The Friction Cone and Convex Contact Optimization

Contact forces come in two parts: a normal force $f_n \ge 0$ that pushes bodies apart, and a friction force $f_t$ that resists sliding. Coulomb's law caps the second by the first. That cap has a geometry, and that geometry decides how hard the contact problem is to solve.

Friction is a cone

Coulomb's law says the tangential force cannot exceed the normal force scaled by the friction coefficient,

$$\lVert f_t \rVert \le \mu\, f_n .$$

In 3D the admissible force $(f_t, f_n)$ traces out a circular second-order (or quadratic) cone: a normal axis, and at each height $f_n$ a disk of allowed tangential force with radius $\mu f_n$.¹ Normalize $f_n = 1$ and look straight down that axis: the limit is a circle of radius $\mu$. Inside the circle the contact sticks, and static friction supplies whatever force is needed. On the boundary it slides, and the force points against the motion at magnitude $\mu f_n$.

A check with numbers. Take $\mu = 0.5$ and $f_n = 10$, so the admissible set is

$$\lVert f_t \rVert \le \mu f_n = 0.5 \times 10 = 5 .$$

Two applied tangential forces:

$$f_t = (3, 4): \quad \lVert f_t \rVert = \sqrt{3^2 + 4^2} = \sqrt{25} = 5.0 ,$$

exactly on the boundary, the onset of sliding; and

$$f_t = (2, 1): \quad \lVert f_t \rVert = \sqrt{2^2 + 1^2} = \sqrt{5} \approx 2.24 \le 5 ,$$

well inside, so the contact sticks.

The friction cone, seen down the normal axisinteractive

tangential-force plane (drag f_t)

friction coefficient μ = 0.50facets N = 6

μ

0.50

|f_t|

0.39

true cone

inside

N-gon

inside

Inside the cone: contact sticks (static friction holds).

Why the cone is inconvenient

The circle is nonlinear, so a contact model built on it is not a linear complementarity problem (LCP), and the fast pivoting machinery from the previous chapter no longer applies directly. There are two ways out.

Linearize the cone. Replace the circle with a regular polygon of $N$ facets, a pyramidal cone.¹ Pick outward unit normals $d_k = (\cos\theta_k, \sin\theta_k)$ with $\theta_k = 2\pi k / N$, and require one linear inequality per facet,

$$d_k \cdot f_t \le \mu f_n , \qquad k = 1, \dots, N ,$$

which folds the contact problem back into an LCP. Keep $\mu = 0.5$, $f_n = 10$ (so $\mu f_n = 5$), and take the facet normal $d = (0.6, 0.8)$ aligned with $f_t = (3, 4)$:

$$d \cdot f_t = 0.6 \times 3 + 0.8 \times 4 = 1.8 + 3.2 = 5.0 = \mu f_n ,$$

exactly on that facet. Rotate the same $\lVert f_t \rVert = 5$ between two facet normals and it scores strictly below $5$, so those directions are handed extra slack: friction now depends on direction, the anisotropy every linearization carries. (The demo inscribes the polygon instead, vertices on the circle, which errs the other way and denies some legal forces; either way the gap closes as $N$ grows.)

Keep the cone, make it convex. The exact limit is nonlinear but still convex, the second-order cone

$$\mathcal{K} = \{\, (f_t, f_n) : \lVert f_t \rVert \le \mu f_n \,\} ,$$

so instead of forcing an LCP you can pose contact as a convex optimization over $\mathcal{K}$. MuJoCo relaxes the strict LCP complementarity to obtain a convex model that is fast and always solvable.² Drake's SAP solver goes further and eliminates the cone constraints analytically, turning the step into an unconstrained convex minimization.³

The trade

Linearization keeps the clean complementarity structure but bakes in a direction bias; convex cone formulations give up strict complementarity for a smooth problem with reliable convergence. That second bargain is why convex and cone-based contact dominate robotics-grade engines, where a step that always returns a sensible answer beats one that is exact but occasionally stalls.

How each of these is actually solved, from LCP pivoting to projected Gauss-Seidel to the convex programs above, is the subject of the solvers chapter.

Footnotes

1. Bender, Erleben, Trinkle & Coumans, Interactive Simulation of Rigid Body Dynamics in Computer Graphics (Computer Graphics Forum 2014). ↩ ↩²

2. MuJoCo docs, Computation. ↩

3. Castro, Permenter & Han, An Unconstrained Convex Formulation of Compliant Contact (2022). ↩