Search

The Miniaturised Asteroid Remote Geophysical Observer (M-ARGO) mission is designed to be ESA’s first stand-alone CubeSat to independently travel in deep space with its own electric propulsion and direct-to-Earth communication systems in order to rendezvous with a near-Earth asteroid. Deep-space Cubesats are appealing owing to the scaled mission costs. However, the operational costs are comparable to those of traditional missions if ground-based orbit determination is employed. Thus, autonomous navigation methods are required to favour an overall scaling of the mission cost for deep-space CubeSats. M-ARGO is assumed to perform an autonomous navigation experiment during the deep-space cruise phase. This paper elaborates on the deep-space navigation experiment exploiting the line-of-sight directions to visible beacons in the Solar System. The aim is to assess the experiment feasibility and to quantify the performances of the method. Results indicate feasibility of the autonomous navigation for M-ARGO with a 3σ accuracy in the order of 1000 km for the position components and 1 m/s for the velocity components in good observation conditions, utilising miniaturized optical sensors.

Jupiter and its moons are a complex dynamical system that include several phenomena like tides interactions, moon’s librations and resonances. One of the most interesting characteristics of the Jovian system is the presence of the Laplace resonance, where the orbital periods of Ganymede, Europa and Io maintain a 4:2:1 ratio, respectively. It is interesting to study the role of the Laplace resonance in the dynamic of the system, especially regarding the dissipative nature of the tidal interaction between Jupiter and its closest moon, Io. The secular orbital evolution of the Galilean satellites, and so the Laplace resonance, is strongly influenced by the tidal interaction between Jupiter and its moons, especially with Io. Numerous theories have been proposed regarding this topic, but they disagree about the amount of dissipation of the system, therefore about the magnitude and the direction of the evolution of the system, mainly because of the lack of experimental data. The future ESA JUICE space mission is a great opportunity to solve this dispute. The data that will be collect during the mission will have an exceptional accuracy, allowing to investigate several aspects of the dynamics the system and possibly the evolution of Laplace Resonance of the Galilean moons. This work will focus on the gravity estimation and orbit reconstruction of the Galilean satellites by precise orbit determination of the JUICE mission during the Jovian orbital phase using radiometric data.

Multi-fidelity approaches to orbit-state probability density prediction reduce computation time, but introduce a systematic error in the single-point prediction of a spacecraft state. An estimate of the systematic error may be quantified using cross-validation. Credibilistic filters based on Outer Probability Measures (OPMs) enable a principled and unified representation of random and systematic errors in object tracking. The quantified error of the multi-fidelity approach defines an OPM-based transition kernel, which is used in a credibilistic filter to account for the systematic error in the orbit determination process. An approach based on automatic domain splitting is proposed to reduce the error beyond what is normally achievable with multi-fidelity methods. A proof-of-concept for the approach is demonstrated for a simulated scenario tracking a newly-detected space object in low-Earth orbit via two ground stations generating radar measurements. An OPM-based definition of the admissible region combined with the multi-fidelity credibilistic filter establishes custody of the object.

The Juno spacecraft has been collecting data to shed light on the planet’s origin and characterize its interior structure. The onboard gravity science experiment based on X-band and Ka-band dual-frequency Doppler tracking precisely measured Jupiter’s zonal gravitational field. Here, we analyze 22 Juno’s gravity passes to investigate the gravity field. Our analysis provides evidence of new gravity field features, which perturb its otherwise axially symmetric structure with a time-variable component. We show that normal modes of the planet could explain the anomalous signatures present in the Doppler data better than other alternative explanations, such as localized density anomalies and non-axisymmetric components of the static gravity field. We explain Juno data by p-modes having an amplitude spectrum with a peak radial velocity of 10–50 cm/s at 900–1200 μHz (compatible with ground-based observations) and provide upper bounds on lower frequency f-modes (radial velocity smaller than 1 cm/s). The new Juno results could open the possibility of exploring the interior structure of the gas giants through measurements of the time-variable gravity or with onboard instrumentation devoted to the observation of normal modes, which could drive spacecraft operations of future missions.

A novel, empirically-validated method for robust initial orbit determination from two or more observations of an object’s angles and angle rates is presented. The method can be applied for all orbital regimes including those where traditional methods, such as Gauss and double-r, fail. It also works for all times-of-flight to produce the corresponding set of feasible orbit solutions using admissible regions. The method uses a cost function that has a unique non-trivial zero at the correct orbit solution for the single revolution case and a set of zeros for the multi-orbit case. Applying admissible regions to both observations allows the state space to be further constrained with a corresponding improvement to the computational efficiency. The proposed method overcomes the shortcomings of many traditional initial orbit determination methods and has been empirically verified to find solutions to cases well-approximated by two-body dynamics. Furthermore, this work includes a comparison of the proposed and existing methods for initial orbit determination to highlight the domains that the various methods excel. While the global optimizer approach of the proposed method sacrifices computation time in comparison to the single iteration of the previously established methods, it consistently converges to the correct solution or set of possible solutions independent of the orbital regime.

Current active satellite maneuver detection techniques can resolve maneuvers as quickly as fifteen minutes post maneuver for large Δv when using angles-only optical tracking. Medium to small magnitude burn detection times range from 6 to 24 h or more. Small magnitude burns may be indistinguishable from natural perturbative effects if passive techniques are employed. Utilizing a photoacoustic signature detection scheme can allow for near real time maneuver detection and spacecraft parameter estimation. We define the acquisition of hypertemporal photometric data as photoacoustic sensing because the data can be played back as an acoustic signal. Studying the operational frequency spectra, profile, and aural perception of an active satellite event such as a thruster ignition or any subsystem operation can provide unique signature identifiers that support resident space object characterization efforts. A thruster ignition induces vibrations in a satellite body which can modulate reflected sunlight. If the reflected photon flux is sampled at a sufficient rate, the change in light intensity due to the propulsive event can be detected. Sensing vibrational mode changes allows for a direct timestamp of thruster ignition and shut-off events and thus makes possible the near real time estimation of spacecraft Δv and maneuver type if coupled with active observations immediately post maneuver. This research also investigates the estimation of other impulse related spacecraft parameters such as mass, specific impulse, exhaust velocity, and mass flow rate using impulse-momentum and work-energy methods. Experimental results to date have not yet demonstrated an operator-correlated detection of a propulsive event; however, the application of photoacoustic sensing has exhibited characteristics unique to hypertemporal photometry that are discussed alongside potential improvements to increase the probability of active satellite event detection. Simulations herein suggest that large, potentially destructive modal displacements are required for optical sensor detection and thus more comprehensive vibration modeling and signal-to-noise ratio improvements should be explored.

Two-line elements are widely used for space operations to predict orbits with a moderate accuracy for 2-3 days. Local optimization methods can estimate a TLE as long as there exists an initial estimate, whereas global optimization methods are computationally intensive, and estimating a large number of them is prohibitive. In this paper, the feasibility of predicting the initial estimates within the radius of convergence of the actual TLEs using machine learning methods is investigated. First, a Monte-Carlo approach to estimate a TLE, when there is no initial estimate that is within the radius of convergence of the actual TLE, is introduced. The proposed Monte-Carlo method is leveraged for demonstrating the behavior of the fitting error between the realistic trajectory and the trajectory propagated by SGP4 theory during the TLE estimation processes and evaluating the unbiased performance of the proposed machine learning models. Second, gradient boosting decision trees and fully-connected neural networks are trained to map the orbital evolution of space objects to the associated TLEs using 9.5 million publicly available TLEs from the US space catalog. The desired precision in the mapping to estimate a TLE is achieved for one of the three test cases, which is a low area-to-mass ratio space object.

