Optimization of low-thrust orbit transfers with constraints

Approximate numerical methods of optimization are presented for multi-orbit noncoplanar orbit transfers of low-thrust spacecraft. The linear representation of derivatives of boundary parameters is used in the vicinity of a reference trajectory with its discretization into small segments. For each segment a set of pseudo-impulses is introduced, representing possible directions of the thrust vector. In order to solve essentially nonlinear problems, the iterative process of upgrading the reference trajectory is used. At each iteration the linear programming problem of high dimensionality is solved, its boundary conditions being represented in the form of a linear matrix equation. Interval constraints are considered in the form of blocking the propulsion system operation on shadow segments of the orbit, as well as cycling constraints, and constraints on total duration of maneuvers at certain trajectory segments. The results of comparison with solutions obtained by other methods are presented together with examples illustrating the convergence of iterative processes. Optimizations of the trajectories for launching geosynchronous satellites to their orbits and of the trajectories of a noncoplanar transfer from low to high-elliptic Molniya orbit are considered under these constraints.     

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.     

The Motion of a Material Point in a Newtonian Field under the Action of Equiangular Thrust

The problems of investigation and optimization of the motion of spacecraft are extensively discussed in the literature. Nevertheless, in many cases a large variety of qualitative characteristics of their motion and of the form of their trajectories are still unclear. In this paper we consider a plane equiangular acceleration of a spacecraft both in a Newtonian field and in its absence (at a large distance from the center of attraction). The general equation of a trajectory of plane acceleration is presented with the introduction of a new variable, an index of an exponent, which allows one to obtain convenient solutions at different values of the time-independent angle of inclination of the vector of thrust to the spacecraft's radius vector (i.e., when equiangular acceleration takes place). Asymptotic solutions are constructed and an interesting fact is revealed. Namely, it is shown that when the center of attraction exists or is absent, for all initial conditions the trajectories appearing at the above equiangular acceleration of a material point tend to the standard logarithmic spirals at a large distance from the center. Specifically, when the value of transverse (perpendicular to the radius vector) thrust is constant, there appears a logarithmic spiral with an angle of inclination to the radius vector equal to 35.264°. Different forms of the trajectory of equiangular acceleration of spacecraft at a low thrust are also studied. The results obtained can be useful for the investigation and choice of optimum space trajectories.     

全太阳系深空探测轨迹全局优化(英文)

针对太阳系中全部的248997颗行星的探测问题,给出了一种关于探测飞行器的深空探测全局四维轨迹(t,x,y,z)优化方案,即飞行器从地球发射进入太阳系并采用小推力控制,优化方案的性能指标为飞行器与太阳系中全部行星中相遇和交会的星的数量最多并且燃料消耗最少。本方案给出了四维飞行轨迹进行全局优化的一套算法,该算法由搜索算法和四维轨迹优化算法组成。此搜索算法从太阳系的248997颗行星中寻找获得尽可能多的经过近地球3维走廊内的行星;而四维轨迹优化算法由改进的动态规划算法、基于最优控制理论的共轭梯度算法和静态参数优化算法组成,其中静态参数优化算法用于搜索最优发射时间窗口。基于该组合算法,通过长时间的大规模的飞行数字仿真,最终计算出探测器的四维最优飞行轨迹,在一年内路过了太阳系中全部行星中的12颗行星。     

非合作空间目标自主交会凸优化制导技术

为解决非合作空间目标机动、自旋、章动等动态特性对自主交会过程带来的困难和挑战,在某飞行器姿轨控一体化推进系统配置的基础上,充分考虑各类约束后构建标准的凸优化自主交会数学模型;着重分析了非合作目标动态特性对制导误差的影响,设计相应的修正策略;以原始对偶内点法作为核心求解器,设计了具备状态预测和更新能力的制导算法;数学仿真验算表明,该方法合理可行,克服了动态特性造成的接近轨迹偏差,良好地适应了非合作交会任务。     

有限推力航天器双主动交会的最优控制

研究了两异面椭圆轨道的有限推力航天器在协同轨道机动以完成交会任务(双主动交会)时的最优控制问题.构造了有限推力航天器双主动交会的数学模型,讨论了其实现最优控制的必要条件.针对推进剂总消耗最少和有限推进剂约束下的最短时间交会2种不同情况,采用直接配置法和序列二次规划法求解了反平方力场中的最优控制数值解.考察了不同初始参数对双主动交会最优控制历程的影响,并将双主动交会形式与主被动交会形式进行了对比.结果表明,当两航天器质量接近时,双主动交会在减少燃料消耗或缩短交会所需时间方面具有明显优势.     

