Cross‐entropy‐based adaptive fuzzy control for visual tracking of road cracks with unmanned mobile robot

Visual tracking of road cracks in unstructured road environment was, is, and remains a crucial and challenging task, which plays a vital role in accurate crack sealing for automated road cracks repair. However, many problems have not been well solved in existing automated road cracks repair, such as the low automation due to partial dependence on manual and the interrupted traffic flow caused by the heavy equipment used. In this article, a cross‐entropy‐based adaptive fuzzy control (CEAFC) method is proposed, which reaches visual tracking with unmanned mobile robot (VT‐UMbot) for road cracks. Specifically, the CEAFC method uses cross‐entropy optimization iteration to tune parameters for the tracking controller, and fuzzy logic is constructed to explore robustness improvement. Moreover, a framework of VT‐UMbot based on a four‐wheel independent differential drive is established, and visual servo and tracking control are integrated into the system. Our experiment shows that the proposed method is extensively evaluated on three road cracks scenarios and achieves state‐of‐the‐art performance with high efficiency.

mon form of failure in pavements, which may be due to repetitive traffic loads and sudden temperature changes (Teltayev & Radovskiy, 2018), surface water and groundwater (Werkmeister et al., 2003) and interaction with other infrastructure systems (Golestani et al., 2018), and can develop into greater damage, which threatens road safety (Cebon, 1986;Henning, 2008;Lorenz, 1987).Timely and adequate maintenance can protect pavement structures and significantly extend the service life of roads (Haider et al., 2020;Pellecuer et al., 2014).At present, the repair of pavement cracks mainly depends on manual recognition and sealing.There are shortcomings such as low repair efficiency, low operation accuracy, and large safety hazards.It is easy to affect the normal traffic flow, resulting in traffic accidents.Therefore, automatic pavement crack repair has become an important demand in the road maintenance industry.The geometric characteristics of pavement cracks are more complex, the width is generally in millimeters, and the characteristics are more subtle.The existing pavement crack repair is less involved in the tracking of pavement crack trajectories.The automatic repair of pavement cracks puts forward higher requirements for the tracking of crack trajectories.
The application of deep learning and neural networks in road disease detection, monitoring, and maintenance has become popular (Yang et al., 2021).Cha et al. (2017) proposed a vision-based method that uses the deep structure of a convolutional neural network (CNN) to detect concrete cracks without calculating defect features, but a large amount of training data is needed to train the robust classifier.An image-based machine learning method was proposed by Müller et al. (2021) to classify cracked and uncracked samples, and the analysis of ductile fracture images was discussed.Melching et al. (2022) proposed a network ParallelNets combining the segmentation and regression of crack tip coordinates for crack tip detection in fatigue crack propagation experiments.B. Kim et al. (2021) proposed a surface concrete crack detection architecture based on shallow CNN, but the system is not completely automatic.The deep learning method for asphalt pavement crack identification is summarized by Nguyen et al. (2022), and the data volume and quality are the key factors for model improvement, mainly supervised learning, which takes a lot of time.It is believed that the model based on deep learning is very suitable for detecting cracks on the pavement (Balaji et al., 2018).However, the computational requirements are significantly increased.Ji et al. (2022) proposed a deep learning network based on Deep Bridge Crack Classification-Net, but frame rate per second (FPS) is reduced.A two-step pavement crack detection and segmentation method based on CNN was proposed by Liu et al. (2020).Guan et al. (2021) developed an integrated stereo vision and deep learning framework to detect road cracks at the pixel level.Guan et al. (2022) combined deep learning and stereo vision for digital reconstruction of pavement.However, the implementation of this method relies on large-scale server equipment, which requires high computing power and is rarely used on movable automation equipment.
