Planar dynamics of variable length multi-tethered spacecraft near collinear Lagrangian points

This paper examines the planar dynamics of a wheel-and-spoke configured multi-spacecraft system, connected together by variable length tethers, near the second Sun–Earth Lagrangian point. The closed form solutions of the system under some simple tether length functions are determined and numerical results for the tether pitch librations under more complex tether length functions are obtained, along with the control effort required to maintain the desired tether librations.     

Stability and control of electrodynamic tethers for de-orbiting applications

Electrodynamic tethers provide a very promising propulsion system for de-orbiting of spent upper stages or LEO satellites. In this application, the Lorentz force generated by the interaction between the current in the wire and the geomagnetic field produces an electrodynamic drag leading to a fast orbital decay. The attractiveness of tether system lies especially in their capability to operate with uncontrollable satellites and in the modest mass requirement.The need for significant along-track forces leads however to the onset of an undesirable torque which, if not controlled, may drive the system into a dangerous instability. The electrodynamic torque determines in-plane and out-of-plane librations whose amplitude depends upon the current in the wire, mass distribution and system dimensions. Even more important, this torque is modulated along the orbit due to the changing magnetic field and ionospheric plasma density, giving rise to forced oscillations. The counteracting (and stabilizing) gravity-gradient torque is generally to small to ensure stability in typical, strongly non-symmetrical mass distributions, where a massive satellite or upper stage is attached at the lower end and a light electron collecting device (or passive ballast mass) is deployed a few kilometers above. Reducing the electron current or increasing the mass at the upper end are both unattractive solutions.In this paper we show how the electrodynamic torque pumps energy into the system (finally leading to large librations angles) and indicate that many proposed configurations are intrinsically unstable. Our results point out the need for a control strategy. Fortunately, the librations amplitudes can be limited by acting on the current flowing in the wire. Our model of a rigid, conductive tether shows that a control based upon timely current switch-off, using energy criteria, is indeed effective and simple to implement. The resultant duty-cycles are satisfactory and affect only marginally the de-orbiting times.     

Control of the deployment of an orbital tether system that consists of two small spacecraft

Control of an orbital tether system that consists of two small spacecraft has been considered. The proposed control laws are based on the modification of well-known programs for the deployment of tether system systems under the assumption that the masses of spacecraft and the tether are comparable in magnitude. To construct nominal deployment programs, we have developed a mathematical model of the motion of the given system in an orbital moving coordinate system taking into account the specific features of this problem. The performance of the proposed deployment programs is assessed by a mathematical model of the orbital tether system with distributed parameters written in the geocentric coordinate system. The test calculations involve a linear regulator that implements feedback on the tether length and velocity.     

YES2 optimal trajectories in presence of eccentricity and aerodynamic drag

YES2 (launching 2007) aims to demonstrate a tether-assisted re-entry concept, whereby payload will be returned to Earth using momentum provided from a swinging tether. Deployment takes place in two phases: (1) deployment of 3.5 km of tether to the local vertical and hold, and (2) deployment to 30 km for a swinging cut. Optimal trajectories are determined for both phases after comparing the effect of different cost functions on the deployment dynamics. Closed-loop control is provided by linearizing the dynamics around the optimal trajectories and solving a receding horizon control problem for a set of linear feedback gains. The controllers are tested in a flexible tether model with large disturbances to the hardware model and environmental variables. Closed-loop simulations show that the system can be controlled quite well using only feedback of length and length rate.     

Orbital and relative motion of a tethered satellite system

A simplified model for the orbital and relative motion of a tethered satellite system is presented. The tether acts as a light elastic string with small structural damping but without bending stiffness. Its mass is taken into account in the calculation of the total kinetic and potential energies of the tethered system. This formulation allows the inclusion of the complete gravity gradient influence on the dynamics of the system. The resulting three-dimensional motion is separated in the centre of mass orbital motion on the one hand and the relative motion of the end-bodies on the other. No restrictions on length of the tether or on mass ratio of the end-masses are imposed. It is found that the frequencies and amplitudes of the longitudinal tether oscillations are realistic as long as the tether mass is less than that of the subsatellite.     