Global navigation satellite systems and positioning, navigation, and timing services, such as the newly developed BeiDou satellite system (BDS), require high-accuracy satellite observations. Satellite-induced code bias is present in BDS-2 satellite observations while it is negligibly present in BDS-3 satellite observations. The traditional method of mitigating BDS code bias involves two steps, first addressing multipath delay and then code bias, and does not obtain optimal results. A one-step strategy is therefore proposed to model and eliminate code bias and multipath delay considering BDS-2 and BDS-3 integrated processing. A combined least-square and autoregressive strategy is selected to estimate the model coefficients of code bias and construct the multipath delay with one solution. Moreover, inter-satellite correlations of BDS-2 and BDS-3 are extracted to improve the weight matrix in the estimation of model coefficients. To verify the proposed strategy, experiments are designed for eight schemes to analyze the coefficients and residuals of modelling code bias. Experimental results indicate that a more stable and accurate code bias model is acquired by introducing inter-satellite correlation; with the one-step strategy, the model residuals are reduced and a more optimal code bias model is output. Meanwhile, the single-frequency precise point positioning (PPP) and real-time PPP of BDS-2 and BDS-3 combined estimations are tested for different stations, frequencies, and code bias models. The results of final positions reveal that accuracy in the Up (U) direction is improved compared with that in the East (E) and North (N) directions; improvements are greater for B1I than for B3I. In general, the one-step strategy enhances the precision of single-frequency (B1I/B3I) PPP, especially in the U direction. However, the effects on BDS ultra-rapid orbit determination are negligible. Equally, the double-frequency real-time PPP solution is improved, compared with adopting the traditional method, by 10.6–64.9%, 0.0–59.1%, and 12.6–67.2% for E, N, and U directions, respectively, through BDS-2/BDS-3 integrated processing. The proposed one-step strategy therefore outperforms the two-step strategy in terms of rapidly processing high-precision BDS observations.

As the gravity formation flying spacecraft jointly developed by National Aeronautics and Space Administration and German Research Centre for Geosciences, the Gravity Recovery and Climate Experiment (GRACE) and GRACE follow-on (GRACE-FO) satellites have adopted the same K-band ranging system and orbit design in order to detect the Earth’s gravity field information. Different from the BlackJack receiver onboard GRACE, a TriG GNSS receiver is loaded on the GRACE-FO satellites. The orbit determination accuracy of GRACE is better than 2.5 cm, but for GRACE-FO there is no comprehensive assessment of orbit accuracy. We discuss the quality of Global Positioning System data, as well as effects of phase center variation (PCV) model and attitude for GRACE and GRACE-FO. The results show that there is no significant difference in terms of rate of change in ionospheric delay (IOD) and multipath effect, which suggests that the performance of TriG receiver is as excellent as that of BlackJack receiver. After using PCV corrections, the root mean square (RMS) errors of kinematic and reduced-dynamic (RD) orbit residuals decrease by 0.4–0.5 and 0.6–0.9 mm, respectively. Satellite laser ranging RMS errors for RD orbit solutions are lower than 2.59 cm whether PCV corrections are used or not. The effect of attitude data on kinematic and RD orbits indicates that nominal attitude data can reliably replace measured attitude data in GRACE-FO orbit determination.

The deep-space multi-object orbit determination system (DMOODS) and its application in the asteroid proximity operation of the Hayabusa2 mission are described. DMOODS was developed by the Japan Aerospace Exploration Agency (JAXA) for the primary purpose of determining the trajectory of deep-space spacecraft for JAXA’s planetary missions. The weighted least-squares batch filter is used for the orbit estimator of DMOODS. The orbit estimator supports more than 10 data types, some of which are used for relative trajectory measurements between multiple space objects including natural satellites and small bodies. This system consists of a set of computer programs running on Linux-based consumer PCs on the ground, which are used for orbit determination and the generation of radiometric tracking data, such as delta differential one-way ranging and doppler tracking data. During the asteroid proximity phase of Hayabusa2, this system played an essential role in operations that had very strict navigation requirements or operations in which few optical data were obtained owing to special constraints on the spacecraft attitude or distance from the asteroid. One example is orbit determination during the solar conjunction phase, in which the navigation accuracy is degraded by the effect of the solar corona. The large range bias caused by the solar corona was accurately estimated with DMOODS by combining light detection and ranging (LIDAR) and ranging measurements in the superior solar conjunction phase of Hayabusa2. For the orbiting operations of target markers and the MINERVA-II2 rover, the simultaneous estimation of six trajectories of four artificial objects and a natural object was made by DMOODS. This type of simultaneous orbit determination of multi-artificial objects in deep-space has never been accomplished before.

Inter-Satellite Links (ISLs) are intended to improve precision of orbit determination and satellite clock estimation. The ISLs provide a precise pseudorange measurements between satellites in a specific constellation. The study is a preparatory assessment of exploitation of seven connectivity schemes in the terms of the precise orbit determination for three types of constellations—Galileo-like with 24 satellites on three orbital planes, GPS-like with 24 satellites on six orbital planes, and GPS with real positions. The first part of the study focused on detailed analysis of the various ISL connectivity schemes, considering the geometry of ISL observations. The selected results of ranging were examined in the context of the precise orbit determination based on weighted least squares adjustment. The second part of the analysis was based on simulated measurements with two approaches. First approach focuses on geometrical dependencies and the second is performed with ISL measurement biases estimation. It was found that the use of the ISL technique with GNSS measurements in orbit determination improves the results by reducing the RMS error in the along-track and cross-track components. Choice of connectivity schemes does not have a significant impact on the total results of orbit determination, but give different contribution to particular components. Introducing constant bias in ISL measurements occurs in slightly worse estimation results. However, the relations between connectivity schemes is very similar to approach without simulation of ISL bias, the differences are at the level of 10%. Satellite and station clock estimation errors are almost equal for all used connectivity schemes. Results of clocks are also not influenced by ISL bias. This study showed that the ISL technique is a highly promising addition for future generations of satellite navigation systems and that sequential and ring connectivity schemes can be recommended for use in future navigation constellations.

Observations of resident space objects generated by sensors are the primary method of maintaining knowledge of the object states. With the increasing number of objects, efficient sensor allocation is becoming integral. This requires the coordination of multiple sensors with different capabilities in an optimized manner. While the optimization is greatly simplified if instantaneous communication between the sensors can be assumed and immediate processing is available, this is not a realistic setup. Information exchange and processing induces time delays that are longer than the time available to plan and start the sensor tasking step, without unnecessarily idling the sensor. In this paper, a method is introduced to form efficient sensor tasking in a multi-sensor system, without immediate communication between the sensors and observation processing, in the following called feedback. The coordination is exemplified using two sensors, with different fields of view in a follow-up scenario of objects in the geosynchronous region. The efficiency of the method is evaluated using the two line element catalog.

This paper presents the mission concept and engineering design of a debris-removing nanosatellite called Deorbiter CubeSat, within the framework of NASA’s Pre-Phase A studies. The spacecraft is designed based on the utilization of an eight-unit form factor, and is intended for the removal of predetermined sizable debris objects from the low Earth orbit. A number of attitude and orbit determination sensors and control actuators are included on the CubeSat, which are employed during the rendezvous, attachment, and deorbiting operations. Upon attaching to a debris, the CubeSat stabilizes the rotational motion of the debris, and then proceeds to reducing the debris orbit size, in order to re-enter Earth’s atmosphere and burn up due to the high atmospheric density. The engineering design of Deorbiter CubeSat is outlined, and the selected components are detailed. The selected components are commercially available and have long space heritage. System’s mass budget is analyzed, and preliminary component costs are estimated. Three scenarios for the Deorbiter CubeSat mission operations are considered, and the spacecraft power budget and components duty cycles are investigated for each scenario. In light of the results, the feasibility of each scenario for the Deorbiter CubeSat mission is discussed.

