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1Professor, Academician of the Russian Academy of Sciences, 2 Professor, Institute for Problems in Mechanics of the Russian Academy of Sciences

Abstract:

The subject matter of the lecture is the control and optimization of multibody mechanical systems. The lecture is based on the results obtained at the Institute for Problems in Mechanics of the Russian Academy of Sciences.

The lecture consists of two parts. The first part deals with the investigation of the motion of a body containing a movable internal mass.

It is well-known that a rigid body can move in a resistive medium, if this body contains internal masses that perform special motions inside the body. This effect is utilized in certain projects of mobile robots and underwater vehicles. A simple mechanical model of this phenomenon is analyzed. A mechanical system is considered that consists of a rigid body of mass and an internal mass
m that can move inside the body. Rectilinear motions of the system along a horizontal line are studied.

Certain periodic motions of the internal mass
m relative to the body are analyzed, namely, two-phase and three-phase motions. In the two-phase motion, the relative velocity of the mass
m is piecewise-constant, and the period includes two intervals of constant velocity. In the three-phase motion, the relative acceleration of the mass
m is piecewise constant, and the period contains three intervals of constant acceleration. It is shown that, under the imposed periodicity conditions, the two-phase and three-phase motions have the least possible number of intervals of constant velocity and acceleration, respectively.

Various kinds of resistance forces acting upon the body are considered: Coulombs dry friction, linear and nonlinear resistance depending on the velocity of the body. These forces can be anisotropic, i.e., different for onward and backward motions of the body .

Optimal parameters of periodic motions are calculated that result in the maximal average speed of the body under the constraints imposed on the displacement, velocity, and acceleration of the mass
m.

The second part is devoted to constructing control laws which are robust against uncertainties in the external disturbances and perturbations of the system’s parameters. A Lagrangian mechanical system of the general form is considered under the assumption that the matrix of inertia of the system is not known exactly, and the system undergoes undetermined bounded disturbances. A continuous feedback bounded control is proposed which steers the system to a prescribed terminal state of rest in a finite (unfixed) time. The elaborated approach employs a linear feedback control with the gains which are functions of the phase variables. The gains increase and tend to infinity as the phase variables tend to zero; nevertheless, the control forces are bounded and meet the imposed constraint.

To construct the control law and justify it, the Lyapunov direct method is used.

Biagraphies
：

Felix Leonidovich Chernousko was born in 1938 in St. Petersburg, Russia. He graduated from the Moscow Institute of Physics and Technology (MIPT) (was honor) as an Engineer-Physicist in 1961 and entered the postgraduate course in the same Institute. In 1961, he was awarded the First Prize at the Russian Contest of students research works. In 1964, he received the Ph.D. degree at MIPT for the dissertation on dynamics of the angular motion of satellites. Since 1963 until 1968, F.L. Chernousko worked as a researcher at the Computational Center of the USSR Academy of Sciences. Since 1968, he is a Head of Laboratory at the Institute for Problems in Mechanics of the Russian Academy of Sciences. In 1969, F.L. Chernousko received the Doctor of Sciences degree for the dissertation on dynamics of rigid bodies containing liquid masses. In 2004, F.L. Chernousko was elected as a Director of the Institute for Problems in Mechanics of the Russian Academy of Sciences.

Prof. F.L. Chernousko has made major contributions to mechanics, applied mathematics, and control sciences.

In applied mathematics and control, F .L. Chernousko proposed and developed new efficient numerical methods for optimal control and variational problems. As early as in 1961, he proposed the first numerical method of optimal control based on Pontryagins maximum principle (max-H) method. Later, he improved and modified this method and made it an efficient tool widely used for trajectory optimization of spacecraft, aircraft, machines, robotic systems etc. He also proposed the method of local variations which was applied to variational problems in solid mechanics. He developed analytical methods of optimal control based on asymptotic techniques and investigated optimal control of oscillations in mechanical systems. Chernousko made a valuable contribution to control and estimation in uncertainty conditions by developing the guaranteed (set-membership) approach to dynamical systems in the presence of disturbances and measurement errors. Especially efficient is his method of two-sided optimal ellipsoidal estimation for reachable sets of dynamical systems. Chernousko proposed optimal methods for search of roots and extrema for certain classes of functionS.

In mecha11ics, Chernousko investigated nonlinear oscillations and rotations of spacecraft about its center of mass under the action of external and internal perturbations, including the influence of elastic parts and fluid masses. He created a theory of motions of rigid bodies containi11g tanks of arbitrary shape carrying viscous fluid. Chernousko analyzed equilibrium, stability, and oscillations of fluid in low gravity conditions, carried out research on other problems of mechanics and biomechanics. He took an active part in Russian space programs.

In robotics, Chernousko investigated optimal and suboptimal motions of robotic manipulators. He investigated the dynamics of wall-climbing and tube-crawling robots, and studied snake-like motions of multilink systems along a plane in the presence of dry friction.

F.L. Chernousko is an author of 10 books and more than 350 scientific papers. He was an invited lecturer at a number of international scientific meetings, delivered lectures at numerous universities.

F.L. Chernousko is a professor of the Moscow institute of Physics and Technology (since 1974), he supervised more than 30 Ph.D. and 15 Dr. of Sciences dissertations.

F.L. Chernousko was awarded the Medal for Valor in Labor, Leninsky Komsomol Prize (the highest Soviet award for scientists under 33), the USSR State Prize for Science and Technology, the Russian State Prize for Science and Technology, Korber Prize for the European Science, A. von Humboldt Research Award), Chaplygin Gold Medal of the Russian Academy of Sciences.

He was elected a Full Member (Academician) of the Russian Academy of Sciences, International Academy of Astronautics, Serbian Academy of Sciences and Arts, European Academy of Sciences, a member of numerous scientific committees, societies, and editorial boards of leading Russian and international scientific journals.

Ananievskiy Igor Mikhailovich, born on 16 October 1956 in Zlatoust, graduated from Leningrad State University in 1979 (the Dept. of Mathematics and Mechanics). He studied qualitative theory of ordinary differential equations as postgraduate student at the same Department and received Ph.D. (Physics & Mathematics) degree in 1983. Since 1984 I. M. Ananievskiy has worked at the Institute for Problems in Mechanics of the Russian Academy of Sciences. He is Leading Researcher at the Laboratory of Control in Mechanical Systems. In 1999, he received the Doctor of Sciences degree for the dissertation on control of mechanical systems. I. M. Ananievskiy is a specialist in control theory and applied mathematics.