Independent Mars spacecraft precise orbit ...

Astrophys Space Sci (2017) 362:123 DOI 10.1007/s10509-017-3105-0

O R I G I N A L A RT I C L E

Independent Mars spacecraft precise orbit determination software development and its applications Jianguo Yan1 · Xuan Yang1 · Mao Ye1 · Fei Li1 · Weitong Jin1 · Jean-Pierre Barriot1,2

Received: 26 March 2017 / Accepted: 19 May 2017 © Springer Science+Business Media Dordrecht 2017

Abstract In this paper, we present an independent software for Mars spacecraft precise orbit determination and gravity field recovery we call the Mars Gravity Recovery and Analysis Software (MAGREAS), which is aimed to analyze tracking data from the Chinese Mars exploration mission and similar NASA and ESA Mars-related projects. The design structure, module distribution, and functions of the software are described in this manuscript. A detailed cross validation with the mature precise orbit determination platform Geodyn-II was done. Additionally, we use MAGREAS to process the MEX orbital tracking data with two-way and three-way tracking modes separately. Measurement residuals and the difference from the reconstructed ephemeris provided by Royal Observatory of Belgium indicate that our software is reliable. In addition to describe of our software and validate with Geodyn-II, we give a simulation case close to Chinese Mars exploration mission to indicate the application of our software. We present a simulation of a fourway tracking mode between Earth tracking station, Mars orbiter, and Mars lander to validate the effectiveness of our MAGREAS-based approach for Mars orbiter determination and lander positioning. Experimental results show that our proposed tracking mode significantly improves positioning accuracy. This work will provide a reference for the design of the Chinese Mars exploration mission as well as for the processing of Chinese Mars mission orbital tracking data.

B J. Yan

[email protected]

1

State Key Laboratory of Engineering in Surveying, Mapping and Remote Sensing, Wuhan University, 129 Luoyu Road, 430070, Wuhan, China

2

Observatoire géodésique de Tahiti, BP 6570, 98702 Faa’a, Tahiti, French Polynesia

Keywords Precise orbit determination · Independent · Software · MEX

1 Introduction Mars is close to planet Earth and is thought to have once had a similar environment as Earth. Up to now, Mars is the planet most visited by spacecraft and is a continuous hot spot for deep space exploration. Whether it once had life and liquid water has long been a research interest. For the last sixty years, NASA and ESA have launched numerous Mars exploration missions, with many scientific and engineering achievements. Typical NASA Mars exploration missions include the Mars Global Surveyor (MGS), Odyssey, and the Mars Reconnaissance Orbiter (MRO). These missions contributed significantly to our understanding of the topography of Mars, its gravity and chemical element composition (Smith et al. 2001; Zuber et al. 2000). Mars Express (MEX) was ESA’s first Mars exploration mission. It was launched on June 2003 and successfully operated for more than ten years. During its mission, many high-resolution Mars surface images were transmitted, and contributed to the discovery of methane in Mars atmosphere and water ice in Mars’ south polar cap (Pätzold et al. 2016). This mission also improved the determination of the mass of Phobos as well as the low degree gravity field coefficients (Andert et al. 2010). China announced that its ambitious Mars exploration mission would launch in 2020. This mission will include a Mars orbiter, a lander, and rover. Its science goals include studying the Martian topography, soil, environment, atmosphere and water ice, as well as the internal structure of the planet and a search for possible signs of life. In order to guarantee the operation and scientific achievements of this mission, a highly accurate and reliable software for Mars spacecraft precise orbit determination and gravity field recovery

