On the estimation of an average rigid body motion

This paper investigates the definition and the estimation of the Fréchet mean of a random rigid body motion in ℝ-super-p. The sample space SE(p) contains objects M=(R,t) where R is a p×p rotation matrix and t is a p×1 translation vector. This work is motivated by applications in biomechanics where the posture of a joint at a given time is expressed as M∈SE(3), the rigid body displacement needed to map a system of axes on one segment of the joint to a similar system on the other segment. This posture can also be reported as M-super- - 1=(R-super-T, - R-super-Tt) by interchanging the role of the two segments. Several definitions of a Fréchet mean for a random motion are proposed using weighted least squares distances. A special emphasis is given to a Fréchet mean that is equivariant with respect to the inverse transform; this means that if P is the Fréchet mean for M then P-super- - 1 is the Fréchet mean for M-super- - 1, where M is a random SE(p) object. The sampling properties of moment estimators of the Fréchet means are studied in a large concentration setting, where the scatter of the random Ms around their mean value P is small, and as the sample size goes to ∞. Some simple exponential family models for SE(p) data that generalize Downs' (1972) Fisher--von Mises matrix distribution for rotation matrices are introduced and the least squares mean values for these distributions are calculated. Asymptotic comparisons between the estimators presented in this work are carried out for a particular model when p=2. A numerical example involving the motion of the ankle is presented to illustrate the methodology. Copyright 2012, Oxford University Press.