In the last video, we learned that rigid-body
velocities can be represented as a 6-vector
twist.
The twist can be represented in any arbitrary
frame; for example, the twist could be represented
as V_a in frame {a} or as V_b in frame {b}.
If we want to change the frame of representation
of a twist, it is tempting to try a subscript
cancellation rule, V_a equals T_ab times V_b
but this doesn't work due to dimension mismatch:
transformation matrices are 4 by 4 but twists
are 6-vectors.
It is apparent that we need to premultiply
V-b by a 6 by 6 matrix.
The 6 by 6 matrix we need is called the adjoint
representation of a transformation matrix,
and it is defined as you see here.
Now we can apply a modified version of our
subscript cancellation rule to change the
frame of representation of a twist.
By analogy to the matrix representation of
angular velocity, we would like to find a
matrix representation for twists.
Recall that, for angular velocities, we had
the 3 by 3 skew-symmetric matrix representations
of angular velocities bracket omega_b equals
R-inverse times R-dot and bracket omega_s
equals R-dot times R-inverse
Similarly, if T represents the body frame
{b} in the space frame {s}, we have 4 by 4
matrix representations of the twists bracket
V_b equals T inverse times T-dot and bracket
V_s equals T-dot times T-inverse where little
se(3) is the space of 4 by 4 matrix representations
of twists.
Little se(3) gets its name from its relationship
with big SE(3).
The top left 3 by 3 submatrix is the skew-symmetric
matrix representation of the angular velocity,
as we've seen before, and the top right 3
by 1 vector is the linear velocity of a point
at the origin of the frame, expressed in that
frame.
The bottom row is 4 zeros.
Notice that we are overloading the bracket
notation.
In one case it means the matrix representation
of an angular velocity.
In this case it means the matrix representation
of a twist.
These matrix representations will be used
in the next video when we develop the matrix
exponential and log for rigid-body motions,
analogous to the matrix exponential and log
for rotations that we've already seen.