Experiments and numerical simulations of an electrodynamic tether deployment from a spool-type reel using thrusters

The amount of space debris is ever increasing, and pollution of the space environment has become a serious problem that can no longer be ignored. Consequently, the active removal of large space debris from crowded economically useful orbits should begin as soon as possible. The Japan Aerospace Exploration Agency has been investigating an active debris removal system that employs highly efficient electrodynamic tether (EDT) technology for orbital transfer. This study investigates the tether deployment from a spool-type reel using thrusters by means of numerical simulations of an EDT system. The thrusters are used in order to ensure the deployment of a tether with the length of several kilometers. In the simulations using a multiple mass tether model, the key parameters are estimated from various on-ground experiments. By means of the numerical simulations, the dynamics of tether deployment is studied and requirements of thruster needed for the deployment, such as the thrust forces and the periods of thruster activation, are clarified.     

Tether-platform coupled control

From the control point of view, tethered systems pose several challenges, the major one pertaining to the regulation of the unstable system dynamics during the retrieval phase. On the other hand, the system configuration permits design of controllers using length rate, tension and offset schemes, which are not feasible with other satellites. Here “offset” refers to the time dependent variation of the tether attachment point at the platform end. The present paper studies several applications of the offset scheme in controlling the tethered systems. To that end, planar equations of motion of a space platform based Tethered Satellite System (TSS) are derived by the Lagrangian procedure. This is followed by representative results aimed at the offset control of platform pitch, tether attitude and vibration motions. The offset scheme is used for simultaneous control of platform and tether pitch motion. Finally the attention is directed towards simultaneous regulation of the platform pitch and longitudinal tether vibration. The numerical results clearly show considerable promise for the offset control scheme in regulating tether, platform and combined tether-platform dynamics.     

TETHERED SATELLITE SYSTEM ANALYSIS (1) — TWO-DIMENSIONAL CASE AND REGULAR DYNAMICS

The study on tethered satellite system (TSS) in two-dimensional in-planar motion is restricted in that the tether is assumed to be massless. The equations of motion are given in a spherical coordinate system to describe the magnitude (tether length) and direction angle of the position vector between the satellites. A length rate control algorithm is adopted, and the controlled motion of the directional angle by the algorithm will have a stable equilibrium state. The equilibrium state is a fixed point if the orbit of the base-satellite is circular, and a limit cycle if the orbit is elliptic. The value and stability of the equilibrium state are determined by the parameters of the control algorithm, and the bifurcation analysis is also given. Two typical TSS missions have been simulated.     

Optimal control of the Shuttle-Tethered-Subsatellite system

Alternate control laws based on an application of the linear regulator problem are developed for possible use in the operation of the Shuttle-Tethered-Subsatellite system. Control is assumed to be provided only by modulating the tension level in the tether as a function of the difference between actual and commanded tether line length, length rate, in (orbital) plane swing angle and swing angle rate. Necessary and sufficient conditions for stability of the (linear) system motion in the vicinity of its nominal local vertical orientation are developed. By proper selection of the state and control penalty matrices it is possible to obtain faster responses with no increase in maximum power levels for use in station keeping when compared with alternate control strategies. The weighting matrices are adjusted in a piecewise adaptive manner to provide control law gains in order to achieve a smooth deployment history. Successful retrievel is dependent mainly on the initial conditions and the rate at which the commanded length is reduced.     

Theory of physical libration of the Moon with the liquid core: Forced librations

For a two-layer model of the Moon that consists of a solid nonspherical mantle and an ellipsoidal homogeneous liquid core, a theory of forced librations under the effect of gravitational Earth’s moments has been developed. The motion of the Moon over its orbit has been described by the high-accuracy theory of DE/LE-4 orbital motion. Tables have been constructed that present forced librations of the Moon caused by the second harmonic of its force function, in the neighborhood of its motion according to the generalized Cassini laws. Disturbances of the first-order with respect to dynamic compressions of the Moon and its core are obtained in analytical form for Andoyer variables and Poincare variables and for the projection of the angular velocity vector of Moon’s mantle rotation and the Poincare coordinate system (relative to which core’s liquid accomplishes simple motion) on its major central axes of inertia, as well as for the classical variables in the Moon libration theory, etc. Constructed tables of the forced librations theory give the amplitudes and periods of librations and combinations of arguments of the orbital motion theory that correspond to libration parameters. The interpretation of basic variations has been given and a comparison with the previous theories has been carried out, in particular with the modern empirical theory constructed based on the laser observation data.     

