High order optimal control of space trajectories with uncertain boundary conditions

A high order optimal control strategy is proposed in this work, based on the use of differential algebraic techniques. In the frame of orbital mechanics, differential algebra allows to represent, by high order Taylor polynomials, the dependency of the spacecraft state on initial conditions and environmental parameters. The resulting polynomials can be manipulated to obtain the high order expansion of the solution of two-point boundary value problems. Since the optimal control problem can be reduced to a two-point boundary value problem, differential algebra is used to compute the high order expansion of the solution of the optimal control problem about a reference trajectory. Whenever perturbations in the nominal conditions occur, new optimal control laws for perturbed initial and final states are obtained by the mere evaluation of polynomials. The performances of the method are assessed on lunar landing, rendezvous maneuvers, and a low-thrust Earth–Mars transfer.     

Optimization of Multi-Orbit Transfers between Noncoplanar Elliptic Orbits

Using the maximum principle formalism, the problem of optimizing interorbital transfer between two noncoplanar elliptic orbits is reduced to solution of a boundary value problem for a system of ordinary differential equations. In order to solve the resulting boundary value problem numerically, the numerical homotopic method or modified Newton's method is used. When solving the boundary value problem, the right-hand sides of differential equations of motion are averaged numerically. Efficient software is developed, and a large number of optimal trajectories are calculated using it. As a result of analysis of these numerical data, new high-quality results are obtained. Specifically, a bifurcation of optimal solutions is found, the existence of critical inclination is demonstrated, and a partial classification of the structure of optimal control is performed.     

刚弹性复杂系统在重力场中的定点转动

本文研究了轴对称刚－弹系统在重力场中的定点运动。首先利用模态分析方法将偏微分方程转化为常微分方程组，然后引进小参数将藕合的长周期运动和短周期运动分离。得到了这个非线性刚弹藉合系统的一次近似解，并利用它研究了刚－弹系统运动特性。     

从地球到日-火系平动点轨道的转移

平动点轨道特殊的空间位置及动力学特征,使其在深空探测中具有重要的应用。以日-火系平动点轨道(Lissajous与Halo轨道)任务为目标,结合平动点轨道的不变流形理论,研究了小推力转移问题。首先给出了圆型限制性三体动力学模型下平动点附近不变流形(稳定和不稳定流形)高阶分析解以及相应的计算实例。接着以流形分析解为基础,建立了初始小推力轨道优化模型,并利用改进的协作进化算法求解初始小推力轨道。最后将初始轨道离散,采用多点打靶法将最优控制问题转化为参数优化问题,并用序列二次规划方法(SQP)求解。仿真结果证明轨道设计方法的有效性。     

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: II

As examples of application of the technique suggested in the first part of this work, the problems of optimizing the trajectories of spacecraft transfers between circular coplanar orbits are considered in this second part. During the transfer the spacecraft is controlled by the vector of thrust of a limited-thrust jet engine. The mass consumption is minimized for a limited time of transfer. Extreme trajectories with two and three powered sections (Homan-type and bi-elliptic transfer trajectories) are numerically determined. The solution of these well-studied problems allows one to compare the results of applying the suggested technique with the results of application of other previously used techniques.     

非凸二次约束下航天器姿态机动路径迭代规划方法

针对复杂约束下航天器姿态机动路径规划问题,首先描述和分析了航天器姿态机动过程中面临的动力学和运动学约束、有界约束、姿态指向约束,把姿态指向约束利用非凸二次型进行表述;其次从能量最优角度出发,将该约束机动问题归纳为非凸二次约束二次规划问题;然后引入线性松弛技术,将该问题转化成双线性规划问题,求出其中一个变量的凸包络和凹包络,降低求解复杂度,从而求出原问题的一个线性松弛。同时为了提高求解精度,提出一种基于评价函数的迭代规划算法,利用线性松弛求出的解作为初值,通过评价函数进行迭代规划,最终求出原问题的最优解。仿真结果表明该方法不仅可以满足复杂的姿态约束,得到全局姿态优化路径,而且能够降低能量消耗。     

Exact analytic solution for the spin-up maneuver of an axially symmetric spacecraft

The problem of spinning-up an axially symmetric spacecraft subjected to an external torque constant in magnitude and parallel to the symmetry axis is considered. The existing exact analytic solution for an axially symmetric body is applied for the first time to this problem. The proposed solution is valid for any initial conditions of attitude and angular velocity and for any length of time and rotation amplitude. Furthermore, the proposed solution can be numerically evaluated up to any desired level of accuracy. Numerical experiments and comparison with an existing approximated solution and with the integration of the equations of motion are reported in the paper. Finally, a new approximated solution obtained from the exact one is introduced in this paper.     

