Orbit and attitude control of spacecraft formation flying

Formation flying is a novel concept of distributing the functionality of large spacecraft among several smaller, less expensive, cooperative satellites. Some applications require that a controllable satellite keeps relative position and attitude to observe a specific surface of another satellite among the cluster. Specially, the target space vehicle is malfunctioning. The present paper focuses on the problem that how to control a chaser satellite to fly around an out-of-work target satellite closely in earth orbit and to track a specific surface. Relative attitude and first approximate relative orbital dynamics equations are presented. Control strategy is derived based on feedback linearization and Lyapunov theory of stability. Further, considering the uncertainty of inertia, an adaptive control method is developed to obtain the correct inertial ratio. The numerical simulation is given to verify the validity of proposed control scheme.     

卫星编队飞行中相对轨道的J2摄动分析

详细分析了$J_{2}$摄动对编队卫星相对轨道构形的影响. $J_{2}$ 摄动对相对轨道的影响分为相对轨道构形的漂移、相对轨道平面的章动和进动. 首先, 分析了相对轨道构形漂移速度、章动角速度和进动角速度的一阶近似表达式的数量级及其影 响因素. 其次,给出一个准则,来判断同一相对轨道的漂移和转动之间的关系. 最后, 利用该准则,分析了主、从星的轨道根数差对相对轨道的漂移和转动的影响.     

USEFUL RELATIVE MOTION DESCRIPTION METHOD FOR PERTURBATIONS ANALYSIS IN SATELLITE FORMATION FLYING

Nomenclature OXYZEarth’sequatorialinertialreferenceframeωArgumentofperigee SlxyzLeadingsatelliteorbitframeMMeananomaly SfxyzFollowingsatelliteorbitframefTrueanomalyaSemi majoraxisθ=ω fArgumentoflatitude eEccentricitynMeanmotion iOrbitinclinationrSatel…     

Stability and control of tethered satellite with chemical propulsion in orbital plane

The tethered satellite with chemical propulsion has broad application prospects in the disposal of abandoned satellites, the orbital rescue of spacecrafts, and the transportation of space supplies, which is completely different from the traditional applications of tethered satellites. Therefore, new research on its dynamics, stability, and control becomes useful and interesting. In this article, based on a dumbbell model of tethered satellite, the dynamics equations of tethered system in orbital maneuvering are established. Furthermore, according to the definitions of transversal and radial propulsive coefficients, analytical solutions of the equilibrium position for librational angle are derived during maneuvering in orbital plane; meanwhile, the effects of propulsive coefficients on librational stability are analyzed, which provides a basis for a selection of expected attitude trajectory. Then, a method of hierarchical sliding-mode tension control is presented to track the expected in-plane angle. This method can address the underactuated problem of tethered systems without either complex coordinate transformation for the system state model or constraint equation restrictions. During orbital flight, in-plane and out-of-plane angles are decoupled, so the tether tension control cannot be conducted to inhibit the out-of-plane angle. To solve this problem, the binormal component of thrust acceleration normal to the orbital plane is adopted as a control variable, and a feedback linearization-based thrust controller is designed to damp out the out-of-plane angle. Afterwards, orbital transfer cases between two circular orbits are studied to demonstrate the effectiveness of the tethered satellite with chemical propulsion. Numerical simulation results indicate that the stability of librational angles has a close relation to propulsive coefficients, and distributions of stable centers and unstable saddle points are totally different on both sides of bifurcation point. In addition, tracking control requirements for tethered satellite are guaranteed by designed controllers, which ensure flight safety in orbital maneuvering.     

