# A Comparative Analysis of Multi-Epoch Double-Differenced Pseudorange Observation and Other Dual-Satellite Lunar Global Navigation Systems

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## Abstract

**:**

## 1. Introduction

## 2. Assumptions

- ●
- The bias of the satellite clock is not ignorable due to the limited capacity of micro-satellites;
- ●
- The bias of the rover clock is not ignorable due to the limited capacity of micro-rovers;
- ●
- The satellite orbit determination error is not ignorable due to the limitation of the availability and number of Earth ground stations.

## 3. Mathematical Models of the Three Navigation Methods

#### 3.1. Pseudorange Observation

#### 3.2. Pseudodoppler Observation

#### 3.3. Multi-Epoch Double-Differenced Pseudorange Observation

#### 3.4. Double-Differenced TOA–FOA

#### 3.5. Single-Differenced Two-Way Ranging

#### 3.6. Systematic Errors

#### 3.6.1. Satellite Orbit Determination Error

#### 3.6.2. Time Tag Error

#### 3.6.3. Signal Processing Delay Time Uncertainty

#### 3.6.4. DEM Information Error

#### 3.6.5. Other Systematic Errors

#### 3.7. Design Parameters

#### 3.7.1. DOP and Availability

#### 3.7.2. Satellite Orbital Parameters and Systematic Errors Related to DEM

## 4. Comparative Analysis of Three Navigation Methods

#### 4.1. Overview of Numerical Simulation

#### 4.2. Other User-Set Conditions

#### 4.3. Numerical Simulation Result

#### 4.4. Numerical Simulation Result with Increased Reveiver Observation Noise

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 6.**The simulated rover trajectories and user position errors. (

**a**,

**b**) correspond to the MDPO, (

**c**,

**d**) correspond to the double-differenced TOA–FOA, (

**e**,

**f**) correspond to the single-differenced two-way ranging.

Method | Observation Type | Number of Supported Users | Observation Time | Navigation Accuracy |
---|---|---|---|---|

Multi-epoch double-differenced pseudorange observation | Passive ranging | Multi-user | Using observations from at least two epochs ^{2} | 50 m under the condition used in Section 4.3. |

Double-differenced time of arrival (TOA)-frequency of arrival (FOA) | Passive ranging and Doppler | Multi-user | Using observations from a single epoch ^{3} | 100 m under the condition used in Section 4.3. |

Single-differenced two-way ranging | Active ranging | Single-user at a time ^{1} | Using observations from a single epoch ^{3} | 30 m under the condition used in Section 4.3. |

^{1}The second user needs another set of radio signals separately.

^{2}For example, 1 min (0.5 min × 2 epochs) such as set in this research.

^{3}For example, 0.5 min (0.5 min × 1 epoch) such as set in this research.

**Table 2.**The Total GDOP and availability comparison among three navigation methods under different orbital conditions: two satellites are placed in 300 km circular low lunar orbit with different phase differences (5, 10, 15, and 25 deg). The rover/lander position were fixed to the south pole ($-90$ deg latitude), and the satellite orbit inclination was fixed to 110 deg.

Navigation Methods | Phase Difference [deg] | Total GDOP | Availability [%] |
---|---|---|---|

MDPO | 5 | 217.8 | 4.1 |

10 | 63.1 | 3.2 | |

15 | 38.5 | 2.3 | |

25 | 34.3 | 1.4 | |

Double-differenced TOA–FOA | 5 | 891.9 | 6.2 |

10 | 253.5 | 4.7 | |

15 | 159.5 | 3.3 | |

25 | 144.6 | 2.1 | |

Single-differenced Two-way Ranging | 5 | 8.1 | 6.2 |

10 | 2.5 | 4.7 | |

15 | 1.6 | 3.3 | |

25 | 1.4 | 2.1 |

**Table 3.**The Total GDOP and availability comparison among three navigation methods under different orbital conditions: two satellites were placed in 2100 km circular low lunar orbit with different phase differences (5, 10, 15, and 25 deg). The rover/lander positions were fixed to the south pole ($-90$ deg latitude), and the satellite orbit inclination was fixed to 110 deg.

