Multi-objective optimization method for reducing mutual interference in cockpit illumination

Li Zhou; Li Zhou; Li Zhou; Liangzhuang Wei; Liangzhuang Wei; Jun Song; Chao Ruan; Huishuang Wang; Yandan Lin; Yandan Lin; Yandan Lin

doi:10.1364/OE.451828

1. Introduction

The pilot receives more than 90% of information through vision [1], and visual tasks are an essential part for a safe flight. Visual tasks in the aircraft cockpit include subjective visual perception and visual performance tasks. Improper lighting can affect subjective perception by causing visual fatigue and emotional irritability for pilots [2]. Tasks on the aircraft comprise observation of the visual interface, which is an important part of human-computer interaction. Execution of flight crew visual performance tasks is determined by accuracy, speed, and comfort. Visual performance factors include task types, luminance, illuminance, contrast, color and glare of the whole lighting [3]. Thresholds for several safety requirements are designed for luminescence parameters in the cockpit. For example, SAE ARP1048 provides standards for illumination parameters in aircraft cockpits, including typical area illumination, brightness and brightness contrast and color coordinates [4]. In addition, SAE AIR1151 provides the requirements for glare [5].

The standards for the dark ambient mainly focus on the area lighting. Area lighting provides illumination on surfaces to aid visibility under dark or low surrounding lighting conditions. Area lighting should not interfere with a clear visualization of or the legibility of instruments, displays, switches and controls. Light distribution and shielding should eliminate direct or indirect glare [3].

Therefore, there is need to set strict optical requirements in the flight. The optical modulation should be consistent with other rigid standards of the aircraft under the conventional design of a flight. For instance, the shape of the windshield should meet the requirements of aerodynamics before considering the optical requirements first [6]. Design of parameters of a single light-emitting and display is conducted by the manufacturers. The optical environment of the entire flight comprises various types of light-emitting and display equipment. Selection and assembly of hardware is already completed during the evaluation phase of the overall optical environment of the flight. A study on cockpit lighting by the U.S. air force reported that the pilots were not comfortable with illumination and spot of the read lighting [7]. The floodlights were turned on as auxiliary lighting to get a comfortable lighting for reading, resulting in new glare sources into the pilot’s field of vision. Moreover, the overall brightness of the flight was increased which thus affect the pilot's dark-adapted conditions [8].

Therefore, the optimal value of single target surface is set to match a set of parameters of the lamp during the design of cockpit lighting, representing a single-objective optimization. However, the parameters may affect the result of the other target surfaces, thus these data cannot be directly used in the cockpit. Multiple optical parameters of different target surfaces should be considered as multiple optimized targets in the early stages of the optical design. A comprehensive search of indicators of all optical devices should then be conducted, which is a multi-objective decision-making process. A framework for formulating merit-based multi-objective optimization was introduced as constrained optimization problems for the synthesis of compounds [9]. The method sought to optimize one main goal, while taking other goals as constraints and keeping them within the desired range. A multi-objective optimization method based on simulation results, at the stage of cockpit optical simulation design, should be a probable approach with the industry standards set as constraints. The approach can find a combination of ideal optical indicators, including the parameters of different lamps and the optical indicators of light-emitting devices. Non-dominant sorting genetic algorithm II (NSGA-II) [10] is one of the most effective multi-objective genetic algorithms for solving Pareto optimal solution, NSGA-II was proposed on the basis of NSGA [11] by Deb in 2000 [12].

Notably, the values of constraints and optimization objectives for a compact cockpit lighting are not strictly linear related to the input optical variables. Therefore, a complex mathematical relationship exists between input and output variables of lighting optimization objectives. It is difficult to find an expression to meet this mapping relationship. Studies should explore an appropriate mapping relationship as the objective function of multi-objective optimization limitations. Gaussian process regression (GPR) is a novel machine learning method developed based on Bayesian theory and statistical learning theory [13]. GPR is used to effectively solve complex and small samples regression problems. Therefore, a kernel based on GPR model was developed in the current study to predict the illuminance and uniformity of the target surface under different lighting conditions, which are important indicators in cockpit illumination and well-defined requirements in the standards [4].

Several tests are carried out to determine the parameter combination of lamps for rapid determination of the optimal lighting design combination. This trial and error method is time consuming, and it is computationally expensive to effectively obtain the optimal combination for several optimization objectives. In order to meet the requirements of multiple target surfaces at the same time, a GPR-NSGA-II framework for cockpit lighting design optimization to optimize the parameters of lighting lamps in the cockpit is proposed in the present study. Different combinations of lamp parameters were explored through orthogonal experiment. The corresponding GPR prediction model of target surface illumination and uniformity was established based on the orthogonal test results. In addition, the multi-objective genetic optimization and Pareto optimal front of GPR-NSGA-II were described in detail.

2. Computational scheme

The proposed framework can be divided into two parts: prediction model training and NSGA-II method, as shown in Fig. 1. Analysis of a specific case can be conducted following the procedure presented in the flow chart, respectively starting from the training part and the NSGA-II parameter setting, thus getting the optimization results.

