# Dual-Satellite Lunar Global Navigation System Using Multi-Epoch Double-Differenced Pseudorange Observations

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## Abstract

**:**

## 1. Introduction

## 2. Algorithm

#### 2.1. Multi-Epoch Double-Differenced Pseudorange Observations (MDPO) Algorithm

#### 2.2. Two-Dimentional MDPO Algorithm Using a Pre-Known User Altitude

#### 2.3. Other Systematic Errors

#### 2.3.1. Satellite Orbit Determination Error

#### 2.3.2. Time Tag Error

#### 2.3.3. DEM Information Error

#### 2.3.4. Other System Errors

#### 2.4. Design Parameters

## 3. Simulation

#### 3.1. Simulation Overview

#### 3.2. Simulation Results

#### 3.3. Discussions

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Availability and Total GDOP under different orbital conditions: circular orbits with four different satellite altitudes (300 km (

**a**), 600 km (

**b**), 900 km (

**c**), 2100 km (

**d**)) and five different orbital phase differences $\Delta \Omega $ between two satellites (5 deg, 15 deg, 25 deg, 35 deg). Rover/lander position were fixed to the south-pole (−90 deg, 90 deg) and satellite orbital inclination was fixed to 110 deg.

**Figure 4.**Example of simulation result: (

**Red dots**) true rover trajectory; (

**Blue dots**) rover positions estimated by MDPO (receiver observation errors $\Delta \nabla \mathsf{\omega}$ ($\mathbf{2}\times {\sigma}_{\Delta \nabla \mathsf{\omega}}$ ) = 0.4 m).

**Figure 5.**Example of position error distributions (receiver observation errors $\Delta \nabla \mathsf{\omega}$ ($\mathbf{2}\times {\sigma}_{\Delta \nabla \mathsf{\omega}}^{}$ ) = 0.4 m).

**Figure 6.**GDOP history over the course of the simulation period. (

**a**) for 1000 min (closer look), (

**b**) for 15,000 min (overall). Unless both satellites are in view, DOP is not calculated and not shown in the figures.

Method | User (Rover) Segment Burden | Space (Satellite) Segment Burden | Ground Segment Burden |
---|---|---|---|

Visual Sensor-based Navigation | Visual Sensor-based navigation does not work when the lunar surface is flat with no landmarks. | - | - |

Accelerometers and Star Tracker Navigation | Sensor alignment precision becomes outrageous to achieve high position accuracy. | - | - |

Lunar Global Navigation Satellite Systems using TOA | Use a passive ranging receiver. | At least four satellites in view with a stable satellite clock are required. | Frequent satellite clock bias estimation by the ground segment is required. |

Single Satellite AOA Navigation | Use a passive ranging receiver. User position accuracy is very sensitive to AOA error. | Single satellite in view with a stable satellite clock is required. | Frequent satellite clock bias estimation by the ground segment is required. |

Dual Satellite TDOA/FDOA Navigation | Use a passive ranging and/or Doppler receiver. | Two satellites in view with a stable satellite clock and/or frequency are required. | Frequent satellite clock bias estimation by the ground segment is required. |

Law of Cosines | Use a passive Doppler receiver with a static reference station. The frequency of the receiver must be stable. | Single satellite in view is required, with no need for a stable satellite frequency. | No need for frequent satellite clock bias estimation by the ground segment. |

Joint Doppler and Ranging (single satellite case) | Use a passive ranging and Doppler receiver with a static reference station. The clock and frequency of the receiver must be stable or must be compensated by two-way ranging. | Single satellite in view is required, with no need for a stable satellite clock. | No need for frequent satellite clock bias estimation by the ground segment. |

Two-way Ranging based Navigation | Active ranging between the satellite and user is required. | Two satellites in view are required, with no need for a stable satellite clock. | No need for frequent satellite clock bias estimation by the ground segment. |

Dual Satellite MDPO Navigation (This research) | Use a passive ranging receiver with a static reference station. | Two satellites in view are required, with no need for a stable satellite clock. | No need for frequent satellite clock bias estimation by the ground segment. |

Items | Value | Unit | Remarks |
---|---|---|---|

Simulation Period | 15,000 | min | Approximately two weeks in Earth time |

Range measurement resolution of the user pseudorange receivers | 0.4 | m | Minimum observable range by the rover and lander receivers |

Latitude of Initial Rover/Lander Position | $-$90 | deg | |

Longitude of Initial Rover/Lander Position | 90 | deg | |

Interval of pseudorange observations | 0.5 | min | Total observation period of one MDPO estimation is equivalent to 1 min when the number of multi-epoch observations is 2. |

Rover traveling distance between MDPO observations | 3.75 | m | The rover travels at 7.5 m/min for 0.5 min between MDPO estimations |

Rover traveling direction | Random | deg | Heading direction is selected from three values (+$\frac{\pi}{3},-\frac{\pi}{3},0$) randomly. |

