Velcocity of robots in 3D and Jacobian matrix

We know  the inverse of the rotation matrix is the transpose of it.

Now we can get the vector omega since we know about the structure of skew symmetric matrix.

Combine the translation part and rotation part.

... for all joints.

Now this is a Jacobian matrix.

Each column means the effect on the spatial velocity of the robot due to each joint velocity.

* ref

[1] https://moocs.qut.edu.au/courses/814/learn