Nonlinear Control and Synchronization with Time Delays of Multiagent Robotic Systems

Abstract

We investigate the cooperative control and global asymptotic synchronization Lagrangian system groups, such as industrial robots. The proposed control approach works to accomplish multirobot systems synchronization under an undirected connected communication topology. The control strategy is to synchronize each robot in position and velocity to others robots in the network with respect to the common desired trajectory. The cooperative robot network only requires local neighbor-to-neighbor information exchange between manipulators and does not assume the existence of an explicit leader in the team. It is assumed that network robots have the same number of joints and equivalent joint work spaces. A combination of the lyapunov-based technique and the cross-coupling method has been used to establish the multirobot system asymptotic stability. The developed control combines trajectory tracking and coordination algorithms. To address the time-delay problem in the cooperative network communication, the suggested synchronization control law is shown to synchronize multiple robots as well as to track given trajectory, taking into account the presence of the time delay. To this end, Krasovskii functional method has been used to deal with the delay-dependent stability problem.

1. Introduction

Nowadays, much research has been focusing on group coordination, cooperative control, and synchronization problems. In fact, motivated by the profit acquired by using multiple inexpensive systems working together to achieve complex tasks exceeding the abilities of a single agent, cooperative synchronization control has received significant attention. Distributed coordination and decentralized synchronization of multiagent systems have recently been studies extensively in the context of cooperative control [1–5], to name a few. In particular, design based on graph theory and Laplacian matrix produce interesting results [6–9]. Agreement, consensus problems in the area of cooperative control of multiagent systems have been studied in [7, 8, 10–12]. The coordination control strategies are closely related to the synchronization problem in which control laws are coupled and each agent robot control is updated using local rule based on its own sensors and the states of its neighbors. In this context, one recent representative work [13] shows that we can synchronize the multicomposed system in the case of partial knowledge, that is, only position measuring. A decentralized tracking control law globally exponentially synchronizes an arbitrary number of robots and represents a generalization of the average consensus problem. This has been presented in [5]. A synchronization approach to trajectory tracking of multiple mobile robots while maintaining time-varying formations has been presented in [14]. Adaptive control strategy to position synchronization of multiple motion axes using cross-coupling technology has been developed in [15]. In many engineering applications, communication delays between subsystems cannot be neglected. Therefore, the problem of time-delayed communication in control of multirobot systems is important in numerous practical applications. Indeed, without control measures of time delays in cooperative task may even cause instability. The problem of time-delayed communication in control of multiagent systems has been studied in several references [7, 16–18]. The consideration of time-delayed communication in control of multirobot systems is a mainly practical necessity. In particular, this need occurs when addressing areas which require real-time applications such as operations in unsafe environments and robotic surgery.

The objective of this paper is to design a control approach that can achieve both synchronization of the robot movements and asymptotic stable tracking of a common desired trajectory. The proposed controller relies principally on a consensus algorithm for systems modeled by nonlinear second-order dynamics and applies the algorithm to 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 a robot manipulator positions and different positions of its neighbors. In the literature, most of earlier works on multiagent coordination and consensus [3, 4, 7, 19] mainly deal with very simple dynamic models such as linear systems and focuses on an algorithm taking the form of first-order dynamics [11, 20, 21]. In particular, most previous works on consensus and coordination of multiagent systems using the graph theory and laplacian [3, 4, 7–9] 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. Therefore, the aforementioned algorithms cannot give solutions for robot networks, where a desired trajectory is required. In contrast, the present work deals with highly nonlinear systems. Moreover, the developed approach achieves not only global asymptotic synchronization of the configuration variables, but also global asymptotic convergence to the desired trajectory. Notable works have focused on highly nonlinear systems. Their developed strategy requires the coupling feedback of the most adjacent robots [5] or axis [15] for the algorithm. However, the proposed strategy is based in partial mesh topology in which there are interconnections between all robots, such that all robots have direct influence in the combined dynamics. We provide by the use of partial mesh topology a high degree of reliability due to the presence of multiple paths for data between robots. On the other hand, it is not a fully connected mesh topology and consequently we avoid the expense and the complexity required for a connection between every robot in the network. In this paper, we study the problem of mutual synchronization when there are communication delays in the network. The delays are assumed to be bounded.

