Revisiting the general periodic relative motion in elliptic reference orbits

A general periodicity condition is presented by analyzing the relative motion between two spacecraft performing formation flight in Keplerian elliptic orbits. The Tschauner–Hempel equation is used to describe the relative motion, and the general periodicity condition is derived through a state transition matrix with a true anomaly as a free variable. The general periodicity condition is also derived by using the energy matching condition, and the resulting periodic conditions by two approaches are compared to each other. Moreover, the zero offset condition is presented to locate the leader spacecraft at the center of the formation geometry. Then, the periodic relative motion in the elliptic reference orbit is expressed using the periodicity condition and the zero offset condition. Numerical simulations demonstrate the periodic relative motion in the elliptic reference orbit, and the results show that the general periodicity condition guarantees the bounded periodic relative motion in arbitrary elliptic orbits, and the zero offset condition makes the formation center coincide with the leader spacecraft.     

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.     

Control of satellite imaging formations in multi-body regimes

Libration point orbits may be ideal locations for satellite imaging formations. Therefore, control of these arrays in multi-body regimes is critical. A continuous feedback control algorithm is developed that maintains a formation of satellites in motion that is bounded relative to a halo orbit. This algorithm is derived based on the dynamic characteristics of the phase space near periodic orbits in the circular restricted three-body problem (CR3BP). By adjusting parameters of the control algorithm appropriately, satellites in the formation follow trajectories that are particularly advantageous to imaging arrays. Image reconstruction and coverage of the (u, v) plane are simulated for interferometric satellite configurations, demonstrating potential applications of the algorithm and the resulting motion.     

圆型限制性三体问题相对运动解析研究

针对圆型限制性三体问题共线平动点附近周期/拟周期轨道下的相对运动问题,提出一种新的、通用的解析研究方法。在周期/拟周期轨道近似解析解的基础上,结合微分修正方法,获得了精确的周期/拟周期轨道。对周期/拟周期轨道的单值矩阵进行分析,同时借鉴Floquet理论核心思想,建立了六个相对运动模态,并将相对运动表示为六个相对运动模态的线性组合,获得了相对运动的近似解析解。最后在地-月系统圆型限制性三体问题下,以L1点作为研究对象,分别以Halo轨道、Lissajous轨道和Lyapunov轨道为参考轨道,对相对运动模态和相对运动进行仿真分析,说明了相对运动模态的正确性以及相对运动近似解析解的有效性。     

Two-craft tether formation relative equilibria about circular orbits and libration points

The relative equilibria of a two spacecraft tether formation connected by line-of-sight elastic forces moving in the context of a restricted two-body system and a circularly restricted three-body system are investigated. For a two spacecraft formation moving in a central gravitational field, a common assumption is that the center of the circular orbit is located at the primary mass and the center of mass of the formation orbits around the primary in a great-circle orbit. The relative equilibrium is called great-circle if the center of mass of the formation moves on the plane with the center of the gravitational field residing on it; otherwise, it is called a nongreat-circle orbit. Previous research shows that nongreat-circle equilibria in low Earth orbits exhibit a deflection of about a degree from the great-circle equilibria when spacecraft with unequal masses are separated by 350 km. This paper studies these equilibria (radial, along-track and orbit-normal in circular Earth orbit and Earth–Moon Libration points) for a range of inter-craft distances and semi-major axes of the formation center of mass. In the context of a two-spacecraft Coulomb formation with separation distances on the order of dozens of meters, this paper shows that the equilibria deflections are negligible (less than 10^?6°) even for very heterogeneous mass distributions. Furthermore, the nongreat-circle equilibria conditions for a two spacecraft tether structure at the Lagrangian libration points are developed.     

