# Trajectory Predictor and Conflict Detection Figures of Merit for a Performance-Based Adaptive Air Traffic Monitoring System

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## Abstract

**:**

## 1. Introduction

#### 1.1. The State of the Art

## 2. Methodology

#### 2.1. Horizontal Trajectory Predictor

- aircraft characteristics: wake vortex (WV);
- information collected at the current position: current position (${\gamma}_{p}$), current flight level ($F{L}_{p}$), current speed (${v}_{p}$) and elapsed time from the entry (${t}_{p}$);
- target point: distance from current location to the target distance for prediction ($\delta \gamma $).

- Extreme gradient boosting regressor: a tree-based algorithm capable of providing reasonable performance with a small training sample. This HTP is referred to as XGB-HTP.
- Multi-layer perceptron regressor: a feed-forward artificial neural network (ANN). This HTP is referred to as MLP-HTP.

#### 2.2. Conflict Detector

#### Conflict Detection Process

- 1.
- Select the concurrent flights of ${f}_{2}$; these are the flights that are also in the sector at ${\gamma}_{p}$. Figure 2A shows an example of the instant when ${f}_{2}$ enters the sector. The triangles represent the flights and the three coloured lines represent the recurrent patterns of ${f}_{2}$’s flow; these are possible trajectories that the flight might take. ${f}_{2}$ has five concurrent flights at the entry (A, B, D, E, and F), and flight C is discarded because it is still out of the sector.
- 2.
- Select from the concurrent flights those that belong to flows that interact with ${f}_{2}$’s flow and do not belong to the same flow as ${f}_{2}$. In the example of Figure 2A, A and F are discarded because A belongs to ${f}_{2}$’s flow and F does not interact with ${f}_{2}$.
- 3.
- Estimate the present location of the concurrent flights: ${\gamma}_{p1}$.
- 4.
- Select those flights that have not reached any of their CA with ${f}_{2}$’s flow, i.e., flights whose ${\gamma}_{p1}<min({\gamma}_{C{P}_{1}}^{i,{f}_{1}},{\gamma}_{C{P}_{2}}^{i,{f}_{1}},\dots )$. The flights that have already overflown the critical areas will not lose separation with ${f}_{2}$. In the example of Figure 2A, B has already overflown CA.

- 1.
- (a)
- Predict the time each flight crosses $C{P}_{k}^{i,j}$ for both flights: ${\widehat{t}}_{{\gamma}_{p},{\gamma}_{C{P}_{k}}}^{i,{f}_{1}}$ and ${\widehat{t}}_{{\gamma}_{p},{\gamma}_{C{P}_{k}}}^{j,{f}_{2}}$. Figure 2B takes the pattern 1 (blue lines) of flight D and ${f}_{2}$, as an example. The HTP of each flow will predict the time each flight crosses the corresponding ${\gamma}_{C{P}_{k}}^{i,{f}_{1}}$ (${\gamma}_{C{P}_{k}}^{j,{f}_{2}}$) of each $C{P}_{k}^{i,j}$.
- (b)
- Determine the standard deviation of the following predictions: ${\sigma}_{{\gamma}_{p},{\gamma}_{C{P}_{k}}}^{i,{f}_{1}}$ and ${\sigma}_{{\gamma}_{p},{\gamma}_{C{P}_{k}}}^{j,{f}_{2}}$.
- (c)
- Obtain $\Delta {\widehat{t}}_{res,{\gamma}_{p},C{P}_{k}}^{i,j}$ and the corresponding prediction error ${\sigma}_{res,{\gamma}_{p},C{P}_{k}}^{i,j}$ (see Equations (3) and (4), as follows).$$\mu =\Delta {\widehat{t}}_{res,{\gamma}_{p},C{P}_{k}}^{i,j}={\widehat{t}}_{{\gamma}_{p},{\gamma}_{C{P}_{k}}}^{i,{f}_{1}}-{\widehat{t}}_{{\gamma}_{p},{\gamma}_{C{P}_{k}}}^{j,{f}_{2}}$$$$\sigma ={\sigma}_{res,{\gamma}_{p},C{P}_{k}}^{i,j}=\sqrt{{\left({\sigma}_{{\gamma}_{p},{\gamma}_{C{P}_{k}}}^{i,{f}_{1}}\right)}^{2}+{\left({\sigma}_{{\gamma}_{p},{\gamma}_{C{P}_{k}}}^{j,{f}_{2}}\right)}^{2}}$$
- (d)
- Based on the selected $\Delta {t}_{THR}$, evaluate ${P}_{{\gamma}_{p},C{P}_{k}}^{i,j,\Delta {t}_{THR}}$ using Equation (2).$$\begin{array}{c}\hfill {P}_{{\gamma}_{p},C{P}_{k}}^{i,j,\Delta {t}_{THR}}=P(x\le \Delta {t}_{THR})\end{array}$$

