Inter-satellite Range Measurements Research Articles

Spatiotemporal Characterization of Geophysical Signal Detection Capabilities of GRACE‐FO

AbstractThe intersatellite range measurements between the Gravity Recovery and Climate Experiment Follow‐On (GRACE‐FO) satellite pair enable mapping time variations in Earth's gravity field. The Laser Ranging Interferometer (LRI) and Microwave Instrument (MWI) provide independent range measurements. We compare gravity fields derived from the two instruments. The LRI captures time‐variable gravity field signals up to 21 mHz (165 km half‐wavelength) globally, while MWI reaches frequencies up to 15 mHz (230 km). Time‐variable mass variations free from annual cycles and trends are sensed up to 15 mHz (LRI) and 11 mHz (320 km) (MWI). Both instruments are shown to be excellent tools for detection and monitoring of sub‐monthly geophysical signals. Along‐track global maps of sub‐monthly signals are expected to shed light on the high‐frequency behavior of a large number of geophysical processes.

Orbit determination for a space-based gravitational wave observatory

It is necessary to keep the constellation with high accuracy and high stability in a space-based gravitational wave observatory. Few orbital maneuvers are carried out in the science mode, which puts forward higher requirements for the initial constellation condition. The orbit determination method for the space-based gravitational wave observatory is mainly studied in this paper. Firstly, the orbit determination accuracy requirement of the space-based gravitational wave observatory is analyzed. And then, an orbit determination method based on an extended Kalman filter is designed to improve the orbit determination accuracy by using inter-satellite range and angle measurement information.

Fully Decentralized Cooperative Navigation for Spacecraft Constellations

This article proposes a method to decentralize the navigation burden and improve the fault tolerance for a spacecraft constellation. The constellation body reference system is introduced, which is the perifocal frame of one satellite in the constellation. The structure of the proposed navigation method is constructed to enable each spacecraft to estimate its own orbit in this body reference system. This step is essentially the relative orbit determination (OD) based on inter-satellite range measurements. Thereafter, the approach to transfer an orbit from the constellation body reference system to an inertial reference system is developed. The essential requirements on absolute measurements to realize the coordinate transfer are presented. By dividing the absolute OD into relative OD and coordinate transfer, each navigation subsystem operated in a spacecraft can be independent with others, and the absolute measurements collected by any spacecraft can contribute to the absolute OD of the whole constellation. The proposed method applies to constellations in any geometric configuration. A Walker constellation is taken as an example for numerical simulations. The results show that the proposed method has a lower computation burden compared to an integrated navigation system. With the same type of absolute measurements, the proposed method has higher accuracy and convergence velocity than conventional decentralized algorithms. When a spacecraft occurs with fault, the orbit results of other spacecraft are not affected using the proposed method, which is beyond the ability of conventional methods.

Hybrid-precision arithmetic for numerical orbit integration towards future satellite gravimetry missions

The precision of numerical orbit integration is crucial in precise orbit determination and gravity field recovery, especially in the context of future satellite gravimetry missions where nanometer-level inter-satellite measurements are provided by laser ranging interferometer. In this paper, we propose a hybrid-precision arithmetic in both Cowell’s and Encke’s formulations for numerical orbit integration after comprehensive analyses on the computational time costs and numerical errors by dividing the integration process into two parts, namely the increment calculation and the orbit propagation. Based on such a separation, we find that the round-off errors basically occur in the least time-consuming orbit propagation part while the more time-demanding increment calculation part is less sensitive to the round-off due to its small numerical magnitude. Therefore, it is natural in the hybrid-precision arithmetic to use the double- and quadruple-precision arithmetic in the increment calculation part and orbit propagation part, respectively. Numerical results demonstrate that the hybrid-precision arithmetic is as efficient as the double-precision method, while far better than its double-precision counterparts in terms of both accuracy and relative precision. In consideration of the ultra-high precision inter-satellite range and range rate measurements of future satellite gravimetry missions, the currently widely-used double-precision arithmetic cannot meet the numerical requirements of one-day arc orbit integration for the inter-satellite range measurement, in which numerical errors should be at least the same level of the measurement precision and had better be one-order magnitude smaller to fully avoid their contaminations, while for the inter-satellite range rate the double-precision Gauss-Jackson integrator is qualified. Instead, by implementation of hybrid-precision arithmetic the range rate precision is easily achieved at 10−12 m/s in terms of the maximum absolute error of the one-day arc by either Cowell’s or Encke’s formulation. Furthermore, the sub-nanometer-level range precision is obtainable in the Encke’s formulation with reference orbit selected as the best-fit one. Therefore, it is recommended to adopt the hybrid-precision arithmetic for orbit integration to fully exploit the ultra-high precise measurement from the pure numerical data processing point of view.

