Optimal ballistically captured Earth–Moon transfers

The optimality of a low-energy Earth–Moon transfer terminating in ballistic capture is examined for the first time using primer vector theory. An optimal control problem is formed with the following free variables: the location, time, and magnitude of the transfer insertion burn, and the transfer time. A constraint is placed on the initial state of the spacecraft to bind it to a given initial orbit around a first body, and on the final state of the spacecraft to limit its Keplerian energy with respect to a second body. Optimal transfers in the system are shown to meet certain conditions placed on the primer vector and its time derivative. A two point boundary value problem containing these necessary conditions is created for use in targeting optimal transfers. The two point boundary value problem is then applied to the ballistic lunar capture problem, and an optimal trajectory is shown. Additionally, the problem is then modified to fix the time of transfer, allowing for optimal multi-impulse transfers. The tradeoff between transfer time and fuel cost is shown for Earth–Moon ballistic lunar capture transfers.     

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.     

从月球逃逸探测小行星的发射机会搜索

从月球逃逸探测小行星的发射机会搜索因需考虑日、地、月引力的影响而使问题变得复杂。针对该多体系统的发射机会搜索问题,提出了一种分层渐近的搜索方法。该方法首先通过分析地月系质心与小行星的几何关系,搜索从地月系质心到小行星的发射机会,进而以地月运动为研究对象,推导出了从月球轨道切向逃逸机会的判别条件,并基于此判别条件及等高线图法对逃逸机会进行了搜索。同时,为提高所得发射机会在多体模型下的轨道修正收敛性,给出了基于月心逃逸轨道参数为终端约束的日-地与日-地-月动力学模型的轨道渐近修正方法。最后,以近地小行星(3908)Nyx和(190491)2000 FJ20为例,搜索其从月球逃逸的发射机会,仿真计算表明了该方法的有效性。     

Approximate solutions to minimax optimal control problems for aeroassisted orbital transfer

This paper considers minimax problems of optimal control arising in the study of aeroassisted orbital transfer. The maneuver considered involves the coplanar transfer from a high planetary orbit to a low planetary orbit. An example is the HEO-to-LEO transfer of a spacecraft, where HEO denotes high Earth orbit and LEO denotes low Earth orbit. In particular, HEO can be GEO, a geosynchronous Earth orbit.The basic idea is to employ the hybrid combination of propulsive maneuvers in space and aerodynamic maneuvers in the sensible atmosphere. Hence, this type of flight is also called synergetic space flight. With reference to the atmospheric part of the maneuver, trajectory control is achieved by means of lift modulation. The presence of upper and lower bounds on the lift coefficient is considered.The following minimax problems of optimal control are investigated: (i) minimize the peak heating rate, problem P1; and (ii) minimize the peak dynamic pressure, problem P2. It is shown that problems P1 and P2 are approximately equivalent to the following minimax problem of optimal control: (iii) minimize the peak altitude drop occurring in the atmospheric portion of the trajectory, problem P3.Problems P1–P3 are Chebyshev problems of optimal control, which can be converted into Bolza problems by suitable transformations. However, the need for these transformations can be bypassed if one reformulates problem P3 as a two-subarc problem of optimal control, in which the first subarc connects the initial point and the point where the path inclination is zero, and the second subarc connects the point where the path inclination is zero and the final point: (iv) minimize the altitude drop achieved at the point of junction between the first subarc and the second subarc, problem P4. Note that problem P4 is a Bolza problem of optimal control.Numerical solutions for problems P1–P4 are obtained by means of the sequential gradient-restoration algorithm for optimal control problems. Numerical examples are presented, and their engineering implications are discussed. In particular, it is shown that, from an engineering point of view, it is desirable to solve problem P3 or P4, rather than problems P1 and P2.     

Contingency and recovery options for the European Student Moon Orbiter

This paper presents an overview of the analysis performed on the lunar orbit and some of the possible contingencies for the European Student Moon Orbiter (ESMO). Originally scheduled for launch in 2014 –2015 as a piggyback payload, it was the only ESA planned mission to the Moon. By way of a weak stability boundary transfer, ESMO is inserted into an orbit around the Moon. Propellant use is at a premium, so the operational orbit is selected to be highly eccentric. In addition, an optimization is presented to achieve an orbit that is stable for 6 months without requiring orbit maintenance. A parameter study is undertaken to study the sensitivity of the lunar orbit insertion. A database of transfer solutions across 2014 and 2015 is used to study the relation between the robustness of weak capture and the planetary geometry at lunar arrival. A number of example recovery scenarios, where the orbit insertion maneuver partially or completely fails, are also considered.     

