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.     

Optimization of low-thrust transfers in the three body problem

We consider transfers with low thrust in an arbitrary field of forces. The modified method of transporting trajectory [1–4] is used for optimization of the transfers. The complexity of finding the transporting trajectory of a preset type can be the main obstacle to application of this method. This challenge is solved for the three-body problem in the Hill motion model. Numerical analysis of the method is performed using an example of the transfers to halo-orbits around the solar-terrestrial libration points.     

Optimum low-thrust limited power transfers between neighbouring elliptic non-equatorial orbits in a non-central gravity field

A complete first-order analytical solution is developed for the problem of optimum low-thrust limited power transfers between neighbouring elliptic non-equatorial orbits in a non-central gravity field. The optimization problem is formulated as a Mayer problem of optimal control with Cartesian elements as state variables. After applying the Pontryagin maximum principle and determining the optimal thrust acceleration, an intrinsic canonical transformation is performed: the Cartesian elements are changed by suitable orbital elements. Hori's method is applied in determining a first-order analytical solution. Simple analytical solutions are obtained explicitly for long-time transfers.     

Optimal low-thrust transfers between close near-circular coplanar orbits

Four types of optimal solutions are demonstrated to exist for transfers (time of flight is not fixed) between close near-circular coplanar orbits. One solution is realized with the help of fixed orientation of the propulsion system (PS) along a transversal in the orbital coordinate system. Another is reached at fixed orientation of the PS in the inertial coordinate system. The third and fourth types of solutions change the PS orientation in the process of executing the maneuver. Regions of existence are established for all types of solutions, and algorithms for determination of parameters of these maneuvers are suggested. The algorithms were used to calculate parameters of the maneuvers of transfer from a launching orbit to a working Sun-synchronous orbit, and to calculate the maneuvers of supporting the parameters of such an orbit in a specified range.     

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.     

Nonlinear feedback control of low-thrust orbital transfer in a central gravitational field

This paper investigates the problem of continuous-thrust orbital transfer using orbital elements feedback from a nonlinear control standpoint, utilizing concepts of controllability, feedback stabilizability and their interaction. Gauss's variational equations (GVEs) are used to model the state-space dynamics of motion under a central gravitational field. First, the notion of accessibility is reviewed. It is then shown that the GVEs are globally accessible. Based on the accessibility result, a nonlinear feedback controller is derived which asymptotically steers a spacecraft form an initial elliptic orbit to any given elliptic orbit. The performance of the new controller is illustrated by simulating an orbital transfer between two geosynchronous Earth orbits. It is shown that the low-thrust controller requires less fuel than an impulsive maneuver for the same transfer time. Closed-form, analytic expressions for the new orbital transfer controller are given. Finally, it is proven, based on a topological nonlinear stabilizability test, that there does not exist a continuous closed-loop controller that can transfer a spacecraft onto a parabolic escape trajectory.     

Optimal low-thrust transfer between neighbouring quasi-circular orbits around an oblate planet

The problem of optimal low-thrust, limited power transfer between quasi-circular orbits (e 0) around an oblate planet is analysed. It is assumed that the orbital changes due to thrust acceleration and Earth oblateness are of the same order. A first order solution to the problem is obtained by application of Pontryagin's Maximum Principle. Subsequently, by application of Hori's method for generalized canonical systems, a first order solution in a small parameter ε is derived. Finally, three particular cases of long-time transfer and the orbit maintenance manoeuvre are considered. The results obtained are in agreement and represent an extension of the work done by Marec.     

Alternating magnetic field forces for satellite formation flying

Selected future space missions, such as large aperture telescopes and multi-component interferometers, will require the precise positioning of a number of isolated satellites, yet many of the suggested approaches for providing satellite positioning forces have serious limitations. In this paper we propose a new approach, capable of providing both position and orientation forces, that resolves or alleviates many of these problems. We show that by using alternating fields and currents that finely-controlled forces can be induced on the satellites, which can be individually selected through frequency allocation. We also show, through analysis and experiment, that near field operation is feasible and can provide sufficient force and the necessary degrees of freedom to accurately position and orient small satellites relative to one another. In particular, the case of a telescope with a large number of free mirrors is developed to provide an example of the concept. We also discuss the far field extension of this concept.     

Method of continuation for optimization of interplanetary low-thrust trajectories

The problem of local optimization of interplanetary low-thrust trajectories is considered with the use of the maximum principle and continuation numerical methods. Two types of problems are analyzed: problems with limited power and problems with limited thrust. The latter problem is generalized by introducing the dependence of thrust and specific impulse on available electric power. In order to reduce the problem of optimal control to a boundary value problem, the Pontryagin maximum principle is used, and then, using the continuation method, this boundary value problem is reduced to the Cauchy problem. Variants of the continuation method for optimizing low-thrust trajectories are presented in the paper, including a new method of continuation for the limited thrust problem, which does not require any choice of the initial approximation for boundary values of conjugate variables.     