Navigation Methods | Phase Difference [deg] | Total GDOP | Availability [%] |
---|---|---|---|

MDPO | 5 | 1336.6 | 10.9 |

10 | 447.8 | 10.0 | |

15 | 274.2 | 9.0 | |

25 | 203.6 | 8.1 | |

Double-differenced TOA–FOA | 5 | 5669.0 | 16.3 |

10 | 1899.0 | 15.0 | |

15 | 1162.9 | 13.4 | |

25 | 863.5 | 12.1 | |

Single-differenced Two-way Ranging | 5 | 21.6 | 16.3 |

10 | 6.7 | 15.0 | |

15 | 3.8 | 13.4 | |

25 | 2.7 | 12.1 |

**Table 4.**One example of correlation between the satellite altitude and Total UPE: two satellites were placed in different circular LLO, 300, 600, 900, and 2100 km. The simulation takes an average of 100 simulation cases for each.

Navigation Methods | Satellite Altitude (km) | Total UPE (2drms) (m) |
---|---|---|

MDPO | 300 | 57.9 |

600 | 121.4 | |

900 | 197.3 | |

2100 | 1038.3 | |

Double-differenced TOA–FOA | 300 | 119.8 |

600 | 245.7 | |

900 | 454.3 | |

2100 | 2049.6 | |

Single-differenced Two-way Ranging | 300 | 27.0 |

600 | 40.1 | |

900 | 56.3 | |

2100 | 217.8 |

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 resolution by the rover and lander receivers. |

Doppler measurement resolution of the user pseudodoppler receivers | 0.2 | Hz | Minimum observable resolution by the rover and lander receivers. |

Latitude of initial rover/lander position | $-$90 | deg | |

Interval of pseudorange/doppler observations | 0.5 | min | |

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

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

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

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

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

Systematic error ${c}_{OD-Along}$ | 200.0 | m | Systematic error ${c}_{OD}$ is an output of the sinusoidal function $A\times sin\left(2\pi x/T\right)$: the argument $x$ is epoch time, the period $T$ was set equal to the satellite orbital period, and the amplitude $A$ is randomly selected between $-Value$ and $Value$ at the beginning of each simulation. | |

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

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

Systematic error ${c}_{OD-Radial}$ | 20.0 | m | ||

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

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

Systematic error ${c}_{OD-Cross}$ | 200.0 | m |

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

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

Offset error ${c}_{timetag}$ | 1.0 | ms | Offset error ${c}_{timetag}$ is randomly selected between $-Value$ and $Value$ after the time synchronization and fixed until the next time synchronization. The time synchronization takes place in every orbital period. | |

Random walk ${x}_{timetag}$ | $1.0\times {10}^{-8}$ | ms/min | A random walk is a time series model ${x}_{timetag\left(t\right)}$ such that ${x}_{timetag\left(t\right)}={x}_{timetag\left(t-1\right)}+{\omega}_{t}$ where ${\omega}_{t}$ is a discrete white noise series. Random walk noise is reset to zero after the time synchronization and increases until the next time synchronization. The time synchronization takes place in every orbital period. |

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

DEM Error | $d{z}_{RDEM}^{}={\omega}_{DEM}$ + ${c}_{DEM}$ | |||

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

Offset error ${c}_{DEM}$ | 5.0 | m | Offset error ${c}_{DEM}$ is randomly selected between $-Value$ and $Value$ at the beginning of each simulation and fixed during the simulation. |

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

Receiver Observation Error | Range white Gaussian random error ${\omega}_{r}$ | 0.2 | m | ${\omega}_{r}=Value\times $ a random scalar drawn from the standard normal distribution each time, i.e., ${\sigma}_{\omega r}^{}$ = 0.2 m. |