Optimization of the trajectory of the spacecraft insertion into the system of heliocentric orbits

The possibility of the spacecraft insertion into the system of operational heliocentric orbits has been analyzed. It has been proposed to use a system of several operational heliocentric orbits. On each orbit, the spacecraft makes one or more revolutions around the Sun. These orbits are characterized by a relatively small perihelion radius and relatively high inclination, which allows one to investigate the polar regions of the Sun. The transition of the spacecraft from one orbit to another has been performed using an unpowered gravity assist maneuver near Venus and does not require the cruise propulsion operation. Each maneuver transfers the spacecraft into the sequence of operational heliocentric orbits. We have analyzed several systems of operational heliocentric orbits into which the spacecraft can be inserted by means of the considered transportation system with electric propulsion (EP). The mass of the spacecraft delivered to these systems of operational orbits has been estimated.     

地月高能共振循环轨道的快速计算及最优选择

针对现有高能共振循环轨道计算方法存在计算量大、有可能改变轨道共振特性和不能构造共振比大于2.3的地月循环轨道等缺点,本文提出了一种地月圆型限制性三体问题下高能共振循环轨道的快速计算方法。首先根据轨道在月球附近的组成弧段对高能共振循环轨道进行分类;然后根据轨道类型构建二体开普勒椭圆轨道;再进一步计算圆型限制性三体问题下的地月高能共振循环轨道;最后根据能量、稳定性、时间周期、近地点高度和近月点高度对所计算出的地月高能共振循环轨道进行最优选择。仿真结果表明,本文所提出的方法简单有效,能够计算出共振比为5:2的地月高能共振循环轨道。     

On the Precession of Saturn

The precession of Saturn under the effect of the gravity of the Sun, Jupiter and planet’s satellites has been investigated. Saturn is considered to be an axisymmetric (A = B) solid body close to the dynamically spherical one. The orbits of Saturn and Jupiter are considered to be Keplerian ellipses in the inertial coordinate system. It has been shown that the entire set of small parameters of the problem can be reduced to two independent parameters. The averaged Hamiltonian function of the problem and the integrals of evolutionary equations are obtained disregarding the effect of satellites. Using the small parameter method, the expressions for the precession frequency and the nutation angle of the planet’s axis of rotation caused by the gravity of the Sun and Jupiter are obtained. Considering the planet with satellites as a whole preceding around the normal to the unmovable plane of Saturn’s orbit, the satellites effect on the Saturn rotation is taken into account via the corrections in the formula for the undisturbed precession frequency. The satellites are shown to have no effect on the nutation angle (in the framework of the accepted model), and the disturbances from Jupiter to make the main contribution to the nutation angle evolution. The effect of Jupiter on the nutation angle and the precession period is described with regard to the attraction of satellites.     

Applications of multi-body dynamical environments: The ARTEMIS transfer trajectory design

The application of forces in multi-body dynamical environments to permit the transfer of spacecraft from Earth orbit to Sun–Earth weak stability regions and then return to the Earth–Moon libration (L₁ and L₂) orbits has been successfully accomplished for the first time. This demonstrated that transfer is a positive step in the realization of a design process that can be used to transfer spacecraft with minimal Delta-V expenditures. Initialized using gravity assists to overcome fuel constraints; the ARTEMIS trajectory design has successfully placed two spacecrafts into Earth–Moon libration orbits by means of these applications.     

