Quaternion regularization and trajectory motion control in celestial mechanics and astrodynamics: II

Problems of regularization in celestial mechanics and astrodynamics are considered, and basic regular quaternion models for celestial mechanics and astrodynamics are presented. It is shown that the effectiveness of analytical studies and numerical solutions to boundary value problems of controlling the trajectory motion of spacecraft can be improved by using quaternion models of astrodynamics. In this second part of the paper, specific singularity-type features (division by zero) are considered. They result from using classical equations in angular variables (particularly in Euler variables) in celestial mechanics and astrodynamics and can be eliminated by using Euler (Rodrigues-Hamilton) parameters and Hamilton quaternions. Basic regular (in the above sense) quaternion models of celestial mechanics and astrodynamics are considered; these include equations of trajectory motion written in nonholonomic, orbital, and ideal moving trihedrals whose rotational motions are described by Euler parameters and quaternions of turn; and quaternion equations of instantaneous orbit orientation of a celestial body (spacecraft). New quaternion regular equations are derived for the perturbed three-dimensional two-body problem (spacecraft trajectory motion). These equations are constructed using ideal rectangular Hansen coordinates and quaternion variables, and they have additional advantages over those known for regular Kustaanheimo-Stiefel equations.     

The Use of Quaternions in the Optimal Control Problems of Motion of the Center of Mass of a Spacecraft in a Newtonian Gravitational Field: II

The problem of a rendezvous in the central Newtonian gravitational field is considered for a controlled spacecraft and an uncontrollable spacecraft moving along an elliptic Keplerian orbit. For solving the problem, two variants of the equations of motion for the spacecraft center of mass are used, written in rotating coordinate systems and using quaternion variables to describe the orientations of these coordinate systems. In the first variant of the equations of motion a quaternion variable characterizes the orientation of an instantaneous orbit of the spacecraft and the spacecraft location in the orbit, while in the second variant it characterizes the orientation of the plane of the spacecraft instantaneous orbit and the location of a generalized pericenter in the orbit. The quaternion variable used in the second variant of the equations of motion is a quaternion osculating element of the spacecraft orbit. The problem of a rendezvous of two spacecraft is formulated as a problem of optimal control by the motion of the center of mass of a controlled spacecraft with a movable right end of the trajectory, and it is solved on the basis of Pontryagin's maximum principle.     

Trajectory Control for a Spacecraft with an Inflatable Braking Device in the Martian Atmosphere

The possibility of using an inflatable braking device for controlled descent in the Martian atmosphere of large-capacity cargoes is analyzed. The most complicated version of the trajectory control problem is considered, namely, the injection of a spacecraft at hyperbolic velocity into a parking orbit after braking in the atmosphere.     

近距离航天器相对轨道的鲁棒自适应控制

针对近距离航天器的相对轨道提出了一种鲁棒自适应控制律。在追踪星本体坐标系中考虑航天器的相对运动。首先,在转动惯量未知的情形下提出了自适应控制律,保证系统的全局渐近稳定性。其次,将两星地心引力加速度之差作为干扰加速度,并假设干扰有未知上界,对自适应控制律进行修正,提出了鲁棒自适应律,使得系统是全局一致最终有界稳定的。控制律的设计不需要绝对轨道信息,适用于任意轨道。对航天器编队飞行和空间交会两种情形分别进行了仿真分析,结果表明所设计的控制律是合理有效的。     

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.     

访问多个特定相对位置的航天器轨道设计

在航天器轨道设计问题中,将惯性空间中经典的吉布斯三矢量定轨方法拓展到相对运动空间中,给出了一种相对运动条件下的三矢量定轨方法。针对已知轨道的目标航天器,以及二个或三个给定的空间相对位置,基于相对运动方程,提出了设计跟随航天器飞行轨道的数值方法。以轨道面共面或异面,以及目标航天器轨道形状为椭圆或圆,将问题分为四种情况进行约束条件和自由变量个数的分析讨论。对于自由变量个数多于约束方程的情况,额外给定周期重访约束,将各种情况下的特定相对位置访问问题转化为一至二维的非线性方程(组)求解问题。对一维方程求解采用分段黄金分割+割线法进行快速求解;对二维方程组通过网格法搜索迭代初值并通过牛顿迭代快速求解。进一步基于线性模型的解,采用微分修正方法求解了各情况下J2摄动模型下的结果。数值算例验证了提出方法的正确性及有效性。     

The Use of Quaternions in the Optimal Control Problems of Motion of the Center of Mass of a Spacecraft in a Newtonian Gravitational Field: I

