Grasp matrix

Combine contact actions into an object wrench#

Each fingertip or support contact applies a force and, for some contact models, a moment. The grasp matrix transforms the stacked contact quantities into one resultant wrench on the object. Its blocks depend on contact positions, contact frames, and which force or moment components the contact model allows.

The grasping chapter by Prattichizzo and Trinkle presents the grasp matrix together with the hand Jacobian as the two principal maps for grasp kinematics and force transmission. The grasp matrix acts on the object side, while the hand Jacobian relates joint motion and effort to contact motion and force.

Equilibrium, closure, and internal forces#

Static equilibrium requires the contact wrench combination to balance gravity and other external wrenches. Force closure asks whether admissible contact forces can oppose every wrench direction. The grasp matrix is part of that test, but the friction cones and unilateral-contact constraints determine which contact forces are admissible.

Different contact-force vectors can produce the same object wrench when the matrix has a null space. Their difference is an internal force: it changes squeezing or load sharing without changing the net wrench on the object. A controller can use that freedom to maintain friction margins or reduce excessive fingertip force.

The contact model controls the result#

A point contact without friction, a frictional point contact, and a soft-finger contact permit different wrench components and therefore different grasp matrices. A rank or closure conclusion is only valid for the stated model and coordinate convention.

The matrix also says nothing by itself about actuator limits, contact compliance, uncertain friction, or whether the fingers can reach the assumed contact locations. A mathematically adequate grasp can still slip, deform the object, or overload a joint. Real grasp evaluation must combine the map with physical limits and sensing.