Mikhail P. Yushkov

On the relationship between control theory and nonholonomic mechanics

This paper concerns transitions of a mechanical system, in a given time, from one state, in which the generalized coordinates and velocities are given, into another one, in which the system should have the required coordinates and velocities. It is assumed that such a transition may be effected using a single control force. It is shown that if one determines the force using the Pontryagin maximum principle (from the minimality condition of the time integral of the force squared during the time of motion), then a nonholonomic high-order constraint occurs in the defined process of motion of the system. As a result, this problem can be attacked by the theory of motion of nonholonomic systems with high-order constraints. According to this theory, in the set of different motions with a constraint of the same order, the optimal motion is the one for which the generalized Gauss principle is fulfilled. Thus, a control force that is chosen from the set of forces that provide the transition of a mechanical system from one state to another during a given time can be specified both on the basis of the Pontryagin maximum principle and on the basis of the generalized Gauss principle. Particular attention is paid to the comparison of the results that are obtained by these two principles. This is illustrated using the example of a horizontal motion of a pendula cart system to which a required force is applied. To obtain the control force without jumps at the beginning and the end of the motion, an extended boundary problem is formulated in which not only the coordinates and velocities are given at these times, but also the derivatives of the coordinates with respect to time up to the order n ≥ 2. This extended boundary-value problem cannot be solved via the Pontryagin maximum principle, because in this case the number of arbitrary constants will be smaller than the total number of the boundary-value conditions. At the same time, the generalized Gauss principle is capable of solving the problem, because in this case it is only required to increase the order of the principle to a value that agrees with the number of given boundary-value conditions. The results of numerical calculations are presented.

On Vector Form of Differential Variational Principles of Mechanics

We introduce variation of a vector δx which can be interpreted either as a virtual displacement of a system, or as variation of the velocity of a system, or as variation of the acceleration of a system. This vector is used to obtain a unified form of differential variational principles of mechanics from the scalar representative equations of motion. Conversely, this notation implies the original equations of motion, which enables one to consider the obtained scalar products as principles of mechanics. Using the same logical scheme, one constructs a differential principle on the basis of the vector equation of constrained motion of a mechanical system. In this form of notation, it is proposed to conserve the zero scalar products of reactions of ideal constraints and the vector δx. This enables one to obtain also the equations involving generalized constrained forces from this form of notation.

Dynamics of a Stewart platform

The kinematics and dynamics of a Stewart platform supported by six pneumocylinders are considered. The differential equations of motion are written, and the forces providing the fulfillment of the formulated law of motion are calculated. The inertia and weight of pneumocylinders are introduced into consideration to refine the equations of motion. The obtained equations are used to study the motion of a loaded Stewart platform, providing the stability of this motion by means of feedback control. Some numerical examples are given.

A New approach to finding the control that transports a system from one phase state to another

In their previous papers, the authors have considered the possibility of applying the theory of motion for nonholonomic systems with high-order constraints to solving one of the main problems of the control theory. This is a problem of transporting a mechanical system with a finite number of degrees of freedom from a given phase state to another given phase state during a fixed time. It was shown that, when solving such a problem using the Pontryagin maximum principle with minimization of the integral of the control force squared, a nonholonomic high-order constraint is realized continuously during the motion of the system. However, in this case, one can also apply a generalized Gauss principle, which is commonly used in the motion of nonholonomic systems with high-order constraints. It is essential that the latter principle makes it possible to find the control as a polynomial, while the use of the Pontryagin maximum principle yields the control containing harmonics with natural frequencies of the system. The latter fact determines increasing the amplitude of oscillation of the system if the time of motion is long. Besides this, a generalized Gauss principle allows us to formulate and solve extended boundary problems in which along with the conditions for generalized coordinates and velocities at the beginning and at the end of motion, the values of any-order derivatives of the coordinates are introduced at the same time instants. This makes it possible to find the control without jumps at the beginning and at the end of motion. The theory presented has been demonstrated when solving the problem of the control of horizontal motion of a trolley with pendulums. A similar problem can be considered as a model, since when the parameters are chosen correspondingly it becomes equivalent to the problem of suppression of oscillations of a given elastic body some cross-section of which should move by a given distance in a fixed time. The equivalence of these problems significantly widens the range of possible applications of the problem of a trolley with pendulums. The previous solution of the problem has been reduced to the selection of a horizontal force that is a solution to the formulated problem. In the present paper, it is offered to seek an acceleration of a trolley with which it moves by a given distance in a fixed time, as a time function but not a force applied to the trolley, while the velocities and accelerations are equal to zero at the beginning and end of motion. In this new problem, the rotation angles of pendulums are the principal coordinates. This makes it possible to find a sought acceleration of a trolley on the basis of a generalized Gauss principle according to the technique developed before. Knowing the motion of a trolley and pendulums it is easy to determine the required control force. The results of numerical calculations are presented.

Relationship between the Udwadia-Kalaba Equations and the Generalized Lagrange and Maggi Equations

In their paper “A New Perspective on Constrained Motion,” F. E. Udwadia and R. E. Kalaba propose a new form of matrix equations of motion for nonholonomic systems subject to linear nonholonomic second-order constraints. These equations contain all of the generalized coordinates of the mechanical system in question and, at the same time, they do not involve the forces of constraint. The equations under study have been shown to follow naturally from the generalized Lagrange and Maggi equations; they can be also obtained using the contravariant form of the motion equations of a mechanical system subjected to nonholonomic linear constraints of second order. It has been noted that a similar method of eliminating the forces of constraint from differential equations is usually useful for practical purposes in the study of motion of mechanical systems subjected to holonomic or classical nonholonomic constraints of first order. As a result, one obtains motion equations that involve only generalized coordinates of a mechanical system, which corresponds to the equations in the Udwadia–Kalaba form.

Pafnutii lvovich chebyshev as a mechanician

In this paper we present an original sketch about the works of an Academician of St. Petersburg Imperial Academy of Science (and Academies of other countries), the world-famous mathematician and mechanician P.L. Chebyshev (1821-1894). This sketch was made by a famous Russian specialist in nonholonomic mechanics, professor M.P. Yushkov. During several years (in the middle of 1990s) M.P. Yushkov was an invited speaker in Moscow State University, seminars devoted to the works of P.L. Chebyshev and the assessment and development of the ideas of P.L. Chebyshev. These seminars were organized by professor S.N. Kruzhkov. Here we present a small modern introduction and a sketch from one of this seminars.