# Proposed Orbital Products for Positioning Using Mega-Constellation LEO Satellites

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Processing Procedures

#### 2.1. Level A Products

- The GNSS data collected onboard the LEO satellites are downlinked to the GMSs with a time interval ∆T between subsequent downloads, and then transferred to the MPC.
- High-accuracy reduced-dynamic orbits are then processed with comprehensive dynamic models (as will be described in Section 2.1.2), having the Keplerian elements at the initial condition and certain dynamic parameters estimated to compensate for the deficiencies in the dynamic models.
- The orbits are then predicted for several hours into the future with numerical integration, which will be discussed in Section 2.1.1. The prediction interval should cover at least a period of ∆T + ∆t
_{p}, where ∆t_{p}is the time needed for the data downloading, processing and uploading during the next GMS–LEO contact. - Next, LEO-specific ephemeris parameters (Section 2.1.3) are estimated using the least-squares adjustment to describe the predicted orbits within a pre-defined fitting interval ∆t
_{F}(Section 2.1.3), where ∆t_{F}is normally selected much shorter than the prediction interval. - The fitted ephemeris parameters are updated with an interval of ∆t
_{U}< ∆t_{F}, so that overlapping time exists between two subsequent sets of ephemeris parameters. - All the estimated ephemeris parameters are then uplinked to the LEO satellites, and the LEO satellites broadcast their ephemeris parameters to the users.

#### 2.1.1. Prediction Interval

#### 2.1.2. Orbit Estimation and Prediction

#### 2.1.3. Ephemeris Parameters

#### 2.1.4. Alternative Approaches

- Upload the initial conditions and the estimated dynamic parameters with the numerical integration performed onboard, or directly uplink the predicted orbits to the LEO satellite [12].
- With enough computational power onboard the LEO satellites, it is also possible to make use of the GNSS broadcast ephemeris and observations, directly compute the real-time LEO orbits onboard, and extrapolate them for a short time in the future—i.e., tens of seconds [27].
- Make use of the GNSS broadcast ephemeris and observations, directly compute the SPP solutions onboard in real time, and broadcast the epoch-wise positions to the users.

- No heavy burden on the LEO onboard computational power, where fast processing is carried out using high-performance computers in the MPC.
- Comprehensive dynamic models can be used for the POD.
- High-accuracy real-time GNSS products can be continuously obtained via Internet links.
- Only a limited amount of parameters (for each LEO satellite) are transferred during the uploading process.
- The navigation information can be downlinked to users with a relatively low sampling rate—e.g., 10 min.
- The LEO orbits derived from the broadcast ephemeris are smooth, which is suitable for the polynomial fitting of the precise Level B orbits (see Section 2.2).

- Multiple GMSs might be needed to guarantee the upload intervals of several hours.
- With the rapidly increasing number of the LEO satellites, a heavy burden is put on the downlink and uplink systems at the GMSs.
- A high-grade GNSS receiver is required onboard the LEO satellite.

#### 2.2. Level B Products

#### Alternative Approaches

- No heavy burden on the LEO onboard computational power; comprehensive dynamic models can be used for the POD; fast processing with high-performance computers.
- High-accuracy real-time GNSS products can be continuously obtained with Internet links free of charge.

## 3. Test Results

#### 3.1. Level A Products

#### 3.2. Level B Products

^{−1}s (see Figure 11) with a sampling interval of 30 s (see Figure 14). Within the 60 s polynomial fitting interval, one consistent set of the ephemeris parameters was used for the Level A orbits.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Ascending time of GRACE FO-1 for the day 14 August 2018, (

**a**) from 0:00 to 1:30 in GPS time (GPST) and (

**b**) from 6:00 to 7:30 in GPST. The almost equal colors around the North and the South Poles are caused by the spherical shape of the Earth and the form of the map.

**Figure 3.**(

**a**) The mean and (

**b**) the maximum time gaps between subsequent satellite visible periods for GRACE FO-1 on 14 August 2018. The almost equal colors around the North and the South Poles are caused by the spherical shape of the Earth and the form of the map.

**Figure 4.**Example of the ground-based ground monitoring stations (GMSs) distributed in latitudes within $\pm {60}^{\xb0}$, so that a maximum observation time gap not exceeding 1 h can be ganranteed for GRACE FO-1 on 14 August 2018. Note that the GMS located around ${70}^{\xb0}$ E and ${50}^{\xb0}$ S is above an island in the Indian Ocean.