Observability analysis of autonomous navigation using inter-satellite range: An orbital dynamics perspective

This paper focuses on the observability of the autonomous navigation for multi-satellite systems using only inter-satellite range (ISR), especially under complicated orbital dynamical model. According to previous studies, the autonomous is unobservable because the Right Ascension of Ascending Node (RAAN) can be determined, which is derived under reduced dynamics model only considering J2 non-spherical perturbation. In order to analyze this observability issue further, the ISR measurement model with respect to orbital elements and the orbital dynamical model considering several kinds of perturbations are built. Through calculating the value of the partial derivative of ISR with respect to the RAAN, we find that the RAAN is observable under complicated dynamical model considering the non-spherical, the solar pressure and third-body perturbations. This is because that the solar pressure and third-body perturbations make the RAAN change rates of satellites different, so the overall rotation does not exist. These conclusions are verified by simulation cases. Furthermore, the simulation results show that the perturbation model error has remarkable impact on the RAAN estimation accuracy.

System Design for the Event Horizon Imaging Experiment Using the PECMEO Concept

The concept for space interferometry from Polar or Equatorial Circular Medium Earth Orbits (the PECMEO concept) is a promising way to acquire the image of the “shadow” of the event horizon of Sagittarius A^* with an angular resolution of circa 5 microarcseconds. The concept is intended to decrease the size of the main reflector of the instrument to about 3 m using a precise orbit reconstruction based on Global Navigation Satellite System (GNSS) navigation, inter-satellite range and range-rate measurements, and data from the Attitude and Orbit Determination System (AODS). The paper provides the current progress on the definition of the subsystems required for the concept on the basis of simulations, radio regulations, and available technology. The paper proposes the requirement for the localization of the phase centre of the main reflector. The paper provides information about the visibility of GNSS satellites and the needed accuracies of the AODS. The paper proposes the frequency plan for the instrument and its Inter-Satellite Links (ISLs). The concepts for measurement of range and range rate using ISLs (as well as for the data exchange at these ISLs) are presented. The block diagram of the interferometer is described and its sensitivity is estimated. The link budget for both ISLs is given as well as their critical components. The calculated measurement quality factors are given. The paper shows the expected performance of the sub-systems of the interferometer. The requirements for the localization of the main reflectors and the information about the availability of the GNSS satellites are based on the simulations results. The frequency plan is obtained according to the PECMEO concept and taking into account the radio regulations. The existing technology defines the accuracies of the AODS as well as the link budgets and the measurement accuracies for both ISLs and the sensitivity of the instrument. The paper provides input information for the development of the orbit reconstruction filter and the whole PECMEO system.

Line of sight calibration for the laser ranging interferometer on-board the GRACE Follow-On mission: on-ground experimental validation.

The laser ranging interferometer (LRI) on board of the GRACE follow-on spacecraft, launched in May 2018, is the first laser interferometer to perform an inter-satellite range measurement. It is designed for ranging noise levels of 80 nm Hz-1/2 for frequencies above 20 mHz, i.e., about a ten-fold improvement with respect to the GRACE follow-on main microwave ranging instrument. One of the most critical steps during the commissioning phase of the instrument is the so-called initial line of sight calibration procedure (or initial acquisition). This process is required to quantify large uncertainties with respect to laser beam pointing angles and laser frequency, which must be known to establish the interferometer link. It is a nine hour scan of five degrees of freedom, which all need to match simultaneously at least once. Here we report on laboratory tests to further validate the calibration procedure using a mock-up LRI and a set-up, the so-called laser link simulator, that creates conditions similar to those with ~220 km distance between the SC. The experiments presented here made use of LRI-like hardware and software and were carried out recreating critical conditions such as received laser powers on the pico-Watt level and their dependence on the SC misalignments, flat-top beams as receiving beams and Doppler frequency shifts. Several configurations were tested, including a full line of sight calibration with angular scans in both mock-up SC and frequency scan in one of the lasers. Results are well in agreement with the expectations and confirm, well before the LRI commissioning phase, the robustness of the procedure under realistic conditions, which had not yet been fully tested experimentally.