Deep Space Gateway is a NASA program planned to support deep space human exploration and prove new technologies needed to achieve it. One Gateway requirement is the ability to operate in the absence of communications with the Deep Space Network (DSN) for a period of at least three weeks. In this paper, three types of onboard sensors (a camera for optical navigation, a GPS receiver, and X-ray navigation) are considered to enhance its autonomy and reduce the reliance on DSN. A trade study is conducted to explore alternatives on how to achieve autonomy and how to reduce DSN dependency while satisfying navigation performance requirements. Using linear covariance analysis, error budgets, and sensitivity analysis, the performance of navigation systems using combinations of DSN with the aforementioned onboard sensors is shown.

This paper presents the optical navigation results of the asteroid explorer Hayabusa2 during the final rendezvous approach phase with the asteroid Ryugu. The orbit determination of Hayabusa2 during the cruising phase uses a triangulation-based method that estimates the probe and asteroid orbits using the directions from which they are observed. Conversely, the asteroid size is available as optical information just prior to arrival. The size information allows us to estimate the relative distance between the probe and the asteroid with high accuracy, that is strongly related to the success or failure of the rendezvous. In this study, the relative distance and asteroid size in real space are simultaneously estimated in real time by focusing on the rate of change of the asteroid size observed in sequential images. The real-time estimation results coincided with those of precise analyses performed after arrival.

The estimation and modeling of drag effects on low altitude satellites and debris is a limiting problem in the prediction of their orbits over time. Independent of the stochastic variations in atmospheric density, which drive the magnitude of the drag force, there are systematic variations in the drag coefficient due to satellite specific factors and atmospheric conditions. These include satellite attitude shifts, variations in ambient atmospheric parameters such as temperature, molecular composition, and the satellite wall temperature. Thus, even with accurate empirical models for the atmospheric density, there remain systematic model uncertainties and time variations in the effective coefficient of drag that affect the satellite’s motion, which are not captured using the standard constant drag coefficient model. We report on our recent research on developing Fourier expansion based models of an arbitrary object’s coefficient of drag with the overall purpose to develop corrections to the standard model. If the attitude profile of a satellite or object is known, a Fourier series expansion of the overall coefficient of drag as a function of wind vector direction in the body-frame can be introduced. For bodies with unknown or more complex attitude profiles, we introduce a periodically varying drag coefficient tied to a satellite’s position in orbit. This model enables the estimation of higher-order temporal variations in drag that would be correlated with variations in altitude, temperature and density. An improved prediction capability over the standard cannonball model is shown and the formulation of how it can be incorporated into standard filtering techniques is presented.

Two fundamentally different approaches of determining normality of the probability density function of the state estimation error are compared by application to a range of test cases. The first method is the Henze-Zirkler test, which operates on a random particle sample. The variability of its result is quantified. Using this method, departure from normality has been found to occur in three stages which are detailed. The second test compares the offset in whitened space of the predicted state to the predicted covariance mean obtained from the unscented transform. This test is much more efficient than the random particle based approach and can be applied using any perturbations model. The comparison is performed on the state estimation error in Cartesian space and using two-body motion without process noise. The more efficient, unscented transform based approach shows excellent agreement with the Henze-Zirkler test for constructed test cases. Application to orbit determination results from passive optical observations assessed with a Batch-Least-Squares orbit determination however reveals some discrepancies which have yet to be understood and underline the importance of rigorous testing.

Precise orbit products of Global Navigation Satellite Systems (GNSS) are an essential precondition for precise positioning. Ambiguity resolution (AR) can enhance the orbit accuracy in precise orbit determination (POD). To improve the quality of orbits, we propose a method of combined POD for GPS and GLONASS with AR. Firstly, GLONASS wide-lane and narrow-lane fractional cycle biases (FCBs) are daily estimated. Then, by applying the estimated FCBs, GLONASS and GPS double-differenced wide-lane and narrow-lane ambiguities are successfully resolved, even for the baselines of up to several thousand kilometers. Finally, the ambiguity-resolved solutions are achieved by introducing the constraints of the resolved ambiguities into the real-valued solutions. To prove the contribution of the AR to GPS and GLONASS POD, a network including 141 sites is processed over 2018. The results show that the receiver types and firmware versions seriously affect the stability of the daily wide-lane FCBs. The fluctuation of the inter-system biases between two adjacent days is obviously larger than a half narrow-lane wavelength, causing an irregular change of the daily narrow-lane FCBs. After FCB calibration, the success rate of GLONASS can reach up to 90% over the whole year, which is at the same level compared with that of GPS. The improvements of GLONASS and GPS orbits after AR are confirmed by the orbit comparison with the International GNSS Service final products, the orbit misclosures at day boundaries and satellite laser ranging residuals. Due to some other issues, such as the GLONASS frequency-division multiple access and the high noise of observations, the improvement of GLONASS orbit is still less obvious than that of GPS orbit.

This paper explores an alternative interpolation approach for the line-of-sight measurements in Laplace’s method for angles-only initial orbit determination (IOD). The classical implementation of the method uses Lagrange polynomials to interpolate three or more unit line-of-sight (LOS) vectors from a ground-based or orbiting site to an orbiting target. However, such an approach does not guarantee unit magnitude of the interpolated line-of-sight path except at the measurement points. The violation of this constraint leads to unphysical behavior in the derivatives of the interpolated line-of-sight history, which can lead to poor initial orbit determination (IOD) performance. By adapting a spherical interpolation method used in the field of computer graphics, we can obtain an interpolated line-of-sight history that is always unit norm. This new spherical interpolation method often leads to significant performance improvements in Laplace’s IOD method, with comparable robustness to measurement noise as the traditional interpolation methods. This paper also demonstrates the benefit of interpolating through more than three data points, which is easily enabled up to an arbitrary number of measurements by recursive definitions of the interpolations.

One of the most valuable regions for telecommunication and space science is the geostationary earth orbit (GEO). On 27 November 2019, Egypt successfully launched its telecommunications satellite TIBA-1 into space. This paper aims to effectively identify and concurrently track this satellite from our Kottamia observatory station, National Research Institute of Astronomy and Geophysics (NRIAG) - Egypt. The images of the satellite are processed by the Apex II astronomical image processing package to obtain optical measurements, the apparent magnitude, and the phase angle. The initial orbit of the TIBA-1 satellite is determined using angle-only optical measurements for 4 months. The Double-r iteration method is used to determine the state distance vectors for orbital elements under different perturbation forces. Then, the results are compared with the positions predicted using the North American Aerospace Defense Command (NORAD) reference catalog.

