Adaptive control for autonomous rendezvous of spacecraft on elliptical orbit

A strategy for spacecraft autonomous rendezvous on an elliptical orbit in situation of no orbit information is developed. Lawden equation is used to describe relative motion of two spacecraft. Then an adaptive gain factor is introduced, and an adaptive control law for auton- omous rendezvous on the elliptical orbit is designed using Lyapunov approach. The relative motion is proved to be ultimately bounded under this control law, and the final relative position error can achieve the expected magnitude. Simulation results indicate that the adaptive control law can realize autonomous rendezvous on the elliptical orbit with relative state information only.

An ergodic algorithm for long-term coverage of elliptical orbits

This paper deals with the coverage analysis problem of elliptical orbits. An algorithm based on ergodic theory, for long-term coverage of elliptical orbits, is proposed. The differential form of the invariant measure is constructed via the perturbation on mean orbital elements resulted from the J2 term of non-spherical shape of the earth. A rigorous proof for this is then given. Different from the case of circular orbits, here the flow and its space of the dynamical system are defined on a physical space, and the real-value function is defined as the characteristic function on station mask. Therefore, the long-term coverage is reduced to a double integral via Birkhoff-Khinchin theorem. The numerical implementation indicates that the ergodic algorithm developed is available for a wide range of eccentricities.

Angles-only relative navigation in highly elliptical orbits

For angles-only relative navigation system only measures line-of-sight information,there are inherent problems in the ability to determine the range between Chaser and Target. Angles-only relative navigation is an attractive alternative for inspecting or rendezvous with noncooperative target,if adequate accuracy can be achieved. Angles-only relative navigation model considering J2 perturbation is presented for tracking and rendezvous with noncooperative target in highly elliptical orbit. Impulsive out-of-plane maneuvers of the Chaser are used to improve the navigation accuracy. The first impulse burns in cross-track directions to change the orbit inclination of the Chaser. The second impulse burns after one orbit period to change the orbit of the Chaser back. The simulation results show that the relative navigation system without maneuvers can't correct the initial state errors,while impulsive out-ofplane maneuvers of the Chaser improves the navigation accuracy. Angles-only relative navigation with chaser vehicle maneuvers to improve observability is effective when the spacecrafts are in highly elliptical orbits.

Randomization-based algorithms for servicing spacecraft motion planning on elliptical orbit

Randomization-based motion planning algorithms are presented to solve problems of servicing spacecraft maneuvering in proximity to servicing targets on an elliptical orbit.The feasible trajectories of position and attitude for spacecraft are obtained by these algorithms under a variety of constraints.The state transition matrix is applied to computation of relative motion on elliptical orbits without performing numerical integration.The pseudo body coordinate system is built for identifying the planners on three coordinate axes with different functions.Finally,motion planning algorithm for translation and attitude taking account of the dependent variable (i.e.time) is used to obtain feasible trajectories.As the simulation examples indicate,the effectiveness of these methods is verified for relative motion while getting close to large structures,and the paper concludes with a detailed analysis of the results.

Simulation for MSS-2 low-perigee elliptical orbit satellites:an example of lithospheric magnetic field modelling

A future constellation of at least four geomagnetic satellites(designated Macao Scientific Satellite-1(MSS-1)and Macao Scientific Satellite-2(MSS-2))was recently proposed,to continue high-quality geomagnetic observations in the post-Swarm period,focusing especially on collecting data that will provide a global,three-dimensional survey of the geomagnetic field.In this paper,we present a simulation of two years of orbits(2020.01.01-2022.01.01)of two satellites(tentatively denoted as MSS-2)that are constellated in elliptical(200×5,300 km)low-perigee orbits.By comparing error variances of Gauss coefficients,we investigate the sensitivity of lithospheric magnetic field modelling to data collected from various satellite orbits,including a near circular reference orbit of 300×350km,and elliptical orbit of 180×5,300 km,220×5,300 km,200×3,000 km and 200×1,500 km.We find that in two years the two MSS-2 satellites can collect 35,000 observations at altitude below 250 km,data that will be useful in advancing the quality of lithospheric magnetic field modelling;this number of observations reflects the fact that only 4.5%of the flight time of these satellites will be below250 km(just 6.4%of their flight time below 300 km).By combining observations from the MSS-2 satellites’elliptical orbits of 200×5,300km with observations from a circular reference orbit,the variance of the geomagnetic model can be reduced by a factor of 285 at spherical harmonic degree n=200 and by a factor of 1,300 at n=250.The planned lower perigee of their orbits allows the new satellites to collect data at unprecedentedly lower altitudes,thus dramatically improving the spatial resolution of satellite-derived lithospheric field models,(up to 80%at n=150).In addition,lowering the apogee increases the time interval during which the satellites fly at near-Earth altitudes,thus improving the model predictions at all spherical harmonic degrees(around 52%-62%at n=150).The upper limit of the expected improvement to the field model at the orbital apogee is not as good as at the perigee.However,data from the MSS-1 orbit can help fill the gap between data from the MSS-2 orbits and from the circular reference orbit for the low-degree part of the model.The feasibility of even lower-altitude flight requires further discussion with satellite engineers.

