A Nonlinear Rebalanced Control Compensation Model for Visual Information of Drivers in the Foggy Section of Expressways

Abstract

To obtain the optimal driving visual guidance methods in sudden low-visibility fog environments, it is crucial to analyze the changes in visual characteristics and information demand under low-visibility foggy conditions. The paper constructs a driving visual information demand model for foggy environments based on visual information input and output, using Shannon’s theory and feedback control theory. Two types of foggy road sections with the same visibility, one with guidance lights and one without, were selected for real-vehicle experiments based on the driver’s blood pressure, heart rate, and driving gaze domain tests. The study found the following: (1) In sudden foggy environments, the amount of driving information obtained by drivers decreases instantly with a sudden drop in visibility, failing to meet the information demand for driving cognition, thereby disrupting the dynamic balance state of driving based on speed, visibility, and other road environment factors. The experiment also found that in low-visibility environments, the radius of the human eye’s visual gaze domain becomes smaller, with the gaze range mainly concentrated directly in front of the vehicle, and the lower the visibility, the smaller the gaze domain range; (2) Foggy conditions affect changes in drivers’ blood pressure and heart rate. Installing guidance lights with sufficient illumination at foggy sections to compensate for drivers’ visual information can effectively supplement the visual information required for safe driving; (3) The experiment indicates that the guidance effect of the lights is most pronounced when visibility is within the range of [50 m, 150 m]; however, when visibility is above 500 m, the presence of guidance lights can, to some extent, affect driving safety and increase the risk of accidents.

1. Introduction

The human visual system is a complicated and partially known multidimensional nonlinear information processing system. Zhang et al. [1] pointed out that human eyes perceive information, process effective parts of the perceived information, and then send the result to the brain. Serving as the basis for subsequent decision-making, this process is a complicated and ubiquitous information feedback control process. Down the ages, humans have always been exploring the inherent relation of decision control between vision, the brain, and the limbs. As an information receiver, eyes are one of the organs receiving significant attention. Besides biological, psychological, and medical methods, physical methods and multidisciplinary integration are also effective approaches to explore the mystery of visual information processing by human eyes. Stark [2] first carried out feedback analysis of human pupils’ response to light based on control theories. Afterwards, in order to better analyze the information input, perception, and feedback processes of human vision, scholars have divided the visual attention mechanism of human eyes in perceiving objects into three time stages: (1) pre-attention stage, namely the stage before the focus of attention shifts to the object; (2) attention stage, namely the stage after the focus of attention locks onto the object; (3) post-attention stage, namely the stage after the focus of attention leaves the object. With the introduction of biocybernetics, Sun et al. [3,4,5,6,7,8] have thoroughly researched pupil response to light and the characteristics of the visual biocybernetic system based on experiments and thus perfected the theoretical system of biocybernetics. Qualitative analysis of human eyes’ visual features based on biocybernetic theories is the main approach used in visual biocybernetics. By quantifying visual information through pupils, effective visual information can be calculated by using mathematical models. This method can be applied in the design of roadway lighting, especially visual guidance in low-visibility road environments. However, there are few theoretical and engineering studies on this application, and it is necessary to carry out relevant studies to complement human eye biocybernetics.

Based on computational studies on visual information, Weber [9] proposed the Weber–Fechner law that was developed on the basis of visual psychophysics and cognitive psychology by measuring human eyes’ perception of incremental threshold [10]. Marr [11] first proposed a relatively complete visual computing theory, which is of great significance in studying the influence of visual psychology on driving safety. Through driving experiments on physiologic feedback, Jing et al. [12] studied cognitive mechanisms of drivers with different characteristics in the process of information interpretation, providing a theoretical basis for appropriate allocation of information content conveyed by guiding signs. Nasiri et al. [13] and GE et al. [14] conducted eye-movement experiments with EMR-HM8 eye trackers to explore visual cognition during driving, and based on the experimental results, found out the general rules of the visual cognition as well as the quantitative relation between the information content conveyed by guiding signs and the response time of the visual cognition. Huang et al. [15] found that the characteristic of the driver receiving information during driving is a decrease in the time of each regard and an increase in the number of regards. The NHTSA concluded the following characteristics regarding human eyes during car following: The duration of regards away from the front during car following is shorter than when the vehicle is not following a car; during driving, the duration of a driver’s regard on a specific object is about 1.6 s~2 s [16]. Besides the visual input information, other types of external information received by various drivers, including auditory information and tactile information, are also required for them to make driving decisions. Based on the integration of all these types of information [17,18], drivers will perform cognition, generate corresponding psychological feelings, and finally make the driving decisions that they think are optimal [19]. Through vehicle experiments, the study [20,21] has shown that the main risk factor in foggy environments is visibility, which can lead to missing or even wrong visual information in drivers, namely visual illusions. However, regardless of the differences between drivers in individual culture, driving proficiency, and visual acuity, road space and climatic environments have consistent effects on humans’ psychological feelings. When studying the factors influencing drivers’ driving safety in the area of road traffic, scholars have focused on the influence of road alignment, driving visibility, illuminance, and adverse climate conditions (such as rain, fog, ice, and snow) on road traffic safety. However, there are few studies that explore road traffic safety from the perspectives of information content and driving information demand, and even fewer studies that analyze the influence of required driving information content and changes of the information content caused by harsh climate conditions (such as fog) on drivers’ psychology and driving safety.