2. Background and Preliminaries

2.1. Modelling Multi-Lagrangian System Network

The 𝑛 degree-of-freedom robot manipulator composed of rigid bodies is expressed based on Newton’s and Euler’s equations as follows: 𝑀_𝑖𝑞_𝑖𝑞_𝑖̈+𝐶_𝑖𝑞_𝑖,𝑞_𝑖̇𝑞_𝑖̇+𝑔_𝑖𝑞_𝑖=𝜏_𝑖, (1) where 𝑞_i∈ℝ^𝑛 denotes the joint angles of the 𝑖th manipulator, 𝑞_𝑖̇∈ℝ^𝑛, and 𝑞_𝑖̈∈ℝ^𝑛 are the vectors of joint velocity and joint acceleration, respectively. 𝑀_𝑖(𝑞_𝑖)∈ℝ^𝑛×𝑛 represents 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. 𝑔_𝑖(𝑞_𝑖)∈ℝ^𝑛 is a vector function consisting of gravitational forces. Although the above equations of motion are coupled and nonlinear, they exhibit certain fundamental properties due to their Lagrangian dynamic structure. The most important property is the well-known skew symmetry of the matrix 𝑀̇−2𝐶[22].

2.2. Multiagent Communication Topology

Since we based on our coordination algorithm conception on consensus strategy and concepts of graph theory, we present several basic properties of these technology. Let 𝐺=(𝑉,𝐸) a digraph with 𝑁 nodes, the set of nodes 𝑉=1,2,…,𝑛, and edges 𝐸⊆𝑉×𝑉. Each node is labeled by 𝑣_𝑖∈𝑉 and each edge is denoted by 𝑒_𝑖𝑗=(𝑣_𝑖,𝑣_𝑗). Neighbors of agent 𝑣_𝑖 are denoted by 𝑁={𝑣_𝑗∈𝑉/(𝑣_𝑖,𝑣_𝑗)∈𝐸}. The adjacency matrix 𝐴=[𝑎_𝑖𝑗]∈ℝ^𝑛×𝑛 of a weighted digraph is defined as 𝑎_𝑖𝑗>0if𝑣_𝑖,𝑣_𝑗𝑎∈𝐸,_𝑖𝑗=0,elsewhere. (2) Agent 𝑖 communicates with agent 𝑗 if 𝑗 is a neighbor of 𝑖 or if 𝑎_𝑖𝑗≠0. Note that an edge 𝑒_𝑖𝑗 in a directed graph means that robot 𝑗 can reach information from robot 𝑖, but not necessarily vice versa. In contrast, in an undirected graph, pairs of node are unordered and an edge 𝑒_𝑖𝑗 implies that robots 𝑖 and 𝑗 can get information from one another. The adjacency matrix of an undirected graph has the same meaning as that of the directed graph except that 𝑎_𝑖𝑗=𝑎_𝑗𝑖. The degree matrix of the digraph 𝐺=(𝑉,𝐸) is a diagonal matrix defined as 𝑑_𝑖𝑗=deg_out𝑣_𝑖if𝑖=𝑗∈𝐸,0,otherwise, (3) where deg_out(𝑣_𝑖)=∑^𝑛_𝑗=1𝑎_𝑖𝑗.

The graph laplacian of 𝐺 is defined as 𝐿=𝐷−𝐴, where 𝐷=[𝑑_𝑖𝑖] the degree matrix of 𝐺. In the undirected graph case, 𝐿 is symmetric positive semidefinite. In the present topology, the edge represents bidirectional communication links. This consists on a group of 𝑛 manipulators interchanging information that can be viewed as an undirected graph (see Figure 1).

Figure 1 

Undirected graph topology structure.

