# Contact-Implicit Trajectory Optimization using an Analytically Solvable Contact Model for Locomotion on Variable Ground

arXiv: https://arxiv.org/abs/2007.11261 · HTML: https://mlsanity.com/paper/2007.11261/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Jul 2020.

## Built on by (most cited first, 3 of 3)

- [Robust and Versatile Bipedal Jumping Control through Reinforcement Learning](https://mlsanity.com/paper/2302.09450/) (2023)
- [Quadruped Locomotion on Non-Rigid Terrain using Reinforcement Learning](https://mlsanity.com/paper/2107.02955/) (2021)
- [Meta-Reinforcement Learning for Adaptive Motor Control in Changing Robot Dynamics and Environments](https://mlsanity.com/paper/2101.07599/) (2021)

## Abstract (arXiv)

This paper presents a novel contact-implicit trajectory optimization method using an analytically solvable contact model to enable planning of interactions with hard, soft, and slippery environments. Specifically, we propose a novel contact model that can be computed in closed-form, satisfies friction cone constraints and can be embedded into direct trajectory optimization frameworks without complementarity constraints. The closed-form solution decouples the computation of the contact forces from other actuation forces and this property is used to formulate a minimal direct optimization problem expressed with configuration variables only. Our simulation study demonstrates the advantages over the rigid contact model and a trajectory optimization approach based on complementarity constraints. The proposed model enables physics-based optimization for a wide range of interactions with hard, slippery, and soft grounds in a unified manner expressed by two parameters only. By computing trotting and jumping motions for a quadruped robot, the proposed optimization demonstrates the versatility for multi-contact motion planning on surfaces with different physical properties.