Given a set of frictional contacts acting on a body, it is in force closure if the positive span of the wrench cones is the entire wrench space. Another way of saying this is that the contacts can theoretically resist any wrench applied to the body.

The test for this condition is essentially the same as for first-order form closure: First, we construct the matrix F, whose columns are the j friction cone edges of the contacts. Spatial friction cones are approximated by polyhedral friction cones with a finite number of edges. The F matrix has j columns, one for each friction cone edge, and either 3 rows or 6 rows, depending on whether the body is planar or spatial.

The contacts yield force closure if and only if the F matrix is full rank and there is a vector k of positive coefficients multiplying the friction cone edges such that F times k equals zero. This ensures that the positive span of the friction cone edges is the entire wrench space.

Note that our definition of force closure depends only on the contact locations, contact normals, and friction coefficients. A force-closure grasp does not mean that the contacts necessarily resist all wrenches. For example, the fingertip contacts of a hand might satisfy our definition of force closure, but the joints of the fingers might not be able to generate the squeezing forces necessary to create a contact wrench in an arbitrary direction.

If the friction coefficient at each contact is zero, then the friction cone is just along the contact normal, and frictionless force closure is therefore equivalent to first-order form closure.

If there is nonzero friction at the contacts, however, force closure is possible with as few as 2 contacts in the plane or 3 contacts in space.

As an example, this figure shows a triangular object grasped by two disks. The composite wrench cone due to the 2 frictional contacts is shown using moment labels. If this external wrench is applied, then the fingers would need to be able to create the opposing red wrench to prevent the triangle from moving. Since the line of action of the red wrench passes through the region labeled minus, it cannot be generated by the two frictional contacts. Therefore, the triangle would move because of this external wrench.

If we increase the friction coefficient and move the fingers, however, then there is no consistent moment label. This means that the frictional contacts can generate any wrench. As an example, imagine the triangle is subjected to the same external wrench.
Then the wrench shown in green has to be generated by the fingers to maintain static balance. This wrench can be obtained as a positive linear combination of one friction cone edge with a force inside the other friction cone. The parallelogram vector sum rule shows us that the two fingers have to squeeze very hard, however.

Two fingers are not enough for force closure of a spatial body. There is no way to resist moments about the axis between the two fingers. If the fingertip is soft, however, it can deform to create a contact patch with the body. The contact patch can provide frictional moment about the normal vector, and two soft fingers can create force closure. If the contacts are just points, however, at least 3 contacts are needed to satisfy force closure, as shown in the book.

When planning a grasp by a robot hand, force closure is a good minimum requirement. Form closure is usually too strict, requiring too many contacts. If you're machining a workpiece, however, and that workpiece will have significant forces applied to it, a form-closure fixture is a good idea.

In the final videos of this chapter, we'll apply what we've learned about contact kinematics and contact forces to solve manipulation problems which don't involve grasping.