3.1 Optical deviation fuzzification

During carrier-based aircraft landing, pilots determine the height deviation relative to the reference glide path by observing the relative position between the `meatball’ and the datum lights in the Fresnel Lens Optical Landing System (FLOLS). This visual perception process relies heavily on subjective judgement and pilot experience, which introduces inherent uncertainty and ambiguity. Therefore, fuzzy sets are employed to model the optical deviation information, mapping continuously varying optical positions into computational fuzzy states.

As the aircraft deviates from the glide slope, the meatball position varies continuously (Figure 3). Its relative position with respect to the datum lights produces several typical visual states, such as `one ball high’, `between one and two balls high’ and `two balls high’. These states represent the pilot’s fuzzy perception intervals of altitude deviation.

Figure 3.

Fuzzy perception process of optical deviation in the FLOLS: (a) one-ball high; (b) between one-ball and two-balls high; and (c) two-balls high.

For a five-cell FLOLS system, the input space can be discretised into five representative optical states: (i) the meatball located at the second reference light above the datum light; (ii) the first reference light above the datum light; (iii) aligned with the datum light; (iv) the first reference light below the datum light; and (v) the second reference light below the datum light.

Correspondingly, the output space is quantised into five levels of height deviation: (i) two-balls high deviation; (ii) one-ball high deviation; (iii) on-glide (zero deviation); (iv) one-ball low deviation; and (v) two-balls low deviation.

This input–output relationship forms the basis of the fuzzy inference system.

3.2 Fuzzy perception modelling

To represent the continuous variation of optical deviation, the input variable is modelled using triangular membership functions, consistent with the approximately linear relationship between the optical cue and altitude deviation. The corresponding linguistic variable set is defined as

(1) \begin{equation} \mathcal{O} = \{\mathrm{OB}, \mathrm{SB}, \mathrm{FB}, \mathrm{HB}, \mathrm{DL}, \mathrm{HA}, \mathrm{FA}, \mathrm{SA}, \mathrm{OA}\} \end{equation}

Here, $\mathcal{O}$ denotes the linguistic variable set of the IFLOLS optical position, where $\mathrm{OB}$ , $\mathrm{SB}$ , $\mathrm{FB}$ , $\mathrm{HB}$ , $\mathrm{DL}$ , $\mathrm{HA}$ , $\mathrm{FA}$ , $\mathrm{SA}$ and $\mathrm{OA}$ denote outer below, second below, first below, halfway below, datum line, halfway above, first above, second above, and outer above, respectively.

Considering the subjective uncertainty in pilot perception, the output variable is modelled using Gaussian membership functions to represent the continuously varying perceptual state. The linguistic variable set is defined as

(2) \begin{equation} \mathcal{P} = \{\mathrm{EL}, \mathrm{TL}, \mathrm{OL}, \mathrm{HL}, \mathrm{CB}, \mathrm{HH}, \mathrm{OH}, \mathrm{TH}, \mathrm{EH}\} \end{equation}

Here, $\mathcal{P}$ denotes the linguistic variable set of the perceived height-deviation state, where $\mathrm{EL}$ , $\mathrm{TL}$ , $\mathrm{OL}$ , $\mathrm{HL}$ , $\mathrm{CB}$ , $\mathrm{HH}$ , $\mathrm{OH}$ , $\mathrm{TH}$ , and $\mathrm{EH}$ denote extremely low, two-balls low, one-ball low, half-ball low, centre ball, half-ball high, one-ball high, two-balls high and extremely high, respectively.

The membership functions of the input and output variables are illustrated in Figure 4. The triangular membership functions on the input side reflect the linear geometric distribution of optical cues, whereas the Gaussian membership functions on the output side describe the nonlinear decay of perceptual sensitivity as the deviation increases.

Figure 4.

Membership functions used in pilot perception modelling: (a) input membership functions; (b) output membership functions.