The utilization of Global Navigation Satellite System (GNSS) is becoming an attractive approach for the orbit determination of geostationary orbit (GEO) satellites. As a flight test for the feasibility of GNSS-based orbit determination at high orbit altitude, a GEO satellite named TJS-2 is launched. This satellite is equipped with a high-gain antenna, a pre-amplifier, and a high-sensitivity receiver. This study investigates the methods of high-sensitivity processing for the GNSS side lobe signals and the onboard orbit determination filter to improve navigation performance. In accordance with flight data, the GNSS signal characteristics, including availability, position dilution of precision (PDOP), carrier-to-noise ratio density (C/N0), and quality of observations, are analyzed. The mean number of GPS, GLONASS, and BDS satellites tracked is 7.6, 4.6, and 0.3, respectively. The mean PDOP of GPS, GPS + GLONASS, and GPS + BDS satellites tracked is 10.8, 8.4, and 8.9, respectively. The distribution of C/N0 and the number of observations with respect to the nadir angles are illustrated. For GPS, GLONASS, and BDS, the corresponding standard deviation of the pseudorange noise is 7.7, 16.1, and 5.2 m, and that of the carrier-phase noise is 37.6, 41.8, and 53.7 mm, respectively, in terms of C/N0

In this simulation study we analyze the benefit of ground-space optical two-way links (OTWL) for Galileo precise orbit determination (POD). OTWL is a concept based on continuous wave laser ranging and time transfer with modulated signals from and to ground stations. The measurements are in addition to Global Navigation Satellite System (GNSS) observations. We simulate the measurements with regard to 16 Galileo Sensor Stations. In the simulation study we assume that the whole Galileo satellite constellation is equipped with terminals for OTWL. Using OTWL together with Galileo L-band, in comparison with an orbit solution calculated with L-band-only, demonstrates the advantage of combining two ranging techniques with different influences of systematic errors. The two-way link allows a station and satellite clock synchronization. Furthermore, we compare the ground-space concept with the satellite-to-satellite counterpart known as optical two-way inter-satellite links (OISL). The advantage of OTWL is the connection between the satellite system and the solid Earth as well as the possibility to synchronize the satellite clocks and the ground station clocks. The full network, using all three observation types in combination is simulated as well. The possibility to estimate additional solar radiation pressure (SRP) parameters within these combinations is a clear benefit of these additional links. We paid great attention to simulate systematic effects of all observation techniques as realistically as possible. For L-band these are measurement noise, tropospheric delays, phase center variation of receiver and transmitter antennas, constant and variable biases as well as multipath. For optical links we simulated colored and distance-dependent noise, offsets due to the link repeatability and offsets related to the equipment calibration quality. In addition, we added a troposphere error for the OTWL measurements. We discuss the influence on the formal orbit uncertainties and the effects of the systematic errors. Restrictions due to weather conditions are addressed as well. OTWL is synergetic with the other measurement techniques like OISL and can be used for data transfer and communication, respectively., Deutsche Forschungsgemeinschaft http://dx.doi.org/10.13039/501100001659, Technische Universität München (1025)

Ionosphere Associate Analysis Centers (IAACs) of the International GNSS Service (IGS) independently produce global ionosphere maps (GIMs) of the total electron content (TEC). The GIMs are based on different modeling techniques, resulting in different TEC levels and accuracies. In this study, we evaluated the accuracy and consistency of the IAAC GIMs during high (2014) and low (2018) solar activity periods of the 24th solar cycle. In our study, we applied two different evaluation methods. First, we carried out a comparison of the GIM-derived slant TEC (STEC) with carrier phase geometry-free combination of GNSS signals obtained from 25 globally distributed stations. Second, vertical TEC (VTEC) from GIMs was compared to altimetry-derived VTEC obtained from the Jason-2 and Jason-3 satellites and complemented for plasmaspheric TEC. The analyzed GIMs obtained STEC RMS values reaching from 1.98 to 3.00 TECU and from 0.96 to 1.29 TECU during 2014 and 2018, respectively. The comparison to altimetry data resulted in VTEC STD values that varied from 3.61 to 5.97 TECU and from 1.92 to 2.78 TECU during 2014 and 2018, respectively. The results show that among the IAACs, the Center for Orbit Determination in Europe global maps performed best in low and high solar activity periods. However, the highest accuracy was obtained by a non-IGS product—UQRG GIMs provided by Universitat Politècnica de Catalunya. It was also shown that the best results were obtained using a modified single layer model mapping function and that the map time interval has a relatively small influence on the resulting map accuracy.

Compared to the BeiDou regional navigation satellite system (BDS-2), the BeiDou global navigation satellite system (BDS-3) includes the newly designed B1C and B2a signals, which are compatible with the L1 and L5 frequencies of the global positioning system (GPS). Considering that the precise orbit determination (POD) of the BDS-3 constellation is currently restricted to the legacy B1I and B3I signals, we reported the POD performance of BDS-3 satellites using B1C and B2a dual-frequency measurements. Nine globally distributed Multi-GNSS Experiment (MGEX) stations were selected to determine the orbits of BDS-3 satellites during the period of July 2019. The results show that B1C/B2a-based POD enables an average three-dimensional root-mean-square error (3D RMS) of 24.2 cm, and the precision is better than 6 cm in the radial component in a comparison of two-day overlapping arcs. Satellite laser ranging (SLR) validation achieves an overall precision of 6.8 cm in RMS differences. Compared to the B1I/B3I-based POD results, the quality of B1C/B2a-based orbits is improved by approximately 9% across the whole BDS-3 constellation, indicating that the new B1C/B2a signals can be employed for the superior POD performance of BDS-3 satellites. Moreover, we investigated the behavior of the solar radiation pressure (SRP), which is generally considered one of the primary error sources in BDS-3 medium earth orbit (MEO) dynamic orbit determination. The ECOM7 SRP model has a better POD performance in continuous yaw steering (CYS) mode than the ECOM5 and ECOM9 SRP models. The results also show that there is no degradation in the orbit precision of BDS-3 MEO satellites when the elevation angle of the sun above the orbital plane (β angle) varies within the range of − 4° to + 4°.

Delta-DOR (differential one-way ranging) is a technique which addresses the problem of orbit determination of a target deep-space spacecraft, in particular by determining its angular position in the sky. This can be achieved by means of analyzing both quasar signals, using this radio source as a calibrator, and a proper signal transmitted by the target spacecraft. These signals should be received by two or more stations, possibly set at a large distance from each other. Nowadays, European Space Agency (ESA) can count only on a few stations, more precisely the largest ones, since large signal-to-noise ratios are required to acquire the faint transmissions involved. This study has the aim to exploit a different set of stations, the ones belonging to the very long baseline interferometry (VLBI) network, to perform Delta-DOR measurements. VLBI antennas have in general the right hardware requirements but lack a proper recording output; therefore, a software translation of the recorded signal is required. As a proof-of-concept of the proposed technique, a shadow pass, involving an Italian VLBI station, of a standard ESA Delta-DOR session has been scheduled. The recorded data were successfully translated and analyzed by means of the ESA’s software correlator, showing the feasibility of this innovative procedure.

Accurate quantification of geocenter motion is important for maintaining the reference frame and estimating large-scale mass variations using the Gravity Recovery and Climate Experiment (GRACE) time-variable gravity solutions. Geocenter motion (equivalent to the variations in the degree-1 spherical harmonics of the gravity field) can be determined from different geodetic techniques and approaches, and the results generally show reasonable agreement, but significant differences still exist. To be more consistent with GRACE gravity solutions, here we present an improved approach to solve geocenter motion using GRACE GPS data. Over the past 15 years, GRACE satellites have acquired enough GPS data for studying the geocenter motion. In the meantime, data processing methods, reference system, and background geophysical models for GRACE precise orbit determination have also been significantly improved. Those aspects are very important for accurate determination of geocenter motion from GRACE GPS observations. For comparison, geocenter motion is also derived from LAGEOS satellite laser ranging (SLR) observations. With these independent geocenter motion solutions from GRACE GPS and LAGEOS SLR, we explore the reasons that lead to the differences between the solutions, and try to resolve these discrepancies. Daily geocenter motion time series from GRACE GPS data and 28-day geocenter variations from LAGEOS SLR observations for the time span 2003–2016 have been derived. Internal comparisons between the GRACE-A and GRACE-B geocenter motion time series and external comparisons between GRACE and LAGEOS show good agreements after using the improved approach. To verify the results, the annual geocenter motion from this study is compared with other recent geocenter motion solutions as well as predictions from geophysical models. The comparisons show reasonable agreements in both amplitude and phase with our improved approach.

