# Adaptive Multi-Sensor Joint Tracking Algorithm with Unknown Noise Characteristics

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## Abstract

**:**

## 1. Introduction

## 2. Target Motion and Measurement Model

- The ECEF coordinate system is a typical Cartesian coordinate system, which is fixed relative to the Earth. The origin of the coordinate system O is located in the center of the Earth, the OX axis is in the equatorial plane and points to the meridian where the Greenwich Observatory is located, the OZ axis is perpendicular to the equatorial plane and points toward the North Pole direction when combined with the axis of the rotation of the Earth, and the relationship between the OX, OY, and OZ axes satisfies the right-hand rule.
- The ECI coordinate system is also a typical Cartesian coordinate system, which is stationary relative to the fixed star. It is usually considered to be in the inertial space, with the geocentric point O as the coordinate origin. OX is in the equatorial plane pointing to the equinox point. OZ is perpendicular to the equatorial plane and points to the North Pole, and the orientation of the OY axis conforms to the right-hand rule.
- In the orbital coordinate system in which the coordinate origin O of the sensor body coordinate system is the geometric center of the sensor, the OX axis is the extension of the line connecting the Earth’s center and the coordinate origin, the OZ axis is located in the plane formed by the OX axis and the Earth’s rotation axis and is perpendicular to the OX axis, and the orientation of the OY axis satisfies the right-hand rule.

## 3. Adaptive Multi-Sensor Joint Tracking Algorithm

Algorithm 1: EKF |

$1.\mathbf{Inputs}:\mathrm{initial}\mathrm{estimates}{\widehat{X}}_{0},{P}_{0}$ |

2. Set k = 0 |

3. Repeat: |

k = k + 1 |

Prediction process of extended Kalman filtering: ${\widehat{X}}_{k+1|k}=f\left({\widehat{X}}_{k|k}\right)$ ${P}_{k+1|k}=F{P}_{k}{F}^{T}+Q$ |

Kalman gain update: $K={P}_{k+1|k}{F}^{T}{\left(F{P}_{k+1|k}{F}^{T}\right)}^{-1}$ |

Residuals of predictions: ${v}_{k}={Z}_{k+1}-{H}_{k+1}{\widehat{X}}_{k+1|k}$ |

Update process of extended Kalman filtering: ${X}_{k}={X}_{k}+K{v}_{k}$ ${P}_{k+1|k+1}=(I-K{H}_{k+1}){P}_{k+1|k}$ |

Exit the loop when there is no new measured state quantity $4.\mathbf{Output}:{X}_{k}$ |

^{th}moments that have been obtained are processed iteratively again, and the outcome after each iteration is used as the input for the next iteration so that a better performance can be achieved for the filtering results for each state ${X}_{k}$.

## 4. Simulation and Validation

_{2}perturbation term is adopted. The initial state of the two satellites and the space target is shown in the following Table 2 in ECEF coordinates.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**A geometric diagram of multiple space-based optical sensors for orbital target observations.

**Figure 8.**(

**a**) Diagram of the inserted system noise Q change over time. (

**b**) Comparison of RMSE after IEKF and AMSJTA filtering when Q decreases over time.

Parameters | Notation | Value |
---|---|---|

Equatorial radius of the Earth | ${R}_{e}$ | 6.37814 × 10^{6} (m) |

Earth’s gravitational constant | ${\mu}_{e}$ | 3.986006 × 10^{14} (m^{3}/s)^{2} |

Angular velocity of the Earth’s rotation | ${\mathrm{\omega}}_{\mathrm{e}}$ | 7.292115 × 10^{−5} (rad/s) |

J_{2} constant | J_{2} | 1.082626836 × 10^{−3} |

Unit (of Measure) | x/m | y/m | z/m |
---|---|---|---|

Reference satellite (Sensor #1) | $3.23\times {10}^{5}$ | $3.51\times {10}^{6}$ | $6.54\times {10}^{6}$ |

Companion satellite (Sensor #2) | $5.18\times {10}^{5}$ | $3.21\times {10}^{6}$ | $6.79\times {10}^{6}$ |

Space target | $2.30\times {10}^{-9}$ | $2.42\times {10}^{6}$ | $6.65\times {10}^{6}$ |

Unit (of measure) | ${v}_{x}/(\mathrm{m}\xb7{\mathrm{s}}^{-1})$ | ${v}_{y}/(\mathrm{m}\xb7{\mathrm{s}}^{-1})$ | ${v}_{z}/(\mathrm{m}\xb7{\mathrm{s}}^{-1})$ |

Reference satellite (Sensor #1) | $-6.78\times {10}^{3}$ | $-3.20\times {10}^{3}$ | $1.30\times {10}^{3}$ |

Companion satellite (Sensor #2) | $-6.35\times {10}^{3}$ | $-3.11\times {10}^{3}$ | $1.56\times {10}^{3}$ |

Space target | $-5.76\times {10}^{3}$ | $-4.54\times {10}^{3}$ | $1.64\times {10}^{3}$ |

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**MDPI and ACS Style**

Sun, W.; Wang, Y.; Diao, W.; Zhou, L.
Adaptive Multi-Sensor Joint Tracking Algorithm with Unknown Noise Characteristics. *Sensors* **2024**, *24*, 3314.
https://doi.org/10.3390/s24113314

**AMA Style**

Sun W, Wang Y, Diao W, Zhou L.
Adaptive Multi-Sensor Joint Tracking Algorithm with Unknown Noise Characteristics. *Sensors*. 2024; 24(11):3314.
https://doi.org/10.3390/s24113314

**Chicago/Turabian Style**

Sun, Weihao, Yi Wang, Weifeng Diao, and Lin Zhou.
2024. "Adaptive Multi-Sensor Joint Tracking Algorithm with Unknown Noise Characteristics" *Sensors* 24, no. 11: 3314.
https://doi.org/10.3390/s24113314