Wrench cones

Wrench cones are 6D polyhedral convex cones that characterize feasible contact wrenches, that is to say, wrenches that can be realized during contact-stable motions. They are a 6D generalization of Coulomb friction cones, and can also be used to encode other power limitations such as maximum joint torques.

Contact Wrench Cone

Illustration of the Contact Wrench Cone in

The contact wrench cone (CWC) is the frictional wrench cone of the (net) contact wrench acting on the robot in multi-contact, i.e. the sum of all contact wrenches. It can be computed analytically in simple cases, or using numerical polyhedral projection algorithms.

By construction, a solution can be found to the contact force optimization QP if and only if the centroidal momentum \((\pdd_G, \dot{\bfL}_G)\) lies in the CWC. When there is a large number of momenta to check, it is computationally more efficient to compute the CWC as a matrix \(\bfU\) first, and then perform each check as \(\bfU [\pdd_G\ \dot{\bfL}_G] \leq \boldsymbol{0}\) rather than solving a large number of quadratic programs.

On an advanced note: in the figure above, the CWC is represented by a red force cone and a green moment cone. This is a drawing convenience: in practice, the CWC is a 6D cone where force and moment are not independent. If you choose a resultant force in the red cone, it will affect the shape of the green one. And conversely, fixing the resultant moment to a given value (e.g. zero, see below) will affect the shape of the red cone.

Contact-stability areas and volumes

When additional constraints are imposed on the centroidal motion, the 6D centroidal wrench cone reduces to lower-dimensional areas and volumes that can be used for planning or control:

  • When the robot is not moving, contact stability is characterized by the static equilibrium polygon: the configuration of the robot is feasible (sustainable) if and only if the center of mass lies in a specific polygon, which can be computed efficiently.

  • When the robot moves in the Linear Inverted Pendulum Mode (LIPM, i.e. with conserved angular momentum and keeping the COM in a plane), the CWC reduces to a ZMP support area.

  • When the robot moves with a conserved angular momentum \((\Ld_G = 0)\), the CWC reduces to a 3D cone over COM accelerations that can be used e.g. for multi-contact locomotion.

To go further

Stability conditions are used extensively in motion generation, for instance in multi-contact walking pattern generation. A description of the computation of the CWC can be found in this paper. The algorithms themselves are implemented in:

  • The ContactSet class of the pymanoid library (Python).
  • WrenchConeLib: a C++ library with Python bindings.

Wrench cones can represent all sorts of power limitations, including but not restricted to friction. One example is given by the torque-limited wrench cones that we developed in a predictive controller for impact absorption.

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