Fig. 1. A flow chart showing the key steps for GPR-NSGA-II model

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2.1 Orthogonal experimental method

Orthogonal experimental method is an effective method for exploring experimental design for multi-factor and multi-level studies developed by Taguchi and Konishi [14,15]. Several design factors and levels are considered in the lighting design. Moreover, interaction occurs among the experimental design factors. Use of trial and error method results in a large workload which is difficult to implement. Therefore, the orthogonal experimental design method should be initially used for collection and analysis of experimental data based on the specific case for the framework proposed in this study.

2.2 Prediction model – Gaussian process regression (GPR)

In this section, the detailed derivation of Gaussian process is introduced firstly. Then, in order to explore the influence of different covariance functions on prediction accuracy, different covariance functions used in the subsequent experiments are introduced. And finally the hyperparametric optimization method of GPR is described.

2.2.1 Estimation using the Gaussian process

It is assumed that a training set (target of interest) of the following form is available:

(1)$$T = \{{({{x_1},{y_1}} ), \cdots ,({{x_m},{y_m}} )} \}\subset {{\mathbb R}^k} \times {{\mathbb R}^n}$$

where, m represents the number of the training data, x and $T$ y represent the input and output values of the training data, respectively.

Traditional regression methods, such as linear regression, and polynomial regression methods, seek to obtain the corresponding transformation matrix based on the training set. The parametric model is abandoned in GPR method and the prior probability distribution is directly defined based on the function. The distribution of a function can be described by a Gaussian process $GP({\cdot} )$, which is determined by the mean function $m(x)$ and covariance function ${\textbf k}(x,x^{\prime})$ :

(2)$$f(x)\sim GP(m(x),{\textbf k}(x,x^{\prime}))$$

where,

(3) $$m(x) = E(f(x))$$

(4)$${\textbf k}(x,x^{\prime}) = E((f(x) - m(x))(f(x^{\prime}) - m(x^{\prime})))$$

For the convenience of calculation, the mean function is usually set to 0.

The output data of GPR method includes the target function $f(x)$ and background noise, as shown below:

(5)$${\textbf y} = f({\textbf x}) + {\mathbf \varepsilon }$$

where, ${\mathbf \varepsilon }$ represents the independently and identically distributed noise. Then, we assume that:

(6)$${\mathbf \varepsilon } \sim N({\textbf 0},\sigma _{\mathbf \varepsilon }^2)$$

where ${\sigma _{\mathbf \varepsilon }}^2$ indicates the variance.

The prior distribution of the observed value ${\textbf y}$ is thus expressed as follows:

(7)$${\textbf y} \sim N(0,{\textbf k}({\textbf x},{\textbf x}) + \sigma _{\mathbf \varepsilon }^2{I_m})$$

where ${\textbf k}({\textbf x},{\textbf x})$ represents the covariance matrix obtained from the training set and ${I_m}$ indicates identity matrix.

The joint prior distribution of the observed value ${\textbf y}$ and the predicted value $f(x^\ast )$ is given by the expression below:

(8)$$\left( {\begin{array}{{c}} {\textbf y}\\ {f(x^\ast )} \end{array}} \right) \sim N\left( {0,\left( {\begin{array}{*{20}{c}} {{\textbf k}({\textbf x},{\textbf x}) + \sigma_{\mathbf \varepsilon }^2{I_m}}&{{\textbf k}({\textbf x},x^\ast )}\\ {{\textbf k}(x^\ast ,{\textbf x})}&{k(x^\ast ,x^\ast )} \end{array}} \right)} \right)$$

where $x^\ast{\in} x$ represents the test input data; ${\textbf k}({\textbf x},x^\ast ) = {\textbf k}{(x^\ast ,{\textbf x})^T}$ indicates the covariance matrix of the test input data and the training input data set; $k(x^\ast ,x^\ast )$ indicates the covariance matrix of the test input data.

Then, the posterior distribution of the predicted value $f(x^\ast )$ can be calculated as follows:

(9)$$f(x^\ast |{\textbf y} ,{\textbf x}) \sim N(\overline {f(x^\ast )} ,Var(f(x^\ast )))$$

where, the mean is calculated as follows:

(10)$$\overline {f(x^\ast )} = {\textbf k}(x^\ast ,{\textbf x}){[{\textbf k}({\textbf x},{\textbf x}) + \sigma _{\mathbf \varepsilon }^2{I_m}]^{ - 1}}{\textbf y}$$

and the variance is calculated as follows:

(11)$$Var(f(x^\ast )) = k(x^\ast ,x^\ast ) - {\textbf k}(x^\ast ,{\textbf x}){[{\textbf k}({\textbf x},{\textbf x}) + \sigma _{\mathbf \varepsilon }^2{I_m}]^{ - 1}}{\textbf k}({\textbf x},x^\ast )$$

The predicted values ($\widehat {{\textbf y^\ast }}$) of the test input data set ${\textbf x^\ast }$ can be calculated as shown below:

(12)$$\widehat {{\textbf y^\ast }} = {{\textbf y}^T}{({\textbf k}({\textbf x},{\textbf x}) + \sigma _{\mathbf \varepsilon }^2{I_m})^{ - 1}}{\textbf k}({\textbf x^\ast },{\textbf x})$$

2.2.2 Covariance functions

Covariance functions in GPR method are also known as kernel functions. Selection of covariance functions [16] can significantly affect the prediction accuracy as shown in Eq. (12). Therefore, for the subsequent experiments, as an initial definition a first order polynomial kernel is applied as shown below:

(13)$${k_1}({\textbf x^\ast },{\textbf x}) = {\textbf x}^{{^\ast }^T}{\textbf x} + 1$$

Second, an isotropic squared exponential kernel is expressed as follows:

(14)$${k_\textrm{2}}({\textbf x^\ast },{\textbf x}) = \textrm{exp} ( - \frac{{{{||{{\textbf x^\ast } - {\textbf x}} ||}^2}}}{{2{\sigma _s}^2}})$$

Third, a more flexible anisotropic squared exponential kernel is expressed as shown in the equation below:

(15)$${k_3}({\textbf x^\ast },{\textbf x}) = \textrm{exp} ( - {({\textbf x^\ast } - {\textbf x})^T}{{\mathbf \sigma }_{\textbf s}}^{ - 1}({\textbf x^\ast } - {\textbf x}))$$

Fourth, an isotropic Matérn kernel is expressed as follows:

(16)$${k_4}({\textbf x^\ast },{\textbf x}) = (1 + \frac{{\sqrt 3 ||{{\textbf x^\ast } - {\textbf x}} ||}}{{{\sigma _s}}})\textrm{exp} ( - \frac{{\sqrt 3 ||{{\textbf x^\ast } - {\textbf x}} ||}}{{{\sigma _s}}})$$

Fifth, an anisotropic Matérn kernel is expressed as shown in the equation below:

(17)$${k_\textrm{5}}({\textbf x^\ast },{\textbf x}) = {\varsigma _\textrm{1}}^2(\sqrt 3 \times \sqrt {{{({\textbf x^\ast } - {\textbf x})}^T}{{\mathbf \sigma }_{\textbf s}}^{ - 1}({\textbf x^\ast } - {\textbf x})} )\textrm{exp} (\sqrt 3 \times \sqrt {{{({\textbf x^\ast } - {\textbf x})}^T}{{\mathbf \sigma }_{\textbf s}}^{ - 1}({\textbf x^\ast } - {\textbf x})} )$$

where ${\textbf x}$ represents the design input variable matrix; ${\sigma _s}$ indicates the positive parameters corresponding to the kernel function; ${\varsigma _\textrm{1}}$ is a scaling parameter; ${{\mathbf \sigma }_{\textbf s}} \in {{\mathbb R}^{k \times k}}$ represents a diagonal matrix where the elements are specific length scales ${\sigma _1},{\sigma _2} \cdots {\sigma _k} > 0$ for the component.

Finally, a combined kernel function [16] is proposed for comparative study as shown below:

(18)$${k_6}({\textbf x^\ast },{\textbf x}) = {k_1}({\textbf x^\ast },{\textbf x}) + {\varsigma _\textrm{2}}^2{k_5}({\textbf x^\ast },{\textbf x})$$

where ${\varsigma _\textrm{2}}$ indicates a scaling parameter for the ${k_5}$.

2.2.3 Parameter optimization

Maximum Likelihood Estimation (MLE) is commonly used to estimate hyper parameters in GPR models. The maximum likelihood estimation in determination of the hyper parameters in GPR is mainly for construction of a likelihood function with training data and parameters. Generally, it is important to calculate the logarithm of the likelihood function, and then use the log-likelihood function to obtain the partial derivative of the hyper parameters. Further, the optimization algorithm is used to find the optimal solution for the hyper parameter. The most commonly used optimization method is the gradient descent method. The process of this method is introduced below.

The marginal likelihood function is expressed as shown in the following equation:

(19)$$p(\theta |{y,{\textbf x}} ) = \frac{{p(y|{\textbf x} ,\theta )p(\theta )}}{{p(y|{\textbf x} )}}$$

where $\theta \textrm{ = \{ }{\varsigma _\textrm{1}},{\varsigma _\textrm{2}}\textrm{,}{\sigma _1}\textrm{,}{\sigma _2} \cdots {\sigma _k}{\} }$ represents a vector of all hyper parameters and $p(y|{\textbf x} )$ the probability as $p(y|{\textbf x} ) = \sum {(\textrm{y}|{\textbf x} ,{\theta _i})} p({\theta _i}),i = 1,2, \cdots ,n$. The logarithm of the marginal likelihood function is calculated to obtain the log likelihood function shown below:

(20)$$\log (p(\textrm{y}|{\textbf x} ,\theta )) ={-} \frac{1}{2}{y^T}{({\textbf k}({\textbf x},{\textbf x}) + \sigma _{\mathbf \varepsilon }^2I)^{ - 1}}y - \frac{1}{2}\log |{{\textbf k}({\textbf x},{\textbf x}) + \sigma_{\mathbf \varepsilon }^2{I_\textrm{m}}} |- \frac{n}{2}\log (2\pi )$$

where ${\textbf k}({\textbf x},{\textbf x})$ represents the corresponding covariance matrix.

