Unperturbed Motion Research Articles

Self-excited yawing vibrations of railway cars

The site [1] presents a video recording of the motion of empty freight railway cars. The video demonstrates that some railway cars experience angular self-excited vibrations in the horizontal plane (i.e., yawing). In the same reference, the main causes of such a “negative dynamics” of railway cars are listed. All causes are related to some defects in railway car manufacturing. We show that such self-excited vibrations are possible even in the case of absolutely ideal railway car without any defects at all. It is shown that, for a certain combination of parameters, the rectilinear (unperturbed) motion of a railway car may experience flutter instability. This type of self-excited vibrations of railway cars has not been studied earlier [2, 3].

Averaging of incompressible flows on two-dimensional surfaces

We consider a process on a compact two-dimensional surface which consists of the fast motion along the stream lines of an incompressible periodic vector field perturbed by white noise. It gives rise to a process on the graph naturally associated to the structure of the stream lines of the unperturbed flow. It has been shown by Freidlin and Wentzell that if the unperturbed motion is periodic for almost all the initial points, then the corresponding process on the graph weakly converges to a Markov process. We consider the situation when the unperturbed motion is not periodic and the flow has ergodic components of positive measure. We show that the process on the graph still converges to a Markov process, which spends a positive proportion of time in the vertices corresponding to the ergodic components of the flow. As shown in a companion paper (Deterministic and stochastic perturbations of area preserving flows on a two-dimensional torus, by D. Dolgopyat, M. Freidlin, and L. Koralov), these results allow one to describe the viscosity regularization of dissipative deterministic perturbations of incompressible flows. In the case of surfaces of higher genus, the limiting process may exhibit intermittent behavior in the sense that the time axis can be divided into the intervals of random lengths where the particle stays inside one ergodic component separated by the intervals where the particle moves deterministically between the components.

On the orbital stability of pendulum-like oscillations and rotations of a symmetric rigid body with a fixed point

We deal with the problem of orbital stability of planar periodic motions of a dynamically symmetric heavy rigid body with a fixed point. We suppose that the center of mass of the body lies in the equatorial plane of the ellipsoid of inertia. Unperturbed periodic motions are planar pendulum-like oscillations or rotations of the body around a principal axis keeping a fixed horizontal position.

Coupled dynamical system based arm–hand grasping model for learning fast adaptation strategies

Performing manipulation tasks interactively in real environments requires a high degree of accuracy and stability. At the same time, when one cannot assume a fully deterministic and static environment, one must endow the robot with the ability to react rapidly to sudden changes in the environment. These considerations make the task of reach and grasp difficult to deal with. We follow a Programming by Demonstration (PbD) approach to the problem and take inspiration from the way humans adapt their reach and grasp motions when perturbed. This is in sharp contrast to previous work in PbD that uses unperturbed motions for training the system and then applies perturbation solely during the testing phase. In this work, we record the kinematics of arm and fingers of human subjects during unperturbed and perturbed reach and grasp motions. In the perturbed demonstrations, the target’s location is changed suddenly after the onset of the motion. Data show a strong coupling between the hand transport and finger motions. We hypothesize that this coupling enables the subject to seamlessly and rapidly adapt the finger motion in coordination with the hand posture. To endow our robot with this competence, we develop a coupled dynamical system based controller, whereby two dynamical systems driving the hand and finger motions are coupled. This offers a compact encoding for reach-to-grasp motions that ensures fast adaptation with zero latency for re-planning. We show in simulation and on the real iCub robot that this coupling ensures smooth and “human-like” motions. We demonstrate the performance of our model under spatial, temporal and grasp type perturbations which show that reaching the target with coordinated hand–arm motion is necessary for the success of the task.

Fourier Transform Mass Spectrometry

This article provides an introduction to Fourier transform-based mass spectrometry. The key performance characteristics of Fourier transform-based mass spectrometry, mass accuracy and resolution, are presented in the view of how they impact the interpretation of measurements in proteomic applications. The theory and principles of operation of two types of mass analyzer, Fourier transform ion cyclotron resonance and Orbitrap, are described. Major benefits as well as limitations of Fourier transform-based mass spectrometry technology are discussed in the context of practical sample analysis, and illustrated with examples included as figures in this text and in the accompanying slide set. Comparisons highlighting the performance differences between the two mass analyzers are made where deemed useful in assisting the user with choosing the most appropriate technology for an application. Recent developments of these high-performing mass spectrometers are mentioned to provide a future outlook.

