Constrained motion control on a hemispherical surface: path planning.

Surface-constrained motion, i.e., motion constraint by a rigid surface, is commonly found in daily activities. The current work investigates the choice of hand paths constrained to a concave hemispherical surface. To gain insight regarding paths and their relationship with task dynamics, we simulated various control policies. The simulations demonstrated that following a geodesic path (the shortest path between 2 points on a sphere) is advantageous not only in terms of path length but also in terms of motor planning and sensitivity to motor command errors. These stem from the fact that the applied forces lie in a single plane (that of the geodesic path). To test whether human subjects indeed follow the geodesic, and to see how such motion compares to other paths, we recorded movements in a virtual haptic-visual environment from 11 healthy subjects. The task comprised point-to-point motion between targets at two elevations (30° and 60°). Three typical choices of paths were observed from a frontal plane projection of the paths: circular arcs, straight lines, and arcs close to the geodesic path for each elevation. Based on the measured hand paths, we applied k-means blind separation to divide the subjects into three groups and compared performance indicators. The analysis confirmed that subjects who followed paths closest to the geodesic produced faster and smoother movements compared with the others. The "better" performance reflects the dynamical advantages of following the geodesic path and may also reflect invariant features of control policies used to produce such a surface-constrained motion.