I am wondering how (if) one can translate trajectories of translation matrices to joint velocities?

I'm reading the notes of Russ Tedrake's class on manipulation. One part of the notes concerns a discussion of the joint velocities, calculates as function of the end-effector frame velocity: For a given end-effector frame velocity $V$ and jacobian of the joint positions $J(\theta)$, one can calculate the joint velocities as:

$$ d\theta = J(\theta)^{-1} V $$ (equation 18). Now suppose the end-effector follows the trajectory:

$$ X(s) = X_{start} \exp(\log(X_{start}^{-1} X_{End} ) s)) $$ The derivative of $X'(s)$ seem to be to me $X'(s) = X_{start} \exp(\log(X_{start}^{-1} X_{End} ) s)) X_{start}^{-1} X_{End}$. This is however a $4x4$ matrix and I do not understand how to use it to determine the joint velocities $d\theta$.

I do not understand how (if) one can use any information about the trajectory $X(s)$ to determine appropriate joint velocities $d\theta$ to follow that trajectory. There seems to be a relationship to me, but I cannot figure the mathematical formula to relate them. Does somebody else know? I am grateful for any hints or suggestions!


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