  In this paper we study the problem of designing periodic orbits for a special
class of hybrid systems, namely mechanical systems with underactuated
continuous dynamics and impulse events. We approach the problem by means of
optimal control. Specifically, we design an optimal control based strategy that
combines trajectory optimization, dynamics embedding, optimal control
relaxation and root finding techniques. The proposed strategy allows us to
design, in a numerically stable manner, trajectories that optimize a desired
cost and satisfy boundary state constraints consistent with a periodic orbit.
To show the effectiveness of the proposed strategy, we perform numerical
computations on a compass biped model with torso.
