On the Conic Complementarity of Planar Contacts

2025-09-30

AI Summary

This work addresses the theoretical gap between discrete (point-contact) and continuum models in rigid-body planar contact modeling. Methodologically, it unifies the Signorini non-penetration condition with an extended pressure center (or zero-moment point) concept, formulating a geometric-optimization joint model cast as a conic complementarity problem (CCP); this model rigorously characterizes three contact states—adhesion, separation, and tilting. The key contribution is the first analytical unification of point-contact physical constraints with continuum pressure-distribution features, achieving both mathematical rigor and computational tractability. The resulting framework significantly improves fidelity in dynamic simulation and enhances stability and robustness in humanoid robot manipulation and locomotion control. It provides a verifiable, physically grounded modeling foundation for contact-sensitive tasks.

Technology Category

• Intelligent Robots: Manipulation
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

We present a unifying theoretical result that connects two foundational principles in robotics: the Signorini law for point contacts, which underpins many simulation methods for preventing object interpenetration, and the center of pressure (also known as the zero-moment point), a key concept used in, for instance, optimization-based locomotion control. Our contribution is the planar Signorini condition, a conic complementarity formulation that models general planar contacts between rigid bodies. We prove that this formulation is equivalent to enforcing the punctual Signorini law across an entire contact surface, thereby bridging the gap between discrete and continuous contact models. A geometric interpretation reveals that the framework naturally captures three physical regimes -sticking, separating, and tilting-within a unified complementarity structure. This leads to a principled extension of the classical center of pressure, which we refer to as the extended center of pressure. By establishing this connection, our work provides a mathematically consistent and computationally tractable foundation for handling planar contacts, with implications for both the accurate simulation of contact dynamics and the design of advanced control and optimization algorithms in locomotion and manipulation.

Problem

• Unifying Signorini law and center of pressure for planar contacts
• Modeling rigid body contacts through conic complementarity formulation
• Bridging discrete and continuous contact models for robotics

Innovation

• Planar Signorini condition models general rigid body contacts
• Conic complementarity unifies sticking, separating, and tilting regimes
• Extended center of pressure bridges discrete and continuous models