# Research on Engine Thrust and Load Factor Prediction by Novel Flight Maneuver Recognition Based on Flight Test Data

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## Abstract

**:**

## 1. Introduction

## 2. Real-Time Flight Maneuver Recognition (FMR) Method

#### 2.1. PLR–PIP Algorithm

_{1}, x

_{1}, …, x

_{n}) and the important series set the important points (IPs). The first point of the sequence is added to the set IPs. Then, each pair of adjacent points is taken as a sub-sequence and the PIP points are calculated and added to the set IPs. Adjacent PIP points form a new sub-sequence, which is linearly interpolated, and the root mean square error (RMSE) is calculated with respect to the original sequence. Sequences are extracted with RMSE values that do not meet the fitting error threshold condition. These steps are repeated until all sequences meet the threshold requirements ${\Delta}_{PIP}$. The algorithm flow is shown in Figure 3. According to this description of the improved strategies, PLR–PIP is given in Algorithm 1.

Algorithm 1 PLP-PIP |

Input: Row data, normal overload value sequence, threshold Δ_{PIP} |

1: x_{b} = dataMap.get(P_{b}); //get the value of begin point |

2: x_{e} = dataMap.get(P_{e}); //get the value of begin point |

3: dist = 0; // Initial piecewise fitting error |

4: for i = 1→n do |

5: x_{i}; = dataMap.get(i); //get the value of point i |

6: dist_{sin}[i] = cal(x_{b}, P_{b}, x_{e}, P_{e}, x_{i}, P_{i}); //calculate the piecewise fitting error |

7: if dist_{sin} > Δ_{PIP} then //If the piecewise fitting error ≥ Δ_{PIP} |

8: continue //Add Perceptually Important Point |

9: else |

10: return (P_{b}, P_{1}, P_{2}, …, P_{i}, P_{e}); //get PIP set |

11: end if |

12: end for//Select the maneuver Sequences between PIP |

_{PIP}= 0.0001. The calculation process is demonstrated in Figure 4, where the dashed line represents the original sequence, the red dots represent PIP points, and the straight line represents the shape described using PIP. As the number of PIP increases, Δ

_{PIP}gradually decreases, and the data compression efficiency decreases as well. Therefore, the threshold Δ

_{PIP}needs to be set according to the actual situation.

#### 2.2. Algorithm 2—Proposed DTW Distance-Based Algorithm

#### 2.2.1. Method and Principle by Sequence Re-Description

_{PIP}between two adjacent points (the ratio of the normal load factor difference between two points to the distance between two sample points) and the average normal load factor ${\overline{n}}_{y}$ between two points are calculated in sequence. The threshold values of the slope and the average normal load factor are set as $\Delta {R}_{PIP}$ and $\Delta {\overline{n}}_{y}$, respectively. When the $\left|{\mathrm{R}}_{PIP}\right|<\Delta {\mathrm{R}}_{PIP}$ and ${\overline{n}}_{y}<\Delta {\overline{n}}_{y}$, the segment between these two points is defined as a level flight maneuver. The results are shown in Figure 6, where the red curve represents the identified maneuver segments, and the black segments represent the level flight segments.

#### 2.2.2. Improved Hierarchical Clustering Algorithm

#### 2.2.3. Results of DTW Distance-Based Algorithm

_{Norm}= (D − D

_{min})/(D

_{max}− D

_{min}). The MFR results based on the proposed DTW distance-based algorithm are shown in Figure 8; the straight flight segments formed during the middle partition process are defined as one class (i.e., class 10) without hierarchical clustering. Finally, 9 classes of maneuvers are formed, with a total of 22 maneuver segments. It can be seen that within a maneuver class, the trend and magnitude of the flight parameter data for each maneuver segment are similar, while significant differences exist between different maneuver classes.

