In the previous video we saw that a cone of planar twists can be represented as a region of centers of rotation. In this video, we'll learn a simple rotation center representation for the feasible twist cone of a planar body subject to multiple stationary contacts.

This figure shows a stationary triangle contacting a planar body. A center of rotation at the contact point, whether it is clockwise or counterclockwise, causes rolling at the contact. Therefore, we label this rotation center R.

Rotation centers to the left of the contact normal are feasible if they have a plus sign, for counterclockwise rotation. Feasible rotation centers to the right of the contact normal have a minus sign. All of these rotation centers are labeled B, for breaking contact. An example is this rotation center labeled minus. Rotation about this center causes breaking contact.

Finally, rotation centers on the contact normal are labeled S, for sliding. For planar problems, sliding contacts can be further classified as left-sliding, where the body slides left relative to the constraint, or right-sliding, where the body slides right relative to the constraint. With this distinction, we can refine the labels of rotation centers on the contact normal to be Sl, where the body slides left relative to the constraint, or Sr, where the body slides right relative to the constraint. For example, this positive rotation center causes the body to slide to the right relative to the triangular constraint, while this positive rotation center causes the body to slide to the left relative to the constraint.

Putting everything together, we get this picture of the twists that are feasible when there is a single contact. Feasible rotation centers to the left of the contact normal have a plus label and cause breaking contact. Feasible rotation centers to the right of the contact normal have a minus label and cause breaking contact. Rotation centers at the contact location cause rolling. Finally, rotation centers along the contact normal line, but not at the contact, cause sliding. Positive rotation centers above the contact and negative rotation centers below the contact cause right sliding, and negative rotation centers above the contact and positive rotation centers below the contact cause left sliding.

Now consider a body with three points of contact. Contacts 1 and 2 are with a table, and contact 3 is a robot finger. Contact 1 allows the twists shown here: plus or minus for rotation centers along the normal, plus for rotation centers to the left of the normal, and minus for rotation centers to the right of the normal. The rotation centers that satisfy contact 2 are intersected with those for contact 1, yielding this smaller set of rotation centers. Finally, the third contact reduces the set of feasible rotation centers even further. This set of rotation centers is a graphical representation of the feasible twist cone for the three contacts.

We could also write the contact mode for each group of rotation centers. The contact mode has 3 labels, one for each of the three contacts. Rotation about any of the rotation centers labeled BBB causes breaking at all three contacts.

Let's focus on one particular positive rotation center and illustrate it on the body. According to the contact mode, this rotation center causes slipping to the right at contacts 1 and 3 and breaking contact at contact 2. If we set the body in motion, though, we see that it immediately penetrates the finger. So in fact this rotation center is not possible because of contact 3. Our prediction was wrong because our first-order analysis considers only the contact normal, not the full details of the local contact geometry.

In general, if a first-order kinematic analysis concludes that a twist causes breaking or penetrating of a contact, then so will a higher-order analysis. But if a first-order analysis indicates rolling or sliding, a higher-order analysis may change the conclusion.

In the next video we conclude our purely kinematic analysis of contact by studying form closure, which occurs when the contacts completely immobilize the body.