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is required. Such software can be used to accurately map the ephemeris, which will be necessary for laser ranging and image processing (Smith et al. 2001). A reliable model of the Martian gravity field as derived from the software is essential for analysis of the interior structure of Mars (Zuber et al. 2000). The development of a highly accurate and reliable Mars spacecraft POD and gravity field recovery software has been a focus of work at several deep space research centers. Currently the well-known software for these purposes include Geodyn-II from Goddard/NASA (Lemoine et al. 2001), ODP (Moyer 1971, 2000) and updated MONTE (Evans et al. 2016) from JPL/NASA, and GINS (Marty et al. 2009) originally developed by GRGS and later adapted for Mars by the Royal Observatory of Belgium (ROB). These three software products achieve similar accuracy for Mars spacecraft orbital tracking reduction. Geodyn-II has been a powerful tool for Goddard/NASA since the 1970s, and it has been employed in many planetary exploration missions, including the GRAIL mission to the Moon (Lemoine et al. 2013), the MESSENGER mission to Mercury (Smith et al. 2012), and the Dawn mission to Vesta (Centinello et al. 2013). It has also been extensively applied in Mars exploration to process MGS, Odyssey and MRO data for POD and Martian gravity field recovery (Lemoine et al. 2001; Genova et al. 2016). MGS reaches a robust orbital accuracy of 10 m (Lemoine et al. 2001). The ODP software is a classic platform used for planetary spacecraft orbital tracking data processing in JPL for long time and has been deployed in all JPL planetary exploration missions. Such as the LP and GRAIL mission to the Moon (Konopliv et al. 2013), the Magellan mission to Venus (Konopliv et al. 1999), the NEAR mission to Eros (Miller et al. 2002), the Dawn mission to Vesta (Konopliv et al. 2014), with extensive application in several Mars missions (Konopliv et al. 2006, 2011, 2016). In absorbing rapidly evolving compute technologies and software development practices, JPL designed MONTE (Mission Design and Operations Navigation Toolkit Environment) in 1998 and began to replace ODP (Evans et al. 2016). MONTE has been successfully applied to Cassini and MESSENGER missions (Iess et al. 2014; Verma and Margot 2016). In order to prepare the NetLander mission (Dehant et al. 2004), ROB adapted Géodésie par Intégrations Numériques Simultanées (GINS) software from Centre National d’Etudes Spatiales (CNES) for application in Mars spacecraft and lander tracking data processing. It has been extensively used in investigations of the contribution of radio links between Mars spacecraft and lander to constrain the Mars rotation dynamic parameters solution, as well as in interior structure analysis of Mars using simulations (Duron et al. 2003; Karatekin et al. 2005; Dehant et al. 2009, 2011).

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In order to validate the reliability of the GINS application in Mars missions, it was used to process orbital tracking data of MGS and Odyssey to derive a Martian gravity field and time variations. The POD results from GINS were similar to those published by Goddard (Marty et al. 2009). GINS is also applied in the processing of MEX orbital tracking data to obtain precise ephemeris and mass of the Martian moons (Rosenblatt et al. 2008). GINS has been and will be used to various planetary exploration missions like Rosetta for comet 67P/Churyumov-Gerasimenko, BepiColombo for Mercury, and Laplace for Europa Jupiter system. Accurate and reliable Mars POD software is critical for the incoming Chinese Mars mission as well as the scientific goals. It was necessary to develop independent software because of the limited access to the mature software. Based on Yinghuo-1 mission, the Shanghai Astronomical Observatory and Beijing Aerospace Control Center each separately built Mars spacecraft POD software, and tested their software with MEX orbital tracking data. The difference from the reconstructed MEX ephemeris from ESA was at the level of 600 meters (Huang et al. 2009; Hu and Tang 2010; Cao et al. 2010). Another independent software is now developed at Wuhan University. It is paying more attention to Mars gravity field recovery and other dynamical solutions for dynamic Mars rotation parameters, in addition to POD for Mars spacecraft. This software is based on the Lunar Gravity Recovery and Analysis Software (LUGREAS) (Li et al. 2016) and was adapted to Mars in a new module. To improve the ability of Mars Gravity Recovery and Analysis Software (MAGREAS) and investigate the contributions of various tracking modes, we compared the proposed four-way tracking mode to traditional two-way tracking modes in a simulation of an Earth tracking station, Mars orbiter, and Mars lander. The four-way tracking mode was applied for the first time in the SELENE mission to retrieve lunar farside gravity field information (Namiki et al. 2009). This approach however, has not been considered for Mars orbital navigation. Nevertheless, the lunar four-way tracking mode for an Earth tracking station, lunar orbiter, and lunar lander was investigated to assess the contribution of this tracking mode in precise lunar orbit determination and lander positioning (Li et al. 2016). Since the first Chinese Mars exploration mission will launch an orbiter, a lander, and a rover at same time and it is possible to employ a fourway tracking mode to improve the positional accuracy of the Mars orbiter and lander. We implemented this tracking mode in MAGREAS and executed a simulation to provide reference for the design of the Chinese Mars exploration mission. In this paper, we introduce our independently developed Mars POD and gravity field recovery software MAGREAS and compared the performance of MAGREAS to

Independent Mars spacecraft precise orbit determination software development and its applications

the Geodyn-II software. Three-way MEX Doppler tracking data from Chinese VLBI (Very Long Baseline Interferometry) stations and two-way Doppler tracking data from the ESA New Norcia tracking station collected in same time period were processed using MAGREAS. The POD results and analysis were presented. In Sect. 2, we describe the structure and design of MAGREAS; in Sect. 3, the comparative results with Geodyn-II are presented. In Sect. 4, we present the POD results and analysis of the two-way and three-way tracking data of MEX. In Sect. 5, we give a simulation case of four-way tracking mode between Earth tracking station, Mars orbiter and lander. Conclusions are drawn in Sect. 6.

