Research on safety assessment methods based on tasks for aircraft takeoff pitch attitude

Abstract

During the aircraft’s takeoff process, the pitch attitude is the key factor determining takeoff safety. It is crucial to objectively and accurately evaluate the safety level of the aircraft’s takeoff pitch attitude. This study quantitatively evaluates the safety level associated with aircraft takeoff pitch attitude. Firstly, based on the task-oriented system safety assessment theory, a safety assessment mathematical model for aircraft takeoff pitch attitude is constructed using key parameters such as the pitch angle and its changing rate in Quick Access Recorder (QAR) data. Secondly, the model was quantified using Flight Operational Quality Assurance (FOQA) and industry-wide attitude data of the A319 fleet in 2023. Then, the model’s effectiveness was verified by comparing the evaluation results with actual flight exceedance situations, using QAR data from a Chinese airline’s A319 fleet. Finally, an in-depth analysis was conducted on the correlation between the pitch attitude at the moment of the main wheels off the ground and the process of lifting the nose wheel, identifying the key moments in the formation of the takeoff pitch attitude. By conducting sensitivity analysis on the safety level of takeoff pitch attitude and the rate at which the nose wheel is lifted, the study proposes suggestions for the optimal range of nose wheel lifting rates for the aircraft. The research has ultimately developed a quantitative evaluation method and formation law for aircraft takeoff pitch attitude, offering new insights and ideas for QAR data-driven flight safety risk management.

Introduction

Aircraft takeoff pitch attitude refers to the whole flight pitch attitude of the aircraft from the start of taxiing to the main wheel off the ground. Aircraft leaving the ground is a key process in the takeoff phase, which has a decisive impact on flight safety¹. The takeoff pitch attitude is related to whether the aircraft can take off successfully and involves safety and efficiency during the takeoff process. Using QAR data to monitor the aircraft takeoff pitch attitude can identify non-compliant takeoff exceedances as early as possible². Although existing FOQA systems effectively identify significant exceedances³, they focus only on the aftermath of an event rather than prevention. It can only use QAR data to determine whether an exceedance event has occurred according to the monitoring criteria, cannot quantitatively assess the flight safety level before the exceedance occurs, does not explore the hidden sources of danger behind the exceedance event, and lacks the prospective identification and judgment of trending risks.

To achieve continuous and absolute safety, it is necessary to study the safety assessment method of aircraft takeoff pitch attitude based on all QAR data (including normal data and exceedance data), to objectively and quantitatively assess the aircraft takeoff pitch attitude, and to reveal potential safety hazards by assessing the key indexes and their development trends to achieve a continuous and comprehensive guarantee of flight safety.

At present, the research on aircraft takeoff pitch attitude mainly focuses on two aspects: first, the flight dynamics analysis based on a physical model; researchers simulate the takeoff process of aircraft through the establishment of an aircraft dynamics model to analyze the aircraft takeoff pitch attitude under different operation strategies and environmental conditions, which is mainly used in the stage of aircraft manufacturing and test flights; and second, the use of QAR data to carry out empirical analysis to study the actual aircraft takeoff pitch attitude in the process of flight after the aircraft has been put into service. The change in rules and safety levels of off-ground attitude during flight and related research techniques have made significant progress. Zhou proposed an adaptive quaternion particle filtering (QPF) method based on the generalized likelihood ratio test (GLRT), which effectively dealt with the attitude assessment problem under anomalous measurement conditions through the introduction of the GLRT⁴. Chen investigated the adaptive decay memory unscented Kalman filtering algorithm, which enhances the accuracy and robustness of estimation by refining the weight allocation during the filtering process⁵. C and Jain compare the multiplicative error state Kalman filter (MEKF) with nonlinear complementary filtering⁶. Yang et al. (2019) and Ping-an et al.(2021) on the study of exploring the aircraft by using the long short-term memory (LSTM) deep neural network and the Unscented Kalman filter (UKF), respectively. Trajectory prediction and attitude measurement provide a new way to deal with nonlinear and complex flight data^7,8. Han focuses on using the Iterative Centre Difference Kalman Filter (ICDKF) to provide a low-cost aircraft attitude assessment system solution⁹. Pozo Dominguez proposed a novel approach to develop up-to-date, tail-specific performance models based on the use of Quick Access Recorder (QAR) data and machine-learning techniques¹⁰.

The above techniques have significantly improved the accuracy and adaptability of attitude assessment. However, most of them focus on analyzing the consequences of a single event and lack a comprehensive assessment of the event formation process. There are still some limitations in the quantitative assessment methods and model building, and the complexity of the aircraft takeoff pitch process requires us to consider the pitch attitude assessment problem from a broader perspective.