_{a}), and altitude (H). The positional expression for any point on the trajectory is as follows:

Algorithm 2 Proposed DTW distance-based algorithm |

Input: Row data, normal overload value sequence, threshold Δ_{PIP} |

1: x_{b} = dataMap.get(P_{b}); //get the value of begin point |

2: x_{e} = dataMap.get(P_{e}); //get the value of begin point |

3: dist = 0; // Initial piecewise fitting error |

4: for i = 1→n do |

5: x_{i}; = dataMap.get(i); //get the value of point i |

6: dist_{sin}[i] = cal(x_{b}, P_{b}, x_{e}, P_{e}, x_{i}, P_{i}); //calculate the piecewise fitting error |

7: if dist_{sin} > Δ_{PIP} then //If the piecewise fitting error ≥ Δ_{PIP} |

8: continue //Add Perceptually Important Point |

9: else |

10: return (P_{b}, P_{1}, P_{2}, …, P_{i}, P_{e}); //get PIP set |

11: end if |

12: end for//Select the maneuver Sequences between PIP |

13: r_{0,0} = 0; r_{i}_{,0} = r_{0,j} = ∞; i ∈ [n], j ∈ [m] |

14: for j = 1, 2, …, m do //Sequence 1 |

15: for i = 1, 2, …, n do //Sequence 2 |

16: r_{i}_{,j} = δ(x_{i}, y_{i}) + min {r_{i}_{−1,j−1}, r_{i}_{−1,j}, r_{i}_{,j−1}} //Calculate Sequence similarity |

17: Hierarchical clustering |

18: end for |

19: end for |

Output: Mancuver |

#### 2.3. Algorithm 3—Proposed Sequence Important Point-Based Method

#### 2.3.1. Method and Principle of Sequence Important Point

Algorithm 3 Proposed Perceptually important point-based method |

Input: Row data, n, d_{PIP}, k_{PIP} |

1: Divide the raw data into sequences of length n |

2: Perform PLR-PIP segmentation on the horizontal projection of the sequences |

3: for i = 1→m do |

4: d = |(y_{1} + (y_{2} − y_{1})[(x_{i} − x_{1})/(x_{2} − x_{1})] − y_{i})|; //get the value of VD |

5: D = (d_{1}, d_{2}, …, d_{m})//get the distance set |

6: if d_{i} ≤ d_{PIP} then |

7: Straight sequence |

8: else |

9: Bending sequence |

10: end if |

11: end for |

12: Perform PLR-PIP segmentation on the vertical projection of the sequences |

13: for j = 1 → ω do |

14: Calculate the slope of adjacent, PIP points |

15: D = (k_{1}, k_{2}, …, k_{ω})//get the slopes set |

16: if k_{j} ≤ −k_{PIP} then |

17: Descend sequence |

18: else if −k_{PIP} < k_{j} < k_{PIP} then |

19: Level sequence |

20: else |

21: Ascend sequence |

22: end if |

23: end for |

24: Combine into maneuver |

Output: Maneuver |

#### 2.3.2. Trend Recognition and Subdivision/Merge Rules

_{1}, x

_{2}, …, x

_{n}), Y = (y

_{1}, y

_{2}, …, y

_{n}), and Z = (z

_{1}, z

_{2}, …, z

_{n}). On the projection of the sequence onto the horizontal plane, PLR–PIP is performed to obtain a set of sub-sequence segments (PLR

_{1}and IPs

_{1}sets) where the PIP point sequence is denoted by the x-coordinate $a=({a}_{1},{a}_{2},\cdots ,{a}_{m})$ and y-coordinate $b=({b}_{1},{b}_{2},\cdots ,{b}_{m})$. Another PIP point is selected from each sub-sequence segment, and the VD distance of the individual sub-sequence is calculated, with the distance set denoted as $D=({d}_{1},{d}_{2},\cdots ,{d}_{m-1})$. A distance threshold is defined as ${d}_{PIP}$, where a sub-sequence is classified as a straight primitive if ${d}_{i}\le {d}_{PIP}$, and as a curved primitive if ${d}_{i}>{d}_{PIP}$. Upon projecting the sequence onto the height coordinate system, PLR–PIP segmentation is performed to obtain a set of sub-sequence segments. Note $k=({k}_{1},{k}_{2},\cdots ,{k}_{w-1}$) as the gradient set between each PIP and ${k}_{PIP}$ as a gradient threshold. When ${k}_{i}\le -{k}_{PIP}$, the interval is marked as a descent element; when $-{k}_{PIP}<{k}_{i}<{k}_{PIP}$, the interval is marked as a horizontal element; when ${k}_{PIP}\le {k}_{i}$, the interval is marked as an ascent element.

#### 2.3.3. Results of Sequence Important Point-Based Method

_{pip}= 5 and Δ

_{pip}= 0.001. The resulting partition of the aircraft flight trajectory on the horizontal plane can be obtained, as shown in Figure 14.

