Collision-free formation control of multi-agent systems on Lie groups with input bias

This research investigates the formation control problem of multi-agent systems on Lie groups, subject to the challenges arising from collision avoidance and input bias. While current studies predominantly focus on flat Euclidean spaces, extending these traditional models to nonlinear Lie groups often encounters severe difficulties due to inherent complexities, such as the problem of multiple equilibria and challenging convergence behaviors. To address this, we propose an intrinsic collision-free formation control model that leverages a PID-like (Proportional-Integral-Derivative) controller and a potential term for collision avoidance, thereby ensuring robust formation control. Furthermore, we establish that formation on direct product Lie groups can only be achieved if it is successfully realized in each factor space. To validate the generality and backward compatibility of the proposed intrinsic framework, we evaluate our method on four representative Lie groups, ranging from the flat Euclidean space as a baseline to nonlinear manifolds including the unit circle, the infinite cylinder, and the special orthogonal group. Simulation results are also provided to verify the theoretical conclusions for all the considered Lie groups.