The safety assessment of aircraft takeoff pitch attitude requires high accuracy and reliability and has unique task attributes. From the start of taxiing to the departure of the main wheel from the ground, the aircraft must adopt various flight operation strategies to ensure that the task is completed with a safe ground departure attitude under variable flight scenarios and random environmental factor perturbations. The safety assessment of an aircraft’s takeoff pitch attitude cannot be limited to the pitch attitude changes only but must also comprehensively consider the support of the pitch attitude to the task, which is strongly correlated with the task requirements of the aircraft’s takeoff. Therefore, it becomes essential to conduct a safety assessment of aircraft takeoff pitch attitude from a task-oriented perspective and ultimately provide an objective and quantitative assessment of the degree of satisfaction of aircraft takeoff pitch attitude for aircraft takeoff tasks.

In summary, this paper aims to study the quantitative assessment of aircraft takeoff pitch attitude and the formation law. Firstly, take the pitch angle of the aircraft’s main wheel off the ground and the rate of lifting the nose wheel in the process of lifting the nose wheel in the QAR data as the key parameter, and construct the mathematical model of aircraft takeoff pitch attitude safety assessment based on the theory of task-oriented system safety assessment; secondly, based on the specifications of FOQA and the A319 fleet, we will develop an industry-wide attitude category model for aircraft takeoff pitch. 2023 industry-wide attitude class data for model quantification; then, based on the actual flight QAR data of the A319 fleet of a Chinese airline, the accuracy and reliability of the model are verified by comparing the model assessment results with the actual flight data; finally, by analyzing the change rule of the key parameters, the key moments in the process of takeoff pitch attitude formation are further analyzed, and the formation of the aircraft’s takeoff pitch attitude is explored. The model accuracy and reliability of the model are discussed. The correlation between the takeoff pitch attitude at different safety levels and the rate of the nose wheel lift and its actual values are discussed, relevant suggestions on the values of the key parameters during the actual flight operation are put forward, and theoretical analyses are made on the formation of the key moments of the nose wheel lift. Finally, the quantitative assessment method and formation law of aircraft takeoff pitch attitude are derived, providing new insights and ideas for QAR data-driven flight safety risk management.

Aircraft takeoff pitch attitude analysis

The aircraft takeoff pitch attitude analysis includes two parts: the analysis of the moment of the main wheel off the ground and the analysis of the process of lifting the nose wheel. The aircraft’s air-to-ground switch status is GND when on the ground and AIR when in flight. The moment of the main wheel off the ground is the moment when the main wheel air-ground switch changes from ground (GND) to air (AIR)¹¹. The process of lifting the nose wheel refers to the process from the aircraft nose wheel off the ground to the main wheel off the ground, starting from the moment when the nose wheel air-ground switch changes from GND to AIR and ending at the moment when the main wheel air-ground switch changes from GND to AIR, which usually lasts for 4s and corresponds to the four moment points of $\:{t}_{{v}_{R1}}$, $\:{t}_{{v}_{R2}}$, $\:{t}_{{v}_{R3}}$and $\:{t}_{{\nu\:}_{D}}$¹². As shown in Fig. 1, the pitch angle directly characterizes the aircraft takeoff pitch attitude at the moment when the main wheel are off the ground, and its formation process is related to the rate of takeoff of the nose wheel of the nose wheel lifting process, which is then determined by the combination of$\:R\left({t}_{vR1}\right)$, $\:R\left({t}_{vR2}\right)$and $\:R\left({t}_{vR3}\right)$. Table 1 shows the explanations of basic concept variable.

Fig. 1

Process of takeoff pitch attitude formation.

Table 1 Basic concept variables and their definitions.

Time correlation of aircraft takeoff pitch attitude safety

The safety of aircraft takeoff pitch attitude is closely related to time¹³. The assessment criteria and indexes of the pitch attitude are different between the process of lifting the nose wheel and the moment of the main wheel off the ground. The influence of different moment points in the process of lifting the nose wheel on the pitch attitude is also different¹⁴. Therefore, task-oriented safety assessment of aircraft takeoff pitch attitude needs to be able to reflect both the safety level of the pitch attitude at any moment during the ground departure process and the comprehensive safety level of the ground departure process rather than only the safety level of the pitch attitude at the main wheel off the ground¹⁵. The pitch angle $\:P\left({t}_{vD}\right)$ at the moment of lifting the main wheel off the ground is used as the indicator of the safety level of the aircraft takeoff pitch attitude to quantitatively assess the degree of support for the pitch attitude for the aircraft takeoff task. The rate of lifting the nose wheel in the process of lifting the nose wheel $\:R\left({t}_{vR}\right)$ is used as the process indicator to analyze the formation law of the pitch attitude quantitatively.