According to investigations, most agglomerate fog road sections on China’s expressways cannot be timely restricted and closed to traffic, so installing guiding lamps along the road sections where agglomerate fogs take place frequently is an effective method to reduce safety risks in low-visibility foggy sections and also a commonly used engineering measure in China. The literature [22] claims that guiding facilities for traffic safety in foggy weather should be turned on when the visibility is less than 200 m. The study [23] has pointed out that it is unable to implement real-time monitoring and very hard to perform timely control on small-scale agglomerate fogs, because the length of an agglomerate fog road section is just several kilometers, and the fog occurs often unexpectedly. The main measures taken in some expressway sections in China are to set up notice boards to warn drivers [24] and install a guiding system. The current intelligent guiding system in foggy areas mainly contains two guiding methods: (1) setting up static warning signs along road sections in foggy areas and (2) installing guiding system along the road sections [25]. The two methods have the following problems in guiding traffic in agglomerate fog environments. The first method only serves as a reminder, and in low-visibility, dense fog environments, drivers cannot clearly see the prompt messages from a distance during driving. Practically, the second method is implemented as double-color strobe edge flashers installed on both sides of expressway sections, which play a role in lighting road alignment and warning against collision to some extent. However, due to the scattering effect of fog on light, the traffic-guiding effect of only lighting road alignment still needs to be verified.

For now, although there are industry standards on the spacing of guiding lamps, there is no national standard in China. In addition, guiding lamps only supplement partial visual information to a certain extent, so the impact of foggy environments on driving still exists. Therefore, after installing guiding lamps, how to achieve a new human–vehicle–road balance, namely how to determine the speed limits for a foggy environment with different visibility to achieve a theoretical safe state, still needs to be solved. Integrating bio-cybernetics with road traffic engineering to investigate visual guiding mechanisms in foggy environments is the key to implementing traffic guidance in low-visibility fog or haze environments. Road traffic guidance in low-visibility environments can be achieved by setting up roadside accessory equipment from the perspective of actively improving visual range, and rules of visual information compensation in road traffic under driving conditions can be studied from the perspective of visual physiology and physics, specifically from the point of visual contrast increase [26].

Confronted with the pervasive challenges of absence or sub-optimal deployment of guidance apparatus on expressways during foggy conditions, this scholarly work initiates with an analytical framework grounded in feedback control theory and Shannon’s theory to elucidate the ocular feedback control mechanisms of drivers within dynamic environmental contexts. After this theoretical underpinning, the paper advances to perform a nonlinear feedback control analysis specific to the visual information dynamics within expressway fog segments. This analytical progression culminates in the formulation of an informational requirement model for an expressway fog zone guidance system and a compensatory control equation for the visual information quantity in fog-enshrouded expressway driving scenarios. The empirical validation of these theoretical models is executed through a series of vehicular experiments under real-world conditions. The scholarly contributions of this research are poised to furnish substantive theoretical underpinnings for the engineering and deployment of visual guidance apparatus in the context of expressways subjected to sudden reductions in visibility.

2. Nonlinear Feedback Control Analysis of Visual Information in Foggy Section of Expressway

The preliminary research has studied the impact of the road design elements (road alignment, road space, and climate environment) on drivers’ driving behaviors, and thus it is a passive analysis of the potential risks of adverse factors like climate linearity to driving safety. In view of the impact of global air pollution, urban heat islands and wet islands on the current climate, the influence of haze and dense winter fog, especially agglomerate fog, on expressway traffic safety is increasingly severe [27]. Speed limits, road closures, and large-scale and super-large-scale traffic accidents caused by low-visibility climates such as fog, agglomerate fog, and haze happen occasionally, which have a serious impact on road use efficiency and result in great losses of life and property. When determining design speeds for roads, road designers often do not take much account of the influence of sudden climate disasters such as agglomerate fog but perform this work based on road traffic environments under ideal climate conditions. Therefore, it can be preliminarily assumed that, without considering traffic flow conditions, the balance state of the expressway driving system proposed by the road designer can be achieved when the driver has a good visual physiological state and drives the vehicle within the design speed range in good weather. However, after the vehicle enters a low-visibility agglomerate fog section, the amount of visual information received by human eyes will reduce with the decrease in visibility [28]. Meanwhile, the fog acts on the road surface and further on the vehicle and driver through changes in the friction coefficient of the road surface and the mirror surface effect, finally causing disturbances to the speeds of the traffic flow and the vehicle. Changes in vehicle speed will react upon the driver’s visual acuity, thus leading to an imbalance of the human visual decision-feedback system.

In the field of communications, signals are divided into digital signals and analog signals. Analog signals are continuous waveform signals, while digital signals are discrete signals, such as computer signals. Digital signals are more conducive to the calculation of visual information received by the driver during driving. According to the Shannon theorem [29], the visual information received by human eyes during communication can be converted to codes. Therefore, by expressing physical information in mathematical form, the information communication process can be transformed into a mathematical operation process [30]. When the vehicle initially enters an agglomerate fog road section, the view of the road section in the visual field of human eyes can be seen as a circular light curtain vertically, and digital codes can be used to represent visual information. In this way, the light curtain diagram of visual information received by human eyes during driving in the agglomerate fog road section can be obtained.

The diameter of the field of regard is used as the research parameter in this paper. Thus, the below-mentioned field of regard is expressed by the width of the field. In the initial emergency braking phase, the diameter of the field of regard is the minimum, and the least amount of road information is received.

The control theory is used to analyze the input and output of human eye information process [31]. We set the difference between input $x\left(t\right)$ and output $y_i\left(t\right)$ as $e\left(t\right)$, calculated as follows:

$$e\left(t\right)=x\left(t\right)-y_i\left(t\right)=e^{-t/{\tau }_1}$$

(1)

$$x={\tau }_1{\displaystyle \frac{dy}{dt}}+y$$

(2)

Visualizing the information input and output from the driver in foggy environments can be described by a first-order system, which can be approximately abstracted as the input–output control system model.

