On the generation of satellite position (and velocity) by a mixed analytical-numerical procedure

The generation of accurate Earth-satellite ephemerides by numerical integration, over a period of perhaps weeks, can consume an inordinate amount of computer time. No satisfactory purely analytical procedure exists, but if short-period components of the standard elliptic elements are removed analytically, the resulting mean elements can be integrated with a step time that is longer than the satellite's orbital period.The definition of the mean elements depends on the particular perturbations included in the orbit generator and regarded as non-resonant. It is best if short-period perturbations are not applied to the orbital elements themselves but to the satellite's position (and velocity if required), expressed in a system of cylindrical polar coordinates, and the paper shows how mean elements can be recovered from position and velocity.A computer program has been written to test the proposed procedure for generating ephemerides, using a truncated potential field. Some results from this program are presented.     

A high order method for orbital conjunctions analysis: Sensitivity to initial uncertainties

A high order method to quickly assess the effect that uncertainties produce on orbital conjunctions through a numerical high-fidelity propagator is presented. In particular, the dependency of time and distance of closest approach to initial uncertainties on position and velocity of both objects involved in a conjunction is studied. The approach relies on a numerical integration based on differential algebraic techniques and a high-order algorithm that expands the time and distance of closest approach in Taylor series with respect to relevant uncertainties. The modeled perturbations are atmospheric drag, using NRLMSISE-00 air density model, solar radiation pressure with shadow, third body perturbation using JPL’s DE405 ephemeris, and EGM2008 gravity model. The polynomial approximation of the final position is used as an input to compute analytically the expansion of time and distance of closest approach. As a result, the analysis of a close encounter can be performed through fast, multiple evaluations of Taylor polynomials. Test cases with objects ranging from LEO to GEO regimes are considered to assess the performances and the accuracy of the proposed method.     

Symplectic integration of space debris motion considering several Earth’s shadowing models

In this work, we present a symplectic integration scheme to numerically compute space debris motion. Such an integrator is particularly suitable to obtain reliable trajectories of objects lying on high orbits, especially geostationary ones. Indeed, it has already been demonstrated that such objects could stay there for hundreds of years. Our model takes into account the Earth’s gravitational potential, luni-solar and planetary gravitational perturbations and direct solar radiation pressure. Based on the analysis of the energy conservation and on a comparison with a high order non-symplectic integrator, we show that our algorithm allows us to use large time steps and keep accurate results. We also propose an innovative method to model Earth’s shadow crossings by means of a smooth shadow function. In the particular framework of symplectic integration, such a function needs to be included analytically in the equations of motion in order to prevent numerical drifts of the energy. For the sake of completeness, both cylindrical shadows and penumbra transitions models are considered. We show that both models are not equivalent and that big discrepancies actually appear between associated orbits, especially for high area-to-mass ratios.     

Analysis of radial orbit errors of ERS-1, and the development of super-tailored gravity models

The altimetry mission of the future ESA remote sensing satellite ERS-1 requires very accurate orbit solutions, of which in particular the radial position component should have an accuracy of approximately 10 cm. This paper presents some investigations into the possibility of reducing the radial position error due to the earth's gravity field, which is by far the largest contributing error source.With a detailed harmonic analysis of the ERS-1 orbit a number of gravity field model terms are identified which produce the major radial orbit perturbations. These dominant terms are adjusted in a least-squares orbit determination and parameter estimation procedure using actual SEASAT laser tracking observations and altimeter height measurements. The initial gravity model is the NASA GEM-L2 model derived from satellite tracking data only, with an emphasis on LAGEOS data. The resulting super-tailored model yields a significantly improved radial accuracy relative to GEM-L2, but fails to reach the accuracy of the SEASAT-tailored model PGS-S4.Finally, the SEASAT altimeter residuals and the residuals of the cross-over differences are analyzed in the frequency domain by applying a special filtering technique which separates the major radial orbit error and geoid error contributions.     

Gravity field models from kinematic orbits of CHAMP,GRACE and GOCE satellites

The aim of our work is to generate Earth’s gravity field models from GPS positions of low Earth orbiters. Our inversion method is based on Newton’s second law, which relates the observed acceleration of the satellite with forces acting on it. The observed acceleration is obtained as numerical second derivative of kinematic positions. Observation equations are formulated using the gradient of the spherical harmonic expansion of the geopotential. Other forces are either modelled (lunisolar perturbations, tides) or provided by onboard measurements (nongravitational perturbations). From this linear regression model the geopotential harmonic coefficients are obtained.     