3. Controller Design

3.1. Tracking and Synchronization Errors

In this paper, we consider the synchronization of multiple robots following a common time-varying trajectory. We will design decentralized control laws for 𝑛 robots 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 positions 𝑞_𝑗 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 motions of 𝑛 robots in communication so that the synchronization error converges to zero. To this end, we define the measure of the position tracking error of the 𝑖th manipulator as 𝑒_1𝑖(𝑡)=𝑞_𝑖(𝑡)−𝑞_𝑑(𝑡)+^𝑡_𝑡0Λ_𝑖𝑞_𝑖(𝜆)−𝑞_𝑑(𝜆)𝑑𝜆, (4) where Λ_𝑖 is a diagonal positive definite matrix. Information on the vector 𝑒_1𝑖 will give insight on the convergence of the joint positions to the desired trajectory. 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 [13], authors include the error information of all systems involved in the synchronization. Our approach will make use of the cross-coupling technique to propose a feasible and efficient synchronization error, which consists on a measure of the synchronization for robot manipulator as defined as follows: 𝑒_2𝑖(𝑡)=_𝑗≠𝑖𝛽_𝑖𝑗𝑞_𝑖−𝑞_𝑗, (5) where, 𝛽_𝑖𝑗 is a diagonal positive definite matrix which gives insight on the weighted communication among the robot network.

3.2. Feedback Control Design

The objective of this paper is to design individual tracking controller for 𝑛 manipulators such that they coordinate their motions and track synchronously a desired trajectory. To this end, we define the global error which encompasses both synchronization error and trajectory tracking error for manipulator 𝑖 as 𝑒_𝑖=𝑒_1𝑖+^𝑡_𝑡0𝑒_2𝑖(𝜆)𝑑𝜆. (6) Under the above strategy, motions of all manipulators are synchronized. The control of each manipulator considers motion responses of the other manipulators for synchronization. It takes into account only robots which make the exchange of information with it. The objective is to design a control law such that the coupling errors, that is, the 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,𝑞_𝑖̇𝑞_𝑖̇+𝑔_𝑖𝑞_𝑖+𝑀_𝑖𝑞_𝑖𝑞_𝑑̈−𝐾_𝑝𝑖𝑒_𝑖−𝐾_𝑑𝑖𝑒_𝑖̇−Λ_𝑖𝑞_𝑖̇−𝑞_𝑑̇+𝑀_𝑖𝑞_{𝑖𝑗≠𝑖}𝐾_𝑖𝑗𝑒_2𝑖−𝑒_2𝑗+Λ_𝑖𝑒_2𝑖, (7) 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 derivatives exist for all 𝑡≥0. 𝐾_𝑑𝑖 is a symmetric positive definite matrix. 𝐾_𝑖𝑗 is a matrix that reflects the quality of communication channels; it is a symmetric positive matrix.

3.3. Stability Analysis

Substituting (7) into (1) yields 𝑀_𝑖𝑞_𝑖𝑞_𝑖̈=𝑀_𝑖𝑞_𝑖𝑞_𝑑̈−𝐾_𝑝𝑖𝑒_𝑖−𝐾_𝑑𝑖𝑒_𝑖̇−Λ_𝑖𝑞_𝑖̇−𝑞_𝑑̇+𝑀_𝑖𝑞_{𝑖𝑗≠𝑖}𝐾_𝑖𝑗𝑒_2𝑖−𝑒_2𝑗+Λ_𝑖𝑒_2𝑖. (8) This results in 𝑞_𝑖̈=𝑞_𝑑̈−𝐾_𝑝𝑖𝑒_𝑖−𝐾_𝑑𝑖𝑒_𝑖̇−Λ_𝑖𝑞_𝑖̇−𝑞_𝑑̇+_𝑗≠𝑖𝐾_𝑖𝑗𝑒_2𝑖−𝑒_2𝑗+Λ_𝑖𝑒_2𝑖, (9) which can be written as follows: 𝑒_𝑖̈=−𝐾_𝑝𝑖𝑒_𝑖−𝐾_𝑑𝑖𝑒_𝑖̇+_𝑗≠𝑖𝐾_𝑖𝑗𝑒_2𝑖−𝑒_2𝑗+Λ_𝑖𝑒_2𝑖+𝑒̇_2𝑖. (10) Using the expression of the synchronization error 𝑒_2𝑖 and its first derivative gives 𝑒_𝑖̈=−𝐾_𝑝𝑖𝑒_i−𝐾_𝑑𝑖𝑒_𝑖̇+_𝑗≠𝑖𝐾_𝑖𝑗𝑒_2𝑖−𝑒_2𝑗+_𝑗≠𝑖𝐾_𝑖𝑗𝑞̇_𝑖−𝑞̇_𝑑−𝑞_𝑗̇−𝑞_𝑑̇+_𝑗≠𝑖Λ_𝑖𝐾_𝑖𝑗𝑞_𝑖−𝑞_𝑑−𝑞_𝑗−𝑞_𝑑. (11) Further calculation, will result in 𝑒_𝑖̈=−𝐾_𝑝𝑖𝑒_𝑖−𝐾_𝑑𝑖𝑒_𝑖̇+_𝑗≠𝑖𝐾_𝑖𝑗𝑒_𝑖̇−𝑒_𝑗̇. (12) Equation (12) 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 to show that control goals: the position errors, velocity errors, and synchronization errors, all converge to zero. To prove the stability of the overall synchronized system, we define𝑒^𝑇=𝑒^𝑇₁,𝑒^𝑇₂,…,𝑒^𝑇_𝑛^𝑇. (13) Using (12) we obtain the synchronized error dynamics 𝑒̈=−𝐾_𝑝𝑒−𝐾_𝑑𝑒̇+𝐾_𝑐𝑒,̇ (14) 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. (15) Note that 𝐾_𝑐 is symmetric and positive semidefinite matrix, since we have an undirected graph, that is, 𝐾_𝑖𝑗=𝐾_𝑗𝑖.

