Stability and control of transversal oscillations of a tethered satellite system

The tethered satellite system is characterized by weak nonlinearities but it practically works in conditions of internal resonance which produces unstable oscillations. The effect of a longitudinal control force is investigated. Since the displacement component in the orbit plane is always present in the motion due to the nonlinear coupling, the control force is assumed depending only on this component and also when a prevailing out-of-plane oscillation is considered. The harmonic balance method and numerical solutions of amplitude modulated equations are used to obtain stationary and nonstationary oscillations, respectively; the Floquet theory is followed in the stability analysis. The assumed control force is shown to be effective in reducing the primary and secondary instability regions of oscillations perturbed by internally resonant disturbance components.     

A targeting strategy for the deployment of a tethered satellite system

^** Email: hans.troger{at}tuwien.ac.at One of the most important operations during a tethered satellitesystem mission is the deployment of a subsatellite from a spaceship.We restrict to the simple, but practically important case thatthe system is moving on a circular Keplerian orbit around theEarth. The main problem during deployment due to the local gravitygradient vector is that due to the Coriolis acceleration thetwo satellites do not move along the straight radial relativeequilibrium position in the orbital frame. Instead, a strongdeviation from the local vertical direction occurs, which afterthe deployment process is finished results in weakly dampedlarge-amplitude oscillations, which in some cases are even transientchaotic. This chaotic dynamics will be used to steer the satellitewith small control actions into the final radial relative equilibriumposition far away from the spaceship. Both deployment time andenergy input are computed and compared to other deployment strategies.     

Center manifold approach to the control of a tethered satellite system

It is well known that in a linearized analysis the in-plane oscillation of a tethered satellite system about the radial earth pointing position decouples from the out-of-plane oscillation. By tension control, therefore, only the in-plane but not the out-of-plane oscillation can be affected. Hence, using tension control linearization of the equations of motion cannot be used and a nonlinear problem must be treated. For a simple mechanical model of a tethered satellite system we show by means of center manifold theory that for the nonlinear system the out-of-plane oscillations can be stabilized by tension control.     

Fractional order attitude stability control for sub-satellite of tethered satellite system during deployment

This paper investigates the sub-satellite attitude stabilization control of a tethered satellite system (TSS) during deployment stage. The dynamic model of sub-satellite motion is first established using Euler equations. During the tether deployment stage, the stability characteristics of attitude motion are analyzed with dissymmetric junction points between tether and sub-satellite. Then, a fractional-order attitude controller with memory ability is proposed to achieve stable control of the sub-satellite attitude, in which a dynamic model is linearized by using the feedback linearization method. Finally, validity of the fractional order controller and the advantages over an integer order controller are examined using numerical simulation. Comparing with the corresponding integer order controller, numerical simulation results indicate that the proposed sub-satellite attitude controller based on fractional order can not only stabilize the sub-satellite attitude, but can also respond faster with smaller overshoot.     

Quadrature discretization method in tethered satellite control

A pseudospectral method for solving the tethered satellite retrieval problem based on nonclassical orthogonal and weighted interpolating polynomials is presented. Traditional pseudospectral methods expand the state and control trajectories using global Lagrange interpolating polynomials based on a specific class of orthogonal polynomials from the Jacobi family, such as Legendre or Chebyshev polynomials, which are orthogonal with respect to a specific weight function over a fixed interval. Although these methods have many advantages, the location of the collocation points are more or less fixed. The method presented in this paper generalizes the existing methods and allows a much more flexible selection of grid points by the arbitrary selection of the orthogonal weight function and interval. The trajectory optimization problem is converted to set of algebraic equations by discretization of the necessary conditions using a nonclassical pseudospectral method.     

Dynamics of variable-length tethers with application to tethered satellite deployment

The dynamics of variable-length tethers are studied using a flexible multibody dynamics method. The governing equations of the tethers are derived using a new, hybrid Eulerian and Lagrangian framework, by which the mass flow at a boundary of a tether and the length variation of a tether element are accounted for. The variable-length tether element based on the absolute nodal coordinate formulation is developed to simulate the deployment of satellite tethers. The coupled dynamic equations of tethers and satellites are obtained using the Lagrangian multiplier method. Several tethered satellite systems involving large displacements, rotations, and deformations are numerically simulated, where the tethers are released from several meters to about 1 km. A control strategy is proposed to avoid slackness of the tethers during deployment. The accuracy of the modeling and solution procedures was validated on an elevator model.     