The problem can be normalized through Laplace transform. When $\mathrm{t}\in \left[0,+\infty \right)$, we multiply both ends of Equation (2) with $e^{-\mathrm{s}t}$, and the following equation can be obtained through the Laplace transform [31]:

$$Y\left(s\right)\left({\tau }_{1}s+1\right)=X\left(s\right)+{\tau }_1{y}_0$$

(3)

The deformation is as follows:

$$Y\left(s\right)={\displaystyle \frac{X\left(s\right)}{{\tau }_{1}s+1}}+{\displaystyle \frac{{\tau }_1{y}_0}{{\tau }_{1}s+1}}={\displaystyle \frac{1}{{\tau }_{1}s+1}}\left[X\left(s\right)+{\tau }_1{y}_0\right]$$

(4)

Supposing the following:

$$F\left(s\right)={\displaystyle \frac{1}{{\tau }_{1}s+1}}$$

(5)

Equation (4) can be expressed as follows:

$$Y\left(s\right)=F\left(s\right)\left[X\left(s\right)+{\tau }_1{y}_0\right]$$

(6)

$F\left(s\right)$ is the transfer function of the input and output control system. ${\displaystyle \frac{X\left(s\right)}{{\tau }_{1}s+1}}$ represents the output produced by the input, abbreviated as $Y_i\left(s\right)$; and ${\displaystyle \frac{{\tau }_1{y}_0}{{\tau }_{1}s+1}}$ represents the output generated by the initial condition, abbreviated as $Y_c\left(s\right)$. Equation (4) can be expressed as follows:

$$Y\left(s\right)=Y_i\left(s\right)+Y_c\left(s\right)$$

(7)

$Y_c\left(s\right)$ represents the visual information input to the driver driving on the road section, where agglomerate fog takes place frequently, in normal weather at a speed within the design range. $Y_i\left(s\right)$ represents the visual information input resulting from changes in visibility, driving speed, and other factors in agglomerate fog environments. For the purpose of accuracy, $Y_c\left(s\right)$ is substituted by $Y_n\left(s\right)$, $Y_i\left(s\right)$ is substituted by $Y_{\mathrm{f}}\left(s\right)$, Equation (7) can be transformed into the following:

$$Y\left(s\right)=Y_f\left(s\right)+Y_n\left(s\right)$$

(8)

The free flow state of vehicles running on the expressway at the design speed in normal weather is regarded as the equilibrium state of the system. According to the relative motion theorems, the relative motion of the driver’s eyes and the lane in this equilibrium state can be regarded as a dynamic process where the eyes are stationary, tracking a moving object. The linear feedback system observed by the eyes during driving is expressed as follows:

$$h(x_t-y_t)=y_t+1$$

(9)

In the Equation (9), $x_t$ represents the target position, $y_t$ represents eye’s position, $h$ represents filter or system gain with time invariant. When the transfer function is $Y(z)/X(z)$, the $z$ transformation [31] of a linear system is equal to the following:

$$\left\{\begin{array}{l}H(z)\left(X(z)-Y(z)\right)=Y(z)\\ H(z)X(z)=Y(z)\left(1+H(z)\right)\\ {\displaystyle \frac{H(z)}{1+H(z)}}={\displaystyle \frac{Y(z)}{X(z)}}\end{array}\right.$$

(10)

From Equations (3) and (4), obtained through Laplace transform, it can be seen that the visual information received by the eyes during driving in an agglomerate fog environment and the feedback system of the eyes form a nonlinear system. In other words, the visibility-based speed control by the driver would further act on the visual information. Speed, visibility, and visual information constitute a dynamic nonlinear system in which the three interact with each other. It is necessary to perform the analysis according to the input–output relation equation of the feedback servomechanism, and nonlinear transformation should be performed for the equation.

From the perspective of systematology, when the driver driving on the expressway receives information through multiple senses such as vision, audition, and touch, establishes a personal comprehensive anticipation system based on the driving environment, and eventually makes decisions on acceleration or deceleration and lane selection, which he/she believes are safe and optimal. After the vehicle enters the foggy section, the visual information received by the driver will be incomplete due to the fog, or an illusion may even come into being due to light scattering by the fog. As a result, the balance of the personal comprehensive anticipation system formed in a fog-free environment is upset, thus impairing the correctness of the driver’s decision-making and leading to decision delays or erroneous judgments. In addition, the driver may be distracted from driving and become nervous, which would finally result in disordered driving behaviors and faulty operations. The nonlinear compensation of visual information [32] consists of physiological photoelectric conversion, visual signal transmission and processing, and other processes. Receiving, transmitting, and processing of visual information is a typical process of information feedback control. The view seen by the driver is input to the eyes in the form of visual information. At this point, the information of people, vehicles, and road conditions on the expressway acts as visual influencing factors on the visual information input to the eyes and is received by the eyes after filtration. The visual receiving and feedback system of nonlinear information disturbances can be abstracted as the basic feedback control model of visual information input to human eyes on the expressway, as shown in Figure 1.

The driving feedback control equation can be obtained based on the feedback control model from Figure 1.

$$Y\left(s\right)=W\left(s\right)E\left(s\right)=\left[F\left(s\right)U\left(s\right)+V\left(s\right)\right]E\left(s\right)$$

(11)

According to Equation (3), the $Y\left(s\right)$ can be obtained through Laplace transform from $y\left(t\right)$; $W\left(s\right)$ represents other information input based on the human system; $V\left(s\right)$ represents an arbitrary disturbance caused by the driver for other reasons; $L\left(s\right)$ represents the transfer function of each measuring instrument; and $C\left(s\right)$ represents the transfer function of control factors in fog environment [31].

$$U\left(s\right)=C\left(s\right)\left[X\left(s\right)-Z\left(s\right)\right]=C\left(s\right)\left[X\left(s\right)-L\left(s\right)Y\left(s\right)\right]$$

(12)