Acquiring and processing astrometric measurements of a spacecraft’s target using on-board images, generically referred to as optical navigation, is an integral function of the orbit determination and navigation of NASA’s New Horizons spacecraft. Since New Horizons’ reconnaissance of the Pluto system in July 2015, many preparations have been completed to further enhance the optical navigation system and prepare for the reconnaissance of New Horizons’ next target, Kuiper Belt Object (486958) 2014 MU69 (unofficially nicknamed Ultima Thule). Due to its low relative brightness compared to most planetary exploration targets, Ultima Thule presents several unique challenges to the optical navigation system. The optical navigation system design, imaging schedule, and technical algorithms that were developed and tailored to these challenges are explored in detail. Additionally, several operational readiness tests, simulation methods, and test results are presented and analyzed to assess the optical navigation system performance and implications to flight operations. Lastly, a first look at Ultima as viewed from the New Horizons LORRI imager is presented.

We present a method to estimate geocenter motion through single low-earth orbiter (LEO) precise orbit determination (POD) using global positioning system (GPS) tracking data and accelerometer data from the GRACE satellites. We fix the values of the GPS ephemerides and time-varying clock offsets to precise estimates from the definitive constellation product produced by the Jet Propulsion Laboratory. As part of the POD process of the LEOs, we estimate a translation of the reference coordinate system realized by the GPS orbit and clock product. Doing so accounts for the inconsistency between the Earth’s center of mass coordinate system used for the orbit integration of the LEOs, and the terrestrial reference frame produced by the GPS orbit and clock product. The resulting translation parameters estimated separately from the GRACE-A and GRACE-B satellites show very similar variations from day to day. They represent the geocenter motion, as realized from the difference between the origin of the terrestrial reference frame represented by the GPS orbit and clock product and the Earth’s instantaneous center of mass defined by the GRACE satellite dynamical orbital motion. Comparisons with geocenter motion observations from other techniques show that our daily estimates of geocenter motion agree well, when smoothed, in both the amplitude and phase of the annual signal. This validates both the high sensitivity of the GRACE GPS measurement type to geocenter motion and the high precision of the GRACE force model enabled by accelerometer measurements, the two essential components for estimation of geocenter motion with this technique.

Eight selected geopotential models obtained through the International Center for Global Earth Models were used in the dynamic orbit determination process of the satellite of the Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) mission. For the estimation of various GOCE orbital arc variants, the following gravity models were taken into account: HUST-GRACE2016S, ITU_GRACE16, ITSG-GRACE2014S, ITSG-GRACE2014K, TONGJI-GRACE01, EIGEN-51C, EIGEN5S, EGM2008. The official kinematic and reduced-dynamic precise science orbit (PSO) of the GOCE satellite received via the European Space Agency was adopted as the reference orbit. Cartesian coordinates of the GOCE satellite in this orbit were treated as pseudo-observations in the estimation process using the classical least-squares method. The estimated orbital arcs were fitted to the corresponding arcs of the reference orbit. This allowed the values of 3D root-mean-square (RMS) of the distance between the estimated and reference arcs to be computed. The averages for these 3D RMS values, computed for ten and fifty orbital arcs, made it possible to compare the orbital performance of selected gravity models. Additionally, the fit to both types of the GOCE PSO, i.e., the kinematic orbit and the reduced-dynamic orbit made it possible to compare their quality. Investigation of the gravity model performance was also the opportunity to describe the effectiveness of dynamic orbit determination solutions, depending on the estimated arc lengths, type of reference orbit and the use (or not) of the background models.

On September 20th, 2015, twenty satellites were successfully deployed into a near-polar circular orbit at 520 km altitude by the Chinese CZ-6 test rocket, which was launched from the TaiYuan Satellite Launch Center. Among these satellites, a set of 4 CubeSats conform the atmospheric density detection and precise orbit determination (APOD) mission, which is projected for atmospheric density estimation from in-situ detection and precise orbit products. The APOD satellites are manufactured by China Spacesat Co. Ltd. and the payload instruments include an atmospheric density detector (ADD), a dual-frequency dual-mode global navigation satellite system (GNSS) receiver (GPS and Beidou), a satellite laser ranging (SLR) reflector, and an S/X-band very long baseline interferometry (VLBI) beacon. In this paper, we compare the GNSS precise orbit products with co-located SLR observations, and the 3D orbit accuracy shows better than 10 cm RMS. These results reveal the great potential of the onboard micro-electro-mechanical system (MEMS) GNSS receiver. After calibrating ADD density estimates with precise orbit products, the accuracy of our density products can reach about 10% with respect to the background density. Density estimates from APOD are of a great importance for scientific studies on upper atmosphere variations and useful for model data assimilation.

The accurate modeling of tropospheric path delay is significant for data processing of radio space-geodetic techniques such as Global Navigation Satellite System (GNSS) and Very-Long-Baseline Interferometry (VLBI). The Vienna Mapping Function 1 (VMF1) model, based on continuous update of Numerical Weather Prediction (NWP) data from the European Centre for Medium-Range Weather Forecasts (ECMWF), is recommended for this purpose in post-processing. The VMF1 coefficients determined from forecast data of the ECMWF are now readily and freely available. However, little or no implementation of this forecast VMF1 (VMF1-FC) model in real-time GNSS or VLBI applications has occurred. This study investigates the performance of the VMF1-FC model in terms of its three components which are critical for the modeling of tropospheric path delay: the Zenith Hydrostatic Delay (ZHD), the Zenith Wet Delay (ZWD) and mapping functions. All three components are assessed in the context of GNSS Precise Point Positioning (PPP) using 28 global stations over a 70-day period. The Zenith Total Delays (ZTD) derived with the VMF1-FC (implemented in real-time PPP) are shown to agree well with the tropospheric delay product from the Center for Orbit Determination Europe (CODE). Root mean square (RMS) errors associated with these ZTD estimates are

Doppler frequency shift which is defined as “the observed changes in frequency of transmitted waves when relative motion exists between the source of waves and an observer”, has wide applications in science, technology and different industries including communications, medicine, aerospace, electronics and etc. One of its common applications in aerospace is its use in tracking satellites and orbit determination. In low earth orbiting satellites, because of their low altitude to earth and their high rotation period around it, Doppler shift is appropriate to be used for tracking. In this paper, satellite distance and its derivative in time which is parameter in obtaining Doppler shift is derived. Position of the satellite is firstly derived using equations governing on them and TLE which contains orbital elements and essential information in different coordinate systems such as ECI, ECEF and topocentric. Then satellite distance to ground station and its changes in time is calculated via transformations of coordinate systems to each other. Having the distance, its derivative and beacon signal frequency, Doppler curve can be estimated in each pass of the satellite. Doppler shift can be used in an appropriate and precise satellite tracking technique.

Classical techniques for initial orbit determination (IOD) require the analyst to find a body’s orbit given only observations such as bearings, range, and/or position. In these cases, one of the goals is often to solve for the unknown velocity vector at one or more of the observation times to fully define the orbit. Recently, however, a new class of IOD problems has been proposed that switches the knowns and unknowns in these classic IOD problems. Specifically, the objective is to find the unknown position vectors given only velocity measurements. This paper presents a detailed assessment of the geometric properties of this new family of velocity-only IOD problems. The primary tool for this geometric analysis is Hamilton’s orbital hodograph, which is known to be a perfect circle for all orbits obeying Keplerian dynamics. This framework is used to produce intuitive and efficient algorithms for IOD from three velocity vectors (similar structure to Gibbs problem) and for IOD from two velocity vectors and time-of-flight (similar structure to Lambert’s problem). Performance of these algorithms is demonstrated through numerical results.

