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.     

Rapid generation of entry trajectories with waypoint and no-fly zone constraints

Waypoints are positions for multiple payload deployments or reconnaissance missions, and no-fly zones are exclusion zones that cannot be passed for threat avoidance or due to geopolitical restrictions. This paper proposes a rapid entry trajectory generation approach satisfying waypoint and no-fly zone constraints for entry vehicles with relatively high lift-to-drag ratio. A lateral planning algorithm based on a Newton iteration scheme is developed to simultaneously design both the magnitude and sign of the control variable according to waypoints and no-fly zones. The algorithm converts the highly constrained trajectory planning problem into a series of one-parameter search problems based on a reduced-order system. Then, the quasi-equilibrium glide phenomenon is employed to extract the remaining state variables corresponding to longitudinal motion. The algorithm is tested using the Common Aero Vehicle model, and the results demonstrate that the algorithm can generate flyable entry trajectories rapidly within allowable tolerances while satisfying all the flight constraints.     

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.     

Missions to the asteroid anteros and the space of true anomalies

Three opportunities for missions to rendezvous ballistically with the Earth-crossing asteroid Anteros are studied to illustrate the requirements for a trip to a near-Earth minor planet. The rationale, sample payload, spacecraft requirements and trajectory characteristics of these opportunities are typical of a rendezvous mission to an accessible near-Earth object. Round trip ballistic trajectories to return small samples of the asteroid with launch dates between 1985 and 2000 are also presented. Contours of minimum total ΔV drawn in the space of launch and arrival true anomalies, given the designation Prime Rib curves, are introduced as a useful tool for mission design.     

Are aircraft parabolic flights really parabolic?

Aircraft parabolic flights provide repetitively short periods of reduced gravity by flying a ballistic trajectory, preceded and followed by periods of acceleration up to 2g. The ballistic part has a shape of an arc of a parabola and is quite commonly referred to as a parabolic flight.In this short note, we want to show that, although the wording parabolic is stricto sensu incorrect, it is though not far from reality and we show how to calculate simply the approximation committed by calling this trajectory parabolic.     

Design of Interplanetary Transfers with Passive Gravity Assists and Deep Space Maneuvers

In the paper, the problem of designing interplanetary trajectories with several swing-bys and deep-space maneuvers is solved using the method of virtual trajectories developed by the authors. The algorithms for the calculation of both heliocentric and planetocentric trajectory arcs are presented, including the case of resonant trajectories. The results of applying the method of virtual trajectories to the problem of designing an interplanetary transfer to Jupiter are given and compared with the baseline trajectories for the Juno, Europa Clipper, and Laplace missions.     

A method of calculation of vertical cutoff rigidity in the geomagnetic field

On the basis of generalization of the results of extensive trajectory calculations for trial charged particles moving in the geomagnetic field the method of calculation of effective vertical cutoff rigidity, taking into account the values of K _p -index and local time, is developed. The IGRF and Tsyganenko-89 models are used for the geomagnetic field. A comparison of the results of model simulations with the experimental data on penetration of charged particles into near-Earth space is made, and penetration functions for typical spacecraft orbits are calculated.     

Problems of control of a re-entry vehicle with a moderate lift-to-drag ratio during its entry into the atmosphere

We consider the problems of control of the angular and trajectory motion of the Kliper re-entry vehicle. This spacecraft with a moderate hypersonic lift-to-drag ratio is designed according to the load-carrying frame scheme. Gas-dynamic engines, a split balancing flap, and an air brake are used as mounting devices of control.     

