The Mountain Car Problem with a Dynamical and Finite Energy System

One of the most well known benchmark problems in autonomous robot locomotion and navigation control is the Mountain Car Problem (MCP), in which a point mass vehicle is to be taken from the bottom of a sinusoidal valley to its top. This article proposes a generalization of the MCP with energy constraints on locomotion: Moving the vehicle consumes energy from a dynamical storage of finite capacity, as described by Keller’s theory of competitive running. We address both minimum time and minimum energy consumption optimal control for this generalized MCP. We demonstrate in numerical simulations that, for both of these control problems, the optimal solutions employ, perhaps surprisingly, vehicle operation at maximum forces. It is the aggressiveness (rate of change) of the force profiles that is different between them. The key future applications of this study are in battery electric vehicle design and autonomous systems.