Minimum Energy-Time Optimal Control of Wheeled Mobile Robots: Application to Parallel Parking

Optimal controller design for a kinematic model of a wheeled mobile robot (WMR) to minimize input energy and maneuver time in parallel parking maneuvers in an obstacle-free environment is the focus of this paper. A multi-objective optimal control problem is formulated using a weighted combination of maneuver time and energy. The necessary conditions for optimality are derived using the Hamiltonian of the system, resulting in a two point boundary value problem. The analytical expression for the states, costates, and controls are derived using the Jacobi Elliptic functions, which transforms the optimal control problem to a nonlinear optimization problem. It is noted that the controls required for the first half of the maneuvers can be used to derive the second half with sign changes. Although the optimal control problem is formulated without imposing constraints on control inputs, it turns out that the inputs are bounded to lie on a circle, the radius of which is a function of the weighting factor. The unique observations are analytically proven and illustrated experimentally.