Locomotion and Perception as Communication over Principal Fiber Bundles

This chapter can be viewed as a demonstration of applications of the Lie-theoretic methods presented in Chapters 10–12, the inequalities in Chapter 19, and the stochastic processes on Lie groups in Chapter 20. As in Chapter 1, the simple system used to illustrate these concepts is the nonholonomic kinematic cart. When any trajectory of the cart is discretized into smaller segments which are drawn from a set of intended maneuvers, then this set serves as an alphabet of basic moves. As the cart moves and noise is added to these intended moves, it will not move exactly as planned. This corruption of the resulting output position and orientation can be viewed as an injection of noise through the combined space of pose and wheel angles. This space is an example of the differential geometric structure called a

principal fiber bundle

.1 An external observer (which might be a human or another robot) watching the motion of the robot can then attempt to infer the robot’s intent and functionality. The combination of stochastic models, information theory, and Lie groups is helpful in studying such scenarios.