Relative Equilibrium Positions Research Articles

On the motion stabilization of a three-link robotic manipulator with incomplete measurement

Abstract. This paper considers a mathematical model of a manipulator which consists of a vertical column, two links, connected to it in series, and a gripper with a load. The column resting on a fixed base can rotate around its vertical axis. The links are connected by cylindrical hinges allowing them to rotate in the same vertical plane. The column and the links are modeled as rigid bodies with the links having unequal principal moments of inertia. The position of the manipulator in space is determined by three rotation angles of the column and the links. The manipulator can have several types of steady-state program movements. When gravitational torques are compensated by control torques applied in the cylindrical hinges, the manipulator has a program equilibrium position. The manipulator can also have a program motion when the column rotates at a given constant angular velocity, and the links have given relative equilibrium positions in their plane. The stabilization problem of manipulator motion is investigated by means of control torques with feedback when only the rotation angles of the column and links are measured. The problem posed is solved in the form of a nonlinear proportional-integral controller taking into account the cylindrical phase space of the manipulator's mathematical model. The solution includes construction of a Lyapunov functional with a semi-definite derivative and application of the corresponding theorems on the asymptotic stability of non-autonomous functional differential retarded-type equations. The obtained conditions for the program motion stabilization are robust with respect to the mass-inertial parameters of the manipulator. The numerical simulation results demonstrate global attraction to its given position in cylindrical phase space.

On the equilibriums stability in an approximate problem of the dynamics of a rigid body with a suspension point vibrating along an inclined straight line

We consider the heavy rigid body dynamics under the assumption that one of the body points (the suspension point) performs the specified high-frequency vibrations of small amplitude along an inclined straight line, and the body mass centre lies on the principal axis of inertia for the suspension point. In the framework of an approximate autonomous system of canonical equations of motion, the question of existence, number, and stability of the body relative equilibrium positions is solved. It is shown that for all such equilibria, the mass centre radius-vector lies in the vertical plane containing the vibration axis. The number of equilibria is four, six or eight depending on the vibration intensity. Sufficient and necessary conditions for their stability are found. The existence of high-frequency periodic motions of the initial non-autonomous system, which are generated by the investigated equilibria, is justified using Poincare’s method. Conclusions about stability (in linear approximation) of these periodic motions are drawn.

Relative equilibria of dynamically symmetric CubeSat nanosatellite under the action of aerodynamic and gravitational torques

Motion of a dynamically symmetric CubeSat nanosatellite around the mass center on the circular orbit under the action of aerodynamic and gravitational torques is considered. We determined the nanosatellite equilibrium positions in the flight path axis system. We took into account the fact that the CubeSat nanosatellite has a rectangular parallelepiped shape and, therefore, the aerodynamic drag force coefficient depends on the angles of attack and proper rotation. We obtained formulae which allow calculating the values of the angles of attack, precession and proper rotation that correspond to the equilibrium positions, depending on the mass-inertia and geometric parameters of the nanosatellite, the orbit altitude, and the atmospheric density. It is shown that if the gravitational moment predominates over the aerodynamic one, there are 16 equilibrium positions, if the aerodynamic moment predominates over the gravitational one, there are 8 equilibrium positions, and in the case when both moments have comparable values there are 8, 12 or 16 equilibrium positions. Using the formulae obtained, we determined the equilibrium positions of the SamSat-QB50 nanosatellite. We calculated the ranges of altitudes where SamSat-QB50 nanosatellite has 8, 12, or 16 relative equilibrium positions.

The Stability of Relative Equilibrium Positions of a Pendulum on a Mobile Platform

The motion of a pendulum mounted on a platform that rotates around a vertical line with a constant velocity and simultaneously makes specified harmonic oscillations along this vertical line is considered. The platform angular velocity is assumed to be precisely equal to the frequency of small oscillations of the pendulum in the case of an immobile platform. There are relative equilibrium positions of the pendulum when it is oriented along the platform rotation axis (in the hanging or inverted state). The nonlinear problem of stability of these relative equilibria is considered.

