On the stability of relative programmed motion of satellite- gyrostat

This paper is devoted to study the asymptotic stability of the relative programmed motion of a satellite-gyrostat with the help of the three rotors attached to the principal axes of inertia of the satellite. The programmed control moments are obtained. The control moments on the rotors using the condition which impose the asymptotic stabilization of the programmed motion are obtained.     

On the exponential stability of the permanent rotational motion of a gyrostat

This paper is devoted to the study of the problem of exponential asymptotic stability of the rotational motion of a gyrostat using servo-control moments which are applied to the internal rotors. The servo-control moments which impose the rotational motion are obtained. The stabilizing servo-control moments are obtained from the conditions to ensure exponential asymptotic stability of the desired motion. Estimations of the phase coordinations as exponential functions are presented. The method based on a choice of the structural form of the servo-control moments such that the equations of motion reduce to a system of differential equations with exponential asymptotic stability of an special solution.     

The control of a permanent rotational motion and equilibrium position of a spacecraft

This article has adopted an analytical method to obtain a non-linear control law to reach the exponential asymptotic stablity of the permanent rotational motion of a spacecraft. The control moments achieving this rotational motion are obtained. The control moments to establish exponential asymptotic stablity of the mentioned motion are obtained as non-linear functions of the phase coordinates of the spacecraft. The general solution of the equations of perturbed motion is derived. Furthermore, analysis and numerical simulation study of this solution are presented. For numerical examples the time needed for control is calculated. An equilibrium position of the spacecraft is proved to be exponentially asymptotically stable as a special case of the above-studied problem.     

Biquaternion solution of the kinematic control problem for the motion of a rigid body and its application to the solution of inverse problems of robot-manipulator kinematics

The problem of reducing the body-attached coordinate system to the reference (programmed) coordinate system moving relative to the fixed coordinate system with a given instantaneous velocity screw along a given trajectory is considered in the kinematic statement. The biquaternion kinematic equations of motion of a rigid body in normalized and unnormalized finite displacement biquaternions are used as the mathematical model of motion, and the dual orthogonal projections of the instantaneous velocity screw of the body motion onto the body coordinate axes are used as the control. Various types of correction (stabilization), which are biquaternion analogs of position and integral corrections, are proposed. It is shown that the linear (obtained without linearization) and stationary biquaternion error equations that are invariant under any chosen programmed motion of the reference coordinate system can be obtained for the proposed types of correction and the use of unnormalized finite displacement biquaternions and four-dimensional dual controls allows one to construct globally regular control laws. The general solution of the error equation is constructed, and conditions for asymptotic stability of the programmed motion are obtained. The constructed theory of kinematic control of motion is used to solve inverse problems of robot-manipulator kinematics. The control problem under study is a generalization of the kinematic problem [1, 2] of reducing the body-attached coordinate system to the reference coordinate system rotating at a given (programmed) absolute angular velocity, and the presentedmethod for solving inverse problems of robotmanipulator kinematics is a development of the method proposed in [3–5].     

On the stability of an equilibrium position and rotational motion of a gyrostat

This article is devoted to study the compulsory stability of equilibrium position and rotational motion of a rigid body containing fluid with the help of three rotors carried on the body. The control moments on the rotors using that condition which impose the stabilization of equilibrium position of the rigid body and rotational motion are obtained.     

Control of the rotational motion of the rigid body with the help of internal rotors using Rodrigues-Caley parameters

The purpose of this paper is to study the control of the rotational motion of the rigid body with the help of three rotors attached to the principal axes of the body. In such study the asymptotic stability of this motion is proved by using the Lyapunov technique. As a particular case of our problem, the equilibrium position of the rigid body, which occurs when the principal axes of inertia of the body coincide with the inertial axes, is proved to be asymptotically stable. The control moments that impose the stabilization of the rotational motion and equilibrium position are obtained.     

Dynamic collision avoidance and constraint violation compensation

This paper uses concepts in multibody dynamics, together with a collision detection algorithm to study the dynamics of collision avoidance. Obstacle avoidance of a mechanical system in motion is expressed in terms of distances, relative velocities and relative accelerations between potentially colliding bodies. The generalized control forces (constraint forces) used to adjust the system dynamics are based on an n-timestep collision avoidance algorithm. Constraint violations resulting from sudden changes in motion direction are compensated for by feeding back the errors of position and velocity constraints to assure asymptotic stability. The procedures developed are illustrated through a maneuver in space of a robotic manipulator used for grasp and deployment.     

Optimal control of a rigid spacecraft programmed motion without angular velocity measurements

This paper considers the problem of optimal controlling a spacecraft programmed motion without its angular velocity measurements. An optimal control law that stabilizes this programmed motion and minimizes the cost that transfers the spacecraft from arbitrary initial state to the programmed state is obtained as a function of the kinematics attitude parameters and their estimates as well as the angle of programmed rotation. The stabilizing properties of the proposed controllers are proved using Liapunov techniques. Numerical simulation study is presented.     

Optimal control of a rotational motion of a gyrostat on circular orbit

A control scheme is proposed to guarantee an optimal stabilization of a given rotational motion of a symmetric gyrostat on circular orbit. The gyrostat controlled by the control action generated by rotating internal rotors. In such study the asymptotic stability of this motion is proved using Barbachen and Krasovskii theorem's and the optimal control law is deduced from the conditions that ensure the optimal asymptotic stability of the desired motion. As a particular case, the equilibrium position of the gyrostat, which occurs when the principal axes of inertia coincide with the orbital axes, is proved to be asymptotically stable. The present method is shown to more general than previous ones.     