The synchronized error dynamics (14) is a linear time invariant system described by a second-order linear differential equation. A sufficient condition for the error dynamics to be 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. (16) To analyze the stability properties of the closed-loop error dynamics (19), we take the following definite and radially unbounded Lyapunov function candidate: 𝑉=𝑒^𝑇̇𝑒̇+𝑒^𝑇𝐾_𝑝𝑒. (17) Its derivative to respect to time is 𝑉̇=−𝑒^𝑇̇𝐾_𝑑−𝐾_𝑐,𝑒̇≤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 (6): 𝑒_𝑖=𝑞_𝑖−𝑞_𝑑+^𝑡_𝑡0Λ𝑞_𝑖(𝜆)−𝑞_𝑑(𝜆+)𝑑𝜆_{𝑡𝑡0𝑗≠𝑖}𝛽_𝑖𝑗𝑞_𝑖−𝑞_𝑗 (19) as 𝑒̇=0 we have 𝑞_𝑖̇−𝑞_𝑑̇=−Λ_𝑖𝑞_𝑖−𝑞_𝑑−_𝑗≠𝑖𝛽_𝑖𝑗𝑞_𝑖−𝑞_𝑗. (20) Setting 𝜀_𝑖=𝑞_𝑖−𝑞_𝑑, then (20) can be written as 𝜀_𝑖̇=−Λ_𝑖𝜀_𝑖−_𝑗≠𝑖𝛽_𝑖𝑗𝜀_𝑖−𝜀_𝑗. (21) Our objective is to show that lim𝜀_𝑖=0 for 𝑡→∞. To this end, we define 𝜀=[𝜀₁⋯⋯⋅𝜀_𝑛]^𝑇 and Λ=[Λ₁⋯⋯⋯⋯Λ_𝑛]^𝑇. Then (21) can be written as 𝜀̇=𝐴⋅𝜀, (22) where matrix A is given by −Λ₁−_𝑗≠1𝛽_1𝑗⋯𝛽_1𝑗⋯𝛽_1𝑛𝛽⋯⋯⋯⋯⋯_𝑖1⋯−Λ_𝑖−_𝑗≠𝑖𝛽_𝑖𝑗⋯𝛽_𝑖𝑛𝛽⋯⋯⋯⋯⋯_𝑛1⋯𝛽_𝑛𝑗⋯−Λ_𝑛−_𝑗≠𝑛𝛽_𝑛𝑗. (23) We set the Lyapunov function candidate as 𝑣(𝑡)=𝜀^𝑇𝜀. (24) Differentiating 𝑣(𝑡) with respect to time yields 𝑣̇=2^𝑛_𝑖=1𝜀_𝑖𝜀_𝑖̇,𝑣̇=2^𝑛_𝑖=1𝜀_𝑖−Λ_𝑖𝜀_𝑖−_𝑗≠𝑖𝛽_𝑖𝑗𝜀_𝑖−𝜀_𝑗,𝑣̇=−2^𝑛_𝑖=1Λ_𝑖𝜀²_𝑖−2^𝑛_{𝑖=1𝑗≠𝑖}𝛽_𝑖𝑗𝜀_𝑖−𝜀_𝑗𝜀_𝑖,𝑣̇=−2^𝑛_𝑖=1Λ_𝑖𝜀²_𝑖−2^𝑛_{𝑖=1𝑗≠𝑖}𝛽_𝑖𝑗𝜀²_𝑖+2^𝑛_{𝑖=1𝑗≠𝑖}𝛽_𝑖𝑗𝜀_𝑖𝜀_𝑗, (25) knowing that ^𝑛_{𝑖=1𝑗≠𝑖}𝛽_𝑖𝑗𝜀²_𝑖=^𝑛_{𝑗=1𝑖≠𝑗}𝛽_𝑗𝑖𝜀²_𝑗. (26) Consequently, 𝑣̇=−2^𝑛_𝑖=1Λ_𝑖𝜀²_𝑖−^𝑛_{𝑖=1𝑗≠𝑖}𝛽_𝑖𝑗𝜀_𝑖−𝜀_𝑗²≤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 (20), we show that 𝑞_𝑖→𝑞_𝑗 for 𝑡→∞.

