Statement and solution of a generalized Chebyshev problem. I

Abstract

The paper gives a review of the research performed at the department of theoretical and applied mechanics for many years and devoted to solving the generalized Chebyshev problems. By a generalized Chebyshev problem is meant the problem in which the solution of a system of motion equations should simultaneously satisfy an additional system of highorder n ≥ 3 differential equations. Therefore, a new class of control problems is introduced positions can be named as an example of such devices. For solving a generalized Chebyshev problem it is offered to apply two theories of motion of nonholonomic systems with high-order constraints developed at the department of theoretical and applied mechanics of Saint Petersburg State University. In this case an additional system of differential equations is considered as a set of high-order programming constraints, the reaction forces of which prove to be the required control forces solving a generalized Chebyshev problem. The first theory is based on constructing a compatible system of differential equations with respect to the unknown generalized coordinates and Lagrange multipliers, the second one uses a generalized Gauss principle. Application of the theories is demonstrated by solving the problem on motion of the Earth satellite (a spacecraft) with a fixed value of acceleration. In the next paper with the same title the second theory will be applied for solving one of the most important problems of the control theory, the problem considering a mechanical system being transferred from one phase state to another in a given time. into consideration. A generalized Chebyshev problem is considered as an extension of the Chebyshev problem from the theory of synthesis of mechanisms in which it is required to construct a device the certain links of which should perform a required motion to some accuracy. The well-known Chebyshev mechanisms with stoppings of certain links at the given positions can be named as an example of such devices. For solving a generalized Chebyshev problem it is offered to apply two theories of motion of nonholonomic systems with high-order constraints developed at the department of theoretical and applied mechanics of Saint Petersburg State University. In this case an additional system of differential equations is considered as a set of high-order programming constraints, the reaction forces of which prove to be the required control forces solving a generalized Chebyshev problem. The first theory is based on constructing a compatible system of differential equations with respect to the unknown generalized coordinates and Lagrange multipliers, the second one uses a generalized Gauss principle. Application of the theories is demonstrated by solving the problem on motion of the Earth satellite (a spacecraft) with a fixed value of acceleration. In the next paper with the same title the second theory will be applied for solving one of the most important problems of the control theory, the problem considering a mechanical system being transferred from one phase state to another in a given time.