The problem of optimal control is considered for the motion of the center of mass of a spacecraft in a central Newtonian gravitational field. For solving the problem, two variants of the equations of motion for the spacecraft center of mass are used, written in rotating coordinate systems. Both the variants have a quaternion variable among the phase variables. In the first variant this variable characterizes the orientation of an instantaneous orbit of the spacecraft and (simultaneously) the spacecraft location in this orbit, while in the second variant only the instantaneous orbit orientation is specified by it. The suggested equations are convenient in the respect that they allow the general three-dimensional problem of optimal control by the motion of the spacecraft center of mass to be considered as a composition of two interrelated problems. In the first variant these problems are (1) the problem of control of the shape and size of the spacecraft orbit and (2) the problem of control of the orientation of a spacecraft orbit and the spacecraft location in this orbit. The second variant treats (1) the problem of control of the shape and size of the spacecraft orbit and the orbit location of the spacecraft and (2) the problem of control of the orientation of the spacecraft orbit. The use of quaternion variables makes this consideration most efficient. The problem of optimal control is solved on the basis of the maximum principle. Several first integrals of the systems of equations of the boundary value problems of the maximum principle are found. Transformations are suggested that reduce the dimensions of the systems of differential equations of boundary value problems (without complicating them). Geometrical interpretations are given to the transformations and first integrals. The relation of the vectorial first integral of one of the derived systems of equations (which is an analog of the well-known vectorial first integral of the studied problem of optimal control) with the found quaternion first integral is considered. In this paper, which is the first part of the work, we consider the models of motion of the spacecraft center of mass that employ quaternion variables. The problem of optimal control by the motion of the spacecraft center of mass is investigated on the basis of the first variant of equations of motion. An example of a numerical solution of the problem is given.     

The Use of Quaternions in the Optimal Control Problems of Motion of the Center of Mass of a Spacecraft in a Newtonian Gravitational Field: III

The results of numerical solution of the problem of a rendezvous in the central Newtonian gravitational field of a controlled spacecraft with an uncontrollable spacecraft moving along an elliptic Keplerian orbit are presented. Two variants of the equations of motion for the spacecraft center of mass are used, written in rotating coordinate systems and using quaternion variables to describe the orientations of these coordinate systems. The problem of a rendezvous of two spacecraft is formulated [1, 2] as a problem of optimal control by the motion of the center of mass of a controlled spacecraft with a movable right end of the trajectory, and it is solved on the basis of Pontryagin's maximum principle. The paper is a continuation of papers [1, 2], where the problem of a rendezvous of two spacecraft has been considered theoretically using the two above variants of the equations of motion for the center of mass of the controlled spacecraft.     

相邻两航天器相对运动近似方程的解析解及比动能方程

在地心引力场中,当目标航天器沿近圆轨道作无动力运动时,与目标航天器相邻的受控航天器相对于目标航天器的运动可以近似地用Hill方程描述。文章给出了受控航天器对目标航天器运动的推力加速度随时间线性变化时Hill方程的解析解。并根据Hill方程导出了受控航天器相对目标航天器运动的比动能方程。还讨论了比动能方程在上述两航天器轨道相遇和轨道交会问题中的应用。     

航天员舱外相对航天器运动轨迹摄动研究

用轨道根数描述载人航天器运动,在其轨道坐标系(LVLH)中,建立了含地球J2项引力和大气阻力摄动加速度的航天员质心相对航天器的运动模型。在参考轨道存在小偏心率时,对基于圆参考轨道假设推导的航天器编队飞行的线性时变系统状态矩阵进行了修正。仿真结果表明:对偏心率为0.002的航天器轨道,修正后模型所得航天员相对轨迹摄动量在离舱300 s后达到0.8%,是修正前模型的5倍,但计算量增加不超过5%,更适于航天员舱外飞行和近距离编队飞行建模。     

太阳帆日心定点悬浮转移轨道设计

研究了太阳帆航天器日心定点悬浮轨道(HFDO)的转移轨道设计问题,以球坐标形式建立了太阳帆的动力学模型,基于该模型给出在日心悬浮轨道基础上实现定点悬浮的条件,提出了一种实现日心定点悬浮的转移轨道设计方法。首先,确定定点悬浮的位置;然后,设计经过该位置的绕日极轨轨道;最后,实施轨道减速实现定点悬浮,并给出了解析形式的轨道控制律。结合太阳极地观测任务,设计了定点悬浮在太阳北极1AU处的太阳帆转移轨道。仿真结果表明:该轨道转移方案总耗时3.5年,太阳帆定点到黄北极距日心1AU处,此后只要保持太阳光垂直照射帆面,即可维持稳定的悬浮状态。     

