Iterative Estimation of Rotation and Translation using the Quaternion

In this paper, we consider some practical issues when using the quaternion to represent rotation in conjunction with gradientor Jacobian-based search algorithms. Iterative estimation techniques often incorporate gradient or Jacobian information to simultaneously solve for the parameters with respect to many non-linear constraints. We derive a simple form for the Jacobian of the rotation matrix with respect to its quaternion parameters and then use this form to predict some practical numerical problems of using gradient and Jacobian information for iterative search. These problems can be eliminated by some straightforward steps which we describe. This research has been supported in part by the Advanced Research Projects Agency under the Department of the Army, Army Research O ce grant number DAAH04-94-G-0006, and in part by the O ce of Naval Research grant number N00014-93-1-1220. Views and conclusions contained in this document are those of the authors and should not be interpreted as necessarily representing o cial policies or endorsements, either expressed or implied, of the Department of the Army, the Department of the Navy or the United State Government.