Friction Pyramid

Approximates the round friction cone with a polyhedral pyramid, turning the friction constraint into linear inequalities.

Under Coulomb friction, the tangential force at a contact point can't exceed the friction coefficient μ times the normal force, and all the allowed contact forces form a cone in 3D — the friction cone. Because a cone is a nonlinear constraint, it's often approximated with a polyhedral pyramid instead, which is the friction pyramid: the simplest 4-sided pyramid is spanned by the four edges (μ,0,1), (−μ,0,1), (0,μ,1), and (0,−μ,1), and more edges get closer to the true cone. An inscribed pyramid underestimates the available friction, while a circumscribed one overestimates it; grasp analysis usually takes the conservative, inscribed option. The payoff is that the friction constraint becomes a set of linear inequalities, so contact problems can be solved with linear programming, quadratic programming, or a linear complementarity problem. The |fx| ≤ μfz, |fy| ≤ μfz constraints common in convex MPC for quadrupeds are a circumscribed 4-sided pyramid; MuJoCo also offers two global settings, pyramidal and elliptic, and the pyramidal option turns the solver's dual problem into a convex quadratic program with simple box constraints.

ExampleWith μ = 0.5 and a normal force fz = 100 N, the true friction cone allows at most 50 N of tangential force in any direction; with the circumscribed 4-sided pyramid constraint |fx| ≤ 50, |fy| ≤ 50, a solution with fx = fy = 50 N is also allowed, giving a combined force of about 70.7 N — beyond the true limit — which is why a conservative design often substitutes μ/√2 for μ.

Also called

Linearized Friction Cone, Polyhedral Friction Cone, Pyramidal Friction Cone

Related

Friction Cone · Coulomb Friction · Friction Coefficient (Coefficient of Friction) · Convex MPC · Linear Complementarity Problem · MuJoCo (Multi-Joint dynamics with Contact)

Sources

Lynch & Park, Modern Robotics（预印本 PDF，12.2.1 节 Figure 12.18） (Chinese)
MuJoCo Documentation: Computation（Friction cones）