Optimization of transfer trajectories to the Apophis asteroid for spacecraft with high and low thrust

The problem of optimization of a spacecraft transfer to the Apophis asteroid is investigated. The scheme of transfer under analysis includes a geocentric stage of boosting the spacecraft with high thrust, a heliocentric stage of control by a low thrust engine, and a stage of deceleration with injection to an orbit of the asteroid’s satellite. In doing this, the problem of optimal control is solved for cases of ideal and piecewise-constant low thrust, and the optimal magnitude and direction of spacecraft’s hyperbolic velocity “at infinity” during departure from the Earth are determined. The spacecraft trajectories are found based on a specially developed comprehensive method of optimization. This method combines the method of dynamic programming at the first stage of analysis and the Pontryagin maximum principle at the concluding stage, together with the parameter continuation method. The estimates are obtained for the spacecraft’s final mass and for the payload mass that can be delivered to the asteroid using the Soyuz-Fregat carrier launcher.     

The Aster project: Flight to a near-Earth asteroid

The information on the project being developed in Brazil for a flight to binary or triple near-Earth asteroid is presented. The project plans to launch a spacecraft into an orbit around the asteroid and to study the asteroid and its satellite within six months. Main attention is concentrated on the analysis of trajectories of flight to asteroids with both impulsive and low thrust in the period 2013-2020. For comparison, the characteristics of flights to the (45) Eugenia triple asteroid of the Main Belt are also given.     

Analysis of the orbital motion of the asteroid Apophis’ satellite

We have analyzed the orbital disturbed spacecraft motion near an asteroid. The equations of the asteroidocentric spacecraft motion have been used with regard to three perturbations from celestial bodies, the asteroid’s nonsphericity, and solar radiation pressure. It has been shown that the orbital parameters of the main spacecraft and a small satellite with a radio beacon can be selected such that the orbits are rather stable for a fairly long period of time, i.e., a few weeks for the main spacecraft with an orbit initial radius of ~0.5 km and a few years before approaching Apophis with the Earth in 2029, for a small satellite at an orbit initial radius of ~1.5 km. The initial orientation of the spacecraft orbital plane perpendicular to the sunward direction is optimal from the point of view of the stability of the spacecraft flight near an asteroid.     

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.     

About one problem of trajectory optimization

The optimization problem for trajectories of spacecraft flight from the Earth to an asteroid is considered in this paper. The flight is realized in the central Newtonian gravitational field of the Sun with a possibility of gravitational maneuvers near planets. Perturbation maneuvers are taken into account using the method of point area of action with a limitation on the flyby altitude. The spacecraft is controlled by changing the value and direction of the engine thrust. The problem is solved taking into account constraints on the launch time, flight duration, and minimum distance to the Sun.     

The use of solutions to problems of spacecraft trajectory optimization in impulse formulation when solving the problems of optimal control of trajectories of a spacecraft with limited thrust engine: I

In this first part of our paper, it is suggested to use solutions to boundary value problems in the optimization problems (in impulse formulation) for spacecraft trajectories in order to obtain the initial approximation, when boundary value problems of the maximum principle are solved numerically by the shooting method. The technique suggested is applied to the problems of optimal control over motion of the center of mass of a spacecraft controlled by the thrust vector of jet engine with limited thrust in an arbitrary gravitational field in a vacuum. The method is based on a modified (in comparison to the classic scheme) shooting method computation together with the method of continuation along a parameter (maximum reactive acceleration, initial thrust-to-weight ratio, or any other parameter equivalent to them). This technique allows one to obtain the initial approximation with a high precision, and it is applicable to a wide range of optimal control problems solved using the maximum principle, if the impulse formulation makes sense for these problems.     