Missions to the asteroid anteros and the space of true anomalies

Three opportunities for missions to rendezvous ballistically with the Earth-crossing asteroid Anteros are studied to illustrate the requirements for a trip to a near-Earth minor planet. The rationale, sample payload, spacecraft requirements and trajectory characteristics of these opportunities are typical of a rendezvous mission to an accessible near-Earth object. Round trip ballistic trajectories to return small samples of the asteroid with launch dates between 1985 and 2000 are also presented. Contours of minimum total ΔV drawn in the space of launch and arrival true anomalies, given the designation Prime Rib curves, are introduced as a useful tool for mission design.     

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.     

气动辅助空间交会最优轨道设计与闭环导引律

给出了同平面HEO-LEO实现空间交会的必要条件;研究了基于气动辅助轨道转移技术实现同平面HEO-LEO的空间交会方案,通过设计一条标准的同平面HEO-LEO气动辅助轨道转移的最优轨迹,得到了轨道转移飞行器(OTV)与目标实现交会必须满足的标准相角;最后对大气飞行段设计了非线性最优闭环导引律,通过引入Lyapunov最陡下降函数,对函数中相应参数进行适当调整,使应用闭环导引律得到的大气内飞行轨迹与最优轨迹充分接近,仿真结果表明该气动辅助空间交会方法正确、可行。     

Fuel optimum low-thrust elliptic transfer using numerical averaging

Low-thrust electric propulsion is increasingly being used for spacecraft missions primarily due to its high propellant efficiency. As a result, a simple and fast method for low-thrust trajectory optimization is of great value for preliminary mission planning. However, few low-thrust trajectory tools are appropriate for preliminary mission design studies. The method presented in this paper provides quick and accurate solutions for a wide range of transfers by using numerical orbital averaging to improve solution convergence and include orbital perturbations. Thus, preliminary trajectories can be obtained for transfers which involve many revolutions about the primary body. This method considers minimum fuel transfers using first-order averaging to obtain the fuel optimum rates of change of the equinoctial orbital elements in terms of each other and the Lagrange multipliers. Constraints on thrust and power, as well as minimum periapsis, are implemented and the equations are averaged numerically using a Gausian quadrature. The use of numerical averaging allows for more complex orbital perturbations to be added in the future without great difficulty. The effects of zonal gravity harmonics, solar radiation pressure, and thrust limitations due to shadowing are included in this study. The solution to a transfer which minimizes the square of the thrust magnitude is used as a preliminary guess for the minimum fuel problem, thus allowing for faster convergence to a wider range of problems. Results from this model are shown to provide a reduction in propellant mass required over previous minimum fuel solutions.     

极坐标系连续常值推力机动分析

连续常值推力是空间飞行常用的轨道机动方式，在空间交会与星际航行使命中具有重要的应用价值。其中，小推力适合于地球轨道航天器交会机动，而切向或周向推力以及较大的正径向推力可用于脱离地球引力场的逃逸飞行，执行星际交会使命。应用常推力作用下的极坐标系质心运动方程，对机动推力的量值没有限制；在航天器交会应用中，对相对距离也无要求。这种方法可直接获得向径与速度等轨道参数随时间或极角（绕地心的转动角）的变化，便于分析轨道转移与逃逸运动，有助于飞行使命与运动轨迹的设计。特别是，若机动转移的初轨为圆轨道，在推力较小、飞行时间不长的情况下，应用无量纲形式运动方程，可获得具有工程应用价值的近似解。文章给出一些有关的结果与应用案例。     

Trajectory of spacecraft with photonic laser propulsion in the two-body problem

The photonic laser propulsion (PLP) system can produce continuous and constant thrust. This paper reviews its basics and then studies its application in the two-body interplanetary trajectory. This study also derives the equations of motion (EOM) for a spacecraft in an inertial frame and rotational frame for the two-body problem, and uses the Jacobi integral to investigate the spatial restrictions on trajectory. This study proposes a constraint on the smallest thrust at the launching stage, and finds the in-plane and out-of-plane equilibria. Numerical simulations confirm the validity of the derivations. The results of this paper are directly applicable to future PLP thrust missions, and particularly interplanetary travel.     