NUMERICAL-SYMBOLIC METHODS FOR SEARCHING RELATIVE EQUILIBRIA IN THE RESTRICTED PROBLEM OF FOUR BODIES

We discuss here the problem of solving the system of two nonlinear algebraic equations determining the relative equilibrium positions in the planar circular restricted four-body problem formulated on the basis of the Euler collinear solution of the three-body problem. The system contains two parameters $\mu_1$, $\mu_2$ and all its solutions coincide with the corresponding solutions in the three-body problem if one of the parameters equals to zero. For small values of one parameter the solutions are found in the form of power series in terms of this parameter, and they are used for separation of different solutions and choosing the starting point in the numerical procedure for the search of equilibria. Combining symbolic and numerical computation, we found all the equilibrium positions and proved that there are 18 different equilibrium configurations of the system for any reasonable values of the two system parameters $\mu_1$, $\mu_2$. All relevant symbolic and numerical calculations are performed with the aid of the computer algebra system Wolfram Mathematica.

Swimming at small Reynolds number of a planar assembly of spheres in an incompressible viscous fluid with inertia

Translational and rotational swimming at small Reynolds numbers of a planar assembly of identical spheres immersed in an incompressible viscous fluid is studied on the basis of a set of equations of motion for the individual spheres. The motion of the spheres is caused by actuating forces and forces derived from a direct interaction potential, as well as hydrodynamic forces exerted by the fluid as frictional and added mass hydrodynamic interactions. The translational and rotational swimming velocities of the assembly are deduced from momentum and angular momentum balance equations. The mean power required during a period is calculated from an instantaneous power equation. Expressions are derived for the mean swimming velocities and the mean power, valid to second order in the amplitude of displacements from the relative equilibrium positions. Hence these quantities can be evaluated for prescribed periodic displacements. Explicit calculations are performed for three spheres interacting such that they form an equilateral triangle in the rest frame of the configuration.

Relative equilibria of a massive point on a uniformly rotating asteroid

The motion of a massive point (a bead) over the surface of a uniformly rotating asteroid is considered. It is assumed that the force of dry friction acts between the point and the asteroid surface. The sets of nonisolated positions of relative equilibrium of the bead on the asteroid are described, and their dependence on the parameters of the problem is investigated. The results are presented as bifurcation diagrams.

Swimming at small Reynolds number of a collinear assembly of spheres in an incompressible viscous fluid with inertia

Swimming at small Reynolds number of a collinear assembly of identical spheres immersed in an incompressible viscous fluid is studied on the basis of a set of equations of motion for the individual spheres. The motion of the spheres is caused by actuating forces and forces derived from a direct interaction potential, as well as hydrodynamic forces exerted by the fluid as frictional and added mass hydrodynamic interactions. The swimming velocity is deduced from the momentum balance equation for the assembly of spheres, and the mean power required during a period is calculated from an instantaneous power equation. Expressions are derived for the mean swimming velocity and the mean power, valid to second order in the amplitude of displacements from the relative equilibrium positions. Hence these quantities can be evaluated in terms of prescribed periodic displacements. Explicit calculations are performed for a linear chain of three identical spheres.

An estimate for the number of relative equilibria in the motion of a plane rigid body and a material point under mutual attraction

The plane motion of a rigid body with a discrete mass distribution and a material point under mutual attraction is considered. The stationary configurations of this mechanical system are studied in the case when the mass of the material point can be ignored and the body rotates about its mass center at a nonzero angular velocity and in the general case of mutual interaction between the body and the material point. It is shown that in this mechanical system there always exist at least two different positions of relative equilibrium.

Bifurcations of the relative equilibria of a heavy bead on a hoop uniformly rotating about an inclined axis with dry friction

The sliding of a heavy bead threaded onto a thin circular hoop rotating at a constant angular velocity about an inclined axis located in its plane and passing through its centre is considered. A dry friction force acts between the bead and the hoop. The sets of the non-isolated positions of relative equilibrium of the bead on the hoop are found and their dependence on the parameters of the problem is investigated. The results are presented in the form of bifurcation diagrams.

