Title: On the mean and variance of planar displacements

Authors: Qiaode Jeffrey Ge; Zihan Yu; Mona Arbab; Mark P. Langer

Addresses: Department of Mechanical Engineering, Stony Brook University, Stony Brook, NY, USA ' Department of Mechanical Engineering, Stony Brook University, Stony Brook, NY, USA ' Department of Radiation Oncology, UT Southwestern Medical Center, Dallas, TX, USA ' Department of Radiation Oncology, Indiana University, Indianapolis, IN, USA

Abstract: This paper studies the problem of computing the mean (or the average displacement) and variance of a set of planar displacements. It presents three parametric representations of a planar displacement and studies how they influence the computations of means and variances. In particular, it advocates the use of the kinematic concept of relative displacements rather than simple difference in kinematic parameters in formulating the variance and for computing the mean of planar displacements. It is shown that the optimal average displacement is obtained by combining planar-quaternion representation with the concept of relative displacements. Examples are provided to illustrate the advantages of the latter approach.

Keywords: averaging displacements; planar quaternions; mean and variance; radiation therapy.

DOI: 10.1504/IJMRS.2023.137478

International Journal of Mechanisms and Robotic Systems, 2023 Vol.5 No.4, pp.308 - 325

Received: 06 Jul 2022
Accepted: 05 Jan 2023
Published online: 19 Mar 2024 *

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