝑖𝑞_𝑖,𝑞̇_𝑖𝑞̇_𝑖+𝑔_𝑖𝑞_𝑖=𝑌_𝑖𝑞_𝑖,𝑞̇_𝑖,𝑞̈_𝑖𝜃_𝑖, (29) where 𝜃_𝑖∈ℝ^𝑎 is the vector of structural parameters of the manipulator and 𝑎 is the number of unknown parameters. The estimated parameter 𝜃_𝑖 is subject to the adaptation law: ̇𝜃_𝑖=−Γ_𝑖⁻¹𝑀_𝑖⁻¹𝑌_𝑖^𝑇𝑒_𝑖̇, (30) where Γ_𝑖 is a diagonal positive-definite matrix. Since the value of the dynamic parameter 𝜃_𝑖 is hard to be known exactly in practice, one defines 𝜃_𝑖(𝑡) as the estimate of 𝜃_𝑖. 𝑀_𝑖, 𝐶_𝑖, and 𝑔_𝑖 are estimates of 𝑀_𝑖, 𝐶_𝑖, and 𝑔_𝑖, respectively. 𝑌_𝑖(𝑞_𝑖,𝑞̇_𝑖,𝑞̈_𝑖) denotes a regression matrix. Define 𝜃_𝑖=𝜃_𝑖−𝜃_𝑖 as a vector containing model estimation errors. Then the adaptation law (30) can be written as 𝜃̇_𝑖=Γ_𝑖⁻¹𝑀_𝑖⁻¹𝑌_𝑖^𝑇𝑒_𝑖̇. (31)

Theorem 2. If 𝐾_𝑑𝑖>2∑_𝑗≠𝑖𝐾_𝑖𝑗, the proposed adaptive coupling controllers (28) and (31) guarantee the asymptotic convergence to zero of the joint position, velocity, and synchronization errors.