The stability of the relative equilibrium in an Earth–Moon orbital station system with a small reactive acceleration

The problem of stabilizing the relative equilibrium of an orbital station in an Earth–Moon system by imparting a small constant-modulus acceleration with constant orientation of its vector with respect to the body of the station, which is assumed to be a rigid body of variable mass, is considered. It is shown that, in the case of a small displacement of the centre of mass of the station (by means of a small reactive acceleration) with respect to the collinear libration point beyond the Moon, its relative equilibrium position can become stable by virtue of the equations of the first approximation.

Bifurcations of the relative equilibria of a heavy bead on a rotating hoop with dry friction

The sliding of a heavy bead, threaded on a thin circular hoop, rotating with a constant angular velocity around a vertical axis, situated in its plane and, in the general case, not passing through its vertical diameter, is considered. It is assumed that dry friction acts between the bead and the hoop. A set of unisolated positions of relative equilibrium of the bead on the hoop is obtained, and their dependence on the problem parameters is investigated. The results are presented in the form of bifurcation diagrams. The stability properties of the unisolated relative equilibria obtained are discussed.

Relative equilibrium stability of a space station consisting of two solid bodies connected by an elastic rod

The problem is considered for stability of a relative equilibrium of a mechanical system in orbit that consists of two solid bodies connected by a thin inextensible elastic rod. The problem for stability of a relative equilibrium position reduces to the problem of a minimum of a changed potential system energy comprising the potential energy of elastic, gravitational and centrifugal forces.

Poiseuille flow-induced vibrations of two cylinders in tandem

Laminar flows past two tandem cylinders which are free to move transversely in a parallel-wall channel were studied numerically by the lattice Boltzmann method. With fixed Reynolds number Re=100, blockage ratio β=1/4 and structural damping ξ=0, the effect of streamwise separation between two cylinders at a range of S/D=[1.1, 10] on the motions of cylinders and fluids was studied for both mass ratios of m(⁎)=1 and m(⁎)=0.1. A variety of distinct vibration regimes involving periodic, quasi-periodic and non-periodic vibrations with corresponding flow patterns were observed. A detailed analysis of the vibration amplitudes, vibration frequencies and relative equilibrium positions for both mass ratios demonstrated that as S/D increases, the interaction of the two cylinders first enhances and then reduces. In the strong coupling regime, both cylinders oscillate periodically around the centerline of the channel with large vibration amplitudes and high vibration frequencies. By comparing with the case of an isolated cylinder, a further study indicated that the gap flow plays an important role in such a dynamic system, and the vortex cores formation behind the front cylinder causes the interaction of the cylinders decouple rapidly. Based on the present observations, such a dynamic model system can be considered as a novel type of vortex-induced vibrations (VIV) and is expected to find applications in fluid mixing and heat transfer.

The stability in a strict non-linear sense of a trivial relative equilibrium position in the classical and generalized versions of Sitnikov's problem

Stability, in a strict non-linear sense, of a trivial relative equilibrium position is investigated in the classical and generalized versions of Sitnikov's problem in the case of small eccentricities of the orbits of bodies of finite dimensions. In the classical version (n=2) of the problem, it is proved that there are no second-, third- and fourth-order resonances and a degenerate case. In the generalized version (2<n≤5·105), it is proved that there are no second- and third-order resonances and a degenerate case. A fourth-order resonance occurs in versions of the problem in which the number of finite size bodies satisfies the inequality 45000≤n≤62597 and the orbital eccentricities e < 0.25. Use of the Arnold–Moser and Markeyev theorems enables one to establish the Lyapunov stability of the trivial positions of relative equilibrium in the above-mentioned versions of Sitnikov's problem.

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.

