Trajectory Optimization of Contact-Rich Motions Using Implicit Differential Dynamic Programming

This work presents a Differential Dynamic Programming (DDP) approach for systems characterized by implicit dynamics using sensitivity analysis, such as those modelled via inverse dynamics, variational, and implicit integrators. It leads to a more general formulation of DDP, enabling the use of the faster recursive Newton-Euler inverse dynamics. We leverage the implicit formulation for precise and exact contact modelling in DDP, where we focus on two contributions: (1) contact dynamics at the acceleration level; (2) formulation using an invertible contact model in the forward pass and a closed-form solution in the backward pass to improve the numerical resolution of contacts. The performance of the proposed framework is validated by comparing implicit versus explicit DDP for the swing-up of a double pendulum, and by planning motions for two tasks using a single leg model making multi-body contacts with the environment: standing up from ground, where a priori contact enumeration is challenging, and maintaining balance under an external perturbation.