2 Software structure and function The starting point of Mars Gravity Recovery and Analysis Software (MAGREAS) was to solve Martian gravity field, Love number, and Mars rotation parameters in addition to the Mars spacecraft POD. Since two Mars POD software platforms have been developed in China: The MarsODP created by the Shanghai Astronomical Observatory (Huang et al. 2009) and the BODAS developed by the Beijing Aerospace Control Center (Hu and Tang 2010). Our aim was to provide data and models for further investigations of the Mars interior when developing MAGREAS. By comparing with these two platforms, MAGREAS applied more precise force models, and has the ability to provide Mars dynamical parameters like gravity field coefficients, Love number and rotation information. MAGREAS software was developed using Fortran 90. This software was designed for expandability, and can be run both in Linux and Windows environments. The MAGREAS software has more than 60,000 lines of code. We use classic dynamic method (Tapley et al. 2004) to calculate the precise orbit and solve the gravity field coefficients and other global parameters. The estimation for parameters are divided into two kinds: the first set includes local parameters, which are spacecraft initial orbital elements; solar radiation coefficient, drag force coefficient, general acceleration, and station measurement bias; the second set includes global parameters, like Martian gravity field coefficients, Mars solid tidal Love number, and Mars lander position. Currently the software can support one-way Doppler, two-way and three-way Range/Doppler, four-way Doppler, VLBI delay and delay rate, and SBI (Same Beam Interferometry). All the formulas of the force models and observable models follow strictly the algorithms described in Moyer (1971, 2000). MAGREAS uses a 12-degree Adams prediction-correction numerical integrator to integrate the Newton kinematical equation and variational equation, which are usually

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written as (Montenbruck and Gill 2000): GM r r¨ = − 2 + Fε r r ψ¨ = Aψ + Bψ˙ + C

123

(1) (2)

where r is the distance of the spacecraft to the center of mass M, r is the state vector of the spacecraft, Fε is the perturbative forces of the spacecraft, and r¨ indicates the total forces of the spacecraft. In Eq. (2), ψ is called the transfer matrix, ∂ r¨ ], and p indicates the where A = ∂∂rr¨ , B = ∂∂ rr¨˙ , and C = [ 0 ∂p parameter solution. After integration, we can obtain spacecraft ephemeris, state transfer matrix, and parameter sensitivity matrix. By combining tracking station information and geometrical measurement model, we can compute a theoretical observation C. A single arc normal matrix can be obtained from the parameter sensitivity matrix after applying the geometric partials G, and the residuals L, multiplied by the transposed sensitivity matrix and proper weights W. O−C=ψ ·G·x+ε

(3)

where O is the observation, x are the solved parameters. Further, it can be written as: L = Hx + ε

(4)

Weighted least square adjustment solves the local parameters as. −1 (5) x = HT WH HT WL The error of each parameter is derived from: −1 σ = HT WH

(6)

By combining multiple single arc normal matrices, global parameters such as the Martian gravity field coefficients, Love numbers, and lander positions can be determined in batch processing (Kaula 1966). The optimal local and global parameters are achieved when the iterations converge after matching a proper convergence criterion. Figure 1 shows a flowchart of the POD and gravity field solution. In MAGREAS, the Main Program calls and starts the Execution and Initialization modules. The Measurement Models Module, Orientation Module, Global Parameters Module, and Input Parameters Module are required when running the Execution Module. The Orientation Module must be running to start the Initialization Module. The relationships between modules are shown in Fig. 2. The arrows indicate a calling relationship. Each module is independent and flexible, and they can call other modules as needed. For the convenience when using MAGREAS, there are four input cards, the Station, Simulation, Local, and Global cards, to

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Fig. 1 Computation flow for the MAGREAS

set the software. The Station card provides station parameters; the Local and the Global cards deliver local as well as global input parameters; the Simulation card generates simulated observations. The Measurement Models Module includes a two-way Doppler model, three-way Doppler model, four-way Doppler model, VLBI model, and samebeam VLBI model. The Orientation Module describes the coordinates of celestial bodies; therefore, this module calls the Planetary Ephemeris Module. The Force Module includes the gravity field, solar radiation, drag force, general acceleration, and the solid tide models. The Global Parameter Module stores global parameter information required during program execution. There are constant parameters

in Constant Module. The Fault-tolerant Module shows error messages as the program runs.

