Transpose the matrix A and stores the adjoint of the elements in the. 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. The fields c and s represent the cosine and sine of the rotation angle, respectively. In this case it means the matrix representation of a twist.
In one case it means the matrix representation of an angular velocity. Notice that we are overloading the bracket notation. 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. Little se(3) gets its name from its relationship with big SE(3). The twist can be represented in any arbitrary frame for example, the twist could be represented as V_a in frame, 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. In the last video, we learned that rigid-body velocities can be represented as a 6-vector twist.