- 2.
- The final encounter probability between patterns [i,j] (${P}_{{\gamma}_{p},CA}^{i,j,\Delta {t}_{THR}}$) is calculated using the mean of the probabilities obtained from each $C{P}_{k}^{i,j}$ (see Figure 2C).

#### 2.3. Evaluation Metrics and Figures of Merits

- System dynamic range (SDR): an element to support the possible encounter classification and to prioritise resolution, as well as evaluating the performance of the CD tool. It mainly represents the impact of the HTP accuracy on the outputs of the CD tool.
- System tuning envelope (STE): an element to facilitate the tuning of the CD tool using the parameters $\Delta {t}_{THR}$ and ${P}_{THR}$. It assesses the performance of the CD tool, supporting the selection of operating points.

#### 2.3.1. Horizontal Trajectory Predictor Performance Metrics

- Heatmap of residuals’ mean (${\mu}_{res,{\gamma}_{p},\gamma}$), which represents the residual mean for each current position and possible target point. Ideally the mean is zero for every combination of $\gamma $ and ${\gamma}_{p}$.
- Heatmap of residuals’ standard deviation (${\sigma}_{res,{\gamma}_{p},\gamma}$): ideally the deviation is zero for every combination of $\gamma $ and ${\gamma}_{p}$.

#### 2.3.2. System Dynamic Range

- L: the maximum value of the curve (b = 0) when the logistic curve is increasing, or the minimum value when it is decreasing.
- k: the steepness of the curve.
- b: the displacement of the curve over the y-axis.
- ${x}_{0}$: the x-axis value of the midpoint.

#### 2.3.3. System Tuning Envelope

- False alerts or nuisance: incorrectly notified alerts. Flight pairs whose real minimum separation is
**above**the threshold ($|\Delta {t}_{min}|>\Delta {t}_{THR}$) but are notified as potential conflicts because the estimated probability is**above**the decision threshold (${P}_{{\gamma}_{p},C{P}_{maxP}}^{i,j}>{P}_{THR}$). - Missed alerts or missed: incorrectly hidden alerts. Flight pairs whose real minimum separation is
**below**the threshold ($|\Delta {t}_{min}|<\Delta {t}_{THR}$) but the estimated probability is**below**the decision threshold (${P}_{{\gamma}_{p},C{P}_{maxP}}^{i,j}\le {P}_{THR}$). - True alerts (TA): correctly notified alerts. Flight pairs whose real minimum separation is
**below**the threshold ($|\Delta {t}_{min}|<\Delta {t}_{THR}$) and estimated probability is**above**the decision threshold (${P}_{{\gamma}_{p},C{P}_{maxP}}^{i,j}>{P}_{THR}$). - True negative (TN): correctly filtered-out alerts. Flight pairs whose real minimum separation is
**above**the threshold ($|\Delta {t}_{min}|>\Delta {t}_{THR}$) and estimated probability is**below**the decision threshold (${P}_{{\gamma}_{p},C{P}_{maxP}}^{i,j}\le {P}_{THR}$). - Real positive (RP) or real conflict: flight pairs whose real minimum separation is below the threshold ($|\Delta {t}_{min}|<\Delta {t}_{THR}$).

#### 2.4. Use Cases

#### Evaluation Contextual Parameters

- ${d}_{THR}$: context threshold that influences the resulting critical area.
- –
- 1 NM.

- $\Delta {t}_{THR}$: internal parameter of CD that impacts the encounter probability.
- –
- 10 s, 30 s, 60 s, 90 s, and 120 s.

- ${P}_{THR}$: internal parameter of CD that affects the notified alerts.
- –
- from 0.1 to 0.99, incremented by 0.1.