Assessment on the feasibility of future shepherding of asteroid resources

Most plausible futures for space exploration and exploitation require a large mass in Earth orbit. Delivering this mass requires overcoming the Earth's natural gravity well, which imposes a distinct obstacle to any future space venture. An alternative solution is to search for more accessible resources elsewhere. In particular, this paper examines the possibility of future utilisation of near Earth asteroid resources. The accessibility of asteroid material can be estimated by analysing the volume of Keplerian orbital element space from which Earth can be reached under a certain energy threshold and then by mapping this analysis onto an existing statistical near Earth objects (NEO) model. Earth is reached through orbital transfers defined by a series of impulsive manoeuvres and computed using the patched-conic approximation. The NEO model allows an estimation of the probability of finding an object that could be transferred with a given Δv budget. For the first time, a resource map provides a realistic assessment of the mass of material resources in near Earth space as a function of energy investment. The results show that there is a considerable mass of resources that can be accessed and exploited at relatively low levels of energy. More importantly, asteroid resources can be accessed with an entire spectrum of levels of energy, unlike other more massive bodies such as the Earth or Moon, which require a minimum energy threshold implicit in their gravity well. With this resource map, the total change of velocity required to capture an asteroid, or transfer its resources to Earth, can be estimated as a function of object size. Thus, realistic examples of asteroid resource utilisation can be provided.     

火星卫星的冻结轨道研究

冻结轨道是一种稳定的轨道,地球、火星、月球的卫星因引力场的南北不对称,都存在冻结轨道.由于主星体引力场的不同,它们卫星的冻结轨道也有不同的特性.地球卫星的冻结执道的偏心率非常小,对卫星遥感非常有利,国内外已有相当多的近地遥感卫星采用这种轨道.月球卫星的冻结轨道偏心率随轨道倾角的不同有很大的变化,对月球卫星冻结轨道的研究...     

嫦娥五号平动点拓展任务轨道方案研究

中国嫦娥五号探测器成功实现月球采样返回任务,为最大限度利用任务资源,研究了利用嫦娥五号轨道器的平动点拓展任务轨道方案,设计了平动点轨道及其转移轨道.首先,给出了任务轨道设计的轨道动力学模型,包括圆型限制性三体问题模型和精确力模型.其次,基于嫦娥二号和嫦娥5T1平动点拓展任务设计经验,介绍了平动点轨道直接转移与入轨等轨道...     

载人月球极地探测地月转移轨道设计

针对载人月球极地探测任务,采用一种自由返回轨道与三脉冲机动轨道相结合的地月转移轨道方案。关于自由返回轨道部分的设计,建立了基于近月点伪参数的两段拼接模型,采用一种考虑地球扁率修正的改进多圆锥截线法进行求解,仿真结果显示改进的多圆锥截线法具有更高的求解精度,可为精确设计提供更好的初值;关于三脉冲机动轨道部分的设计,基于混合轨道模型,采用特殊点变轨和Lambert算法相结合的方法进行计算,仿真结果显示该方法能够有效地降低速度增量的消耗。最后,通过大量的仿真计算,对轨道的速度增量特性进行了分析。研究结论可为未来载人月球极地探测地月转移轨道方案的设计提供重要参考。     

Low-energy Earth–Moon transfers involving manifolds through isomorphic mapping

Analysis and design of low-energy transfers to the Moon has been a subject of great interest for decades. Exterior and interior transfers, based on the transit through the regions where the collinear libration points 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 concerned with a geometrical approach for low-energy Earth-to-Moon mission analysis, based on isomorphic mapping. The isomorphic mapping of trajectories allows a visual, 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. Two types of Earth-to-Moon missions are considered. The first mission is composed of the following arcs: (i) transfer trajectory from a circular low Earth orbit to the stable invariant manifold associated with the Lyapunov orbit at L₁ (corresponding to a specified energy level) and (ii) transfer trajectory along the unstable manifold associated with the Lyapunov orbit at L₁, with final injection in a periodic orbit around the Moon. The second mission is composed of the following arcs: (i) transfer trajectory from a circular low Earth orbit to the stable invariant manifold associated with the Lyapunov orbit at L₁ (corresponding to a specified energy level) and (ii) transfer trajectory along the unstable manifold associated with the Lyapunov orbit at L₁, with final injection in a capture (non-periodic) orbit around the Moon. In both cases three velocity impulses are needed to perform the transfer: the first at an unknown initial point along the low Earth orbit, the second at injection on the stable manifold, the third at injection in the final (periodic or capture) orbit. The final goal is in finding the optimization parameters, which are represented by the locations, directions, and magnitudes of the velocity impulses such that the overall delta-v of the transfer is minimized. This work proves how isomorphic mapping (in two distinct forms) can be profitably employed to optimize such transfers, by determining in a geometrical fashion the desired optimization parameters that minimize the delta-v budget required to perform the transfer.     

