Analysis of the minimum required coefficient of sliding friction at brachistochronic motion of a nonholonomic mechanical system

Abstract

The paper analyzes the problem of brachistochronic motion of a nonholonomic mechanical system, using an example of a simple car model. The system moves between two default positions at an unaltered value of the mechanical energy during motion. Differential equations of motion, containing the reaction of nonholonomic constraints and control forces, are obtained on the basis of general theorems of dynamics. Here, this is more appropriate than some other methods of analytical mechanics applied to nonholonomic systems, where the provision of a subsequent physical interpretation of the multipliers of constraints is required to solve this problem. By the appropriate choice of the parameters of state as simple a task of optimal control as possible is obtained in this case, which is solved by the application of the Pontryagin maximum principle. Numerical solution of the two-point boundary value problem is obtained by the method of shooting. Based on the thus acquired brachistochronic motion, the active control forces are determined as well as the reaction of constraints. Using the Coulomb laws of friction sliding, the minimum value of the coefficient of friction is determined to avoid car skidding at the points of contact with the ground.

Keywords

• brachistochrone
• nonholonomic mechanical system
• pontryagin's maximum principle
• coulomb friction
• optimal control