Based on these membership functions, fuzzy input–output mapping rules are constructed to describe the relationship between the input variables and the trajectory stability state. The fuzzy output variable is defined as the trajectory stability state

(3) \begin{equation} S_h \in [\!-1,1] \end{equation}

where $S_h$ denotes the fuzzy trajectory stability index, and its linguistic variable set is defined as $\{TH, SH, SS, SL, TL\}$ , corresponding to too high, slightly high, stable, slightly low and too low, respectively. The fuzzy inference process adopts the min–max composition operator, and defuzzification is performed using the centroid method. The defuzzified quantitative output is denoted by $h$ , which serves as the reference label for subsequent data-driven modelling and evaluation. It should be noted that $h$ is a model-based cognitive proxy rather than a directly observed physiological variable. In this study, $h$ is introduced to provide a unified and quantitatively tractable representation of the pilot’s integrated perception of glide-path deviation under multi-source cues. The resulting fuzzy rule base is listed in Table 1. For brevity, Table 1 presents representative rules under a fixed optical-deviation condition, while the complete fuzzy rule base is constructed using the three inputs $\Delta d$ , $\Delta s$ and $\Delta l$ . Here, the linguistic labels of $\Delta s$ and $\Delta l$ denote representative fuzzy subsets of height deviation and the correction rate of height deviation, respectively; only representative rules under a fixed $\Delta d$ condition are listed for brevity.

Table 1.

Representative partial fuzzy inference rules under a fixed $\Delta d$ condition; only representative rules are shown for brevity

Table 1. Long description

A table titled ‘Representative partial fuzzy inference rules under a fixed condition’ with five rows and four columns. The columns are labeled ‘Rule’, ‘Δs’, ‘Δl’, and ‘S_h’. The rows are labeled with rules R1 to R5. The values in the table are as follows: Row 1: Rule R1, Δs PU, Δl SD, S_h TH; Row 2: Rule R2, Δs SU, Δl LD, S_h SH; Row 3: Rule R3, Δs LL, Δl ST, S_h SS; Row 4: Rule R4, Δs SD, Δl LA, S_h SL; Row 5: Rule R5, Δs PD, Δl SA, S_h TL.

To achieve a unified semantic representation of multi-source information, the optical deviation, height deviation relative to the reference glide path and the correction rate of height deviation are fuzzified to construct the pilot perception input space. As illustrated in Figure 5, the fuzzy perception system consists of three main modules: fuzzification, fuzzy rule inference and defuzzification. The IFLOLS and HUD information are first processed through fuzzification and then fed into the rule inference module. Fuzzy reasoning is performed based on the fuzzy rule base and membership functions, yielding the fuzzy output state $S_h$ . The defuzzified pilot perception proxy index $h$ is then obtained through centroid defuzzification.

Figure 5.

Architecture of the fuzzy perception system.

The inference rule can be expressed as

(4) \begin{equation} \begin{aligned} R_i\,:\quad &\text{IF } \Delta d \text{ is } A_i,\ \Delta s \text{ is } B_i,\ \text{and } \Delta l \text{ is } C_i, \\ &\text{THEN } S_h \text{ is } D_i \end{aligned} \end{equation}

where $A_i$ , $B_i$ and $C_i$ denote the fuzzy subsets of the input variables, and $D_i$ represents the fuzzy subset of the fuzzy output state $S_h$ .

The centroid method is used for defuzzification to obtain the quantitative estimate of the perception proxy index $h$ , which provides structured inputs for the subsequent perception fusion modelling based on a BRNN.

3.3 BRNN-based perception fusion model

Although fuzzy inference provides a structured representation of multi-source perceptual information, the relationship between the input variables and the perception proxy index $h$ remains highly nonlinear. To capture this nonlinear mapping, a BRNN is adopted for perception fusion modelling.

Compared with conventional neural networks, BRNN introduces Bayesian regularisation into the training process. By incorporating prior distributions of network weights into the loss function, the model automatically balances the trade-off between data fitting and network complexity. This mechanism effectively prevents overfitting and improves generalisation performance, particularly when dealing with noisy or limited data.

The optimisation objective of the BRNN can be expressed as

(5) \begin{equation} F = \beta E_D + \alpha E_W \end{equation}

where $E_D$ denotes the data error term and $E_W$ represents the weight penalty term. The hyperparameters $\alpha$ and $\beta$ correspond to the precision parameters associated with the weight prior and the data error, respectively. During training, these hyperparameters are automatically updated according to the posterior probability distribution, which balances model complexity and data fitting capability:

(6) \begin{equation} \alpha = \frac {\gamma }{2E_W}, \qquad \beta = \frac {N - \gamma }{2E_D} \end{equation}

where $N$ denotes the total number of samples and $\gamma$ represents the effective number of network parameters.

The BRNN architecture consists of three input neurons, two hidden layers with 32 and 16 neurons respectively, and one output neuron. The hidden layers employ the tansig activation function, while the output layer adopts a purelin activation function. The network is trained using the Levenberg–Marquardt (LM) algorithm.