3 Cross validation with Geodyn-II Validation against a well-established software platform is necessary to demonstrate the reliability of a new software. We selected the mature software Geodyn-II to test the accuracy of MAGREAS. Geodyn-II has been broadly applied to Mars exploration missions analysis and is wellknown for its accuracy and solidity (Lemoine et al. 2001; Genova et al. 2016).


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Fig. 2 The logical relationship of each module in MAGREAS

Table 1 The configuration of the cross validation test setting Force models Martian gravity field

GMM-2B (Lemoine et al. 2001)

N-body perturbation

DE430 (Willimas et al. 2013)

Phobos-Deimos perturbation

SPICE ephemeris

Solar radiation

Canon ball model

Atmosphere drag

Stewart 1987 atmospheric density model

Mars solid tidal perturbation

k2 = 0.169

Relativity perturbation

Schwarzschild (Sun only)

Constant parameter Solar radiation coefficient

1.20

Atmosphere drag coefficient

0.4

Coordinate system Earth

J2000

Mars

IAU1991

Spacecraft Orbit character

Polar orbit, with an altitude of 400 km

In Table 1, we provide the configuration settings employed in MAGREAS and Geodyn-II in validation tests. In order to get credible comparative results, we made all the inputs (force models, coordinate system, et al.) exactly the same. We set the comparison time period as one month with a starting date of 1 Oct, 2007. In Fig. 3, we see that the ephemeris differences between MAGREAS and Geodyn-II are quite small. In the test pe-

riod of one month, the positional difference between results from the two software products was less than 1.2 mm. Position difference in the R and N direction was close to zero (less than 0.2 mm), while the biggest difference was along the T direction, and there is a trend in the T direction differences. At first, we think it is suspicious because there seems to be a turning point after seven days, even the differences are small. After some experiments, we conclude that the trend is mainly due to the integrator. The integrators in MAGREAS and Geodyn-II do not yield the exact same results as they use two distinctive integrators. Our tests show that the force differences between MAGREAS and Geodyn-II is at the 10−14 m/s2 level; the difference in ephemeris will increase with integration time. However, the discrepancy shown in Fig. 3 is still is still extremely small (less than 1.2 mm for arc length in one month) and indicating consistency between these two software products. Furthermore, when processing data of Mars spacecraft, researcher usually used arc-length of two days to seven days (Konopliv et al. 2011; Rosenblatt et al. 2008; Marty et al. 2009; Lemoine et al. 2001). Therefore, this does not affect or reveal the accuracy of MAGREAS. The velocity differences were also similar. The biggest difference was along the R direction, but the value was less than 1.5 × 10−3 mm/s. The velocity differences in T and N direction were less than 1.0 × 10−3 mm/s. Therefore, the two software perfectly agree, as the typical external accuracy for current Mars PODs is of the order of 10 m (Lemoine et al. 2001). If we consider that the average tracking arc length for a Mars spacecraft is normally two days (Lemoine

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Fig. 3 Ephemeris differences between MAGREAS and Geodyn-II. The top figure shows the orbital position and the bottom figure shows velocity. Position and velocity are both displayed in the R (Radial), T (Along Track), and N (Normal) directions

et al. 2001; Konopliv et al. 2011), or seven days at most (for MEX, Rosenblatt et al. 2008), the orbital difference between MAGREAS and Geodyn-II will be even much smaller. In Fig. 4, we present the difference between simulated observations generated by MAGREAS and Geodyn-II. We set the arc length to seven days (1 Oct, 2007–7 Oct, 2007). From Fig. 4, we can see that the simulated two-way range differences are less than 0.4 mm and the two-way range rate differences are less than 2 × 10−6 m/s. The difference is smaller than the current observation noise, and meets the requirement for further high precision POD and gravity field modeling (Budnik et al. 2005).

