Distributed Time-Varying Formation and Optimization with Inequality Constraints of a Multi-Robot System

In this paper, we consider a distributed time-varying formation and optimization problem for a group of robots with uncertain Euler-Lagrange dynamics. The robots are required to keep a time-varying formation as well as optimizing a quadratic objective function that is composed of the state information of all the robots with linear inequality constraints. The well known formation tracking problem can be viewed as a special case of the proposed optimization based framework, provided that the objective function is properly selected and each robot has access to the formation tracking leader's trajectory. The problem is mathematically reformulated as a distributed time-varying optimization problem, where the time-varying formation task is viewed as a time-varying equality constraint. A quadratic penalty function is used to deal with the equality constraint and a third-order differentiable penalty function is introduced to deal with the inequality constraint. A numerical example is provided to show the effectiveness and efficiency of the proposed method.