4.1 Data description and preliminary analysis

The experimental data were obtained from a self-developed carrier-based aircraft–carrier landing environment simulation system. The dataset was generated using a self-developed carrier-based landing simulation environment based on an F/A-18 model. In addition, an IFLOLS is embedded to reproduce the visual guidance signals during landing.

The simulation generates multi-source data including flight attitude, velocity, optical deviation and control commands, resulting in a total of 110,137 samples. After synchronisation and normalisation, $\Delta d$ , $\Delta s$ and $\Delta l$ were selected as the input features, while the defuzzified pilot perception proxy index $h$ , generated by the fuzzy inference system, was taken as the output variable. Here, $h$ serves as a model-based reference label for data-driven modelling, rather than a directly observed physiological or subjective pilot-state measurement. The statistical characteristics of the main variables are summarised in Table 2.

Table 2.

Statistical characteristics of key variables during the carrier-landing process

Table 2. Long description

A table with four rows and five columns summarizing the statistical characteristics of key variables during the carrier-landing process. The columns are labeled Variable, Mean, Std, Min, and Max. Row 1: Variable, Mean, Std, Min, Max. Row 2: Δd, 0.13308, 14.248, -225, 268.5. Row 3: Δs, 0.33421, 34.153, -553.8, 609.04. Row 4: Δl, 0.1787, 10.144, -96.448, 72.383. Row 5: h, 0.00060477, 0.195, -0.92961, 0.90575.

To analyse the relationship between the input features and the perception proxy index $h$ , a Pearson correlation analysis was conducted, as shown in Figure 6(a). The results indicate that the height deviation $\Delta s$ shows the strongest positive linear correlation with $h$ , while the optical deviation $\Delta d$ also exhibits a moderate positive correlation. In contrast, the correction rate of height deviation $\Delta l$ is negatively correlated with $h$ , suggesting that the correction trend affects the perception index with an opposite sign. These findings imply that $\Delta s$ and $\Delta d$ provide more direct linear information about pilot perception, whereas the effect of $\Delta l$ is sign-dependent and should be further interpreted through nonlinear interaction analysis.

Figure 6(b) presents a typical time-series comparison between $\Delta s$ and $\Delta l$ , showing a clear synchronous variation trend. This suggests that height deviation and its correction trend are dynamically coupled during the landing process, and therefore should be jointly considered in subsequent perception modelling.

Figure 6.

Correlation of inputs and $\Delta s$ – $\Delta l$ dynamic trends: (a) correlation matrix of input features and output; (b) time-series comparison of $\Delta s$ and $\Delta l$ .

4.2 Model performance and feature analysis

To investigate the influence of multi-source features on pilot perception, both a linear regression baseline model and a nonlinear BRNN model are constructed.

A multiple linear regression model is first established as

(7) \begin{equation} h = a_0 + a_1 \Delta d + a_2 \Delta s + a_3 \Delta l + \varepsilon \end{equation}

The regression results show that the overall correlation coefficient of the model is $R = 0.753$ ( $R^2 = 0.566$ ), indicating that the linear model can capture the main trend of the perception proxy index $h$ but still presents noticeable residual errors.

Figure 7(a) shows that most samples lie close to the diagonal line. Figure 7(b) illustrates the residual–fitted value distribution, showing that the residuals are generally centred on zero with increased dispersion at higher fitted values. Figure 7(c) presents the normal probability plot of the residuals, indicating that the residuals approximately follow a normal distribution. Overall, the linear model is able to capture the global trend of the perception proxy index $h$ but fails to adequately describe the nonlinear relationships among multi-source perceptual variables.

Figure 7.

Linear regression fit and diagnostics: (a) predicted versus reference labels with regression line; (b) residuals versus fitted values; (c) Q-Q plot of residuals.

The training results are summarised as follows. Training set: $R^2 = 0.8797$ , RMSE = 0.0674, MAE = 0.0446. Test set: $R^2 = 0.8780$ , RMSE = 0.0691, MAE = 0.0455. Overall performance: $R^2 = 0.882$ , MSE = 0.0045.

The prediction performance of the BRNN model is illustrated in Figure 8. Figure 8(a) shows that the predicted values closely follow the reference proxy labels generated by fuzzy inference and are concentrated on the ideal diagonal line. Figure 8(b) further demonstrates that the model accurately captures the dynamic variation of the reference proxy index in the time series.