Properties of the high area-to-mass ratio space debris population at high altitudes

In the framework of its space debris research activities ESA established an optical survey program to study the space debris environment at high altitudes, in particular in the geostationary ring and in the geostationary transfer orbit region. The Astronomical Institute of the University of Bern (AIUB) performs these surveys on behalf of ESA using ESA’s 1-m telescope in Tenerife. Regular observations were started in 1999 and are continued during about 120–140 nights per year. Results from these surveys revealed a substantial amount of space debris at high altitudes in the size range from 0.1 to 1 m. Several space debris populations with different dynamical properties were identified in the geostationary ring. During the searches for debris in the geostationary transfer orbit region a new population of objects in unexpected orbits, where no potential progenitors exist, was found. The orbital periods of these objects are clustered around one revolution per day; the eccentricities, however, are scattered between 0 and 0.6. By following-up some of these objects using the ESA telescope and AIUB’s 1-m telescope in Zimmerwald, Switzerland, it was possible to study the properties of this new population. One spectacular finding from monitoring the orbits over time spans of days to months is the fact that these objects must have extreme area-to-mass ratios, which are by several orders of magnitudes higher than for ‘normal-type’ debris. This in turn supports the hypothesis that the new population actually is debris generated in or near the geostationary ring and which is in orbits with periodically varying eccentricity and inclination due to perturbations by solar radiation pressure. In order to further study the nature of these debris, multi-color and temporal photometry (light curves) were acquired with the Zimmerwald telescope. The light curves show strong variations over short time intervals, including signals typical for specular reflections. Some objects exhibit distinct periodic variations with periods ranging from 10 to several 100 s. All this is indicative for objects with complicated shapes and some highly reflective surfaces.     

Semi-analytical investigations of high area-to-mass ratio geosynchronous space debris including Earth’s shadowing effects

This paper investigates the long-term perturbations of the orbits of geosynchronous space debris influenced by direct radiation pressure including the Earth’s shadowing effects. For this purpose, we propose an extension of our homemade semi-analytical theory [Valk, S., Lemaître, A., Deleflie, F. Semi-analytical theory of mean orbital motion for geosynchronous space debris under gravitational influence. Adv. Space Res., submitted for publication], based on the method developed by Aksnes [Aksnes, K. Short-period and long-period perturbations of a spherical satellite due to direct solar radiation. Celest. Mech. Dyn. Astron. 13, 89–104, 1976] and generalized into a more convenient non-singular formalism. The perturbations accounting for the direct radiation pressure with the Earth’s shadow are computed on a revolution-by-revolution basis, retaining the original osculating Hamiltonian disturbing function. In this framework, we compute the non-singular mean longitude at shadow entry and shadow exit at every orbital revolution in opposition to classical approaches where the singular eccentric anomalies at shadow entry and shadow exit are computed. This new algorithm is developed using non-singular variables. Consequently, it is particularly suitable for both near-circular and near-equatorial orbits as well as orbits which transit periodically around null eccentricities and null inclinations.The algorithm is tested by means of numerical integrations of the equations, averaged over the short periods, including radiation pressure, J₂, the combined Moon and Sun third body attraction as well as the long-term effects of the 1:1 resonance occurring for geosynchronous objects. As an extension of [Valk, S., Lemaître, A., Anselmo, L. Analytical and semi-analytical investigations of geosynchronous space debris with high area-to-mass ratios influenced by solar radiation pressure. Adv. Space Res., doi:10.1016/j.asr.2007.10.025, 2007b], we especially apply our analysis to space debris with area-to-mass as high as 20 m²/kg. This paper provides numerical and semi-analytical investigations leading to a deep understanding of the long-term evolution of the semi-major axis. Finally, these semi-analytical investigations are compared with accurate numerical integrations of the osculating equations of motion over time scales as high as 25 years.     

