Trajectory optimization for contact-rich motions using implicit differential dynamic programming

Abstract

This paper presents a novel approach using sensitivity analysis for generalizing Differential Dynamic Programming (DDP) to systems characterized by implicit dynamics, such as those modelled via inverse dynamics and variational or implicit integrators. It leads to a more general formulation of DDP, enabling for example 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 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 (1) by comparing implicit versus explicit DDP for the swing-up of a double pendulum, and (2) 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.

Implicit Differential Dynamic Programming

“…All these methods require the dynamic equations to be perfectly solved, following a bilevel programming strategy. While explicit dynamics can be efficiently evaluated with high numerical accuracy [25], implicit dynamics typically require several steps of the Newton-Raphson algorithm in order to accurately compute x t+1 , as done in [26], which in turn may slow down the whole solving process. Implicit transcription.…”