Analysis of Attitude Errors in GRACE Range-Rate Residuals—A Comparison Between SCA1B and the Fused Attitude Product (SCA1B + ACC1B)

For missions such as GRACE, precise knowledge of the satellite's attitude is a fundamental requirement for the realization of the intersatellite ranging principle. It is not only essential for the realization of the precise in-orbit intersatellite pointing but also for the recovery of accurate temporal gravity field models. Here, we present a comparative study of two attitude datasets. One of them is the standard SCA1B RL02 datasets provided by JPL NASA, and another is a fused attitude dataset computed at TU Graz, based on the combination of ACC1B angular accelerations and SCA1B quaternions. Furthermore, we also present the impact of the attitude datasets on the intersatellite range measurements by analyzing their residuals. Our analysis reveals the significant improvement in the attitude due to the reprocessed product and reduced value of residuals computed from the reprocessed attitude.

PRECISE ESTABLISHMENT OF THE NEXT‐GENERATION EARTH GRAVITY FIELD MODEL FROM HIP‐3S BASED ON COMBINATION OF INLINE AND PENDULUM SATELLITE FORMATIONS

AbstractBecause the sensitivity of the disturbing intersatellite range measurements to the recovery accuracy of the Earth's gravitational field is superior to that of the intersatellite range measurements, the observation equation of a new Disturbing Intersatellite Range Method (DIRM) is created in this study. Then the orbital stability of the next‐generation Hybrid‐Inline‐Pendulum‐Three‐Satellite (HIP‐3S) formation is efficiently verified. The research results indicate that the HIP‐3S formation is sufficiently steady for further enhancing the accuracy of the future Earth gravity field model. Finally, the Earth's gravitational field complete up to degree and order 120 is precisely recovered by the current GRACE‐2S inline formation and the next‐generation HIP‐3S combined formation based on the new DIRM, and the cumulative geoid height errors are 2.271×10−1 m and 1.923×10−3 m at degree 120, respectively. The study results show that the future HIP‐3S formation is helpful for producing the next‐generation Earth gravity field model with higher accuracy and spatial resolution.

Spacecraft autonomous navigation using multiple model adaptive estimator

Purpose The purpose of this paper is to present a variable structure multiple model adaptive estimator (VSMMAE) for liaison navigation system. Liaison navigation is an autonomous navigation method where inter-satellite range measurements are used to estimate the orbits of all participating spacecrafts simultaneously.

Retroreflector for GRACE follow-on: Vertex vs. point of minimal coupling.

The GRACE Follow-On mission will monitor fluctuations in Earth's geoid using, for the first time, a Laser Ranging Interferometer to measure intersatellite distance changes. We have investigated the coupling between spacecraft rotation and the intersatellite range measurement that is incurred due to manufacturing and assembly tolerances of the Triple Mirror Assembly (TMA), a precision retroreflector to ensure alignment between in- and outgoing laser beams. The three TMA mirror planes intersect in a virtual vertex to which satellite displacements are referenced. TMA manufacturing tolerances degrade this ideal vertex, however, a Point of Minimal Coupling (PMC) between spacecraft rotation and displacement exists. This paper presents the experimental location of the PMC under pitch and yaw rotations for a prototype TMA. Rotations are performed using a hexapod, while displacements are monitored with heterodyne laser interferometry to verify the PMC position. Additionally, the vertex of the three TMA mirror planes is measured using a Coordinate Measuring Machine and compared to the PMC position. In the pitch and yaw axes, the biggest deviation between TMA vertex and PMC was 50 ± 64 μm. Thus, within the measurement uncertainties, no difference between TMA vertex and PMC could be observed. This is a key piece of information for integration of the TMA into the spacecraft: It is sufficient to use the readily-available TMA vertex location to ensure minimal rotation-to-displacement coupling during the mission.

Relative State Estimation and Observability Analysis for Formation Flying Satellites

TOAUTONOMOUSLY control a satellite formation, the relative position and velocity of the satellites needs to be estimated. For tight formations, all the required information needs to be obtained and processed onboard the satellites. In case absolute position measurements, such as provided by the Global Positioning System (GPS), cannot be obtained, estimation of the absolute or relative state requires the satellites to autonomously measure the intersatellite range and/or line-of-sight angles. Markley [1], Herklotz [2], Psiaki [3], Liu and Liu [4], and Yim et al. [5] have analyzed the problem of autonomous absolute orbit determination using intersatellite vector measurements whereas Markley [1], Woffinden and Geller [6–8], Chen and Xu [9], Doolittle et al. [10], Chavez and Lovell [11], Holt and Lightsey [12], Kang et al. [13], and Matko et al. [14] studied the problem of autonomous relative orbit determination using intersatellite measurements. Several authors furthermore studied the effect of intersatellite ranging to augment external measurements like those from GPS and ground stations for either absolute or relative orbit determination. Amongst these are Holt and Lightsey [15], Huxel [16], and Huxel and Bishop [17,18]. Although previous work has been quite extensive, the understanding of the results obtained in some of these works needs to be improved by performing a detailed study of the observability of the system as a function of the relative state and of sensor suite properties, such as sensor accuracy and sensor placement. In addition, as no reference has been found in the literature, a further objective of this research is to provide a derivation of the expected value and variance, in three dimensions, in the estimation of the relative position of two objects. To isolate the basic properties of interest, a simple setting is implemented. The rationale behind this is that more advanced and realistic effects add only minor contributions and effects to the findings while they add considerable complexity to the analysis of the results. Two satellites, a chief and a deputy, areflying in formation in a low-Earth orbit (LEO) and perform intersatellite range measurements using locally generated one-way radio frequency ranging signals. The chief uses either one or three receiver (Rx) antennas to acquire and track the ranging signal transmitted by the deputy and to determine the relative range(s). The range measurements are treated together with a linearized dynamic model of the satellites’ relative motion to estimate their relative orbit using an iterative batch leastsquares (LSQ) algorithm. An observability analysis is performed to gain a deeper insight in the obtained results.

