# Improved Joint Probabilistic Data Association (JPDA) Filter Using Motion Feature for Multiple Maneuvering Targets in Uncertain Tracking Situations

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## Abstract

**:**

## 1. Introduction

## 2. Uncertain Models and Fusion of Measurements and Observed Angles

#### 2.1. Uncertain Models of Measurements and Observed Angles

**Definition 1.**

**Definition 2.**

#### 2.2. Analyzing the Influence of Clutters on Measurements and Observed Angles

#### 2.3. Uncertain Fusion of Measurements and Observed Angles

## 3. Fuzzy Recursive Least Squares Filter (FRLSF)

## 4. Improved Joint Probabilistic Data Association-Fuzzy Recursive Least Squares Filter (IJPDA-FRLSF)

#### 4.1. Calculating the Generalized Joint Association Probability

_{k}is a number of these measurements, and ${Z}_{k}=\{{Z}_{l}{\}}_{l=1}^{k}$ denotes the cumulative set of validated measurements up to time k. According to Equation (11), the generalized joint association probability is composed of the statistical probability and the fuzzy membership degree. Next, we further derive the expression of the generalized joint association probability in the JPDA frame as follows.

#### 4.2. The propsed IJPDA-FRLSF

**Step 1.**Initialize state ${\widehat{\mathit{x}}}_{2}^{t}$ and filter covariance ${P}_{2}^{t}$ of target $t$ for $t=1,2,\cdots ,{n}_{k}$ using Equations (26) and (27), and start the recursive formulas at time $k=3$.**Step 2.**Compute predicted innovation ${V}_{k,i}^{t}$ on measurement ${\mathit{z}}_{k,i}$ using Equation (18).**Step 3.**Compute innovation covariance ${S}_{k}^{t}$ using Equation (19).**Step 4.**Compute gain matrix ${K}_{k}^{t}$ using Equation (20).**Step 5.**Reconstruct the generalized joint association probability ${\rho}_{k,i}^{t}$ using Equation (34).**Step 6.**Compute the fuzzy fading factor ${\tilde{\lambda}}_{k}^{t}$ using Equation (21).**Step 7.**Update the target state ${\widehat{\mathit{x}}}_{k}^{t}$ and filter covariance ${P}_{k}^{t}$ by FRLSF using Equations (35) and (36)$${\widehat{\mathit{x}}}_{k}^{t}={\widehat{\mathit{x}}}_{k-1}^{t}+{K}_{k}^{t}{V}_{k}^{t}$$$${P}_{k}^{t}={({\tilde{\lambda}}_{k}^{t})}^{-1}{P}_{k-1}-(1-{\rho}_{k,0}^{t}){K}_{k}^{t}{S}_{k}^{t}{({K}_{k}^{t})}^{\mathrm{T}}+{\displaystyle \sum _{i=0}^{{m}_{k}}{\rho}_{k,i}^{t}\left[{\widehat{\mathit{x}}}_{k,i}^{t}{({\widehat{\mathit{x}}}_{k,i}^{t})}^{\mathrm{T}}-{\widehat{\mathit{x}}}_{k}^{t}{({\widehat{\mathit{x}}}_{k}^{t})}^{\mathrm{T}}\right]}$$**Step 8.**Repeat the steps 2–7 for the next iterations.

## 5. Experimental Results and Analysis

#### 5.1. An Example of a Simulation Data Set: Two Crossing Targets

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#### 5.2. An Example of a Real Data Set: Three Crossing Targets

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Target I | Target II | ||
---|---|---|---|

Periods | Time | Periods | Time |

constant velocity (CV) | 14 s | constant velocity (CV) | 14 s |

constant turn (CT) | 1 s | constant turn (CT) | 1 s |

constant acceleration (CA) | 14 s | constant acceleration (CA) | 14 s |

constant turn (CT) | 1 s | constant turn (CT) | 1 s |

constant velocity (CV) | 5 s | constant velocity (CV) | 6 s |

**Table 2.**The average root-mean-square (RMS) position error for Target I (unit: m). Improved joint probabilistic data association-fuzzy recursive least squares filter (IJPDA-FRLSF), interacting multiple model-joint probabilistic data association filter (IMM-JPDAF).

Filter | CV | CT | CA | CT | CV |
---|---|---|---|---|---|

IJPDA-FRLSF | 21.5 | 22.0 | 22.1 | 22.5 | 22.0 |

IMM-JPDAF(II) | 14.8 | 13.3 | 32.0 | 37.3 | 37.0 |

IMM-JPDAF(IIIA) | 17.5 | 16.9 | 26.2 | 35.3 | 25.0 |

IMM-JPDAF(IIIB) | 22.7 | 23.5 | 35.3 | 47.2 | 38.5 |

Filter | CV | CT | CA | CT | CV |
---|---|---|---|---|---|

IJPDA-FRLSF | 22.1 | 21.9 | 22.1 | 22.6 | 21.8 |

IMM-JPDAF(II) | 15.0 | 15.0 | 30.9 | 36.8 | 37.2 |

IMM-JPDAF(IIIA) | 16.5 | 15.9 | 24.6 | 32.4 | 24.8 |

IMM-JPDAF(IIIB) | 23.7 | 23.4 | 36.0 | 46.5 | 38.1 |

Filter | CV | CT | CA | Total |
---|---|---|---|---|

IJPDA-FRLSF | mean | good | good | fair |

IMM-JPDAF(II) | good | mean | mean | mean |

IMM-JPDAF(IIIA) | fair | fair | fair | good |

IMM-JPDAF(IIIB) | poor | poor | poor | poor |

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**MDPI and ACS Style**

Fan, E.; Xie, W.; Pei, J.; Hu, K.; Li, X.; Podpečan, V.
Improved Joint Probabilistic Data Association (JPDA) Filter Using Motion Feature for Multiple Maneuvering Targets in Uncertain Tracking Situations. *Information* **2018**, *9*, 322.
https://doi.org/10.3390/info9120322

**AMA Style**

Fan E, Xie W, Pei J, Hu K, Li X, Podpečan V.
Improved Joint Probabilistic Data Association (JPDA) Filter Using Motion Feature for Multiple Maneuvering Targets in Uncertain Tracking Situations. *Information*. 2018; 9(12):322.
https://doi.org/10.3390/info9120322

**Chicago/Turabian Style**

Fan, En, Weixin Xie, Jihong Pei, Keli Hu, Xiaobin Li, and Vid Podpečan.
2018. "Improved Joint Probabilistic Data Association (JPDA) Filter Using Motion Feature for Multiple Maneuvering Targets in Uncertain Tracking Situations" *Information* 9, no. 12: 322.
https://doi.org/10.3390/info9120322