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.
