# Adaptive Tracking Method for Non-Cooperative Continuously Thrusting Spacecraft

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## Abstract

**:**

## 1. Introduction

## 2. Basic Models for Maneuvering Target Tracking

#### 2.1. Relative Dynamics Model

#### 2.2. Coordinate System and Observation Model

#### 2.3. Maneuvering Acceleration Model

## 3. Adaptive Tracking Algorithm for Continuously Thrusting Spacecraft

#### 3.1. Introduction to CSJerk-Based EKF Filtering

- (1)
- The prediction of relative state $\mathit{X}(k+1,k)$$$\mathit{X}(k+1,k)=\mathit{\Phi}(k+1,k)\widehat{\mathit{X}}(k)+U(k)$$
- (2)
- The covariance matrix $\mathit{P}(k+1,k)$$$\mathit{P}(k+1,k)=\mathit{\Phi}(k+1,k)\widehat{\mathit{P}}(k)\mathit{\Phi}{(k+1,k)}^{\mathrm{T}}+Q(k)$$

- (1)
- The gain matrix $K(k+1)$$$K(k+1)=\mathit{P}(k+1,k)H{(k+1)}^{\mathrm{T}}[H(k+1)\mathit{P}(k+1,k))H{(k+1)}^{\mathrm{T}}+R(k+1){]}^{-1}$$$$H(k+1)=\frac{\partial h}{\partial \mathit{X}}{|}_{\mathit{X}(k+1,k)}$$
- (2)
- The estimations $\widehat{\mathit{X}}(k+1)$ and $\widehat{\mathit{P}}(k+1)$.$$\widehat{\mathit{X}}(k+1)=\mathit{X}(k+1,k)+K(k+1)v(k+1)$$$$v(k+1)=z(k+1)-h(\mathit{X}(k+1,k))$$$$\widehat{\mathit{P}}(k+1)=(I-K(k+1)H(k+1))\mathit{P}(k+1,k)$$

#### 3.2. Improved CSJerk Filtering Algorithm Based on Residual-Normalized Orthogonalization

#### 3.3. IMM Algorithm Based on RCSJF

## 4. Results

#### 4.1. Simulation Setups

#### 4.2. Analysis of the Residual-Normalized Orthogonalization

#### 4.3. Analysis of the IMM Algorithm Based on RCSJF

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## References

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**Figure 2.**Comparison of the estimation errors before and after the improvement of residual-normalized orthogonalization.

Parameters | a (km) | e | i (°) | Ω (°) | w (°) | f (°) |
---|---|---|---|---|---|---|

Observing satellite | 42,175.14 | 0.002 | 1.37 | 359.12 | −113.12 | 184.52 |

Maneuvering target | 42,165.14 | 0.008 | 1.38 | 359.12 | −113.12 | 184.80 |

Conditions | $\mathbf{Size}({\mathbf{m}/\mathbf{s}}^{2})$ | Time (s) | $\mathit{x}({\mathbf{m}/\mathbf{s}}^{2})$ | $\mathit{y}({\mathbf{m}/\mathbf{s}}^{2})$ | $\mathit{z}({\mathbf{m}/\mathbf{s}}^{2})$ |
---|---|---|---|---|---|

1 | 0.141 | 1000–2000 | −0.1 | 0 | 0.1 |

2 | 0.707 | 1000–2000 | −0.5 | 0 | 0.5 |

3 | 1.414 | 1000–2000 | −1 | 0 | 1 |

Conditions | Time (s) | $\mathit{x}({\mathbf{m}/\mathbf{s}}^{2})$ | $\mathit{y}({\mathbf{m}/\mathbf{s}}^{2})$ | $\mathit{z}({\mathbf{m}/\mathbf{s}}^{2})$ |
---|---|---|---|---|

1 | 1000–2000 | $-0.1\mathrm{sin}(0.3t-200)$ | 0 | $0.1\mathrm{sin}(0.3t-200)$ |

2 | 1000–2000 | $-0.5\mathrm{sin}(0.2t-100)$ | 0 | $0.5\mathrm{sin}(0.2t-100)$ |

3 | 1000–2000 | $-\mathrm{sin}(0.1t-100)$ | 0 | $\mathrm{sin}(0.1t-100)$ |

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**MDPI and ACS Style**

Yin, J.; Yang, Z.; Luo, Y.
Adaptive Tracking Method for Non-Cooperative Continuously Thrusting Spacecraft. *Aerospace* **2021**, *8*, 244.
https://doi.org/10.3390/aerospace8090244

**AMA Style**

Yin J, Yang Z, Luo Y.
Adaptive Tracking Method for Non-Cooperative Continuously Thrusting Spacecraft. *Aerospace*. 2021; 8(9):244.
https://doi.org/10.3390/aerospace8090244

**Chicago/Turabian Style**

Yin, Juqi, Zhen Yang, and Yazhong Luo.
2021. "Adaptive Tracking Method for Non-Cooperative Continuously Thrusting Spacecraft" *Aerospace* 8, no. 9: 244.
https://doi.org/10.3390/aerospace8090244