General dynamics of a large class of flexible satellite systems

The paper presents a general formulation for librational dynamics of satellites with an arbitrary number, type and orientation of deploying flexible appendages. In particular, the case of beam-type flexible appendages deploying from a satellite in an arbitrary orbit is considered. The governing nonlinear, nonautonomous and coupled equations for vibration of the appendages and libration of the satellite are integrated numerically. Several cases of practical importance are considered making the system progressively more general and hence complex: (i) planar case representing pitch and appendage oscillations in the orbital plane; (ii) general attitude motion with planar vibrations of flexible members; (iii) above two cases together with the out-of-plane component of vibrations. Results show that under critical combinations of the system parameters the combined effect of flexibility and deployment can be substantial.     

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.     

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.     

Motion of a descent capsule relative to its center of mass when deploying the orbital tether system

The spatial motion relative to the center of mass is considered for a capsule on an elastic tether, when it is unrolled from a spacecraft by a special program. The spacecraft is in a circular orbit and oriented relative to the local vertical, which is guaranteed by operation of its own stabilization system. Angular motion of the capsule relative to the tether direction is studied, and the main factors influencing the stability of this motion are analyzed. An approximate quasi-linear mathematical model of the capsule attitude motion is obtained, which allows one to estimate the influence of major disturbances of its motion. The results of numerical simulations are presented for characteristic cases of the capsule motion.     

绳系卫星系统的运动与控制

绳系卫星系统的运动与控制分析,是绳系卫星应用的基础。本文的数学模型考虑了作用在子星、主星及系绳上的空气阻力以及在伸展与收回阶段主星与系绳之间的质量传递,给出了相应的具体的计算式。应用该数学模型,本文对现有的几种主要控制法,进行了模拟计算,作出评审意见。     

Optimal control of a spinning double-pyramid Earth-pointing tethered formation

The dynamics and control of a tethered satellite formation for Earth-pointing observation missions is considered. For most practical applications in Earth orbit, a tether formation must be spinning in order to maintain tension in the tethers. It is possible to obtain periodic spinning solutions for a triangular formation whose initial conditions are close to the orbit normal. However, these solutions contain significant deviations of the satellites on a sphere relative to the desired Earth-pointing configuration. To maintain a plane of satellites spinning normal to the orbit plane, it is necessary to utilize “anchors”. Such a configuration resembles a double-pyramid. In this paper, control of a double-pyramid tethered formation is studied. The equations of motion are derived in a floating orbital coordinate system for the general case of an elliptic reference orbit. The motion of the satellites is derived assuming inelastic tethers that can vary in length in a controlled manner. Cartesian coordinates in a rotating reference frame attached to the desired spin frame provide a simple means of expressing the equations of motion, together with a set of constraint equations for the tether tensions. Periodic optimal control theory is applied to the system to determine sets of controlled periodic trajectories by varying the lengths of all interconnecting tethers (nine in total), as well as retrieval and simple reconfiguration trajectories. A modal analysis of the system is also performed using a lumped mass representation of the tethers.     

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.     

Modifying the atlas of low lunar orbits using inert tethers

For long enough tethers, the coupling of the attitude and orbital dynamics may show non-negligible effects in the orbital motion of a tethered satellite about a central body. In the case of fast rotating tethers the attitude remains constant, on average, up to second order effects. Besides, for a tether rotating in a plane parallel to the equatorial plane of the central body, the attitude–orbit coupling effect is formally equal to the perturbation of the Keplerian motion produced by the oblateness of the central body and, therefore, may have a stabilizing effect in the orbital dynamics. In the case of a tethered satellite in a low lunar orbit, it is demonstrated that feasible tether lengths can help in modifying the actual map of lunar frozen orbits.     

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.     

Qualification and in-flight demonstration of a European tether deployment system on YES2

This paper highlights the design, qualification and mission performance of the tether deployer system on the second Young Engineers’ Satellite (YES2), that featured a tethered momentum transfer. The deployer is designed with a broad range of near-term tether applications in mind. The system contains the tether, including features to enhance safety and wound up in controlled manner onto a spool core, optical deployment sensors, a “barberpole” friction brake controlled by a stepper motor and a triple tether cutter system. To initiate the deployment a spring-based ejection system was developed, and to apply accurate momentum transfer a timer and release system is present on the subsatellite side. A small, 6 kg re-entry capsule was developed as subsatellite. On September 25th, 2007, YES2 deployed a 32 km tether in orbit and gathered a wealth of data. Confidence is gained from the mission results for use of the deployer in future missions.     