Equilibrium attitude and stability of a spacecraft on a stationary orbit around an asteroid

The stationary orbits around an asteroid, if exist, can be used for communication and navigation purposes just as around the Earth. The equilibrium attitude and stability of a rigid spacecraft on a stationary orbit around a uniformly-rotating asteroid are studied. The linearized equations of attitude motion are obtained under the small motion assumption. Then, the equilibrium attitude is determined in both cases of a general and a symmetrical spacecraft. Due to the higher-order inertia integrals of the spacecraft, the equilibrium attitude is slightly away from zero Euler angles. Then necessary conditions of stability of this conservative system are analyzed based on the linearized equations of motion. The effects of different parameters, including the harmonic coefficients C₂₀ and C₂₂ of the asteroid and higher-order inertia integrals of the spacecraft, on the stability are assessed and compared. Due to the significantly non-spherical shape and rapid rotation of the asteroid, the effects of the harmonic coefficients C₂₀ and C₂₂ are very significant, while effects of the third- and fourth-order inertia integrals of the spacecraft can be neglected. Considering a spacecraft on a stationary orbit around an example asteroid, we show that the classical stability domain predicted by the Beletskii–DeBra–Delp method on a circular orbit in a central gravity field is modified due to the non-spherical mass distribution of the asteroid. Our results are confirmed by a numerical simulation.     

On a possibility of delivery of samples of primordial matter from asteroids of the main belt by a spacecraft with electric propulsion

The possibilities of an expedition to asteroids of the Main belt with the goal of obtaining samples of their matter using a spacecraft with a small-thrust electric propulsion, which is under development as a part of the Fobos-Grunt project, are considered. Obtaining matter from different regions of the Solar system is necessary in order to understand its origin and the structure of the Earth. Approximate estimates are made for the flights of spacecraft with electric propulsion and different power of solar batteries, and with perturbation maneuver near Mars for expeditions to the Main asteroid belt, including its middle part.     

Asteroidal resources for space manufacturing

Earth-approaching asteroids (Apollos and Amors) may be competitive candidates as raw materials for space manufacturing. The total energy per unit mass required to transfer material from some of these bodies to high Earth orbit is comparable to that for lunar material. Recent optical studies suggest ordinary and carbonaceous chondrite compositions for these asteroids, with some containing large quantities of metallic iron and nickel, and others, carbon, hydrogen and nitrogen. Discoveries of several new candidate asteroids over the next few years will allow for a better selection of materials and mission possibilities. Material from one of these asteroids, either in raw or manufactured form, could be returned to the vicinity of the Earth by a solar-powered mass-driver reaction engine. With a requirement of ～60 shuttle flights, and with minimal development costs, an automated mission to a 200-m dia. (10⁷ ton) metal-rich asteroid could be carried out by a mass-driver tug assembled in low Earth orbit using shuttle tankage as reaction mass. Such a tug could, within a few years, move the asteroid into high Earth orbit for the manufacturing of ～ 20 satellite power stations using a portion of the asteroid itself as reaction mass. In the next few years over 100 asteroids in this size range could be discovered, orbits determined and composition types classified using existing earthbased and spaceborne search techniques.     

航天器环绕小行星Ivar的运动分析

将小行星Ivar近似为三轴椭球体，给出了非球形引力势函数，建立了航天器环绕小行星Ivar的轨道动力学方程。利用Jacobi积分常数绘制了航天器在Ivar周围的零速度曲线，并分析了航天器的可能运动区域，给出了航天器不碰撞小行星Ivar的边界条件及不同偏心率下的近拱点半径。分析了小行星Ivar扁率和椭率对环绕轨道的影响，数学仿真结果表明：在一个轨道周期内，顺行轨道的开普勒能量、轨道角动量、偏心率和近拱点半径变化较大，而逆行轨道的相应参数变化较小。     

One optimization problem for trajectories of spacecraft rendezvous mission to a group of asteroids

The optimization problem is considered for the trajectory of a spacecraft mission to a group of asteroids. The ratio of the final spacecraft mass to the flight time is maximized. The spacecraft is controlled by changing the value and direction of the jet engine thrust (small thrust). The motion of the Earth, asteroids, and the spacecraft proceeds in the central Newtonian gravitational field of the Sun. The Earth and asteroids are considered as point objects moving in preset elliptical orbits. The spacecraft departure from the Earth is considered in the context of the method of a point-like sphere of action, and the excess of hyperbolic velocity is limited. It is required sequentially to have a rendezvous with asteroids from four various groups, one from each group; it is necessary to be on the first three asteroids for no less than 90 days. The trajectory is finished by arrival at the last asteroid. Constraints on the time of departure from the Earth, flight duration, and final mass are taken into account in this problem.     