Understanding data noise in gravity field recovery on the basis of inter-satellite ranging measurements acquired by the satellite gravimetry mission GRACE

Spectral analysis of data noise is performed in the context of gravity field recovery from inter-satellite ranging measurements acquired by the satellite gravimetry mission GRACE. The motivation of the study is two-fold: (i) to promote a further improvement of GRACE data processing techniques and (ii) to assist designing GRACE follow-on missions. The analyzed noise realizations are produced as the difference between the actual GRACE inter-satellite range measurements and the predictions based on state-of-the-art forcemodels. The exploited functional model is based on the so-called “range combinations,” which can be understood as a finite-difference analog of inter-satellite accelerations projected onto the line-of-sight connecting the satellites. It is shown that low-frequency noise is caused by limited accuracy of the computed GRACE orbits. In the first instance, it leads to an inaccurate estimation of the radial component of the inter-satellite velocities. A large impact of this component stems from the fact that it is directly related to centrifugal accelerations, which have to be taken into account when the measured range-accelerations are linked with inter-satellite accelerations. Another effect of orbit inaccuracies is a miscalculation of forces acting on the satellites (particularly, the one described by the zero-degree term of the Earth’s gravitational field). The major contributors to the noise budget at high frequencies (above 9 mHz) are (i) ranging sensor errors and (ii) limited knowledge of the Earth’s static gravity field at high degrees. Importantly, we show that updating the model of the static field on the basis of the available data must be performed with a caution as the result may not be physical due to a non-unique recovery of high-degree coefficients. The source of noise in the range of intermediate frequencies (1–9 mHz), which is particularly critical for an accurate gravity field recovery, is not fully understood yet. We show, however, that it cannot be explained by inaccuracies in background models of time-varying gravity field. It is stressed that most of the obtained results can be treated as sufficiently general (i.e., applicable in the context of a statistically optimal estimation based on any functional model).

A New Algorithm for Onboard Autonomous Orbit Determination of Navigation Satellites

Autonomous orbit determination of a navigation constellation is the process by which the orbit parameters of navigation satellites are autonomously calibrated onboard the satellites without the need for external aids. It commonly uses a satellite onboard data processing unit and a filtering method to process the measurements of inter-satellite ranges. The onboard data processing unit is the main module of autonomous navigation systems. In this paper, the two main factors that affect the accuracy of autonomous orbit determination for a navigation constellation are discussed first, and then a distributed onboard algorithm for autonomous orbit determination of navigation satellites is proposed. This method is based on a long-term ephemeris prediction and is suitable for the satellite hardware capability. The main feature of this method is that both the distributed computing method and an onboard analytical state transition matrix are used to process inter-satellite range measurements. One of the main advantages of this approach is high-speed computing since the amount of calculations needed is significantly less than that of the centralised computing method and those distributed methods that need to use an onboard numerical integrator. Another advantage of this approach is that the use of the onboard analytical state transition matrix algorithm can save a great amount of resources for both ground-to-satellite data transmissions and data storage units in satellites’ hardware. This could result in substantial cost reduction for space missions. Finally, a simulation method used for testing the proposed algorithm is presented. Results of tests over a period of 90 days show that the user range error of autonomous orbit determination derived from the proposed method is less than three metres.

Study of observability of motion of an orbital group of navigation space system using intersatellite range measurements. II

Study of observability of motion of an orbital group of navigation space system using intersatellite range measurements. II

Study of observability of motion of an orbital group of navigation space system using intersatellite range measurements. I

The problem of specification of reference ephemeris (reference motion) using intersatellite range measurements on a given interval between spacecrafts of the space navigation system is formulated; observability for this problem is analyzed. Observability is the possibility of unambiguous determination of motion parameters of the orbital group of the space navigation system using known inter-satellite measurements on a given measurement interval. Linear approximate model describing the dependence between measurements and specified parameters are obtained for investigation of observability properties. In the framework of linear model problems of observability of motion of two any spacecrafts of the navigation system are studied using intersatellite range measurements.