Gravitational stabilization of a satellite using a movable mass

The plane motion of an axisymmetric satellite with a small movable mass on its axis of symmetry is examined, and the equation of the motion of this system in an elliptical orbit is derived. Problems regarding the gravitational stabilization of two diametrically opposite relative equilibrium positions of the satellite in a circular orbit to in-plane perturbations are investigated. A continuous law for controlling the movable mass, which ensures stabilization of the axis of symmetry of the satellite to the local vertical and reorientation of the satellite by moving it from one stable equilibrium position to the other, is constructed using the swing-by technique. A solution is obtained by using the second method of classical stability theory and constructing the corresponding Lyapunov functions. The asymptotic convergence of the solutions with the control obtained is confirmed by the results of numerical simulation of the motion of the system.

Thermo-electric-induced dichroism in ion-exchanged glasses: a candidate mechanism for the alignment of silver nanoparticles

In the present work, the alignment mechanism of silver nanoparticles on the surface of a heated ion-exchanged glass, in presence of an external uniform DC electric field $(\vec{E}_{0})$ parallel to the surface of the sample, is studied. At high temperature, the ionic silver clusters reduce to neutral ones and move toward the surface. Simultaneously, due to the external electric field the clusters interact with other ones as induced electrical dipoles. This leads to alignment of nanoparticles along $\vec{E}_{0}$ and formation of a chain-like conductive structure, which makes the sample dichroic. Taking into account the matrix surface viscosity and using the method of image dipoles to model the influence of the substrate on the dipole interactions, we give an interpretation about the relative equilibrium positions of generated nanoclusters and consequently the formation mechanism of the chain-like structure on the surface of the ion-exchanged glass.

Psychopathologie du syndrome d’Asperger et « aspérigisation » de la société contemporaine

The author has attempted here to point out, just for a start, the characteristics of Asperger syndrome from the point of view of psychopathology through a rereading of Hans Asperger's original paper (1944). This thesis merits reevaluation, if for no other reason than to fill the gaps in operational diagnostics based on the DSM. It is found by rereading that Asperger's view of the principal disturbances of autistic psychopathy include a “disturbance of natural evidence” or a “crisis of common sense”. This question of natural evidence that he evokes with regard to autistic psychopathy corresponds to W. Blankenburg's natural evidence, which constitutes a key concept for comprehending schizophrenia in the form poor-symptom (“symptomarme Schizophrenie”) that he observes in the speech of his patient Anne Rau. One can deduce from this that in terms of fundamental disturbances, Asperger syndrome and this “symptom-poor” schizophrenia overlap at the level of loss of natural evidence. It is moreover possible to classify Asperger syndrome among the disturbances of spacing in the sense meant by the evolutionary psychiatry of A. Stevens and J. Price. The author then develops our comprehension of Asperger syndrome from the point of view of the perspective proposed by the notion of resilience in people with Asperger syndrome and of the possibility for them, through these mechanisms of adaptation, to find in the organization of the personality of the “as if” type a position of relative equilibrium. They concur or overlap in the creation of crutches, of borrowed personalities secondarily legitimated by the reaction of the socius. This will end up in the production of inventions and œuvres (works). Clearly, one rarely encounters several cases that one could consider pertinently to be “successful” Asperger syndrome. Finally, the author notes that one can find a sort of isomorphism between Asperger syndrome and contemporary society when he proposes the term “asperigisation” to characterize our society, given that the equilibrium between emotion and logic is strongly disturbed in these patients, in whom logic undergoes hypertrophy while emotion is impoverished. From this perspective, the author hopes to suggest reasons for the increase in the number of cases of Asperger syndrome in the clinical setting and in society in general in our contemporary era.

On the motions of a double pendulum with vibrating suspension point

We consider the motions of a system consisting of two pivotally connected physical pendulums rotating about horizontal axes. We assume that the system suspension point, which coincides with the suspension point of one of the pendulums, performs harmonic vibrations of high frequency and small amplitude along the vertical. We also assume that the system has four relative equilibrium positions in which the suspension points and the pendulum centers of mass lie on one vertical line.