The Equations (11) and (12) are combined and when eliminated can generate the following:

$$\begin{array}{l}Y\left(s\right)=W\left(s\right)E\left(s\right)\\ =\left\{F\left(s\right)C\left(s\right)\left[X_0\left(s\right)-L\left(s\right)Y\left(s\right)\right]+V\left(s\right)\right\}E\left(s\right)\\ ={\displaystyle \frac{E\left(s\right)F\left(s\right)C\left(s\right)}{1+E\left(s\right)F\left(s\right)C\left(s\right)L\left(s\right)}}{X}_0\left(s\right)+{\displaystyle \frac{E\left(s\right)}{1+E\left(s\right)F\left(s\right)C\left(s\right)L\left(s\right)}}V\left(s\right)\end{array}$$

(13)

Considering that the vehicle is in the process of driving, if the driver’s i visual output is $Y_i\left(s\right)$, and there are other $n$ information inputs based on the human system, then the $n$ information input can be expressed as $W_n\left(s\right)$, and the expansion of Equation (11) can be obtained as follows:

$$\left\{\begin{array}{l}{Y}_1\left(s\right)=W_1\left(s\right)E_{11}\left(s\right)+W_2\left(s\right)E_{12}\left(s\right)+\cdots +W_n\left(s\right)E_{1n}\left(s\right)\\ Y_2\left(s\right)=W_1\left(s\right)E_{21}\left(s\right)+W_2\left(s\right)E_{22}\left(s\right)+\cdots +W_n\left(s\right)E_{2n}\left(s\right)\\ \cdots \cdots \\ Y_i\left(s\right)=W_1\left(s\right)E_{i1}\left(s\right)+W_2\left(s\right)E_{i2}\left(s\right)+\cdots +W_n\left(s\right)E_{in}\left(s\right)\end{array}\right.$$

(14)

where each $E_{ij}\left(s\right)$ in the formula corresponds to the transfer function of the output $Y_k\left(s\right)$ when entering each $W_m\left(s\right)$, and the acquisition frequency of $E_{ij}\left(s\right)$ is determined by the speed of driving. Equation (14) is decomposed into the transfer function matrix $E\left(s\right)$, which can be constructed as follows:

$$E\left(s\right)=\left(\begin{array}{l}{E}_{11}\left(s\right)E_{12}\left(s\right)\cdots \cdots E_{1n}\left(s\right)\\ E_{21}\left(s\right)E_{22}\left(s\right)\cdots \cdots E_{2n}\left(s\right)\\ \dots \dots \\ E_{j1}\left(s\right)E_{j2}\left(s\right)\cdots \cdots E_{jn}\left(s\right)\\ \dots \dots \\ E_{i1}\left(s\right)E_{i2}\left(s\right)\cdots \cdots E_{in}\left(s\right)\end{array}\right)$$

(15)

During driving, there are more dimensions of information input to the driver than those output from the driver, so the number of input dimensions is greater than that of output dimensions. Namely, if each column of the matrix represents an input dimension and each row represents an output dimension, then $n>i$. In foggy environments, constantly changing visibility leads to correspondingly changing visual information input to the driver. When there are no obstacles in the front, the driver would constantly adjust driving speed to ensure the visual information input is balanced. This process is achieved by using feedback signals to correct input signals. If there are $n$-biased input signals at this time, $n$ input correction signals will be generated. The input equation of the control system [31] can be expanded to the following:

$$\left\{\begin{array}{ll}{U}_1\left(s\right)=& C_{11}\left(s\right)\left[X_1\left(s\right)-Z_1\left(s\right)\right]+C_{11}\left(s\right)\left[X_2\left(s\right)-Z_2\left(s\right)\right]+\cdots +C_{1i}\left(s\right)\left[X_i\left(s\right)-Z_i\left(s\right)\right]\\ & +{C^{\prime }}_{1,i+1}\left(s\right)\left[{\mathsf{\Xi }}_{i+1}\left(s\right)-{{\rm M}}_{i+1}\left(s\right)\right]+\cdots +{C^{\prime }}_{1n,i+1}\left(s\right)\left[{\mathsf{\Xi }}_n\left(s\right)-{{\rm M}}_n\left(s\right)\right]\\ U_2\left(s\right)=& C_{21}\left(s\right)\left[X_1\left(s\right)-Z_1\left(s\right)\right]+C_{22}\left(s\right)\left[X_2\left(s\right)-Z_2\left(s\right)\right]+\cdots +C_{2i}\left(s\right)\left[X_i\left(s\right)-Z_i\left(s\right)\right]\\ & +{C^{\prime }}_{2,i+1}\left(s\right)\left[{\mathsf{\Xi }}_{i+1}\left(s\right)-{{\rm M}}_{i+1}\left(s\right)\right]+\cdots +{C^{\prime }}_{2n,i+1}\left(s\right)\left[{\mathsf{\Xi }}_n\left(s\right)-{{\rm M}}_n\left(s\right)\right]\\ \cdots \cdots \\ U_n\left(s\right)=& C_{n1}\left(s\right)\left[X_1\left(s\right)-Z_1\left(s\right)\right]+C_{n2}\left(s\right)\left[X_2\left(s\right)-Z_2\left(s\right)\right]+\cdots +C_{ni}\left(s\right)\left[X_i\left(s\right)-Z_i\left(s\right)\right]\\ & +{C^{\prime }}_{n,i+1}\left(s\right)\left[{\mathsf{\Xi }}_{i+1}\left(s\right)-{{\rm M}}_{i+1}\left(s\right)\right]+\cdots +{C^{\prime }}_{nn,i+1}\left(s\right)\left[{\mathsf{\Xi }}_n\left(s\right)-{{\rm M}}_n\left(s\right)\right]\end{array}\right.$$

(16)

