This is the equation of motion for a single rigid body subject to frictional contacts. The right-hand side is the rigid-body dynamics we derived in Chapter 8. F_contact is the total wrench from all of the contacts, and F_ext is the wrench due to gravity or other forces.

The procedure to analyze a rigid-body mechanics problem with friction is as follows. Given the state of the body and manipulator and manipulator motions or forces, first enumerate the potential contact modes that could hold at this instant, and for each contact mode, determine whether there is a contact wrench and body motion, consistent with Coulomb's law and the contact mode, satisfying the equation of motion.

If the acceleration and velocity-product terms are negligible, we can replace the right-hand side with a zero, meaning that there is always force balance between the external wrench and the contact wrench. The assumption of force balance is called the quasistatic assumption.

This procedure may sound strange. We don't simply specify the states and controls and solve some equations for the change of state. Instead, we have to test different contact modes for a possible solution. It is not hard to show that this procedure sometimes tells us more than one contact mode is possible. Conversely, for some problems there may be no solution at all. This is one of the prices we pay to use the rigid-body and Coulomb friction assumptions. Fortunately, for many realistic problems there is a single consistent solution.

Let's return to the meter-stick trick from the beginning of this chapter. We balance the meter stick on two fingers, with one finger close to the center of mass. If we move this finger slowly toward the other, the stick doesn't fall; instead, it slides to keep its center of mass between the fingers.

Let's use the procedure I just described to prove this. I'll assume that motion is slow so the quasistatic assumption is satisfied.

Here's an image of the stick balanced on two fingers, with the friction cones of the two fingers illustrated. To balance gravity, the fingers must create a contact force mg upward through the center of mass. To check if the fingers can create this force when they are stationary, we assign the contact label R to each finger and use moment-labeling to find a graphical representation of the composite wrench cone from the two contacts. Because the upward wrench mg creates negative moment about all points labeled minus and positive moment about all points labeled plus, it can be generated by the two fingers. Therefore, the stick can stay at rest on the stationary fingers.

In general, each of the two contacts could be breaking, sliding left, sliding right, or rolling, for a total of 16 possible contact modes between the fingers and the stick. Some of these contact modes are not possible kinematically. For example, the contact mode RR is not possible; there is no way for the stick to remain stationary relative to both fingers as the fingers move toward each other.

For the other contact modes, which are not ruled out solely because of kinematics, we have to undertake a quasistatic force analysis to see if they are possible solutions.

As a first example, let's consider the contact mode where both contacts break, the contact mode called BB. Because the contacts are breaking, no forces can be applied, so the entire plane gets the moment label plus-minus. The required force mg cannot be generated by the two contacts, so we reach the obvious conclusion that the stick does not simply float away.

Now assume that there is no sliding at the left finger but the right finger slides. Then the left finger can apply any force in its friction cone, while the right finger can only apply forces on the left edge of its cone, as dictated by Coulomb's law. The wrenches that can be generated by the contacts are indicated by the moment labels. Since the upward force mg passes through the region labeled minus, the fingers cannot balance gravity, and this contact mode is not possible.

If both fingers slide, as shown here, each contact force lies on the inner edge of its friction cone. Again, the moment labels show that the fingers cannot quasistatically balance gravity.

Finally, if the left finger slides but the right finger does not, the moment labels show us that the fingers can generate a wrench to balance gravity. Therefore, this contact mode satisfies quasistatic balance and is a feasible solution. Technically, we should check that no other modes are possible and that this is a unique solution.

So this explains why the center of mass always stays between the two fingers. Our analysis would show that once the center of mass is centered between the two fingers, then both contacts slide at equal speed until both fingers are directly under the center of mass. In practice, you can see slipping starting and stopping at each finger; this can be explained by a static friction coefficient that is larger than the kinetic friction coefficient.