Motor Compositionality and Timing: Combined Geometrical and Optimization Approaches

Human movements are characterized by their invariant spatiotemporal features. The kinematic features and internal movement timing were accounted for by the mixture of geometries model using a combination of Euclidean, affine and equi-affine geometries. Each geometry defines a unique parametrization along a given curve and the net tangential velocity arises from a weighted summation of the logarithms of the geometric velocities. The model was also extended to deal with geometrical singularities forcing unique constraints on the allowed geometric mixture. Human movements were shown to optimize different costs. Specifically, hand trajectories were found to maximize motion smoothness by minimizing jerk. The minimum jerk model successfully accounted for a range of human end-effector motions including unconstrained and path-constrained trajectories. The two modeling approaches involving motion optimality and the geometries’ mixture model are here further combined to form a joint model whereby specific compositions of geometries can be selected to generate an optimal behavior. The optimization serves to define the timing along a path. Additionally, new notions regarding the nature of movement primitives used for the construction of complex movements naturally arise from the consideration of the two modelling approaches. In particular, we suggest that motion primitives may consist of affine orbits; trajectories arising from the group of full-affine transformations. Affine orbits define the movement’s shape. Particular mixtures of geometries achieve the smoothest possible motions, defining timing along each orbit. Finally, affine orbits can be extracted from measured human paths, enabling movement segmentation and an affine-invariant representation of hand trajectories.