Session: MR-02-02 - Theoretical & Computational Kinematics (A.T. Yang Symposium)

Paper Number: 90156

90156 - On the Computation of the Average of Spatial Displacements 

Many applications in biomechanics and medical imaging calls for the analysis of the kinematic errors in a group of  patients statistically using the average displacement and the standard deviations from the average.
This paper studies the problem of computing the average displacement from a set of given spatial displacements using four representations: Euler angles and translation vectors, unit quaternions and translation vectors, and dual quaternions. 
Two types of distance measures are studied in the spaces of quaternions and dual quaternions, one is based on the Euclidean norm and the other is a kinematic measure based on the relative displacement. It has been shown that the use of Euclidean norm leads to simple algorithms that treat each kinematic parameter separately and independently. The novel kinematic measure captures the amount of relative displacements from the average displacement to all other given displacements. The  average displacement as a constrained least squares minimization problem.  Through a dual quaternion formulation, it is found that the problem decouples and decomposes into that of finding the optimal translation vector and the optimal  unit quaternion. The former is simply the centroid of the set of given translation vectors and the latter  can be obtained as  the eigenvector corresponding to the smallest eigenvalue of a $4\times 4$  symmetric matrix. It is found that the weight factor used in combining rotations and translations in the formulation does not play a role in the final outcome.
Examples are provided to show the comparisons of these methods.

Presenting Author: Qiaode Jeffrey Ge Stony Brook University

Presenting Author Biography: Jeff Ge received his Ph.D. in Mechanical Engineering in 1990 from the University of California, Irvine. He is currently Professor and Chair, Department of Mechanical Engineering, Stony Brook University, SUNY.

Authors:

Qiaode Jeffrey Ge Stony Brook University
Zihan Yu Stony Brook University
Mona Arbab Indiana University
Mark Langer Indiana University