Further, we simulated a Mars spacecraft POD with MAGREAS and Geodyn-II using the two-way range-rate simulated observations. The full configuration setting used in this simulation is given in Table 1. A Gaussian noise with sigma of 1 mm/s was added to the simulated range-rate observations. We also added a 100 m bias along each direction of the spacecraft initial orbit. We solved the six initial orbital elements and a solar radiation coefficient. We compared the recovered parameters and found that the differences in the initial position were at the level of several centimeters, and nearly the same solar radiation coefficient. In Fig. 5, we show the reconstructed ephemeris after POD using MAGREAS and Geodyn-II. The differences in posi-


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Fig. 4 The simulated two-way range (top) and range-rate difference (bottom)

tion are less than 3 cm, and 0.03 mm/s for velocity. Such a small value indicates the consistency between simulated MAGREAS and Geodyn-II results.

4 MEX orbital data reduction The MEX tracking data used in this comparison include three-way and two-way Doppler measurements. From 7 August to 8 August, 2009 the Shanghai Astronomical Observatory set up a tracking campaign of MEX by using three VLBI tracking stations (Shanghai, Kunming, Urumqi). They received MEX signals in the X band; the transmitting station was New Norcia, Australia. The MEX three-way Doppler

measurement was an experiment to validate the new receiver device installed in the three VLBI stations. The original observable is a frequency with a five seconds sampling rate. We use Eq. (7) to transfer frequency to velocity: fR ·c (7) ρ˙ = 1 − Mfs in which fR is the received frequency, fS is the transmitted frequency, and M is transponder ratio (for the X band frequency the value is 880/749). The other kind of tracking data used was two-way Doppler data from the New Norcia antenna. The noise level of the two-way Doppler data from New Norcia varies with the Sun-Earth-Mars angle. For one-second data rate the noise is in the range of 0.05 to

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Fig. 5 The simulated POD reconstructed orbit differences (top figure for the position and bottom figure one for the velocity) between MAGREAS and Geodyn-II

0.37 mm/s, with an average value of 0.13 mm/s, and for 10 s data rate the average value of the noise is 0.051 mm/s (Pätzold et al. 2016). The force models include a Mars gravity field, N-body perturbations, solar radiation, atmospheric drag, and relativity perturbations, Mars solid tidal perturbation, and the perturbations of the two Mars moons (Table 2). Initial orbital elements were retrieved from a reconstructed precise ephemeris of MEX provided by ROB. The parameters to be solved for two-way Doppler data were six initial orbital elements, the solar radiation coefficient (Cr), and the scale factor of the atmospheric drag (Cd). As for three-way Doppler data, we additionally estimated bias to account for unknown exact reference frequencies or drifts between the two different stations.

Table 2 The configuration of MAGREAS in MEX POD Martian gravity field

MRO120D

N-body perturbation

Sun, planets, Phobos, and Deimos

Solar radiation

Fixed ratio of area to mass

Relativity perturbation

Schwarzschild (Sun only)

Mars solid tidal perturbation

k2 = 0.169

Atmosphere drag

Stewart 1987 atmospheric density model

Initial coordinate

Mars ICRF

Mars-centered coordinate

Pathfinder model with updated orientation parameters (Konopliv et al. 2016)

Tracking station coordination correction

Earth solid tide, ocean tide and polar tide correction

Earth tropospheric correction

Hopfield model


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Fig. 6 Residuals of two-way Doppler (the left is pre-fit RMS, the right is post-fit RMS)

Fig. 7 Residuals of three-way Doppler (pre-fit RMS on left, post-fit RMS on right)

Figure 6 and Fig. 7 give prefit and postfit residuals for the two-way and three-way Doppler measurement separately. Table 3 gives the RMS of the two-way and threeway Doppler measurement residuals and bias. From Fig. 6, Fig. 7, and Table 3 we can see that the three-way Doppler residual distribution before POD was close to the two-way Doppler distribution, indicating that the three-way Doppler measurements are reliable. After POD, we can see no trends in the postfit residuals. Table 3 shows that before POD, the RMS value of the two-way Doppler residuals is at the same level to the corresponding three-way Doppler residuals. In Table 3 the MEX POD for each tracking station

is shown separately; the RMS residuals for New NorciaKunming and New Norcia-Urumqi are close to the two-way Doppler. Therefore, the three-way Doppler in Kunming and Urumqi have similar accuracy to that of two-way Doppler. It indicates however, that during this measurement experiment, Kunming and Urumqi stations showed a better accuracy than the Shanghai station. If we combine the three-way Doppler measurement from three tracking stations the RMS drop to 0.079 mm/s. In Table 4, we give the POD parameters solution for the two-way and three-way Doppler measurements. This table shows that the solar radiation coefficient and the drag coeffi-

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Table 3 Statistical information of pre-fit and post-fit RMS (unit: mm/s) Prefit

Postfit

difference was larger than 600 m (Cao et al. 2010; Hu and Tang 2010).