The likelihood function must be maximized to make the observed value infinitely close to the training sample data. Therefore, the partial derivative of the probability likelihood function with respect to all hyper parameters $\theta $ should be established to find the optimal hyper parameters. The partial derivative is shown in Eq. (21) below:

(21)$$\frac{\partial }{{\partial {\theta _i}}}\log (p(\textrm{y}|{\textbf x} ,\theta )) = \frac{1}{2}\textrm{Tr}\left\{ {[\alpha {\alpha^T} - {{({\textbf k}({\textbf x},{\textbf x}) + \sigma_{\mathbf \varepsilon }^2{I_m})}^{ - 1}}]\frac{{\partial ({\textbf k}({\textbf x},{\textbf x}) + \sigma_{\mathbf \varepsilon }^2{I_m})}}{{\partial {\theta_i}}}} \right\}$$

where $\alpha = {({\textbf k}({\textbf x},{\textbf x}) + \sigma _{\mathbf \varepsilon }^2{I_\textrm{m}})^{ - 1}}y$, and $\textrm{Tr(} \cdot \textrm{)}$ represents the matrix trace.

An optimization algorithm is then used to find the optimal value of $\theta $. The commonly used method is the Conjugate Gradient method [17]. Conjugate Gradient method is used to determine the extreme value by calculation of the derivative of each hyper parameter sequentially through multiple derivatives to obtain the fastest descending direction. Multiple iterations are used until the convergence is achieved to obtain the optimal hyper parameters.

2.3 Multi-objective optimization – NSGA-II

The cockpit lighting design optimization can be described as a multi-objective optimization problem as mentioned above, Section 1. The mathematical model for multi-objective optimization can be expressed as shown below:

(22)$$\mathop {Minimize}\limits_x \begin{array}{{c}} {} \end{array}f(x) = {[{f_1}(x),{f_2}(x), \cdots ,{f_n}(x)]^T}$$

Subject to the constraints:

(23)$$\begin{array}{l} {g_i}(x) \le 0,\begin{array}{{c}} {} \end{array}i = 1,2, \cdots ,m\\ {h_j}(x) = 0,\begin{array}{*{20}{c}} {} \end{array}j = 1,2, \cdots ,k \end{array}$$

where, $g(x)$ denotes the inequality constraints, and $h(x)$ denotes the equality constraints. The constraints in this case should be analyzed and selected according to the specific cases.

In performing the multi-objective optimization, a nondominated solution is superior compared to a dominated solution. It improves one objective and causes a degradation in another or has the same superior effect on both objectives than the dominated solution. The set of all the nondominated solutions is called the Pareto front [18]. A simplified illustration of GPR-NSGA-II and its inputs and outputs are presented in Fig. 2. This figure illustrates the basic process of generating the Pareto front, where the boundaries of the feasible domain (the area with point) and the Pareto front position are presented as the curve with the red points and the optimization results are presented (the red point). The figure indicates a simplified process for genetic optimization. The findings indicate that the Pareto front of two optimization objectives should be obtained using the proposed framework. The corresponding parameter design values can be obtained using the Pareto front results [19].

Fig. 2. Illustration of the multi-objective optimization process. The red triangles represent the Pareto optimal solutions set and black dots represent the feasible solution set.

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3. Case study

3.1 Experiment

3.1.1 Experimental design

A driving system was established using TracePro model to simulate the driving process under a dark environment. The optical machine structure of the system was effectively and reasonably simplified to ensure accuracy and tracking efficiency. The simplified structure is presented in the Fig. 3. The optical system mainly comprised the shell structure of the driver’s cab and the lighting system. The lighting system generally comprises the instrument panel flood lights and task lights when the cab system is under a dark working environment [3]. As the main parameters of lamps, flux and beam angle of two kinds of lamps were selected as input variables in this simulation model. Moreover, the illuminance (E) and uniformity of the different task surfaces were selected as the optical quality characteristics of the model. Setting of task surfaces and other simulation parameters is presented in Table 1.

Fig. 3. Cockpit TracePro model and a schematic diagram showing light rays on the target surfaces. The red light originates from the task lights L₁₋₂ and the green light originates from the instrument panel flood lights L₃₋₄. Area lighting is key feature for the cockpit optical design which includes the task surface (the height H is set as 0.5 m and the size S is set as 300mm*300 mm) and the dashboard surface. Stray light propagated to the eye position and the information display location should be minimized.