The Solution of Optimal Two-Impulse Transfer between Elliptical Orbits with Plane Change

The optimizing total velocity increment Δv needed for orbital maneuver between two elliptic orbits with plane change is investigated. Two-impulse orbital transfer is used based on a changing of transfer velocities concept due to the changing in the energy. The transferring has been made between two elliptic orbits having a common centre of attraction with changing in their planes in standard Hohmann transfer with the terminal orbit which is elliptic orbit and not circular. We develop a treatment based on the elements of elliptic orbits a1,e1, a2,e2, and?aT,eT of the initial orbit, final orbit and transferred orbit respectively. The first impulse Δv1 at the perigee induces a rotation of the orbital plane by ?which will be minimized. The second impulse Δv2 at apogee is induced an angle ?to product the final elliptic orbit. The total plane change required . We calculate the total impulse Δv and minimize by optimizing angle of plane’s variation . We obtain a polynomial equation of six degrees on the two transfer angles between neither two elliptic orbits ?and . The solution obtained numerically, using programming code of MATHEMATICA V10, with no condition on the eccentricity or the semi-major axis of the initial, transformed, and the final orbits. We find that there are constrains on the transfer angles and α. For αit must be between 40°and 160°, and there is no solution if αis less than 40°and bigger than 160°and ?takes the values less than 40°. The minimum total velocity increments obtained at the value of ?less than 25°and& alpha;equal to 160°. This is an interesting result in orbital transfer problem in which the change of orbital plane is necessary for the transferring.

Design and control of multiple spacecraft formation flying in elliptical orbits

Spacecraft formation flying is an attractive new concept in international aeronautic fields because of its powerful functions and low cost. In this paper, the formation design and PD closed-loop control of spacecraft formation flying in elliptical orbits are discussed. Based on two-body relative dynamics, the true anomaly is applied as independent variable instead of the variable of time. Since the apogee is considered as the starting point, the six integrating constants are calculated. Therefore, the algebraic solution is obtained for the relative motion in elliptical orbits. Moreover, the formation design is presented and both circular formation and line formation are provided in terms of an algebraic solution. This paper also discusses the PD-closed loop control for precise formation control in elliptical orbits. In this part, the error-type state equation is put forward and the linear quadratic regulator （LQR） method is used to calculate PD parameters. Though the gain matrix calculated from LQR is time-variable because the error-type state equation is time variable, the PD parameters are also considered as constants because of their small changes in simulation. Finally, taking circular formation as an example, the initial orbital elements are achieved for three secondary spacecraft. And the numerical simulation is analyzed under PD formation control with initial errors and J2 perturbation. The simulation results demonstrate the validity of PD closed-loop control scheme.

Resolving multi-orbital effects on high harmonic generation from aligned N2 molecules in linearly and elliptically polarized intense laser fields

We perform an experimental study of the multi-orbital effect on the high-order harmonic generation(HHG) from aligned N_2 molecules in both linearly and elliptically polarized intense laser fields.Measured by a home-built extreme ultraviolet(XUV) flat grating spectrometer with the pump-probe method, the angular distributions of different orders of HHG are obtained, which show distinctive behaviors for harmonics in the plateau and the cut-off regions.The ellipticity dependence of HHG is investigated by aligning the molecular axis parallel or perpendicular to the laser polarization.Our results indicate that both the highest occupied molecular orbital(HOMO) as well as the lower one(HOMO-1) contribute to the HHG of N2 molecules, in either linearly or elliptically polarized intense laser field.The study paves the way for understanding the ultrafast electron dynamics of molecules exposed to an intense laser field.

Mathematical Rotordynamic Model Regarding Excitation Due to Elliptical Shaft Journals in Electrical Motors Considering the Gyroscopic Effect

The paper presents a mathematical rotordynamic model regarding excitation due to elliptical shaft journals in sleeve bearings of electrical motors also considering the gyroscopic effect. For this kind of excitation, a mathematical rotordynamic model was developed considering the influence of the oil film stiffness and damping of the sleeve bearings, the stiffness of the end-shields and bearing housings, the stiffness of the rotor, the electromagnetic stiffness in the air gap of the electrical motor and the mass moment of inertia of the rotor and therefore also considering the gyroscopic effect. The solution of the linear differential equation system leads to the mathematical description of the absolute orbits of the shaft centre, the shaft journals and the bearing housings and to the relative orbits between the shaft journals and the bearing housings. Additionally, the bearing housing velocities can also be derived with this mathematical rotordynamic model.