Bias

Mean

RMS

Mean

RMS

New Norcia

0.065

0.156

0.000

0.067

New Norcia-Shanghai

0.165

0.190

0.000

0.104

5 Mars four-way tracking mode simulation

Two-way 0.005

Three-way New Norcia-Kunming

0.026

0.169

0.000

0.064

−0.139

New Norcia-Urumqi

0.052

0.147

0.000

0.059

−0.131

Combination

0.083

0.182

0.000

0.079

cient are stable, indicating the solar radiation model and the drag model in our software are reliable. After MEX POD, we can extrapolate the MEX ephemeris for five hours and then compare with the reconstructed precise ephemeris from ROB (with a claimed accuracy of 20 to 25 m). In Figs. 8 and 9, the ephemeris differences from the two-way Doppler and three-way Doppler are given in R, T and N directions separately. In Fig. 8 and Fig. 9, although we estimated bias to account for unknown exact reference frequencies or drifts in three-way data, the ephemeris differences from the threeway Doppler are much bigger than those from the two-way Doppler. This is because two-way Doppler is by essence a close-loop mode, and that three-way Doppler is an openloop mode. In contrast to two-way Doppler measurements, drifts or bias in the uplink frequency generation of three-way Doppler measurements are not cancelled by corresponding errors in the frequency standard of the Doppler measurement unit (Weischede et al. 1999). Figure 8 shows the difference is bigger in the T direction, and the maximum value is less than 8 m. In Fig. 9, we can see that the maximum difference in the T direction is less than 100 m. The smaller difference in the two-way Doppler is consistent with a better measurement accuracy (Table 3). From Fig. 8, we get an ephemeris difference within an accuracy of the ROB ephemeris, meaning that for this arc we were able to achieve an orbital accuracy similar to those of ROB. Because we apply a precise light time solution, precise force models, and an orientation model, we obtained a smaller difference in the T direction when compared with previous work, in which the maximum

We implemented a four-way tracking mode between Earth ground tracking station, Mars orbiter and Mars lander in MAGREAS. In Fig. 10, we present the geometry of this tracking mode. The ground tracking station on Earth sends a signal to the Mars orbiter at time Ti ; the Mars orbiter receives the signal at Sj and transponds the signal to the lander. The lander forwards the signal back to the orbiter at Lk , and the orbiter receives and sends the signal back to Earth at Sm . Finally, the ground tracking station receives signal at Tn . The whole computation process was carried out in the Barycentric Celestial Reference System (BCRS) and in the Barycentric Dynamic Time (TBD) and General relativity delay was considered. The range measurement model can be expressed as: R = (R1 + c · RLT nm ) + R2 + R3 + (R4 + c · RLT ij ) + c · TDB(i) − UTC(i) − c · TDB(n) − UTC(n) = c · UTC(n) − UTC(i)

(8)

where R1 , R2 , R3 and R4 represent the geometric distance shown in Fig. 10, X(Sm ) is the orbiter position state at time m, X(Sj ) represents the position state of the obiter at time j , and X(Tn ) describes the location of the Earth station at time i. Since the distance from orbiter to the lander is relatively close, the relativistic delay between the lander and the orbiter was not considered. The calculation of the range rate measurement is expressed in the integral Doppler form. In one Doppler integration period, the range rate measurement can be described as the rate of the distance change between the start time Ts and the end time Te . At time Ts and Te , Rs and Re are obtained by Eq. (8). The range rate can be expressed as: (Re − Rs ) R˙ = Te − Ts

(9)

In our simulation, we validate the contribution of the four-way tracking data for Mars orbiter positioning in comparison with traditional two-way tracking data, and we also

Table 4 The results of parameter solution X (km)

Y (km)

Z (km)

V x (m/s)

V y (m/s)

V z (m/s)

Cr

Cd

Initial value

−6620.000

9111.514

7910.211

449.235

−488.001

941.007

1.200

0.400

Two-way

−6620.000

9111.512

7910.214

449.235

−488.001

941.007

1.204

0.305

Three-way

−6620.035

9111.554

7910.138

449.225

−488.001

941.012

1.192

0.340


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Fig. 8 The Differences between precise orbit results from ROB and reconstructed orbits from two-way Doppler (the left is differences in position, the right is differences in velocity)