探月返回飞行器跳跃式再入轨迹优化

基于节点自适应稀疏配点法,提出一种高精度求解探月返回飞行器跳跃式再入轨迹优化问题的方法。该方法的基本策略是：首先,应用节点自适应稀疏配点法对完整的跳跃式再入轨迹进行优化;然后,根据优化得到的控制变量对再入动力学方程进行数值积分;当积分至跳跃轨迹的最高点时,以积分得到的状态变量值作为新的初始条件,对二次再入轨迹重新优化。仿真结果表明：1)对二次再入轨迹重新优化能够显著提高跳跃式再入轨迹的优化精度,否则轨迹优化精度低,终端误差较大;2)在跳跃轨迹的最高点进行的二次优化是一种准实时优化,在跳跃式再入轨迹的制导领域具有潜在应用价值。     

Manifold dynamics in the Earth–Moon system via isomorphic mapping with application to spacecraft end-of-life strategies

Recently, manifold dynamics has assumed an increasing relevance for analysis and design of low-energy missions, both in the Earth–Moon system and in alternative multibody environments. With regard to lunar missions, exterior and interior transfers, based on the transit through the regions where the collinear libration points L₁ and L₂ 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 focused on the definition and use of a special isomorphic mapping for low-energy mission analysis. A convenient set of cylindrical coordinates is employed to describe the spacecraft dynamics (i.e. position and velocity), in the context of the circular restricted three-body problem, used to model the spacecraft motion in the Earth–Moon system. This isomorphic mapping of trajectories allows the identification and 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. Heteroclinic connections, i.e. the trajectories that belong to both the stable and the unstable manifolds of two distinct periodic orbits, can be easily detected by means of this representation. This paper illustrates the use of isomorphic mapping for finding (a) periodic orbits, (b) heteroclinic connections between trajectories emanating from two Lyapunov orbits, the first at L₁, and the second at L₂, and (c) heteroclinic connections between trajectories emanating from the Lyapunov orbit at L₁ and from a particular unstable lunar orbit. Heteroclinic trajectories are asymptotic trajectories that travels at zero-propellant cost. In practical situations, a modest delta-v budget is required to perform transfers along the manifolds. This circumstance implies the possibility of performing complex missions, by combining different types of trajectory arcs belonging to the manifolds. This work studies also the possible application of manifold dynamics to defining suitable, convenient end-of-life strategies for spacecraft orbiting the Earth. Seven distinct options are identified, and lead to placing the spacecraft into the final disposal orbit, which is either (a) a lunar capture orbit, (b) a lunar impact trajectory, (c) a stable lunar periodic orbit, or (d) an outer orbit, never approaching the Earth or the Moon. Two remarkable properties that relate the velocity variations with the spacecraft energy are employed for the purpose of identifying the optimal locations, magnitudes, and directions of the velocity impulses needed to perform the seven transfer trajectories. The overall performance of each end-of-life strategy is evaluated in terms of time of flight and propellant budget.     

Applications of multi-body dynamical environments: The ARTEMIS transfer trajectory design

The application of forces in multi-body dynamical environments to permit the transfer of spacecraft from Earth orbit to Sun–Earth weak stability regions and then return to the Earth–Moon libration (L₁ and L₂) orbits has been successfully accomplished for the first time. This demonstrated that transfer is a positive step in the realization of a design process that can be used to transfer spacecraft with minimal Delta-V expenditures. Initialized using gravity assists to overcome fuel constraints; the ARTEMIS trajectory design has successfully placed two spacecrafts into Earth–Moon libration orbits by means of these applications.     

一种利用PCA的导弹轨迹重建方法

非同时刻成像将大幅恶化立体视觉方法对导弹的轨迹点重建精度和射向估计精度,针对这一问题,该文提出了应用主成分分析(PCA)重建导弹轨迹并且估计导弹射向的方法。论文在同步轨道双星观测条件下推导了将PCA方法应用到导弹轨迹重建及射向估计的原理,论证了目标三维轨迹最小方差投影直线在像面上的投影即为目标成像轨迹的最小方差投影直线,给出了通过求取目标成像轨迹的最小方差投影直线重建导弹发射面和轨迹点的算法。仿真实验表明了该方法可以有效地进行弹道轨迹的三维重建及射向估计,与现有方法相比,PCA方法重建精度更高,相机定标误差在一定范围内时,射向估计误差更小。     