Rotations of a near-symmetrical satellite in an elliptical orbit with Mercury-type resonance

The motion of a satellite, i.e., a rigid body, about to the centre of mass under the action of the gravitational moments of a central Newtonian gravitational field in an elliptical orbit of arbitrary eccentricity is investigated. It is assumed that the satellite is almost dynamically symmetrical. Plane periodic motions for which the ratio of the average value of the absolute angular velocity of the satellite to the average motion of its centre of mass is equal to 3/2 (Mercury-type resonance) are examined. An analytic solution of the non-linear problem of the existence of such motions and their stability to plane perturbations is given. In the special case in which the central ellipsoid of inertia of the satellite is almost spherical, the stability to spatial perturbations is also examined, but only in a linear approximation. ©2008.     

Stabilization of the radial equilibrium of a tethered satellite with respect to out-of-plane initial perturbations

We investigate the stabilization of the radial equilibrium of a tethered satellite by tension control, if both in-plane and out-of-plane deviations from the vertical position are taken into account. While in-plane perturbations can be eliminated in finite time, the length rate change of the tether acts as parametric control on the out-of-plane deviations. Therefore the control becomes less effective, if the perturbation decreases. In order to improve the convergence we investigate the optimal control problem assuming no costs for the applied tension force but restricting the control force to a finite interval. For tether cconfigurations close to the vertical target position this approach yields a solution with singular control, which leads to a faster decay of the deviations. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Maximizing the value of an Earth observation satellite orbit

Earth observation satellites are platforms equipped with optical instruments that orbit the planet. During the course of an orbit, they take photographs of some regions of the Earth at the request of customers. Each photograph generates a profit but, due to the presence of several constraints, not all requests can be satisfied. The problem is to select a subset of requests of maximal profit for a given orbit. The problem is solved by means of a tabu search heuristic and computational results are reported. This work was initiated as part of a challenge organized by the French Operational Research Society. The algorithm won the second prize in the final round of the competition.     

The oscillations of a body with an orbital tethered system

The motion about a centre of mass of a rigid body with a tethered system, designed to launch a re-entry capsule from a circular orbit is considered. In the deployment of the tethered system the direction and value of the tensile strength of the tether vary and, if the point of application of the tensile strength does not coincide with the centre of mass of the body, a moment occurs which leads to oscillations of the body with variable amplitude and frequency. A non-linear equation of the perturbed motion of the body about the centre of mass under the action of the tensile force of the tether and the gravitational moment is derived. Assuming that the change in the value and direction of the tensile force is slow and also that the gravitational moment is small, approximate and exact solutions of the non-linear differential equation of the unperturbed motion are obtained in terms of elementary functions and elliptic Jacobi functions. For perturbed motion, the action integral is expressed in terms of complete elliptic integrals of the first and second kind.     

On the stability of resonant rotation of a symmetric satellite in an elliptical orbit

We deal with the stability problem of resonant rotation of a symmetric rigid body about its center of mass in an elliptical orbit. The resonant rotation is a planar motion such that the body completes one rotation in absolute space during two orbital revolutions of its center of mass. In [1–3] the stability analysis of the above resonant rotation with respect to planar perturbations has been performed in detail.In this paper we study the stability of the resonant rotation in an extended formulation taking into account both planar and spatial perturbations. By analyzing linearized equations of perturbed motion, we found eccentricity intervals, where the resonant rotation is unstable. Outside of these intervals a nonlinear stability study has been performed and subintervals of formal stability and stability for most initial data have been found. In addition, the instability of the resonant rotation was established at several eccentricity values corresponding to the third and fourth order resonances.Our study has also shown that in linear approximation the spatial perturbations have no effect on the stability of the resonant rotation, whereas in a nonlinear system they can lead to its instability at some resonant values of the eccentricity.     

On a closed orbit of some constrained system

We study on what one calls a constrained system of ODEs on It consists of two ordinary differential equations and an algebraic equation with respect to three unknown functions. We seek closed orbits of such a system. A necessary and sufficient condition for the system to have non-trivial closed orbits is given. Elementary tools such as Lyapunov functions and Poincaré’s index theory are used in the proof of the result. As an application we consider a constrained system associated with a non-constraint system of ODEs called the modified Bonhöffer-van der Pol system.     

