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Consensus Complementarity Control for Multi-Contact MPC

Alp Aydinoglu, Adam Wei, Wei-Cheng Huang, Michael Posa

TL;DR

Consensus Complementarity Control (C3) addresses the challenge of real-time multi-contact MPC by formulating local multi-contact dynamics as a linear complementarity system (LCS) and solving the resulting problem with an ADMM-based consensus approach. The framework decouples time steps through a consensus reformulation and provides multiple projection strategies (MIQP, LCP, ADMM, and a novel convex projection) to enforce the LCP constraints efficiently. By leveraging two local contact-dynamics formulations, Stewart–Trinkle and Anitescu, C3 enables mode-synthesis online without precomputed mode schedules, demonstrated across five numerical scenarios and hardware experiments. The results show substantial speedups over traditional MIQP-based methods while maintaining robust performance in high-dimensional, frictional contact tasks, highlighting C3’s potential for real-time dexterous manipulation and locomotion in contact-rich environments.

Abstract

We propose a hybrid model predictive control algorithm, consensus complementarity control (C3), for systems that make and break contact with their environment. Many state-of-the-art controllers for tasks which require initiating contact with the environment, such as locomotion and manipulation, require a priori mode schedules or are too computationally complex to run at real-time rates. We present a method based on the alternating direction method of multipliers (ADMM) that is capable of high-speed reasoning over potential contact events. Via a consensus formulation, our approach enables parallelization of the contact scheduling problem. We validate our results on five numerical examples, including four high-dimensional frictional contact problems, and a physical experimentation on an underactuated multi-contact system. We further demonstrate the effectiveness of our method on a physical experiment accomplishing a high-dimensional, multi-contact manipulation task with a robot arm.

Consensus Complementarity Control for Multi-Contact MPC

TL;DR

Consensus Complementarity Control (C3) addresses the challenge of real-time multi-contact MPC by formulating local multi-contact dynamics as a linear complementarity system (LCS) and solving the resulting problem with an ADMM-based consensus approach. The framework decouples time steps through a consensus reformulation and provides multiple projection strategies (MIQP, LCP, ADMM, and a novel convex projection) to enforce the LCP constraints efficiently. By leveraging two local contact-dynamics formulations, Stewart–Trinkle and Anitescu, C3 enables mode-synthesis online without precomputed mode schedules, demonstrated across five numerical scenarios and hardware experiments. The results show substantial speedups over traditional MIQP-based methods while maintaining robust performance in high-dimensional, frictional contact tasks, highlighting C3’s potential for real-time dexterous manipulation and locomotion in contact-rich environments.

Abstract

We propose a hybrid model predictive control algorithm, consensus complementarity control (C3), for systems that make and break contact with their environment. Many state-of-the-art controllers for tasks which require initiating contact with the environment, such as locomotion and manipulation, require a priori mode schedules or are too computationally complex to run at real-time rates. We present a method based on the alternating direction method of multipliers (ADMM) that is capable of high-speed reasoning over potential contact events. Via a consensus formulation, our approach enables parallelization of the contact scheduling problem. We validate our results on five numerical examples, including four high-dimensional frictional contact problems, and a physical experimentation on an underactuated multi-contact system. We further demonstrate the effectiveness of our method on a physical experiment accomplishing a high-dimensional, multi-contact manipulation task with a robot arm.
Paper Structure (34 sections, 1 theorem, 55 equations, 18 figures, 3 tables, 1 algorithm)

This paper contains 34 sections, 1 theorem, 55 equations, 18 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Background
  4. Linear Complementarity Problem
  5. Linear Complementarity System
  6. Multi-Contact Dynamics and Local Approximations
  7. Stewart-Trinkle Formulation
  8. Anitescu Formulation
  9. Model Predictive Control of Multi-Contact Systems
  10. Mixed Integer Formulation
  11. Consensus Complementarity Control (C3)
  12. Consensus Formulation
  13. ADMM Steps for the Consensus Formulation
  14. Quadratic Step
  15. Projection Step
  16. ...and 19 more sections

Key Result

Lemma 3

The complementarity error is bounded linearly with $\alpha$, i.e. $(\delta_k^\lambda)^T (E \delta_k^x + F \delta_k^\lambda + H \delta_k^u + c) \rightarrow 0$ as $\alpha \rightarrow 0$.

Figures (18)

  • Figure 1: Manipulating a rigid object (sphere) with a spherical end-effector attached to the Franka Emika Panda arm.
  • Figure 2: Lifting an object using two grippers indicated by red circles. The limits where the grippers should not cross are indicated by yellow lines.
  • Figure 3: Finger gaiting with $s = 10$. The upper blue shading implies that gripper 1 is applying normal force to the object whereas the lower red shading implies that gripper 2 is applying normal force.
  • Figure 4: Pivoting a rigid object with two fingers (blue). The object can make and break contact with the ground and gray areas represent the friction cones.
  • Figure 5: For the pivoting example, Gaussian process noise is added with with standard deviations $\sigma$. The upper blue shading implies that gripper 1 is applying normal force to the object whereas the lower red shading implies that gripper 2 is applying normal force.
  • ...and 13 more figures

Theorems & Definitions (3)

  • Definition 1
  • Definition 2
  • Lemma 3