Grasp Matrix

The linear map that combines each contact point's force into the total force and torque acting on the grasped object.

The grasp matrix G (called the grasp map in the textbook by Murray, Li, and Sastry) describes the relationship between the forces fingers apply at their contact points and the resultant wrench on the object: F_o = G·f_c. Here f_c is the stacked vector of all contact forces, and F_o is the 6-dimensional wrench (3D force plus 3D torque) acting on the object. Each contact point occupies a block within G, determined by the contact's position, its surface normal, and its contact type (frictionless point contact, point contact with friction, or soft-finger contact). In the other direction, Gᵀ maps the object's velocity to the velocity of each contact point. The null space of G corresponds to internal forces: forces where several fingers squeeze against each other but contribute zero net force to the object — this is how fingers grip an object firmly without pushing it around. Force closure can be stated as: under the friction-cone constraints, G can generate a wrench in any direction; combined with the hand's Jacobian J_h (via τ = J_hᵀ f_c), the required joint torques can be worked backward from the force the object needs.

ExampleTwo fingers squeeze a box horizontally from opposite sides with 10 N each: the two contact forces sum to zero net force through G, so this pair of forces lies in G's null space and is a pure internal force. To actually hold the box up without dropping it, an upward friction-force component, within the friction cone, also has to be added so that G·f_c cancels gravity.

Also called

G, Grasp Map

Related

Force Closure · Form Closure · Wrench · Friction Cone · Contact Jacobian · Grasp Quality Metric

Sources

Murray, Li & Sastry, A Mathematical Introduction to Robotic Manipulation（第 5 章 The grasp map） (Chinese)