To evaluate model stability, residual analysis is conducted. The residual is defined as

(8) \begin{equation} e_i = h_i - \hat {h}_i \end{equation}

where $h_i$ denotes the reference proxy label generated by fuzzy inference and $\hat {h}_i$ is the predicted value.

Figure 8.

BRNN prediction performance and residual diagnostics: (a) predicted versus reference labels; (b) comparison of reference and predicted results for the first 2,000 samples; (c) residual distribution with normal fit; (d) residual sequence with mean and zero reference lines.

As shown in Figure 8(c)–(d), the residuals approximately follow a zero-mean normal distribution and fluctuate randomly around the zero line, indicating the absence of systematic bias. The cross-validation and feature-combination results are shown in Figure 9. As illustrated in Figure 9(a), the coefficient of determination $R^2$ obtained from five-fold cross-validation ranges from 0.876 to 0.883, indicating stable prediction performance across different data partitions.

To further analyse the influence of multi-source inputs, feature ablation experiments were conducted using different combinations of $\Delta d$ , $\Delta s$ and $\Delta l$ while keeping the network architecture unchanged. As shown in Figure 9(b), single-feature models provide limited predictive capability. Introducing the posture-related feature $\Delta s$ significantly improves prediction accuracy ( $R^2=0.835$ ), while the fusion of all three features achieves the best performance ( $R^2=0.879$ , RMSE $=0.068$ , MAE $=0.045$ ). These results indicate that the three perceptual variables exhibit strong complementarity and synergistic effects in modelling the pilot perception index.

Figure 9.

Model validation and feature-combination evaluation: (a) five-fold cross-validation performance in terms of $R^2$ ; (b) performance comparison under different feature combinations based on normalised metrics.

Permutation importance analysis shows that the height deviation $\Delta s$ exhibits the highest contribution (0.845), followed by the correction rate of height deviation $\Delta l$ (0.153), while the optical offset $\Delta d$ shows a negligible contribution (0.003). These results indicate that pilot perception is primarily dominated by height deviation, whereas the correction trend plays a secondary role and the direct influence of optical offset is relatively weak. Although $\Delta d$ shows a moderate pairwise Pearson correlation with $h$ , its marginal contribution becomes negligible after $\Delta s$ and $\Delta l$ are jointly considered in the BRNN model, indicating that the information carried by $\Delta d$ is partly redundant with the other state variables rather than independently dominant. To further reveal the interaction effects between variables, bivariate response surfaces of $\Delta s$ – $\Delta l$ and $\Delta d$ – $\Delta s$ were constructed, as shown in Figure 10(a) and (b). The results indicate that when $\Delta s$ is positive, the output $h$ increases correspondingly. Under this condition, a negative deviation in $\Delta l$ further amplifies the response, suggesting that pilots become more sensitive to altitude variations during the posture correction phase. Furthermore, the $\Delta d$ – $\Delta s$ surface exhibits a tilted band-like distribution. When $\Delta d$ and $\Delta s$ vary in the same direction, the predicted pilot perception index $\hat {h}$ increases; otherwise, the response decreases.

Figure 10.

Feature importance and nonlinear interaction results: (a) $\Delta s$ – $\Delta l$ response surface; (b) $\Delta d$ – $\Delta s$ response surface.

Overall, pilot perception during carrier landing follows a multi-source fusion pattern characterised by altitude-deviation dominance, correction-trend assistance and weak independent optical contribution. These contribution patterns may provide a candidate basis for parameterising control-authority regulation in future human–machine shared-control design. As an application-oriented interpretation of the feature-contribution results, a candidate control-authority allocation function can be formulated as

(9) \begin{equation} \lambda (t)=\frac {1}{1+\exp\! [\!-k(\Delta u-\Delta u_0)]} \end{equation}

where $k$ is the sensitivity coefficient, $\Delta u$ represents the pilot control input deviation and $\Delta u_0$ denotes the takeover threshold.

According to the feature-importance weights ( $w_s=0.845$ , $w_l=0.153$ and $w_d=0.003$ ), the sensitivity coefficient $k$ is defined as

(10) \begin{equation} { k=k_0\left (0.845|\Delta s|+0.153|\Delta l|+0.003|\Delta d|\right )\!. } \end{equation}

where $k_0$ is a normalisation coefficient.

Under this formulation, larger perceived deviation levels would lead to a higher sensitivity of control-authority adjustment. This relationship is intended as a physically interpretable application of the perception model, rather than a fully validated shared-control law in the present study.