Inverse Transformation of Elliptical Relative State Transition Matrix

A new set of relative orbit elements (ROEs) is used to derive a new elliptical formation flying model in previous work. In-plane and out-of-plane relative motions can be completely decoupled, which benefits elliptical formation design. In order to study the elliptical control strategy and perturbation effects, it is necessary to derive the inverse transformation of the relative state transition matrix based on relative orbit elements. Poisson bracket theory is used to obtain the linear transformations between the two representations: the relative orbit elements and the geocentric orbital frame. In this paper, the details of these transformations are presented.

Approximate Kepler’s Elliptic Orbits with the Relativistic Effects

Beginning with a Lagrangian, we derived an approximate relativistic orbit equation which describes relativistic corrections to Keplerian orbits. The critical angular moment to guarantee the existence of periodic orbits is determined. An approximate relativistic Kepler’s elliptic orbit is illustrated by numerical simulation via a second-order perturbation method of averaging.

Optimal four-impulse rendezvous between coplanar elliptical orbits

Rendezvous in circular or near circular orbits has been investigated in great detail, while rendezvous in arbitrary eccentricity elliptical orbits is not sufficiently explored. Among the various optimization methods proposed for fuel optimal orbital rendezvous, Lawden＇s primer vector theory is favored by many researchers with its clear physical concept and simplicity in solu- tion. Prussing has applied the primer vector optimization theory to minimum-fuel, multiple-impulse, time-fixed orbital ren- dezvous in a near circular orbit and achieved great success. Extending Prussing＇s work, this paper will employ the primer vec- tor theory to study trajectory optimization problems of arbitrary eccentricity elliptical orbit rendezvous. Based on linearized equations of relative motion on elliptical reference orbit （referred to as T-H equations）, the primer vector theory is used to deal with time-fixed multiple-impulse optimal rendezvous between two coplanar, coaxial elliptical orbits with arbitrary large ec- centricity. A parameter adjustment method is developed for the prime vector to satisfy the Lawden＇s necessary condition for the optimal solution. Finally, the optimal multiple-impulse rendezvous solution including the time, direction and magnitudes of the impulse is obtained by solving the two-point boundary value problem. The rendezvous error of the linearized equation is also analyzed. The simulation results confirmed the analyzed results that the rendezvous error is small for the small eccentric- ity case and is large for the higher eccentricity. For better rendezvous accuracy of high eccentricity orbits, a combined method of multiplier penalty function with the simplex search method is used for local optimization. The simplex search method is sensitive to the initial values of optimization variables, but the simulation results show that initial values with the primer vector theory, and the local optimization algorithm can improve the rendezvous accuracy effectively with fast convergence, because the optimal results obtained by the primer vector theory are already very close to the actual optimal solution.

Universal method for designing periodic orbits by homotopy classes in the elliptic restricted three-body problem

The current methods for designing periodic orbits in the elliptic restricted three-body problem(ERTBP)have the disadvantages of targeting limited orbits and ergodic searches and considering only symmetric orbits.A universal method for designing periodic orbits is proposed in this paper.First,the homotopy classes of orbits are structured based on their topological structures.Second,a dynamic model based on homotopy classes,ranging from the circular restricted three-body problem(CRTBP)to the ERTBP,can be built using the homotopy method.Third,a multi-and a single-period orbit were selected based on the resonance ratios.Finally,the corresponding orbit in the ERTBP was computed by modifying the initial condition of the orbit in the CRTBP.This method,without an ergodic search,can extend to any orbit,including an asymmetric orbit in the CRTBP,to the ERTBP model,and the two orbits are of the same homotopy class.Examples of the Earth–Moon ERTBP are presented to verify the efficiency of this method.

Elliptical formation control based on relative orbit elements

A new set of relative orbit elements（ROEs）is used to derive a new elliptical formation flying model.In-plane and out-of-plane motions can be completely decoupled,which benefts elliptical formation design.The inverse transformation of the state transition matrix is derived to study the relative orbit control strategy.Impulsive feedback control laws are developed for both in-plane and out-of-plane relative motions.Control of in-plane and out-of-plane relative motions can be completely decoupled using the ROE-based feedback control law.A tangential impulsive control method is proposed to study the relationship of fuel consumption and maneuvering positions.An optimal analytical along-track impulsive control strategy is then derived.Different typical orbit maneuvers,including formation establishment,reconfguration,long-distance maneuvers,and formation keeping,are taken as examples to demonstrate the performance of the proposed control laws.The effects of relative measurement errors are also considered to validate the high accuracy of the proposed control method.