The first pair of satellites belonging to the European Global Navigation Satellite System (GNSS)—Galileo—has been accidentally launched into highly eccentric, instead of circular, orbits. The final height of these two satellites varies between 17,180 and 26,020 km, making these satellites very suitable for the verification of the effects emerging from general relativity. We employ the post-Newtonian parameterization (PPN) for describing the perturbations acting on Keplerian orbit parameters of artificial Earth satellites caused by the Schwarzschild, Lense–Thirring, and de Sitter general relativity effects. The values emerging from PPN numerical simulations are compared with the approximations based on the Gaussian perturbations for the temporal variations of the Keplerian elements of Galileo satellites in nominal, near-circular orbits, as well as in the highly elliptical orbits. We discuss what kinds of perturbations are detectable using the current accuracy of precise orbit determination of artificial Earth satellites, including the expected secular and periodic variations, as well as the constant offsets of Keplerian parameters. We found that not only secular but also periodic variations of orbit parameters caused by general relativity effects exceed the value of 1 cm within 24 h; thus, they should be fully detectable using the current GNSS precise orbit determination methods. Many of the 1-PPN effects are detectable using the Galileo satellite system, but the Lense–Thirring effect is not.

The differential code biases (DCBs) of the global positioning system (GPS) receiver onboard low-Earth orbit (LEO) satellites are commonly estimated by a local spherical symmetry assumption together with the known GPS satellite DCBs from ground-based observations. Nowadays, more and more LEO satellites are equipped with GPS receivers for precise orbit determination, which provides a unique chance to estimate both satellite and receiver DCBs without any ground data. A new method to estimate the GPS satellite and receiver DCBs using a network of LEO receivers is proposed. A multi-layer mapping function (MF) is used to combine multi-LEO satellite data at varying orbit heights. First, model simulations are conducted to compare the vertical total electron content (VTEC) derived from the multi-layer MF and the reference VTEC obtained from the empirical ionosphere model International Reference Ionosphere and Global Core Plasmasphere Model. Second, GPS data are collected from five LEO missions, including ten receivers used to estimate both the satellite and receiver DCBs simultaneously with the multi-layer MF. The results show that the GPS satellite DCB solutions obtained from space-based data are consistent with ground-based solutions provided by the Centre for Orbit Determination in Europe. The proposed normalization procedure combining topside observations from different LEO missions has the potential to improve the accuracies of satellite DCBs of Global Navigation Satellite Systems as well as the receiver DCBs onboard LEO satellites, although the number of LEO missions and spatial–temporal coverage of topside observations are limited.

High-precision orbits are essential for low Earth orbit (LEO) satellite applications. Instead of determinating orbit by the ground monitoring stations, LEO missions are trending to incorporate spaceborne global positioning system receivers for autonomous navigation. Focusing on this issue, we propose a method of real-time orbit determination based on ionosphere combination observation of dual-frequency data using adaptive extended Kalman filter (AEKF). The traditional Kalman filter heavily relies on the initial values of measurement and process noise covariance matrix. However, the environment of the satellite is time-variant, and it is hard to keep the error covariance matrix in the optimal condition all the time. To overcome this disadvantage, the Sage–Husa Kalman filter is introduced. The performance of the proposed algorithm is compared with the conventional EKF, and the results show that the proposed method can guarantee the precision and stability of orbit determination. With the proposed algorithms, we developed a software to solve the orbit suitable for one-day Gravity Recovery and Climate Experiment (GRACE) observations. The obtained results show that the accuracy of orbit determination error is improved with unknown Kalman process noise compared with traditional method. Additionally, the performances of AEKF for different gravity field models are analyzed and compared.

Single-difference (SD) ambiguity resolution (AR) and track-to-track (T2T) AR are two typical AR methods in precise orbit determination (POD) for Low Earth Orbit (LEO) satellites, which could improve the accuracy of orbits greatly. In this study, SD AR and T2T AR methods are introduced and analyzed. The performance of these two methods is assessed by three months of GPS observations from the Gravity Recovery and Climate Experiment Follow On (GRACE-FO) twin satellites. Results show that T2T AR is highly dependent on the stability of receiver hardware delays, while SD AR requires Fractional Cycle Bias (FCB) or Integer Recovery Clock (IRC) products. We find that these two methods have comparable performance in Reduced Dynamic Precise Orbit Determination (RDPOD), while SD AR slightly outperforms T2T AR in Kinematic Precise Orbit Determination (KPOD). We also find that SD AR has a higher AR success rate than T2T AR. Therefore, we recommend SD AR as the top choice in LEO orbit determination, and T2T AR can be a good alternative when FCB or IRC products are not available.

The BeiDou navigation satellite system (BDS) tracks medium earth orbit (MEO) satellites using only regional tracking stations in China. As a result, the broadcast clock accuracy of the MEO satellites decreases rapidly during the invisible arcs because of the lack of available observations. The inter-satellite link (ISL) technology of the third generation of BDS (BDS-3) can be used to extend the visible arcs of MEO satellites and to measure the relative inter-satellite clock in nearly real time. We propose a broadcast clock approach for BDS-3 by integrating observations from regional tracking stations and ISLs. The clock error between satellites is obtained through centralized estimation based on ISLs. The Ka-band hardware delay is calibrated by taking the double difference between ISL-centralized clock and the Multi-satellite Precise Orbit Determination clock. The deviation between the ISL-centralized clock and the BeiDou time is obtained using only one Two-way Satellite Time Comparison station or anchor station. To validate the algorithms, we analyze clock estimation and prediction accuracy, hardware delay stability, and time synchronization accuracy. The results show that the frequency stability of the BDS-3 onboard passive hydrogen maser (PHM) and rubidium atomic frequency standard (RAFS) is competitive to those of the GPS IIF RAFS and Galileo FOC PHM and better than those of GPS IIR RAFS. The root-mean-square error of the 2-h clock prediction is better than 0.25 ns, and the validation result relative to the post-processed precise clock product is better than 0.4 ns. The time synchronization accuracy of better than 1 ns can be obtained based on only one TSTC station or an anchor station, and the standard deviation of Ka-band hardware delay is about 0.12 ns. It is believed that the ISL and the proposed algorithms will bring a significant upgrade in the estimation of BDS-3 broadcast clock; the broadcast clock accuracy will be greatly improved, and reliance on the ground segment will also be reduced significantly.

For real-time precise orbit determination (POD) of low earth orbit (LEO) satellites, high-accuracy global navigation satellite system (GNSS) orbit and clock products are necessary in real time. Recently, the Japanese multi-GNSS advanced demonstration of orbit and clock analysis precise point positioning (PPP) service and the new generation of the Australian/New Zealand satellite-based augmentation system (SBAS)-aided PPP service provide free and precise GNSS products that are directly broadcast through the navigation and geostationary earth orbit satellites, respectively. With the high quality of both products shown in this study, a 3D accuracy of centimeters can be achieved in the post-processing mode for the reduced-dynamic orbits of small LEO satellites having a duty cycle down to 40% and at sub-dm to dm level for the kinematic orbits. The results show a promising future for high-accuracy real-time POD onboard LEO satellites benefiting from the precise free-of-charge PPP corrections broadcast by navigation systems or SBAS.

