Optimization of multi-mode rendezvous trajectories with constraints

Approximate numerical methods of optimization of spacecraft rendezvous trajectories are presented that make use of interior point algorithms for problems of linear programming of high dimensionality (tens to hundreds of thousands of variables). The basis of the methods is discretization of a trajectory into small segments in which maneuvers are allowed to be executed; for all segments sets of pseudo-impulses are introduced that determine the possible directions of the spacecraft thrust vector. The terminal conditions are presented in the form of a linear matrix equation. A matrix inequality for the sums of characteristic velocities of pseudo-impulses on each segment is used to make a transformation to the linear programming form. Spacecraft rendezvous trajectories are considered in the neighborhood of circular orbits with the use of multi-mode propulsion systems (including those with low thrust) and existence of boundary conditions at interior points and constraints on the time of operation of the propulsion system at separate segments of the trajectory.     

载人登月绕月自由返回轨道与窗口精确快速设计

针对载人登月任务背景及工程约束,提出一种轨道与窗口一体化设计方法。通过两次坐标转化,将自由返回轨道设计参数解耦为近月点独立变量。在双二体假设下,通过4段二体轨道拼接完成自由返回轨道初值快速搜索及匹配近地停泊轨道(LEO)面的月窗口,其结果作为下一步采用序列二次规划（SQP）迭代求解高精度动力学模型轨道参数的初值,在该条精确轨道近月点时刻90 min邻域内产生可以匹配LEO地月转移入轨相位的零窗口轨道。算例表明,该流程能够精确快速地完成具有复杂任务背景及苛刻工程要求的载人登月绕月自由返回轨道与窗口设计问题。     

不同月球借力约束下的地月Halo轨道转移轨道设计

针对地月系L2点不同任务需求下的低耗能转移轨道设计问题,基于不变流形理论与混合优化技术,深入研究了不同月球借力约束与不同幅值Halo轨道的入轨点（简称HOI点）对转移轨道飞行时间与燃料消耗的影响,给出了HOI点选择策略。首先结合任务要求并考虑月球引力影响,在月球借力点施加不同约束条件,通过微分修正算法调整Halo轨道的稳定流形,设计月球到Halo轨道的转移轨道。采用遗传算法与微分修正算法相结合的混合优化策略,在同时考虑地球停泊轨道高度、倾角、升交点赤经与航迹角等多约束条件下,对燃料最优的地月转移轨道进行研究。最后,分析月球借力高度、借力方位角和不同HOI点对平动点转移轨道飞行时间与燃耗变化量的影响,对于考虑月球借力的地月平动点转移轨道设计与应用具有重要的参考价值。     

Conditions of the Maximum Principle in the Problem of Optimal Control over an Aggregate of Dynamic Systems and Their Application to Solution of the Problems of Optimal Control of Spacecraft Motion

The necessary first-order conditions of strong local optimality (conditions of maximum principle) are considered for the problems of optimal control over a set of dynamic systems. To derive them a method is suggested based on the Lagrange principle of removing constraints in the problems on a conditional extremum in a functional space. An algorithm of conversion from the problem of optimal control of an aggregate of dynamic systems to a multipoint boundary value problem is suggested for a set of systems of ordinary differential equations with the complete set of conditions necessary for its solution. An example of application of the methods and algorithm proposed is considered: the solution of the problem of constructing the trajectories of a spacecraft flight at a constant altitude above a preset area (or above a preset point) of a planet's surface in a vacuum (for a planet with atmosphere beyond the atmosphere). The spacecraft is launched from a certain circular orbit of a planet's satellite. This orbit is to be determined (optimized). Then the satellite is injected to the desired trajectory segment (or desired point) of a flyby above the planet's surface at a specified altitude. After the flyby the satellite is returned to the initial circular orbit. A method is proposed of correct accounting for constraints imposed on overload (mixed restrictions of inequality type) and on the distance from the planet center: extended (nonpointlike) intermediate (phase) restrictions of the equality type.     

奔月轨道的一种求解方法

月球探测器转移轨道的计算通常归结为对商点边值问题的求解，而找到一种计算效率高、迭代次数少、同时具有较好收敛性的算法对于设计者来说是一件很重要的事。本文采用建立在B平面上的参数作为目标轨道参数，改进了状态转移矩阵迭代方法，并与状态转移矩阵方法相结合使用，给出了一种奔月轨道的求解方法。该方法具有计算效率高、迭代次数少的特点，并且对轨道状态量具有良好的线性关系(即对初始条件具有良好的收敛性)。计算算例表明了这种方法的有效性。     

