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.     

Low-energy Earth–Moon transfers involving manifolds through isomorphic mapping

Analysis and design of low-energy transfers to the Moon has been a subject of great interest for decades. Exterior and interior transfers, based on the transit through the regions where the collinear libration points are located, have been studied for a long time and some space missions have already taken advantage of the results of these studies. This paper is concerned with a geometrical approach for low-energy Earth-to-Moon mission analysis, based on isomorphic mapping. The isomorphic mapping of trajectories allows a visual, intuitive representation of periodic orbits and of the related invariant manifolds, which correspond to tubes that emanate from the curve associated with the periodic orbit. Two types of Earth-to-Moon missions are considered. The first mission is composed of the following arcs: (i) transfer trajectory from a circular low Earth orbit to the stable invariant manifold associated with the Lyapunov orbit at L₁ (corresponding to a specified energy level) and (ii) transfer trajectory along the unstable manifold associated with the Lyapunov orbit at L₁, with final injection in a periodic orbit around the Moon. The second mission is composed of the following arcs: (i) transfer trajectory from a circular low Earth orbit to the stable invariant manifold associated with the Lyapunov orbit at L₁ (corresponding to a specified energy level) and (ii) transfer trajectory along the unstable manifold associated with the Lyapunov orbit at L₁, with final injection in a capture (non-periodic) orbit around the Moon. In both cases three velocity impulses are needed to perform the transfer: the first at an unknown initial point along the low Earth orbit, the second at injection on the stable manifold, the third at injection in the final (periodic or capture) orbit. The final goal is in finding the optimization parameters, which are represented by the locations, directions, and magnitudes of the velocity impulses such that the overall delta-v of the transfer is minimized. This work proves how isomorphic mapping (in two distinct forms) can be profitably employed to optimize such transfers, by determining in a geometrical fashion the desired optimization parameters that minimize the delta-v budget required to perform the transfer.     

General solution for the optimal trajectory of an AFE-type spacecraft

The aeroassisted flight experiment (AFE) refers to an experimental spacecraft to be launched and then recovered by the Space Shuttle. It simulates a transfer from a geosynchronous Earth orbit (GEO) to a low Earth orbit (LEO). In this paper, with reference to an AFE-type spacecraft, an actual GEO-to-LEO transfer is considered under the following assumptions: the GEO and LEO orbital planes are identical; both the initial and final orbits are circular; the initial phase angle is given, while the final phase angle is free. The aeroassisted orbital transfer trajectory involves three branches: a preatmospheric branch, GEO-to-entry; an atmospheric branch, entry-to-exit; a post-atmospheric branch, exit-to-LEO. The optimal trajectory is determined by minimizing the total characteristic velocity. The optimization is performed with respect to the velocity impulses at GEO, LEO, and the time history of the angle of bank during the atmospheric pass. It is assumed that the entry path inclination is free and that the angle of attack is constant, = 17.0 deg. The sequential gradient-restoration algorithm is used to compute the optimal trajectory and it is shown that the best atmospheric pass is to be performed with constant angle of bank. The resulting optimal trajectory constitutes an ideal nominal trajectory for the generation of guidance trajectories for two reasons: the fact that the low value of the characteristic velocity is accompanied by relatively low values of the peak heating rate and the peak dynamic pressure; and the simplicity of the control distribution, requiring constant angle of bank.     

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.     

Optimal interplanetary rendezvous combining electric sail and high thrust propulsion system

The aim of this paper is to study, from a mission analysis point of view, the performance of a hybrid propulsion concept for a two-dimensional transfer towards a planet of the Solar System. The propulsion system is obtained by combining a chemical thruster, used for the phases of Earth escape and/or target planet capture, with an electric sail, which provides a continuous thrust during the heliocentric transfer. Two possible mission scenarios are investigated: in the first case the sailcraft reaches the target planet with zero hyperbolic excess velocity, thus performing a classical rendezvous mission in a heliocentric framework. In the second mission scenario, a given final hyperbolic excess velocity relative to the planet is tolerated in order to decrease the mission flight time. The amount of final hyperbolic excess velocity is used as a simulation parameter for a tradeoff study in which the minimum flight time is related to the total velocity variation required by the chemical thruster to accomplish the mission, that is, for Earth escape and planetary capture.     