Global navigation satellite system (GNSS) differential code bias (DCB) is a significant error source in ionosphere modeling that uses GNSS observation data. Given that the orbit altitudes of low earth orbit (LEO) satellites are above the F layer of ionosphere, ionized electrons come from the upper ionosphere or plasmasphere, so observations suffer from small signal delays. DCBs can be estimated using LEO data instead of that from ground stations. In most studies, LEO-based DCB estimation employs onboard data from either a single LEO or from two that are at the same orbit height. In our study, we applied four LEO satellites data from two different orbit heights to estimate GPS satellite DCBs, receiver DCBs and LEO-based vertical total electron content (VTEC) parameters simultaneously. Before DCB estimation, a data preprocessing strategy suitable for LEO onboard data was applied to obtain clear pseudorange data. We developed a processing method for the upper ionosphere using LEO onboard data from different orbit heights, by introducing LEO-based VTEC models in advance to remove its effects; two introduced VTEC models were derived from modeling results using paired data from the same orbital height, respectively. We draw some conclusions as follows. For the LEO satellites at different orbit heights, the more the number of LEO satellites involved, the more stable estimation results achieved. We also noted that the GPS satellite DCBs stability and accuracy with established LEO-based VTEC models, using multi-LEOs data from different orbit heights, were better than the achievable results using onboard data from the same orbital height or using single LEO satellite schemes. Compared with the estimation results of single LEO solutions, the monthly stabilities and accuracies for multi-LEOs solutions from different orbit heights are improved by 25–35% approximately. Among all the tested schemes, superior stability derived from simultaneous estimation using four LEO data at different orbit heights is 0.064 ns; the optimal accuracies for DCB estimation for different GPS satellites are the scheme GRACE-A and JASON-2 (two LEOs) of 0.146 ns, approximately, compared with Center of Orbit Determination of Europe products, respectively. The stability and accuracy of LEO-based DCB estimations were approximately similar to those achieved using ground solutions.

The precise orbit determination (POD) of Global Navigation Satellite System (GNSS) satellites and low Earth orbiters (LEOs) are usually performed independently. It is a potential way to improve the GNSS orbits by integrating LEOs onboard observations into the processing, especially for the developing GNSS, e.g., Galileo with a sparse sensor station network and Beidou with a regional distributed operating network. In recent years, few studies combined the processing of ground- and space-based GNSS observations. The integrated POD of GPS satellites and seven LEOs, including GRACE-A/B, OSTM/Jason-2, Jason-3 and, Swarm-A/B/C, is discussed in this study. GPS code and phase observations obtained by onboard GPS receivers of LEOs and ground-based receivers of the International GNSS Service (IGS) tracking network are used together in one least-squares adjustment. The POD solutions of the integrated processing with different subsets of LEOs and ground stations are analyzed in detail. The derived GPS satellite orbits are validated by comparing with the official IGS products and internal comparison based on the differences of overlapping orbits and satellite positions at the day-boundary epoch. The differences between the GPS satellite orbits derived based on a 26-station network and the official IGS products decrease from 37.5 to 23.9 mm ($$34\%$$ 34 % improvement) in 1D-mean RMS when adding seven LEOs. Both the number of the space-based observations and the LEO orbit geometry affect the GPS satellite orbits derived in the integrated processing. In this study, the latter one is proved to be more critical. By including three LEOs in three different orbital planes, the GPS satellite orbits improve more than from adding seven well-selected additional stations to the network. Experiments with a ten-station and regional network show an improvement of the GPS satellite orbits from about 25 cm to less than five centimeters in 1D-mean RMS after integrating the seven LEOs.

The gravity recovery and climate experiment follow-on (GRACE-FO) satellites, launched in May of 2018, are equipped with geodetic quality GPS receivers for precise orbit determination (POD) and gravity recovery. The primary objective of the GRACE-FO mission is to map the time-variable and mean gravity field of the Earth. To achieve this goal, both GRACE-FO satellites are additionally equipped with a K-band ranging (KBR) system, accelerometers and star trackers. Data processing strategies, data weighting approaches and impacts of observation types and rates are investigated in order to determine the most efficient approach for processing GRACE-FO multi-type data for precise orbit determination and gravity recovery. Two GPS observation types, un-differenced (UD) and double-differenced (DD) observations in general can be used for GPS-based POD and gravity recovery. The GRACE-FO KBR observations are mainly used for gravity recovery, but they can be also used for POD to improve the relative orbit accuracy. The main purpose of this paper is to study the impacts of the DD, UD and KBR observations on GRACE-FO POD and gravity recovery. The precise orbit accuracy is assessed using several tests, which include analysis of orbital fits, satellite laser ranging residuals, KBR range residuals and orbit comparisons. The gravity recovery is validated by comparing different gravity solutions through coefficient-wise comparison, degree difference variances and water height variations over the whole Earth and selected area and river basins.

The Tianping-1B is a 20-kg low earth orbit nanosatellite with a commercial multi-GNSS receiver based on a microelectromechanical system. This receiver collects concurrent code and phase dual-frequency measurements from the global positioning system (GPS) and the second and third generations of the BeiDou Global Navigation Satellite System (i.e., BDS-2 and BDS-3). However, BDS-3 signals with pseudorandom noise code numbers greater than 32 cannot be received. In this study, onboard GPS and BDS measurements from Tianping-1B are collected for days 133–147 of 2019. The performance of the onboard BDS-3 measurements is analyzed, and the potential of the BDS-3-based precise orbit determination (POD) for Tianping-1B is assessed. The carrier-to-noise-density ratio of the BDS-3 is higher than that of the BDS-2 and approaches that of the GPS. The BDS-3 has a smaller code multipath error than the BDS-2 and the GPS and therefore a higher quality of code measurements. The results of the overlap comparison show a GPS-based orbit consistency below 3.5 cm in three dimensions (3D) and below 1.2 cm in the radial direction. The mean of satellite laser ranging validation residual RMS is 1.7 cm. The orbit obtained using onboard BDS measurements is assessed using the GPS-based orbit as a reference: The mean 3D root mean square (RMS) difference between the BDS-3-only-based POD and the reference orbit is 4.57 cm. Thus, a sub-dm-level orbit can be achieved using only onboard BDS-3 measurements. A POD with slightly higher precision than the BDS-3-only-based POD is obtained using both BDS-3 and BDS-2 measurements, using higher weights for the BDS-3 data than the BDS-2 data. Then, the combined BDS-3/GPS POD is performed: The RMS difference between this joint orbit and the GPS-based orbit is 1 cm in 3D. This study can be used as a reference for the development of BDS-3 based POD, which will approve in accuracy and precision upon completion of BDS-3.

We deal with the orbit determination problem for a class of maps of the cylinder generalizing the Chirikov standard map. The problem consists of determining the initial conditions and other parameters of an orbit from some observations. A solution to this problem goes back to Gauss and leads to the least squares method. Since the observations admit errors, the solution comes with a confidence region describing the uncertainty of the solution itself. We study the behavior of the confidence region in the case of a simultaneous increase of the number of observations and the time span over which they are performed. More precisely, we describe the geometry of the confidence region for solutions in regular zones. We prove an estimate of the trend of the uncertainties in a set of positive measure of the phase space, made of invariant curve. Our result gives an analytical proof of some known numerical evidences.