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Table 1. Summary of the main input parameters for optical simulation

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The requirements of the SAE 4103 report for cockpit lighting indicate that it is important to divide the lighting area and the area that does not need illumination and contains self-luminous display information [3]. In the simulation model reported in this study, the lighting area corresponding to the task light and floodlight represents the location of the task surface above the pilot’s lap and the dashboard, which is the area lighting specified in the standard. Areas that should not be illuminated include the position of the human eye, the monitor directly in front of the pilot and the information display on the center console. After determining the position and range of the receiving surface in the model, the light rays on all target surfaces are shown in the model (Fig. 3). In the present study, the eye location was chosen as a typical surface which should not be illuminated by direct light.

The three-dimensional model is given specified optical properties according to other requirements of SAE 4103 for the cockpit [3]. The settings of target surfaces and other parameters in the simulation are resented in Table 1.

3.1.2 Optical characteristics of task surfaces

Flood lighting and task lighting systems should be designed to minimize light in areas other than those intended to be illuminated. Glare, unwanted reflections, and stray light should be eliminated as part of an acceptable and certifiable flight deck lighting design. All information displays should have surfaces that minimize reflections.

The simulation results of the two sub-models indicate that the beam angle is significantly correlated with the uniformity of illumination on the target surface. Illumination uniformity is an important light distribution index, characterized by the expression shown below:

(24)$$U = {{{E_{\min }}} / {{E_{eva}}}}$$

where ${E_{\min }}$ and ${E_{eva}}$ are the minimum and average illuminance on the target surface, respectively. Low illuminance uniformity of the target surface forms a visual division of light and dark, which affects the visual mission of the pilot. Therefore, the illumination uniformity of the target surfaces should be as high as possible. In this model, the uniformity is set as one of the optical characteristics.

In addition to various internal surfaces, the human eye is an important light receiver in the cockpit. Veiling luminance of scattered light will have a significant effect on visibility when intense light sources are present in the peripheral visual field resulting in low contrast of visible objects. Reduction of contrast caused by the veiling luminance may reduce the contrast of an object below the threshold thus affecting visibility of the object. In addition, it can lead reduction of the contrast to near threshold making it difficult to see the target object. The magnitude of the veiling luminance depends on the intensity and distance of the glare source. The distance and intensity of the glare source together synergistically determine the relevant parameter ${E_{glare}}$, the illumination at the eye caused by the glare source, and the angle between the glare source and the line of sight θ. Therefore, the average illumination of the eye position surface is selected as the other optical characteristics in this model.

3.2 Orthogonal experiment

The luminous flux and beam angle of the lamps work synergistically on the target surface. The illumination requirement on the target surface shows that the parameter combination of the lamps can be determined by using the controlled variable method. Each type of luminaire has specific lighting requirements in the single-object lighting model, thus the input parameter ranges of the two luminaires can be determined separately. The parameter combination is then used as the input variable of the entire model. The single-target lighting model is regarded as a sub-model of the entire cockpit, whereby only the specified lamps work, and only the matching target surface is considered.

The flight deck crew panels of the instrument panel flood lights should be illuminated with at least 100 lx. The task lights provide diffused illumination adequate for information reading on the lap without interfering with other visual tasks. The light is mounted such that a fixed writing pad, map or chart is maintained. The illumination on the task surface under full bright position of the lighting control is 300 lx ± 100 lx [21]. These requirements indicate that the range of model input variables should meet the following criteria: illuminance of the dashboard ranging from 100–200 lx, and illuminance of the task surface ranging from 200–400 lx.

The task lighting in the sub-model of the cockpit task lighting comprises two symmetrical reading lamps. The typical beam angle of the reading light was set as (10°, 20°, 30°, 40°, 50°) for simulation. The results showed that the luminous flux utilization rate of the target surface varied with the conditions of different beam angles for task lighting. More focused light was associated with high utilization rate of the luminous flux, and requirement of low power (flux). The range of luminous flux values required for different beam angles for meeting the task lighting of 200–400 lx is shown in Fig. 4. The beam angle range of task lighting ranged from 20° to 50°and the luminous flux range is 30–65 lm owing to uniformity of the target side. Similarly, the flood lighting is derived from the sub-model of the cockpit instrument panel flood lighting and the range of the lamp is shown in Fig. 4. The beam angle range was 20°−50°, and the luminous flux range was 5–20 lm.

Fig. 4. Overall process for determination of the input range in each sub-model. The principle of optical path reversibility states that the relationship between luminaire parameters and the light distribution on the target surface can be obtained by ray tracing from the task surface to the lamps.

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Different precision can be set in the algorithm, designers can choose the appropriate accuracy based on the realistic situations. In this case, considering the practical situation of cockpit lighting, the luminous flux step was set at 5 lm and the beam angle step was set at 10°. Further, an orthogonal L32 design of simulation was established, and the levels of each control factor are presented in Table 2. At least 8*4*4*4 experiments are performed under the full experiment method, whereas orthogonal experimental method requires design of only 32 experiments. Therefore, orthogonal experimental method significantly reduces the number of experiments designed. Table 3 shows the setting of simulation output parameters. The results with a L32 orthogonal array for simulated data are presented in Table 4.

