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.