On the global minimum time in the brachistochronic motion of Chaplygin sleigh

Abstract

This paper presents a procedure for determining the global minimum time in the brachistochronic motion of Chaplygin sleigh [3,4] between two specified positions, with unchanged value of mechanical energy during motion. For this case, the problem is formulated as the simplest problem of optimal control theory that is solved by applying Pontryagin’s Maximum Principle [1]. The corresponding two-point boundary value problem of the system of ordinary nonlinear differential equations is obtained that is necessary, in a general case, to solve numerically [2]. The numerical procedure is based on the shooting method, with the requirement for the assessment of the intervals in which the missing initial conditions can be found. The assessment is provided of the intervals of initial values of the conjugate variables, so that the TPBVP solution does not exist for sure outside those intervals. Graphic representation is given for corresponding surfaces in 3D space of the missing initial conditions, of which each surface corresponds to satisfying the missing conditions. A number of examples are provided for multiple solutions of the Maximum Principle, of which the global minimum is the one corresponding to the minimum time.