同步双小行星系统共振轨道设计

研究了同步双小行星系统中共振轨道的设计方法及演化规律。首先，基于双椭球模型建立探测器运动方程，并给出共振轨道初值选取方法。然后，利用改进并行打靶法，提出一种双小行星系统平面共振轨道两步修正方法。同时结合稳定性理论及分岔理论，给出双小行星系统三维共振轨道生成和延拓方法；最后，以双小行星系统1999KW4为例，设计了共振比为1∶1,1∶2,1∶3,1∶4,2∶3的平面和空间共振轨道族，并分析了共振轨道的特性及轨道周期和轨道能量的变化规律。给出的双小行星系统中共振轨道的设计方法具有普适性，对未来双小行星系统探测任务中的轨道设计具有一定的参考意义与借鉴价值。     

航天器相对运动建模及周期性相对运动求解

面向航天器编队飞行的需求,对椭圆参考轨道航天器非线性周期相对运动条件进行研究,提出了确定椭圆参考轨道编队航天器非线性周期性相对运动条件的新方法。首先,考虑非线性、椭圆轨道等因素,通过哈密尔顿-雅可比(HJ)方程和正则摄动理论,推导了在任意非线性摄动下相对运动的模型和获得不需消耗任何燃料的周期性相对运动轨道的条件;然后,采用时域配点法,结合改进的列文伯格-马夸尔特(LM)法对周期性相对运动的初值进行求解;最后,设计数值仿真算例,利用上述条件,得到不消耗任何燃料的周期性绕飞轨道,由此验证了本文所提模型和方法的正确性。     

航天飞行器相对运动动力学及其控制方法

采用本地轨道从标系对两邻飞船间相对运动动力学展开研究，指出在相对运动中也存在平衡状态，用相平面方法分析了其稳定性，基于此分析，综合出相对运动控制方法，即距离速率控制方法，受控运动轨迹是一条稳定的稳态直线，进而建立了全方位距离速率控制方法。最后以系绳卫星系统和飞船安全为例完成了计算机模拟。     

A vector method for synthesis of orbits and the structure of satellite constellations for multiswath periodic coverage of the Earth

Single satellites and multisatellite constellations for the periodic coverage of the Earth are considered. The main feature is the use of several cameras with different swath widths. A vector method is proposed which makes it possible to find orbits minimizing the periodicities of coverage of a given area of Earth uniformly for all swaths. Their number is not limited, but the relative dimensions should satisfy the Fibonacci series or some new numerical sequences. The results apply to constellations of any number of satellites. Formulas were derived for calculating their structure, i.e., relative position in the constellation. Examples of orbits and the structure of constellations for the Earth’s multiswath coverage are presented.     

Periodic relative orbits of two spacecraft subject to differential gravity and electrostatic forcing

Coulomb forces between charged close-flying satellites can be used for formation control, and constant electric potentials enable static equilibria solutions. In this work, open-loop time-varying potential functions, which produce periodic, two-craft, Coulomb formation motions are demonstrated for the first time. This is done in the rotating Hill-Frame, with linearized gravity, and craft position components assumed in the form of simple harmonic oscillators. Substitution of the oscillatory functions into the dynamics, further constrains these functions, and yields necessary potential histories, to produce the periodic flow. The assumed position functions, however, are not arbitrary, since the dynamical model restricts what oscillatory trajectories are allowed. Specifically, a Hill-Frame integral of motion is derived, and this is used to show certain candidate periodic functions to be inadmissible. The system dynamics are then linearized to expose stability properties of the solutions, and it is established that asymptotic stability is impossible for all orbit families. Finally, the degree of instability in the assumed motions, over free parameter ranges, is determined numerically via the Floquet multipliers of the associated full-cycle state-transition matrices.     

On halo orbits spacecraft stabilization

The paper deals with the application of recent non-linear control techniques to the problem of tracking and maintaining a given satellite on prescribed orbits around the so-called translunar libration point L₂. Such orbits, known in literature as Halo Orbits, have the property of ensuring visibility both from the dark side of the Moon and from Earth at any time. Their importance is strictly related to the placement of a base situated on the dark side of the Moon for advanced space missions as deep space observation, solar system exploration and scientific researches in a low gravity enviroment.

Because of the instability of the equilibrium L₂, such orbits cannot be maintained without an active control. In this paper we investigate the application of nonlinear control techniques to solve the problem. A comparison between linear and nonlinear methods is developed and simulation results are discussed.     