Analysis of the dynamics of the deployed aerodynamic space tether system

An analysis of the motion of a deployed space system that consists of two end bodies connected by a tether has been considered. One of the bodies has a relatively large ballistic coefficient that ensures aerodynamic braking or the stabilization of the motion of the entire system in relatively low near-Earth orbits. The deployment of this system mainly occurs due to the action of aerodynamic forces. Several ways of deploying the system have been analyzed, including (1) the uncontrolled release of the tether with hardly any braking; (2) deployment with constant braking force; (3) the dynamic control law without feedback, when the resistance force varies according to a set program; (4) a kinematic control law with feedback when programs are set for varying the velocity and length of the tether release. To analyze the dynamics of the system, a mathematical model of motion has been constructed in which the motion of the end bodies relative to their centers of mass is taken into account.     

Ground assisted rendezvous with geosynchronous satellites for the disposal of space debris by means of Earth-oriented tethers

Previous studies have shown that extended length Earth-oriented tethers in the geosynchronous (GEO) region can be used to re-orbit satellites to disposal orbits. One such approach involves the extension of a GEO based tether, collection of a debris object, and retraction of the tether, which transfers the retracted configuration to a higher energy orbit for debris disposal. The re-extension of the tether after debris disposal returns the configuration to the near-GEO altitude. The practical feasibility of such a system depends on the ability to collect GEO debris objects, attach them to a deployed tether system, and retract the tethers for transfer to the disposal orbits.This study addresses the collection and delivery of debris objects to the deployed tether system in GEO. The investigation considers the number, type and the characteristics of the debris objects as well as the collection tug that can be ground controlled to detect, rendezvous and dock with the debris objects for their delivery to the tethers system.A total of more than 400 objects are in drift orbits crossing all longitudes either below or above the geostationary radius. More than 130 objects are also known to librate around the stable points in GEO with periods of libration up to five or more years. A characterization of the position and velocity of the debris objects relative to the collection tug is investigated. Typical rendezvous performance requirements for uncooperative GEO satellites are examined, and the similarities with other approaches such as the ESA's CX-OLEV commercial mission proposal to extend the life of geostationary telecommunication satellites are noted.     

Electrodynamic Tethered Deorbit System Dynamics and Performance Analysis

Electrodynamic tethered deorbit technology is a novel way to remove abandoned spacecrafts like upper stages or unusable satellites. This paper investigates and analyses the deorbit performance and mission applicability of the electrodynamic tethered system. To do so, the electrodynamic tethered deorbit dynamics with multi-perturbation is firstly formulated, where the Earth magnetic field, the atmospheric drag, and the Earth oblateness effect are considered. Then, the key system parameters, including payload mass, tether length and tether type, are analyzed by numerical simulations to investigate their influences on the deorbit performance and to give the setting principles for choosing system parameters. Based on this and given an appropriate group of system parameters, numerical simulations are undertaken to study the impact of the mission parameters, including orbit height and orbit inclination, and thus to investigate the mission applicability of the electrodynamic tethered deorbit technology.     