Fig. 9 The Differences between the precise orbit from ROB and reconstructed orbit of three-way Doppler (the left is differences in position, the right is differences in velocity)

validated the precision of the four-way tracking mode in Mars lander positioning. The force models and coordinate systems employed in the simulation are shown in Table 1. Table 5 gives the orbital elements close to the orbit of Chinese Mars mission. The initial epoch was set to November 1, 2021. Since the exact landing site of Chinese Mars mission has not yet been decided, we therefore chose a lander site located in the Amazonis Planitia (30°N, 165°W), due to its low altitude and flatness. The tracking station consists of three stations, Kashi (36.41°N, 76.13°E), Jiamusi (46.4°N 136.7°E) in China and Bajada del Agrio (BDA) in Argentina. The tracking stations at Kashi and BDA transmit and receive two-way signals; Jiamusi transmits and receives four-way signals. During the simulation, our aim was to investigate the contribution of the four-way tracking mode; we ignored the disturbance effect of the Mars atmosphere and ionosphere on radio signal. The effect can be estimated by systematic bias in actual data processing. There were seven arcs in this simulation; each arc length was one earth day, and the time span was from November 1 to November 8 2021. We generated the true orbit by integrating the motion equation with the initial orbit elements as shown in Table 5. The data-sampling rate was set to five seconds. The two-way range rate and four-way tracking data were obtained and Gauss white noise was added with 0.1 mm/s as actual observations. We randomly added 100 m of deviation to the initial orbit position. For the Mars lander, 100 m of deviation was added to three directions, X, Y , and Z in the Mars body-fixed coordinate. These simulated observations were used for precise orbit determination of the Mars spacecraft and precise positioning of the Mars lander. The results are presented in Table 6 and Table 7.

Fig. 10 A schema of the Mars four-way tracking mode

From Table 6 we can see that after we included the fourway Doppler data, the accuracy of the orbit determination was significantly improved as compared with the results obtained when only two-way Doppler was used. In general, after we included the four-way Doppler data, the accuracy of the Mars spacecraft orbit determination improved about one order of magnitude. In Table 7, we show the results of the Mars lander positioning. It can be seen that by using the four-way tracking data of seven arcs, the accuracy of the lander positioning can be solved to the centimeter level after the iterations were convergent. In this simulation, we ignored the effect of the Mars atmosphere and ionosphere on the four-way Doppler link, however, this simulation still indicated the potential contribution of this link for precise positioning of the Mars lander. Figure 11 presents the orbital differences of two arcs, 2021-11-01 and 2021-11-06, to illustrate the improvements gained from four-way Doppler data. The orbital differences

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Table 5 The orbital elements used in the simulation Semi-major axis (m)

Eccentricity

Inclination (degree)

RAAN (degree)

Argument of perigee (degree)

Mean anomaly (degree)

7,617,819

0.495

15.666

119.478

323.181

87.152

Table 6 Initial orbit position difference between solution and true one Arc No. YYYY-MM-DD

Two-way range rate only

Two-way range rate and four-way range rate

x (m)

y (m)

z (m)

Errors (m)

x (m)

y (m)

z (m)

2021-11-01

0.219

−0.338

−0.170

0.437

−0.015

0.010

0.014

0.023

2021-11-02

−0.684

0.348

2.085

2.221

−0.078

0.041

0.220

0.237

2021-11-03

−0.012

−0.023

0.169

0.171

−0.015

−0.028

0.128

0.132

2021-11-04

−0.118

0.788

−1.364

1.579

−0.001

0.055

−0.107

0.120

2021-11-05

−0.566

0.508

0.999

1.255

−0.104

0.091

0.182

0.229

2021-11-06

0.525

0.052

−2.233

2.294

−0.070

−0.010

0.252

0.262

2021-11-07

0.171

−0.332

0.180

0.414

0.014

−0.025

0.009

0.030

Table 7 The process of lander position calculation and estimating lander position errors Number of iterations

x (m)

y (m)

z (m)

1

−49.683

−49.691

49.645

2

−24.784

−24.819

24.721

3

−12.355

−12.387

12.244

4

−6.154

−6.237

6.089

5

−3.229

−3.251

3.313

6

−1.734

−1.725

1.827

7

−0.937

−0.914

1.012

8

−0.515

−0.494

0.549

9

−0.281

−0.258

0.295

10

−0.158

−0.139

0.148

11

−0.085

−0.0654

0.076

12

−0.044

−0.037

0.037

13

−0.021

−0.017

0.019

14

−0.010

−0.006

0.009

15

−0.005

−0.004

0.005

16

−0.003

−0.004

0.002

Total

−99.998

−100.048

99.991

Errors

−0.002

0.048

0.009

were the difference between the true orbit and reconstructed orbit from the two-way Doppler data only, and the differences from the two-way and four-way Doppler data together. The left row shows the results from the two-way Doppler data, and the right row is the results from the two-way and four-way Doppler data together. Differences were projected in three directions (R, radial; T, tangential and N, normal). These results reveal that the difference in directions T and N