According to Equation (16), $\mathsf{\Xi }\left(s\right)$ represents a predetermined value; ${\rm M}\left(s\right)$ represents the measured value of different instrument measurements; $\mathsf{\Xi }\left(s\right)-{\rm M}\left(s\right)$ represents different error values due to different instrument measurements; and $C_{\mathrm{a},b}\left(s\right)$ and ${C^{\prime }}_{\alpha ,\delta }\left(s\right)$ represent two different control influencing factors. Equation (16) is decomposed into the control factor transfer function [31] matrix $C\left(s\right)$, written as follows:

$$C\left(s\right)=\left(\begin{array}{l}{C}_{11}\left(s\right)C_{12}\left(s\right)\cdots \cdots C_{1i}\left(s\right){C^{\prime }}_{1,i+1}\left(s\right)\cdots \cdots {C^{\prime }}_{1n}\left(s\right)\\ C_{21}\left(s\right)C_{22}\left(s\right)\cdots \cdots C_{2i}\left(s\right){C^{\prime }}_{1,i+1}\left(s\right)\cdots \cdots {C^{\prime }}_{2n}\left(s\right)\\ \dots \dots \\ C_{k1}\left(s\right)C_{k2}\left(s\right)\cdots \cdots C_{ki}\left(s\right){C^{\prime }}_{k,i+1}\left(s\right)\cdots \cdots {C^{\prime }}_{kn}\left(s\right)\\ \dots \dots \\ C_{n1}\left(s\right)C_{n2}\left(s\right)\cdots \cdots C_{ni}\left(s\right){C^{\prime }}_{n,i+1}\left(s\right)\cdots \cdots {C^{\prime }}_{mn}\left(s\right)\end{array}\right)$$

(17)

The relationship between the $Z\left(s\right)$ & $M\left(s\right)$ and the measured values of $Y\left(s\right)$ & $W\left(s\right)$ are determined by the transfer function $L\left(s\right)$ of the measuring instrument, written as follows:

$$Z\left(s\right)=L\left(s\right)Y\left(s\right)$$

(18)

$$M\left(s\right)=L\left(s\right)W\left(s\right)$$

(19)

3. Construction of Visual Information Rebalancing Compensation Model

3.1. The Information Demand Model of Expressway Fog Induction System

Some fog-prone road sections in China are currently provided with a roadside guiding system, which is designed to improve visibility in the fog sections by using active light-emitting devices such as fog lights and light-emitting road signs to help drivers make correct judgments on road conditions. Assume the height of the car is 1.5 m, and the drivers’ fields of regard during driving in the conditions with and without guiding lamps turned on (or without guiding lamps). The view fields of the drivers are divided into eight sections. In the fog-free environment, the front view fields of the drivers cover all the eight sections. However, in the agglomerate fog environments with visibility less than 200 m, the front view fields are mainly concentrated in the fields in the agglomerate fog environments with visibility less than 50 m that are mainly distributed in the directly front sections.

According to the GB/T 24965-2010 [32] regulation on dynamic visual recognition distance, the driving speed of 60 ± 5 km/h is regarded as the standard speed in the dense fog state, in which the driver’s vision is blocked with a smaller visual range than in the normal state at the same driving speed. $R_f$ represents the visual range in the condition without guiding lamps or with damaged guiding lamps, and $R_{fg}$ represents the visual range in the condition with functional guiding lamps. $R_d$ represents the difference between $R_{fg}$ and $R_f$, $R_d$ can be expressed as follows:

$$R_d=R_{fg}-R_f$$

(20)

It can be concluded that the blind spot of the driver is reduced by $R_d$($R_d\ge 0$) in the uninduced condition of the foggy environment compared to that of the induced condition.

$S_f$ represents the braking distance of the vehicle after it enters the agglomerate fog section without guiding lamps or with damaged guiding lamps, and $S_{fo}$ represents the braking distance of the vehicle after it enters the agglomerate fog section with guiding lamps turned on. $△S_L$ represents the difference between $S_f$ and $S_{fo}$, and can be expressed as follows:

$$△S_L=S_{fo}-S_f\ge 0$$

(21)

3.2. The Compensation Equation for Control of Visual Information of Expressway Driving in Foggy Environment

According to JT/T 1032-2016 [33] and CJJ 45-2015 [34], the induction lamp in the fog area is arranged symmetrically at an interval of 30 m, so the brightness coefficient angle of P at a certain point in the lane is shown in Figure 2.

According to the CJJ 45-2015 [34], the formula of road brightness coefficient is as follows:

$$L=r\left(\beta ,\gamma \right)I\left(c,\gamma \right)/H^2$$

(22)

$$r\left(\beta ,\gamma \right)=q\left(\beta ,\gamma \right){\cos }^3\gamma$$

(23)

In Equations (22) and (23), the brightness of a certain point on the road surface is denoted $L$. $r\left(\beta ,\gamma \right)$ denotes the simplified brightness coefficient; $\mathrm{q}$ denotes the brightness coefficient; $I\left(c,\gamma \right)$ represents the light intensity of the lamp pointing in the specified direction determined by $c,\gamma$; $H$ represents the height of the lamp from the ground; $\gamma$ represents the vertical angle of incident light; $\beta$ represents the angle between the incident plane of the light and the observation plane.

The formula can be used to calculate the brightness in a normal, fog-free daytime environment. However, the expressway speed limits are designed based on specific road features and under the assumption that the driver has normal visual acuity and drives in normal weather with a nice visual range. In good weather, the higher the speed is, the smaller the dynamic visual acuity will be. The relation of the two is shown in Figure 3. From Figure 3, it can be seen that as the speed increases, the visual information required for human driving increases, which is consistent with the research conclusion of reference [35].

In the case that the driver has normal visual acuity and drives in good weather, the higher the design speed is, the larger the size of the object that the driver can see clearly will be. Meanwhile, the higher the design speed is, the greater the distance from which the driver is required to see objects clearly will be; namely, the demand for visual range will be higher. The relation between the design speed and the size and distance of the objects that the driver can see clearly is shown in Figure 4.