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.     

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.     

Transient attitude dynamics of satellites with deploying flexible appendages

The paper presents a general formulation for librational dynamics of satellites with an arbitrary number, types, and orientation of deploying flexible appendages. The generalized force term is incorporated making the formulation applicable to a wide variety of situations where aerodynamic forces, solar radiation, earth's magnetic field, etc. become significant. In particular, the case of a beam-type flexible appendage deploying from a satellite in an arbitrary orbit is considered. The corresponding nonlinear, non-autonomous equations for in-plane and out-of-plane vibrations are derived, allowing for the variation of mass density and flexural rigidity along the length with time dependent deployment velocity and spin rate. Next, the attention is focused on the linearized analysis of the in-plane vibrational equation using the assumed-mode method and its substantiation through numerical integration. Finally, the paper presents results for both steady-state and transient attitude behaviour for a representative gravity gradient configuration for a range of initial conditions and system parameters. Results show the combined effect of flexibility and deployment on the dynamics of the system to be substantial. Disturbance of the appendage can excite large amplitude librations. On the other hand, the converse situation is not necessarily true. Furthermore, Coriolis loading, induced by the extending appendages, can become a limiting factor in arriving at a deployment strategy; an effect not pointed out in the literature.     

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.     

Tether satellite system collision study

A study was performed to determine the probability of collision with resident space objects and untrackable debris for the tether component of the Tethered Satellite System (TSS) after it broke away from the space shuttle orbiter (mission STS-75) in February 1996. Both an analytical and a numerical approach were used in this study, and the results obtained with these two methods were found to be in good agreement. These results show that the deployed tether is expected to have been impacted by several particles 0.1 mm or larger in size. The probability of collision with objects 10 cm in size or larger was on the order of 10⁻³ per month. Since the severed tether reentered within one month after deployment, the collision hazard to other objects while in orbit was small. The analytical methods used in this study are general and can be applied to future tether collision evaluations.     

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。     

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.     

大型空间电站在轨展开与组装动力学与控制

针对大型空间电站在轨展开与组装过程中复杂的动力学环境,将桁架类大型空间电站视为一个刚柔多体系统,采用自然坐标法对刚性构件建模,基于绝对节点坐标方法描述柔性桁架结构,编写了一套动力学仿真软件,实现了大型空间电站在轨展开与组装动力学的精确仿真。仿真结果表明,采用摆线运动插值函数作为控制方程有利于提高多级展开过程的稳定性;在结构组装过程中,减小组装速度,增大组装机构阻尼系数与劲度系数,有利于提高结构稳定性。最终集成了一套针对大型空间电站的动力学仿真平台搭建方法,为大型空间电站在轨展开与组装动力学预示提供了理论依据和方法指导。     

绳系卫星实施安全捕获的能量最优控制

考虑系绳质量、系统质心变化及状态、控制约束,基于Lagrange方程给出了圆轨道条件下空间绳系网捕系统三维动力学模型.推导得到了零相对速度条件下的安全捕获末端条件,研究了面内安全捕获策略下的非线性能量最优控制问题.为保证方法的适用性,基于Legendre伪谱法将连续时间最优控制问题离散为标准的非线性动态规划问题.最后在...     

基于三阶解析解的小行星平衡点附近halo轨道确定方法研究

尽管周期解的存在性已经被证明,但要在给定的动力学系统中寻找到满足一定精度要求的周期解依然是一件极富挑战性的工作.提出如下方法确定小行星平衡点附近精确的周期轨道(halo轨道).首先扩展运动方程:将小行星平衡点附近轨道运动方程的右端项在平衡点处展成三阶幂级数.从而将非线性运动学方程扩展为拟线性微分方程.然后求近似解析解:应用Lindstedt-Poincaré方法求解扩展后的运动方程组,将周期解和其运动频率展开成三阶幂级数,并将二者代人扩展后的拟线性微分方程中.这样就可以得到三个不同阶的线性运动方程,逐次求解三个微分方程并消除解中的永年项即可得到hal.轨道的三阶解析解.最后微分校正:将周期轨道在三阶解析解附近线性化,得到状态转移矩阵,并使用状态转移矩阵和轨道终端状态的偏差修正轨道初值,从而得到满足精度要求的精确引力场中的halo轨道.