# Can I relate trajectory generation to differential inverse kinematics?

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!