For the convenience of subsequent equation model verification, the equation of the relationship between friction coefficient and speed in fog environment is [36]:

$$f=1.89\times {10}^{-5}{V}^2-4.16\times {10}^{-3}V+0.513$$

(24)

where f represents the road friction coefficient; $V$ represents the speed; the value of vehicle braking efficiency is set as 1.0; the longitudinal sliding friction coefficient between the vehicle and the road surface is set as 0.3; and the driver’s braking reaction time is 2.5 s. According to Equations (20), (22) and (23), and brought into Equations (14) and (16), the feedback control compensation equation of driving speed $V^{\prime }\left(s\right)$ in a fog environment and brightness coefficient $L^{\prime }\left(s\right)$ at a point in front of the road can be obtained, written as follows:

$$\left\{\begin{array}{l}{V^{\prime }}_1\left(s\right)={L^{\prime }}_1\left(s\right)E_{11}\left(s\right)+{L^{\prime }}_2\left(s\right)E_{12}\left(s\right)+\cdots +{L^{\prime }}_n\left(s\right)E_{1n}\left(s\right)\\ {V^{\prime }}_2\left(s\right)={L^{\prime }}_1\left(s\right)E_{21}\left(s\right)+{L^{\prime }}_2\left(s\right)E_{22}\left(s\right)+\cdots +{L^{\prime }}_n\left(s\right)E_{2n}\left(s\right)\\ \cdots \cdots \\ {V^{\prime }}_i\left(s\right)={L^{\prime }}_1\left(s\right)E_{i1}\left(s\right)+{L^{\prime }}_2\left(s\right)E_{i2}\left(s\right)+\cdots +{L^{\prime }}_n\left(s\right)E_{in}\left(s\right)\end{array}\right.$$

(25)

4. The Experimental Test

In this experiment, the drivers’ blood pressure and heart rate in good weather were first measured and used as the standard values for the subsequent visual information compensation. After that, the drivers’ blood pressure and heart rate measured in normal traffic environments with the same visibility and good weather were compared. The vehicle speed corresponding to the same blood pressure and heart rate was used as the reference value of the vehicle speed in foggy environments.

4.1. The Experiment Equipment

The test apparatus used in this study includes the following: a physiology instrument, a blood pressure tester, an eye tracker, and two sedans (1.5 L, automatic gearshift), which were selected as the test vehicles. Other equipment includes two laptops, a vehicle-mounted power supply, a walkie-talkie, digital cameras, video recorders, and automobile data recorders.

4.2. The Test Conditions of the Expressway

The tests were all outdoor tests with the focus on studying the drivers’ visual information compensation, taking no account of the influence of traffic flow and road wetness on the driving speed. To improve the safety of the tests, relatively straight road sections on the expressway were selected as the test sections. The testing time period for the agglomerate fog environment was selected from 8:00 to 9:30 AM when the expressway was closed to traffic due to dense winter fog, so as to obtain test data in the foggy environments with different visibility and ensure sufficient brightness. The G5013 expressway section from the Damiao toll station to Yunwushan on the Chongqing-to-Sichuan side was chosen as the test section.

The parameters of the test section are shown in

Table 1. The parameters of test section [37].

Expressway Number	Design Speed	Number of Lanes	Width of Roadbed	Test Sections	Induction Condition	Length of Test Section
G5013	120 km/h	Two-way 6 lane	33.5 m	Damiao to Yunwushan	Good	4 km

, the climate and environmental indicators of the day of the test are shown in

Table 2. The experimental climatic and environmental indicators.

Date	Time	Inducing Equipment	Temperature /°C	Wind Speed	Relative Humidity/%	Minimum Visibility/m	Surface Friction Coefficient	Warning Level
23 December 2021	8:30–10:30	Functioning	5~11	Northwest wind grade 2	94	≤100	0.5	Yellow
13 December 2021	8:00–9:30		3~7	Westerly wind grade 2	92	≤50	0.5	Orange
21 December 2021	8:00–9:30		3~6	north wind grade 1	95	≤200	0.4	Yellow
24 December 2021	8:00–9:30		4~8	Northwest wind grade 2	94	≤200	0.4	Yellow
15 February 2022	8:00–9:30		4~8	Westerly wind grade 2	92	≤50	0.4	Yellow
27 January 2022	8:00–9:30		3~7	Northwest wind grade 1	96	≤200	0.5	Orange

and the parameters of the fog induction lamp in the test section are shown in

Table 3. The parameters of induced light in fog section of experimental section.

Power Supply Mode			Frequency/Hz	Luminance (Cd/m2)	The Effective Area of the Luminous Bead/m2	Tolerant Environmental Temperature/°C	Height of Node/m	Layout Spacing/m
Alternating current 120 V~240 V	Solar energy	Storage battery	50~60	5000~7000	0.02	−40~50	1.0~1.5	30

.

4.3. Basic Information About the Experimenters

A total of 40 drivers were selected and divided into eight groups for the tests to guarantee both sample size and data volume. The inclusion criteria included more than 3 years or 30,000 km of driving tests, no drunk or drugged driving records, adequate sleep and good mood before the tests, and no history of cardiovascular or cerebrovascular diseases. The number of male drivers is significantly higher than that of female drivers, according to the sex ratio of drivers in China. Therefore, the male-to-female ratio in this test was set as 5:3. This test was designed to determine the impact of different visibility in foggy environments on the driver and investigate the driving information compensation after providing supplementary lighting on both sides of the road. The eight drivers were 38 years old on average, having held the driver’s license for an average of 6 years, and their corrected visual acuity all exceeded 1.0, so the influence of the drivers’ individual visual acuity could be ignored.

