Nonlinear Analysis of Directional Motion
Rasquinha, Brian J.
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Directional data can be represented as unit vectors. This representation defines a nonlinear geometry, in which directions can be considered as points on the unit sphere. A consistent analysis of directional data would measure distances, calculate means, and describe variations along the surface of the sphere, rather than deviating from the surface using more common Euclidean measures. Some nonlinear analysis methods developed for statistical shape models use a mapping from a nonlinear geometry to a linear tangent space, where familiar principal component analysis can be applied. In biomechanics, linear principal component analysis has been used to analyze directional motions encoded as high-dimensional points; however, this approach does not account for the spherical structure of these data. In this work, these concepts are combined in a novel motion analysis method, Nonlinear Analysis of Directional Motion. The method was applied to quasi-elliptical motions in a set of one-parameter simulations; it was also applied to wrist circumduction of healthy subjects. The small-circle model, which fits circles smaller than the diameter of a sphere, was used as the comparative standard. In simulation, the nonlinear method out-performed small-circle fitting using one component; this method also accurately captured the number of parameters of the data. Analyzing wrist circumduction, the method produced a five-parameter model, with lower fitting error than the small-circle model after two components. Nonlinear directional analysis also described differences between clockwise and counter-clockwise senses of circumduction in these healthy subjects. Nonlinear analysis of directional motion was demonstrated to provide an accurate model of circumduction with few parameters. This method may be useful for describing kinematic differences in any mechanism that has variable, multi-axial motion.