_{PIP}= 0.01, the PIP points of the vertical trend sequence are extracted, and the slope between each PIP point is calculated. The rules for superimposing states are shown in Table 3. This paper utilizes criteria based on the changes in flight parameters, such as velocity, height, pitch angle, and roll angle. For detailed division criteria, refer to Table 4. Finally, 14 classes of refined maneuver states are formed. The results of the simple maneuvers and task segment division are shown in Figure 15, where the altitude data have been normalized.

## 3. Comparison of the Two Proposed Maneuver Classification Algorithms

- (1)
- Acquisition of maneuver segments

- (2)
- Hierarchical maneuver segments

- (3)
- Normalization of maneuver segments

## 4. Prediction of Engine Thrust and Load Factor

#### 4.1. Flight Load Calculation of Aircraft Engine

_{i}represents the current minute segment of drag, ρv

^{2}/2 is dynamic pressure, S represents the wing area, C

_{X}is the aircraft’s drag coefficient, C

_{X}

_{0}is the zero-lift drag coefficient, A is the lift-induced drag factor, ΔC

_{X}

_{h}is the drag coefficient’s altitude correction value, and C

_{Yi}is the lift coefficient for the current minute segment.

_{i}represents the available thrust for the current minor segment of the aircraft, m

_{i}is the mass of the aircraft in the current minor segment; v

_{i}is the flight vector velocity for the current minor segment of the aircraft; R

_{i}is the engine thrust; η is the thrust efficiency conversion ratio; φ

_{p}is the angle between the engine thrust vector and the aircraft’s axis Ox

_{b}at specific times; α

_{i}is the angle between the projection of airflow axis Ox

_{a}on the aircraft’s symmetry plane and the body axis Ox

_{b}; β

_{i}is the angle between the airflow axis Ox

_{a}and the aircraft’s symmetry plane; θ

_{i}is the angle between the trajectory axis Ox

_{k}and the horizontal plane that is positive upwards relative to the terrain.

_{i}represents the lift along the Oy

_{a}direction; Z

_{i}represents the lateral force along the Oz

_{a}direction; γ

_{si}represents the angle between the airflow Oy

_{a}-axis and the flight path Oy

_{k}, known as the roll angle; the Ψ

_{ai}track angle, also referred to as the heading angle, is the angle between the projection of the flight path Ox

_{k}-axis on the horizontal plane and the ground Ox

_{g}-axis, with the convention that rightward is positive.

_{xi}, n

_{yi}, n

_{zi}are the engine load factor in the current path coordinate frame, respectively.

#### 4.2. Engine Flight Load and Thrust Prediction

_{1}corresponds to the start time of the action, x

_{2}corresponds to the time of the roll angle reaching its extreme value, x

_{4}corresponds to the end time of the action, y

_{1}corresponds to the initial roll angle at the start of the action, and y

_{2}corresponds to the roll angle at the extreme point of the action. In Figure 18, the black curve represents the observed changes in the aircraft’s roll angle during maneuvers, and the red curve (dashed line) represents the linear fitting result.

_{1}, H

_{0}is the initial altitude, v is the flight speed, and ϑ is the pitch angle.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviation

CDTW | Context-based dynamic time warping |

CNN | Convolutional Neural Network |

CTW | Context-Tree Weighting |

DBSCAN | Density-based spatial clustering of applications with noise |

DTW | Dynamic time warping |

ED | Euclidean distance |

FAA | Federal Aviation Administration |

FDR | Flight data recorder |

FMR | Flight Maneuver Recognition |

IC | Inconsistency coefficient |

IPs | Important points |

LDA | Latent Dirichlet Allocation |

LSM | Logistic Sigmoid Model |

PCA | Principal Component Analysis |

PD | Perpendicular distance |

PIP | Perceptually important point |

PLR | Piecewise linear representation |

RMSE | Root mean square error |

SVD | Singular Value Decomposition |

VD | Vertical distance |

1-NN | 1-nearest neighbors |

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**Figure 1.**Flow chart of engine thrust and load factor prediction method in this paper (The red item is the method applied in this paper).

**Figure 2.**Three methods to calculate distance: Perpendicular distance (PD), Vertical distance (VD), and Euclidean distance (ED).

**Figure 6.**The results of FMR (The black line represents level flight maneuvers, while the dotted line represents other maneuver characteristics.).

**Figure 9.**Classification of maneuver with motion locus into 9 class based on the flight data, (

**a**–

**i**) shows 9 different maneuvering types that are manually defined, where the red dots represent turning points in maneuvers.

