Motion Feasibility Conditions for Multi-Agent Control Systems on Lie Groups
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We study motion feasibility conditions of decentralized multi-agent control systems on Lie groups with collision avoidance constraints, modeled by an undirected graph. We first consider agents modeled by a kinematic left invariant control systems (single integrator) and next as dynamical control systems (double integrator) determined by a left-trivialized Lagrangian function. In the kinematic approach, we study the problem of determining whether there are nontrivial trajectories of all agent kinematics that maintain the collision avoidance constraints. Solutions of the problem give rise to linear combinations of the control inputs in a linear subspace annihilating the constraints. In the dynamical problem, first order necessary conditions for the existence of feasible motions are obtained using techniques from variational calculus on manifolds and by introducing collision avoidance constraints among agents into an augmented action functional by using the Lagrange multipliers theorem.
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