Task requirements correlation of aircraft takeoff pitch attitude safety

The successful completion of the aircraft takeoff task requires that the pitch attitude meets the corresponding requirements at every moment of the process. Only when the pitch attitude consistently satisfies these task requirements can the takeoff be considered safe. The higher the percentage of time that the pitch attitude meets the task requirements in the process of aircraft takeoff, the higher the safety level. According to the literature¹⁶ traditional assessment methods cannot consider the influence brought by the task demand In the task context, the pitch attitude safety assessment not only needs to take into account the variation of the aircraft takeoff pitch attitude during the takeoff pitch process but also should take into account the task demand, task time, and the stochastic characteristics of the perturbation¹⁷. Different from the description of task demand characteristics in the literature, the aircraft takeoff pitch attitude is not based on the lowest threshold and the higher, the better, but fluctuates up and down around the optimal value, and the smaller the fluctuation, the higher the safety level.

The framework for safety assessment of aircraft takeoff pitch attitude

The safety assessment framework of aircraft takeoff pitch attitude is shown in Fig. 2, including 4 steps:

• (1) Aircraft takeoff pitch attitude analysis: determine two types of key parameters for safety assessment, namely the moment of the main wheel off the ground of the takeoff pitch angle and the rate of lifting the nose wheel of the process of lifting the nose wheel.

• (2) Safety assessment model construction: According to the task-oriented system safety assessment theory, construct the takeoff pitch attitude safety assessment model for the moment of the main wheel off the ground and the pitch attitude safety assessment model for the process of lifting the nose wheel.

• (3) Quantification of safety assessment model: Based on the specifications of the flight operation quality assurance project and the 2023 industry-wide attitude class data of the A319 fleet, determine the optimal values and safety intervals of the takeoff pitch angle and the rate of lifting the nose wheel and quantify the model parameters.

• (4) Case study: Use the QAR data of a Chinese airline’s A319 fleet to conduct model validation analysis, correlation analysis between the moment of the main wheel off the ground and the process of lifting the nose wheel, and key moment analysis of the process of lifting the nose wheel. Analyze the rationality of the key moment of lifting the nose wheel and the sensitivity of the safety level of the pitch attitude.

Fig. 2

Framework for safety assessment of aircraft takeoff pitch attitude.

Methodology

In this section, the aircraft takeoff pitch attitude safety assessment model was constructed and quantified based on the FOQA standard and the 2023 industry-wide attitude class data of the A319 fleet.

Safety assessment model construction

Distinguishing from the literature¹⁶, which characterizes the task requirements in terms of a minimum threshold, the aircraft takeoff pitch attitude safety assessment quantifies the level of support for the task requirements in terms of fluctuations within the safety interval and around the optimum value¹⁸. The fluctuation of the parameter value of the takeoff pitch attitude during the aircraft takeoff pitch is shown in Fig. 3.

Fig. 3

Flight attitude parameter value in $\:{t}_{{\nu\:}_{R1}}-{t}_{\nu\:D}$

Where $\: \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{X}\left(t\right)$ the lower limit of the safety interval of the key parameters of aircraft takeoff pitch attitude; $\:\overline{X}\left(t\right)$ is the upper limit of the safety interval of the key parameters of aircraft takeoff pitch attitude; $\:OX\left(t\right)$ is the optimal value of the key parameters of aircraft takeoff pitch attitude; and $\:X\left(t\right)$ is the actual value of the key parameters of aircraft takeoff pitch attitude at the time t. According to the task-oriented system safety assessment theory, the mathematical model of aircraft takeoff pitch attitude safety assessment is constructed as Eq. (1):

$$\:\begin{array}{c}{S}_{\alpha\:}\left(t\right)=\left\{\:\:\:\:\begin{array}{c}\begin{array}{c}\frac{{\int\:}_{{t}_{{\nu\:}_{R1}}}^{{t}_{{\nu\:}_{D}}}{\left|X\left(t\right)-\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{X}\left(t\right)\right|}^{\alpha\:}\left[X\left(t\right)-\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{X}\left(t\right)\geq 0\right]\left[\overline{X}\left(t\right)-X\left(t\right)\geq 0\right]\text{d}t}{{{\int\:}_{{t}_{{\nu\:}_{R1}}}^{{t}_{{\nu\:}_{D}}}\left(\text{O}X\right(t)-\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{X}\left(t\right))}^{\alpha\:}\text{d}t},X\left(t\right)\le\:OX\left(t\right)\\\:\frac{{\int\:}_{{t}_{{\nu\:}_{R1}}}^{{t}_{{\nu\:}_{D}}}{\left|\overline{X}\left(t\right)-X\left(t\right)\right|}^{\alpha\:}\left[X\left(t\right)-\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{X}\left(t\right)\geq 0\right]\left[\overline{X}\left(t\right)-X\left(t\right)\geq 0\right]\text{d}t}{{{\int\:}_{{t}_{{\nu\:}_{R1}}}^{{t}_{{\nu\:}_{D}}}(\overline{X}\left(t\right)-OX\left(t\right))}^{\alpha\:}\text{d}t},X\left(t\right)>OX\left(t\right)\end{array}\end{array}\right.\end{array}$$

(1)

Where $\:{S}_{\alpha\:}\left(t\right)\:$is the aircraft takeoff pitch attitude safety level; $\:{S}_{\alpha\:}\left(t\right)\in\:\left[\text{0,1}\right]$; [P] is the Iverson bracket; [P] = 1 only if it is P true, otherwise, [P] = 0; $\:\alpha\:\:$is the switching parameter between the time metric and the safety level metric, $\:\alpha\:\in\:\left(\text{0,1}\right)$.