Errors (m)

were reduced by one order of magnitude after the four-way Doppler data were combined. Two-way Doppler data is only sensitive to line-of-sight direction, while four-way data has a more rigorous geometric configuration. The link line in the inertial space between the Lander and Orbiter is at an angle to the orbit plane. Due to this intersection geometry, the extra link of four-way lander–orbiter tracking mode could have an additional constraint in the tangential and normal directions, thus making this mode a complement to the traditional two-way tracking mode. In general, our simulation of the four-way tracking mode indicates this mode has the potential to improve orbital accuracy when compared with traditional two-way tracking mode. The four-way lander–orbiter tracking mode can also help to constrain the Mars lander position. In consideration of this ability, the four-way link can be also employed when calculating the Mars orientation parameter solution as in the proposed ExoMars mission (Dehant et al. 2009).

6 Conclusion A comparison of MAGREAS and Geodyn-II after initial processing of MEX two-way and three-way Doppler tracking data showed that the MAGREAS software developed at Wuhan University achieved reliable initial results. The orbital ephemeris difference between the MAGREAS and Geodyn-II results was less than 1.2 mm and the velocity difference was less than 1.5 × 10−3 mm/s for an arc length with one month. In two-way range and range rate measurements, the differences were less than 0.4 mm and 2 × 10−6 m/s, respectively. In the simulated POD the differences between the reconstructed orbits were less than 3 cm


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Fig. 11 The Differences between true orbit and reconstructed orbits from two-way Doppler and four-way Doppler (the left is differences using two-way Doppler, the right shows differences using two-way Doppler and four-way Doppler; the top is the result of arc 2021-11-01, the bottom is the result of arc 2021-11-06)

for the position and less than 0.03 mm/s for velocity. Such small differences between MAGREAS in orbital prediction, measurement, and simulated POD and results from GeodynII validate the accuracy and reliability of MAGREAS software, and indicate that it can be applied to POD for Mars spacecraft. In order to test the accuracy of MAGREAS, we processed the MEX two-way and three-way Doppler tracking data compare with the reconstructed orbit provided by ROB. We obtained similar residual levels as those from ROB, and found that the accuracy of two-way Doppler tracking was better than three-way Doppler tracking. Besides, the results showed that the receivers in Kunming and Urumqi performed better than that the one in Shanghai. The maximum postfit orbital difference with ROB by using the threeway Doppler was less than 100 m and better than previously published results. This was possible because we employed better force models and a more accurate observation model. The maximum difference for the two-way Doppler was less than 10 m, indicating that for this arc we achieved a similar POD accuracy as the ROB group. Furthermore, we implemented a four-way tracking mode for Mars orbiter and lander. Our simulation of this fourway tracking mode indicated it had the potential to improve orbital accuracy when compared with traditional two-way tracking mode. For the seven arcs we considered, the improvement of orbit determination accuracy was about one order of magnitude. For the Mars lander position solution obtained from the seven arcs, the accuracy of the lander position reached the centimeter level. This four-way Doppler link could be considered for the Chinese Mars mission.

Acknowledgements We appreciate Dr. Huang Yong from Shanghai Astronomical Observatory to provide the MEX three-way Doppler data. Dr. Huang Yong, Dr. Jian Nianchuan, Dr. Hu Songjie, Dr. Chen Ming and Dr. Cao Jianfeng are acknowledged for their instruction in software development and data process. We are grateful to NASA and ESA to provide models and data to make this research possible. Royal Observatory of Belgium kindly provides the precise ephemeris of MEX used in this work. Our deepest gratitude goes to the anonymous reviewers for their careful work and thoughful suggestions that have helped improve this paper substantially. The work is supported by National Scientific Foundation of China (41374024, 41404021, 41174019) and Innovation Group of Natural Fund of Hubei Province (2015CFA011). This work is also supported by open funding of Astro Dynamic Laboratory (No. 2016ADL-DW0103) and the Open Research Fund of Key Laboratory of Space Object Measurement. The Geodyn-II is authorized by GSFC/NASA and we run it in workstation in Shanghai Astronomical Observatory.

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