12.1.6. Planar Graphical Methods (Part 1 of 2)

This video demonstrates how a polyhedral convex cone of planar twists (velocities of a rigid body restricted to the plane) can be represented as a set of centers of rotation (CoRs).

Any planar twist can be visualized as a rotation about an axis out of the plane. This twist can be represented by a point (x_c, y_c) and the angular velocity omega_z about the point. We define the center of rotation to be the point (x_c, y_c) plus a label that gives the sign of the angular velocity.

The motion could also be expressed as a planar twist V in the fixed space frame. Given this twist, we can calculate the point (x_c, y_c), and given (x_c, y_c) and omega_z, we can calculate the twist.

The center of rotation is a convenient graphical representation of a planar twist when we only need to know the sign of the angular velocity.

Let's visualize the mapping from twists to centers of rotation.

This is the three-dimensional space of twists of a planar rigid body and the sphere of unit twists. We draw planes equipped with coordinate frames at omega_z equals 1 and omega_z equals -1. The top plane is the plane of rotation centers with a plus sign label, for a positive angular velocity. The bottom plane is at omega_z equals -1, and it is the plane of rotation centers with a minus sign label, for a negative angular velocity. A twist V with a positive angular component can be intersected with the positive plane, scaling V by a positive coefficient if need be. The plane a twist intersects with determines the plus or minus label associated with the twist, and the location of the intersection determines (x_c,y_c). If the twist has no angular component, then it can be thought of as a rotation center at infinity, as we'll see in a moment.

Now consider three unit twists, written a, b, and c, and their mapping to three rotation centers. The rotation centers a and b have positive labels, while c has a negative label.

As we saw in the previous video, the set of feasible twists of a body in contact with stationary constraints is a polyhedral convex cone, and it will be convenient to be able to represent such cones using rotation centers. As an example, let's construct a polyhedral convex twist cone as the positive span of the three unit twists, a, b, and c. The intersection of that twist cone with the unit sphere is indicated. Using the center of rotation mapping, the twist cone can be represented as this region of rotation centers. This hatched region is properly interpreted as a single convex region, just like the twist cone. It is connected by rotation centers at infinity, which correspond to pure translational motion without rotation.

To see this, let's move a twist from twist b, as indicated by the green dot on the unit sphere, to twist c. The path of the twist is shown in green. The corresponding rotation center also moves from b, through infinity, to c along the path shown here. This green segment, passing through infinity, is the positive span of the rotation center b with a plus label and the rotation center c with a minus label.

In other words, the positive span of two rotation centers of opposite signs consists of a ray of positive rotation centers, a ray of negative rotation centers, and a point at infinity corresponding to a pure translation. All the rotation centers are on the same line. The positive span of two rotation centers of the same sign is the line segment between the two centers, with the same sign.

We can generalize these examples to find the positive span of three rotation centers labeled plus, or the positive span of two rotation centers labeled plus and one rotation center labeled minus.

In the next video we'll use the center of rotation representation to analyze planar contact kinematics.