Equation (1) is switched to $\:{S}_{0}\left(t\right)$ and $\:{S}_{1}\left(t\right)$ by switching the parameter $\:\alpha\:$. When $\:\alpha\:=0$ ,as in Eq. (2), $\:{S}_{0}\left(t\right)$ denotes the ratio of the sum of the time for the takeoff pitch to satisfy the takeoff task requirements to the current takeoff duration in the time period from $\:{t}_{{\nu\:}_{R1}}$ to $\:{t}_{{\nu\:}_{D}}$. When $\:X\left(t\right)$ is not within the safety interval, $\:{S}_{0}\left(t\right)=0$, which can be used to determine whether an aircraft takeoff exceedance event is triggered.

$$\:\begin{array}{c}{S}_{0}\left(t\right)=\frac{{\int\:}_{{t}_{{\nu\:}_{R1}}}^{{t}_{{\nu\:}_{D}}}\left[X\left(t\right)-\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{X}\left(t\right)\geq 0\right]\left[\overline{X}\left(t\right)-X\left(t\right)\geq 0\right]\text{d}t}{{t}_{{\nu\:}_{D}}-{t}_{{\nu\:}_{R1}}}\end{array}$$

(2)

When $\:\alpha\:=1$, as in Eq. (3), $\:{S}_{1}\left(t\right)$ represents the distance between the actual value of the pitch attitude key parameter $\:X\left(t\right)$ and the optimal value $\:OX\left(t\right)$ during the effective task time period (when $\:X\left(t\right)$ is within the safety interval). The closer the distance between $\:X\left(t\right)$ and $\:OX\left(t\right)$, the higher the value of$\:\:{S}_{1}\left(t\right)$. Table 2 shows the explanations of model construction variable.

$$\:\begin{array}{c}{S}_{1}\left(t\right)=\left\{\begin{array}{c}\frac{{\int\:}_{{t}_{{\nu\:}_{R1}}}^{{t}_{{\nu\:}_{D}}}\left|\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{X}\left(t\right)-X\left(t\right)\right|\left[X\left(t\right)-\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{X}\left(t\right)\geq 0\right]\left[\overline{X}\left(t\right)-X\left(t\right)\geq 0\right]\text{d}t}{{\int\:}_{{t}_{{\nu\:}_{R1}}}^{{t}_{{\nu\:}_{D}}}(OX\left(t\right)-\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{X}\left(t\right))\text{d}t},X\left(t\right)\le\:OX\left(t\right)\\\:\frac{{\int\:}_{{t}_{{\nu\:}_{R1}}}^{{t}_{{\nu\:}_{D}}}\left|\overline{X}\left(t\right)-X\left(t\right)\right|\left[X\left(t\right)-\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{X}\left(t\right)\geq 0\right]\left[\overline{X}\left(t\right)-X\left(t\right)\geq 0\right]\text{d}t}{{\int\:}_{{t}_{{\nu\:}_{R1}}}^{{t}_{{\nu\:}_{D}}}(\overline{X}\left(t\right)-OX\left(t\right))\text{d}t},X\left(t\right)>OX\left(t\right)\end{array}\right.\end{array}$$

(3)

Table 2 Model construction variables and their definitions.

Quantification of safety assessment model

Based on the monitoring projects of the takeoff pitch attitude category in the ‘Flight operational quality assurance events and deviation limits for Boeing and Airbus aircraft’¹⁹ and the 2023 industry-wide attitude class data of the A319 fleet, the optimal values of the key parameters of the aircraft takeoff pitch attitude, the upper and lower safety intervals are determined, and the quantitative model parameters are quantified, as shown in Tables 3, to obtain the takeoff pitch attitude safety assessment model for the moment of the main wheel off the ground (Eq. (4)) and the pitch attitude safety assessment model for the process of lifting the nose wheel (Eq. (5)).

Table 3 Quantification of model parameters and basis.

