  Trajectory optimization is becoming increasingly powerful in addressing
motion planning problems of underactuated robotic systems. Numerous prior
studies solve such a class of large non-convex optimal control problems in a
hierarchical fashion. However, numerical accuracy issues are prone to occur
when one uses a full-order model to track reference trajectories generated from
a reduced-order model. This study investigates an approach of Alternating
Direction Method of Multipliers (ADMM) and proposes a new splitting scheme for
legged locomotion problems. Rigid body dynamics constraints and other general
constraints such as box and cone constraints are decomposed to multiple
sub-problems in a principled manner. The resulting multi-block ADMM framework
enables us to leverage the efficiency of an unconstrained optimization
method--Differential Dynamical Programming--to iteratively solve the
optimizations using centroidal and whole-body models. Furthermore, we propose a
Stage-wise Accelerated ADMM with over-relaxation and varying-penalty schemes to
improve the overall convergence rate. We evaluate and validate the performance
of the proposed ADMM algorithm on a car-parking example and a bipedal
locomotion problem over rough terrains.