**Figure 11.**Projections of a somersault motion on the vertical plane, where the red dashed arrow indicates the direction of flight.

**Figure 15.**Segment classification: (

**a**) Simple maneuver recognition, (

**b**) Subdivision of simple maneuvers.

**Figure 17.**Classification of pitch angle, where the red dashed line represents the results of linear fitting. (

**a**–

**c**) represent parameter fitting results for three different scenarios, respectively.

**Figure 18.**Roll angle change history, where the red dashed line represents the results of linear fitting.

**Figure 19.**Parameter fitting results (The red dashed line represents the results of fitting). (

**a**) represents the Velocity fitting result, (

**b**) represents the Pitch angle fitting result, and (

**c**) represents the Height fitting result.

Step 1. | Use the PLR–PIP method to identify important points in the normal load factor sequence and use them to describe the sequence. |

Step 2. | Based on the characteristic that maneuvering actions cause changes in the normal load factor, divide all non-level flight segments. |

Step 3. | Select several parameters, such as normal load factor, altitude, pitch angle, roll angle, and heading angle. Use DTW to measure the similarity among segments and use an improved hierarchical clustering algorithm to cluster them. |

Algorithm Parameter | Setting |
---|---|

Fitting error threshold Δ_{PIP} | 0.1 |

Overload slope threshold Δ_{RPIP} | 0.012 |

Average overload threshold $\Delta {\overline{n}}_{y}$ | 1.3 |

IC threshold Δ_{IC} | 0.3 |

Horizontal Projection | Vertical Projection Trend | Superimposing Motion |
---|---|---|

Flat and straight | Level | Level |

Flat and straight | Rise | Climb |

Flat and straight | Lower | Descend |

Curvature movement | Level | Turn |

Curvature movement | Rise | Turn and ascend |

Curvature movement | Lower | Turn and descend |

Maneuver | Speed | Heading Angle | Roll Angle | Superimposing Motion |
---|---|---|---|---|

Level | Maintain | Maintain | Maintain | Uniform level |

Level | Increase | Maintain | Maintain | Accelerated level |

Level | Reduce | Maintain | Maintain | Decelerated level |

Level | Maintain/Increase/Reduce | Maintain | Increase/Reduce | level roll |

Climb | Maintain/Increase | Maintain | Maintain | climb |

Climb | Reduce | Maintain | Maintain | zoom |

Descend | Maintain/Reduce | Maintain | Maintain | descend |

Descend | Increase | Maintain | Maintain | dive |

Descend | Maintain/Increase/Reduce | Increase/Reduce | Reduce | Level left turn |

Descend | Maintain/Increase/Reduce | Increase/Reduce | Increase | Level right turn |

Turn and ascend | Maintain/Increase/Reduce | Increase/Reduce | Reduce | Climbing left Turn |

Turn and ascend | Maintain/Increase/Reduce | Increase/Reduce | Increase | Climbing right Turn |

Turn and descend | Maintain/Increase/Reduce | Increase/Reduce | Reduce | Descent left turn |

Turn and descend | Maintain/Increase/Reduce | Increase/Reduce | Increase | Descent right turn |

Task Segment | Takeoff and Climb | Intermediate Segment | Descent and Landing |
---|---|---|---|

Average fuel consumption kg/h^{−1} | 39.2 | 31.5 | 8.5 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, M.; Xia, S.; Huang, Y.; Tian, J.; Yin, Z.
Research on Engine Thrust and Load Factor Prediction by Novel Flight Maneuver Recognition Based on Flight Test Data. *Aerospace* **2023**, *10*, 961.
https://doi.org/10.3390/aerospace10110961

**AMA Style**

Zhang M, Xia S, Huang Y, Tian J, Yin Z.
Research on Engine Thrust and Load Factor Prediction by Novel Flight Maneuver Recognition Based on Flight Test Data. *Aerospace*. 2023; 10(11):961.
https://doi.org/10.3390/aerospace10110961

**Chicago/Turabian Style**

Zhang, Mengchuang, Shasha Xia, Yongsheng Huang, Jiawei Tian, and Zhiping Yin.
2023. "Research on Engine Thrust and Load Factor Prediction by Novel Flight Maneuver Recognition Based on Flight Test Data" *Aerospace* 10, no. 11: 961.
https://doi.org/10.3390/aerospace10110961