A Spline Smoother in Phase Space for Trajectory Fitting
Many important problems in robotics and related fields can be addressed by fitting smooth trajectories to datasets containing several example trajectories. A smoother operating in phase space rather than configuration space, used alone or with the principal curves algorithm, can thus potentially build better trajectory models. Using velocity information in addition to configuration information enables us to build better trajectory models. The end of the trajectory has a similar problem, but it is less pronounced since the local variance of the sample points is smaller. The trajectory fitted using velocity information more closely tracks a single example trajectory. The formulation is based upon a smoothing spline because the polynomial form of the spline sections expresses the relationship between position and velocity in a straightforward manner. For deriving generalized cross-validation, conventional spline smoothers are generally formulated using the mathematics of reproducing kernel Hilbert spaces.