末端能量管理段迭代校正轨道算法研究

针对航天器进入末端能量管理段接口处时位置和航迹偏角存在大范围摄动的问题，提出一种使用迭代校正法的轨道快速生成算法。该算法可以根据航天器的具体初始状态，自动选择直接进场或者间接进场策略，快速生成可行的参考轨迹。首先通过跟踪轨迹地面投影实现侧向制导；根据末端能量管理段的起始点与终点的高度与速度约束生成参考动压-高度剖面，并跟踪此剖面实现纵向制导。然后采用迭代校正计算快速确定航向校准柱的位置与最终半径两个参数用以调整航程，保证航天器在末端的所有状态满足自动着陆段接口的边界约束。仿真结果校验了该算法可以根据航天器的具体状态快速生成符合约束条件的末端能量管理段飞行轨道，具有很好的鲁棒控制性能。     

地月循环轨道动力学建模与计算研究

根据地月循环轨道的概念,按照生成第二类周期轨道的弧段进行分类,并讨论了其共振性与对称性在轨道设计与应用中的作用。然后归纳了三种循环轨道的动力学建模与计算方法及其轨道延拓策略,最后总结了三种方法的利弊和应用轨道类型。对地月系统循环轨道的研究和分析,能够为我国未来载人登月工程提供一种新的思路与理论支持。     

Optimization of interplanetary trajectories for spacecraft with ideally regulated engines using the continuation method

The problem of optimization of interplanetary trajectories is considered for spacecraft with a small-thrust ideally regulated engine. When the maximum principle is used, determination of the optimal trajectory is reduced to solution of a two-point boundary value problem for a system of ordinary differential equations. In order to solve this boundary value problem, the method of continuation in parameter is used, and with the help of it the formal reduction of the boundary value problem to a Cauchy problem is performed. Different variants of the continuation method are considered, including the method of continuation in the gravitational parameter which allows one to find extreme trajectories with a preset angular distance. The issues of numerical realization of the continuation method are discussed, and numerical examples of its use for solving the problems of optimization of interplanetary trajectories are presented.     

双月旁转向轨道的动力学与建模研究

 双月旁转向轨道是实现“多任务、多目标”探测模式的理想工具,在以往的深空探测任务中得到了广泛的应用。首先,根据双月旁转向轨道的空间构型和特性,将其分为两大类：平面型和Backflip型;其次,基于“零影响球”和“CR3BP”两种模型,阐述了近旁转向轨道的基本原理;同时,讨论了平面型双月旁转向轨道的动力学特性和建模方法,并给出了其两类应用背景,与此相似,归纳了Backflip型双月旁转向轨道的原理和求解方法。对双月旁转向轨道开展的研究和分析工作能够为我国未来开展的行星际探测任务提供新的思路和参考。     

迭代制导情况下姿态控制系统稳定性分析方法研究

为了提高火箭的入轨精度和轨道适应能力,我国在新一代运载火箭末级中采用了迭代制导技术,俯仰、偏航程序角根据火箭运动状态和目标轨道参数实时变化,目前工程设计中仍按照固定程序角方式开展稳定性分析。本文推导出迭代制导程序角与火箭速度、位置之间的线性关系式,在国内首次提出了迭代制导情况下的稳定性分析方法,并以新一代运载火箭为例进行了实例计算,结果表明此分析方法正确、可行,具有一定的参考价值。     

基于Gauss伪谱法的助推-滑翔飞行器多阶段约束轨迹优化(英文)

在同时存在路径约束和边界条件时确定助推-滑翔飞行器的最优运动轨迹是近年来的热点问题。本文讨论了助推-滑翔飞行器的具有最大纵向航程和横向航程的两种最优轨迹。Gauss伪谱法是由Benson和Huntington提出的,在求解复杂的最优控制问题时有较快的收敛速度并能提供高精度的解。文中介绍了该方法并利用其将轨迹优化问题变换为可由NLP求解器进行数值求解的非线性规划问题。文中还介绍了其它相关的技术,例如端点控制的计算方法、处理包含奇异弧段的多阶段问题的处理方法、获得初始猜测值的方法和验证结果的方法等。仿真结果可以说明在给定的热流、过载和动压约束下最优轨迹的若干关键特性。通过优化和仿真计算,同时证明了此方法的实用性和有效性。     

深空机动对运载火箭发射火星探测轨道研究

为解决长征(LM)运载火箭发射火星探测器转移轨道时，因低温入轨级最长允许滑行时间及测控限制，有效发射日期窗口亟需拓展的问题，采用主矢量理论结合序列二次规划算法(SQP)，研究了探测器深空机动(DSM)对优化运载火箭发射火星转移轨道的作用。在发射直接转移火星探测轨道算法基础上，重点研究了包含引力影响球(SOI)内近地及近火飞行段后，采用主矢量获取深空机动最优猜测初值的分析算法，通过直接使用探测器近火点目标轨道参数优化运载火箭发射轨道，研究对比不同优化目标及设计约束下深空机动的分析结果，证实深空机动对降低转移轨道总发射能量需求、拓展发射日期窗口的高效性；该算法已应用于工程设计。     

