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

Iordanis Chatzinikolaidis, Zhibin Li

TL;DR

The paper addresses the challenge of planning through-contact motions for robots by extending Differential Dynamic Programming to systems with implicit dynamics. It introduces an invertible contact model that enables exact contact resolution in the forward pass and a closed-form backward pass for differentiating through contacts, leveraging sensitivity analysis to accommodate implicit integrators. The authors formulate an acceleration-level contact scheme for high-order integration and validate the approach by comparing implicit versus explicit DDP on a double pendulum swing-up and by planning standing-up and balance tasks for a single-leg model with multi-body contacts in a receding-horizon framework. Results demonstrate improved numerical contact resolution and the feasibility of through-contact motions, highlighting the method's potential for scalable, multi-contact locomotion.

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.

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

TL;DR

The paper addresses the challenge of planning through-contact motions for robots by extending Differential Dynamic Programming to systems with implicit dynamics. It introduces an invertible contact model that enables exact contact resolution in the forward pass and a closed-form backward pass for differentiating through contacts, leveraging sensitivity analysis to accommodate implicit integrators. The authors formulate an acceleration-level contact scheme for high-order integration and validate the approach by comparing implicit versus explicit DDP on a double pendulum swing-up and by planning standing-up and balance tasks for a single-leg model with multi-body contacts in a receding-horizon framework. Results demonstrate improved numerical contact resolution and the feasibility of through-contact motions, highlighting the method's potential for scalable, multi-contact locomotion.

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.

Paper Structure

This paper contains 9 sections, 1 figure.

Table of Contents

  1. Introduction
  2. Machine Learning
  3. Model-based Mathematical Optimization
  4. Contributions
  5. Prior Work on Differential Dynamic Programming
  6. Differential Dynamic Programming
  7. Through-contact Motion Planning
  8. Preliminaries
  9. Summary of Differential Dynamic Programming

Figures (1)

  • Figure 1: Complex multi-contact motions of a single leg robot computed by the proposed framework in time-lapsed snapshots: dynamic standing up from the ground (top), and balancing against an external perturbation (bottom).