Solution of Kepler’s equation for rectilinear motion

The properties of the solution of the kinematic equation (Kepler’s equation) presented as a series in powers of a function of time are considered for the case of unperturbed, rectilinear elliptical and rectilinear hyperbolic motion. Kepler’s equation for unperturbed, rectilinear elliptical motion has the form E − sinE = z 3/6, where E is the eccentric anomaly determining the position in the orbit, z 3/6 is the mean anomaly, which is proportional to the time measured from an encounter, and its solution can be represented as a series in powers of z. It is established that the coefficients of the series are positive. The asymptotic for the coefficients in the region of convergence of the series is found, which covers the entire orbit. The series continues to converge over the entire boundary of the circle of convergence. The kinematic equation for unperturbed, rectilinear hyperbolic motion has the form sinhH − H = ζ 3/6, where, as before, ζ 3 is proportional to time. The substitution E = iH, z = iζ reduces one equation to the other. The series for the solution in the hyperbolic case differs from the series for the elliptical solution only in its alternating-sign coefficients. However, the region of convergence covers only part of the orbit in the hyperbolic case.

Universal irreversibility of normal quantum diffusion

Time-reversibility measured by the deviation of perturbed time-reversed motion from the unperturbed motion is examined for normal quantum diffusion exhibited by four classes of quantum maps with contrasting physical nature. Irrespective of the system, there exists a universal minimal quantum threshold above which the system completely loses memory of the past. The time-reversed dynamics as well as the time-reversal characteristics are asymptotically universal curves independent of the details of the systems.

On the orbital stability of pendulum-like motions of a rigid body in the Bobylev-Steklov case

We deal with the problem of orbital stability of pendulum-like periodic motions of a heavy rigid body with a fixed point. We suppose that the geometry of the mass of the body corresponds to the Bobylev-Steklov case. Unperturbed motion represents oscillations or rotations of the body around a principal axis, occupying a fixed horizontal position. The problem of the orbital stability is considered on the basis of a nonlinear analysis.

First-order rotation solution of an oblate rigid body under the torque of a perturber in circular orbit

Aims. We discuss the rotational motion of an axisymmetric rigid body under the torque exerted by a massive point orbiting in a circular Keplerian orbit. The perturbation of the torque-free motion is modelled by applying MacCullagh’s approximation. Methods. Through successive approximations we use the unperturbed motion solution to get a first order solution for the Hamilton equations, which we apply to the complete problem. Results. The solution of the perturbed problem is computed in closed form without the usual series expansions, and shows the main effects on the dynamics of the rigid body that are introduced by the perturbation. Remarkably, special configurations are discovered, for which the perturbation only introduces periodic oscillations in the torque-free motion. An application to the dwarf-planet Ceres shows the reliability of the analytical solution.

Energy transfer in binary collisions of two gyrating charged particles in a magnetic field

Binary collisions of the gyrating charged particles in an external magnetic field are considered within a classical second-order perturbation theory, i.e., up to contributions that are quadratic in the binary interaction, starting from the unperturbed helical motion of the particles. The calculations are done with the help of a binary collision treatment, which is valid for any strength of the magnetic field and involves all harmonics of the particles’ cyclotron motion. The energy transfer is explicitly calculated for a regularized and screened potential, which is both of finite range and nonsingular at the origin. The validity of the perturbation treatment is evaluated by comparing with classical trajectory Monte Carlo (CTMC) calculations which also allow one to investigate the strong collisions with large energy and velocity transfer at low velocities. For large initial velocities, on the other hand, only small velocity transfers occur. There the nonperturbative numerical CTMC results agree excellently with the predictions of the perturbative treatment.

Long-Latency Responses During Reaching Account for the Mechanical Interaction Between the Shoulder and Elbow Joints

Although considerable research indicates that reaching movements rely on knowledge of the arm's mechanical properties and environment to anticipate and counter predictable loads, far less research has examined whether this degree of sophistication is present for on-line corrections during reaching. Here we examine the R2/3 response to mechanical perturbations (45-100 ms, also called the long-latency reflex), which is highly flexible and includes the fastest possible contribution from primary motor cortex, a key neural substrate for self-initiated action. Torque perturbations were occasionally and unexpectedly applied to the subject's shoulder and/or elbow in the course of performing reaching movements. Critically, these perturbations would evoke different patterns of feedback corrections from a shoulder extensor muscle if it countered only the local shoulder displacement relative to unperturbed motion or accounted for the mechanical interactions between the shoulder and elbow joints and countered the underlying shoulder torque. Our results show that the earliest response (R1: 20-45 ms) reflected local shoulder displacement, whereas the R2/3 response (45-100 ms) reflected knowledge of multijoint dynamics. Moreover, the same pattern of feedback occurred whether the shoulder muscle helped initiate the movement (during its agonist phase) or helped terminate the movement (during its antagonist phase). These results contribute to the accumulating evidence that highly sophisticated feedback control underlies motor behavior and are consistent with a shared neural substrate, such as primary motor cortex, for feedforward and feedback control.

