As we learned in the last video, a transformation matrix T can be used to represent the configuration of the body frame {b} relative to the space frame {s}. Now we need to represent the velocity of the body frame. Just as the time derivative of a rotation matrix was not our representation of angular velocity, the time derivative of a transformation matrix is not our representation of a rigid-body velocity.

Let's just jump right to our representation, without deriving it. You can see the details of the derivation in the book. It turns out that any rigid-body velocity, which consists of a linear component and an angular component, is equivalent to the instantaneous velocity about some screw axis. The screw axis is defined by a point q on the axis; a unit vector s in the direction of the axis; and the pitch h of the screw, which is the ratio of the linear speed along the axis to the angular speed about the axis. For now we will assume that the pitch h is finite; later we will return to the case where the pitch is infinite.

Given any linear and angular velocity of a body, there is a corresponding screw axis. It's as if the body's instantaneous motion is twisting about the screw axis. The screw axis defines the direction the body is moving, and theta-dot is a scalar indicating how fast the body rotates about the screw.

Our representation of a screw axis is not a point q, a unit vector s, and a pitch h, however. Instead, we choose a reference frame, and we define the screw axis S as a 6-vector in that frame's coordinates, consisting of S-omega, the 3-dimensional unit angular velocity when the rotational speed theta-dot is 1, and S_v, the 3-dimensional linear velocity of the origin of the frame when the rotational speed is 1. The linear velocity of the origin, as you see in the figure, is a combination of two terms: h times s, which is the linear velocity due to translation along the screw axis if there is a nonzero pitch, and -s cross q, which is the linear velocity due to rotation about the screw axis.

Multiplying our representation of the screw axis S by the scalar rate of rotation theta-dot, we get the twist, a full representation of angular and linear velocity.

Let's look at a simple example, where the screw axis is a zero pitch screw, a pure rotation like a turntable. The axis is pointing toward you, out of your screen.

This is an animation of a turntable moved by the screw axis. We start rotating about the screw at a rate of theta-dot = 1. Defining a reference frame as shown, we see that the angular velocity S-omega is 1 about the z-axis, which is also out of the screen. Since the reference frame is 2 units from the screw axis, the linear velocity at the frame origin is 2 units in the minus y direction, so we get S_v equal to (0,-2,0).

We can choose a reference frame at a different location. In this frame, the angular velocity is the same as before, but S_v is different.

Finally, if we choose a reference frame on the screw axis itself, S_v is zero. Because the frame has a different orientation from before, the angular velocity is now 1 unit in the minus y direction.

We have been focusing on the case where the screw axis has finite pitch, but there are two cases to consider: the pitch is infinite, or the pitch is finite. If the pitch is infinite, the motion is a pure linear motion with no rotation. In this case, S-omega is zero, S_v is a unit vector, and theta-dot indicates the linear speed. If the pitch is finite, S-omega is a unit vector and theta-dot is the rotational speed in radians per second.

If the screw axis S is expressed in coordinates of the body frame {b}, then S-theta-dot is called the body twist V_b. If the screw axis S is expressed in coordinates of the space frame {s}, then S-theta-dot is called the spatial twist V_s.

In summary, a twist is a 6-vector consisting of a 3-vector expressing the angular velocity and a 3-vector expressing the linear velocity. Both of these are written in coordinates of the same frame, and the linear velocity refers to the linear velocity of a point at the origin of that frame. Both the body twist and the spatial twist represent the same motion, just in different coordinate frames. The body twist is not affected by the choice of the space frame, and the spatial twist is not affected by the choice of the body frame.

In the next video we discuss a matrix representation of twists, which will be used in the matrix exponential for rigid-body motion.