Stability of lobed balloons

This paper presents a computational study of the stability of simple lobed balloon structures. Two approaches are presented, one based on a wrinkled material model and one based on a variable Poisson’s ratio model that eliminates compressive stresses iteratively. The first approach is used to investigate the stability of both a single isotensoid and a stack of four isotensoids, for perturbations of infinitesimally small amplitude. It is found that both structures are stable for global deformation modes, but unstable for local modes at sufficiently large pressure. Both structures are stable at any pressure if an isotropic model is assumed. The second approach is used to investigate the stability of the isotensoid stack for large shape perturbations, taking into account contact between different surfaces. For this structure a distorted, stable configuration is found. It is also found that the volume enclosed by this configuration is smaller than that enclosed by the undistorted structure.     

Medium-scale traveling ionospheric disturbances (MSTID) modeling using a dense German GPS network

The MSTIDs are wave-like perturbations of the ionospheric plasma, which cause the most common ionospheric disturbances in mid-latitude regions. Generally the MSTIDs have velocities of several hundred meters per second and wavelengths of several hundred kilometers. The wave-like effect of the MSTID is one of the main obstacles for accurate interpolation of ionospheric corrections in a medium-scale reference GPS network. In this paper we show a new method of detecting and modeling MSTIDs using dense German GPS network. The between-epoch single difference ionospheric delays from a medium scale dense GPS network are used to estimate the parameter of the MSTID e.g. amplitude, wavelength and velocity. The efficiency of the approach is tested with data from about 320 GPS stations in and near Germany. A MSTID wave moving from east to west across Germany was observed at September 27 in 2009. Its wavelength is about 302 km, with a period of ∼7 min and velocity of about 700 m/s.     

广义正交多项式在时变系统跟踪问题中的应用

采用广义正交多项式的展开技术,利用其积分、乘积运算阵,对于时变线性系统最优控制中的跟踪问题,直接从最优控制的性能泛函入手,将最优控制问题转化为代数极值问题,从而避免了求解非线性Riccati方程,得到了一个较为简便的算法,并结合实例说明了方法的可行性.      

Comparative study of plasma wave activity in the plasma sheet boundary and near Earth plasma sheet

The general structure of low frequency wave activity in the Earth's plasma sheet and its boundary layer is studied on the basis of the measurements made by ‘Prognoz-8’ satellite in the northern night and morning parts of the magnetotail. Pronounced wave activity is permanently observed in the high latitude parts of the plasma sheet boundary layer. The level of perturbations diminishes when a spacecraft moves towards tail lobes and drops rather sharply when it moves to the central plasma sheet. The peaks near the low hybrid resonance frequency (correlating with the local strength of the magnetic field) are evident in the electric field fluctuations spectra. A plasma instability of low hybrid type driven by transverse current is though to be the possible candidate for the excitation of these waves. Wave activity in tail lobes is related mainly to the isolated hot and cold plasma streams.     

Statistical analysis of the ionospheric ion density recorded by DEMETER in the epicenter areas of earthquakes as well as in their magnetically conjugate point areas

Results of a statistical variation of total ion density observed in the vicinity of epicenters as well as around magnetically conjugated points of earthquakes are presented in this paper. Two data sets are used: the ion density measured by DEMETER during about 6.5?years and the list of strong earthquakes (M_W?≥?4.8) occurring globally during this period (14,764 earthquakes in total). First of all, ionospheric perturbations with 23–120?s observation time corresponding to spatial scales of 160–840?km are automatically detected by a software (64,287 anomalies in total). Second, it is checked if a perturbation could be associated either with the epicenter of an earthquake or with its magnetically conjugated point (distance?<?1500?km and time?<?15?days before the earthquake). The index Kp?<?3 is also considered in order to reduce the effect of the geomagnetic activity on the ionosphere during this period. The results show that it is possible to detect variations of the ionospheric parameters above the epicenter areas as well as above their conjugated points. About one third of the earthquakes are detected with ionospheric influence on both sides of the Earth. There is a trend showing that the perturbation length increases as the magnitude of the detected EQs but it is more obvious for large magnitude. The probability that a perturbation appears is higher on the day of the earthquake and then gradually decreases when the time before the earthquake increases. The spatial distribution of perturbations shows that the probability of perturbations appearing southeast of the epicenter before an earthquake is a little bit higher and that there is an obvious trend because perturbations appear west of the conjugated point of an earthquake.     