Proof. Substituting (28) into the dynamic model (1) leads to the following closed-loop dynamics: 𝑀_𝑖𝑞_𝑖𝑞̈_𝑖+𝐶_𝑖𝑞_𝑖,𝑞̇_𝑖𝑞̇_𝑖+𝑔_𝑖𝑞_𝑖=𝑀_𝑖𝑞_𝑖𝑞̈_𝑑−𝐾_𝑝𝑖𝑒_𝑖−𝐾_𝑑𝑖𝑒_𝑖̇−Λ_𝑖𝑞̇_𝑖−𝑞̇_𝑑+_𝑗≠𝑖𝐾_𝑖𝑗𝜀_2𝑖−𝜀_2𝑗+Λ_𝑖𝜀_2𝑖+𝐶_𝑖𝑞_𝑖,𝑞̇_𝑖𝑞̇_𝑖+𝑔_𝑖𝑞_𝑖. (32) We define 𝑀_𝑖𝑞_𝑖=𝑀_𝑖𝑞_𝑖−𝑀_𝑖𝑞_𝑖,𝐶_𝑖𝑞_𝑖,𝑞̇_𝑖=𝐶_𝑖𝑞_𝑖,𝑞̇_𝑖−𝐶_𝑖𝑞_𝑖,𝑞̇_𝑖,𝑔_𝑖𝑞_𝑖=𝑔_𝑖𝑞_𝑖−𝑔_𝑖𝑞_𝑖. (33) Taking into account (33), (32) gives: 𝑀_𝑖𝑞_𝑖𝑞̈_𝑖+𝐶_𝑖𝑞_𝑖,𝑞̇_𝑖𝑞̇_𝑖+𝑔_𝑖𝑞_𝑖+𝑀_𝑖𝑞_𝑖𝑞̈_𝑖−𝑞̈_𝑑=𝑀_𝑖𝑞_𝑖−𝐾_𝑝𝑖𝑒_𝑖−𝐾_𝑑𝑖𝑒_𝑖̇−Λ_𝑖𝑞̇_𝑖−𝑞̇_𝑑+_𝑗≠𝑖𝐾_𝑖𝑗𝜀_2𝑖−𝜀_2𝑗+Λ_𝑖𝜀_2𝑖. (34) Using (29), the dynamic (34) is written as 𝑌_𝑖𝑞_𝑖,𝑞̇_𝑖,𝑞̈_𝑖𝜃_𝑖+𝑀_𝑖𝑞_𝑖𝑞̈_𝑖−𝑞̈_𝑑=𝑀_𝑖𝑞_𝑖−𝐾_𝑝𝑖𝑒_𝑖−𝐾_𝑑𝑖𝑒_𝑖̇−Λ_𝑖𝑞̇_𝑖−𝑞̇_𝑑+_𝑗≠𝑖𝐾_𝑖𝑗𝜀_2𝑖−𝜀_2𝑗+Λ_𝑖𝜀_2𝑖. (35) Consequently, 𝑀_𝑖𝑞_𝑖𝑒_𝑖̈+𝑌_𝑖𝑞_𝑖,𝑞̇_𝑖,𝑞̈_𝑖𝜃_𝑖=𝑀_𝑖𝑞_𝑖−𝐾_𝑝𝑖𝑒_𝑖−𝐾_𝑑𝑖𝑒_𝑖̇+_𝑗≠𝑖𝐾_𝑖𝑗𝑒_𝑖̇−𝑒_𝑗̇. (36) To prove the stability of the overall synchronized system, we define 𝑌^𝑇=[𝑌^𝑇₁,…,𝑌^𝑇_𝑖,…,𝑌^𝑇_𝑛]^𝑇, 𝜃^𝑇=[𝜃^𝑇₁,…,𝜃^𝑇_𝑖,…,𝜃^𝑇_𝑛]^𝑇, Γ=diag(Γ^𝑇₁,…,Γ^𝑇_𝑖,…,Γ^𝑇_𝑛)𝐾_𝑑=diag(𝐾_𝑑𝑖), 𝐾_𝑝=diag(𝐾_𝑝𝑖), 𝑀=diag(𝑀_𝑖).
Writing (36) in a compact form gives the following: 𝑀𝑒̈+𝐾_𝑝𝑒+𝐾_𝑑𝑒̇−𝐾_𝑐𝑒̇=−𝑌𝑞_𝑖,𝑞̇_𝑖,𝑞̈_𝑖𝜃. (37) Multiplying by 𝑀⁻¹ both sides yields 𝑒̈=−𝐾_𝑝𝑒−𝐾_𝑑𝑒̇+𝐾_𝑐𝑒̇−𝑀⁻¹𝑌𝑞_𝑖,𝑞̇_𝑖,𝑞̈_𝑖𝜃. (38) Define the Lyapunov function candidate as 1𝑣=2𝑒^𝑇̇𝑒̇+𝑒^𝑇𝐾_𝑝𝑒+𝜃^𝑇Γ𝜃. (39) Differentiating 𝑣(𝑡) with respect to time yields 𝑣̇=𝑒^𝑇̈𝑒̇+𝑒^𝑇𝐾_𝑝𝑒̇+𝜃^𝑇Γ𝜃̇. (40) Substituting (38) into (40) 𝑣̇=−𝐾_𝑝𝑒−𝐾_𝑑𝑒̇+𝐾_𝑐𝑒̇−𝑀⁻¹𝑌𝜃^𝑇𝑒̇+𝑒^𝑇𝑘_𝑝𝑒̇+𝜃^𝑇Γ𝜃̇, (41) consequently 𝑣̇=−𝑒^𝑇̇𝐾_𝑑−𝐾_𝑐𝑒̇+𝜃^𝑇Γ𝜃̇−𝑀⁻¹𝑌^𝑇𝑒̇. (42) Using the adaptation law (31) gives 𝑣̇=−𝑒^𝑇̇𝐾_𝑑−𝐾_𝑐𝑒.̇ (43) Since 𝐾_𝑑𝑖>∑_𝑗≠𝑖𝐾_𝑖𝑗, we have 𝑣(̇𝑡)<0, and this yields that 𝑣(𝑡)<𝑣(0), which gives that 𝑒,𝑒̇, and 𝑀⁻¹𝑌𝜃^𝑇 are bounded.
Differentiating 𝑣(̇𝑡) with respect to time yields 𝑣̈=2𝐾_𝑝+𝐾_𝑑−𝐾_𝑐𝑒̇+𝑀⁻¹𝑌𝜃^𝑇𝐾_𝑑−𝐾_𝑐𝑒.̇ (44) Using Barbalat's lemma, 𝑣̈ is bounded because 𝑒, 𝑒̇, and 𝑀⁻¹𝑌𝜃 are bounded. This implies 𝑣(̇𝑡)→0 for 𝑡→∞ and hence 𝑒̇→0 for 𝑡→∞. The proof pursued the same line reasoning as the proof of Section 3. Consequently, we show that position errors, velocity errors and synchronization errors, converge asymptotically to zero.

