# A Geometric Layout Method for Synchronous Pseudolite Positioning Systems Based on a New Weighted HDOP

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Geometric Precision Factor

#### 2.1. Mathematical Description of Geometric Precision Factor

#### 2.2. Fitting Algorithm of GDOP

#### 2.3. WHDOP Fitting Algorithm

## 3. Simulation Experiment of the Proposed Algorithm

#### 3.1. Simulation for Calculating WHDOP

_{1}through trace(H), and get the value of h

_{2}and h

_{3}through trace(H

^{2}) and trace(H

^{3}). The values of h

_{1}, h

_{2}, h

_{3}, and h

_{4}are 8.889, 74.100, 637.300, $1.701\times {10}^{-4}$. According to the traditional calculation method, we can get the value of WHDOP is 9.797. The WHDOP result obtained by using Equation (22) is 9.449. According to the method of six pseudolites, we also carried out the calculation of four pseudolites and eight pseudolites and summarized the results in Table 1. It should be noted that, when randomly generating the pseudolite distribution, we avoided the case where the pseudolite distribution is concentrated or linear and chose the relatively scattered geometric distribution of pseudolites for calculation. This is also in line with the situation when the ground-based system is deployed, making the overall DOP in the positioning area small.

#### 3.2. Pseudolites Geometric Planning Simulation

#### 3.3. Actual Positioning Experiment

## 4. Experiments by Indoor Navigation System

#### 4.1. Static Point Positioning

#### 4.2. Pearson Correlation Coefficient Verification

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Comparison of the results of brute force search (

**top**) and the proposed algorithm (

**bottom**).

**Figure 9.**(

**top**) The positioning result of the static point A1 (2.6777, 6.059); (

**bottom**) the positioning result of the static point E5 (0, 2.6734).

Number of Pseudolites | 4 | 6 | 8 |
---|---|---|---|

a | −0.295 | −0.259 | −0.656 |

b | 0.401 | 0.173 | 0.394 |

c | −0.052 | 0.132 | 0.387 |

d | 313 | 1064 | 1995 |

Mean WHDOP value | 14.072 | 12.002 | 6.843 |

Average error | 1.134 | 0.743 | 0.643 |

Error percentage | 8.66% | 6.19% | 9.40% |

WHDOP Value | ≤4 | 4–5 | 5–6 | >6 | Mean WHDOP |
---|---|---|---|---|---|

Proposed Method | 41.8% | 21.6% | 13.8% | 22.8% | 4.8 |

Brute Force Search | 39.4% | 26.6% | 14.0% | 20.0% | 4.7 |

**Table 3.**Distribution of WHDOP values using the proposed method under six pseudolites and eight pseudolites.

WHDOP Value | ≤4 | 4–5 | 5–6 | > 6 | Mean WHDOP |
---|---|---|---|---|---|

6 pseudolites | 78.0% | 10.2% | 10.8% | 1.0% | 2.958 |

8 pseudolites | 85.8% | 8.2% | 5.8% | 0.4% | 2.694 |

Pseudolite | X (m) | Y (m) | Z (m) |
---|---|---|---|

1 | 3.5659 | −1.8707 | 11.3553 |

2 | −2.8797 | −2.2747 | 11.3568 |

3 | −5.2303 | 0.464 | 11.3562 |

4 | −5.6214 | 7.344 | 11.3456 |

5 | −3.3354 | 9.1587 | 11.3428 |

6 | 2.6414 | 9.5559 | 11.3459 |

7 | 5.098 | 8.0073 | 11.3476 |

8 | 5.5518 | 1.1654 | 11.3507 |

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**MDPI and ACS Style**

Zhao, X.; Shuai, Q.; Li, G.; Lu, F.; Zhu, B.
A Geometric Layout Method for Synchronous Pseudolite Positioning Systems Based on a New Weighted HDOP. *ISPRS Int. J. Geo-Inf.* **2021**, *10*, 601.
https://doi.org/10.3390/ijgi10090601

**AMA Style**

Zhao X, Shuai Q, Li G, Lu F, Zhu B.
A Geometric Layout Method for Synchronous Pseudolite Positioning Systems Based on a New Weighted HDOP. *ISPRS International Journal of Geo-Information*. 2021; 10(9):601.
https://doi.org/10.3390/ijgi10090601

**Chicago/Turabian Style**

Zhao, Xinyang, Qiangqiang Shuai, Guangchen Li, Fangzhou Lu, and Bocheng Zhu.
2021. "A Geometric Layout Method for Synchronous Pseudolite Positioning Systems Based on a New Weighted HDOP" *ISPRS International Journal of Geo-Information* 10, no. 9: 601.
https://doi.org/10.3390/ijgi10090601