Study on relative orbital configuration in satellite formation flying

In this paper, the relative orbital configurations of satellites in formation flying with non-perturbation and J₂ perturbation are studied, and an orbital elements method is proposed to obtain the relative orbital configurations of satellites in formation. Firstly, under the condition of non-perturbation, we obtain many shapes of relative orbital configurations when the semi-major axes of satellites are equal. These shapes can be lines, ellipses or distorted closed curves. Secondly, on the basis of the analysis of J₂ effect on relative orbital configurations, we find out that J₂ effect can induce two kinds of changes of relative orbital configurations. They are distortion and drifting, respectively. In addition, when J₂ perturbation is concerned, we also find that the semi-major axes of the leading and following satellites should not be the same exactly in order to decrease the J₂ effect. The relationship of relative orbital elements and J₂ effect is obtained through simulations. Finally, the minimum relation perturbation conditions are established in order to reduce the influence of the J₂ effect. The results show that the minimum relation perturbation conditions can reduce the J₂ effect significantly when the orbital element differences are small enough, and they can become rules for the design of satellite formation flying.The project supported by the National Natural Science Foundation of China (10202008) and Specialized Research Fund for the Doctoral Program of Higher Education (20020003024)     

THE CONTROL OF MEGA-CONSTELLATION AT LOW EARTH ORBIT BASED ON TLE

近几年来,美国SpaceX, OneWeb等创新型企业纷纷计划打造低轨巨型卫星星座,引发卫星互联网的发展热潮。本文利用公开的两行轨道根数(two-line element, TLE)对国外星座控制策略进行分析,重点分析了铱星、一网、星链星座的控制规律。通过相对相位偏差分析,反演得到星座中不同卫星之间的平半长轴差,避免了小控制量条件下难以通过平半长轴判断卫星是否进行了轨控的问题。获得了国外星座控制频次和控制精度等重要信息,所得结论能够为我国未来互联网星座的建设提供参考。     

基于TLE的低轨巨星座控制研究

近几年来,美国SpaceX, OneWeb等创新型企业纷纷计划打造低轨巨型卫星星座,引发卫星互联网的发展热潮。本文利用公开的两行轨道根数(two-line element, TLE)对国外星座控制策略进行分析,重点分析了铱星、一网、星链星座的控制规律。通过相对相位偏差分析,反演得到星座中不同卫星之间的平半长轴差,避免了小控制量条件下难以通过平半长轴判断卫星是否进行了轨控的问题。获得了国外星座控制频次和控制精度等重要信息,所得结论能够为我国未来互联网星座的建设提供参考。     

Solution set on the natural satellite formation orbits under first-order earth＇s non-spherical perturbation

Using the reference orbital element approach, the precise governing equations for the relative motion of formation flight are formulated. A number of ideal formations with respect to an elliptic orbit can be designed based on the relative motion analysis from the equations. The features of the oscillating reference orbital elements are studied by using the perturbation theory. The changes in the relative orbit under perturbation are divided into three categories, termed scale enlargement, drift and distortion respectively. By properly choosing the initial mean orbital elements for the leader and follower satellites, the deviations from originally regular formation orbit caused by the perturbation can be suppressed. Thereby the natural formation is set up. It behaves either like non-disturbed or need little control to maintain. The presented reference orbital element approach highlights the kinematics properties of the relative motion and is convenient to incorporate the results of perturbation analysis on orbital elements. This method of formation design has advantages over other methods in seeking natural formation and in initializing formation.     

地球轨道空间碰撞风险区域态势分析

太空中的人造目标数量持续加速增长,地球轨道空间已变得愈发拥挤,在轨卫星运行中面临的碰撞风险日益升高。本文基于公开的大量空间目标编目轨道数据,对当前空间环境下在轨卫星面临的碰撞威胁程度进行了简要计算和分析,给出不同轨道区域中卫星的碰撞风险等级,并指出了几个碰撞风险显著偏高的典型区域。最后,从空间碎片环境安全角度出发,对卫星轨道选取提出建议,可为待发射卫星的轨道部署提供参考。

     

Dynamic analysis of a tethered satellite system with a moving mass

This paper presents a dynamic analysis of a tethered satellite system with a moving mass. A dynamic model with four degrees of freedom, i.e., a two-piece dumbbell model, is established for tethered satellites conveying a mass between them along the tether length. This model includes two satellites and a moving mass, treated as particles in a single orbital plane, which are connected by massless, straight tethers. The equations of motion are derived by using Lagrange’s equations. From the equations of motion, the dynamic response of the system when the moving mass travels along the tether connecting the two satellites is computed and analyzed. We investigate the global tendencies of the libration angle difference (between the two sections of tether) with respect to the changes in the system parameters, such as the initial libration angle, size (i.e. mass) of the moving mass, velocity of the moving mass, and tether length. We also present an elliptic orbit case and show that the libration angles and their difference increase as orbital eccentricity increases. Finally, our results show that a one-piece dumbbell model is qualitatively valid for studying the system under certain conditions, such as when the initial libration angles, moving mass velocity, and moving mass size are small, the tether length is large, and the mass ratio of the two satellites is large.     

