Trajectory Optimization of Contact-rich Motions using Implicit Differential Dynamic Programming

This paper presents a novel approach for generalising Differential Dynamic Programming (DDP) to systems characterised by implicit dynamics, such as those modelled via inverse dynamics, variational and implicit integrators, using sensitivity analysis. This leads to a more general formulation, enabling for example the use of the faster recursive Newton-Euler inverse dynamics. We leverage this formulation for the precise and exact contact modelling in DDP, where we focus on two contributions: 1) Contact dynamics in acceleration level that enables high-order integration schemes; 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. Two tasks of multi-contact motion planning are studied for a single leg model with multi-body contacts with the environment: standing up from ground, where a priori contact enumeration is challenging, and maintaining balance under an external perturbation.