The takeoff pitch attitude safety assessment model for the moment of the main wheel off the ground is as follows:

$$\:\begin{array}{c}{S}_{\alpha\:P}\left(t\right)=\left\{\:\:\:\:\begin{array}{c}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\\\:\begin{array}{c}\frac{{\int\:}_{0}^{1}{\left|P\left({t}_{vD}\right)-4.48\right|}^{\alpha\:}\left[P\left({t}_{vD}\right)-4.48\geq 0\right]\left[12.4-P\left({t}_{vD}\right)\geq 0\right]\text{d}t}{{{\int\:}_{0}^{1}(5.22)}^{\alpha\:}\text{d}t},P\left({t}_{vD}\right)\le\:9.76\\\:\frac{{\int\:}_{0}^{1}{\left|12.4-P\left({t}_{vD}\right)\right|}^{\alpha\:}\left[P\left({t}_{vD}\right)-4.48\geq 0\right]\left[12.4-P\left({t}_{vD}\right)\geq 0\right]\text{d}t}{{{\int\:}_{0}^{1}(2.64)}^{\alpha\:}\text{d}t},P\left({t}_{vD}\right)>9.76\end{array}\end{array}\right.\end{array}$$

(4)

Where $\:P\left({t}_{vD}\right)\:$is the pitch angle of the aircraft main wheel off the ground.

The pitch attitude safety assessment model for the process of lifting the nose wheel is as follows:

$$\:\begin{array}{c}{S}_{\alpha\:R}\left(t\right)=\left\{\:\:\:\:\:\:\begin{array}{c}\:\:\:\:\\\:\begin{array}{c}\frac{{\int\:}_{0}^{3}{\left|R\left({t}_{vR}\right)-1.3\right|}^{\alpha\:}\left[R\left({t}_{vR}\right)-1.3\geq 0\right]\left[3.5-R\left({t}_{vR}\right)\geq 0\right]\text{d}t}{{{\int\:}_{0}^{3}(0.22)}^{\alpha\:}\text{d}t},R\left({t}_{vR}\right)\le\:1.52\\\:\frac{{\int\:}_{0}^{3}{\left|3.5-R\left({t}_{vR}\right)\right|}^{\alpha\:}\left[R\left({t}_{vR}\right)-1.3\geq 0\right]\left[3.5-R\left({t}_{vR}\right)\geq 0\right]\text{d}t}{{{\int\:}_{0}^{3}(1.98)}^{\alpha\:}\text{d}t},R\left({t}_{vR}\right)>1.52\end{array}\end{array}\right.\:\:\:\end{array}$$

(5)

Where $\:R\left({t}_{vR}\right)$ is the rate of lifting the nose wheel in the process of lifting the nose wheel.

Case study

This study analyzed the A319 QAR data of a single Chinese airline, focusing on flights taking off from a Class 4E international civil airport. Given the diverse topographical conditions, data collection was restricted to this single location, totaling 453 flights (1812 segments).

The criteria for data selection are designed to minimize the influence of external conditions on pilot operations, which can be affected by various factors such as the flight phase, aircraft model conditions, airport conditions, and flight weather conditions. To ensure data reliability, flights with complete takeoff phase data were selected, and takeoff airports with non-special flight conditions were prioritized. Additionally, the study focused on the A319 aircraft model, with a consistent takeoff configuration determined for all selected flights. Furthermore, the wind speed and direction during the takeoff phase were limited to reduce variability.

The takeoff attitude develops within 4 s of lifting the nose wheels of the aircraft, with the dataset focusing on QAR data from this phase. Organized, merged, and cleared 453 flights (1812 segments), resulting in 379 flights (1516 segments). In fact, the number of pitch exceedance flights is far less than the number of normal flights. In order to avoid sample imbalance, a comprehensive comparison is made between normal flights and exceedance flights. We focus on the same time window and select 1200 normal segments and 316 exceedance segments from the corresponding groups.

The dataset contains variables such as numerical and categorical variables; for example, pitch angle is a numerical variable, while air-ground switch is a categorical variable. Key parameters included in the dataset are pitch angle and the rate of lifting the nose wheel. Collect key parameters $\:R\left({t}_{vR1}\right)$, $\:R\left({t}_{vR2}\right)$, $\:R\left({t}_{vR3}\right)\:$, and$\:P\left({t}_{vD}\right)$from the raw QAR data during the takeoff phase (4 s of the process of lifting the nose wheel) to comprehensively reflect the pitch attitude of the aircraft. The description and statistics of key parameters are shown in Table 4 below.

Table 4 Description and statistics of key parameters.

