On the Conic Complementarity of Planar Contacts
Yann de Mont-Marin, Louis Montaut, martial hebert, Jean Ponce, Justin Carpentier
AI summary
Problem
Robotics simulation relies on punctual Signorini conditions for point contacts, while locomotion control depends on the center of pressure constraint for planar patches, leaving a theoretical and computational gap for general planar contacts.
Approach
The authors derive a conic complementarity formulation that relates the integrated contact wrench to the relative twist, using dual cones defined by the geometry of the contact patch.
Key results
- Derivation of the planar Signorini condition as a conic complementarity problem
- Proof of equivalence between punctual Signorini conditions and the planar formulation
- Geometric interpretation extending the center of pressure to capture sticking, separating, and tilting regimes
- A unified, computationally tractable framework for planar contact modeling
Why it matters
Provides a mathematically consistent foundation for accurate contact simulation and advanced optimization-based control in robotics and locomotion.
Abstract
We present a unifying theoretical result that con- nects 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 inter- pretation 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.