Consensus of Mobile Robots Under Markovian Communication

For mobile robots moving in a plane, the mean square consensus problem is investigated under Markovian communication of partly known transition probabilities. Based on linear matrix inequalities, bisection search and numerical optimization, a design method is presented of feedback gains guaranteeing mean square consensus.

For systems whose dynamics change in various possible scenarios, the Markovian jump system is an effective description. Recently, some interesting results with regard to Markovian jump systems have been presented on uncertain transition probabilities [10][11][12]. Noticing that the Markov chain can be used to describe the stochastic switching of communication among mobile robots, this paper addresses the mean square consensus problem of mobile robots under the Markovian communication of partly known transition probabilities. An sufficient condition of this problem is provided and the corresponding design algorithm is given.
In this paper,   is used to denote the set of all nonnegative integers. The n n  real identity matrix is denoted by n I . The Euclidean norm is denoted by  . If a matrix P is positive (negative) definite, it is denoted by P>0(<0). The Kronecker product is represented by  and the expected value is represented by E      .
The remainder of the paper is organized as follows. Section 2 describes a mobile robot system and its consensus problem. The condition and algorithm for the mean square consensus problem is derived in Section 3. Section 4 provides the numerical simulation results and Section 5 draws conclusions.

Mobile robots system and its consensus problem
Consider n mobile robots moving in a plane. At time represent position and velocity, respectively. Accordingly, the dynamics of the ith robot is modelled as The communication situation among these n robots is described by matrix where '?' represents the inaccessible entries. For The task of these n mobile robots is to go to a prescribed target point   * * , x y p p . Among the n robots, only the 1st robot knows   * * , x y p p . As the leading robot, the 1st robot adopts the following control law The other robots unaware of   * * , x y p p have to utilize a control law without * z The mobile robots system is said to reach mean square consensus if . This paper aims to design 1 k and k such that the n robots reach mean square consensus.

A Sufficient Condition of Mean Square Consensus
For the mobile robot system, from (2), (4), (5) and (6), we can easily see that its discrete time dynamics and control in x-direction are the same as that in the y-direction. Therefore an investigation in x-direction is enough. For From (2), (4) and (7), we can obtain From (2), (6) and (7), we can obtain Now, consider all the n robots together. Define A combination of (8), (9) and (10) Proof: Conditions (12) and (13) Since the y-direction model is the same as the x-direction model, the motion in y-direction is also convergent. Thus we conclude

Design Method
Given a pair   Step c) If    . An arrow from the ith to the jth one in Figure 1 Figure 3 and Figure 4 show the motion. It can be seen that these robots move to the target point under either 1  or 2  .

Conclusion
The consensus problem of a mobile robot system has been studied under a Markovian switching communication topology of partly known transition probabilities. The control input of each robot depends on the information exchange among robots. A stochastic Lyapunov function has been employed to investigate the mean square consensus of mobile robots, and the controller design