Optimal trajectories for the aeroassisted flight experiment

This paper deals with the determination of optimal trajectories for the aeroassisted flight experiment (AFE). The intent of this experiment is to simulate a GEO-to-LEO transfer, where GEO denotes a geosynchronous Earth orbit and LEO denotes a low Earth orbit. Specifically, the AFE spacecraft is released from the Space Shuttle and is accelerated by means of a solid rocket motor toward Earth, so as to achieve atmospheric entry conditions identical with those of a spacecraft returning from GEO. During the atmospheric pass, the angle of attack is kept constant, and the angle of bank is controlled in such a way that the following conditions are satisfied: (a) the atmospheric velocity depletion is such that, after exiting, the AFE spacecraft first ascends to a specified apogee and then descends to a specified perigee; and (b) the exit orbital plane is identical with the entry orbital plane. The final maneuver, not analyzed here, includes the rendezvous with and the capture by the Space Shuttle. In this paper, the trajectories of an AFE spacecraft are analyzed in a 3D space, employing the full system of 6 ODEs describing the atmospheric pass. The atmospheric entry conditions are given, and the atmospheric exit conditions are adjusted in such a way that requirements (a) and (b) are met, while simultaneously minimizing the total characteristic velocity, hence the propellant consumption required for orbital transfer. Two possible transfers are considered: indirect ascent (IA) to a 178 NM perigee via a 197 NM apogee; and direct ascent (DA) to a 178 NM apogee. For both transfers, two cases are investigated: (i) the bank angle is continuously variable; and (ii) the trajectory is divided into segments along which the bank angle is constant. For case (ii), the following subcases are studied; 2, 3, 4 and 5 segments; because the time duration of each segment is optimized, the above subcases involve 4, 6, 8 and 10 parameters, respectively. It is shown that the optimal trajectories of cases (i) and (ii) coalesce into a single trajectory: a two-subarc trajectory, with the bank angle constant in each subarc (bang-bang control). Specifically, the bank angle is near 180° in the atmospheric entry phase (positive lift projection phase) and is near 0° in the atmospheric exit phase (negative lift projection phase). It is also shown that, during the atmospheric pass, the peak values of the changes of the orbital inclination and the longitude of the ascending node are nearly zero; hence, the peak value of the wedge angle (angle between the instantaneous orbital plane and the initial orbital plane) is nearly zero. This means that the motion of the spacecraft is nearly planar in an inertial space.     

Online generation of quasi-optimal spacecraft rendezvous trajectories

A direct method for rapid generation of combined time-propellant near-optimal trajectories of proximity maneuvers of a chaser spacecraft required to dock a target one, with predetermined thrust history along a master direction, is presented. The predetermined thrust history is generated by applying the Pontryagin maximum principle. The new direct method, already implemented and tested on board real aircraft, is based on three concepts: high-order polynomials as reference functions, preset on–off sequence of a master control, and reduction of the optimization problem to the determination of a small set of parameters. Presetting the master control, the remaining controls act as slaves, guarantying the chaser to move along the desired path. Seeking of the optimum strategy is transformed into a nonlinear programming problem, and then numerically solved through an ad hoc algorithm in accelerated time scale. Examples are reported to prove the rapidness of the approach to generate a sub-optimal docking trajectory.     

The Use of Quaternions in the Optimal Control Problems of Motion of the Center of Mass of a Spacecraft in a Newtonian Gravitational Field: III

The results of numerical solution of the problem of a rendezvous in the central Newtonian gravitational field of a controlled spacecraft with an uncontrollable spacecraft moving along an elliptic Keplerian orbit are presented. Two variants of the equations of motion for the spacecraft center of mass are used, written in rotating coordinate systems and using quaternion variables to describe the orientations of these coordinate systems. The problem of a rendezvous of two spacecraft is formulated [1, 2] as a problem of optimal control by the motion of the center of mass of a controlled spacecraft with a movable right end of the trajectory, and it is solved on the basis of Pontryagin's maximum principle. The paper is a continuation of papers [1, 2], where the problem of a rendezvous of two spacecraft has been considered theoretically using the two above variants of the equations of motion for the center of mass of the controlled spacecraft.     