4. Coordination 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 timedelay due to the short-range communication channels. To take into account the time delay produced during the communication among the robots, we introduced in a 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 delayed model of multiagent network: 𝑒_2𝑖(𝑡)=_𝑗≠𝑖𝛽_𝑖𝑗𝑞_𝑖(𝑡−𝜏)−𝑞_𝑗(𝑡−𝜏). (28) Consequently, the controller implanted in each lagrangian system among the network take the following expression: 𝜏_𝑖=𝐶_𝑖𝑞_𝑖,𝑞_𝑖̇𝑞_𝑖̇+𝑔_𝑖𝑞_𝑖+𝑀_𝑖𝑞_𝑖𝑞_𝑑̈−𝐾_𝑝𝑖𝑒_𝑖(𝑡)−𝐾_𝑑𝑖𝑒_𝑖̇(𝑡)−Λ_𝑖𝑞_𝑖̇−𝑞_𝑑̇+𝑀_𝑖𝑞_{𝑖𝑗≠𝑖}𝐾_𝑖𝑗𝑒_2𝑖(𝑡−𝜏)−𝑒_2𝑗(𝑡−𝜏)+Λ_𝑖𝑒_2𝑖(𝑡−𝜏). (29) It will be shown that the behavior of the coordinated system under the effect of time delay changes significantly.