5. Synchronization with Time Delays

In this section, we study the coordination control problem taking into account time delays of communication channels. As a first assumption, we suppose that these delays can be justified by the fact that data information sent by the neighboring vehicles 𝑗≠𝑖 reaches vehicle 𝑖 after a certain time delay due to the short-range communication channels. To take into account time delays produced during the communication among robots, we introduce, in the coordination error expression, a term 𝜏 which represents the same time delay due to the short-range communication channels. Therefore, a coordination error, in the time delay context, will be presented as the well-known classical time delay model of multiagent network [9]: 𝜀_2𝑖(𝑡)=_𝑗≠𝑖𝐾_𝑖𝑗𝑞_𝑖(𝑡−𝜏)−𝑞_𝑗(𝑡−𝜏). (45) It will be shown that the behavior of the coordinated system under the effect of time delay changes significantly. Consequently, the controller implanted in each lagrangian system among the network takes the following expression.

Theorem 3. Implemented controller taking the following expressions (𝑖=1,…,𝑝): 𝜏_𝑖=𝐶_𝑖𝑞_𝑖,𝑞̇_𝑖𝑞̇_𝑖+𝑔_𝑖𝑞_𝑖+𝑀_𝑖𝑞_𝑖×𝑞̈_𝑑−𝐾_𝑝𝑖𝑒_𝑖(𝑡)−𝐾_𝑑𝑖𝑒_𝑖̇(𝑡)−Λ_𝑖𝑞̇_𝑖−𝑞̇_𝑑+𝑀_𝑖𝑞_{𝑖𝑗≠𝑖}𝐾_𝑖𝑗𝜀_2𝑖(𝑡−𝜏)−𝜀_2𝑗(𝑡−𝜏)+Λ_𝑖𝜀_2𝑖(𝑡−𝜏) (46) stabilize the behavior of the robot network.