Constant-thrust glideslope guidance algorithm for time-fixed rendezvous in real halo orbit

This paper presents a fixed-time glideslope guidance algorithm that is capable of guiding the spacecraft approaching a target vehicle on a quasi-periodic halo orbit in real Earth–Moon system. To guarantee the flight time is fixed, a novel strategy for designing the parameters of the algorithm is given. Based on the numerical solution of the linearized relative dynamics of the Restricted Three-Body Problem (expressed in inertial coordinates with a time-variant nature), the proposed algorithm breaks down the whole rendezvous trajectory into several arcs. For each arc, a two-impulse transfer is employed to obtain the velocity increment (delta-v) at the joint between arcs. Here we respect the fact that instantaneous delta-v cannot be implemented by any real engine, since the thrust magnitude is always finite. To diminish its effect on the control, a thrust duration as well as a thrust direction are translated from the delta-v in the context of a constant thrust engine (the most robust type in real applications). Furthermore, the ignition and cutoff delays of the thruster are considered as well. With this high-fidelity thrust model, the relative state is then propagated to the next arc by numerical integration using a complete Solar System model. In the end, final corrective control is applied to insure the rendezvous velocity accuracy. To fully validate the proposed guidance algorithm, Monte Carlo simulation is done by incorporating the navigational error and the thrust direction error. Results show that our algorithm can effectively maintain control over the time-fixed rendezvous transfer, with satisfactory final position and velocity accuracies for the near-range guided phase.     

Electric transfer optimization for mars sample return mission

The aim of this paper is to analyse an alternative scenario for Mars Sample Return Orbiter mission, where electric propulsion is used for Earth-Mars and Mars-Earth heliocentric cruises and for Mars orbit insertion / escape transfers, whereas chemical propulsion is used for final Mars rendezvous. The problem consists in minimizing the initial vehicle mass to obtain a specific final dry mass in reasonable time. The planetocentric phases correspond to continuous low-thrust trajectories, spiraling around Mars between a low orbit and the influence sphere altitude. The heliocentric phases consist of a succession of low-thrust and coasting arcs with specific departure and arrival conditions at the Earth. For these two types of transfer, efficient optimal control tools exist based on Pontryagin's maximum principle. Thanks to the coordination between planetocentric and heliocentric phases, the solution obtained with these two separate tools gives a good upper bound of the optimal solution in terms of propellant consumption and duration. This optimization procedure is described and finally applied to the proposed mission. The numerical results are presented and compared with the baseline chemical mission solution. The electric option could allow to decrease the spacecraft departure mass but may lead to rather long mission duration.     

Optimization of Flights between the Orbits of Artificial Satellites of Two Planets Using a Combination of High and Low Thrusts

The problem of optimization of the trajectory of an interplanetary flight of a multistaged spacecraft using jet engines with high and low thrust is considered. The issues concerning the problem of choosing the main design parameters of a multistage spacecraft are touched upon. A mathematical model of start-to-finish optimization of all segments of the interplanetary flight trajectory is proposed. Using this model the specific features of flights to the orbits of satellites of Jupiter and Mars are studied.     

Radioisotope electric propulsion of sciencecraft to the outer solar system and near-interstellar space

Recent results are presented in the study of radioisotope electric propulsion as a near-term technology for sending small robotic sciencecraft to the outer Solar System and near- interstellar space. Radioisotope electric propulsion (REP) systems are low-thrust, ion propulsion units based on radioisotope electric generators and ion thrusters. Powerplant specific masses are expected to be in the range of 100 to 200 kg/kW of thrust power. Planetary rendezvous missions to Pluto, fast missions to the heliopause (100 AU) with the capability to decelerate an orbiter for an extended science program and prestellar missions to the first gravitational lens focus of the Sun (550 AU) are investigated.