A projected Lagrange-d´Alembert principle for forced nonsmooth mechanics and optimal control

This paper extends the projected Hamilton´s principle (PHP) formulation of nonsmooth mechanics to include systems with nonconservative forcing according to a projected Lagrange-d´Alembert principle (PLdAP). As seen with the conservative PHP, the PLdAP treats mechanical systems on the whole of their configuration space, captures nonsmooth behaviors using a projection mapping onto the system´s feasible space, and offers additional smoothness (relative to classical approaches) in the space of solution curves. Examining implications of the PLdAP for fully actuated optimal control problems, we prove that to identify optimal feasible trajectories it is sufficient to find unconstrained trajectories according to an alternate set of optimality conditions. Focusing on the control problem expressed in the unconstrained space, we approximate optimal solutions with a path planning method that dynamically adds and removes impacts during optimization. The method is demonstrated in determining an optimal policy for a forced particle subject to a nonlinear unilateral constraint.