MESSENGER's use of solar sailing for cost and risk reduction

The MErcury Surface, Space ENvironment, GEochemistry, and Ranging (MESSENGER) mission used six planetary gravity assists in order to enable capture into orbit about Mercury. A key element of MESSENGER's successful trajectory was achieving the proper gravity assist from each planetary flyby. The criticality of the MESSENGER gravity assists levied tight accuracy requirements on the planetary-flyby targeting. Major errors could have precluded Mercury orbit insertion or required modifications to the trajectory that increased mission complexity, cost, and risk by requiring additional Mercury flybys and extending mission duration. Throughout the mission, MESSENGER modified its strategy for achieving accurate planetary flybys. By using solar sailing, the MESSENGER team was able to eliminate all of the flyby approach maneuvers without sacrificing flyby accuracy, thereby saving mission ΔV margin. The elimination of these approach maneuvers also markedly reduced mission risk, as these approach maneuvers were nominally planned during a time of heightened sensitivity to errors and precluded unique flyby science opportunities. The paradigm shift used by MESSENGER may be useful for other interplanetary missions, particularly if their trajectories require gravity assists in the inner solar system.     

RLV再入标准轨道制导与设计

将可重复使用运载器(RLV)再入轨道设计分为纵向轨道和侧向轨道设计两个过程:首先引入形状约束因子,在不进行轨道积分的情况下,在再入走廊内快速优化设计RLV再入飞行剖面;然后结合轨道跟踪控制,设计侧向方位误差走廊,快速生成满足末端能量管理(TAEM)接口和航程等要求的再入轨道。进一步提出修正标准飞行剖面参数的航程更新及制导方法,并采用RLV概念模型进行仿真分析。仿真结果表明这种RLV再入轨道设计方法能够快速生成再入标准轨道,相应的RLV再入制导方法可行,且具有较高的制导精度和可靠性,鲁棒性好。     

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.      

Evolution of the out-of-plane amplitude for quasi-periodic trajectories in the Earth–Moon system

The out-of-plane amplitude along quasi-periodic trajectories in the Earth–Moon system is highly sensitive to perturbations in position and/or velocity as underscored recently by the ARTEMIS spacecraft. Controlling the evolution of the out-of-plane amplitude is non-trivial, but can be critical to satisfying mission requirements. The sensitivity of the out-of-plane amplitude evolution to perturbations due to lunar eccentricity, solar gravity, and solar radiation pressure is explored and a strategy for designing low-cost deterministic maneuvers to control the amplitude history is also examined. The method is sufficiently general and is applied to the L₁ quasi-periodic orbit that serves as a baseline for the ARTEMIS P2 trajectory.     

复杂禁飞区高超声速飞行器路径-轨迹双层规划

针对高超声速飞行器在复杂禁飞区的规避场景,为解决现有轨迹规划方法对任务初值依赖性强的问题,提出一种基于双层规划建模的路径-轨迹规划方法。其中,上层为路径规划,为轨迹提供路径点引导信息,避免轨迹陷入局部解;下层为轨迹规划,利用上层输出的路径点信息,将轨迹分割成多个横向机动小的子段,解析求解横纵向飞行剖面,减小运动模型简化误差。数值仿真表明,与现有轨迹优化方法相比,本方法能够选择指标更优的路径,提高了轨迹规划的全局性能;解析飞行剖面制导误差不超过0.03%,解决了大范围横向机动的剖面解析难题。     

Trajectory correction of the Spektr-R spacecraft motion

The results of refining the parameters of the Spektr-R spacecraft (RadioAstron project) motion after it was launched into the orbit of the Earth’s artificial satellite in July 2011 showed that, at the beginning of 2013, the condition of staying in the Earth’s shadow was violated. The duration of shading of the spacecraft exceeds the acceptable value (about 2 h). At the end of 2013 to the beginning of 2014, the ballistic lifetime of the spacecraft completed. Therefore, the question arose of how to correct the trajectory of the motion of the Spektr-R satellite using its onboard propulsion system. In this paper, the ballistic parameters that define the operation of onboard propulsion system when implementing the correction, and the ballistic characteristics of the orbital spacecraft motion before and after correction are presented.