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