Binary collisions of charged particles in a magnetic field

Binary collisions between charged particles in an external magnetic field are considered within a classical second-order perturbation theory, i.e., up to contributions which are quadratic in the binary interaction, starting from the unperturbed helical motion of the particles. The calculations are done with the help of an improved binary collisions treatment which is valid for any strength of the magnetic field, where the second-order energy and velocity transfers are represented in Fourier space for arbitrary interaction potentials. The energy transfer is explicitly calculated for a regularized and screened potential which is both of finite range and nonsingular at the origin, and which involves as limiting cases the Debye (i.e., screened) and Coulomb potential. Two distinct cases are considered in detail: (i) the collision of two identical (e.g., electron-electron) particles, and (ii) the collision between a magnetized electron and a uniformly moving heavy ion. The energy transfer involves all harmonics of the electron cyclotron motion. The validity of the perturbation treatment is evaluated by comparing with classical trajectory Monte Carlo calculations which also allows to investigate the strong collisions with large energy and velocity transfer at low velocities. For large initial velocities on the other hand, only small velocity transfers occur. There the nonperturbative numerical classical trajectory Monte Carlo results agree excellently with the predictions of the perturbative treatment.

Holomorphy of coordinates with respect to eccentricities in the planetary three-body problem

The boundaries of the domains of holomorphy of the coordinates of unperturbed elliptic motion with respect to the eccentricities of planetary orbits are determined for the cases when any of the five anomalies of one of the planets-eccentric, true, tangential, or one of two mutual anomalies suggested by M.F. Subbotin—is used as an independent variable. The resulting equations are a generalization of the known equations for the boundaries of the domains of the holomorphy of coordinates for the cases when the time is the independent variable and determine the bisymmetric ovals, whose size and shape depend on the eccentricities and on the ratio of the planetary mean motions. The largest domains of holomorphy are obtained when the tangential anomaly or one of the Subbotin mutual anomalies is used. A function was found that conformally maps the domain of holomorphy to the unit disk. It was demonstrated that the application of any anomaly of the outer planet as the independent variable can result in a significant shrinking of the domain of the holomorphy of the coordinates of the inner planet, so that the analytic continuation of the initial power series with the center at the origin of the coordinates of a complex plane becomes impossible.

The motion of a railway wheelset

The rolling of a railway wheelset along rails without slipping is investigated taking the creep hypothesis into account. The wheelset is represented by two cones that have a common base, and the rails are represented by two circular cylinders with parallel axes. The kinematic characteristics of the unperturbed rolling motion of the wheelset, which occurs when the centre of mass moves along a straight line, and of the perturbed motion, which occurs when the centre of mass of the wheelset describes a sinusoidal trajectory, are determined. The constraint reactions are found for the motions investigated up to small second-order values of the perturbed variables. When the elastic properties of the material in the contact area are taken into account, the creep hypothesis is used, averaging over the fast variables is employed, and the value of the critical speed, above which the rectilinear rolling of the wheelset becomes unstable, is found using averaged equations. In the latter case a periodic mode with two time intervals when the wheel flanges come into contact with the rails is investigated. The reaction force, the work of the dry friction force, and the moment of the active forces needed to maintain the periodic mode are found at the flange/rail contact point within the dry friction model. The boundaries of the stability regions, the parameters of the periodic mode and the moment of the resistance forces as functions of the problem parameters are determined from the formulae obtained by analytical methods.

Logarithmic matrix norms in motion stability problems

The problem of the stability of the motions of mechanical systems, described by non-linear non-autonomous systems of ordinary differential equations, is considered. Using the logarithmic matrix norm method, and constructing a reference system, the sufficient conditions for the asymptotic and exponential stability of unperturbed motion and for the stabilization of progammed motions of such systems are obtained. The problem of the asymptotic stability of a non-conservative system with two degrees of freedom is solved, taking for parametric disturbances into account. Examples of the solution of the problem of stabilizing programmed motions – for an inverted double pendulum and for a two-link manipulator on a stationary base – are considered.

Linear problems of the stability of a type of rotation of a satellite about the centre of mass

The stability in the first approximation of the rotation of a satellite about a centre of mass is investigated. In the unperturbed motion the satellite performs, in absolute space, three rotations around the normal to the orbital plane in a time equal to two periods of rotation of its centre of mass in the orbit (Mercury-type rotation). Three cases of such rotations are considered: the rotations of a dynamically symmetrical satellite and a satellite, the central ellipsoid of inertia of which is close to a sphere, in an elliptic orbit of arbitrary eccentricity, and the rotation of a satellite with three different principal central moments of inertia in a circular orbit.

