To the construction of optimal motions of controlled mechanical systems

Abstract

The solution of optimal control problems for mechanical systems is an important practical problem. For the solution of the optimization problem can be used the necessary conditions of the extremum in the form of the maximum principle of L.S. Pontryagin. However, the direct solution of the boundary value problem of maximum principle associated to large computational difficulties. This is due to the nonlinearity of the dynamic system of equations, the need for chose a reasonable first approximation for the conjugate variables at initial time moment, the need for a joint integrate of both the primary and the conjugate system with simultaneous selection of control function from the condition of maximum of the Hamiltonian. The latter circumstance often degrades (or breaks) the properties of continuous dependence of the residuals of the boundary value problem (usually, the values of the conjugate variables in a finite time moment) of variable parameters (typically, the values of the conjugate variables at initial time moment). The effective technology for the study of mechanical systems is developed in the article. The core technology is the integrated use of Direct Optimization Methods for dynamic systems (the method of successive linearization and its modifications); Methods of Solution of Boundary Value Problems (standard methods, based on the many times numerical solution of the system of algebraic equations that provide the required boundary conditions of the maximum principle); Qualitative Methods of study the structure of optimal control functions; Methods for constructing “exact” optimal control function, taking into account the features previously identified properties of the optimal control functions (methods of parametrization of the set of control function); Construction of Simple Techniques to calculate optimal motions of mechanical systems. The results of solution of the following tasks are presented: the problem of optimal control the maximum of angle of rotation of the excavator - dragline on a fixed time interval with finite damping of the oscillations occurring bifilar suspended from the boom of the bucket; the problem of optimal control for movement of foot of the walking machine when it step over through the obstacle.