Lorentz-force-perturbed orbits with application to J2-invariant formation

The Lorentz force acting on an electrostatically charged spacecraft in the Earth's magnetic field provides a new propellantless means for controlling a spacecraft's orbit. Assuming that the Lorentz force is much smaller than the gravitational force, the perturbation of a charged spacecraft's orbit by the Lorentz force in the Earth's magnetic field, which is simplified as a titled rotating dipole, is studied in this article. Our research starts with the derivation of the equations of motion in geocentric equatorial inertial Cartesian coordinates using Lagrange mechanics, and then derives the Gauss variational equations involving Lorentz-force perturbation using a set of nodal inertial coordinates as an intermediate step. Subsequently, the approximate averaged changes in classical orbital elements, including single-orbit-averaged and one-day-averaged changes, are obtained by employing orbital averaging. We have found that the approximate analytic one-day-averaged changes in semi-major axis, eccentricity, and inclination are nearly zero, and those in the other three angular orbital elements are affected by J₂ and Lorentz-force perturbations. This characteristic is applied to model bounded relative orbital motion in the presence of the Lorentz force, which is termed Lorentz-augmented J₂-invariant formation. The necessary condition for J₂-invariant formation is derived when the chief spacecraft's reference orbit is either circular or elliptical. It is shown that J₂-invariant formation is easier to implement if the deputy spacecraft is capable of establishing electric charge. All conclusions drawn from the approximate analytic solutions are verified by numerical simulation.     

Optimal nodal flyby with near-Earth asteroids using electric sail

The aim of this paper is to quantify the performance of an Electric Solar Wind Sail for accomplishing flyby missions toward one of the two orbital nodes of a near-Earth asteroid. Assuming a simplified, two-dimensional mission scenario, a preliminary mission analysis has been conducted involving the whole known population of those asteroids at the beginning of the 2013 year. The analysis of each mission scenario has been performed within an optimal framework, by calculating the minimum-time trajectory required to reach each orbital node of the target asteroid. A considerable amount of simulation data have been collected, using the spacecraft characteristic acceleration as a parameter to quantify the Electric Solar Wind Sail propulsive performance. The minimum time trajectory exhibits a different structure, which may or may not include a solar wind assist maneuver, depending both on the Sun-node distance and the value of the spacecraft characteristic acceleration. Simulations show that over 60% of near-Earth asteroids can be reached with a total mission time less than 100 days, whereas the entire population can be reached in less than 10 months with a spacecraft characteristic acceleration of 1 mm/s².     

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.     

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。     

太阳帆航天器的轨道动力学和轨道控制研究

研究了太阳帆轨道动力学和利用太阳帆推进实现非开普勒轨道的太阳帆控制问题 ,推导了Gauss形式的太阳帆探测器密切轨道六要素微分方程,分析了太阳帆的轨道控制设 计方法,描述了适合太阳帆姿态控制的执行机构。在此理论基础上以SPORT计划作为设计实 例,并进行了设计与仿真,实现了任务要求的目标轨道。     

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.     

运载火箭推力故障下在线计算轻量化任务重构方法

为了避免运载火箭推力下降故障引起发射任务失败,基于径向基神经网络,提出了一种在线计算轻量化的任务重构方法,可快速在线计算最优救援轨道对应飞行轨迹(最优轨迹)的近似解.在离线部分,结合凸优化与自适应配点法产生"故障状态-最优轨迹"数据集.数据集被用来训练径向基神经网络,建立轨迹决策模型来构建故障状态到最优轨迹的动力学关系...     

一种挠性航天器的对偶四元数姿轨耦合控制方法

为解决复杂的挠性航天器的姿轨控制问题,对于挠性航天器的姿轨耦合动力学建模与控制展开研究。基于对偶四元数原理,推导给出一套挠性航天器的姿轨一体化动力学模型。此种模型能够紧凑描述航天器的轨道和姿态,且能够自动引入航天器平动、转动与挠性附件振动三者之间的关联耦合作用。基于此模型设计了一种自适应位置姿态跟踪控制器,该控制器能够在航天器质量特性参数未知的情况下,对其位置和姿态进行轨迹跟踪控制,并使位置和姿态误差收敛。该自适应控制器还可对航天器上挠性附件对系统的耦合作用进行估计,进而在控制输出中对其进行补偿,提高卫星控制系统的稳定性。通过仿真对控制律进行校验,结果表明该控制律对挠性航天器控制效果良好,具有一定的工程应用参考价值。     

Precise modelling of solar and thermal accelerations on Rosetta

This paper presents an analytical approach for the high-fidelity model of the accelerations induced by the Solar Radiation Pressure (SRP) and the Thermal Recoil Pressure (TRP) on ESA’s Rosetta spacecraft. The relevant gravitational forces that are induced by planets, moons, and asteroids can readily be incorporated for predicting interplanetary trajectories. However, there are additional perturbation forces that cause residual errors in the orbit determination process. These are the so-called “small forces”, which are mainly induced by the SRP and TRP effects and are often not modelled adequately or not completely. In the case of deep-space missions, the spacecraft travels a wide range of distances relative to the Sun. This makes the spacecraft exposed to a wide range of solar fluxes and surface temperatures. This paper establishes a high-fidelity acceleration model, which enables more precise orbit predictions for interplanetary spacecraft. The application of the model is demonstrated and validated using the orbit determination data and in-flight temperature data of the Rosetta spacecraft.