# Adaptive IMM-UKF for Airborne Tracking

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## Abstract

**:**

## 1. Introduction

## 2. Tracking Logic

- A Cartesian tracker to estimate the horizontal coordinates and velocity of an aircraft relative to the ownship by means of an augmented UKF;
- A range tracker to estimate the horizontal range and the relative horizontal speed by means of a UKF as well;
- A vertical tracker to estimate the vertical position and velocity of an othership aircraft using a Kalman filter.

- Mode 1 or non-maneuvering mode: in uniform linear motion, this mode is modeled with the same equations previously described.

## 3. Monte Carlo Method

- The RMS errors in each of the state variables (x, y, ${v}_{x}$, ${v}_{y}$). This is the most popular type of error in tracking for measuring accuracy because it is sensitive to outliers (i.e., it penalizes large estimation errors more heavily due to the squared differences).
- The normalized estimation error squared (NEES) error and its 95% probability region for checking the consistency of the filter, as stated in [21]. The consistency analysis studies whether the output covariance matrix corresponds or not to the provided state. If the estimation error is much greater than its corresponding covariance, then the filter is overconfident; otherwise, it is underconfident. This is the preferred metric when the ground truth is available due to its easy interpretation and normalization.
- The settling time against velocity changes to measure the tracker’s lag. This is useful for assessing the velocity of adaptation in dynamic systems during transient periods.
- The noise reduction factor provided by the tracker. This is accomplished by assessing the difference between the measurement error (tracker´s input) and the estimation error (tracker´s output), and it shows how much noise is filtered out by the tracking logic.

## 4. Experiments and Results

- Uniform linear motion applied for both aircraft;
- The ownship aircraft behaved in uniform linear motion, while the othership aircraft made a turn;
- This encounter simulates a relative trajectory with a step function as the velocity.

#### 4.1. Encounter 1

#### 4.2. Encounter 2

#### 4.3. Encounter 3

#### 4.4. Estimation Error Analysis

#### 4.5. Robustness of the Prediction

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**Measurement and estimation RMSE comparison. (

**a**) RMS error comparison for the x coordinate. (

**b**) RMS error comparison for the y coordinate.

Variable | Mean | Median | Std Dev | 75th Percentile | 25th Percentile |
---|---|---|---|---|---|

ACAS Xa-RMSE x (ft) | 426 | 281.3 | 457 | 573.7 | 121 |

ACAS Xa-RMSE y (ft) | 724.1 | 464.6 | 853.4 | 973.7 | 180.8 |

ACAS Xa-RMSE ${v}_{x}$ (ft/s) | 33.17 | 13.1 | 61.3 | 25.8 | 5.9 |

ACAS Xa-RMSE ${v}_{y}$ (ft/s) | 35.7 | 25 | 34.4 | 47.3 | 11.4 |

IMM-RMSE x (ft) | 390.9 | 265.5 | 417.3 | 528 | 115.9 |

IMM-RMSE y (ft) | 749.4 | 490.2 | 862.5 | 1021.4 | 189.9 |

IMM-RMSE ${v}_{x}$ (ft/s) | 33.5 | 16 | 56.9 | 30.2 | 14.1 |

IMM-RMSE ${v}_{y}$ (ft/s) | 41.5 | 31.2 | 36.5 | 57.8 | 14.2 |

Measured RMSE x (ft) | 906.2 | 904.2 | 63.6 | 949 | 860.1 |

Measured RMSE y (ft) | 2145.5 | 2137.8 | 143 | 2241.7 | 2051 |

Variable | Mean | Median | Std Dev | 75th Percentile | 25th Percentile |
---|---|---|---|---|---|

ACAS Xa-RMSE x (ft) | 2515 | 2001 | 2195 | 3407 | 1017 |

ACAS Xa-RMSE y (ft) | 1783 | 1470 | 1655 | 2362 | 736 |

ACAS Xa-RMSE ${v}_{x}$ (ft/s) | 187.6 | 135.8 | 159.2 | 279.6 | 66.3 |

ACAS Xa-RMSE ${v}_{y}$ (ft/s) | 121.5 | 92.9 | 112 | 157 | 44 |

IMM-RMSE x (ft) | 2182 | 1700 | 2049 | 2817 | 866.8 |

IMM-RMSE y (ft) | 1698 | 1390 | 1600 | 2277 | 677.7 |

IMM-RMSE ${v}_{x}$ (ft/s) | 153.7 | 113.5 | 146.4 | 186.1 | 58.7 |

IMM-RMSE ${v}_{y}$ (ft/s) | 122.2 | 83.4 | 107.6 | 166 | 46 |

Measured RMSE x (ft) | 5628 | 5618 | 321 | 5837 | 5415 |

Measured RMSE y (ft) | 4568 | 4561 | 262 | 4741 | 4380 |

Encounter | Hypothesis Test | p-Value | Outcome |
---|---|---|---|

Encounter 1 | Two-sample t-test | 0.18 | Not rejected |

Encounter 2 | Two-sample t-test | 6.8 × 10^{−60} | Rejected |

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**MDPI and ACS Style**

Arroyo Cebeira, A.; Asensio Vicente, M.
Adaptive IMM-UKF for Airborne Tracking. *Aerospace* **2023**, *10*, 698.
https://doi.org/10.3390/aerospace10080698

**AMA Style**

Arroyo Cebeira A, Asensio Vicente M.
Adaptive IMM-UKF for Airborne Tracking. *Aerospace*. 2023; 10(8):698.
https://doi.org/10.3390/aerospace10080698

**Chicago/Turabian Style**

Arroyo Cebeira, Alvaro, and Mariano Asensio Vicente.
2023. "Adaptive IMM-UKF for Airborne Tracking" *Aerospace* 10, no. 8: 698.
https://doi.org/10.3390/aerospace10080698