The data of 1516 valid flight segments are brought into Eq. (4) and Eq. (5), and the assessment results of $\:{S}_{0}\left(t\right)$ are compared with the actual results of the $\:\text{F}\left(t\right)\:$to verify the validity of the model. According to the assessment results of $\:{S}_{1R}\left({t}_{{\nu\:}_{R}}\right)$and $\:{S}_{1P}\left({t}_{{\nu\:}_{D}}\right)$, a correlation analysis is conducted between the process of lifting the nose wheel and the pitch attitude of the moment of the main wheel off the ground to quantitatively assess the safety level of pitch attitude. From the assessment results of $\:{S}_{1R}\left({t}_{{\nu\:}_{R1}}\right)$, $\:{S}_{1R}\left({t}_{{\nu\:}_{R2}}\right)$ and $\:{S}_{1R}\left({t}_{{\nu\:}_{R3}}\right)$, the key moment analysis of the takeoff pitch attitude of the nose wheel is carried out to analyze and derive the aircraft takeoff pitch attitude formation law as shown in Fig. 4. Table 5 shows the explanations of model validation variable.

Table 5 Model validation variables and their definitions.

Fig. 4

Case study structure diagram.

Safety assessment model validation

Based on the ‘Flight Operational Quality Assurance Events and Deviation Limits for Boeing and Airbus Aircraft,’ the actual exceedances for 1516 flight segments were identified. These include the moment of the main wheel off the ground exceedance $\:F\left({t}_{{\nu\:}_{D}}\right)$, lift nose wheel 1st-second exceedance $\:F\left({t}_{{\nu\:}_{R1}}\right)$, lift nose wheel 2nd-second exceedance $\:F\left({t}_{{\nu\:}_{R2}}\right)$, and lift nose wheel 3rd-second exceedance $\:F\left({t}_{{\nu\:}_{R3}}\right)$. and when the value of 0, the exceedance is triggered; when the value of 1, the exceedance is not triggered.

The data of 1516 segments containing the pitch angle of the moment of the main wheel off the ground and the rate of lifting the nose wheel in the process of lifting the nose wheel are brought into Eq. (4) and Eq. (5), and the results of the aircraft takeoff pitch attitude assessment are calculated as $\:{S}_{0P}\left({t}_{{\nu\:}_{D}}\right)$, $\:{S}_{0R}\left({t}_{{\nu\:}_{R1}}\right)$, $\:{S}_{0R}\left({t}_{{\nu\:}_{R2}}\right)$, and $\:{S}_{0R}\left({t}_{{\nu\:}_{R3}}\right)$, and when the value is 0, the safety level is the lowest; when the value is 1, the safety level is highest.

The results of the comparison between the model evaluation value and the actual monitoring value at the same moment are shown in Fig. 5, and the results of the two are consistent. The model evaluation results show that when the safety level is the highest, the actual monitoring results show that the exceedance is not triggered, and vice versa, which verifies the model’s validity. However, $\:{S}_{0}\left(t\right)$ can only indicate whether the exceedance event is triggered, and the degree of support of aircraft takeoff pitch attitude for the takeoff task before the exceedance event is triggered needs to be quantitatively assessed by $\:{S}_{1}\left(t\right)$.

Fig. 5

Comparison between the model evaluation value and the actual monitoring value.

Safety level correlation analysis between the moment of the main wheel off the ground and the process of lifting the nose wheel

The data of the 1516 segments containing the pitch angle of the moment of the main wheel off the ground and the rate of lifting the nose wheel during the process of lifting the nose wheel are brought into Eq. (4) and Eq. (5) to calculate the pitch attitude safety level $\:{S}_{1P}\left({t}_{{\nu\:}_{D}}\right)$ at the moment of the main wheel off the ground and the pitch attitude safety level $\:{S}_{1R}\left({t}_{{\nu\:}_{R}}\right)$ during the process of lifting the nose wheel. When the evaluated actual value $\:{S}_{1}\left(t\right)$ is greater than the evaluated mean value $\:\overline{{S}_{1}\left(t\right)}$ ($\:{S}_{1}\left(t\right)-\overline{{S}_{1}\left(t\right)}\ge\:0$), the level of safety is high, and vice versa. From Fig. 6, $\:{S}_{1P}\left({t}_{{\nu\:}_{D}}\right)$ and $\:{S}_{1R}\left({t}_{{\nu\:}_{R}}\right)$ are positively correlated; the specific comparison data are shown in Table 6. When$\:\:{S}_{1R}\left({t}_{{\nu\:}_{R}}\right)$ has a high-security level, the probability of $\:{S}_{1P}\left({t}_{{\nu\:}_{D}}\right)$having a high-security level is 69%, and when $\:{S}_{1R}\left({t}_{{\nu\:}_{R}}\right)$ has a low-security level, $\:{S}_{1P}\left({t}_{{\nu\:}_{D}}\right)$ has a probability of 85% for a low safety level. It can be seen that the safety level of the pitch attitude at the moment of the main wheel off the ground is determined by the safety level of the process of lifting the nose wheel in the pitch attitude, and the process safety level can determine the result safety level.

Fig. 6

Comparison of safety assessment levels between the process of lifting the nose wheel and the moment of the main wheel off the ground.

Table 6 Probability table of safety assessment metrics for the process of lifting the nose wheel and the moment of the main wheel off the ground.

