Convex Optimization of Bearing Formation Control of Rigid bodies on Lie Group

TL;DR

This paper introduces a convex optimization approach for bearing-based formation control of rigid bodies on Lie groups, enabling decentralized coordination without a global reference frame.

Contribution

It develops a convex relaxation method for rotation matrices on Lie groups to achieve bearing formation control with minimal energy, a novel approach in this context.

Findings

Convex relaxation of rotation matrices enables unique global solutions.

The control law effectively achieves desired formations with minimal energy.

Simulation verifies the theoretical results and control effectiveness.

Abstract

In this paper, the problem of reaching formation for a network of rigid agents over a special orthogonal group is investigated by considering bearing-only constraints as the desired formation. Each agent is able to gather the measurements with respect to other agents in its own body frame. So, the agents are coordinated-free concerning a global reference frame. Attracting to the desired formation is founded on solving an optimization problem for minimizing the difference between the instantaneous bearing between agents and their desired bearing. In order to have a unique global solution, the convex optimization method is implemented. Since the rotation matrices are not convex, the method of convex relaxation of rotation matrices space is used to embed the rotation matrices on the convex hull of the Lie group. Then the control law is designed to achieve the desired bearing with minimum energy consumption. Finally, a simulation example is provided to verify the results.