On the Conic Complementarity of Planar Contacts

TL;DR

This work introduces the planar Signorini condition, a conic complementarity formulation that extends the punctual Signorini nonpenetration to planar contact patches. It defines a patch-dependent cone $K_P$ and its dual $K_P^*$ to capture repulsivity and nonpenetration in a frame-invariant way, and proves that enforcing Signorini at every point of a patch is equivalent to this planar CCP. The framework naturally describes three physical regimes—sticking, separating, and tilting—and yields an extended center of pressure, including a set-valued extension when the normal force vanishes. The approach promises computational benefits for surface contact simulation and integrates cleanly with CoP-based locomotion and manipulation control, while offering a principled path toward frictional extensions via Coulomb models.

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.

Figures (7)

• Figure 1: Illustration of a planar contact patch between two cubes. The contact surface is the intersection between the two faces, and the normal contact force distribution over this contact surface prevents the bodies from interpenetrating.
• Figure 2: Planar contact between two bodies. The planar contact patch of coordinate $P$ in $\mathbb{R}\xspace^2$ is supported by the plane $(O,e_x,e_y)$. The third axis of the frame $\mathcal{R}$ is the normal of the contact. For this figure and the following ones, we use an elliptical patch for illustration purposes. However, the results hold for any planar contact patch, even nonconvex ones.
• Figure 3: The cone $K_P$ is the set of all homogeneous coordinates of any point in $\mathcal{C}(P^\perp)$. In red we illustrate the $\mathcal{C}(P^\perp)\times \{1\}$ in $\mathbb{R}\xspace^3$ in order to visualize well the cone $K_P$ in blue. An example of $[\bm m_T, f_N]\neq 0$ in $K_P$ is depicted in orange, which allows us to identify the center of pressure (up to a rotation) $\bm c_p$ in yellow. $\bm 0$ is also an element of $K_P$ with no associated center of pressure.
• Figure 4: The dual $K_P^*$ is the set of all elements with a positive dot product with all the elements of $K_P$. It is the set of normals of oriented planes that leave $K_P$ on the positive side. This set is depicted in green. When $\bm{\omega}_T \neq \bm 0$, the plane intersects the contact plane ($z=1$) in a unique line, the zero normal velocity line (zero-line for short) $D_z$ (up to a $\pi/2$ rotation). In brown, we illustrate an element $[\bm{\omega}_T,v_N]$ in $K_P^*$ such that $\bm{\omega}_T \neq \bm 0$ with the associated zero-line. For any $v_N\geq 0$, $[\bm 0,v_N]$ is also an element of $K_P^*$ with no associated zero-line.
• Figure 5: The five different regimes in the contact patch problem. If $\bm{\omega}_T$ is not zero, there is a line $D_z$ in the plane where $\nu_N(x) = 0$, and we represent it in brown. If $[\bm{m}_T,f_N]$ is not zero, the center of pressure $\bm c_p$ is well-defined by $\bm c_p^\perp = \bm{m}_T/f_N$ and we represent it as a black dot, together with an arrow representing $f_N$. We observe the three regimes of the cone complementarity of Planar Signorini, either $[\bm{\omega}_T, v_N]$ is zero: the contact is sticking or sliding depending on the friction, $[\bm{m}_T,f_N]$ is zero: the contact is breaking; or the center of pressure $\bm c_p$ must lie on the zero-line $D_z$, the contact is tipping or slide-tipping depending on the friction.

Theorems & Definitions (3)

• Lemma 1: Repulsivity
• Lemma 2: Nonpenetration
• Proposition 1: Planar Signorini Condition