Planar oscillations of a vibrating dumbbell-like body in a central field of forces

The problem of planar motions of a dumbbell-like body of variable length in a central field of Newtonian attraction is considered both in the exact formulation and in satellite approximation. In the satellite approximation the true anomaly of the center of mass is used as an independent variable, which has allowed us to represent the equation of planar oscillations of the dumbbell in the form similar to the Beletskii equation. Some exact solutions to the inverse problem are given both in the complete formulation and in satellite approximation. Under an assumption of small orbit eccentricity and amplitude of the dumbbell vibrations the conditions of existence are found for families of almost periodic motions and splitting separatrices. The phenomena of alternation of regular and chaotic motions are established numerically, as well as chaos suppression with increasing frequency of vibrations. Using the method of averaging the stabilization of tangent equilibria is proved to be impossible.     

Motion of three viscoelastic planets in the field of their mutual attraction forces

Translational-rotational motion of three planets modeled by viscoelastic balls in the gravitational field of mutual attraction is studied in this paper. The system of equations of motion for the mechanical system under consideration is deduced from the d’Alembert-Lagrange variational principle. Using the method of separation of motions, an approximate system of ordinary differential equations, describing the translational-rotational motion of the planets, is obtained with taking into account perturbations caused by elasticity and dissipation. The found steady-state motion of the system is an analog to triangular libration points in the classical three-body problem.     

基于傅立叶平均法下的连续小推力动力学分析

通过将小推力展开为偏近点角的傅立叶级数,并对高斯摄动方程在一个轨道周期上的平均,将原方程的推力转化为仅由14个傅立叶系数表示的控制变量。仿真计算表明,平均化后的高斯方程使计算量与牛顿积分相比显著减少,且对小推力而言有足够的精度。对利用平均化后的高斯方程计算轨道根数时产生误差的原因进行了研究,并进一步分析小推力的范围和小推力近似表达式对上述误差的影响,为今后小推力下非开普勒轨道动力学分析提供了理论依据和参数。     

Evolution of motion of two viscoelastic planets in the field of forces of their mutual attraction

Translational-rotational motion of two viscoelastic planets in a gravitational force field is studied. The planets are modeled by homogeneous isotropic viscoelastic bodies. In their natural undeformed state each of the planets represents a sphere. We investigate a specific case when the planet’s centers of mass move in a fixed plane, the axis of rotation for each planet being directed along the normal to this plane. An equation describing the evolution of a slow angular variable (perihelion longitude) is derived. The observed displacement of the perihelion of Mercury is compared with the results obtained in the considered model problem about motion of two viscoelastic planets. Quite important is the fact that the planet of smaller mass (Mercury) moves not in a central Newtonian field of forces, but rather in the gravitational field of a rotating viscoelastic planet (Sun).     

Novel orbits of Mercury,Venus and Mars enabled using low-thrust propulsion

Exploration of the inner planets of the Solar System is vital to significantly enhance the understanding of the formulation of the Earth and other planets. This paper therefore considers the development of novel orbits of Mars, Mercury and Venus to enhance the opportunities for remote sensing of these planets. Continuous acceleration is used to extend the critical inclination of highly elliptical orbits at each planet and is shown to require modest thrust magnitudes. This paper also presents the extension of existing sun-synchronous orbits around Mars. However, unlike Earth and Mars, natural sun-synchronous orbits do not exist at Mercury or Venus. This research therefore also uses continuous acceleration to enable circular and elliptical sun-synchronous orbits, by ensuring that the orbit's nodal precession rate matches the planets mean orbital rate around the Sun, such that the lighting along the ground-track remains approximately constant over the mission duration. This property is useful both in terms of spacecraft design, due to the constant thermal conditions, and for comparison of images. Considerably high thrust levels are however required to enable these orbits, which are prohibitively high for orbits with inclinations around 90°. These orbits therefore require some development in electric propulsion systems before becoming feasible.     

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.     

Hydrodynamics during melting of an electroconductive sample in microgravity conditions under contactless positioning by electromagnetic forces

MHD-flow during melting of a spherical metal sample under contactless positioning by alternating magnetic field produced by two circular turns with currents, is investigated. It is shown that when the currents supplying the turns are opposite in direction, electromagnetic forces, originating from the interaction of the Foucault currents with alternating magnetic field, induce two torus-like eddies in the interior of the molten metal drop.     

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.     

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.     

某发动机喷管周围流场研究

运用Fluent流体动力学软件,采用结构化网格和RNG k-ε湍流模型研究了热试车条件下,某带高温隔热屏发动机喷管周围及隔热屏上温度场的分布,计算结果与试验结果吻合较好.研究结果表明：推进系统热试车条件下,高温隔热屏各组成部分温度均在各自耐受温度以内;高温隔热屏能有效地将发动机羽流的对流及辐射进行隔离,避免高温燃气对发动机周围的推进系统组件进行再加热.