- ${\gamma}_{p}$: independent variable that indicates LAT and consequently, the level of prediction error.
- –
- 0, 20, 40, 60, 80, 100, and 120 NM from the entry point in the sector of analysis.

## 3. Results

#### 3.1. Horizontal Trajectory Predictor Performance

#### 3.2. Conflict Detection

#### 3.2.1. System Dynamic Range

- L:
**−1**. The negative sign indicates the direction of the function. - ${x}_{0}$: $\Delta {t}_{THR}$.
- k:
**the higher the better**. - b:
**1**.

#### 3.2.2. System Tuning Envelope

#### CD Stability

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**A**) Example of differences between patterns of the same flow. The same $\gamma $ represents different geographical locations depending on the pattern. (

**B**) Prediction when the flight has flown 60 NM (${\gamma}_{p}$ is 60 NM) and the target point is at 140 NM.

**Figure 2.**(

**A**) An example of an instantaneous capture of the situation when ${f}_{2}$ enters the sector featuring candidate interacting flights (E and D) and non-interacting flights (the rest). The red triangle symbolizes ${f}_{2}$, while the black ones indicate other flights; those marked with a cross are non-interacting flights. (

**B**) Each red line connects a pair of points, representing a critical point. Alongside each red line, the associated predicted time of the corresponding ${\gamma}_{C{P}_{k}}^{1,{f}_{1}or{f}_{2}}$ of each $C{P}_{k}^{1,1}$ between pattern combination [1,1] is shown. (

**C**) Encounter probability between ${f}_{2}$ and ${f}_{1}$ (${P}_{CA}^{1,1}$), if they follow their planned trajectory.

**Figure 4.**Conditions to detect nuisance and missed alerts based on predicted and real time separation. The grey lines represent the $\Delta {t}_{THR}$ set at 90 s.

**Figure 5.**Recurrent patterns of the two flows of analysis (pattern 1 refers to the planned route). The two flows interact in LECMPAU.

**Figure 9.**Speed variation in relation to the entry speed-test sample. The vertical lines indicate the positions of region splits in the performance heatmaps.

**Figure 10.**Comparison of ${\widehat{t}}_{{\gamma}_{p},{C{P}_{1}}^{1,1}}$ variation by ${\gamma}_{p}$ of the flights between MLP and XGB models—examples 1 and 2.

**Figure 11.**(

**A**) The SDR and logistic curve fit variation by ${\gamma}_{p}$ when $\Delta {t}_{THR}$ = 60 s and variation by $\Delta {t}_{THR}$ when ${\gamma}_{p}$ = −108 NM. The grey step line represents the ideal performance of the CD tool for the selected $\Delta {t}_{THR}$—MLP-HTPs for F4–F24. (

**B**) The same graphs using XGB-HTPs for F4–F24.

**Figure 12.**Variation of the four parameters of the sigmoid function by $\Delta {t}_{THR}$ and ${\gamma}_{p}$.

**Figure 13.**(A) STE variation by $\Delta {t}_{THR}$ when ${\gamma}_{p}$ = −108 NM, and STE variation by ${\gamma}_{p}$ when $\Delta {t}_{THR}$ = 30 s for MLP-HTPs. (

**B**) Same graphs for XGB-HTPs.

Abbreviation | Definition | Abbreviation | Definition |
---|---|---|---|

ANN | Artificial Neural Network | MLP | Multi-Layer Perceptron Regressor |

AoI/R | Area of Interest/Responsibility | NM | Nautical Miles |

ATCo | Air Traffic Controller | OSED | Operational Service and Environment Definition |

CA/P | Critical Area/Point | RP | Real positive |

CDR | Conflict Detection and Resolution | s | Seconds |

DCB | Demand and Capacity Balancing | SDR | System Dynamic Range |

DST | Decision Support Tool | STE | System Tuning Envelope |

FA/MA | False Alert/Missed Alert | TA | True Alert |

FL | Flight Level | TN | True negative |

HTP | Horizontal Trajectory Predictor | TP | Trajectory Predictor |

LAT | Look-Ahead Time | WV | Wake Vortex |

ML | Machine Learning | XGB | eXtreme Gradient Boosting regressor |

Index | Definition | Index | Definition |
---|---|---|---|

i | Recurrent pattern of flow 1 | k | Critical point within a critical area |

j | Recurrent pattern of flow 2 | p | Present or current |

Notation | Definition | Notation | Definition |
---|---|---|---|

$\gamma $ | Arc-length along the trajectory of each flow pattern [NM] | ${t}_{e}$ | Elapsed time at sector entry [s] |

