Singular Control in Time Minimization of System of Bodies Motion

Program konferencij

Free vibrations of planar serial frame structures in the case of axially functionally graded materials

This paper considers the problem of modal analysis and finding the closed-form solution to free vibrations of planar serial frame structures composed of Euler-Bernoulli beams of variable cross-sectional geometric characteristics in the case of axially functionally graded materials. Each of these beams is performing coupled axial and bending vibrations, where coupling occurs due to the boundary conditions at their joints. The numerical procedure for solving the system of partial differential equations, after the separation of variables, is reduced to solving the two-point boundary value problem of ordinary linear differential equations with nonlinear coefficients and linear boundary conditions. In this case, it is possible to transfer the boundary conditions and reduce the problem to the Cauchy initial value problem. Also, it is possible to analyze the influence of different parameters on the structure dynamic behavior. The method is applicable in the case of different boundary conditions at the right and left ends of such structures, as illustrated by an appropriate numerical example

Mass minimization of an AFG Timoshenko beam with a coupled axial and bending vibrations

The paper considers mass minimization of an axially functionally graded (AFG) Timoshenko beam of a variable cross-sectional area, with a specified fundamental frequency. The analyzed case of coupled axial and bending vibrations involves contour conditions as the cause of coupling. The problem is solved applying Pontryagin’s maximum principle, where the beam cross-sectional area is taken for control. The two-point boundary value problem is obtained, and the shooting method is used to solving it. The property of self-adjoint systems is deployed. The percent saving of the beam mass is determined, achieved by using the beam of an optimum variable square cross-section as compared to the beam of a constant cross-section. The procedure developed by the author in his earlier papers is extended herein to the case of a limited cross-sectional area. The second generalization relates to the general case of contour conditions at the beam ends

Mass minimization of an AFG Timoshenko beam with a coupled axial and bending vibrations

The paper considers mass minimization of an axially functionally graded (AFG) Timoshenko beam of a variable cross-sectional area, with a specified fundamental frequency. The analyzed case of coupled axial and bending vibrations involves contour conditions as the cause of coupling. The problem is solved applying Pontryagin’s maximum principle, where the beam cross-sectional area is taken for control. The two-point boundary value problem is obtained, and the shooting method is used to solving it. The property of self-adjoint systems is deployed. The percent saving of the beam mass is determined, achieved by using the beam of an optimum variable square cross-section as compared to the beam of a constant cross-section. The procedure developed by the author in his earlier papers is extended herein to the case of a limited cross-sectional area. The second generalization relates to the general case of contour conditions at the beam ends

Bio-inspired control of redundant robotic systems: Optimization approach

Osnovni cilj ovog rada je da promoviše pristup biološki inspirisanog sinergijskog upravljanja koji omogućava da se razreši redundansa datog robotizovanog sistema koji se može koristiti i za vojne svrhe. Pokazano je da je moguće razrešiti kinematički redundansu primenom metode lokalne optimizacije i bioloških analogona - sinergijsko upravljački pristup sa uvođenjem logičkog upravljanja i distribuiranog pozicioniranja. Takođe, mogućnost prebacivanja između sinegrija u okviru jedne trajektorije je razmatrano. Na kraju, problem aktuatorske redundanse je postavljen i rešen primenom Pontrjaginovog principa maksimuma. Upravljačka sinergija je ustanovljena primenom postupka optimizacije na koordinacionom nivou. Na kraju, efikasnost predložene biološki inspirisane optimalne upravljačke sinergije je demonstriran na pogodno usvojenom robotskom sistemu sa tri stepena slobode i četiri upravljačke promenljive, kao ilustrativnog primera.The major aim of this paper is to promote a biologically inspired control synergy approach that allows the resolution of redundancy of a given robotized system which can be used for military purposes. It is shown that it is possible to resolve kinematic redundancy using the local optimization method and biological analogues - control synergy approach, introducing hypothetical control and distributed positioning. Also, the possibility of switching synergies within a single trajectory is treated, where the control synergy approach applying logical control is used. The actuator redundancy control problem has been stated and solved using Pontryagin's maximum principle. Control synergy as a class of dynamic synergy is established by the optimization law at the coordination level. Finally, the effectiveness of the suggested biologically inspired optimal control synergy is demonstrated with a suitable robot with three degrees of freedom and four control variables, as an illustrative example.

Brachistochronic rigid body general motion

Razmatra se minimizacija vremena kretanja krutog tela uz neizmenjenu vrednost mehaničke energije. Za generalisane koordinate uzete su koordinate centra masa i Ojlerovi uglovi, čije su vrednosti zadate na početku i kraju intervala kretanja. Zadatak je rešen primenom Pontrjaginovog principa maksimuma. Numeričko rešenje dvotačkastog graničnog problema dobijeno je metodom konačnih razlika za sisteme običnih diferencijalnih jednačina. .The time interval minimization of rigid body motion with constant mechanical energy has been considered in this paper. Generalized coordinates are Cartesian's coordinates of mass center and the Euler's angles, which are specified at the initial and the final position. The problem has been solved by the application of the Pontryagin's principle. Finite difference method has been applied in order to obtain the solution of the two-point boundary value problem.