系统参数对电动力绳系动力学的影响

将电动力绳系(EDT)的主星质量、子星质量、绳系质量以及绳系中的电流视为系统参数，研究这些参数对系统的摆动动力学和轨道动力学的影响。哑铃模型下的电动力绳系摆动动力学方程存在不稳定的周期解，通过Floquet理论来衡量周期解的不稳定程度，从而研究各系统参数对摆动动力学的影响。建立了用春分点轨道元素的形式描述的电动力绳系轨道动力学方程，并以降轨时间来衡量电动力绳系的降轨效率，从而研究系统参数对轨道动力学的影响。运用算例对周期解迁移矩阵的特征值、降轨时间随各系统参数的变化关系进行了仿真，分别得出了各系统参数对系统摆动动力学和轨道动力学的影响。综合本文的仿真结果，并考虑实际发射及空间运行中的其它因素，对电动力绳系的设计和降轨策略提出了建议。     

Satellite pitch and roll attitude maneuvers through very short tethers

Explored here is the feasibility of achieving satellite pitch and roll attitude maneuvers through tethers. The proposed tethered satellite system (TSS) comprises of four identical tethers connecting the auxiliary mass to the satellite at its four distinct off-centered and equiangularly spaced points. The open-loop tether length control laws have been developed in order to achieve arbitrary pitch and roll attitude slewing maneuvers. Numerical simulation of the nonlinear governing equations of motion for these tether length variations establishes the feasibility of executing fixed as well as chase-slewing maneuvers. Nearly passive nature of the proposed mechanism using very short tethers along with small auxiliary mass needed makes the concept particularly attractive for future space missions.     

空间绳系机器人抓捕非合作目标的质量特性参数辨识

针对空间绳系机器人抓捕非合作目标/空间垃圾后需要对其进行回收/拖曳的精确控制问题,提出了一种利用抓捕后保持阶段的振动特性辨识目标参数的方法。首先,根据质量特性参数辨识的需要,推导了系统的动力学模型。不同于以往将本体卫星和被抓捕目标简化为质点的动力学模型,本文针对任意的目标抓捕位置,在考虑重力梯度影响的基础上,利用拉格朗日法获得系统各广义坐标的动力学公式。然后,分析非合作目标和系绳在后抓捕保持阶段的姿态运动。最后,在非合作目标与本体卫星没有任何信息交互的情况下,利用后抓捕阶段目标卫星和系绳特有的振动,并使用具有鲁棒性可遗忘因子的递推最小二乘法,提出了包括转动惯量和质心到任意抓捕点距离在内的质量特性参数辨识算法。     

Stability of a ring of connected satellites

The motion of a large number of artificial satellites connected in a ring one after another by tethers of variable length is considered. Every satellite is supposed to have a control system programmed according to some tether tension law as a function of the distance between tethered satellites. The effect of the tension control law on the stability of stationary rotation of this ring is investigated. The final stability condition includes two requirements: 1) the nominal tether tension should be less than a definite limit equal, up to numerical coefficient, to one satellite weight divided by the number of satellites; 2) tether tension should decrease (or remain constant) with the increase of the distance between tethered satellites. In dynamics the artificial rings of this kind are much like their natural prototype—meteor rings. On the other hand, the investigation of the artificial rings contributes to developing an unexpected view upon meteor rings, suggesting a model of an imaginary equivalent string.     

Chaotic oscillations of spacecraft with an elastic radially oriented tether

A mechanical system consisting of a spacecraft with weightless elastic tether and load is considered in the paper. The spacecraft motion under the action of tensile force of a radially oriented tether is investigated. It is shown that elastic tether oscillations can result in appearance of chaotic modes of spacecraft motion. By means of the Melnikov method a condition is obtained, allowing one to determine the measure of damping sufficient for prevention of these chaotic modes. The influence of system’s mass-geometric and elastic characteristics on the form of phase portrait and on the value of periodic disturbance, caused by oscillations of the elastic vertical tether, is studied.     

Mechanics of very long tethered systems

A space elevator has been proposed as an alternate method for launching satellites; however, the materials available now are not strong enough to support the stress generated in the structure. On the other hand, with the existing technology, a partial elevator is feasible. In this paper, the mechanics of a very long tethered system that functions as a partial elevator is studied. For such a system, the center of mass, center of gravity, and center of orbit are not coincident; disregarding this distinction can lead to erroneous results. A relation between these three points is presented in this paper. A consistent stress distribution along the tether is obtained by taking into account the distinction between these points. Dynamics of the system consisting of two end bodies, the tether (with mass), and a climber is examined. The equations of motion are derived using the Lagrangian formulation and analyzed numerically.