Band-Limited Trajectory Planning and Tracking for Certain Dynamically Stabilized Mobile Systems

In this paper, a general framework for trajectory planning and tracking is formulated for dynamically stabilized mobile systems, e.g., inverted wheeled pendulums and autonomous helicopters. Within this framework, the system state is divided into slow and fast substates. The fast substate is used as a pseudocontrol for tracking a desired slow substate trajectory. First, an exponential fast substate controller is designed to track a fast substate reference trajectory. This fast substate reference trajectory is, in turn, planned so that the slow substate follows its desired trajectory. To ensure that the fast substate reference trajectory is feasible for the exponential controller, it is designed using band-limited “Sinc” functions whose maximum frequency is less than the inverse of the time constant of the exponential controller. To illustrate the procedure, the dynamic model of an inverted wheeled pendulum is reformulated by a partial feedback linearization such that it is amenable to the separation into slow and fast components. The planning and tracking controller design is explained using simulation results. This technique is shown to be easily embedded inside a modified nonlinear model predictive control framework for the slow subsystem. This framework tries to explicitly take the computational delay into account. The computation time required for this technique is encouraging from a real-world implementation perspective.