The key moment analysis of the process of lifting the nose wheel

The data of 1516 segments containing the rate of lifting the nose wheel for the process of lifting the nose wheel are brought into Eq. (5), and $\:{S}_{1R}\left({t}_{{\nu\:}_{R}}\right)$, $\:{S}_{1R}\left({t}_{{\nu\:}_{R1}}\right)$,$\:{S}_{1R}\left({t}_{{\nu\:}_{R2}}\right)$, and$\:{S}_{1R}\left({t}_{{\nu\:}_{R3}}\right)$ are calculated. The correlation between the pitch attitude safety level $\:{S}_{1R}\left({t}_{{\nu\:}_{R1}}\right)$, $\:{S}_{1R}\left({t}_{{\nu\:}_{R2}}\right)$,and$\:{S}_{1R}\left({t}_{{\nu\:}_{R3}}\right)$and the pitch attitude integrated safety level $\:{S}_{1R}\left({t}_{{\nu\:}_{R}}\right)$of the process of lifting the nose wheel at each moment is analysed separately, and the key moment of lifting the nose wheel has been identified through screening. The safety level is high when the evaluated actual value$\:{\:S}_{1R}\left(t\right)$ is greater than the evaluated mean value $\:\overline{{S}_{1R}\left(t\right)}$and vice versa. Figure 7 shows the direct comparison results, $\:{S}_{1R}\left({t}_{{\nu\:}_{R1}}\right)$ has the highest correlation with $\:{S}_{1R}\left({t}_{{\nu\:}_{R}}\right)$, and the probability of both being the same as high and low is higher than that of $\:{S}_{1R}\left({t}_{{\nu\:}_{R2}}\right)$ and $\:{S}_{1R}\left({t}_{{\nu\:}_{R3}}\right)$, which indicates that the 1st-second of the process of lifting the nose wheel is the key moment.

Fig. 7

Probability plots related to the safety level of pitch attitude for each second of lifting the nose wheel and the overall rate of lifting the nose wheel.

The correlation of$\:\:{S}_{1R}\left({t}_{{\nu\:}_{R}}\right)$ with $\:{S}_{1R}\left({t}_{{\nu\:}_{R1}}\right)$, $\:{S}_{1R}\left({t}_{{\nu\:}_{R2}}\right)$, and $\:{S}_{1R}\left({t}_{{\nu\:}_{R3}}\right)$is analyzed in-depth by using the confusion matrix, and the results of the key moment analysis of the lifted nose wheel are further verified. The confusion matrix is constructed by taking $\:{S}_{1R}\left({t}_{{\nu\:}_{R}}\right)$ as the actual value of the confusion matrix, and $\:{S}_{1R}\left({t}_{{\nu\:}_{R1}}\right)$, $\:{S}_{1R}\left({t}_{{\nu\:}_{R2}}\right)$, and$\:{\:S}_{1R}\left({t}_{{\nu\:}_{R3}}\right)$ as the predicted values of the confusion matrix, as shown in Table 7.

Table 7 Confusion matrix analysed at the key moment of lifting the nose wheel.

• accuracy: $\:accuracy=\frac{TP+TN}{P+N}$, denotes the proportion of correct predictions. It measures how well a classify predicts a condition²².

• precision: $\:precision=\frac{TP}{TP+FP}$, denotes the proportion of examples classified as positive examples that are actually positive examples.

• (3) recall: $\:recall=\frac{TP}{TP+FN}$, indicates the proportion between truly positive predictions and the set of all the positive predictions²².

As can be seen from Table 8, the accuracy, precision, and recall of$\:\:{S}_{1R}\left({t}_{{\nu\:}_{R1}}\right)$ are higher than $\:{S}_{1R}\left({t}_{{\nu\:}_{R2}}\right)$ and $\:{S}_{1R}\left({t}_{{\nu\:}_{R3}}\right)$, the correlation between $\:{S}_{1R}\left({t}_{{\nu\:}_{R1}}\right)$and $\:{S}_{1R}\left({t}_{{\nu\:}_{R}}\right)$ is the highest. The specific correlation data are shown in Fig. 8, which further validates that the 1st second of the ground release process is the key moment of lifting the nose wheel, and the process safety level is mainly determined by the safety level at the key moment.

Table 8 Confusion matrix calculation results.

Fig. 8

Comparison of pitch attitude safety levels between the 1st-second process of lifting the nose wheel and the combined process of lifting the nose wheel.

