I'm trying to find the Jacobian for an inverse kinematics system, using derivatives of the end effector's angles with respect to the system's joint angles. My system actually has two subsystems, one where I calculate, successfully, the Jacobian in the standard way, ie, by finding the partial derivatives of the end-effector w.r.t. the joint angles, and this second subsystem where I am instead interested in using the orientation, not the position, of the end-effector. I'm using the method outlined in this paper, math.ucsd.edu/~sbuss/ResearchWeb/ikmethods/iksurvey.pdf, for the first subsystem (page 5). Each entry in the Jacobian is given by (delta)s_i / (delta)Theta_j = v_j X (s_i - p_j). Theta_j is the angle of the jth joint. p_j is the position of the joint, and v_j is a unit vector pointing along the current axis of rotation for the joint. s__i is a vector denoting the position of the end effector. So, what I would like to do is to find a Jacobian based on the end-effector's orientation instaed of position, ie with entries (delta)a_i / (delta)Theta_j, where a_i is a rotation of the end-effector. My question is specifically, what is the expression that yields each entry in such a Jacobian? Currently I am using the same expression as in the previous paragraph, but this results in some very strange behavior.