**Figure 5.**Relationships between the estimation, the prediction intervals of the orbits, the fitting and the update intervals ($\mathsf{\Delta}{t}_{U}$) of the ephemeris parameters.

**Figure 7.**Areas observing GRACE FO-1 at (

**a**) 4:15:00 and (

**b**) 0:10:00 on 14 August 2018, with an elevation mask angle of 5°.

**Figure 8.**(

**a**) The orbital user range error (OURE) and (

**b**) the 3D root mean square error (RMSE) of the predicted orbits applying different dynamic models (Table 2).

**Figure 9.**(

**a**) RMSE of the predicted orbits in the radial, along-track and cross-track directions and (

**b**) the 50%, 70% and 90% percentile lines of the 3D prediction errors. The dynamic model used applies stochastic velocity pulses set up for every 2 h (Table 2).

**Figure 10.**The along-track errors of the predicted and the ephemeris orbits from 23:00 on 15 August 2018, to 5:00 on 16 August 2018.

**Figure 11.**The OURE and the 3D RMSE of the ephemeris fitting errors within a 20-min fitting interval using 20 ephemeris parameters. The dashed lines mark the time interval in which the ephemeris is used by the users.

**Figure 12.**(

**a**) The OURE and (

**b**) the 3D RMSE of the prediction errors, the fitting errors and the total errors of the Level A orbital products.

**Figure 14.**Test strategy for the low-order polynomial corrections merging the Level A and Level B orbits.

**Figure 15.**The OUREs of the polynomial fitting errors when the beginning of the fitting interval corresponds to (

**a**) different prediction times and (

**b**) different ephemeris fitting times of the Level A products.

**Figure 16.**The OURE of (

**a**) the polynomial fitting errors and (

**b**) the total Level B orbital errors concerning the polynomial fitting time.

**Table 1.**Minumum number of the land-based GMSs required to guarantee an observation gap not exceeding $\mathsf{\Delta}{T}_{gap}$. The data of GRACE FO-1 on 14 August 2018, were used for the calculation.

Latitude (B) Range | $\mathit{\Delta}{\mathit{T}}_{\mathit{g}\mathit{a}\mathit{p}}=1\mathbf{h}$ | $\mathit{\Delta}{\mathit{T}}_{\mathit{g}\mathit{a}\mathit{p}}=2\mathbf{h}$ | $\mathit{\Delta}{\mathit{T}}_{\mathit{g}\mathit{a}\mathit{p}}=3\mathbf{h}$ | $\mathit{\Delta}{\mathit{T}}_{\mathit{g}\mathit{a}\mathit{p}}=6\mathbf{h}$ |
---|---|---|---|---|

$\left|B\right|\le 60\xb0$ | 8 | 5 | 2 | 2 |

$\left|B\right|\le 70\xb0$ | 5 | 2 | 2 | 1 |

$\left|B\right|\le 80\xb0$ | 2 | 1 | 1 | 1 |

All | 2 | 1 | 1 | 1 |

Category | Parameters | Estimation Interval | |
---|---|---|---|

Keplerian elements | $a$, $e$, $I$, $\mathsf{\Omega}$, $\omega $, ${u}_{0}$ | 24 h | |

Dynamic parameters | Option A | ${R}_{0}$, ${S}_{0}$, ${W}_{0}$ $\delta \mathit{a}$ | 24 h 6/15/30 min |

Option B | ${R}_{0}$, ${S}_{0}$, ${W}_{0}$, ${R}_{c}$, ${R}_{s}$, ${S}_{c}$, ${S}_{s}$, ${W}_{c}$, ${W}_{s}$ $\delta \mathit{v}$ | 24 h 15/30 min, 1/2/3/4/6/12/24 h |

**Table 3.**Processing parameters and the dynamic models used in the processing. The following abbreviations are used: CNES for the National Centre for Space Studies in France, JPL for the Jet Propulsion Laboratory in the USA, and IERS for the International Earth Rotation and Reference Systems Service.

