A phase space spline smoother for fitting trajectories

This paper presents a phase space spline smoother, which is especially useful for finding a best-fit trajectory from multiple examples of a given physical motion. Unlike conventional spline smoothers, the phase space spline smoother can simultaneously fit position and velocity information. The use of velocity information is important for modeling the dynamic motion of physical systems because the state space of these systems typically includes both position and velocity variables. A detailed description of the computational procedure is presented, along with a discussion of computational expense and practical guidelines for variance estimation, preprocessing of the target dataset, smoothing of multidimensional datasets, and cross-validation for selection of smoothing weights. The smoother is demonstrated on a dataset of handwriting motions.