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This paper presents a Differential Dynamic Programming (DDP) framework fortrajectory optimization (TO) of hybrid systems with state-based switching. Theproposed Hybrid Systems DDP (HS-DDP) approach is considered for application towhole-body motion planning with legged robots. Specifically, HS-DDPincorporates three algorithmic advances: an impact-aware DDP step addressingthe impact event in legged locomotion, an Augmented Lagrangian (AL) methoddealing with the switching constraint, and a Switching Time Optimization (STO)algorithm that optimizes switching times by leveraging the structure of DDP.Further, a Relaxed Barrier (ReB) method is used to manage inequalityconstraints and is integrated into HS-DDP for locomotion planning. Theperformance of the developed algorithms is benchmarked on a simulation model ofthe MIT Mini Cheetah executing a bounding gait. We demonstrate theeffectiveness of AL and ReB for handling switching constraints, frictionconstraints, and torque limits. By comparing to previous solutions, we showthat the STO algorithm achieves 2.3 times more reduction of total switchingtimes, demonstrating the efficiency of our method.

Hybrid Systems Differential Dynamic Programming for Whole-Body Motion Planning of Legged Robots