基于精密星历的星链轨道特性研究

当前,星链巨型星座受到国际航天业内广泛关注。由于其运营方SpaceX公司的商业性质以及星链卫星潜在军事用途,涉及星链卫星技术细节的资料比较匮乏。为减小星链卫星星座运行过程中对国际有关行业的不良影响,SpaceX公司定期公开全部在轨星链卫星的精密星历。这为研究星链控制策略及其运行规律提供了难得的一手资料。本文尝试分析SpaceX公司发布星历的精度,利用星链卫星的精密星历分析星链卫星工作轨道维持段的站位保持策略和卫星自动碰撞规避策略,为巨型星座卫星设计和管控提供一定参考。

     

Optimal control of the deployment (and retrieval) of a tethered satellite under small initial disturbances

The deployment and retrieval processes of satellites from a space station are demanding tasks during the operations of tethered satellite systems. The satellite should be steered into its working state within a reasonable amount of time and without too much control efforts. For the pure in-plane oscillation we have found time-optimal solutions with bang–bang control strategy for the deployment and retrieval process. In our working group we have also investigated different stabilization methods of the vertical equilibrium configuration, for example parametric swing control and chaotic control. In this article we concentrate on the final stage of the operation, when the oscillations around the vertical configuration should be brought to halt. While this task is quite simple for a motion of the satellite in the orbital plane, it is considerably more difficult, if the satellite has been perturbed out of that plane. We first analyze the control for a purely out-of-plane oscillation, which is governed by a Hamiltonian Hopf bifurcation, and then investigate the combined control for the spatial dynamics. Using a center manifold ansatz for the in-plane oscillations, we can show, that it is possible to diminish the oscillations of the tethered satellite in both directions, but the decay is extremely slow.     

Chaotic motion in a class of asymmetrical Kelvin type gyrostat satellite

This work is an investigation on the roots of chaotic attitudinal motion in a class of asymmetrical gyrostat satellites. The result shows that for a class of Kelvin type gyrostat satellite, there is an equivalent rigid spinning satellite with the same attitude dynamics. Finding some constants of motion and eliminating the cyclic coordinates, the rotational kinetic energy is changed to a quadratic form and using Jordan canonical form of the associated inertia tensor and transforming the coordinate system, the Hamiltonian has been changed to those of a rigid satellite. The Hamiltonian has been split into integrable and non-integrable parts. Using Deprit canonical transformation and Andoyer variables the integrable part has been reduced to a one-dimensional form. The reduced Hamiltonian shows that the regular dynamics of the satellite can be chaotic, under the influence of gravitational effects. To demonstrate various attitudinal dynamics of the satellite, a second-order Poincaré map is employed. This research shows firstly, that the attitudinal dynamics of Kelvin type gyrostat satellites and rigid satellites follow the same dynamical patterns, secondly, for non-linear analysis of dynamics of gyrostat satellite based on the perturbation methods, there is a preferable form for Hamiltonian of the system in the near-integrable fashion and thirdly the chaotic motion is originated from the gravitational field effects that can be suppressed by increasing the attitudinal energy of the satellite in comparison with the translational energy.     