Stability of wave forms of motion of a viscous fluid layer under tangential stress

We study the stability of wave flow of a viscous incompressible fluid layer subjected to tangential stress and an inclined gravity force with respect to long-wave disturbances.An asymptotic solution is constructed for the equations of the disturbed motion and the problem is reduced to the study of a second-order ordinary differential equation. It is shown that after loss of stability by a Poiseuille flow the laminar nature of the flow is not destroyed, but the form of the free surface acquires a wave-like profile. The Poiseuille regime is stable for low Reynolds numbers. The critical Reynolds number for wave flow is found, and the stability and instability regions are determined.     

The Onset of Convection in an Anisotropic Porous Layer Using a Thermal Non-Equilibrium Model

The stability of a horizontal fluid saturated anisotropic porous layer heated from below and cooled from above is examined analytically when the solid and fluid phases are not in local thermal equilibrium. Darcy model with anisotropic permeability is employed to describe the flow and a two-field model is used for energy equation each representing the solid and fluid phases separately. The linear stability theory is implemented to compute the critical Rayleigh number and the corresponding wavenumber for the onset of convective motion. The effect of thermal non-equilibrium and anisotropy in both mechanical and thermal properties of the porous medium on the onset of convection is discussed. Besides, asymptotic analysis for both very small and large values of the interphase heat transfer coefficient is also presented. An excellent agreement is found between the exact and asymptotic solutions. Some known results, which correspond to thermal equilibrium and isotropic porous medium, are recovered in limiting cases.     

Asymptotic solutions of mathieu equation with damping

This paper first reduces the motion equation of a collapsible tube to the Mathieu equation with damping. Then the stability charts correcting the accuracy to each order are obtained with the method of asymptotic expansions. The accuracy of the results obtained with the average variational method is shown. And some phenomena observed in the experiment are also explained.     

Eulerian prediction of the fluid/particle correlated motion in turbulent two-phase flows

An approximate equation governing the turbulent fluid velocity encountered along discrete particle path is used to derive the fluid/particle turbulent moments required for dispersed two-phase flows modelling. Then, closure model predictions are compared with results obtained from large-eddy simulation of particle fluctuating motion in forced isotropic fluid turbulence.     

New control laws for stabilization of a rigid body motion using rotors system

This paper presents a new class of globally asymptotic stabilizing control laws for dynamics and kinematics attitude motion of a rotating rigid body. The rigid body motion is controlled with the help of a rotor system with internal friction. The Lyapunov technique is used to prove the global asymptotic properties of the stabilizing control laws. The obtained control laws are given as functions of the angular velocity, Cayley–Rodrigues and Modified-Rodrigues parameters. It is shown that linearity and nonlinearity of the control laws depend not only upon the Lyapunov function structure but also the rotors friction. Moreover, some of the results are compared with these obtained in the literature by other methods. Numerical simulation is introduced.     

Decentralized coordinated attitude control for satellite formation flying via the state-dependent Riccati equation technique

The goal of the present study is to develop a decentralized coordinated attitude control algorithm for satellite formation flying. To handle the non-linearity of the dynamic system, the problems of absolute and relative attitude dynamics are formulated for the state-dependent Riccati equation (SDRE) technique. The SDRE technique is for the first time utilized as a non-linear controller of the relative attitude control problem for satellite formation flying, and then the results are compared to those from linear control methods, mainly the PD and LQR controllers. The stability region for the SDRE-controlled system was obtained using a numerical method. This estimated stability region demonstrates that the SDRE controller developed in the present paper guarantees the globally asymptotic stability for both the absolute and relative attitude controls. Moreover, in order to complement a non-selective control strategy for relative attitude error in formation flying, a selective control strategy is suggested. This strategy guarantees not only a reduction in unnecessary calculation, but also the mission-failure safety of the attitude control algorithm for satellite formation. The attitude control algorithm of the formation flying was tested in the orbital-reference coordinate system for the sake of applying the control algorithms to Earth-observing missions. The simulation results illustrate that the attitude control algorithm based on the SDRE technique can robustly drive the attitude errors to converge to zero.     

Stability of the relative equilibrium of three vortex charges with zero total intensity

The relative motion of three vortex charges with zero total intensity in a rotating baroclinic fluid is considered in the quasi-geostrophic approximation. The geometry and the linear stability of the relative equilibrium positions of the charges are analyzed. The motion parameter values are classified with respect to the number, type, and stability of equilibrium positions.     

Particle collisions in a turbulent flow

An equation for the two-point probability density function of the two-particle the coordinate and velocity distribution is obtained. A closed system of equations for the first and second two-point moments of the velocity fluctuations of a pair of particles with allowance for the turbulent flow inhomogeneity is given. Boundary conditions for the equations of the particle concentration and the intensity of the relative random velocity during particle collision are obtained. A unified formula describing the interparticle collision process as a result of turbulent motion and the average relative particle velocity slip is obtained for the kernel of the coagulation equation. The effect of the average velocity slip of the particles and the carrier phase on the parameters of motion of the dispersed admixture and its coagulation is investigated on the basis of a two-point two-time velocity fluctuation autocorrelation function with two time and space scales representing the energy-bearing and small-scale motion of the fluid phase.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 104–116, March–April, 1996.     

Unsteady motion of a source in a fluid under a floating plate

This paper studies the three-dimensional unsteady problem of the hydroelastic behavior of a floating infinite plate under the impact of waves generated by horizontal straight motion of a point source of mass in a fluid of infinite depth. The solution is carried out using known integral and asymptotic methods. The formulas obtained are used to numerically analyze the effect of plate thickness, depth of submergence of the source, and its acceleration, deceleration, and velocity of uniform motion on the deflection of the floating plate.     

RESEARCH OF THE PERIODIC MOTION AND STABILITY OF TWO-DEGREE-OF-FREEDOM NONLINEAR OSCILLATING SYSTEMS

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