In the previous video we learned that a contact between two rigid bodies divides the space of twists of one body relative to another into two halves: a half-space of feasible relative twists and a half-space of relative twists that violate the rigid-body assumption. Twists on the dividing hyperplane result in rolling or sliding contact.

In this video we study the case where a single rigid body is subject to multiple contacts.

This is a simple example from the previous video. The hexagon can translate in the plane, but not rotate, so its twist has only two linear components. The hexagon is in contact with the stationary triangle. The contact defines the line of twists S, which separates the half-space of twists that break contact from the half-space of twists that cause penetration. The zero twist implies no sliding, called a rolling contact.

If we place a different contact, it defines a different dividing line of sliding twists.

If both contacts are present, the two half-spaces of feasible twists intersect to create a cone of feasible twists. Twists along the top bounding ray are labeled SB, because these twists cause sliding along contact 1 and breaking at contact 2. Twists along the bottom bounding ray are labeled BS, because they cause breaking at contact 1 and sliding at contact 2. A zero twist is labeled RR, because the contacts are maintained and there is no sliding. Twists strictly inside the cone are labeled BB, as they cause breaking at both contacts. The concatenation of the labels for each contact is called the contact mode. The two contacts in this example allow four possible contact modes.

If each fixture contacting the body is stationary, then each contact constraint separating the penetrating and feasible twists passes through the origin of the body's twist space. The intersection of these feasible half-spaces creates a polyhedral convex cone. It is polyhedral, because faces of the cone are flat lines, planes, or hyperplanes, depending on the dimension of the twist space, and it is convex because the line between any two points in the cone is also in the cone.

Now, beginning with our two contacts, let's add a third contact. If we intersect the feasible twist half-spaces, we find that the only allowed twist is the zero twist. The object is immobilized, and all contacts have the label R.

If the third fixture is set into motion, then the contact's constraint surface does not pass through the origin. Instead it passes through the twist of the moving fixture, V_3. Intersecting the half-spaces for the three contacts, we get this triangular region of feasible twists for the hexagon. Any twist strictly inside the triangle results in breaking contact at all contacts. There are also six other possible contact modes, depending on the hexagon's twist. Since the sliding constraint surfaces do not all pass through the origin, the set of feasible twists is no longer a cone, but a more general polyhedral convex set.

The examples we just looked at were for the case of a planar body that can only translate, to make it easy to draw the feasible twist regions, but the same principles apply when the twist space is 3-dimensional for a general planar body or 6-dimensional for a general spatial body. In summary, if the body A is in contact with moving bodies, the set of feasible twists is the polyhedral convex set satisfying each of the half-space constraints.

If we assume that all external fixtures are stationary, then the set of feasible twists is a polyhedral convex cone.

This image shows an example of a polyhedral convex cone for three stationary fixtures acting on a planar body. Each contact defines a wrench and therefore a constraint plane in the three-dimensional twist space. Those planes form the faces of the cone. Twists strictly inside the cone cause breaking at all contacts. Twists on a face of the cone cause sliding or rolling at one of the contacts. Twists on an edge of the cone cause sliding or rolling at two of the contacts.

In the case that the only feasible twist is the zero twist, the body is immobilized. We call this form closure, the subject of a later video.

In the next video I'll introduce a representation of a planar twist called the center of rotation.