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.