Cubature Kalman filtering for relative spacecraft attitude and position estimation

A novel relative spacecraft attitude and position estimation approach based on cubature Kalman filter is derived. The integrated sensor suit comprises the gyro sensors on each spacecraft and a vision-based navigation system on the deputy spacecraft. In the traditional algorithm, an assumption that the chief?s body frame coincides with its Local Vertical Local Horizontal (LVLH) frame is made to construct the line-of-sight observations for convenience. To solve this problem, two relative quaternions that map the chief?s LVLH frame to the deputy and chief body frames are involved. The general relative equations of motion for eccentric orbits are used to describe the positional dynamics. The implementation equations for the cubature Kalman filter are derived. Simulation results indicate that the proposed filter provides more accurate estimates of relative attitude and position over than the extended Kalman filter.     

Design of transfer trajectories between resonant orbits in the Earth–Moon restricted problem

The application of dynamical systems techniques to mission design has demonstrated that employing invariant manifolds and resonant flybys enables previously unknown trajectory options and potentially reduces the ΔV$\mathrm{\Delta }V$ requirements. In this investigation, planar and three-dimensional resonant orbits are analyzed and cataloged in the Earth–Moon system and the associated invariant manifold structures are computed and visualized with the aid of higher-dimensional Poincaré maps. The relationship between the manifold trajectories associated with multiple resonant orbits is explored through the maps with the objective of constructing resonant transfer arcs. As a result, planar and three-dimensional homoclinic- and heteroclinic-type trajectories between unstable periodic resonant orbits are identified in the Earth–Moon system. To further illustrate the applicability of 2D and 3D resonant orbits in preliminary trajectory design, planar transfers to the vicinity of L₅ and an out-of-plane transfer to a 3D periodic orbit, one that tours the entire Earth–Moon system, are constructed. The design process exploits the invariant manifolds associated with orbits in resonance with the Moon as transfer mechanisms.     

Stable spatial orbits around collinear libration points

A technique of generation of spatial periodic solutions to the restricted circular three-body problem from periodic orbits of the planar problem has been used for the families of orbits around collinear libration points L ₁ and L ₂. Developing the families obtained at the 1: 1 resonance, we have obtained stable solutions both in the Earth-Moon system and in the Sun-Earth system. Of course, the term “around the libration point” is rather conventional; the obtained orbits become more similar to the orbits around the smaller attracting body. The further development of the family of orbits “around” the libration point L ₂ in the Sun-Earth system made it possible to find the orbits satisfying the new, much more rigorous constraints on cooling the spacecraft of the Millimetron project.     

非线性条件下编队卫星周期性相对运动条件

利用编队卫星机械能守恒原理，提出了非线性条件下求解编队卫星周期性相对运动条件的新方法，给出了非线性周期相对运动的初始条件。编队卫星相对距离较近时，利用非线性周期运动条件，可修正Hill方程的初始条件，抑制编队卫星的长期漂移。编队卫星相对距离较大，非线性因素不可忽略时，利用非线性周期运动条件，可找到不需消耗任何燃料的周期性相对运动轨道。最后的数值仿真结果验证了该方法的正确性。     

Halo orbits in the vicinity of the L 2 libration point in the Sun-Earth system

Methods are proposed for constructing the orbits of spacecraft remaining for long periods of time in the vicinity of the L ₂ libration point in the Sun-Earth system (so-called halo orbits), and the trajectories of uncontrolled flights from low near-Earth orbits to halo orbits. Halo orbits and flight trajectories are constructed in two stages: A suitable solution to a circular restricted three-body problem is first constructed and then transformed into the solution for a restricted four-body problem in view of the real motions of the Sun, Earth, and Moon. For a halo orbit, its prototype in the first stage is a combination of a periodic Lyapunov solution in the vicinity of the L ₂ point and lying in the plane of large-body motion, with the solution for the linear second-order system describing small deviations of the spacecraft from this plane along the periodic solution. The desired orbit is found as the solution to the three-body problem best approximating the prototype in the mean square. The constructed orbit serves as a similar prototype in the second stage. In both stages, the approximating solution is constructed by continuation along a parameter that is the length of the approximation interval. Flight trajectories are constructed in a similar manner. The prototype orbit in the first stage is a combination of a solution lying in the plane of large-body motion and a solution for a linear second-order system describing small deviations of the spacecraft from this plane. The planar solution begins near the Earth and over time tends toward the Lyapunov solution existing in the vicinity of the L ₂ point. The initial conditions of both prototypes and the approximating solutions correspond to the spacecraft’s departure from a low near-Earth orbit at a given distance, perigee, and inclination.     