Analysis of the brachistochronic motion of a variable mass nonholonomic mechanical system

The paper considers the brachistochronic motion of a variable mass nonholonomic mechanical system [3] in a horizontal plane, between two specified positions. Variable mass particles are interconnected by a lightweight mechanism of the 'pitchfork' type. The law of the time-rate of mass variation of the particles, as well as relative velocities of the expelled particles, as a function of time, are known. Differential equations of motion, where the reactions of nonholonomic constraints and control forces figure, are created based on the general theorems of dynamics of a variable mass mechanical system [5]. The formulated brachistochrone problem, with adequately chosen quantities of state, is solved, in this case, as the simplest task of optimal control by applying Pontryagin's maximum principle [1]. A corresponding two-point boundary value problem (TPBVP) of the system of ordinary nonlinear differential equations is obtained, which, in a general case, has to be numerically solved [2]. On the basis of thus obtained brachistochronic motion, the active control forces, along with the reactions of nonholonomic constraints, are determined. The analysis of the brachistochronic motion for different values of the initial position of a variable mass particle B is presented. Also, the interval of values of the initial position of a variable mass particle B, for which there are the TPBVP solutions, is determined

Analysis of the brachistochronic motion of a variable mass nonholonomic mechanical system

The paper considers the brachistochronic motion of a variable mass nonholonomic mechanical system [3] in a horizontal plane, between two specified positions. Variable mass particles are interconnected by a lightweight mechanism of the 'pitchfork' type. The law of the time-rate of mass variation of the particles, as well as relative velocities of the expelled particles, as a function of time, are known. Differential equations of motion, where the reactions of nonholonomic constraints and control forces figure, are created based on the general theorems of dynamics of a variable mass mechanical system [5]. The formulated brachistochrone problem, with adequately chosen quantities of state, is solved, in this case, as the simplest task of optimal control by applying Pontryagin's maximum principle [1]. A corresponding two-point boundary value problem (TPBVP) of the system of ordinary nonlinear differential equations is obtained, which, in a general case, has to be numerically solved [2]. On the basis of thus obtained brachistochronic motion, the active control forces, along with the reactions of nonholonomic constraints, are determined. The analysis of the brachistochronic motion for different values of the initial position of a variable mass particle B is presented. Also, the interval of values of the initial position of a variable mass particle B, for which there are the TPBVP solutions, is determined

Analysis of the minimum required coefficient of sliding friction at brachistochronic motion of a nonholonomic mechanical system

Analizira se problem brahistohronog kretanja mehaničkog sistema na primeru jednog uprošćenog modela vozila. Sistem se kreće između dva zadata položaja pri neizmenjenoj vrednosti mehaničke energije u toku kretanja. Diferencijalne jednačine kretanja, u kojima figurišu reakcije neholonomnih veza i upravljačke sile, dobijaju se na osnovu opštih teorema dinamike. Ovde je to podesnije umesto nekih drugih metoda analitičke mehanike primenjenih na neholonomne sisteme, u kojima je neophodno dati naknadno fizičko tumačenje množitelja veza da bi se ovaj problem rešio. Podesnim izborom veličina stanja, dobija se, najprostiji moguć u ovom slučaju, zadatak optimalnog upravljanja, koji se rešava primenom Pontrjaginovog principa maksimuma. Numeričko rešavanje dvotačkastog graničnog problema vrši se metodom šutinga. Na osnovu tako dobijenog brahistohronog kretanja određuju se aktivne upravljačke sile, a ujedno i reakcije veza. Koristeći Kulonove zakone trenja klizanja, određuje se minimalno potrebna vrednost koeficijenta trenja klizanja, da ne bi došlo do proklizavanja vozila u tačkama kontakta sa podlogom.The paper analyzes the problem of brachistochronic motion of a nonholonomic mechanical system, using an example of a simple car model. The system moves between two default positions at an unaltered value of the mechanical energy during motion. Differential equations of motion, containing the reaction of nonholonomic constraints and control forces, are obtained on the basis of general theorems of dynamics. Here, this is more appropriate than some other methods of analytical mechanics applied to nonholonomic systems, where the provision of a subsequent physical interpretation of the multipliers of constraints is required to solve this problem. By the appropriate choice of the parameters of state as simple a task of optimal control as possible is obtained in this case, which is solved by the application of the Pontryagin maximum principle. Numerical solution of the two-point boundary value problem is obtained by the method of shooting. Based on the thus acquired brachistochronic motion, the active control forces are determined as well as the reaction of constraints. Using the Coulomb laws of friction sliding, the minimum value of the coefficient of friction is determined to avoid car skidding at the points of contact with the ground