  In this paper, we present an efficient Dynamic Programing framework for
optimal planning and control of legged robots. First we formulate this problem
as an optimal control problem for switched systems. Then we propose a
multi--level optimization approach to find the optimal switching times and the
optimal continuous control inputs. Through this scheme, the decomposed
optimization can potentially be done more efficiently than the combined
approach. Finally, we present a continuous-time constrained LQR algorithm which
simultaneously optimizes the feedforward and feedback controller with $O(n)$
time-complexity. In order to validate our approach, we show the performance of
our framework on a quadrupedal robot. We choose the Center of Mass dynamics and
the full kinematic formulation as the switched system model where the switching
times as well as the contact forces and the joint velocities are optimized for
different locomotion tasks such as gap crossing, walking and trotting.