The automatic repair of pavement cracks is an effective method to prolong the service life of pavement and has always been a research hotspot in the road maintenance industry.Some progress has been made in pavement crack repair research.Lee et al. (2006) designed a manual remote operation pavement crack filling equipment (analogue process control services (APCS)) and developed a machine vision algorithm composed of noise elimination, crack network mapping, and modeling modules, which also requires six workers to operate.Y. S. Kim et al. (2009) introduced the development history of pavement crack sealing based on the x-y table.Most of the crack-sealing equipment is semi-automatic.Lee et al. (2012) made detailed design drawings and hardware (manipulator and end effector) of an automatic pavement crack-sealing machine, but the single-machine action is not smooth, and the economic feasibility is low.Pavement crack sealing equipment has been developed by many research institutes.The actual work and production of the equipment, most of which are heavy vehicles and a trailer hung behind, work on the temporarily closed lane, resulting in the interruption of traffic flow.The driver needs to control the nozzle to fill the road cracks, and the degree of automation is low.It is impossible to accurately identify and track pavement cracks.A manipulator system for crack sealing was developed by Zhu et al. for indoor simulation experiments, in which the manipulator is fixed on the bracket, and the moving range is limited (Zhu et al., 2019).An experimental three-dimensional (3D) printer based on an automated pavement crack sealing system was designed by Liu et al. (2021), but there is a splash of asphalt sealant and overfilling phenomenon.Awuah and Garcia-Hernández (2022) designed a small 3D printing nozzle device to repair cracks in laboratory asphalt concrete.Pavement crack-filling repair based on 3D printing is still in the laboratory stage and has not been integrated with mobile devices.The existing trajectory tracking research based on mobile device motion control is mostly aimed at distribution rules and trajectory obeying certain rules.The scene of the function law is less involved in the tracking of pavement crack trajectories.The tracking of pavement crack trajectory is an important prerequisite for automatic crack repair.
In recent years, autonomous mobile robot (AMR) technology has been widely used in industry and research.Mobile robots are capable of moving from one site to another while performing complex activities (Siegwart et al., 2011).Software with integrated sensors, such as infrared, ultrasonic, camera, Global Positioning System (GPS), and magnetic sensors, are used to drive mobile robots (Ma'arif et al., 2020).Robots are propelled across space by wheels and direct current (DC) motors (Oltean, 2019).Mobile robots are widely utilized in army, firefighting, manufacturing, farming, and search and rescue missions to perform complex tasks (Masuzawa et al., 2017).Many industrial logistics applications, including the transportation of bulky and dangerous items, the agricultural industry, and inventory management systems in libraries, can benefit from the employment of linefollowing robots.AMRs can perceive their surroundings, make judgments, and take appropriate action, enabling them to move around safely and accurately on a predetermined route.These duties call for sophisticated control technology, autonomous and powerful controllers, and best-effort task completion (Hadi Amoozgar et al., 2011).On a ring-climbing robot system, an automatic crack assessment algorithm based on deep learning was created by Jang et al. (2021).An adaptive robotic disinfection based on a material identification network was successfully implemented (Hu & Li, 2022).Jiang and Zhang (2020) developed a wall-climbing unmanned aerial system for real-time crack detection (Jiang & Zhang, 2020).Research interest in the field of robotics has increased as a result of the challenge of route tracking and control of wheeled mobile robots (Gupta et al., 2014;Vans et al., 2014).Pathfollowing problems are often associated with mobile robots in industrial and manufacturing environments and in agricultural applications (Oliveira et al., 2019).In order to realize robot path tracking based on environmental factors, driving path tracking and robot control algorithms are the keys.Kayacan and Chowdhary (2019) developed a tracking error learning control method for mobile robot trajectory tracking in an off-road environment, which leverages the tracking error to dynamically update the feedforward control action (Kayacan & Chowdhary, 2019).Road roller route tracking using a thermal-based technique that integrates intelligent compaction with a global positioning system was proposed (Lu et al., 2021).Zhang et al. (2023) summarized the automatic guided vehicle (AGV) and AMR in civil engineering.It is pointed out that with the development of AGV/AMR technology, the technology will be applied.In the field of civil engineering, disease repair is a research trend, and AGV/AMR recognition accuracy and algorithm processing seed are the two main problems to be studied.There are still challenges in unmanned construction and intelligent disease repair.Motion vision recognition and control based on autonomous mobile devices are of great significance to the automatic repair of pavement cracks.
