Mapping 2D Spatial Velocity Between Coordinate Frames


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.


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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.

Professor Peter Corke

Professor of Robotic Vision at QUT and Director of the Australian Centre for Robotic Vision (ACRV). Peter is also a Fellow of the IEEE, a senior Fellow of the Higher Education Academy, and on the editorial board of several robotics research journals.

Skill level

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.

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