Definition and analysis of properties of optimal three-impulse point-to-orbit transfers under time constraint

This paper deals with energetically optimal multi-impulse transfers of a spacecraft in the central Newtonian gravitational field near a planet. The transfer from a point on initial orbit to the final orbit with the given angular momentum and energy constants is considered. The transfer time is bounded above.With the distance from spacecraft to planet limited and the time free, such parameters of given orbits are chosen that the 3-impulse apsidal transfer T_r is optimal with an intermediate impulse at the maximum distance. On the basis of necessary optimality conditions an algorithm is developed to numerically determine the desired optimal transfer trajectory T_t under time constraint, the apsidal trajectory T_r being taken as initial approach. From the geometry and energy viewpoints, both trajectories T_t and T_r are close to each other. The trajectory T_t is also 3-impulsive, all impulses on it are nonapsidal. The distance from the planet is larger and the sum of impulses is less for this trajectory than for the initial trajectory T_r with the same transfer time.The simplified solution of the problem is constructed producing good approximation to the exact numerical optimization results. The solution asymptotics is found when the transfer time tends to infinity.     

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.     

Optimal control laws for heliocentric transfers with a magnetic sail

A magnetic sail is an advanced propellantless propulsion system that uses the interaction between the solar wind and an artificial magnetic field generated by the spacecraft, to produce a propulsive thrust in interplanetary space. The aim of this paper is to collect the available experimental data, and the simulation results, to develop a simplified mathematical model that describes the propulsive acceleration of a magnetic sail, in an analytical form, for mission analysis purposes. Such a mathematical model is then used for estimating the performance of a magnetic sail-based spacecraft in a two-dimensional, minimum time, deep space mission scenario. In particular, optimal and locally optimal steering laws are derived using an indirect approach. The obtained results are then applied to a mission analysis involving both an optimal Earth–Venus (circle-to-circle) interplanetary transfer, and a locally optimal Solar System escape trajectory. For example, assuming a characteristic acceleration of 1 mm/s², an optimal Earth–Venus transfer may be completed within about 380 days.     

A low-thrust transfer between the Earth–Moon and Sun–Earth systems based on invariant manifolds

A low-energy, low-thrust transfer between two halo orbits associated with two coupled three-body systems is studied in this paper. The transfer is composed of a ballistic departure, a ballistic insertion and a powered phase using low-thrust propulsion to connect these two trajectories. The ballistic departure and insertion are computed by constructing the unstable and stable invariant manifolds of the corresponding halo orbits, and a complete low-energy transfer based on the patched invariant manifolds is optimized using the particle swarm optimization (PSO) algorithm on the criterion of smallest velocity discontinuity and limited position discontinuity (less than 1 km). Then, the result is expropriated as the boundary conditions for the subsequent low-thrust trajectory design. The fuel-optimal problem is formulated using the calculus of variations and Pontryagin's Maximum Principle in a complete four-body dynamical environment. Then, a typical bang–bang control is derived and solved using the indirect method combined with a homotopic technique. The contributions of the present work mainly consist of two points. Firstly, the global search method proposed in this paper is simply handled using the PSO algorithm, a number of feasible solutions in a fairly wide range can be delivered without a priori or perfect knowledge of the transfers. Secondly, the indirect optimization method is used in the low-thrust trajectory design and the derivations of the first-order necessary conditions are simplified with a modified controlled, restricted four-body model.     

Optimum design of power-limited propulsion systems with application to fast Earth-to-Mars transfer

Power-limited systems with variable Isp, which have been studied theoretically since the beginning of astronautics, are getting closer to practical applications thanks to recent technological advances in the field of magnetosplasma rockets, such as Ad-Astra’s VASIMR concept. This type of propulsion system is considered for high-speed interplanetary transfers, such as Mars missions, with demanding payload fractions that would be compatible with manned missions. This paper explores the problem of the optimization of a power-limited propulsion system through simple performance models, and investigates the trade-off between the technological requirements, the transfer time and the payload fraction¹. Following previous works existing in literature, we model the technological characteristics of the vehicle through a small number of parameters, the most important of which being the specific weight (or mass-to-power ratio) of the power generation system. Also, we use in our models the classical “trajectory characteristic” parameter (defined as the integral over time of the squared thrust acceleration) which represents – under certain hypotheses – the propulsion requirements for an orbital or interplanetary transfer with a given time and a given thrust strategy. In this paper, we first give a review of existing methods in literature, then we present the equations of a new class of optimal design which maximizes the payload fraction, for a given transfer time and given technological characteristics. This class of optimal design is described through very simple equations that make possible to study more straightforwardly than existing calculations the links between the main mission requirements (transfer time and payload fraction) and the main technological requirements (specific weight of the power generation and structure mass ratio of the whole vehicle, excluding the power generation system). One important result obtained from these equations is a simple expression which estimates the theoretical upper limit of the power source’s specific weight as a function of transfer time and the payload mass ratio. In the last part of this paper, we apply this simple performance model to discuss the feasibility of a fast Earth-to-Mars transfer using a power-limited system.     

