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

TL;DR

This paper introduces a generalized implicit DDP framework for contact-rich motion planning, enabling precise contact modeling and high-order integration schemes, validated through robotic swing-up and balancing tasks.

Contribution

It extends DDP to implicit dynamics, allowing for accurate contact modeling and the use of advanced inverse dynamics methods.

Findings

Implicit DDP outperforms explicit DDP in swing-up tasks.

The framework effectively plans contact-rich motions without predefined contact sequences.

Validated on robotic tasks involving multi-body contacts and balance.

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.