Trajectory smoothing as a linear optimal control problem

In many areas of science and engineering there is a need for techniques to robustly extract velocity and its derivatives from a finite sample of observed positions. The extracted information can be used to infer related quantities such as curvature and speed, which are important for analysis of strategies and feedback laws associated with the motion. In this work a novel approach is proposed to reconstruct trajectories from a set of discrete observations. A simple linear model is used as the generative model for trajectories, and high values of the jerk (derivative of the acceleration) path integral are penalized during reconstruction. The positions, reconstructed in this way, can be represented as a linear combination of the sample data. The regularization (penalty) parameter plays a very important role in the reconstruction process, and it may be determined from data using ordinary cross validation.