Mapping orbits around the asteroid 2001SN263

The present paper has the goal of mapping orbits, with respect to the perturbations, for a spacecraft traveling around the asteroid 2001SN₂₆₃. This asteroid is a triple system, which center of mass is in an elliptic orbit around the Sun. The perturbations considered in the present model are the ones due to the oblateness of the central body, the gravity field of the two satellite bodies (Beta and Gamma), the Sun, the Moon, the asteroids Vesta, Pallas and Ceres and all the planets of the Solar System. This mapping is important, because it shows the relative importance of each force for a given orbit for the spacecraft, helping to make a decision about which forces need to be included in the model for a given accuracy and nominal orbit. Another important application of this type of mapping is to find orbits that are less perturbed, since it is expected that those orbits have good potential to require a smaller number of station-keeping maneuvers. Simulations under different conditions are made to find those orbits. The main reason to study those trajectories is that, currently, there are several institutions in Brazil studying the possibility to make a mission to send a spacecraft to this asteroid (the so-called ASTER mission), because there are many important scientific studies that can be performed in that system. The results showed that Gamma is the main perturbing body, followed by Beta (10 times smaller) and the group Sun–Mars-oblateness of Alpha, with perturbations 1000 times weaker than the effects of Gamma. The other bodies have perturbations 10⁷ times smaller. The results also showed that circular and polar orbits are less perturbed, when compared to elliptical and equatorial orbits. Regarding the semi-major axis, an internal orbit is the best choice, followed by a larger external orbit. The inclination of the orbit plays an important role, and there are values for the inclination where the perturbations show minimum and maximum values, so it is important to make a good decision on those values.     

Predicting atmospheric delays for rapid ambiguity resolution in precise point positioning

Integer ambiguity resolution in precise point positioning (PPP) can shorten the initialization and re-initialization time, and ambiguity-fixed PPP solutions are also more reliable and accurate than ambiguity-float PPP solutions. However, signal interruptions are unavoidable in practical applications, particularly while operating in urban areas. Such signal interruptions can cause discontinuity of carrier phase arc, which introduces new integer ambiguities. Usually it will take approximately 15 min of continuous tracking to a reasonable number of satellites to fix new integer ambiguities. In many applications, it is impractical for a PPP user to wait for such a long time for the re-initialization. In this paper, a method for rapid ambiguity fixing in PPP is developed to avoid such a long re-initialization time. Firstly, the atmospheric delays were estimated epoch by epoch from ambiguity-fixed PPP solutions before the data gap or cycle slip occurs. A random walk procedure is then applied to predict the atmospheric delays accurately over a short time span. The predicted atmospheric delays then can be used to correct the observations which suffer from signal interruptions. Finally, the new ambiguities can be fixed with a distinct WL-LX-L3 (here LX denotes either of L1, L2) cascade ambiguity resolution strategy. Comprehensive experiments have demonstrated that the proposed method and strategy can fix zero-difference integer ambiguities successfully with only a single-epoch observation immediately after a short data gap. This technique works even when all satellites are interrupted at the same time. The duration of data gap bridged by this technique could be possibly extended if a more precise atmospheric delay prediction is found or on-the-fly (OTF) technology is applied. Based on the proposed method, real-time PPP with integer ambiguity fixing becomes more feasible in practice.     

Virtual baseline method for Beidou attitude determination – An improved long-short baseline ambiguity resolution method

Ambiguity resolution (AR) is a critical step for successful attitude determination using carrier phase measurements of a satellite navigation system such as Beidou. This paper proposes an improved method for AR in support of Beidou attitude determination based on the concept of a “virtual baseline”. In the traditional long-short baseline method, the short baseline is limited to a length less than half of the carrier wave length of the Beidou signals. In the proposed method, a virtual short baseline is formed by differencing two collinear baselines. The AR equations for virtual short and long baselines are derived and the factors impacting the AR accuracy are analysed. Numerical simulation studies were carried out to evaluate the performance of the proposed AR method. The simulation results confirmed that the proposed method is an improvement over the traditional approach -- not only is it easier to deploy collinear antennas but also it keeps the capability of epoch-by-epoch AR, which makes it immune to cycle slips and there is no need for initialisation of ambiguity searching.     

