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What could be the unit ( dimension) of the vectors spanning the Jacobian matrix?

The units of the linear velocity portions of your Jacobian (what you have denoted as $\boldsymbol{s}_{i \parallel} \times \boldsymbol{l}_{i}$) are whatever distance unit you have used to define $\...
1 vote
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angle rates wrt another frame

It sounds like part of the problem here is what frame the RPY derivative, $\dot{\mathbf{x}}_{RPY}$, is in. Considering the fact that the RPY values, lets call them $\psi, \theta, \phi$ respectively, ...
1 vote
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How to control end-effector orientation with vector part of quaternion?

The error with the above code when compared to the axis-angle representation method is that this quaternion method returns its values still in a quaternion representation. In other words, we have: $$ \...

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