大型低轨星座自适应绝对站位保持法

针对大型低轨星座的长期维持控制,提出了一种自适应绝对站位保持法。利用卫星受大气阻力影响引起平半长轴衰减进而使得相位产生漂移的原理,推导了半长轴维持控制量计算公式,给出了星上自主计算维持控制量的流程。利用该算法分析了不同面质比卫星在太阳活动低年和高年的维持控制规律,结果表明使用该算法能够有效使卫星相位保持在维持区间。使用两行轨道根数分析了星链卫星的维持控制规律,采用该算法对其维持情况做出了改进,有效降低了维持控制次数。     

Dynamics of a satellite carrying a point mass moving about it

We study the dynamics of a complex system consisting of a solid and a mass point moving according to a prescribed law along a curve rigidly fixed to the body. The motion occurs in a central Newtonian gravitational field. It is assumed that the orbit of the system center of mass is an ellipse of arbitrary eccentricity.We obtain equations that describe the motion of the carrier (satellite) about its center of mass. In the case of a circular orbit, we present conditions that should be imposed on the law of the relative motion of the mass point carried by the satellite so that the latter preserves a constant attitude with respect to the orbital coordinate system. In the case of a dynamically symmetric satellite, we consider the problem of existence of stationary and nearly stationary rotations for the case in which the carried point moves along the satellite symmetry axis.We consider several problems of dynamics of the satellite plane motion about its center of mass in an elliptic orbit of arbitrary eccentricity. In particular, we present the law of motion of the carried point in the case without eccentricity oscillations and study the stability of the satellite permanent attitude with respect to the orbital coordinate system.     

WHY CAN THE MAJOR PLANETS HAVE THEIR SATELLITES MOVING IN DIRECT AND RETROGRADE ORBITS AS WELL?

Now we use the Jacobian integral of circular restricted three-body problem toestablish a testing function of the stability of satellites.This method of criterion may beapplied to the stability problem of satellites when the six elements of the instantaneous orbitof the satellite with respect to its parent planet are known.By means of an electronic computer,we can find the stable region of a satellite with aquasi-circular orbit.The boundary surface of this region is a nearly oblate ellipsoid.Thevolume of this enclosed space is much smaller than that of binding by Hill surface and thatof“sphere of action”.As the expressions of relative kinetic energy of a satellite with respect to its parentplanet have the same form for the direct as well as the retrograde orbits,they can coexist inthe same region at the same time.     

基于对偶四元数的双星编队相对位姿测量

为了描述编队卫星中主从星的相对位置和姿态信息,提出了基于对偶四元数的编队卫星相对位姿测量算法。以双星编队飞行的位姿运动为主线,运用对偶四元数工具,充分发挥其能以最简洁的形式表示一般性刚体运动的优点,对卫星轨道和姿态进行分析并建立了对偶四元数位姿模型。同时设计类GPS测量技术来测量编队卫星的相对位置和姿态,该技术载波相位波长和伪码码元比GPS的更短,可获得更高精度的相对测量信号。由于状态方程和观测方程的非线性特征,使用UKF滤波来消除随机噪声对量测过程的干扰。实验结果表明,所设计的算法能够有效估计系统误差,卫星的位置误差和四元数误差收敛于零,验证了该算法的有效性。     

Nonlinear analysis of a simple model of temperature evolution in a satellite

We analyze a simple model of the heat transfer to and from a small satellite orbiting round a solar system planet. Our approach considers the satellite isothermal, with external heat input from the environment and from internal energy dissipation, and output to the environment as black-body radiation. The resulting nonlinear ordinary differential equation for the satellite’s temperature is analyzed by qualitative, perturbation and numerical methods, which prove that the temperature approaches a periodic pattern (attracting limit cycle). This approach can occur in two ways, according to the values of the parameters: (i) a slow decay towards the limit cycle over a time longer than the period, or (ii) a fast decay towards the limit cycle over a time shorter than the period. In the first case, an exactly soluble average equation is valid. We discuss the consequences of our model for the thermal stability of satellites.

     

Telemetry-based estimate of orientation accuracy for the Tat’yana-2 satellite

Some results of processing the data registered by the Moscow University satellite named Tat’yana-2 are discussed. An estimate for the course angle in the process of the satellite orientation with respect to the Earth is obtained. The predicted angular motion of the satellite and the telemetry data are compared. An improved algorithm for the orientation with respect to the Earth is proposed for small satellites to enhance the orientation accuracy when the course angle is arbitrary. A possibility of using the micro vibratory gyroscopes to control the orientation of small satellites is estimated.