# Distributed Orbit Determination for Global Navigation Satellite System with Inter-Satellite Link

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## Abstract

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## 1. Introduction

## 2. Overview of Orbit Determination Algorithms

- the ISL communications and measurements in ultra high frequency (UHF) band;
- a high-precision autonomous navigation algorithm which is adapted to the computing capacity of the satellite on-board computers.

## 3. Fundamental Equations for Measurement and Motion

#### 3.1. Equations for Measurement

#### 3.2. Equations for Motion

## 4. Whole-Constellation Centralized Extended Kalman Filter

## 5. Distributed Orbit Determination

#### 5.1. Reduced-Order Iterative Cascade EKF

#### 5.2. Reduced-Order Increased Measurement Covariance EKF

#### 5.3. Balanced Extended Kalman Filter

- The method of BEKF collects the data of ISL measurements, satellite-to-GAS measurements (if they are available), and the satellite state vectors as well as their covariance matrices on both ends of the ISL. After the calculation of the BEKF algorithm, the improved state vectors and their covariance matrices are sent to the other visible satellites. The BEKF method modifies the denominator of the gain matrix ${\mathit{K}}_{k}^{ij}$ to ${\mathit{H}}_{k}^{i}{\overline{\mathit{P}}}_{k}^{i}{\mathit{H}}_{k}^{i\mathrm{T}}+{\mathit{H}}_{k}^{j}{\overline{\mathit{P}}}_{k}^{j}{\mathit{H}}_{k}^{j\mathrm{T}}+{\mathit{R}}_{k}^{i}$, which is similar to the method of IMCEKF. Furthermore, it modifies the gain matrix by a factor of $(I-{M}_{C}^{})$. Therefore, the BEKF algorithm is expected to yield results with higher precision.
- It seems that BEKF requires more ISL processes than the other EKFs. In fact, the state vectors and their covariance matrices on both ends are improved at the same time. It is unnecessary to repeat the ISL process for the same two satellites. The computation load of BEKF is similar to that of IMCEKF.
- The iteration process that is implemented in the ICEFK algorithm is not required in the BEKF method to achieve high accuracy.
- The improving state vectors are balanced in such a way that the satellite with lower-state precision will undergo more increments in accuracy while the satellite with higher-state precision will have fewer adjustments.
- Compared to the other EKFs, in Equation (64), ${M}_{C}^{}\left[\begin{array}{l}{\overline{\mathit{P}}}_{k}^{i}\text{\hspace{1em}}0\\ 0\text{\hspace{1em}}{\overline{\mathit{P}}}_{k}^{j}\end{array}\right]$ is subtracted from the state vectors’ covariance matrices of the two satellites. Therefore, the values in the matrices are reduced and the accuracy of the state vectors is improved.
- The two satellites are correlated by ${\widehat{\mathit{P}}}_{k}^{ij}$ and ${\widehat{\mathit{P}}}_{k}^{ji}$ in Equation (65); however, these two matrices have to be ignored in this distributed filter. As a result, the current method should be categorized as a reduced-order sub-optimal orbit determination method.

## 6. Simulations and Analyses

- (1)
- the earth’s gravitational effects of 70 × 70,
- (2)
- the lunar, solar, and other planetary gravitational perturbations,
- (3)
- the solar radiation pressure, and
- (4)
- the other general relativistic forces.

- (1)
- An analytical orbit was generated and the corresponding ISL and satellite-to-GAS PRs were calculated.
- (2)
- Using the abovementioned PRs, the satellite orbits were calculated by the different methods and compared with the analytical orbit to find out orbit determination precisions.

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Orbit determination errors of SV-01 by the whole-constellation centralized extended Kalman filter (WCCEKF) algorithm. (

**a**) the position error and radial error; (

**b**) along track error and cross track error.

**Figure 2.**Orbit determination errors of SV-01 by the iterative cascade extended Kalman filter (ICEKF) algorithm. (

**a**) the position error and radial error; (

**b**) along track error and cross track error.

**Figure 3.**Orbit determination errors of SV-01 by the increased measurement covariance extended Kalman filter (IMCEKF) algorithm. (

**a**) the position error and radial error; (

**b**) along track error and cross track error.

**Figure 4.**Orbit determination errors of SV-01 by the balanced extended Kalman filter (BEKF) algorithm. (

**a**) the position error and radial error; (

**b**) along track error and cross track error.

Methods | Computation Amount (FLOPS: Floating-Point Operations Per Second) |
---|---|

WCCEKF | $4{N}_{W}^{2}({N}_{W}^{2}-1)+({N}_{W}^{}-1){N}_{W}^{}({N}_{W}^{}+1)/6$ + $(2{N}_{W}^{2}+7{N}_{W}^{}+1)\times m$ |

ICEKF or IMCEKF | $4{N}_{i}^{2}({N}_{i}^{2}-1)+({N}_{i}^{}-1){N}_{i}^{}({N}_{i}^{}+1)/6$ + $(2{N}_{i}^{2}+7{N}_{i}^{}+1)\times (m/n)$ |

BEKF | $4{N}_{i}^{2}({N}_{i}^{2}-1)+({N}_{i}^{}-1){N}_{i}^{}({N}_{i}^{}+1)/6$ + $(2{N}_{ij}^{2}+7{N}_{ij}^{}+1)\times (m/2n)$ |

Algorithm | Description | Computation Amount | Orbital Accuracy |
---|---|---|---|

WCCEKF | Whole-constellation centralized EKF | Maximum | Best |

ICEKF | Iterative cascade EKF | Minimum | Normal |

IMCEKF | Increased measurement covariance EKF | Minimum | Normal |

BEKF | Balanced EKF | Minimum | Better |

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**MDPI and ACS Style**

Wen, Y.; Zhu, J.; Gong, Y.; Wang, Q.; He, X.
Distributed Orbit Determination for Global Navigation Satellite System with Inter-Satellite Link. *Sensors* **2019**, *19*, 1031.
https://doi.org/10.3390/s19051031

**AMA Style**

Wen Y, Zhu J, Gong Y, Wang Q, He X.
Distributed Orbit Determination for Global Navigation Satellite System with Inter-Satellite Link. *Sensors*. 2019; 19(5):1031.
https://doi.org/10.3390/s19051031

**Chicago/Turabian Style**

Wen, Yuanlan, Jun Zhu, Youxing Gong, Qian Wang, and Xiufeng He.
2019. "Distributed Orbit Determination for Global Navigation Satellite System with Inter-Satellite Link" *Sensors* 19, no. 5: 1031.
https://doi.org/10.3390/s19051031