On the Conic Complementarity of Planar Contacts

-- 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. The Signorini law for punctual contact is fundamental to contact modeling in robotics, mechanics, and computer graphics. It formalizes rigid, frictionless, point contact as a nonpenetration condition expressed via complementarity between the gap and the normal contact force [1]. For a given contact point between two objects in contact, this law states that if a force acts on the contact point, it should be repulsive, and the contact velocity can only separate the objects in contact; however, the two cannot occur simultaneously.