Contact wrench cone

From distributed pressure to one resultant wrench#

A foot sole touches the ground over an area, so the underlying model may contain many possible contact forces. Their combined effect on the robot is a six-dimensional wrench: three force components and three moment components. The contact wrench cone collects the resultants that can be produced while contact forces remain compressive and obey the chosen friction model.

This differs from a friction cone, which constrains a force at a point. The contact wrench cone also represents moments created by forces distributed across a finite support area.

What a rectangular foot must satisfy#

For a rigid rectangular surface under Coulomb friction, Caron, Pham, and Nakamura derive closed-form conditions for the contact wrench. The conditions combine a friction bound on the resultant force, a zero-moment point inside the support area, and bounds on yaw torque. The yaw condition matters because a foot may twist even when the force and zero-moment-point tests pass.

A controller or planner can test candidate whole-body forces against this set instead of retaining a separate force variable at every sampled point on the sole. In multi-contact motion, cones or their polyhedral approximations can be combined to reason about hands, feet, and other supports.

Environment feasibility is not actuator feasibility#

The contact wrench cone describes what the modelled environment contact can transmit. It does not by itself prove that the robot's motors can generate the required wrench. Orsolino and colleagues distinguish the contact wrench cone from an actuation wrench polytope and intersect the two to form a feasible wrench polytope.

The result also inherits its modelling assumptions. Friction coefficients are uncertain, real soles deform, terrain is rarely perfectly planar, and contacts can start to slip or lift. Controllers commonly add margins, but a larger margin is not a substitute for validating the contact model on the robot and surface in question.