${\gamma}_{p}$ | Current position of the flight, measured as elapsed distance $\gamma $ of the flight from the entry point [NM] | ${t}_{p}$ | Elapsed time from entry to ${\gamma}_{p}$ [s] |

$\delta \gamma $ | Target distance from the current location to make prediction [NM] | $F{L}_{p}$ | Flight level at ${\gamma}_{p}$ [FL] |

${\widehat{t}}_{\gamma ,{\gamma}_{p}}$ | Predicted required time to reach the target point $\delta \gamma $ from the current position ${\gamma}_{p}$ [s] | ${v}_{p}$ | Ground speed at ${\gamma}_{p}$ [knots (kts)] |

Notation | Definition | Notation | Definition |
---|---|---|---|

${f}_{1}$ | Flight 1 of an interacting flight pair | ${\gamma}_{p}$ | Arc-length of evaluation, indicating ${\gamma}_{p2}$ |

${f}_{2}$ | Flight 2 of an interacting flight pair | ${\gamma}_{C{P}_{k}}^{i,{f}_{1}}\left({\gamma}_{C{P}_{k}}^{j,{f}_{2}}\right)$ | $\gamma $ of pattern i (j) of ${f}_{1}$ (${f}_{2}$) to reach the critical point k [NM] |

${d}_{THR}$ | Horizontal distance threshold to identify CP [NM] | $C{A}_{i,j}$ | Critical point k between patterns [i, j], composed of the location of ${f}_{1}$ and ${f}_{2}$ along patterns i and j, respectively, [NM] |

$\Delta {t}_{THR}$ | Time difference threshold to obtain real conflict and encounter probability [s] | $C{P}_{k}^{i,j}$ | Critical point k between patterns [i, j], composed of the location of ${f}_{1}$ and ${f}_{2}$ along patterns i and j, respectively, [NM] |

${P}_{THR}$ | Probability threshold and operation threshold to decide whether notify an alert | ${\widehat{t}}_{{\gamma}_{p},{\gamma}_{C{P}_{k}}}^{i,{f}_{1}}\left({\widehat{t}}_{{\gamma}_{p},{\gamma}_{C{P}_{k}}}^{j,{f}_{2}}\right)$ | Time predicted to reach ${\gamma}_{C{P}_{k}}^{i,{f}_{1}}\left({\gamma}_{C{P}_{k}}^{j,{f}_{2}}\right)$ at ${\gamma}_{p}$ through pattern i (j) [s] |

$\Delta {t}_{min}$ | Real-time difference [s] | $\Delta {\widehat{t}}_{res,{\gamma}_{p},C{P}_{k}}^{i,j}$ | Difference between the predicted time to reach $C{P}_{k}^{i,j}$ of both flights for the patterns [i, j] [s] |

$\Delta {d}_{min}$ | Real minimum horizontal separation [NM] | ${\sigma}_{{\gamma}_{p},{\gamma}_{C{P}_{k}}}^{i,{f}_{1}}\left({\sigma}_{{\gamma}_{p},{\gamma}_{C}{P}_{k}}^{j,{f}_{2}}\right)$ | Standard deviation of HTP to predict ${\gamma}_{C{P}_{k}}^{i,{f}_{1}}\left({\gamma}_{C{P}_{k}}^{j,{f}_{2}}\right)$ at ${\gamma}_{p}$ through pattern i (j) [s] |

$\Delta {h}_{min}$ | Real minimum vertical separation [ft] | ${\sigma}_{res,{\gamma}_{p},C{P}_{k}}^{i,j}$ | Combined standard deviation of the predictions of both flights for $C{P}_{k}$ [s] |

$\mu $ | Mean | ${P}_{{\gamma}_{p},C{P}_{k}}^{i,j,\Delta {t}_{THR}}$ | Encounter probability at $C{P}_{k}$ of the patterns [i, j] at ${\gamma}_{p}$ |