4.4. Test Process

To compare the test results of different drivers on the same road section, two vehicles of the same model and with basically the same conditions were used to carry out the tests simultaneously. To ensure safety and avoid any influence of the front car on the rear car, the distance between the two cars was kept at 200 m. Each of the cars carried four people, including one driver (subject), two test recorders, and one test operator. The test recorders sat in the passenger seat and back seat, respectively. The recorder sitting in the passenger seat was responsible for recording data, while the other recorder took charge of videoing synchronously and operating the vehicle-mounted monitoring system. The test operator sat in the back seat behind the driver. When the car arrived at the starting point, the test recorder recorded vehicle speed, blood pressure, heart rate, visibility, condition of the guiding device, etc. During the test, the driver should act in close coordination with the recorders. The eye-tracking system would record the entire process of the driver’s eye movement, and the camera would record the driver’s operations and driving status from beginning to end. In order to more accurately monitor the physiological changes of the subjects, physiological status monitoring was performed three times for each driver, including the routine physiological status monitoring before the test, physiological status monitoring in normal driving conditions, and physiological status monitoring during the test in agglomerate fog environments with different visibility.

4.5. Analysis of Test Results

Based on the test data, the values of blood pressure, heart rate, visibility, and brightness of guiding lamps under different visibility were taken as input values, and the average driving speeds under different visibility were taken as the output values. Then, without considering traffic flow and road alignment, these values were substituted in Equation (25), and the calculation results basically satisfy the above findings on control compensation.

To verify the model, according to the visibility in

Table 4. Table of comparison between model calculated values and experimental values [38].

Visibility/m	Induced Light Intensity /mcd	Mean Diastolic Blood Pressure/mmhg		Mean Systolic Pressure/mmhg		Heart Rate (times/min)		Vehicle Speed (km/h)
		Experimental Value	Calculated Value	Experimental Value	Calculated Value	Experimental Value	Calculated Value
0~50	7000	85	86	123	123	74	84	20
50~100	6500	82	85	111	120	75	81	40
100~200	6000	81	79	111	114	73	77	60
200~1500	5500	86	83	113	120	74	75	80

and the matching experimental speed value, put it into the model to calculate and obtain the driver’s blood pressure and heart rate under the fog environment of the corresponding speed and visibility. After the abnormal data were removed from the experimental values, the mean value was calculated according to the group of experimenters, as shown in Table 4. According to the experimental data and the model calculation value characteristic, the model calculation values as the overall mean, experimental personnel mean as a sample mean t-test, and according to the different t-test results to judge two kinds of data, the test results show no significant difference, so the design of the experiment can be thought of as meeting the model validation and building the model in accord with the actual experiment condition. To better analyze the model mechanism based on the experimental conclusions, experimental values are selected for the subsequent mechanism analysis of driving information rebalancing compensation in a fog environment.

Based on four intervals with different visibility, the average saccade amplitude and saccade time of the drivers were analyzed comprehensively [38], and the two indicators were used to determine the visual characteristics under different visibility to further analyze the inner relation between the visual characteristics and visibility. As can be seen from the test results, the drivers’ saccade amplitude and saccade time varied greatly under different visibility. The elliptical areas represent the length of the regard time. The larger the area, the longer the regard time is. The length of the line segment between the two origins represents the drivers’ saccade amplitude. The longer the line segment, the greater the saccade amplitude is; the shorter the line, the smaller the amplitude.

It can be seen that the maximum field of regard was observed in the visibility range of (200 m, 1500 m], in which the saccade amplitude is also the maximum. The second greatest saccade amplitude and second longest saccade time were observed in the visibility range of (100 m, 200 m]. The smallest saccade amplitude and saccade time were observed in the visibility range of (0, 50 m]. Obviously, the saccade amplitude and saccade time decreased successively in the four visibility ranges. The saccade amplitude and saccade time in the four visibility ranges show a similar changing pattern. By comparing, it can be seen that, in the visibility ranges of (50 m, 100 m] and (0 m, 50 m], the drivers’ saccade amplitude in the road section with guiding lamps was larger than that without guiding lamps, and the field of regard was dominated by transverse saccade in the road section with guiding lamps and by vertical saccade in the road section without guiding lamps. It thus can be concluded that the lower the visibility, the less effective visual information the driver would receive, and the longer time it took for the driver to make a decision. To make a more decided driving decision, the driver needed more time to regard, thus resulting in a decrease in the average saccade frequency and saccade amplitude. As shown in Figure 5, the lower the visibility, the more concentrated the field of regard would be, the less the visual information would be acquired accordingly, and the greater the time lag for the acquisition of the information about roadside emergency situations and for decision-making would be. On the contrary, the higher the visibility was, the greater the saccade amplitude and frequency of the driver would be, the shorter the time for acquiring side and front road conditions would be, and the smaller the time lag for making driving decisions would be. After guiding lamps were set up on the roadside, the saccade amplitude increased, thus increasing the amount of roadside visual information to some extent, as shown in Figure 6.

A clustering analysis of the blood pressure and heart rate data obtained from the tests was performed. Through comparative analysis, it was found that the lower the visibility was, the more significant the effect of guiding lamps on both sides of foggy road sections in mitigating the drivers’ vision-induced psychological tension would be, and this effect in the visibility range of (0, 50 m] was more significant than that when the visibility was greater than 200 m. However, the analysis indicates that, in the visibility range of [100 m, 500 m], the lower the visibility was, the more significant the effect of guiding lamps on both sides of foggy road sections in mitigating the drivers’ vision-induced psychological tension would be, with the most significant effect observed in the visibility range of [50 m, 150 m]. When the visibility was above 500 m, the existence of guiding lamps would increase the drivers’ blood pressure and heart rate, namely leading to increases in their psychological stress. The above findings can be described by Figure 7.

As shown in Figure 7, the higher the visibility, the lower the psychological relief effect of guiding lamps on the drivers would be. When the visibility was above 500 m, guiding lamps would even increase the drivers’ psychological stress. The analysis of heart rate data shows that the lower the visibility, the more significant the effect of guiding lamps on both sides of foggy road sections in mitigating the drivers’ psychological stress would be, and this effect in the visibility range of (0, 200 m) was more significant than that when the visibility was greater than 300 m.