Alternative mission architectures for a gravity recovery satellite mission

Since its launch in 2002, the Gravity Recovery and Climate Experiment (GRACE) mission has been providing measurements of the time-varying Earth gravity field. The GRACE mission architecture includes two satellites in near-circular, near-polar orbits separated in the along-track direction by approximately 220 km (e.g. collinear). A microwave ranging instrument measures changes in the distance between the spacecraft, while accelerometers on each spacecraft are used to measure changes in distance due to non-gravitational forces. The fact that the satellites are in near-polar orbits coupled with the fact that the inter-satellite range measurements are directed in the along-track direction, contributes to longitudinal striping in the estimated gravity fields. This paper examines four candidate mission architectures for a future gravity recovery satellite mission to assess their potential in measuring the gravity field more accurately than GRACE. All satellites were assumed to have an improved measurement system, with an inter-satellite laser ranging instrument and a drag-free system for removal of non-gravitational accelerations. Four formations were studied: a two-satellite collinear pair similar to GRACE; a four-satellite architecture with two collinear pairs; a two-satellite cartwheel formation; and a four-satellite cartwheel formation. A cartwheel formation consists of satellites performing in-plane, relative elliptical motion about their geometric center, so that inter-satellite measurements are, at times, directed radially (e.g. parallel to the direction towards the center of the Earth) rather than along-track. Radial measurements, unlike along-track measurements, have equal sensitivity to mass distribution in all directions along the Earth’s surface and can lead to higher spatial resolution in the derived gravity field. The ability of each architecture to recover the gravity field was evaluated using numerical simulations performed with JPL’s GIPSY-OASIS software package. Thirty days of data were used to estimate gravity fields complete to degree and order 60. Evaluations were done for 250 and 400 km nominal orbit altitudes. The sensitivity of the recovered gravity field to under-sampled effects was assessed using simulated errors in atmospheric/ocean dealiasing (AOD) models. Results showed the gravity field errors associated with the four-satellite cartwheel formation were approximately one order of magnitude lower than the collinear satellite pair when only measurement system errors were included. When short-period AOD model errors were introduced, the gravity field errors for each formation were approximately the same. The cartwheel formations eliminated most of the longitudinal striping seen in the gravity field errors. A covariance analysis showed the error spectrum of the cartwheel formations to be lower and more isotropic than that of the collinear formations.

Gravity model determination from the GRACE mission

The Gravity Recovery and Climate Experiment (GRACE) is a collaborative NASA and DLR mission whose satellites were launched on March 17, 2002 as the first of the Earth System Science Pathfinder (ESSP) Missions. The purpose of the GRACE mission is to characterize the spatial and temporal variations in the Earth’s mass, through precise measurements of its gravity field. This is accomplished by using micron level inter-satellite range measurements to determine monthly solutions for the Earth’s global gravity field. The GRACE mission has provided 82 monthly solutions that have led to unprecedented improvements in the mean gravity field accuracy and a seven-year record of monthly gravity signals associated with the global mass flux. This continuous, multi-year record of monthly measurements supports investigations that characterize the seasonal cycle of mass transport between the oceans, land, and atmosphere; observes its inter-annual variability; and monitors the secular trends in mass transport. In the following discussion, the overall mission concept is described, the approach to the gravity model determination is discussed, and the mission progress is partially summarized.

Gravitational perturbation theory for intersatellite tracking

An analytical gravitational perturbation theory for the intersatellite tracking range and range-rate measurement between two satellites is developed. The satellite-to-satellite tracking (SST) range data measure the difference between the position perturbations of two satellites along the direction of the intersatellite range. The SST range-rate data measure the difference between the velocity perturbations along the direction of the intersatellite range, and the difference of the position perturbation along the direction perpendicular to the intersatellite range (cross-range). The SST range and range rate depend on different orbital excitations for mapping the gravity field. For the Gravity Recovery and Climate Experiment (GRACE), approximately 97% of the geopotential coefficient pairs produce perturbations with a root-mean-square larger than 1 m on the range and 0.1 m/sec on the range rate based on the EGM96 gravity field truncated at degree and order 140. Results in this study showed that ocean tides produce significant perturbations in the range and range-rate measurements. An ocean tide field with a higher degree and order (>70) is required to model the ocean tide perturbations on the intersatellite range and range-rate measurement.