Motion Feasibility Conditions for Multi-Agent Control Systems on Lie Groups

We study the problem of motion feasibility for multiagent control systems on Lie groups with collision avoidance constraints. We first consider the problem for kinematic left invariant control systems and next, for dynamical control systems given by a left-trivialized Lagrangian function. Solutions of the kinematic problem give rise to linear combinations of the control inputs in a linear subspace annihilating the collision avoidance constraints. In the dynamical problem, motion feasibility conditions are obtained by using techniques from variational calculus on manifolds, given by a set of equations in a vector space, and Lagrange multipliers annihilating the constraint force that prevents deviation of solutions from a constraint submanifold.