$\sigma $ | Standard deviation | ${P}_{{\gamma}_{p},CA}^{i,j,\Delta {t}_{THR}}$ | Encounter probability of the patterns [i, j] of the flight pair |

${\gamma}_{p1}\left({\gamma}_{p2}\right)$ | Current position of ${f}_{1}$ (${f}_{2}$) [NM] | ${P}_{{\gamma}_{p}}^{\Delta {t}_{THR}}$ | Encounter probability of the flight pair |

${\mathit{\gamma}}_{\mathit{p}\mathbf{1}},{\mathit{\gamma}}_{\mathit{p}\mathbf{2}}$ | ${\mathit{f}}_{\mathbf{1}}$ MLP ${\mathit{\sigma}}_{\mathbf{res},{\mathit{\gamma}}_{\mathit{p}\mathbf{1}},\mathit{\gamma}}$ | ${\mathit{f}}_{2}$ MLP ${\mathit{\sigma}}_{\mathbf{res},{\mathit{\gamma}}_{\mathit{p}2},\mathit{\gamma}}$ | ${\mathit{f}}_{1}$ XGB ${\mathit{\sigma}}_{\mathbf{res},{\mathit{\gamma}}_{\mathit{p}1},\mathit{\gamma}}$ | ${\mathit{f}}_{2}$ XGB ${\mathit{\sigma}}_{\mathbf{res},{\mathit{\gamma}}_{\mathit{p}2},\mathit{\gamma}}$ | MLP $\mathit{\sigma}$ | XGB $\mathit{\sigma}$ | MLP $\mathit{\sigma}$–XGB $\mathit{\sigma}$ |
---|---|---|---|---|---|---|---|

0, 0 | 20.78 | 17.73 | 21.15 | 15.56 | 38.63 | 37.13 | 1.50 |

20, 20 | 17.35 | 14.27 | 17.74 | 11.47 | 31.77 | 29.88 | 1.89 |

40, 40 | 14.06 | 12.91 | 14.30 | 9.69 | 27.00 | 24.42 | 2.57 |

60, 60 | 11.48 | 12.08 | 11.54 | 6.64 | 23.57 | 18.82 | 4.74 |

80, 80 | 9.07 | 11.41 | 9.05 | 5.34 | 20.61 | 14.86 | 5.75 |

100, 100 | 4.18 | 10.65 | 3.87 | 1.64 | 16.18 | 6 | 10.23 |

**Table 6.**Number of flight pairs with $\Delta {t}_{min}<\Delta {t}_{THR}$ at ${\gamma}_{p}$ = −108 NM and ${P}_{THR}$ = 0.5.

$\Delta {\mathit{t}}_{\mathbf{THR}}$ | No. RP | No. FA | No. MA |
---|---|---|---|

10 | 35 | 0 | 35 |

30 | 92 | 26 | 21 |

60 | 173 | 12 | 15 |

90 | 226 | 21 | 23 |

120 | 325 | 18 | 28 |

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## Share and Cite

**MDPI and ACS Style**

Xia, C.; Verdonk Gallego, C.E.; Fabio Bracero, A.; Gómez Comendador, V.F.; Arnaldo Valdés, R.M.
Trajectory Predictor and Conflict Detection Figures of Merit for a Performance-Based Adaptive Air Traffic Monitoring System. *Aerospace* **2024**, *11*, 155.
https://doi.org/10.3390/aerospace11020155

**AMA Style**

Xia C, Verdonk Gallego CE, Fabio Bracero A, Gómez Comendador VF, Arnaldo Valdés RM.
Trajectory Predictor and Conflict Detection Figures of Merit for a Performance-Based Adaptive Air Traffic Monitoring System. *Aerospace*. 2024; 11(2):155.
https://doi.org/10.3390/aerospace11020155

**Chicago/Turabian Style**

Xia, Chen, Christian Eduardo Verdonk Gallego, Adrián Fabio Bracero, Víctor Fernando Gómez Comendador, and Rosa María Arnaldo Valdés.
2024. "Trajectory Predictor and Conflict Detection Figures of Merit for a Performance-Based Adaptive Air Traffic Monitoring System" *Aerospace* 11, no. 2: 155.
https://doi.org/10.3390/aerospace11020155