4.1. Stability Analysis

Substituting (29) into (1) yields 𝑀_𝑖𝑞_𝑖𝑞̈_𝑖=𝑀_𝑖𝑞_𝑖𝑞_𝑑̈−𝐾_𝑝𝑖𝑒_𝑖−𝐾_𝑑𝑖𝑒_𝑖̇−Λ_𝑖𝑞_𝑖̇−𝑞_𝑑̇+𝑀_𝑖𝑞_{𝑖𝑗≠𝑖}𝐾_𝑖𝑗𝑒_2𝑖(𝑡−𝜏)−𝑒_2𝑗(𝑡−𝜏)+Λ_𝑖𝑒_2𝑖(𝑡−𝜏). (30) Multiplying by 𝑀⁻¹ and adding 𝑒̇_2𝑖(𝑡−𝜏) in both sides yields 𝑞̈_𝑖−𝑞_𝑑̈+Λ_𝑖𝑞_𝑖̇−𝑞_𝑑̇+𝑒̇_2𝑖(𝑡−𝜏)=−𝐾_𝑝𝑖𝑒_𝑖−𝐾_𝑑𝑖𝑒_𝑖̇+_𝑗≠𝑖𝐾_𝑖𝑗𝑒_2𝑖(𝑡−𝜏)−𝑒_2𝑗(𝑡−𝜏)+Λ_𝑖𝑒_2𝑖(𝑡−𝜏)+𝑒̇_2𝑖(𝑡−𝜏). (31) Using the expression of the synchronization error 𝑒_2𝑖 and its first derivative gives 𝑒_𝑖̈=−𝐾_𝑝𝑖𝑒_𝑖−𝐾_𝑑𝑖𝑒_𝑖̇+_𝑗≠𝑖𝐾_𝑖𝑗𝑒_2𝑖(𝑡−𝜏)−𝑒_2𝑗(+𝑡−𝜏)_𝑗≠𝑖𝐾_𝑖𝑗𝑞̇_𝑖−𝑞̇_𝑑−𝑞_𝑗̇−𝑞_𝑑̇+_𝑗≠𝑖Λ_𝑖𝐾_𝑖𝑗𝑞_𝑖−𝑞_𝑑−𝑞_𝑗−𝑞_𝑑. (32) Further calculation, we obtain the synchronized error dynamics 𝑒̈=−𝐾_𝑝𝑒−𝐾_𝑑𝑒̇+𝐾_𝑐𝑒̇(𝑡−𝜏), (33) where 𝐾_𝑝, 𝐾_𝑑, and 𝐾_𝑐 are the same matrices already defined (see Section 3.3). By the Leibnitz formula, we have 𝑒̇−𝑒̇(𝑡−𝜏)=^𝑡_𝑡−𝜏𝑒̈(𝜆)𝑑𝜆 (34) substituting (34) into (33) leads to 𝑒̈=−𝐾_𝑝𝑒−𝐾_𝑑𝑒̇+𝐾_𝑐𝑒̇−^𝑡_𝑡−𝜏𝑒̈(𝜆)𝑑𝜆. (35) Setting 𝑒̃=[𝑒,𝑒]̇^𝑇. Therefore (35) can be written as 𝑒̇̃=0𝐼−𝐾_𝑝−𝐾_𝑑+𝐾_𝑐𝑒̃−000𝐾_𝑐^{𝑡𝑡−𝜏}𝑒̇̃(𝜆)𝑑𝜆. (36) This yields the following form: 𝑒̇̃=𝛽₀𝑒̃−𝛽₁^𝑡_𝑡−𝜏𝑒̇̃(𝜆)𝑑𝜆 (37) 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, (38) 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 (37) as 𝑣̇(𝑡)=𝛿^𝑇2𝑁^𝑇𝑃𝑀+𝑁^𝑇𝑅𝑁−𝑄^𝑇𝑅𝑄+𝜏𝑀^𝑇𝛿−𝑍𝑀_{𝑡𝑡−𝜏}𝑒^𝑇̇̃(𝜆)𝑍𝑒̇̃(𝜆)𝑑𝜆, (39) 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 (39). −^𝑡_𝑡−𝜏𝑒^𝑇̇̃(𝜆)𝑍𝑒̇̃(𝜆)𝑑𝜆≤^𝑡_𝑡−𝜏𝑒^𝑇̇̃(𝜆)𝑍𝑑𝜆𝜏^𝑡_𝑡−𝜏𝑒̇̃(𝜆)𝑒̃𝑑𝜆≤−^𝑇𝑇^𝑇𝑍𝜏𝑇𝑒̃ (40) with 𝑇=[𝐼,−𝐼]. The time derivative of the LKF (38) can thus be bounded by 𝑣(̇𝑡)≤𝛿^𝑇𝜉𝛿, where 𝜉=2𝑁^𝑇𝑃𝑀+𝑁^𝑇𝑅𝑁−𝑄^𝑇𝑅𝑄+𝜏𝑀^𝑇𝑍𝑀−(1/𝜏𝑇^𝑇𝑍𝑇). Then if the LMI 𝜉<0 is satisfied, the derivative of the Lyapunov-Krasovskii functional is negative definite. To ensure that matrix 𝜉<0 is negative definite, we select appropriate control gains 𝐾_𝑝>𝐾^∗_𝑝 and 𝐾_𝑑−𝐾_𝑐>𝐾^∗ through processing Matlab’s LMI solver such that 2𝑁^𝑇𝑃𝑀+𝑁^𝑇𝑅𝑁−𝑄^𝑇𝑅𝑄+𝜏𝑀^𝑇1𝑍𝑀−𝜏𝑇^𝑇𝑍𝑇<0. (41) 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 asymptotic convergence of the coordinated tracking error 𝑒̃ is not sufficient to prove the convergence to zero of both error 𝑒₁ and 𝑒₂. Our concern now 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 3.3. 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). (42) Rewriting all states of (42) into a compact representation and applying the Laplace transform leads to 𝑠𝜀(𝑠)−𝜀(0)=−Λ𝜀(𝑠)−𝑒^−𝜏𝑠𝐾_𝑐𝜀(𝑠). (43) This can be written as 𝜀(𝑠)=𝑠𝐼+Λ+𝑒^−𝜏𝑠𝐾_𝑐⁻¹𝜀(0). (44) 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 stable and that 𝑃(𝑠,𝜏) is a continuous function of 𝜏, then using the D-Decomposition, the minimal positive solution to the following equation: 𝜏_𝑐𝑖=arccos−Λ_𝑖/𝐾_𝑐𝑖𝐾_𝑐𝑖²−Λ²_𝑖 (45) would make all the zeros of the characteristic equation in the left half complex plane. Therefore if we select 𝜏∈[0,𝜏_𝑐], where 𝜏𝑐=sup(𝜏_𝑐𝑖) for all 1≤𝑖≤𝑛, solutions of (44) converge to zero and consequently 𝑞_𝑖→𝑞_𝑑 for 𝑡→∞, 𝑞̇_𝑖→𝑞̇_𝑑 for 𝑡→∞, and 𝑞_𝑖→𝑞_𝑗 for 𝑡→∞.