Table 2. Input parameters and control level of each parameter

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Table 3. Simulation output parameter settings

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Table 4. Results with a L32 orthogonal array for simulated data

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3.3 GPR model

No.1 to No.28 simulation results from Section 3.2 are used as the training samples and the last 4 samples are used as the test samples for GPR model. As described in section 2.2.2, the prediction accuracy of GPR model mainly depends on the selection of kernel function. A 4-fold-cross-validation with the training dataset was performed to identify the best kernel function. The mean predicted absolute error of the training samples is presented in Table 5.

Table 5. Prediction results and error analysis of GPR model

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The results showed that the combined kernel function (${k_6}$) has a lower prediction error in each output. Notably, the mean predicted error of the GPR model with ${k_6}$ is the lowest in terms of ${I_1}$, ${I_2}$, ${U_1}$ and G. Results for ${U_2}$ show that, the GPR model with ${k_6}$ is slightly poor compared with that for ${k_5}$.Therefore, ${k_6}$ was selected as the kernel function of GPR model to predict the outputs. The training data were used for training and the four test samples were used for validation to explore the generalization performance of the model. The test results are presented in Table 6.

Table 6. Prediction results and error analysis for GPR model using ${k_6}$ as the kernel function

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Analysis of the same significant figure of outputs showed that the prediction error of test samples could be controlled within an acceptable error range (less than 10%) (Table 6). The maximum absolute prediction error of test samples was 8.50% and the minimum was 0%. This finding indicates that the GPR model with ${k_6}$ kernel function has good generalization performance and can accurately predict the value of outputs within the value range of the input variables.

3.4 Multi-objective optimization

3.4.1 Optimization case of the GPR-NSGA-II

The optimization of task light and instrument panel flood light was consistent with the characteristics of multi-objective optimization problems according to Eq. (22) and (23). The optimization process is as expressed as the mathematical formula shown below:

(25)$$\left\{ {\begin{array}{{c}} {Opt1:\max \{{{U_1}(A,B,C,D)} \}}\\ {Opt2:\max \{{{U_2}(A,B,C,D)} \}}\\ {Opt3:\min \{{G(A,B,C,D)} \}} \end{array}} \right.$$

Subject to the constraints:

(26)$$\left\{ {\begin{array}{{c}} {Con1:200 \le {I_1}( A,B,C,D) \le 400}\\ {Con2:100 \le {I_2}(A,B,C,D) \le 200} \end{array}} \right.$$

where A, B, C, D represent the input parameter, corresponding to flux of task light, beam angle of task light, flux of instrument panel flood light and beam angle of instrument panel flood light, respectively.

The initial population samples of the GPR-NSGA-II algorithm were randomly generated using the GPR model, thus the final return results for each run were not consistent after performing the genetic algorithm. The optimizer executes multiple times and non-poor solutions are filtered from each call. The search ends after 150 generations (iterations) when the iteration reaches the specified maximum. Pareto optimal front and Pareto optimal solution for illuminance uniformity of the task surface (${U_1}$), illuminance uniformity of the instrument panel (${U_2}$) and the average illumination on eye position surface ($G$) are presented in Fig. 5 and Table 7, respectively.

Fig. 5. Pareto front profile

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Table 7. Pareto solutions with GPR-NSGA-II

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Notably, ${U_1}$, ${U_2}$ and G cannot be optimized at the same time, and the user can choose an appropriate solution based on the design requirements. The constraints of the present case were nonlinear, therefore, the Pareto front values presented in Fig. 5 were not continuous. However, the red circle in Fig. 5 can represent the Pareto front of every two optimization objectives. Therefore, an obvious Pareto front is observed in each two optimization objectives in the present case, implying that the GPR-NSGA-II framework is applicable to this model case.

In practical and engineering applications, the Pareto front solutions for the model-specific cockpit can be obtained with GPR-NSGA-II framework. The final parameters can be determined according to different emphasis and design styles (e.g., the most energy-efficient choice or the highest score of subjective visual comfort). For example, the No.3 and No.9 solution can be chosen when the design wants to focus on the uniformity of illumination, which has been shown to be beneficial for job performance [22]. While if the design puts emphasis on visual comfort and the minimum power, the No.1 solution can be selected. Take the three conditions as examples, the corresponding solutions can be attained as following.

(1) For a maximum ${U_1}$ of 0.357 obtained with $A = 4\textrm{5}lm$, $B = 20^\circ $, $C = 1\textrm{0}lm$ and $D = 50^\circ $, the corresponding ${U_2}$ and G were 0.507 and 87, respectively;
(2) For a maximum ${U_2}$ of 0.575 obtained with $A = \textrm{40}lm$, $B = \textrm{3}0^\circ $, $C = \textrm{10}lm$ and $D = 40^\circ $, the corresponding ${U_1}$ and G were 0.301 and 106, respectively;
(3) For a minimum G of 65 obtained with $A = \textrm{40}lm$, $B = 30^\circ $, $C = \textrm{5}lm$ and $D = \textrm{5}0^\circ $, the corresponding ${U_1}$ and ${U_2}$ were 0.290 and 0.511, respectively.