Parameters/Models | Details |
---|---|

Observations | GPS IF combination (L1/L2), code + phase |

Sampling interval | Observations: 30 s; Prediction: 1 s |

Estimation interval | 24 h |

Prediction interval | 6 h |

Elevation mask | 5° |

GPS orbits/clocks | CNES real-time products [20] |

Dynamic models | Earth gravity terms: EGM2008 (degree: 120) [21] |

Gravity terms of other planets: JPL DE405 [22] | |

Solid Earth tides, Pole tides: IERS 2010 [23] | |

Ocean tides: FES2004 [24] | |

General relativistic effects |

Category | Ephemeris Parameters |
---|---|

GPS LNAV ephemeris parameters | ${t}_{oe}$, $\sqrt{{a}_{0}}$, $\mathsf{\Delta}n$, ${\mathsf{\Omega}}_{0}$, ${I}_{0}$, $\dot{I}$, $$, ${C}_{us}$, ${C}_{uc}$, ${C}_{rs}$, ${C}_{rc}$, ${C}_{is}$, ${C}_{ic}$ |

Transformed GPS LNAV ephemeris parameters | ${e}_{x}$, ${e}_{y}$, ${\lambda}_{0}$ |

Additional ephemeris parameters | $\dot{a}$, $\dot{n}$, ${C}_{rs3}$, ${C}_{rc3}$ |

Parameters/Models | Details |
---|---|

Strategy of orbit estimation | Reduced-dynamic orbits (Section 2.1.2) |

Sampling interval of the observations | 30 s |

Strategy of orbit prediction | Dynamic orbits (Table 2 and Table 3) |

Prediction sampling interval | 1 s |

Prediction interval | 60 s |

Polynomial degree | 1, 2, 3 |

**Table 6.**The prediction error budget for different prediction intervals when using the dynamic model having stochastic velocity pulses set up for every 2 h (Table 2).

Prediction Interval | RMSE Radial (m) | RMSE Along-Track (m) | RMSE Cross-Track (m) | 3D RMSE (m) | OURE (m) |
---|---|---|---|---|---|

0.5 h | 0.035 | 0.131 | 0.028 | 0.138 | 0.086 |

1 h | 0.042 | 0.156 | 0.033 | 0.165 | 0.102 |

2 h | 0.052 | 0.359 | 0.039 | 0.365 | 0.228 |

3 h | 0.070 | 0.560 | 0.047 | 0.566 | 0.355 |

4 h | 0.072 | 0.829 | 0.055 | 0.834 | 0.523 |

5 h | 0.083 | 1.203 | 0.064 | 1.207 | 0.758 |

6 h | 0.105 | 1.599 | 0.082 | 1.605 | 1.008 |

Prediction Interval | OURE (m) | 3D RMSE (m) | ||||
---|---|---|---|---|---|---|

Prediction | Fitting | Total | Prediction | Fitting | Total | |

0.5 h | 0.086 | 0.059 | 0.101 | 0.138 | 0.111 | 0.173 |

1 h | 0.102 | 0.058 | 0.118 | 0.165 | 0.112 | 0.200 |

2 h | 0.228 | 0.059 | 0.236 | 0.365 | 0.113 | 0.383 |

3 h | 0.355 | 0.059 | 0.360 | 0.566 | 0.102 | 0.579 |

4 h | 0.523 | 0.059 | 0.527 | 0.834 | 0.114 | 0.842 |

5 h | 0.758 | 0.059 | 0.760 | 1.207 | 0.114 | 1.213 |

6 h | 1.008 | 0.060 | 1.010 | 1.605 | 0.114 | 1.609 |

**Table 8.**Total error budget of the Level B orbital products. The results at the prediction time of 1 and 60 s are separated by “/”.

Fitting Type | Prediction Errors [cm] | Fitting Errors [cm] | Total Errors [cm] | |||
---|---|---|---|---|---|---|

OURE | 3D RMSE | OURE | 3D RMSE | OURE | 3D RMSE | |

Linear polynomial | 2.4/2.5 | 4.3/4.4 | 0.6/0.5 | 1.0/0.9 | 2.52/2.59 | 4.43/4.56 |

Quadratic polynomial | 0.1/0.1 | 0.2/0.2 | 2.45/2.53 | 4.29/4.44 | ||

Cubic polynomial | 0.1/0.1 | 0.2/0.2 | 2.45/2.54 | 4.28/4.45 |

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Wang, K.; El-Mowafy, A.
Proposed Orbital Products for Positioning Using Mega-Constellation LEO Satellites. *Sensors* **2020**, *20*, 5806.
https://doi.org/10.3390/s20205806

**AMA Style**

Wang K, El-Mowafy A.
Proposed Orbital Products for Positioning Using Mega-Constellation LEO Satellites. *Sensors*. 2020; 20(20):5806.
https://doi.org/10.3390/s20205806

**Chicago/Turabian Style**

Wang, Kan, and Ahmed El-Mowafy.
2020. "Proposed Orbital Products for Positioning Using Mega-Constellation LEO Satellites" *Sensors* 20, no. 20: 5806.
https://doi.org/10.3390/s20205806