Nonlinear Dynamic Equations of Satellite Relative Motion Around an Oblate Earth

A J2 dynamic model of satellite relative motion in the form of differential equations with reference to the local vertical local horizontal (LVLH) coordinate is basic and critical to the study of satellite formation flying. Many versions [1–7] of J2 dynamic equations have been published. Thework [1] byKechichian presents an exact nonlinear relative model taking into account both J2 perturbation and air drag. Kechichian applied techniques of Newtonian mechanics and vector calculus to derive the relative dynamics. However, his model is very complex and does not explicitly include some critical components of the J2 acceleration. The calculation of these J2 acceleration components is by a tedious algorithm. The model’s complexity hampers its application in control designs. Other published dynamic models [2–7] were developed assuming the reference orbit as being unperturbed Keplerian motion; thus, modeling errors are introduced. In this Note, the satellite relative dynamics is studied based on the perturbed reference orbit, which is accurately described by a set of differential equations. We express the reference orbit in terms of reference satellite variables (RSV) and obtain a simpler representation than that by Kechichian. Furthermore, we use Lagrangian mechanics to derive the satellite relative dynamics, which is different from Kechichian’s approach. As a result, we establish an exact J2 nonlinear relative model independent of the right ascension of the ascending node. The satellite relativemotion is explicitly expressed in terms of very few physical parameters and a set of simple first-order differential equations. This simplicity is due to the fact that both the spherical and the J2 accelerations are independent of the change of the right ascension of the ascending node. II. J2 Reference Satellite Dynamics in a Rotating Frame

Synthesis of subminimum-time bounded control of system of rigid bodies

An algorithm of control of a system of rigid bodies in a form of feedback in special coordinates is discussed. The control is bounded, and it is also subminimum-time in the sense that it coincides with the optimal in linear approximation and in the case of a system with one degree of freedom. It is shown that the controller endows the unperturbed motion of the system with the property of asymptotic stability in large and is also robust. The results of numerical experiments, demonstrating the closeness of the trajectories to the time-optimal ones, are presented.

Osculating orbits in Schwarzschild spacetime, with an application to extreme mass-ratio inspirals

We present a method to integrate the equations of motion that govern bound, accelerated orbits in Schwarzschild spacetime. At each instant the true worldline is assumed to lie tangent to a reference geodesic, called an osculating orbit, such that the worldline evolves smoothly from one such geodesic to the next. Because a geodesic is uniquely identified by a set of constant orbital elements, the transition between osculating orbits corresponds to an evolution of the elements. In this paper we derive the evolution equations for a convenient set of orbital elements, assuming that the force acts only within the orbital plane; this is the only restriction that we impose on the formalism, and we do not assume that the force must be small. As an application of our method, we analyze the relative motion of two massive bodies, assuming that one body is much smaller than the other. Using the hybrid Schwarzschild/post-Newtonian equations of motion formulated by Kidder, Will, and Wiseman, we treat the unperturbed motion as geodesic in a Schwarzschild spacetime with a mass parameter equal to the system's total mass. The force then consists of terms that depend on the system's reduced mass. We highlight the importance of conservative terms in this force, which cause significant long-term changes in the time dependence and phase of the relative orbit. From our results we infer some general limitations of the radiative approximation to the gravitational self-force, which uses only the dissipative terms in the force.

A study of evolution of spin-up mode of a satellite in the plane of its orbit

We investigate the mode of spinning up a low-orbit satellite in the plane of its orbit. In this mode the satellite rotates around its principal central axis of the minimum moment of inertia which executes small oscillations with respect to the normal to the orbit plane; the angular velocity of the rotation around this axis several times exceeds the mean orbital motion. Gravitational and restoring aerodynamic moments are taken into account in the satellite’s equations of motion. A small parameter characterizing deviation of the satellite from a dynamically symmetric shape is introduced into the equations. A two-dimensional integral surface of the equations of motion, describing quasi-steady-state rotations of the satellite close to cylindrical precession of the corresponding symmetrical satellite in a gravitational field, has been studied by the method of small parameter and numerically. Such quasi-steady-state rotations are suggested to be considered as unperturbed motions of the satellite in the spin-up mode. Investigation of the integral surface is reduced to numerical solution of a periodic boundary value problem of a certain auxiliary system of differential equations and to calculation of quasi-steady-state rotations by the two-cycle method. A possibility is demonstrated to construct quasi-steady rotations by way of minimization of a special quadratic functional.