On relationship between the control theory and nonholonomic mechanics

Abstract

The transition of a mechanical system from one state in which the generalized coordinates and velocities are given to another one in which the required coordinates and velocities are prescribed to the system is considered. It is assumed that this transition can be provided by a single control force. It is shown that if one determines the force with the help of the Pontryagin maximum principle (from the minimality condition of the time integral of the force squared during the time of motion) then a nonholonomic highorder constraint is realized in the defined process of motion of the system. Hence, the theory of motion of nonholonomic systems with high-order constraints can be applied for solving the same problem. According to the theory, in the set of different motions with a constraint of the same order the optimal motion is that one in the process of which a generalized Gauss principle is fulfilled. Thus, a control force chosen from the set of forces providing the transition of a mechanical system from one state to another during the given time can be defined both on the basis of the Pontryagin maximum principle and on the basis of a generalized Gauss principle. The paper focuses on the comparison of the results obtained by these two principles. The material is illustrated with the example of a horizontal motion of a trolley with pendulums to which a required force is applied. To obtain the control force without jumps in the beginning and in the end of motion, an extended boundary problem is formulated in which at these time moments not only the coordinates and velocities are given but also the derivatives of coordinates with respect to time up to the order n. This extended boundary problem can’t be solved with the help of the Pontryagin maximum principle as in this case the number of arbitrary constants is less than the total number of boundary conditions formulated. At the same time the problem can be solved with the help of the generalized Gauss principle, as in this case one should only increase the order of the principle up to the value which is consistent with the number of given boundary conditions. The results of numerical calculations are presented. Refs 6. Figs 5.

Keywords:

control theory, nonholonomic mechanics, high-order constraints, generalized Gauss principle, generalized principle of Hamilton-Ostrogradsky