Items | Value | Unit | Remarks |
---|---|---|---|

Initial Orbital Parameters of Satellite1 | |||

Perilune altitude | 300 | km | |

Apolune altitude | 300 | km | |

Inclination | 110 | deg | |

Right Ascension of the Ascending Node | 0 | deg | |

Argument of Perigee | 0 | deg | |

True Anomaly | 0 | deg | |

Initial Orbital Parameters of Satellite2 | |||

Perilune altitude | 300 | km | |

Apolune altitude | 300 | km | |

Inclination | 110 | deg | |

Right Ascension of the Ascending Node | 0 | deg | |

Argument of Perigee | 0 | deg | |

True Anomaly | $-$15 | deg |

Items | Type | Value | Unit | Remarks |
---|---|---|---|---|

Satellite Orbit Determination Error in the Along direction | $\Delta Alon{g}_{}\left({t}_{i}\right)$= ${\omega}_{OD-Along}\left({t}_{i}\right)+{c}_{OD-Along}$ | |||

White Gaussian noise ${\omega}_{OD-Along}$ | 100.0 | m | ${\omega}_{ODt}=Value\times $a random scalar drawn from the standard normal distribution. | |

$\mathrm{Bias}\text{}\mathrm{noise}\text{}{c}_{OD-Along}$ | 200.0 | m | Bias ${c}_{OD}$ is a random number that is greater than or equal to $-Value$ and less than $Value$ | |

Satellite Orbit Determination Error in the Radial direction | $\Delta Radia{l}_{}\left({t}_{i}\right)$= ${\omega}_{OD-Radial}\left({t}_{i}\right)+{c}_{OD-Radial}$ | |||

White Gaussian noise ${\omega}_{OD-Radial}$ | 10.0 | m | Same as above | |

$\mathrm{Bias}\text{}\mathrm{noise}\text{}{c}_{OD-Radial}$ | 20.0 | m | ||

Satellite Orbit Determination Error in the Cross direction | $\Delta Cros{s}_{}\left({t}_{i}\right)$= ${\omega}_{OD-Cross}\left({t}_{i}\right)+{c}_{OD-Cross}$ | |||

White Gaussian noise ${\omega}_{OD-Cross}$ | 100.0 | m | Same as above | |

$\mathrm{Bias}\text{}\mathrm{noise}\text{}{c}_{OD-Cross}$ | 200.0 | m |

Item | Type | Value | Unit | Remarks |
---|---|---|---|---|

Time Tag Error | $d{\tau}_{R}^{}\left({t}_{i}\right)={\omega}_{timetag}+{x}_{timetag}$ | |||

White Gaussian noise ${\omega}_{timetag}$ | 100.0 | ms | ${\omega}_{timetagt}=Value\times $a random scalar drawn from the standard normal distribution. | |

Random walk ${x}_{timetag}$ | 0.1 | ms/min | A random walk is a time series model ${x}_{timetagt}$ such that ${x}_{timetagt}={x}_{timetagt-1}+{\omega}_{t}$ where ${\omega}_{t}$ is a discrete white noise series. Random walk noise is reset to zero periodically assuming orbit determination takes place every orbital period. |

Item | Type | Value | Unit | Remarks |
---|---|---|---|---|

DEM Error | $\Delta {z}_{R}^{}={\omega}_{DEM}$+ ${c}_{DEM}$ | |||

White Gaussian noise ${\omega}_{DEM}$ | 10.0 | m | ${\omega}_{DEMt}=Value\times $a random scalar drawn from the standard normal distribution. | |

Bias noise ${c}_{DEM}$ | 5.0 | m | Bias ${c}_{DEM}$ is a random number that is greater than or equal to $-Value$ and less than $Value$ |

$\mathbf{Receiver}\text{}\mathbf{Observation}\text{}\mathbf{Errors}\text{}\Delta \nabla \mathsf{\omega}$$(2\times {\mathit{\sigma}}_{\Delta \nabla \mathsf{\omega}}^{})\text{}\left[\mathbf{m}\right]$ | Total GDOP | Total UPE (2drms) [m] |
---|---|---|

0.4 | 44.3 | 45.6 |

0.8 | 44.3 | 55.4 |

1.6 | 44.3 | 89.6 |

3.2 | 44.3 | 172.6 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tanaka, T.; Ebinuma, T.; Nakasuka, S.
Dual-Satellite Lunar Global Navigation System Using Multi-Epoch Double-Differenced Pseudorange Observations. *Aerospace* **2020**, *7*, 122.
https://doi.org/10.3390/aerospace7090122

**AMA Style**

Tanaka T, Ebinuma T, Nakasuka S.
Dual-Satellite Lunar Global Navigation System Using Multi-Epoch Double-Differenced Pseudorange Observations. *Aerospace*. 2020; 7(9):122.
https://doi.org/10.3390/aerospace7090122

**Chicago/Turabian Style**

Tanaka, Toshiki, Takuji Ebinuma, and Shinichi Nakasuka.
2020. "Dual-Satellite Lunar Global Navigation System Using Multi-Epoch Double-Differenced Pseudorange Observations" *Aerospace* 7, no. 9: 122.
https://doi.org/10.3390/aerospace7090122