On the mean and variance of planar displacements

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.