China is currently focusing on the establishment of its BDS-3 system, and a BDS-3 constellation with 18 satellites in medium Earth orbit (MEO) and one satellite in geostationary Earth orbit (GEO) has been able to provide preliminary global services since the end of 2018. These BDS-3 satellites feature the inter-satellite link (ISL) and new high-quality onboard clocks. In this study, we present the analysis of BDS-3 orbits and clocks determined by Ka-band ISL measurements from 18 MEO satellites and one GEO satellite. The ISL data of 43 days from 1 January to 12 February 2019 are used. The BDS-3 ISL measurement is described by a dual one-way ranging model. After converting bidirectional observations to the same epoch, Ka-band clock-free and geometry-free observables are obtained by the addition and subtraction of dual one-way observations, respectively. One anchor station with Ka-band bidirectional observations is introduced into the orbit determination to provide the orientation constraints. Using Ka-band clock-free observables, BDS-3 satellite orbits are determined. The ISL hardware delays are estimated together with orbits, and the resulting hardware delay estimates are quite stable with STD of about 0.03 ns. The Ka-band orbits are evaluated by orbit overlap differences, comparison with L-band precise orbits, and satellite laser ranging validation. The results indicate that the radial orbit errors are on the 2–4 cm level for MEO satellites and 8–10 cm for the GEO satellite. In addition, we investigate the ground anchoring capability by adding one anchor station and reducing the amount of data of the anchor station. Using Ka-band geometry-free observables, BDS-3 satellite clocks are estimated and the RMS of post-fit ISL residuals is about 5 cm. The Ka-band clock offsets are analyzed and compared with L-band precise clocks. Independent of orbit errors, the Allan deviation of Ka-band clocks for averaging interval longer than 5000 s is superior to that of L-band clocks. Furthermore, a pronounced bump, which appears in the Allan deviation of L-band clocks, almost vanishes in Ka-band clocks. Finally, the periodic variations are detected for L-band and Ka-band clocks.

There is increasing concern about the uncombined (UC) observation model in the field of global navigation satellite system (GNSS). Based on the global positioning system (GPS) and the third-generation BeiDou navigation satellite system (BDS-3), this study processed the UC precision orbit determination (POD) for single and dual systems. First, a UC observation model suitable for multi-GNSS POD was derived, and the ionospheric-free (IF) combination observation model was presented. Although the ambiguity parameters of UC and IF strategies were different after reparameterization, the difference could be removed when processing ambiguity resolution, and the equivalence was proved theoretically. To demonstrate the accuracy of BDS-3 orbits fully, the observation data of approximately 1 month were selected for determining the precise orbit for global positioning system (GPS) only, BDS-3 only, and GPS/BDS-3 systems based on the UC and IF models. The orbit precision of BDS-3 satellites was validated by using metrics, including comparison with precision products released by Wuhan University, orbit boundary discontinuity, and satellite laser ranging (SLR) residuals. The results show that the orbit accuracies of the IF and UC models are almost the same, the difference in orbits is approximately several millimeters, and the clock difference is within 0.01 ns. The GPS/BDS-3 combined solution shows better accuracy compared to other solutions. The average accuracies in the R and 3D directions are approximately 4 and 15 cm, and the clock standard deviation is approximately 0.2 ns compared to external orbit product. The root mean square of SLR residuals is approximately 4 cm.

The precise orbit of the Hayabusa2 spacecraft with respect to asteroid Ryugu is dynamically determined using the data sets collected by the spacecraft’s onboard laser altimeter (LIght Detection And Ranging, LIDAR) and automated image tracking (AIT). The LIDAR range data and the AIT angular data play complementary roles because LIDAR is sensitive to the line-of-sight direction from Hayabusa2 to Ryugu, while the AIT is sensitive to the directions perpendicular to it. Using LIDAR and AIT, all six components of the initial state vector can be derived stably, which is difficult to achieve using only LIDAR or AIT. The coefficient of solar radiation pressure (SRP) of the Hayabusa2 spacecraft and standard gravitational parameter (GM) of Ryugu can also be estimated in the orbit determination process, by combining multiple orbit arcs at various altitudes. In the process of orbit determination, the Ryugu-fixed coordinate of the center of the LIDAR spot is determined by fitting the range data geometrically to the topography of Ryugu using the Markov Chain Monte Carlo method. Such an approach is effective for realizing the rapid convergence of the solution. The root mean squares of the residuals of the observed minus computed values of the range and brightness-centroid direction of the image are 1.36 m and 0.0270°, respectively. The estimated values of the GM of Ryugu and a correction factor to our initial SRP model are 29.8 ± 0.3 m3/s2 and 1.13 ± 0.16, respectively.

Autonomous orbit determination (AOD) is the ability of navigation satellites to estimate with accurate satellite orbit parameters using onboard using inter-satellite link (ISL) measurements. To overcome the unobservability of the constellation rotation error in AOD when using only the ISL measurements, the properties that the orbit inclination $$ i $$ and the longitude of the ascending node $$ \varOmega $$ of the medium earth orbit (MEO) navigation satellites, which can be predicted with high accuracy over a long time, are explored. This leads to an onboard extended Kalman filter (EKF) where $$ \left( {i,\varOmega } \right) $$ are subjected to constraints. Three experiments are carried out to assess the effectiveness of the proposed AOD EKF and analyze the causes of the constellation rotation error by processing 30-day ISL measurements of 18 MEO satellites of BDS-3 in a distributed mode. The results verify that the proposed EKF with $$ \left( {i,\varOmega } \right) $$ constraints can resolve the unobservable constellation rotation error issue effectively. When using precise EOP parameters, the 3D orbit errors of BDS-3 AOD in 30 days could be less than 2.30 m. The errors increase to 3.4 m when the predicted EOP parameters are used.

We developed a global empirical model for computing spherical harmonic (SH) coefficients based on the empirical orthogonal function (EOF) method and periodic functions by utilizing global ionospheric SH coefficients data provided by Center for Orbit Determination in Europe (CODE) during the years 2005–2016. Results show that the first four-order base functions and corresponding associated coefficients can represent 97.15% of the basic characteristics of the original data. The first four-order associated coefficients have noticeable periodic variations, and the correlation between these coefficients and the solar activity intensity is high. By fitting the associated coefficients with the periodic function that takes into account the influence of solar activity, it is possible to establish an EOF model to characterize the variations of the ionospheric SH coefficients with few model parameters and further calculate the global vertical total electron content (VTEC). Relative to the existing global VTEC EOF models, this EOF method can achieve high-accuracy modeling of global VTEC with a smaller number of model coefficients. By comparing the SH coefficients and the global VTEC between the EOF model and CODE, results demonstrate that the EOF model can achieve high accuracy and reliability under different solar activities. The difference between the EOF model and CODE is basically at the same level as that between the individual ionospheric analysis centers and CODE. In the extreme event of magnetic storms, the EOF model can also present higher accuracy than the international reference ionosphere (IRI) model.

For low Earth orbit (LEO) satellites, activities such as precise orbit determination, gravity field retrieval, and thermospheric density estimation from accelerometry require modeled accelerations due to radiation pressure. To overcome inconsistencies and better understand the propagation of modeling errors into estimates, we here suggest to extend the standard analytical LEO radiation pressure model with emphasis on removing systematic errors in time-dependent radiation data products for the Sun and the Earth. Our extended unified model of Earth radiation pressure accelerations is based on hourly CERES SYN1deg data of the Earth’s outgoing radiation combined with angular distribution models. We apply this approach to the GRACE (Gravity Recovery and Climate Experiment) data. Validations with 1 year of calibrated accelerometer measurements suggest that the proposed model extension reduces RMS fits between 5 and 27%, depending on how measurements were calibrated. In contrast, we find little changes when implementing, e.g., thermal reradiation or anisotropic reflection at the satellite’s surface. The refined model can be adopted to any satellite, but insufficient knowledge of geometry and in particular surface properties remains a limitation. In an inverse approach, we therefore parametrize various combinations of possible systematic errors to investigate estimability and understand correlations of remaining inconsistencies. Using GRACE-A accelerometry data, we solve for corrections of material coefficients and CERES fluxes separately over ocean and land. These results are encouraging and suggest that certain physical radiation pressure model parameters could indeed be determined from satellite accelerometry data., Deutsches Zentrum für Luft- und Raumfahrt http://dx.doi.org/10.13039/501100002946, ftp://ftp.tugraz.at/outgoing/ITSG/tvgogo/orbits/GRACE/, ftp://podaac-ftp.jpl.nasa.gov/allData/grace/L1B/JPL/