Time-optimal control of rolling bodies

Abstract

The brachistochrone problem is usually solved in classical mechanics courses using the calculus of variations, although it is quintessentially an optimal control problem. In this paper, we address the classical brachistochrone problem and two vehicle-relevant generalisations from an optimal control perspective. We use optimal control arguments to derive closed-form solutions for both the optimal trajectory and the minimum achievable transit time for these generalisations. We then study optimal control problems involving a steerable disc rolling between prescribed points on the interior surface of a hemisphere. The effects of boundary and control constraints are examined. For three-dimensional problems of this type, which involve rolling bodies and nonholonomic constraints, numerical solutions are used.

Keywords:

• optimal control
• mechanics
• Hamiltonian
• rolling bodies
• brachistochrone

Acknowledgements

This work was supported by the UK Engineering and Physical Sciences Research Council. The authors would also like to thank Matthew Arthington for several helpful discussions.

David Limebeer would like to dedicate this paper to Alistair MacFarlane, FRS, who introduced him to many of the important ideas in optimal control and classical mechanics. Alistair MacFarlane was, throughout his career, interested in the interplay between mechanics, optimal control and the action principle.

Notes

The approximate geodesic is the arc of a great circle (on the surface of the sphere) that passes through the intersections of the spherical surface and the lines joining the centre of the sphere and the terminal (start and end) positions of the mass centre of the coin.