Problem of algorithm in precision orbit determination

In orbit determination, the precision ephemeris and state transition matrix are usually obtained by solving two groups of ordinary differential equations with numerical integration method due to the complexity of the force models. A kind of simplified analytical method to compute the state transition matrix is given. The method is not only very efficient for the case where the orbit arc is not too long, but also can avoid the integration of two groups of ordinary differential equations at the same time. Some practical test examples also show the efficiency of the method.     

Continuous orbit transitions in a one-dimensional inelastic particle system

Continuous transitions between different periodic orbits in a one-dimensional inelastic particle system with two particles are investigated. We explain why continuous transitions that occur when adding or subtracting a single collision are, generically, of co-dimension 2. However, we show that there are an infinite set of degenerate transitions of co-dimension 1. We provide an analysis that gives a simple criteria to classify which transitions are degenerated purely from the discrete set of collisions that occur in the orbits.     

A fuzzy clustering application to precise orbit determination

In recent years, fuzzy logic techniques have been successfully applied in geodesy problems, in particular to GPS. The aim of this work is to test a fuzzy-logic method with an enhanced probability function as a tool to provide a reliable criteria for weighting scheme for satellite-laser-ranging (SLR) station observations, seeking to optimize their contribution to the precise orbit determination (POD) problem. The data regarding the stations were provided by the International Laser Ranging Service (ILRS), NASA/Crustal Dynamics Data Information System (CDDIS) provided the satellite data for testing the method. The software for processing the data is GEODYN II provided by NASA/Goddard Space Flight Center (GSFC). Factors to be considered in the fuzzy-logic clustering are: the total number of LAGEOS passes during the past 12 months, the stability measure of short- and long-term biases, the percentage of LAGEOS normal points that were accepted in CSR weekly LAGEOS analysis, and the RMS uncertainty of the station coordinates. A fuzzy-logic statistical method allows classifying the stations through a clear ‘degree of belonging’ to each station group. This degree of belonging translates into a suitable weight to be assigned to each station in the global solution. The first tests carried out showed improvements in the RMS of the global POD solution as well as individual stations, to within a few millimeters. We expect further work would lead to further improvements.     

On the determination of the basin of attraction of a periodic orbit in two-dimensional systems

We consider the general nonlinear differential equation with x∈R² and develop a method to determine the basin of attraction of a periodic orbit. Borg's criterion provides a method to prove existence, uniqueness and exponential stability of a periodic orbit and to determine a subset of its basin of attraction. In order to use the criterion one has to find a function W∈C¹(R²,R) such that LW(x)=W^′(x)+L(x) is negative for all x∈K, where K is a positively invariant set. Here, L(x) is a given function and W^′(x) denotes the orbital derivative of W. In this paper we prove the existence and smoothness of a function W such that LW(x)=−μ‖f(x)‖. We approximate the function W, which satisfies the linear partial differential equation W^′(x)=〈∇W(x),f(x)〉=−μ‖f(x)‖−L(x), using radial basis functions and obtain an approximation w such that Lw(x)<0. Using radial basis functions again, we determine a positively invariant set K so that we can apply Borg's criterion. As an example we apply the method to the Van-der-Pol equation.     

Algorithm of orbit determination using one or two GPS satellites

The problem of orbit determination using one or two GPS satellites is discussed. Methods of getting initial values based on linear translation is presented; the Secant method and the descend Newton iterative procedure and the continuation algorithm are used synthetically to solve the nonlinear equations. Computer simulation shows that this algorithm can give preliminary state of satellite orbit with a certain precision in short time.     

Geometric orbit datum and orbit covers

Vogan conjectured that the parabolic induction of orbit data is independent of the choice of the parabolic subgroup. In this paper we first give the parabolic induction of orbit covers, whose relationship with geometric orbit datum is also induced. Hence we show a geometric interpretation of orbit data and finally prove the conjugation for geometric orbit datum using geometric method.     

The semigroup generated by a similarity orbit or a unitary orbit of an operator

Let be an invertible operator that is not a scalar modulo the ideal of compact operators. We show that the multiplicative semigroup generated by the similarity orbit of is the group of all invertible operators. If, in addition, is a unitary operator, then the multiplicative semigroup generated by the unitary orbit of is the group of all unitary operators.