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.     

Sсheme of rendezvous mission to lunar orbital station by spacecraft launched from Earth

In recent years, great experience has been accumulated in manned flight astronautics for rendezvous in near-Earth orbit. During flights of Apollo spacecraft with crews that landed on the surface of the Moon, the problem of docking a landing module launched from the Moon’s surface with the Apollo spacecraft’s command module in a circumlunar orbit was successfully solved. A return to the Moon declared by leading space agencies requires a scheme for rendezvous of a spacecraft launched from an earth-based cosmodromee with a lunar orbital station. This paper considers some ballistic schemes making it possible to solve this problem with minimum fuel expenditures.     

航天器异面气动力辅助变轨大气飞行段的最优轨迹

本文研究航天器利用气动力辅助变轨实现由远地轨道向近地轨道异面转移问题，就航天器利用大气飞行实现减速及轨道平面机动提出了一种考虑过程约束存在时的优化设计方法。通过方案设计将函数优化问题转换成参数优化问题，并利用美国航天飞机数据进行了模拟计算，得到满意结果。     

地月平动点拟周期轨道探测器自主导航方法研究

地月L2点的拟周期轨道可以用于实现与月球背面的持续通信,具有重要的科学研究价值和广阔的应用前景。针对地月L2点探测器所处的弱稳定拟周期轨道,论证了基于日地月信息的自主导航方法的可行性,并进行了深入分析。首先,推导了会合坐标系下带有星历的精确导航动力学方程;其次,针对弱稳定轨道不同于近地强稳定轨道的特性,在基于日地月方位信息导航的基础上,提出了三种敏感器组合方案。使用迭代最小二乘方法给出导航仿真结果,并结合非线性可辨识性理论对这三种情况下历元状态的可辨识性及可辨识度进行分析。最后,仿真结果表明,使用日地月敏感器以及对地多普勒雷达可以满足历元状态的可辨识性、导航的收敛性以及系统经济性的要求。     

Optimal aeroassisted return from high earth orbit with plane change

This paper gives a complete analysis of the problem of aeroassisted return from a high Earth orbit to a low Earth orbit with plane change. A discussion of pure propulsive maneuver leads to the necessary change for improvement of the fuel consumption by inserting in the middle of the trajectory an atmospheric phase to obtain all or part of the required plane change. The variational problem is reduced to a parametric optimization problem by using the known results in optimal impulsive transfer and solving the atmospheric turning problem for storage and use in the optimization process. The coupling effect between space maneuver and atmospheric maneuver is discussed. Depending on the values of the plane change i, the ratios of the radii, $\text{n =}\text{r}{}_1\text{r}{}_2$ between the orbits and $\text{a =}\text{r}{}_2\text{R}$ between the low orbit and the atmosphere, and the maximum lift-to-drag ratio E1 of the vehicle, the optimal maneuver can be pure propulsive or aeroassisted. For aeroassisted maneuver, the optimal mode can be parabolic, which requires only drag capability of the vehicle, or elliptic. In the elliptic mode, it can be by one-impulse for deorbit and one or two-impulse in postatmospheric flight, or by two-impulse for deorbit with only one impulse for final circularization. It is shown that whenever an impulse is applied, a plane change is made. The necessary conditions for the optimal split of the plane changes are derived and mechanized in a program routine for obtaining the solution.     

Spacecraft Injection into a Heliocentric Terrestrial Orbit

We consider the problem of injection of a spacecraft into the heliocentric Earth's orbit ahead and/or behind the Earth by 60° and 120° in heliographic longitude. The range of solar and astrophysical problems for which these orbits are necessary is reviewed. The variants of injection into heliocentric orbits work from a low around-Earth orbit with one turn-on of the engine in this orbit and one turn-on at the end of the injection trajectory. In this case, it turns out to be more profitable to put spacecraft into orbit for three or even four revolutions of the Earth about the Sun. The velocities necessary for the start from a low around-Earth orbit, the velocities at the final point of injection, and the fuel mass (relative to the spacecraft mass) necessary for injection are estimated. The problems for which injection to similar orbits is executed, using the low-thrust engine and with a combined regime of injection, are also considered.