Here we have something that should be quite familiar to us by now. We have a coordinate frame denoted A and we have a point. Now that point is moving. It’s got a spacial velocity, nu, with respect to frame A and I denote it this way. It’s got translational velocity component, X dot and Y dot and a rotational velocity component theta dot but I represent that by a single arrow.
Now I am going to introduce another coordinate frame, a red coordinated frame denoted B and we have a relative pose that describes the pose of frame B with respect to the pose of frame A. Now I can also describe the velocity of the point with respect to coordinate frame B and I do that by using a leading super script B instead of A.
Now, the velocity of the point with respect to frame B is related to the velocity with respect to frame A and it’s related by a Jacobian matrix, a Jacobian that maps velocity from one frame to another.
This particular Jacobian is a function of the relative pose between the two frames. It’s a three by three matrix that comprises the rotational part of the relative pose and a bunch of zeros and ones. In fact, it is only a function of the relative orientation between the frames. It doesn't depend at all on the distance between the origins of these two frames.
We can also derive a Jacobian which relates the velocity of a point, defined relative to one coordinate frame, to the velocity relative to a different coordinate frame.
This content assumes high school level mathematics and requires an understanding of undergraduate-level mathematics; for example, linear algebra - matrices, vectors, complex numbers, vector calculus and MATLAB programming.