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.     

Mission design and analysis of European astrophysics missions orbiting libration points

The main characteristics of the trajectory design of space observatory missions in the Earth–Sun libration point region is highlighted, based on experiences gained in work performed by the authors on ESA missions. Free transfers always lead to large-amplitude orbits around L₂, their properties (amplitudes, phases, non-linear behaviour) are related to the conditions at perigee. Launch scenarios with different degrees of freedom in the perigee geometry and different strategies of sharing the apogee raising between launcher and spacecraft propulsion for Soyuz (with circular parking orbit or direct injection) and Ariane 5 launches from French Guiana will be discussed. Besides the orbit selection and transfer analysis, an important aspect of libration missions is the maintenance of the operational orbit. For some missions it is required to maximise the time between maintenance manoeuvres, and for some the thrust authority is limited. In both cases the exponential nature of the state transition matrix has to be considered. If the equivalent velocity error in the unstable direction becomes too large, the orbit can become unrecoverable, leading to a departure from the environment of the Lagrange point within a few months.     

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.     

月球探测器转移轨道的中途修正

月球探测器的中途制导指的是在其转移轨道中途对轨道进行修正，使其按预定轨道飞行。本文研究的中途修正问题是确定所需的速度修正脉冲，使探测器不断接近标称轨道，并以预定状态到达月球，完成预定的飞行任务。本文首先建立中途修正的模型，其中月球和太阳的位置由DE405得到。然后，采用精确的数值积分方法找出满足预定条件(近地点高度、近月点高度及转移时间)的转移轨道。以该轨道作为标称轨道，分析中途修正所需要的速度修正脉冲与发射入轨时的初始误差(近地点速度误差、入轨高度误差、发射窗口误差等)和修正时刻的关系。最后分析两次中途修正的速度修正脉冲和修正时刻的关系，并得出适合的中途修正时刻。     

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.     

电动帆航天器谷神星探测任务轨迹优化

针对电动帆航天器谷神星探测任务轨迹优化问题,提出一种基于高斯伪谱法和遗传算法的混合优化算法。为了验证所提出的混合优化算法有效性,并考察任务起始时间和电动帆特征加速度对探测任务的影响,进行了一定数量的数值仿真。仿真结果表明：电动帆航天器自地球至谷神星的飞行时间随着起始时间的变化呈周期性波动,波动周期基本与地球和谷神星的会合周期一致;电动帆航天器的特征加速度越小,完成过渡所需要的飞行时间越长,且一个具有中等特征加速度的电动帆航天器便能在可接受的时间内完成自地球至谷神星的过渡;所提出的混合优化算法是有效的,能够在无任何初值猜测的情况下完成电动帆航天器飞行轨迹的优化。     

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.     

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.     

How we will go to Mars

This article studies the efficiency of ejecting waste generated by the life support system (LSS) of a manned spacecraft to reduce initial mass on low earth orbit. The spacecraft is used for a long-duration interplanetary mission and is equipped with either a chemical or a nuclear-thermal propulsion system. For this study we simulate an optimal control problem for a given spacecraft maneuver. An impulsive approximation of the optimal interplanetary spacecraft trajectory is assumed, which allows us to reduce the general optimal control problem to hierarchic structure of 'outer' and 'inner' subproblems. This structure is analyzed using the Pontryagin's Maximum principle. Numerical results, illustrating the efficiency of waste ejection are shown for typical Earth-Mars transfer trajectories. This results confirm in theory that using a waste ejection system makes an early manned Mars mission possible without having to design and build new, advanced biological LSS.     

Maximum terminal velocity of relativistic rocket

The maximum terminal velocity problem of the classical propulsion is extended to a relativistic rocket assumed broken down into active mass, inert mass and gross payload. A fraction of the active mass is converted into energy shared between inert mass and active mass residual. Significant effects are considered. State and co-state equations are carried out to find the exhaust speed optimal profile.A first major result consists of a critical value of inert mass. Beyond it both true and effective jet speeds increase with time. Below it the true jet speed profile is reversed. At criticality, the best control consists of both velocities constant in time.A second meaningful result is represented by an interval of inert mass outside which no optimal control exists. Numerical results are discussed with particular emphasis to current concepts of antimatter propulsion.