3.4.2 Validation of the GPR-NSGA-II model

Although the accuracy of GPR model was verified as described in section 3.3, the accuracy of the Pareto front obtained using the whole training data should be verified to ensure referencing of the non-inferior solutions. The three solution points on the Pareto front described mentioned in section 3.4.1 were considered as the experimental data for verification of the accuracy of the GPR-NSGA-II model.

The absolute error between the multi-objective values of the model outputs and the simulation experimental results are presented in Table 8. The prediction results of the model had a small absolute error, and the maximum absolute error was only 6.61%. The low absolute error indicates the accuracy of high GPR-NSGA-II model and validity of selected features.

Table 8. Comparison of experimental results and GPR-NSGA-II optimization results.

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4. Conclusions

Simulation experiments were carried out to establish the relationship between luminescent parameters and multiple optical characteristics in the cockpit. GPR-NSGA-II framework was used to predict and optimize light distribution of typical surfaces in the cockpit. The cockpit is a tight and narrow space, and the light-emitting devices inside can affect other areas and easily cause light interference. Multi-objective optimization method can find a balance based on the requirements of multiple objectives, which is not possible with single-objective optimization method. The key findings of the study are summarized as follows:

1. The luminaire parameters obtained from single objective design are sufficient for a single target, however, they introduce stray light to other unlit areas. Therefore, these parameters are not accurate but can be used as the input variables for multi-objective optimization. The light distribution requirements indicate that the parameters range of one lamp can be traced from the target surfaces to the lamp using the lighting model.
2. In an actual cockpit, the areas that need to be illuminated include the task surfaces, instrument panels and the ground, whereas the areas that should not be illuminated include the eye surface, all information display surfaces and the windshield. In addition, the actual input variables include other luminescent devices, such as indicators, light panels, and natural light outside the cockpit. The present study sought to establish a method for multi-targets optimization. Notably, only some typical input variables and target surfaces were selected for analysis in a model case.
3. The prediction accuracy of different kernel functions in the GPR prediction model was carried out using the 4-fold-cross-validation. Moreover, a generalization test was conducted. The best kernel function was selected for prediction of the outputs of the case and the accuracy of GPR model was verified. The maximum absolute prediction error of test samples was 8.50% and the minimum was 0%.
4. A case analysis showed that the absolute error between the output value of the model and the simulation experiment was less than 10%. This indicates that the proposed framework can effectively predict and optimize light distribution of typical surfaces in the cockpit.

Funding

Ministry of Industry and Information Technology of the People's Republic of China.

Acknowledgement

This work was supported by grants from “Project of Civil Aircraft Research” from Ministry of Industry and Information Technology of the People's Republic of China.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Parameters		Value	Source
Parameters of simulation model	Wall reflectance, lambert (%)	10	survey
	Equipment surface reflectance, lambert (%)	10	survey
	Windshield transmittance (%)	10	survey
Parameters of task surface	Height of the task surface (m)	0.75	SAE technical paper [3]
	Instrument panel	/	survey
	Height of the eye position (m)	1.2	CIE technical paper [20]

Code	Output parameters	Target
$I_{1}$	Average illuminance $E_{e v a}$ (lx)	Task surface
$I_{2}$	Average illuminance $E_{e v a}$ (lx)	Instrument panel
$U_{1}$	Illuminance uniformity (-)	Task surface
$U_{2}$	Illuminance uniformity (-)	Instrument panel
$G$	Direct glare $E_{g l a r e}$ (lx)	Eye position surface

No	Codes and levels of simulation inputs				Simulation outputs					No	Codes and levels of simulation inputs				Simulation outputs
No	A	B	C	D	$I_{1}$ (lx)	$I_{2}$ (lx)	$U_{1}$ -)	$U_{2}$ -)	$G$ (lx)	No	A	B	C	D	$I_{1}$ (lx)	$I_{2}$ (lx)	$U_{1}$ -)	$U_{2}$ -)	$G$ (lx)
1	8	3	4	3	208	386	0.325	0.622	158	17	7	4	1	1	216	149	0.301	0.449	129
2	2	3	1	2	98	113	0.321	0.408	64	18	3	2	1	4	203	100	0.290	0.511	65
3	3	1	2	3	198	182	0.349	0.562	93	19	1	1	1	1	144	92	0.346	0.473	59
4	1	2	2	2	166	149	0.312	0.384	57	20	4	3	2	4	139	205	0.336	0.502	89
5	4	4	1	3	164	136	0.307	0.598	103	21	5	1	3	2	254	235	0.359	0.400	105
6	7	3	2	2	172	211	0.326	0.404	113	22	7	2	3	3	320	264	0.301	0.562	125
7	6	2	2	4	288	182	0.295	0.502	94	23	2	4	2	1	144	180	0.316	0.622	88
8	5	2	4	1	289	300	0.300	0.408	108	24	6	4	4	2	235	332	0.333	0.391	139
9	6	1	1	3	252	121	0.335	0.546	105	25	6	3	3	1	175	274	0.343	0.433	117
10	7	1	4	4	317	337	0.372	0.501	130	26	8	2	1	2	320	111	0.285	0.456	100
11	5	4	2	3	193	217	0.316	0.626	126	27	3	3	4	1	151	319	0.355	0.409	101
12	8	1	2	1	309	188	0.344	0.477	126	28	2	2	4	3	213	320	0.329	0.543	120
13	1	4	4	4	157	327	0.365	0.468	90	29	4	2	3	1	252	230	0.296	0.414	92
14	1	3	3	3	110	263	0.321	0.586	88	30	4	1	4	2	245	294	0.374	0.377	104
15	5	3	1	4	137	141	0.314	0.497	89	31	8	4	3	4	261	304	0.325	0.487	153
16	3	4	3	2	172	247	0.334	0.391	102	32	2	1	3	4	194	244	0.384	0.495	81

