We address the problem of developing feedback controllers for a group of robots with second-order dynamics in an obstacle-filled, D-dimensional environment. Our control algorithm takes into account communication constraints, obstacle avoidance, and inter-robot collision avoidance, by synthesizing a piecewise smooth vector field for safe navigation. First, the feasible free joint configuration space is tessellated into polytopes that account for the desired constraints. We search the graph of these polytopes to find a discrete path to the goal polytope. We then use a novel navigation function-based feedback controller that drives the system from one polytope to the next and eventually to the goal. The controller exploits the fact that two adjoining polytopes in the planned discrete path together form a star-shaped object that is obstacle free; this enables the design of navigation function-based controller for kinematic and dynamic fully actuated robots without spurious minima. We sequentially compose these controllers to drive the state to the goal. For a polygonal space, the algorithm we propose is complete. We present successful simulation results of the algorithm on a group of ground vehicles and quadrotors performing a cooperative navigation task in constrained environments.