5. Simulation Results

To show the effectiveness of the proposed synchronizing controller we provide some simulation results. These simulations were proposed for a network of 3 identical robot manipulators interconnected under a cooperative scheme as shown in Figure 2.

Figure 2 

Robot network using bidirectional communication.

Let the communication structure among the robots described by an undirected strongly connected graph topology as shown in Figure 3. We set Joint Initial Conditions (JICs), coupling and control gains for the three robots as discussed below (see Table 1). Simulations are performed on Matlab/Simulink. Figure 4 illustrates the synchronization of robots that follow a common trajectory. This proves that the tracking and synchronization objectives are attained by the proposed controller.

Table 1 

Control gains.

Figure 3 

The topology model of the robots in simulation.

(a)

(b)

Figure 4 

Robots synchronization and trajectory tracking.

Figures 5 and 6, show, respectively, the convergence of error positions to zero and the convergence of synchronizing errors to zero, explaining how robots, while tracking the desired trajectory, synchronize their positions. The effect of time delays on the coordination of robots is shown in the following write-up. First, the delay-free case is presented in Figure 7 in which it is shown how the three angular positions asymptotically synchronize. Next, we consider the time delay in communication. Synchronization while tracking periodic trajectory is shown in Figures 8 and 9. From these figures, it is seen that the robots do not have the same starting positions. The speed for achieving an agreement depends essentially on the time-delay communication channels. Figures 10 and 11 illustrate that the behavior of the coordinated system changes significantly, under the effect of time delay.

Figure 5 

Position errors.

Figure 6 

Synchronization errors.

Figure 7 

Robots synchronization without communication delay.

Figure 8 

Robots synchronization in presence of time delay (0.5 s).

Figure 9 

Robots synchronization in presence of time delay (1 s).

Figure 10 

Robots synchronization without time delay.

Figure 11 

Robots synchronization in presence of time delay.

6. Conclusion

This paper has considered the synchronization problem in distributed multi-Lagrangian systems. The aim of this work was to find out a decentralized controller, which individually applied to each lagrangian system, the synchronization in position and velocity is therefore met. Reaching synchronization stability of highly nonlinear robot dynamics constitutes one of the main contributions of this paper. The proposed control law ensures the robots’ states synchronization while tracking a common desired trajectory. Another aspect of robots coordination and trajectory tracking control was investigated. In the coordination strategy there are practically interconnections between all the systems, such that all systems have influence on the overall dynamics. The proposed algorithm works under cooperative scheme in the sense that it does not require any explicit leaders in the team. The studied topology is connected under an undirected interaction graph. To deal with time-delay problem in communication between robots, the proposed decentralized control guarantees that the information variables of each robot reach agreement even in the presence of communication delay. Illustrative examples have shown the effectiveness of the described strategy. Future work will address the coordination control of under actuated lagrangian systems.