Variations of Augmented Lagrangian for Robotic Multi-Contact Simulation

TL;DR

This work tackles the challenge of accurate and efficient multi-contact simulation in robotics by extending the augmented Lagrangian (AL) framework to robotic contact problems. It introduces two practical solver variants: CANAL, a cascaded Newton-based AL method with exact line-search for robust, high-precision solutions, and SubADMM, a subsystem-based ADMM that exploits kinematic-tree structure for scalable, parallelizable computation. The authors derive closed-form slack-variable updates, prove surrogate problem feasibility, and demonstrate substantial performance gains on dense contact tasks (bolt-nut assembly, dish piling) and high-DOF robotic manipulation, achieving sub-millisecond per-step performance in some cases. Together, CANAL and SubADMM offer complementary strengths—CANAL for accuracy and robustness, SubADMM for speed and scalability—significantly advancing practical, model-based robotic simulation and manipulation planning under contact. The framework also opens avenues for integration with broader contact modeling and differentiable dynamics pipelines, enabling improved data generation and control for robotic systems with complex interactions.

Abstract

The multi-contact nonlinear complementarity problem (NCP) is a naturally arising challenge in robotic simulations. Achieving high performance in terms of both accuracy and efficiency remains a significant challenge, particularly in scenarios involving intensive contacts and stiff interactions. In this article, we introduce a new class of multi-contact NCP solvers based on the theory of the Augmented Lagrangian (AL). We detail how the standard derivation of AL in convex optimization can be adapted to handle multi-contact NCP through the iteration of surrogate problem solutions and the subsequent update of primal-dual variables. Specifically, we present two tailored variations of AL for robotic simulations: the Cascaded Newton-based Augmented Lagrangian (CANAL) and the Subsystem-based Alternating Direction Method of Multipliers (SubADMM). We demonstrate how CANAL can manage multi-contact NCP in an accurate and robust manner, while SubADMM offers superior computational speed, scalability, and parallelizability for high degrees-of-freedom multibody systems with numerous contacts. Our results showcase the effectiveness of the proposed solver framework, illustrating its advantages in various robotic manipulation scenarios.

Figures (13)

• Figure 1: Snapshots of a robotic simulation using our multi-contact solver. Top: bolt-nut assembly. Bottom: dish piling. Although intensive contact formation and stiff interactions make these scenarios challenging to simulate, our solvers successfully complete the simulations less than a $\rm{ms}$ of time budget per step.
• Figure 2: Three cases resulting from the Signorini-Coulomb condition, ranging from open ($\lambda_{i,n}=0$), stick ($\lambda_{i,n}>0, \delta_i=0$), to slip ($\lambda_{i,n}>0, \delta_i>0$), shown from left to right. The blue shape illustrates the friction cone, the green arrow indicates the contact frame velocity, and the yellow arrow represents the contact impulse.
• Figure 3: Comparison of the strict operator (left) and the proximal operator (right) for the friction cone projection. Black dot: operator input; red arrow: projection direction.
• Figure 4: Illustrative example demonstrating division and slack variable definition in SubADMM. Left: A multibody system with contacts comprising 4 subsystems, including 1 articulated body (robot) and 3 rigid bodies. Right: Corresponding graphical representation.
• Figure 5: Iteration structure of SubADMM. $\hat{v}$ is updated independently for each subsystem block, while $z$ is independently updated for each constraint factor.

Theorems & Definitions (7)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Remark 1