A Directional Model for the Estimation of the Rotation Axes of the Ankle Joint

This article is motivated by the estimation of the directions of the two rotation axes of the ankle. These axes carry information on individual ankles; they are useful in the construction of biomechanical models and the treatment of orthopedic problems. In biomechanics, the rotation axes of the ankle often are estimated using optimization techniques. This work investigates a statistical model for carrying out the estimation. The data set for analysis is a time-ordered sequence of 3×3 rotation matrixes giving the ankle's orientations as the foot moves with respect to the lower leg. These rotation matrixes are assumed to follow Fisher–von Mises distributions. The predicted values for the observed rotations feature four angles for the orientation of two rotation axes and two series of time-varying rotation angles about the two axes. Maximum likelihood estimators of the parameters are derived. Approximations to their sampling distributions are obtained when the errors are clustered around the identity matrix. Sandwich variance estimators, accounting for autocorrelation in the errors, are proposed for the estimators of the two axes. An extension of the model that uses the translational motion in the estimation of the two axes is presented. Compared with the traditional biomechanical techniques for estimating the anatomic axes of the ankle, the proposed model has two advantages: It allows the calculation of standard errors for the estimates, and can distinguish the parameters whose estimation is affected by a limited range of motion about the two axes. This is illustrated by the estimation of the rotation axes of the ankles of two subjects.