Generalized Chebyshev problem in nonholonomic mechanics and control theory

Abstract

The paper is based on the talk with the same name given at the International scientific conference on mechanics “The Ninth Polyakhov’s Reading” dedicated to the 200th anniversary of the birth of the prominent Russian scientist Pafnuty Lvovich Chebyshev. The generalized Chebyshev problem is formulated, in which the motion of a system in the presence of given generalized forces should satisfy an additional system of linear differential equations in which the order of each equation exceeds three. These problems constitute a new class of control problems in which the motion program is given in the form of the above additional system of differential equations. These equations can be considered as linear nonholonomic constraints of high order, whose reactions are the desired control forces. To solve such problems, two theories were developed at the Department of Theoretical and Applied Mechanics of St. Petersburg University. In the first theory, we construct a consistent system of differential equations for the generalized coordinates and the Lagrange multipliers, which are considered as equitable unknown functions of time. The second theory is based on the generalized Gauss principle. The application of the theory is illustrated by the solution of a real space mechanics problem about the motion of an Earth satellite after fixing the value of its acceleration at some point in time. Especially efficient is the application of the second theory to the determination of the optimal control force for transferring a mechanical system with a finite number of degrees of freedom from an existing phase state to a new specified state within a specified period of time. The new method is used to solve the model problem of controlled horizontal motion of a cart bearing the axes of several mathematical pendulums. It is shown that the use of the generalized Gauss principle for solving this problem is undoubtedly superior to that of the classical Pontryagin maximum principle.