On solving the problems of optimization of trajectories of many-revolution orbit transfers of spacecraft

The problem of optimal control over many-revolution spacecraft orbit transfers between circular coplanar orbits of satellites is considered. The spacecraft flight is controlled by a thrust vector of a jet engine with restricted thrust (JERT). The mass expenditure is minimized at a limited time of flight. The optimal control problem is solved based on the maximum principle. The boundary value problem of the maximum principle is solved numerically using the shooting method. A modified computation scheme of the shooting method is suggested (multi-point shooting), as well as a method (correlated with the scheme) of choosing the initial approximation with the use of a solution to the optimization problem in the impulse formulation. The scheme and method allow one to construct many-revolution spacecraft orbit transfers.     

General solution for the optimal trajectory of an AFE-type spacecraft

The aeroassisted flight experiment (AFE) refers to an experimental spacecraft to be launched and then recovered by the Space Shuttle. It simulates a transfer from a geosynchronous Earth orbit (GEO) to a low Earth orbit (LEO). In this paper, with reference to an AFE-type spacecraft, an actual GEO-to-LEO transfer is considered under the following assumptions: the GEO and LEO orbital planes are identical; both the initial and final orbits are circular; the initial phase angle is given, while the final phase angle is free. The aeroassisted orbital transfer trajectory involves three branches: a preatmospheric branch, GEO-to-entry; an atmospheric branch, entry-to-exit; a post-atmospheric branch, exit-to-LEO. The optimal trajectory is determined by minimizing the total characteristic velocity. The optimization is performed with respect to the velocity impulses at GEO, LEO, and the time history of the angle of bank during the atmospheric pass. It is assumed that the entry path inclination is free and that the angle of attack is constant, = 17.0 deg. The sequential gradient-restoration algorithm is used to compute the optimal trajectory and it is shown that the best atmospheric pass is to be performed with constant angle of bank. The resulting optimal trajectory constitutes an ideal nominal trajectory for the generation of guidance trajectories for two reasons: the fact that the low value of the characteristic velocity is accompanied by relatively low values of the peak heating rate and the peak dynamic pressure; and the simplicity of the control distribution, requiring constant angle of bank.     

形成三星星座的小推力变轨的时间最短控制

在研究和发展星座技术中，星座的发射是一项关键技术。本文针对形成三星星座，利用最优控制中的极小值原理，解算了用恒值、连接工作、牛顿级小推力变轨的时间最短控制问题。文中建立了最优小推力变轨的数学模型，求得了最优变轨的解析解，并通过牛顿下山法求解了三星星座变轨的小推力工作最优时间、最优方向和最优变轨轨迹。最后对星座变轨小推力最优控制工程实现的途径进行了探讨。为工程应用和研究提供参考。     

Possibilities of Combining High- and Low-Thrust Engines in Flights to Mars

The use of combinations of chemical and electric jet engines in the spacecraft designs results in a multistage vehicle configuration and in related problems of the optimum distribution of masses between the stages, the problems of flight trajectory optimization, and the problems of choosing the design parameters of a spacecraft. The appropriate issues are considered using flights to Mars as an example. The conditions for the optimum matching of high and low thrust trajectory segments are presented. The model of the simultaneous optimization of the geocentric and heliocentric legs of the trajectory is proposed. One- and two-orbit optimum trajectories of flight are investigated and analyzed.     

