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.     

Numerical solutions to exact equations of motion along near-optimal multi-orbit trajectories for a spacecraft in a Newtonian gravitational field

The actual topic of optimization of multi-orbit low-thrust spacecraft inter-orbital transfers is considered. We have developed an original approach to solving this problem, and it is described.     

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.     

Optimization of passive orbit with the use of gravity maneuver

A high-precision method of calculating gravitational interactions is applied in order to determine optimal trajectories. A number of problems, necessary for determination of optimal parameters at a launch of a spacecraft and during its flyby near celestial bodies, are considered. The spacecraft trajectory was determined by numerical integration of the equations of passive motion of the spacecraft and of the equations of motion for planets, the Sun, and the Moon. The optimal trajectory of the spacecraft approaching the Sun is determined by fitting its initial conditions.     

Qualitative Properties of Orbits for Minimum-Fuel Impulsive Transfers between Circular Coplanar Orbits with a Given Launch Time

The coplanar problem of minimizing propellant consumption in impulsive transfer between circular boundary orbits is investigated. The launch time and the initial configuration of objects on the boundary orbits are specified arbitrarily. The qualitative properties of optimal two-impulse trajectories and their optimality in the class of multi-impulse transfers are studied.     

Analysis of evolution of multi-orbit perturbed trajectories of a spacecraft in a central field of attraction

Properties of differential equations of multi-orbit trajectory motion of a spacecraft are investigated analytically. The spacecraft moves under the action of small perturbations (in particular, low thrust) in the plane of a central Newtonian field of attraction. The conditions are specified for existence of a partial singular aperiodic solution, in the neighborhood of which the behavior of osculating elements changes sharply. In this case, phase variables (the angular position of the pericenter and the true anomaly) are found to undergo the sharpest changes. The exact superposition of solutions is suggested for the equations of motion transformed to the form of a quasi-linear, weakly non-stationary system: a partial singular aperiodic solution and fast solutions oscillating around it. Asymptotic representations are obtained for both components of the superposition. They are fairly exact in the region of smallness of perturbing terms at a long variation of the argument.     

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.     

Optimization of Perturbed Spacecraft Trajectories Using Complex Dual Numbers. Part 2: Numerical Results

Cosmic Research - Numerical results are presented for optimizing the perturbed trajectories of spacecraft with finite-thrust. The optimal trajectories are calculated using an indirect approach...     

适用于微纳卫星的微型电离层光学探测器

文章介绍了质量小、体积小的微型电离层光学探测器技术,给出了探测器的探测原理和设计方案,通过探测原子氧远紫外夜气辉(136.5 nm)辐射强度,反演电离层总电子含量(TEC)。采用轻量化铝反射镜、抗干扰电荷前放设计,实现反射镜、工业级远紫外探测器、电子学的一体化和小型化,研制出适用于微纳卫星的1 kg量级微型空间电离层光学探测器。该型探测器能够利用微纳星群多种轨道搭载,可获得丰富数据的优势,获取全球电离层高时空分辨率总电子含量分布,为未来实现卫星编队探测提供载荷研制基础。     

不确定条件下采用协方差描述函数法的再入轨迹鲁棒优化

Uncertainties are taken into account in reentry trajectory optimization and a robust trajectory optimization model is constructed based on the robust optimization theory. There are computational difficulties in solving this robust trajectory optimization problem due to the random variables associated with the uncertainties. To overcome these difficulties, the covariance analysis describing function technique (CADET) is employed to convert the robust trajectory optimization model into an equivalent deterministic trajectory optimization formulation, which is then solvable by using the existing pseudo spectral method. In the case study, a robust reentry trajectory optimization problem considering the uncertainties of aerodynamic parameters is solved to obtain the robust optimal trajectories. By comparing with the traditional deterministic optimal trajectories, the robust optimal trajectories are significantly less sensitive to the uncertainties of aerodynamic parameters, showing the effectiveness of the presented method.      