When the visibility was close to 500 m, the influence of guiding lamps on the drivers’ heart rate became smaller, and the heart rate measured on the road section with guiding lamps was higher than that on the road section without guiding lamps. It can be seen that, in good weather, the higher the visibility, the lower the psychological relief effect of guiding lamps on the drivers would be, and when the visibility was above 500 m, they would even increase the drivers’ psychological load. To describe the experimental findings more vividly, the test results were plotted to be a visibility–heart rate relation graph, as shown in Figure 8. It can be seen that the effect was more significant in the dense fog environment with visibility of 50–100 m, and this finding is basically consistent with that found in blood pressure tests.

The comparisons of blood pressure and heart rate of different testers are shown in Figure 7 and Figure 8 separately. The analysis of the test data indicates that, in the four conditions of static state, driving in a fog-free environment in the daytime, driving in a dense fog environment without guiding lamps, and driving in a dense fog environment with guiding lamps, the drivers generally showed the following characteristics: the highest blood pressure and heart rate values were obtained in the condition of driving in a dense fog environment without guiding lamps, followed by the condition of driving in a dense fog environment with guiding lamps. Thus, it can be seen that dense fog environments have an influence on drivers’ blood pressure and heart rate and that guiding lamps in dense fog environments can alleviate the increase in drivers’ blood pressure and heart rate; it can be assumed that guiding lamps in foggy environments can compensate for drivers’ loss of visual information and increase visual input to their eyes.

5. Conclusions

(1) The visual information received by the eyes during driving in an agglomerate fog environment and the feedback system of the eyes form a nonlinear system. The visibility-based speed control by the driver would further act on the visual information. Speed, visibility, and visual information constitute a dynamic nonlinear system in which the three are coupled with each other. When the braking efficiency of the vehicle, the longitudinal sliding friction coefficient between the vehicle and the road surface, and the drivers’ braking reaction time are given. Based on feedback control theory, to construct the fog environment, driving speed and intensity coefficient of the road in front of some point nonlinear equilibrium compensation model again can be calculated by the model under the condition of driving speed and certain visibility. Making drivers drive to balance the roadside induction lamp brightness can induce lateral files for the engineering practice of equipment layout to provide a reference.

(2) The nonlinear rebalancing compensation model is verified by design experiments and the test results are analyzed:

In the agglomerate fog sections of the expressway, fog may lead to changes in visibility and ambient light along the road, resulting in a decrease in the driver’s field of regard and making the field concentrated in the front of the vehicle and in the center of the road. As a result, the driver may fail to acquire the information about incoming vehicles from the left and right sides and the rear. The higher the fog concentration, the longer the driver looks at the center of the road, the smaller the field of regard will be, and the less the visual information will be acquired during driving. As shown by the test results, the maximum field of regard was observed in the visibility range of (200 m, 1500 m], in which the saccade amplitude was also the maximum. The second greatest saccade amplitude and second longest saccade time were observed in the visibility range of (100 m, 200 m]. The smallest saccade amplitude and saccade time were observed in the visibility range of (0, 50 m]. In the visibility ranges of (50 m, 100 m] and (0 m, 50 m], the drivers’ saccade amplitude in the road section with guiding lamps was larger than that without guiding lamps, and the field of regard was dominated by transverse saccade in the road section with guiding lamps and by vertical saccade in the road section without guiding lamps. The lower the visibility, the less effective visual information the driver will receive, and the longer it takes for the driver to make a decision. The driver needs more time to regard, thus resulting in a decrease in the average saccade frequency and saccade amplitude. The lower the visibility, the less the visual information will be acquired accordingly, and the greater the time lag for the acquisition of the information about roadside emergency situations and for decision-making will be.

(3) A clustering analysis of the blood pressure and heart rate data obtained from the vehicle tests was performed. Through comparative analysis of the blood pressure and heart rate data obtained in the environments with and without guiding lamps at the same visibility, it was found that (a) the lower the visibility was, the more significant the effect of guiding lamps on both sides of foggy road sections in mitigating the drivers’ vision-induced psychological tension would be, with the most significant effect observed in the visibility range of [50 m, 150 m]. However, when the visibility was above 500 m, the existence of guiding lamps would increase the drivers’ blood pressure and heart rate, namely leading to increases in their psychological stress. (b) In the four conditions of static state (driving in a fog-free environment in the daytime, driving in a dense fog environment without guiding lamps, and driving in a dense fog environment with guiding lamps) the drivers generally showed the following characteristics: The highest blood pressure and heart rate values were obtained in the condition of driving in a dense fog environment without guiding lamps, followed by the condition of driving in a dense fog environment with guiding lamps. Thus, it can be seen that dense fog environments have an influence on drivers’ blood pressure and heart rate and that guiding lamps in dense fog environments can alleviate the increase in drivers’ blood pressure and heart rate; it can be assumed that guiding lamps in foggy environments can compensate for drivers’ loss of visual information and increase visual information input to their eyes.

(4) Given the imperative of ensuring experimental safety, the extant research on traffic guidance for foggy expressway sections predominantly employs driving simulators for indoor simulation testing. Such an approach yields data that may not accurately reflect the conditions present on actual foggy expressway sections. This study distinguishes itself by conducting real-vehicle experiments within authentic expressway settings, thereby endowing the research outcomes with heightened credibility. However, due to the unique nature and extended duration of these experiments, it was not feasible to comprehensively account for the influence of road surface parameters and variations in road geometric alignment on driver behavior. Future research endeavors will integrate variables such as expressway surface friction coefficients and road geometric structures. The findings of this paper are poised to offer theoretical substantiation for the necessity of deploying traffic safety guidance devices in response to sudden low-visibility conditions both nationally and internationally. Furthermore, the study provides insights into the psychological impact of low-visibility conditions on drivers and can inform the development of guidelines for the placement of guidance lights.