State propagation in an uncertain asteroid gravity field

Various spacecraft have been and will be sent to asteroids to characterize them. Generally, an asteroid's gravity field is very irregular and not accurately known when compared to the gravity field of a major planet, Earth in particular. It has been well studied that the irregularity significantly affects the trajectory of an orbiting spacecraft, and causes it to impact or to escape from the asteroid. Complementary to that, this paper focuses on the influence of the limited knowledge of this gravity field on the evolution of the spacecraft's orbit. It develops a general method by which this influence can be quantified. This method comprises specific Monte Carlo simulations with a discrete set of low-altitude orbits, taking into account the uncertainties in the gravity-field parameters. For illustration purposes, it is applied to two different asteroids. Already after three revolutions, the gravity-field uncertainties propagate to significant position uncertainties; this specifically holds for prograde orbits, and around the smaller asteroid. Applying this robust and accurate method helps mission designers and planners to assess the risk posed by gravity uncertainties, and take appropriate measures such as choosing the most favorable orbital geometries and/or lowering the orbit more slowly.     

Optimal thrust program for deviating an Earth-pointing asteroid

The purpose of this study is to investigate the optimal thrust program for deviating an asteroid when it is flying directly toward or crossing the Earth. Under some assumptions, the problem can be considered as a two-body problem of Earth-asteroid system. The initial relative speed and distance are specified to be 10 km/s and 150 times of Earth's radius, respectively. We have about 1 day (or, 93681 seconds exactly) to take action. If a single impulse is applied to the asteroid at the specified initial point, the required impulse to obtain a miss distance of 2 times of Earth's radius is 169.5 m/s per kg mass. For an asteroid of 10 m in diameter, the total impulse required is 3.02 × 10⁸ m/s. It needs a typical large launching rocket to provide the total impulse. When the asteroid is larger or the initial distance is shorter, the number of launching rockets required increases rapidly. For further analysis with physical and engineering constraints imposed, we shall have to use the variational formulation method.     

Design of optimal earth pole-sitter transfers using low-thrust propulsion

Recent studies have shown the feasibility of an Earth pole-sitter mission using low-thrust propulsion. This mission concept involves a spacecraft following the Earth's polar axis to have a continuous, hemispherical view of one of the Earth's poles. Such a view will enhance future Earth observation and telecommunications for high latitude and polar regions. To assess the accessibility of the pole-sitter orbit, this paper investigates optimum Earth pole-sitter transfers employing low-thrust propulsion. A launch from low Earth orbit (LEO) by a Soyuz Fregat upper stage is assumed after which solar electric propulsion is used to transfer the spacecraft to the pole-sitter orbit. The objective is to minimize the mass in LEO for a given spacecraft mass to be inserted into the pole-sitter orbit. The results are compared with a ballistic transfer that exploits manifold-like trajectories that wind onto the pole-sitter orbit. It is shown that, with respect to the ballistic case, low-thrust propulsion can achieve significant mass savings in excess of 200 kg for a pole-sitter spacecraft of 1000 kg upon insertion. To finally obtain a full low-thrust transfer from LEO up to the pole-sitter orbit, the Fregat launch is replaced by a low-thrust, minimum time spiral, which provides further mass savings, but at the cost of an increased time of flight.     

Trajectories for missions of low-thrust spacecraft aimed at delivery of soil samples from main belt asteroids and Phobos

Trajectories of spacecraft with electro-jet low-thrust engines are studied for missions planning to deliver samples of matter from small bodies of the Solar System: asteroids Vesta and Fortuna, and Martian moon Phobos. Flight trajectories are analyzed for the mission to Phobos, the limits of optimization of payload spacecraft mass delivered to it are determined, and an estimate is given to losses in the payload mass when a low-thrust engine with constant outflow velocity is used. The model of an engine with ideally regulated low thrust is demonstrated to be convenient for calculations and analysis of flight trajectories of a low-thrust spacecraft.     

小推力轨道保持方法

对小推力轨道保持方法进行了研究。用快、慢变量控制器分别控制轨道要素的快慢变量,基于推导的经典轨道要素与2个推力方向角和最佳变轨位置的关系,给出了最优推力方向角的解析表达式。用Lyapunov反馈控制实现卫星轨道机动的轨道转移,并引入相位调整,实现了卫星的站位保持。仿真结果表明:基于Lyapunov的反馈控制可实现小推力轨道的转移和保持。