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## Qiaode Jeffrey Ge: On the mean and variance of rigid-body displacements

：Qiaode Jeffrey Ge 教授

Many applications in biomechanics and medical imaging call 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 talk studies the problem of computing the average displacement from a set of given spatial displacements using three types of parametric representations: Euler angles and translation vectors, unit quaternions and translation vectors, and dual quaternions.

It has been shown that the use of Euclidean norm in the space of unit quaternions reduces the problem to  that of computing the average for each quaternion component separately and independently. While the resulting algorithm is simple, the change of the sign of a unit quaternion could lead to an incorrect result. A novel kinematic measure based on dual quaternions is introduced to capture the separation between two spatial displacement. This kinematic measure is then used to formulate a constrained least squares minimization problem.  It has been shown that the problem 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 least eigenvalue of a $4\times 4$ positive definite 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.

Qiaode Jeffrey Ge教授美国纽约州立大学石溪分校机械工程系主任，是ASME（美国机械工程师协会）Fellow。主要研究方向包括计算设计方法, 机构学/机器人学, CAD/CAM, 计算机图形学与VR技术的工程应用。一直在ASME设计工程部门执行委员会（DED）任职，并多次担任ASME机构学和机器人学委员会和ASME多个学术会议的主席和重要奖项的评委会成员。他也多次担任IFToMM世界大会（机械设计领域最顶级国际会议）的组织委员会的成员、首席美国代表和IFToMM美国成员组织主席。

Qiaode Jeffrey Ge教授目前是ASME的机械设计杂志（Journal of Mechanical Design）的机构学方向主编，并曾担任 ASME Journal of Mechanisms and Robotics、International Journal of Mechanics Based Design of Structures and Machines等机构学与机器人方向国际著名期刊的副主编。自2002年起，他成为中国机械工程学会的机构学委员会理事。他也是美国国家科学基金会的评审小组成员以及美国国家科研委员会、香港政府科研资助委员会、新加坡政府科研资助委员会和奥地利科学基金的评审人。