Discussion

In summary, it can be concluded that the safety level of pitch attitude at the key moment $\:{S}_{1R}\left({t}_{{\nu\:}_{R1}}\right)$, the safety level of pitch attitude during the process of lifting the nose wheel $\:{S}_{1R}\left({t}_{{\nu\:}_{R}}\right)$ and the safety level of pitch attitude at the moment of the main wheel off the ground $\:{S}_{1P}\left({t}_{{\nu\:}_{D}}\right)$ have a causal-time-series relationship. The following is for a specific analysis of the reasonableness and sensitivity of the causal-time-series relationship between $\:{S}_{1R}\left({t}_{{\nu\:}_{R1}}\right)$, $\:{S}_{1R}\left({t}_{{\nu\:}_{R}}\right)$ and$\:{S}_{1P}\left({t}_{{\nu\:}_{D}}\right)$.

The rationality analysis of the key moment of lifting the nose wheel

Based on the Airbus A319 Flight Crew Operating Manual (FCOM), during the process of lifting the nose wheel, the Pilot Flying (PF) must monitor key parameters such as altitude, airspeed, and heading. At Rotation Speed (VR), the PF gently pulls back on the sidestick to lift the nose wheel, avoiding excessive pitch rates and ensuring a positive climb rate. Meanwhile, the Pilot Monitoring (PM) retracts the landing gear to improve aerodynamics and climb efficiency. The 1st-second of the process of lifting the nose wheel is crucial, as it significantly affects subsequent aircraft attitude. This is supported by both the evaluation model and theoretical analysis, highlighting the importance of the initial pitch attitude during takeoff.

The sensitivity analysis of the safety level of pitch attitude

A sensitivity analysis of the moment of the main wheel off the ground pitch attitude safety level $\:{S}_{1P}\left({t}_{{\nu\:}_{D}}\right)$ and the rate of lifting the nose wheel in the process of lifting the nose wheel of the aircraft $\:R\left({t}_{{\nu\:}_{R}}\right)$ can be seen as shown in Fig. 9. When $\:{S}_{1P}\left({t}_{{\nu\:}_{D}}\right)$ is high (i.e., $\:{S}_{1P}\left({t}_{{\nu\:}_{D}}\right)-\overline{{S}_{1P}\left({t}_{{\nu\:}_{D}}\right)}$≥0), the rate of lifting the nose wheel of the aircraft $\:R\left({t}_{{\nu\:}_{R}}\right)$ takes values concentrated in the interval [1.40°/s-2.81°/s]. Therefore, the pilot lifting the nose wheel of the aircraft at the rate of [1.40°/s-2.81°/s] ensures that the aircraft takeoff pitch attitude at the moment of the main wheel off the ground to obtain a high level of safety.

Fig. 9

Correlation analysis of actual takeoff pitch and rate of lifting the nose wheel at different safety levels.

A sensitivity analysis of the pitch attitude safety level$\:\:{S}_{1R}\left({t}_{{\nu\:}_{R}}\right)$ of the overall rate of lifting the nose wheel and the actual value of the rate of the aircraft’s 1st-second lifting the nose wheel $\:R\left({t}_{{\nu\:}_{R1}}\right)$, shows that, as shown in Fig. 10, when $\:{S}_{1R}\left({t}_{{\nu\:}_{R}}\right)$ is high (i.e., $\:{S}_{1R}\left({t}_{{\nu\:}_{R}}\right)-\overline{{S}_{1R}\left({t}_{{\nu\:}_{R}}\right)}$≥0), the rate of the aircraft’s 1st-seccond lifting the nose wheel is concentrated in the interval [1.41°/s-3.16°/s]. Therefore, the pilot lifting the nose wheel of the aircraft at the 1st-second of lifting the nose wheel with the rate of [1.41°/s-3.16°/s] lifting the nose wheel can ensure that the aircraft takeoff pitch attitude at a higher safety level during the process of lifting the nose wheel, which in turn ensures that the aircraft completes the takeoff pitch attitude at a higher safety level.

Fig. 10

Correlation analysis between the overall rate of lifting the nose wheel and the actual value of the rate of lifting the nose wheel in the 1st-second at different safety levels.

Conclusions

The findings demonstrate that the proposed assessment method considers the lower limit, upper limit, and optimal value of the safety level of the takeoff pitch task, thereby providing a more precise quantification of how the safety level of the takeoff pitch attitude supports operational requirements. The safety assessment model not only facilitates a comprehensive evaluation of pitch attitude safety throughout the takeoff process but also specifically investigates the optimal rate characteristics during both the 1st-second and the overall lifting process of nose wheels.

This study employs the A319 aircraft data standard to determine model parameter values, and respectively the actual A319 flight data was used to validate the rationality and feasibility of the task-oriented safety assessment framework. The establishment idea and structure of the safety assessment model are based on task-oriented theory, which is universal and applicable to all aircraft models. Only by reconfiguring the model parameters based on the data standard of the respective aircraft type.

Pitch attitude is taken as the starting point of the study, which demonstrates the feasibility and effectiveness of the task-oriented assessment model. The next step will involve expanding the research to a comprehensive attitude assessment, incorporating multiple parameters such as airspeed.