Doppler white Gaussian random error ${\omega}_{d}$ | 0.1 | Hz | ${\omega}_{d}=Value\times $ a random scalar drawn from the standard normal distribution each time, i.e., ${\sigma}_{\omega d}^{}$ = 0.1 Hz. |

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

Signal Processing Delay Time Uncertainty | $d{t}_{processing}^{S-R}={\omega}_{process}$ + ${c}_{process}$ | |||

White Gaussian random error ${\omega}_{process}$ | 20.0 | ns | ${\omega}_{DEMt}=Value\times $ a random scalar drawn from the standard normal distribution each time. | |

Systematic error ${c}_{process}$ | 20.0 | ns | Systematic error ${c}_{process}$ is an output of the sinusoidal function $A\times sin\left(2\pi x/T\right)$: the argument $x$ is epoch time, the period $T$ was set equal to the lunar rotation period, and the amplitude $A$ is randomly selected between $-Value$ and $Value$ at the beginning of each simulation. |

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

Satellite 1 perilune altitude | 300 | km |

Satellite 1 apolune altitude | 300 | km |

Satellite 1 inclination | 110 | deg |

Satellite 1 right ascension of the ascending node | 0 | deg |

Satellite 1 argument of latitude | 0 | deg |

Satellite 2 perilune altitude | 300 | km |

Satellite 2 apolune altitude | 300 | km |

Satellite 2 inclination | 110 | deg |

Satellite 2 right ascension of the ascending node | 0 | deg |

Satellite 2 argument of latitude | −15 | deg |

Navigation Methods | Total GDOP | Total UPE (2drms) (m) | Availability (%) | Total Traveling Distance (m) |
---|---|---|---|---|

MDPO | 46.7 | 55.3 | 3.3 | 3753.75 |

Double-differenced TOA–FOA | 193.8 | 109.9 | 5.0 | 5625 |

Single-differenced Two-way Ranging | 1.2 | 26.3 | 5.0 | 5625 |

**Table 13.**The numerical simulation results with increased receiver observation errors and signal processing delay time uncertainty: ${\sigma}_{\omega r}=2.0$ m, ${\sigma}_{\omega d}=1.0$ Hz, the magnitude of ${\omega}_{process}$ is 200.0 ns, and the magnitude of ${c}_{process}$ is 200.0 ns.

Navigation Methods | Total GDOP | Total UPE (2drms) (m) | Availability (%) | Total Traveling Distance (m) |
---|---|---|---|---|

MDPO | 46.7 | 437.9 | 3.3 | 3753.75 |

Double-differenced TOA–FOA | 193.8 | 922.1 | 5.0 | 5625 |

Single-differenced Two-way Ranging | 1.2 | 249.9 | 5.0 | 5625 |

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**MDPI and ACS Style**

Tanaka, T.; Ebinuma, T.; Nakasuka, S.; Malki, H.
A Comparative Analysis of Multi-Epoch Double-Differenced Pseudorange Observation and Other Dual-Satellite Lunar Global Navigation Systems. *Aerospace* **2021**, *8*, 191.
https://doi.org/10.3390/aerospace8070191

**AMA Style**

Tanaka T, Ebinuma T, Nakasuka S, Malki H.
A Comparative Analysis of Multi-Epoch Double-Differenced Pseudorange Observation and Other Dual-Satellite Lunar Global Navigation Systems. *Aerospace*. 2021; 8(7):191.
https://doi.org/10.3390/aerospace8070191

**Chicago/Turabian Style**

Tanaka, Toshiki, Takuji Ebinuma, Shinichi Nakasuka, and Heidar Malki.
2021. "A Comparative Analysis of Multi-Epoch Double-Differenced Pseudorange Observation and Other Dual-Satellite Lunar Global Navigation Systems" *Aerospace* 8, no. 7: 191.
https://doi.org/10.3390/aerospace8070191