Suboptimal,two-burn,midlevel thrust,LEO-GEO transfer: Practical control schemes

This paper presentes the results of an algorithm developed at INTELSAT to (1) synthesize suboptimal, two-burn midlevel thrust, LEO-GEO transfer trajectories; (2) define practical steering laws to approximate the nominal trajectories; and (3) simulate their performance. Capabilities of the algorithm include: independently selectable constant thrust levels for the two burns, constant acceleration, staging, fixing the mass at either ends of the transfer. Figures of inefficiency versus ideally impulsive transfer are plotted for a reference constant thrust case over a range of initial accelerations. The diagram indicates that acceptable inefficiencies are attainable in the initial acceleration range above 0.1 g. A comparison with an optimal two-burn low-thrust transfer indicates negligible degradation in efficiency. The results of an application to INTELSAT VI are included.     

Determination of the width of a corridor of re-entry into the atmosphere for the problem of launching a spacecraft into a circumplanetary orbit after braking in the atmosphere

A method of calculating the optimal trajectories of a spacecraft re-entry into the atmosphere is considered under the condition of minimization of a re-orbit burn at the apocenter of the segment of escape from the atmosphere and taking into account a constraint on the maximum overload.     

Optimal control of reorientation of a spacecraft using free trajectory method

The problem of optimal control over spatial reorientation of a spacecraft is considered. The functional having a sense of propellant consumption is minimized. The analytical solution to the formulated problem is presented. It is shown that the optimal solution can be found in the class of two-impulse control at which the spacecraft’s turn is performed along a free motion trajectory. In order to improve the accuracy of spacecraft guidance into a specified angular position, methods of control are suggested that realize the method of free trajectories. The synthesized controls are invariant with respect to both external perturbations and parametric errors. The results of mathematical modeling are presented that demonstrate high efficiency of developed control algorithms. Propellant consumption for realizing a programmed turn is numerically estimated taking into account considerable gravitational and aerodynamic moments acting upon the spacecraft.     

Optimal flights to near-Earth asteroid

The spacecraft flights to the Near-Earth asteroid in order to give an impact influence on the asteroid, correct its orbit and prevent the asteroid’s collision with the Earth are analyzed.In the first part, the impulse flights are analyzed in the Lambert approach. There are determined the optimal trajectories maximizing the asteroid deviation from the Earth.In the second part, the flights with the chemical and electric-jet engines are analyzed. The high thrust is used to launch the spacecraft from the geocentric orbit, and the low thrust is applied for the heliocentric motion. On the base of optimal impulse transfer, the optimal low thrust trajectories are determined using Pontryagin maximum principle.The numerical results are given for the flight to the asteroid Toutatis. Parameters of the spacecraft impact on the asteroid are determined. The asteroid deviation from the Earth caused by the spacecraft influence is presented.     

Optimal impulse point-to-elliptical orbit transfers of a spacecraft

The paper deals with energetically optimal multi-impulse transfer of a spacecraft in the central Newtonian gravity field near a planet. At the initial state of the transfer the distance from the spacecraft to the center of attraction, its radial and transversal velocity projections are known. At the end of the transfer the spacecraft must be located in the elliptical orbit with the given area and energy constants. The distance from the spacecraft to the center of attraction is bounded above and below, the transfer time being unspecified. The initial orbit intersects the inner boundary of the given ring.All the optimal solutions have been obtained by analytical way. A number of new solutions has been found for the given problem in comparison with the case of the transfer from the orbit at the free initial point.Up to five impulses can be applied on the optimal trajectories. The numerical simulation of the problem is carried out. It shows that all obtained solutions give not only local but global optimal energetic input on the corresponding conditions.     

亚轨道飞行器能量管理段在线轨迹设计及仿真

对亚轨道飞行器能量管理段在线轨迹生成方案进行了研究。该方案将地面投影几何轨迹参数化,通过迭代计算参数生成标准轨迹,验证轨迹可行性并选择最优轨迹。经仿真验证,该方案能够在较短时间内完成轨迹的在线生成,结果满足进场着陆要求。     

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.     

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.     

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。     

伴随卫星轨道保持

文中利用伴随轨道方程推导出伴随卫星相对运动状态转移方程。基于这一状态转移方程，研究了双脉冲校正中的伴随运动状态的变化，并得出需要的脉冲控制量的解析式。利用遗传算法对脉冲控制量进行了优化设计，求得使总脉冲最小的最优变轨时间，脉冲控制量和轨迹。