Kernel function	Mean predicted error (%)
Kernel function	$I_{1}$	$I_{2}$	$U_{1}$	$U_{2}$	$G$
$k_{1}$	21.65	6.44	5.48	14.88	10.34
$k_{2}$	28.74	14.92	5.75	13.26	10.78
$k_{3}$	2.98	13.97	5.08	10.47	9.02
$k_{4}$	26.54	11.75	5.78	13.35	10.71
$k_{5}$	3.53	8.78	3.56	6.96	9.25
$k_{6}$	1.26	5.50	3.55	7.39	7.77

No.	Desired value					Predicted value					Absolute error (%)
No.	$I_{1}$	$I_{2}$	$U_{1}$	$U_{2}$	$G$	$I_{1}$	$I_{2}$	$U_{1}$	$U_{2}$	$G$	$I_{1}$	$I_{2}$	$U_{1}$	$U_{2}$	$G$
29	252	230	0.296	0.414	92	253	228	0.295	0.437	91	0.40	0.87	0.34	5.56	1.09
30	245	294	0.374	0.377	104	245	298	0.371	0.386	107	0	1.36	0.80	2.39	2.88
31	261	304	0.325	0.487	153	261	302	0.321	0.502	140	0	0.66	1.23	3.08	8.50
32	194	244	0.384	0.495	81	192	245	0.361	0.498	78	1.03	0.41	5.96	0.61	3.70

Multi-objective optimization method for reducing mutual interference in cockpit illumination

Abstract

1. Introduction

2. Computational scheme

2.1 Orthogonal experimental method

2.2 Prediction model – Gaussian process regression (GPR)

2.2.1 Estimation using the Gaussian process

2.2.2 Covariance functions

2.2.3 Parameter optimization

2.3 Multi-objective optimization – NSGA-II

3. Case study

3.1 Experiment

3.1.1 Experimental design

3.1.2 Optical characteristics of task surfaces

3.2 Orthogonal experiment

3.3 GPR model

3.4 Multi-objective optimization

3.4.1 Optimization case of the GPR-NSGA-II

3.4.2 Validation of the GPR-NSGA-II model

4. Conclusions

Funding

Acknowledgement

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Tables (8)

Equations (26)

Optics Express

Code	Input parameters	Control levels	Level1	Level2	Level3	Level4	Level5	Level6	Level7	Level8
A	Flux of task light (lm)	8	30	35	40	45	50	55	60	65
B	Beam angle of task light (°)	4	20	30	40	50	/	/	/	/
C	Flux of instrument panel flood light (lm)	4	5	10	15	20	/	/	/	/
D	Beam angle of instrument panel flood light (°)	4	20	30	40	50	/	/	/	/

No.	A	B	C	D	$U_{1}$	$U_{2}$	$G$	No.	A	B	C	D	$U_{1}$	$U_{2}$	$G$
1	40	30	5	50	0.290	0.511	65	6	45	20	5	40	0.339	0.564	91
2	40	30	10	30	0.307	0.387	72	7	45	20	5	50	0.360	0.513	78
3	40	30	10	40	0.301	0.575	106	8	45	20	10	40	0.347	0.556	101
4	40	30	10	50	0.302	0.506	75	9	45	20	10	50	0.370	0.507	87
5	45	20	5	30	0.337	0.425	75	10	50	20	10	50	0.366	0.509	95

	A (lm)	B (°)	C (lm)	D (°)	$U_{1}$	$U_{2}$	$G$ (lx)
Experiment	45	20	10	50	0.357	0.512	90
GPR-NSGA-II	45	20	10	50	0.370	0.507	87
Absolute error (%)	/	/	/	/	3.6	1.0	3.3
Experiment	40	30	10	40	0.301	0.562	105
GPR-NSGA-II	40	30	10	40	0.301	0.575	106
Absolute error (%)	/	/	/	/	0	2.3	1.0
Experiment	40	30	5	50	0.289	0.505	65
GPR-NSGA-II	40	30	5	50	0.290	0.511	65
Absolute error (%)	/	/	/	/	0.3	1.2	0