Proof. Substituting (46) into (1) yields 𝑀_𝑖𝑞_𝑖𝑞̈_𝑖=𝑀_𝑖𝑞_𝑖𝑞̈_𝑑−𝐾_𝑝𝑖𝑒_𝑖−𝐾_𝑑𝑖𝑒_𝑖̇−Λ_𝑖𝑞̇_𝑖−𝑞̇_𝑑+𝑀_𝑖𝑞_{𝑖𝑗≠𝑖}𝐾_𝑖𝑗𝜀_2𝑖(𝑡−𝜏)−𝜀_2𝑗(𝑡−𝜏)+Λ_𝑖𝜀_2𝑖(𝑡−𝜏). (47) By further calculation, we obtain the synchronized error dynamics: 𝑒̈=−𝐾_𝑝𝑒−𝐾_𝑑𝑒̇+𝐾_𝑐𝑒̇(𝑡−𝜏), (48) where 𝐾_𝑝, 𝐾_𝑑, and 𝐾_𝑐 are the same matrices already defined (see Section 3). By the Leibnitz formula, we have 𝑒̇−𝑒̇(𝑡−𝜏)=^𝑡_𝑡−𝜏𝑒̈(𝜆)𝑑𝜆. (49) Substituting (49) into (48) leads to 𝑒̈=−𝐾_𝑝𝑒−𝐾_𝑑𝑒̇+𝐾_𝑐𝑒̇−^𝑡_𝑡−𝜏𝑒̈(𝜆)𝑑𝜆. (50) Setting 𝑒̃=[𝑒^𝑇,𝑒^𝑇̇]^𝑇. Therefore (50) can be written as 𝑒̇̃=0𝐼−𝐾_𝑝−𝐾_𝑑+𝐾_𝑐𝑒̃−000𝐾_𝑐^{𝑡𝑡−𝜏}𝑒̇̃(𝜆)𝑑𝜆. (51) This yields the following form: 𝑒̇̃=𝛽₀𝑒̃−𝛽₁^𝑡_𝑡−𝜏𝑒̇̃(𝜆)𝑑𝜆 (52) with 𝛽₀=^0𝐼_{−𝐾𝑝−𝐾𝑑+𝐾𝑐} and 𝛽₁=⁰⁰_0𝐾𝑐.
To analyze the stability of the global system, we consider the following Lyapunov-Krasovskii functional (LKF): 𝑣(𝑡)=𝑣₁(𝑡)+𝑣₂(𝑡)+𝑣₃(𝑡),𝑣₁(𝑡)=𝑒^𝑇̃(𝑡)𝑃𝑒̃(𝑡),𝑣₂(𝑡)=^𝑡_𝑡−𝜏𝑒^𝑇̃(𝜆)𝑅𝑒̃(𝜆)𝑣𝑑𝜆,₃(𝑡)=⁰_−𝜏^𝑡_𝑡+𝑠𝑒^𝑇̇̃(𝛼)𝑍𝑒̇̃(𝛼)𝑑𝛼𝑑𝜆, (53) where 𝑃=𝑃^𝑇>0, 𝑅=𝑅^𝑇>0, 𝑍=𝑍^𝑇>0 are weighting matrices of appropriate dimensions. A straightforward computation gives the time derivative of 𝑣(𝑡) along the solution of (53) as 𝑣̇(𝑡)=𝛿^𝑇2𝑁^𝑇𝑃𝑀+𝑁^𝑇𝑅𝑁−𝑄^𝑇𝑅𝑄+𝜏𝑀^𝑇𝛿−𝑍𝑀_{𝑡𝑡−𝜏}𝑒^𝑇̇̃(𝜆)𝑍𝑒̇̃(𝜆)𝑑𝜆, (54) where 𝛿=[𝑒^𝑇̃(𝑡),𝑒^𝑇̃(𝑡−𝜏)]^𝑇, 𝑁=[𝛽₀,𝛽₁], 𝑀=[𝐼,0], 𝑄=[0,𝐼]. The Jensen's inequality gives a suitable bound for the last term of (54): −^𝑡_𝑡−𝜏𝑒^𝑇̇̃(𝜆)𝑍𝑒̇̃(𝜆)𝑑𝜆≤^𝑡_𝑡−𝜏𝑒^𝑇̇̃(𝜆)𝑍𝑑𝜆𝜏^𝑡_𝑡−𝜏𝑒̇̃(𝜆)𝑒̃𝑑𝜆≤−^𝑇𝑇^𝑇𝑍𝜏𝑇𝑒̃ (55) with 𝑇=[𝐼,−𝐼]. Thus, the time derivative of the LKF (53) can be bounded by 𝑣(̇𝑡)≤𝛿^𝑇𝜉𝛿, where 𝜉=2𝑁^𝑇𝑃𝑀+𝑁^𝑇𝑅𝑁−𝑄^𝑇𝑅𝑄+𝜏𝑀^𝑇1𝑍𝑀−𝜏𝑇^𝑇𝑍𝑇. (56) Then if the LMI 𝜉<0 is satisfied, the derivative of the Lyapunov-Krasovskii functional is negative definite. To ensure matrix 𝜉 is negative definite, we select appropriate control gains 𝐾_𝑝>𝐾^∗_𝑝 and 𝐾_𝑑−𝐾_𝑐>𝐾^∗ throught processing MATLAB's LMI solver such that 2𝑁^𝑇𝑃𝑀+𝑁^𝑇𝑅𝑁−𝑄^𝑇𝑅𝑄+𝜏𝑀^𝑇1𝑍𝑀−𝜏𝑇^𝑇𝑍𝑇<0. (57) Then, if the LMI 𝜉<0 is satisfied, the derivative of the Lyapunov-Krasovskii functional is therefore negative definite. In consequence, the origin 𝑒̃=0 is asymptotically stable.
This results in 𝑒→0 for 𝑡→∞ and 𝑒̇→0 for 𝑡→∞. The proof for the asymptotic convergence of the coordinated tracking error 𝑒̃ is not sufficient to prove the convergence to zero of both errors 𝑒₁ and 𝑒₂. Now, our concern is is to show that coordination is successfully realized for a specific time delay 𝜏_𝑐.
The proof pursued the same line reasoning as the proof of Section 3. Consequently, we obtain the following equation derived from the global error expression: 𝜀_𝑖̇=−Λ_𝑖𝜀_𝑖−_𝑗≠𝑖𝐾_𝑖𝑗𝜀_𝑖(𝑡−𝜏)−𝜀_𝑗(𝑡−𝜏). (58) Rewriting all states of (58) into a compact representation and applying the Laplace transform lead to 𝑠𝜀(𝑠)−𝜀(0)=−Λ𝜀(𝑠)−𝑒^−𝜏𝑠𝐾_𝑐𝜀(𝑠). (59) This can be written as 𝜀(𝑠)=𝑠𝐼+Λ+𝑒^−𝜏𝑠𝐾_𝑐⁻¹𝜀(0). (60) If the characteristic equation 𝑃(𝑠,𝜏)= det|𝑠𝐼+Λ+𝑒^−𝜏𝑠𝐾_𝑐|=0 has all its zeros in the left half complex plan, then the system is stable and one can easily conclude about the convergence of 𝑞_𝑖 to 𝑞_𝑑. Since the ordinal system, free from time delay (i.e, 𝜏=0), is a continuous function of 𝜏, then using the D-decomposition, the minimal positive solution to the following equation: 𝜏_𝑖=1𝐾²_𝑐𝑖−Λ²_𝑖arccos−Λ_𝑖𝐾_𝑐𝑖 (61) which would make all the zeros of the characteristic equation in the left half complex plane. Therefore if we select 𝜏∈[0,𝜏_𝑐], where 𝜏_𝑐=sup{𝜏_𝑖,∀1≤𝑖≤𝑝}, therefore solutions of (60) converge to zero and consequently 𝑞_𝑖→𝑞_𝑑, 𝑞̇_𝑖→𝑞̇_𝑑, and 𝑞_𝑖→𝑞_𝑗 for 𝑡→∞.