基于序列二次规划算法的再入轨迹优化研究

介绍了序列二次规划算法在飞行器再入轨迹优化问题中的应用.首先引入了能量替代变量对无量纲运动方程进行推导,使得运动方程和优化问题易于处理,考虑严格的过程约束和终端约束,以攻角和倾侧角为控制变量,总加热量最小为性能指标;然后通过直接配点法将最优控制问题转化为非线性规划问题,选取各节点的状态量和控制量作为优化参数;最后应用序列二次规划算法对非线性规划问题进行求解.针对多约束的再入飞行器的轨迹优化时对初值敏感的问题,提出一种参考轨迹快速规划算法,提高了优化速度.仿真结果表明提出的方法能够较快地搜索到最优轨迹,满足所有约束且落点精度高.     

A new treatment of Lambertian mechanics

Lambert's formulas for the time of orbital flight between two points in space are rederived by first establishing a universal differential equation governing the time function, independent of the conic type of the trajectory, the focal characteristics of the trajectory sector, and the range angle. A unified form of Lambert's formulas is then obtained as the general solution of the differential equation, and the various forms of the classical Lambert's formulas are obtained as its particular solutions under different boundary conditions. Following this basic treatment, various hypergeometric expansions for Lambert's time function and its derivatives are developed, and the behavior of the function and its implications in the solution of Lambert's problem and the isochronous trajectories are briefly reviewed. Finally, a short comparison of the present treatment with those found in current literature on Lambertian Mechanics is made and briefly discussed.     

基于基准轨道的多节点协调轨道保持方法

在指定时间内，轨道摄动方程的状态变化可以用柯西标准型的常微分方程表述。基于摄动方程，轨道的修正控制参数的计算问题可以表示为指定区间两点边值问题的参数优化问题。传统的参数计算方法本质上是将边值问题转化为初值问题，强制在一个边界点上满足边值条件，弱化了积分区域中的约束条件，造成不适应多约束条件下的轨道保持。基于基准轨道的多节点轨道控制方法在积分区域内的一组约束条件下，建立基于多约束条件方程的有限差分方程，与传统的升交点重合法相比，该算法将约束条件由目标圈目标升交点扩展到点火圈与目标圈之间所有圈的卫星位置参数约束，从而确定控制量将卫星轨道始终保持在偏差管道内，最大限度的满足对卫星轨道保持的要求。     

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.     

一种多级全固体运载火箭上升段自主制导方法

针对多子级全固体运载火箭在终端多约束下的耗尽关机制导问题,设计了一种基于“助推-滑行-助推”飞行模式的真空段自主制导方法。根据轨道动量矩守恒定律,推导出一种同时具有速度和位置矢量约束的定点制导算法(PA)。在PA理论基础上,建立了满足能量匹配的滑行轨道非线性方程组并降阶至一维迭代求解,解决了多级固定总冲约束的两点边值问题。蒙特卡洛仿真结果表明：该算法对固体运载火箭模型的参数偏差和不确定性具有强鲁棒性,并对多终端轨道任务(不同轨道高度和不同载荷质量)具有较强的自适应能力,因此该算法具有重要的理论意义和工程应用价值。     

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 darboux point and the conjugate point on optimal deorbit for reentry trajectories

The concept of the Darboux point at which an extremal loses its global optimality is extended to the case of discontinuous control. Using Contensou's domain of maneuverability, the condition for optimal switching at a corner is derived and the optimality of the trajectory in the neighborhood of a Darboux point is analyzed. The theory is applied to the problems of minimum-fuel planar and noncoplanar deorbit from elliptical orbits for atmospheric entry at a prescribed angle. In each case, the global optimal trajectory is assessed and it is found that in these nonlinear problems the Darboux point and the conjugate point are distinct. The global optimality is always lost before local optimality.     

高度-速度剖面内再入轨迹快速规划

提出一种升力式再入航天器进入稠密大气后的轨迹规划方法。在预先设定攻角剖面的前提下,利用路径约束(驻点热流、动压和过载)在高度-速度(H-V)剖面内直接获得轨迹下边界;利用终端约束确定以轨迹下边界为基准的高度增量,进而通过下边界与高度增量的加和形成满足要求的再入轨迹。其中,增量的形式选取为分段二次型函数,其大小可通过割线法快速获得。倾斜角大小可根据纵向动力学方程反解得到,其方向依据航向误差角走廊确定。通过对典型工况的仿真,结果表明所提方法能够快速规划出再入轨迹,且适应性好。