脉冲展开式飞网质量分布与展开稳定性优化

针对脉冲展开式飞网的展开稳定性问题，从网面质量分布角度开展研究。首先基于传递矩阵方法、集中质量法和反比例函数推导建立了较优的网面质量分布模型。然后结合遗传算法优化得到了网面最优的质量分布状态，并通过大量数值仿真和回归分析得到了网面质量分布经验公式。提出的网面质量分布模型有效降低了网面质量分布问题中的优化变量个数，提升了网面质量分布最优解的求解效率。网面质量分布最优解所对应的网面展开稳定性得到大幅提升，网面质量分布经验公式对应的网面展开状态与最优解对应的网面展开状态基本一致。结果表明，将网面质量分布模型和网面质量分布经验公式应用到脉冲展开式飞网将有效提升飞网的展开稳定性。     

Numerical solution of nonlinear heat problem with moving boundary

Two phase gas–liquid flow in pipes is widely spread in space applications: bubble flows appear in cryogenic components transport through fuel/oxidant supply lines. Another important application is based on the fact that in liquid flows with small bubbles a close contact between the two phases occurs resulting in high rates of transfer between them. The compactness of a system makes it ideally suited to serve as a space-based two-phase bio-reactor which forms an important unit in environmental control and life support system deployed onboard. A numerical method was developed for solving a nonlinear problem of thermal interaction between a spherical gas bubble and surrounding liquid. The system of equations for describing this interaction was formulated. It includes ordinary and nonlinear partial differential equations. The problem was solved using finite-difference technique by dividing the system into spherical layers inside the bubble and employing the new variable which “freezes” the moving boundary of the bubble. A numerical solution is obtained for the problem of radial bubble motion induced by a sudden pressure change in the liquid—a situation which corresponds to the behavior of bubbles beyond a shock wave front when the latter enters a bubble curtain.     

在轨目标逼近远程导引律设计

利用最优多脉冲方法,对目标航天器逼近过程的远程导引段轨道进行设计.基于Lawden主矢量理论,解决固定时间、燃料最省的逼近轨道问题.通过仿真分析了固定初始条件时燃料消耗量随着转移时间的变化关系.对于燃料和时间均有约束的情况,给出了求解燃料最省和时间最小的多目标优化问题的方法.这一研究对于评估具体任务的燃料消耗和转移时间有重要意义.     

Damage assessment of large space structures through the variational approach

The present investigation is focused on the solution of a dynamic inverse problem which is concerned with the assessment of damage in large space structures by means of measured vibration data. This inverse problem has been presented as an optimization problem and has been solved through the use of the conjugate gradient method with the adjoint equation, also called the variational approach. When a high number of damaged elements has to be found and these elements are also severely damaged, it is shown that the use of an additional method is necessary in order to provide a better initial guess for the conjugate gradient method. A stochastic method, represented by the genetic algorithm method, has been chosen because it provides robust search in complex spaces and also reduces the chance of converging to local optima. The application of this hybrid approach showed that better results can be achieved, although the computational time for the application analyzed here could increase. The damage estimation has been evaluated using noiseless and noisy synthetic experimental data, and the reported results are concerned with a space truss structure.     

嫦娥五号探测器多圈调相地月转移应急轨道设计与分析

针对嫦娥五号任务上升段末期火箭二级发动机可能出现的提前关机故障造成入轨半长轴偏差较大和中途修正速度增量超限问题,提出了多圈调相地月转移轨道应急控制策略.首先,分析了不同入轨半长轴偏差、中途修正时刻与中途修正速度增量消耗之间的关系;其次,针对半长轴偏差较大问题,基于微分改正算法与B平面参数,设计了解析窗口搜索与多圈调相地...     

High order optimal feedback control of space trajectories with bounded control

Optimal feedback control is classically based on linear approximations, whose accuracy drops off rapidly in highly nonlinear dynamics. Several nonlinear optimal feedback control strategies have appeared in recent years. Among them, differential algebraic techniques have been used to tackle nonlinearities by expanding the solution of the optimal control problem about a reference trajectory and reducing the computation of optimal feedback control laws to the evaluation of high order polynomials. However, the resulting high order method could not handle control saturation constraints, which remain a critical facet of nonlinear optimal feedback control. This work introduces the management of saturating actuators in the differential algebraic method. More specifically, the constraints are included in the optimal control problem formulation and differential algebra is used to expand the associated optimal bang–bang solution with respect to the initial and terminal conditions. Optimal feedback control laws for thrust direction and switching times are again computed by evaluating the resulting polynomials. Illustrative applications are presented in the frame of the optimal low-thrust transfer to asteroid 1996 FG₃.     

共面圆轨道最省燃料小推力多次变轨研究

本文研究两个共面圆轨道之间最节省燃料的多次变轨过程。通过分析，得到滑行段起点和终点之间的解析关系式，解决了变轨次数增加带来的计算问题。给出了从近地圆轨道到同面同步轨道的最优多次变轨的计算结果，并进行了分析和讨论     

单差PPP相位偏差估计方法及有效性分析

针对常规浮点解精密单点定位收敛时间长的问题,分析了PPP模型中模糊度不具备整数性质的原因,采用方向数据统计的计算方法,以单差模糊度为研究对象,对相位偏差进行了建模估计。算例从精密单点定位单差模糊度固定、定位精度以及收敛速度等方面对相位偏差改正的有效性进行了分析,结果表明,经相位偏差改正后的PPP模糊度固定成功率、收敛速度有了明显提高,定位精度较浮点解提高了一个量级。     

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.     

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.     

基于正则摄动理论的无动力飞行器末段制导方法

以无动力飞行器末制导问题为研究对象,用正则摄动理论对其纵向弹道方程进行处理。在零阶方程中考虑重力和气动力的主导部分,在一阶方程中考虑重力和气动力的剩余部分,进而获得了弹道方程的解析解,仿真证明了该解析解具有较高的计算精度和计算效率。基于弹道解析解所获得的脱靶量,提出了正则摄动制导方法,仿真证明了该方法具有最优性。     

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.