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This paper suggests a nonlinear mixed effects model for data points in$\mathit{SO}(3)$, the set of $3\times3$ rotation matrices, collected accordingto a repeated measure design. Each sample individual contributes a sequence ofrotation matrices giving the relative orientations of the right foot withrespect to the right lower leg as its ankle moves. The random effects are thefive angles characterizing the orientation of the two rotation axes of asubject's right ankle. The fixed parameters are the average value of theseangles and their variances within the population. The algorithms to fitnonlinear mixed effects models presented in Pinheiro and Bates (2000) areadapted to the new directional model. The estimation of the random effects areof interest since they give predictions of the rotation axes of an individualankle. The performance of these algorithms is investigated in a Monte Carlostudy. The analysis of two data sets is presented. In the biomechanicalliterature, there is no consensus on an in vivo method to estimate the tworotation axes of the ankle. The new model is promising. The estimates obtainedfrom a sample of volunteers are shown to be in agreement with the clinicallyaccepted results of Inman (1976), obtained by manipulating cadavers. Therepeated measure directional model presented in this paper is developed for aparticular application. The approach is, however, general and might be appliedto other models provided that the random directional effects are clusteredaround their mean values.

A nonlinear mixed effects directional model for the estimation of the rotation axes of the human ankle