Robust H∞ controller design for attitude stabilization of flexible spacecraft with input constraints

This paper addresses the attitude stabilization and vibration suppression problem for flexible spacecraft subject to model parameter uncertainty, controller perturbations, external disturbances and input constraints. The attitude model of flexible spacecraft is described and converted into a state space form in terms of passive and active vibration suppression schemes. A novel state feedback controller is proposed based on the exactly available expectation of a new variable, which is introduced to model a randomly occurring controller gain perturbation. Based on Lyapunov stability theory, sufficient conditions for the existence of the nonfragile H_∞ controller considering input constraints are given based on linear matrix inequalities (LMIs) in terms of additive perturbation and multiplicative perturbation. Then, the developed controller subject to required constraints can be obtained, where the nonfragile property is fully considered to improve the tolerance to uncertainties in the controller. Numerical simulations are performed to demonstrate the effectiveness and superiority of the proposed control strategy in attitude stabilization and vibration suppression, where it should be noted that the passive vibration suppression scheme is superior for high natural frequencies while the active vibration suppression scheme is superior for low natural frequencies. Moreover, the low natural frequencies have more influence on the performance of attitude stabilization and vibration suppression.     

Particle filter-based real-time phase line bias estimation for GNSS-based attitude determination with common-clock receivers

Global navigation satellite system (GNSS)-based attitude determination has been widely adopted in a wide variety of terrestrial, sea, air, and space applications. Recently, the emergence of commercial multi-GNSS common-clock receivers has brought new opportunities for high-precision GNSS-based attitude determination with single-differenced (SD) model. However, the key requirement of using this approach is the accurate estimation of the troublesome line bias (LB) in real-time. In this contribution, we propose a particle filter-based real-time phase LB estimation approach that apply to SD model with single-system single-frequency observations from common-clock receiver. We first analyzed the relationship between the integer ambiguity ratio value and the phase LB. It is proved that the accuracy of a given phase LB value can be qualified by the related ambiguity resolution ratio value, and the normalized ratio value can therefore be used to represent the likelihood function of observations. Then, we presented the particle filter-based real-time phase LB estimation procedure, and assessed its performance using GPS L1/BDS B1I observations from two datasets collected with different types of common-clock receivers in terms of the accuracy and convergence time of phase LB estimation, the computation load, and the positioning and attitude determination accuracy with respect to the double-differenced (DD) model. Experimental results demonstrated that the phase LB could be accurately estimated with short convergence time (generally within 15 epochs). Moreover, compared with the classical DD approach, the particle filter-based SD approach delivers comparable positioning root-mean-square (RMS) errors in the North and East components but significantly smaller RMS errors in the Up component. Accordingly, the achievable yaw accuracy is comparable whereas the pitch accuracy is remarkably improved. The improvements of positioning accuracy in the Up component and pitch accuracy are approximately 35.7 % to 63.7 %, and 33.3 % to 63.1 %, respectively. Additionally, the single-epoch computation time with our particle filter-based SD approach is generally 0.08 s, which is obviously larger than the DD approach but could still meet the requirements of real-time applications below 10 Hz sampling.     

基于微多普勒的空间锥体目标微动分类

空间锥体目标在飞行时存在多种微动,具体可分为章动、进动及自旋,准确获取目标微动形式是弹道目标微动及结构参数估算的前提。首先分析了3种微动形式下锥体目标锥顶及锥底滑动型散射源微多普勒及其频谱分布特性,发现自旋锥体目标散射源微多普勒为0 Hz,章动锥体目标任意散射源微多普勒谱的峰值非等间距分布,进动锥体目标任意散射源微多普勒谱的峰值等间距分布。据此提出利用微多普勒阈值识别自旋、利用微多普勒谱峰值是否等间距分布识别章动和进动的分类方法。最后通过仿真说明了本文分类方法的有效性,可为空间锥体目标微动分类提供一定的参考。     

The ionospheric effect of atmospheric gravity waves excited prior to strong earthquake

We have computed perturbations in the nighttime mid-latitude F2 region ionosphere that could be produced by internal atmospheric gravity waves generated before strong earthquakes through ionospheric Joule heating due to the seismogenic electric field of short duration. There is a strong anisotropy of the atmospheric gravity wave effect with respect to the imminent earthquake epicentre, the electron density changes being maximum poleward and equatorward of the epicentre and being minimum eastward and westward of it. It should be noted that the duration of the electron density perturbation in the F2 region ionosphere is much longer than the duration of the primary precursor of an earthquake – the enhancement of the vertical electric field at the Earth’s surface, which initiates the atmospheric gravity wave generation. This fact is important from the practical point of view of predicting catastrophic earthquakes.