If two bodies are in contact, then the contact constrains the possible twists of the bodies. For example, let's say that bodies A and B are in point contact, and that point can be expressed in a space frame as p_A or p_B. Even though they're at the same point in space, I've drawn them so we can see that one is considered to be attached to the body A and the other is considered to be attached to the body B.

The twist of body A is V_A and the twist of body B is V_B. These twists, and by default all twists in this chapter, are expressed in the space frame {s}. The twists of the two bodies result in velocities p-dot_A and p-dot_B of the current contact points on the two bodies. Each velocity is calculated as p-dot equals v plus omega-cross-p.

Now let's define the contact normal n as pointing into body A. By our first-order analysis, the rigid-body assumption says that the velocity of point A relative to point B, in the direction of the normal, must be greater than or equal to zero. In other words, the impenetrability constraint says that the dot product of the normal with p-dot_A minus p-dot_B must be greater than or equal to zero. If this quantity is greater than zero, the two bodies break contact. The impenetrability constraint is a single inequality constraint on the twists of the two bodies.

If n-transpose times p-dot_A minus p-dot_B is equal to zero, we call the contact a first-order roll-slide contact. This means that the contact is maintained by our first-order analysis. The roll-slide constraint is a single equality constraint.

If the stronger condition that p-dot_A equals p-dot_B is satisfied, we call this a first-order rolling contact. We could also call this a sticking contact, emphasizing that there is no sliding. The rolling condition places two equality constraints on planar twists and three equality constraints on spatial twists.

It will be convenient to express the impenetrability and roll-slide constraints directly in terms of twists. To do this, let's define a wrench F with a linear component given by the unit normal vector and a moment given by the vector to the contact crossed with the normal. We don't need wrenches for our kinematic analysis, but we use the wrench notation now because we will see it when we discuss contact forces.

With this notation and a simple derivation, the left-hand sides of the impenetrability and roll-slide constraints can be written as F-transpose times V_A minus V_B.

To further categorize a contact satisfying the impenetrability constraint, let's change this greater-than-or-equal-to sign to a strict greater-than sign. Then we define the contact label B, signifying a breaking contact; the contact label R, for a rolling contact; and the contact label S, for a sliding contact that does not satisfy the more restrictive rolling conditions.

These conditions tell us, to first-order, what happens at the contact if we're given the twists of the two bodies. Often we consider the feasible motions of just one body, such as body A, and assume that the other body is stationary. In this case, we simply set V_B equal to zero.

Let's look at an example where A is a planar hexagon and B is a stationary triangle. For ease of drawing, let's assume that the planar bodies cannot rotate, so the space of twists for A has only x and y linear components. Since B is stationary, the twist of A that satisfies the rolling condition is the zero twist. The contact normal can be expressed as a wrench F drawn in the twist space, and the twists that cause sliding are on the line orthogonal to F. Finally, the set of twists that break contact is the entire half-plane to the right of the S-line. Twists to the left of this line would cause penetration.

Now, if the body B is not stationary, then the twist that maintains rolling contact is not the zero twist. We can again plot the contact normal in the twist space, and the set of twists of A corresponding to sliding is the line orthogonal to F, as shown. The set of breaking twists is the half plane to the left of the sliding line.

In general, if A and B are spatial bodies, then their twist relative to each other, V_A minus V_B, is a 6-vector. To enforce the single constraint of a sliding contact, the relative twist must lie on a 5-dimensional hyperplane of the 6-dimensional relative twist space.

To enforce the 3 constraints of a rolling contact, the relative twist must lie on a 3-dimensional hyperplane of the 6-dimensional relative twist space.

If instead the bodies are restricted to a plane, the relative twist is a 3-vector, which we can draw in a 3-dimensional space. In this case, the plane of twists that cause a sliding contact, marked S, divides the 3-dimensional space into a half-space of twists that break contact, marked B, and a half-space of twists that cause penetration. The line of twists that cause rolling, marked R, lies in the S plane.

In the next video we consider the case where multiple contacts act on a body.