地-月系平动点及Halo轨道的应用研究

地-月系统的平动点L1点及L2点的Halo轨道在探月工程中有重要的应用价值，可分别用于地月连续通信覆盖和月球背面的探测。由于在地-月系统中太阳的引力不可忽略，特别是在长时间作用以后，其动力学行为与摄动力较小的日-地系统有明显的不同。本文分析了如何利用太阳引力进入地-月系统的L1点及L2点的Halo轨道、以及由Halo轨道进入近月轨道的问题，两者综合起来构成了一条完整的地月低能转移轨道。研究结果对探月轨道设计有一定的参考价值。     

Manifold dynamics in the Earth–Moon system via isomorphic mapping with application to spacecraft end-of-life strategies

Recently, manifold dynamics has assumed an increasing relevance for analysis and design of low-energy missions, both in the Earth–Moon system and in alternative multibody environments. With regard to lunar missions, exterior and interior transfers, based on the transit through the regions where the collinear libration points L₁ and L₂ are located, have been studied for a long time and some space missions have already taken advantage of the results of these studies. This paper is focused on the definition and use of a special isomorphic mapping for low-energy mission analysis. A convenient set of cylindrical coordinates is employed to describe the spacecraft dynamics (i.e. position and velocity), in the context of the circular restricted three-body problem, used to model the spacecraft motion in the Earth–Moon system. This isomorphic mapping of trajectories allows the identification and intuitive representation of periodic orbits and of the related invariant manifolds, which correspond to tubes that emanate from the curve associated with the periodic orbit. Heteroclinic connections, i.e. the trajectories that belong to both the stable and the unstable manifolds of two distinct periodic orbits, can be easily detected by means of this representation. This paper illustrates the use of isomorphic mapping for finding (a) periodic orbits, (b) heteroclinic connections between trajectories emanating from two Lyapunov orbits, the first at L₁, and the second at L₂, and (c) heteroclinic connections between trajectories emanating from the Lyapunov orbit at L₁ and from a particular unstable lunar orbit. Heteroclinic trajectories are asymptotic trajectories that travels at zero-propellant cost. In practical situations, a modest delta-v budget is required to perform transfers along the manifolds. This circumstance implies the possibility of performing complex missions, by combining different types of trajectory arcs belonging to the manifolds. This work studies also the possible application of manifold dynamics to defining suitable, convenient end-of-life strategies for spacecraft orbiting the Earth. Seven distinct options are identified, and lead to placing the spacecraft into the final disposal orbit, which is either (a) a lunar capture orbit, (b) a lunar impact trajectory, (c) a stable lunar periodic orbit, or (d) an outer orbit, never approaching the Earth or the Moon. Two remarkable properties that relate the velocity variations with the spacecraft energy are employed for the purpose of identifying the optimal locations, magnitudes, and directions of the velocity impulses needed to perform the seven transfer trajectories. The overall performance of each end-of-life strategy is evaluated in terms of time of flight and propellant budget.     

On plane oscillations of a pendulum with variable length suspended on the surface of a planet’s satellite

The problem of planar oscillations of a pendulum with variable length suspended on the Moon’s surface is considered. It is assumed that the Earth and Moon (or, in the general case, a planet and its satellite, or an asteroid and a spacecraft) revolve around the common center of mass in unperturbed elliptical Keplerian orbits. We discuss how the change in length of a pendulum can be used to compensate its oscillations. We wrote equations of motion, indicated a rule for the change in length of a pendulum, at which it has equilibrium positions relative to the coordinate system rotating together with the Moon and Earth. We study the necessary conditions for the stability of these motions. Chaotic dynamics of the pendulum is studied numerically and analytically.