# Contact-Implicit Model Predictive Control: Controlling Diverse Quadruped Motions Without Pre-Planned Contact Modes or Trajectories

arXiv: https://arxiv.org/abs/2312.08961 · HTML: https://mlsanity.com/paper/2312.08961/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Dec 2023.

## Built on by (most cited first, 8 of 8)

- [10 Open Challenges Steering the Future of Vision-Language-Action Models](https://mlsanity.com/paper/2511.05936/) (2025)
- [Towards Quadrupedal Jumping and Walking for Dynamic Locomotion using Reinforcement Learning](https://mlsanity.com/paper/2510.24584/) (2025)
- [MPC-Injection: Biasing Off-Policy Locomotion RL Toward Controller-Induced Behavior Basins](https://mlsanity.com/paper/2606.26392/) (2026)
- [Agile perceptive multi-skill locomotion for quadrupedal robots in the wild](https://mlsanity.com/paper/2607.13579/) (2026)
- [Learning Impact-Rich Rotational Maneuvers via Centroidal Velocity Rewards and Sim-to-Real Techniques: A One-Leg Hopper Flip Case Study](https://mlsanity.com/paper/2505.12222/) (2025)
- [GeCCo -- a Generalist Contact-Conditioned Policy for Loco-Manipulation Skills on Legged Robots](https://mlsanity.com/paper/2509.17582/) (2025)
- [RL-Augmented MPC for Non-Gaited Legged and Hybrid Locomotion](https://mlsanity.com/paper/2603.10878/) (2026)
- [Elastic ODYN: Differentiable Optimization for Infeasible Control and Learning in Robotics](https://mlsanity.com/paper/2606.16564/) (2026)

## Abstract (arXiv)

This paper presents a contact-implicit model predictive control (MPC) framework for the real-time discovery of multi-contact motions, without predefined contact mode sequences or foothold positions. This approach utilizes the contact-implicit differential dynamic programming (DDP) framework, merging the hard contact model with a linear complementarity constraint. We propose the analytical gradient of the contact impulse based on relaxed complementarity constraints to further the exploration of a variety of contact modes. By leveraging a hard contact model-based simulation and computation of search direction through a smooth gradient, our methodology identifies dynamically feasible state trajectories, control inputs, and contact forces while simultaneously unveiling new contact mode sequences. However, the broadened scope of contact modes does not always ensure real-world applicability. Recognizing this, we implemented differentiable cost terms to guide foot trajectories and make gait patterns. Furthermore, to address the challenge of unstable initial roll-outs in an MPC setting, we employ the multiple shooting variant of DDP. The efficacy of the proposed framework is validated through simulations and real-world demonstrations using a 45 kg HOUND quadruped robot, performing various tasks in simulation and showcasing actual experiments involving a forward trot and a front-leg rearing motion.