#### 12.2.4. Duality of Force and Motion Freedoms

This video describes the duality of force and motion freedoms in rigid-body contact with friction. A contact with a rigid body provides the same total number of equality constraints on force and motion regardless of the contact label (sliding, rolling, or breaking contact). Rolling contacts provide the most motion equality constraints and fewest force equality constraints, while breaking contacts provide the most force equality constraints (zero force) and the fewest motion equality constraints (no equality constraints on motion).

To analyze a manipulation task involving contacts, we combine our modeling of contact kinematics with the Coulomb friction model. At any instant, a contact could be breaking, sliding, or rolling, and each of these cases implies different constraints on the feasible motions and forces at the contact. Importantly, though, the total number of constraints on the motions and forces is the same, regardless of the contact label.

For simplicity let's assume first-order dynamics dealing with forces and velocities, but similar arguments apply when we consider second-order dynamics with forces and accelerations.

As an example, consider a single contact between a stationary finger and a rectangular object that moves in a plane. The friction cone is indicated in yellow. If we assume the contact is breaking, then there are no equality constraints on the velocity of the rectangle, and there are 2 equality constraints on the force applied by the finger, namely, that the force is zero.

Let's begin to construct a table of our observations. For a breaking contact B, the velocity at the contact point on the moving body can be any linear velocity in a 2-dimensional set. In other words, there are no equality constraints on the linear velocity of the point.

Now considering the forces at the contact, there are 2 equality constraints, namely that the force in the normal and tangential directions must be zero, so there are zero force freedoms.

If the contact is sliding, then the contact force is constrained to be somewhere on the edge of the friction cone resisting sliding. Referring again to the table, there is 1 constraint on the velocity, that the normal velocity at the contact is zero, and 1 freedom to choose the magnitude of the sliding velocity. Similarly, there is 1 constraint on the contact force, that the angle of the force must be on an edge of the friction cone, and 1 freedom in choosing the magnitude of the friction force.

If the contact is a rolling contact, then the instantaneous relative velocity at the contact is zero, and the contact force can be anywhere inside the friction cone. This cone is a 2-dimensional space of force vectors with bases at the contact point and tips somewhere inside the shaded region.

Referring again to our table, the zero relative velocity at the contact means 2 constraints and zero freedoms for the relative velocity. The contact force has zero equality constraints and 2 freedoms.

So the full table for planar contacts looks like this. Notice that when we solve for the forces and velocities of rigid bodies in contact, the total number of equality constraints on motion and force is 2 for each contact label.

If the contacts are in 3-dimensional space instead of a plane, each contact label provides 3 total constraints when we solve for the velocities and forces, and the full table looks like this. Breaking contacts provide the fewest constraints on velocity and the most constraints on forces, while rolling contacts provide the most constraints on velocity and the fewest constraints on forces.