HEO轨道卫星相对运动建模与编队构形设计

随着全球卫星组网编队飞行技术成熟应用,通过卫星之间基本运动方程进行相对运动求解,各种编队构型设计已经成为一种发展趋势.建立了基于相对轨道根数偏差的相对运动模型,理论推导了二阶近似条件下的适用于大偏心率椭圆轨道的相对运动方程,并进一步分析了二阶项的精度.在二体假设下,使用近地点真近点角差取代平近点角差作为编队构型设计输入,对多种不同初始轨道根数差条件下的典型编队构型进行了设计以及构型特征分析.对编队构型设计的正确性进行了仿真验证,并分析了从二体假设推广到含摄动情况的适用性,结果表明本文推导的椭圆轨道相对运动方程和编队构型设计正确有效.

Switching LPV control for electromagnetic formation flying on highly elliptical orbit

The electromagnetic force generated by the interaction of electromagnetic coils can be used to replace the conventional propellant consumption mode in close relative motion control,thereby promoting the application of formation flight technology for long-term and continuous space missions.Herein,a hysteresis-switching logic-based switching linear parameter varying(LPV)controller synthesis technique with guaranteed performance for electromagnetic formation flying on a highly elliptical orbit is proposed.First,considering that the relative dynamics model of an elliptical orbit is characterized by time-varying uncertainty,the LPV model is described.By introducing switching LPV controllers among different scheduled parameter subsets,conservativeness can be reduced.Second,the system modeling error,the uncertainty caused by a simplified electromagnetic coil model,and external disturbance are considered to derive switching LPV controller synthesis conditions based on the guaranteed H_(∞)performance.Derivation analysis shows that the proposed switching LPV controller not only ensures the robustness of the system against uncertainties,but also realizes the control input constraints.Finally,numerical simulations and comparative analyses are performed to demonstrate the effectiveness and advantages of the proposed control method.

基于轨道动力学的椭圆轨道悬停方法

连续有限推力条件下,基于动力学原理设计了伴随卫星相对于椭圆轨道的参考卫星在任意位置实现悬停的方法。给出了对任意椭圆参考轨道实现悬停的开环控制律,推导了一个周期内的速度增量计算公式。特别分析了参考卫星为"Molniya"轨道时,实现悬停需要的控制推力及速度增量。仿真结果表明,"Molniya"轨道正下方1km的悬停伴飞,一个轨道周期时间内连续有限推力发动机需要产生的速度增量为10.317m/s。文章提出的方法也可用于椭圆轨道的空间圆或水平圆等非自然编队构型设计。

中国区域卫星导航系统星座的方案探讨

首先简要介绍了欧洲、日本发展卫星定位导航系统的情况。阐述了我国第二代区域卫星导航系统的必要性。指出了采用园轨道同步卫星和静止星的星座方案的优缺点。用计算机仿真详细研究了倾斜椭圆轨道同步卫星的轨道参数与定位性能关系。设计了一个静止卫星和椭圆同步卫星组成的 7星星座 ,并且评估了该星座在我国及周边地区的性能。仿真计算表明 。

基于椭圆轨道的Geo-SAR精确多普勒参数解析计算方法

该文综合考虑了轨道偏心率、地球扁率和地球自转三方面的因素,推导得出高精度星载SAR多普勒中心频率和多普勒调频率解析式。鉴于2维姿态导引对于Geo-SAR的必要性,进一步分析了姿态导引后和零姿态导引两种情况下的多普勒参数,为Geo-SAR的多普勒参数计算提供了实用且有效的方法。该文得到的多普勒参数解析计算式适用于任意轨道高度和任意姿态的星载SAR,且具备很高的精度。基于该结论,研究了运行于椭圆轨道的Geo-SAR多普勒中心和多普勒调频率特性。

双作用椭圆轨道滚柱泵输出特性

基于椭圆面积弧长特性分析,推导出双作用椭圆轨道滚柱泵的容积公式,继而求得该泵的流量及脉动;通过仿真,研究了该泵结构尺寸对流量脉动的影响,并对脉动程度作出分析。结果表明:该泵的流量输出是相同尺寸径向柱塞泵的5倍以上,是相同尺寸叶片泵的1.5倍以上,而脉动率只是微增;该泵配流角度与一般柱塞泵或叶片泵的配流角度之间存在一个相位差,差值为13°;脉动主要受滚柱个数影响,一般趋势是随个数增加而降低,但滚柱个数为4的倍数时脉动较大。该泵输出特性的分析结果为滚柱泵结构的优化及减振降噪等提供了理论依据。