Nonlinear dimensionality reduction for kinematic cartography with an application toward robotic locomotion

Planning robot motions often requires a notion of the “distance” between configurations or the “length” of a trajectory connecting them in the configuration space. If these quantities are defined so as to correspond to the effort required to change configurations, then they would likely differ from the Euclidean distance or arclength in the system´s configuration parameters, distorting the visual representation of the relative costs of executing the motions. This problem is fundamentally similar to that of producing map projections with minimal distortion in cartography. A separate problem is that of nonlinear dimensionality reduction (NLDR), which, given a set of data, projects it into a lower-dimensional space while seeking to retain the geometric relationship between data points. In this paper, we show that NLDR can be applied to the kinematic cartography problem, allowing us to generate system parameterizations in which distance and arclength correspond to the effort of motion.