6. Simulation Results

To show the effectiveness of the proposed synchronizing controller, we run two types of simulation.

Simulation 1
Simulations are performed on MATLAB/Simulink. These simulations were performed on identical three manipulators with two degrees of freedom. Robots do not have the same starting positions, and they start their motion at 𝑡₀=0. First, we assume that no parameter uncertainty is present and that the transmission delay can be neglected. Figure 2 illustrates the robots synchronization tracking a common trajectory. This proves that tracking and synchronization objectives are attained by the proposed controller which explains how robots, while tracking the desired trajectory, synchronize their position with others. Next, we assume that the length and the mass of manipulators are unknown. Consequently, mass, Coriolis, and gravity matrices are ill known. Figure 3 plots an adaptive position tracking while in Figures 4 and 5 tracking errors and synchronizing errors are shown, respectively. Finally, we consider the time delay in communication. Figures 6 and 7 illustrate that the behavior of the coordinated system changes significantly, under the effect of time delays in communication channels. The speed for achieving an agreement depends essentially on the time delay of communication channels.

Figure 2 

Synchronization of robots without uncertain parameters.

Figure 3 

Synchronization of robots in the presence of uncertain parameters.

Figure 4 

Position errors: adaptive synchronization.

Figure 5 

Synchronization errors: adaptive synchronization.

Figure 6 

Robots synchronization without time delay.

Figure 7 

Robots synchronization in presence of time delay.

Simulation 2
One effective solution is to develop a platform to simulate the complex system design before hardware implementation. To design this virtual environment, we apply several tools, such as the modeling package Cinema 4D, MATLAB VRML (Virtual Reality Modeling Language) toolbox, and MATLAB/Simulink. Robot CAD (Concept Architecture Design) models created using 4D cinema are extracted in VRML97 (.wrl) format. The VRML file is edited by V-Realm Builder 2.0 before being imported into MATLAB-Simulink by the VR (Virtual Reality) Toolbox. A VRML file uses a standard text format which can be read with any text editor. For MATLAB, the Virtual Reality Toolbox software includes the Ligos V-Realm Builder application as a native 3D editor. But the 3D editing tools remain the most efficient. In fact, the CAD packages offer the power and versatility to create many types of practical models and techniques. In the developed real-time software simulator we used the 4D Cinema as CAD package where the robots are designed. Next, the 3D objects are imported. Simulink model generates signal data which is used to control and animate the virtual world. Reality Toolbox block is added to the Simulink model in order to ensure communication with the virtual world. The 3D virtual reality environment is presented in Figure 8.

(a)

(b)

(c)

(d)

(e)

Figure 8 

The 3D virtual environment.

7. Conclusions

This paper has considered the synchronization problem in distributed Multirobot systems under cooperative schemes. The aim of this work is to find out a decentralized controller, applied to each manipulator; the synchronization is therefore met. It has been shown that the proposed strategy can coordinate manipulator articulations to track a given time-varying trajectory. As an extension, we moved to the step which entails the fact that mass, Coriolis, and gravity matrices are badly known. In this case, the proposed controller has encompassed an adaptive version. The obtained simulation results from Multirobot motion control system demonstrate the effectiveness of the synchronization approach. To deal with time delays problem in communication between robots, the proposed decentralized control guarantees that information variables of each robot reach agreement even in the presence of communication channels delays. A new package for simulation of coordinated robot manipulators is developed. Simulations which have been applied to an illustrative example have shown the effectiveness of the described strategy.