Friction Cone

The cone-shaped set of every force a contact point can apply on an object without slipping.

Under Coulomb's law of friction, a contact force's component tangential to the surface can't exceed its normal component (perpendicular to the surface) times the friction coefficient μ: √(f_x² + f_y²) ≤ μ·f_z, where f_z is the normal force and f_x, f_y are the two tangential components. Every force satisfying that inequality forms, in 3D, exactly a cone centered on the contact normal, with half-angle θ satisfying tanθ = μ. A force inside the cone won't slip; one outside it will. The friction cone is a basic constraint in contact computation: grasp analysis uses it to check force closure, model predictive control and whole-body control for legged robots require each foot's ground reaction force to stay inside its cone, and physics engines use it to constrain contact forces too. Because a cone is a nonlinear constraint, optimization commonly approximates it with a polyhedral cone (a friction pyramid) to get linear inequalities instead; MuJoCo offers both a pyramidal and an elliptical cone option.

ExampleA quadruped robot walking on ground with μ around 0.6 has its controller cap each foot's horizontal force at 0.6 times its vertical force. On slippery tile, μ drops and the cone narrows, so the robot has to shorten its stride and reduce the horizontal push-off force.

Related

Friction Coefficient (Coefficient of Friction) · Coulomb Friction · Friction Pyramid · Force Closure · Ground Reaction Force (GRF) · Contact Force Optimization (Force Distribution)

Sources

Robotic Manipulation (MIT, Russ Tedrake) - Bin Picking: friction cone
MuJoCo Documentation - Computation (pyramidal / elliptic cones)
Friction - Wikipedia