  We extend Gaussian perturbation models in classical functional data analysis
to the three-dimensional rotational group where a zero-mean Gaussian process in
the Lie algebra under the Lie exponential spreads multiplicatively around a
central curve. As an estimator, we introduce point-wise extrinsic mean curves
which feature strong perturbation consistency, and which are asymptotically
a.s. unique and differentiable, if the model is so. Further, we consider the
group action of time warping and that of spatial isometries that are connected
to the identity. The latter can be asymptotically consistently estimated if
lifted to the unit quaternions. Introducing a generic loss for Lie groups, the
former can be estimated, and based on curve length, due to asymptotic
differentiability, we propose two-sample permutation tests involving various
combinations of the group actions. This methodology allows inference on gait
patterns due to the rotational motion of the lower leg with respect to the
upper leg. This was previously not possible because, among others, the usual
analysis of separate Euler angles is not independent of marker placement, even
if performed by trained specialists.