This research is responsible for the development of visual tracking with unmanned mobile robot (VT-UMbot) that integrates a four-wheel differential drive chassis, sensors, and controller control.It can track the trajectory of road cracks.To improve the efficiency of tracking, road cracks are traced manually with chalk.Even in actual crack sealing, the time required for tracing road cracks with chalk is short, and VT-UMbot can significantly improve tracking efficiency.It lays a foundation for the future road crack sealing with unmanned mobile robots.A VT-UMbot framework integrating recognition and tracking is proposed.A VT-UMbot kinematics model for crack trajectory is constructed to track and control.A crossentropy-based adaptive fuzzy control (CEAFC) method is proposed, which effectively reduces the crack tracking error and improves the robustness of the control.
In summary, this research has the following contributions: 1.An overall framework of VT-UMbot for visual tracking is proposed to track subtle and complex road cracks trajectory, in which VT-UMbot is designed by sliding steering kinematics modeling.Such a structure uses four-wheel independent drive and differential control, which only maintains flexibility for unstructured scenes such as road cracks but also achieves precise control of the motor.2. A visual servo system is constructed to ensure strong feature extraction ability for crack trajectory while maintaining a small computing power.Specifically, the VT-UMbot camera system model is designed based on position and angle to ensure recognition accuracy, and the density filtering algorithm is introduced for smoothing and denoising.The trajectory error correction provides the basis for subsequent tracking control.3. A CEAFC method is proposed.The cross-entropy (CE) optimization algorithm is introduced to tune the proportional integral differential (PID) parameters online, and the membership function is designed to perform fuzzy reasoning on the input and output of the controller, which is suitable for the tracking of unstructured road cracks trajectory.4. Extensive experiments are conducted on three road cracks scenarios to demonstrate the effectiveness of our proposed method, which achieves competitive tracking ability with low error accuracy.
The rest of this paper is organized as follows: Section 2 introduces the structure and system of VT-UMbot, and establishes the dynamic model of VT-UMbot; Section 3 presents the methods for road crack recognition and tracking control; Section 4 reports the field test of VT-UMbot on three road cracks scenarios.

METHODOLOGY
In this work, a VT-UMbot with CEAFC for tracking road cracks trajectory is presented.The overall framework, along with technological components, is illustrated in Figure 1.
Overall framework of our proposed method.

VT-UMbot kinematic modeling
VT-UMbot is designed based on the four-wheel skidsteering mobile robot that is a type of differential drive robot.Its four wheels are independent driving wheels (Fareh et al., 2020;Khatoon et al., 2021), providing strong power and great flexibility so that it can well adapt to the subtle and complex road cracks scene.A kinematic model of VT-UMbot is used to communicate the motion of the individual wheels.
In order to simplify the motion model (Wang et al., 2015), two assumptions are made here: (1) No idling phenomenon occurs when the wheels of VT-UMbot roll; (2) the center of mass (COM), while not necessarily being on the geometric horizontal line of symmetry, is situated on the VT-UMbot's longitudinal line of symmetry.The mass distribution of the VT-UMbot body is uniform.As Figure 2 shows, the geometric center of VT-UMbot serves as the origin of the coordinate system X-CENTER-Y, while the COM serves as the origin of the coordinate system X-COM-Y.The two coordinate systems' x axes coincide, whereas their y axes are parallel, and the distance from COM to CENTER is dcc.The VT-UMbot moves ahead along the positive x-axis (red arrow), the positive y-axis (green arrow) is vertical to the left, and the positive z-axis is vertical to the plane of the paper, satisfying the right-hand rule, and there is an instantaneous center of rotation (ICR).
VT-UMbot describes its motion only by the linear velocity and angular velocity [  ,   ]  ; only the angular and linear velocities are present along the x-axis at the point COM, and the ICR is on the y-axis in the coordinate system X-COM-Y.The VT-UMbot travels in a two-dimensional (2D) plane, and its local frame's linear velocity can be expressed as the following Equation (1): where  = (  ,   ) is the linear velocity of VT-UMbot in relation to VT-UMbot's local frame, and the angular velocity of VT-UMbot is in Equation ( 2): where   is the angular velocity of rotation around the zaxis.
Through kinematic analysis, the velocity law is transformed into the following Equation (3): where   and   indicate the left and right wheels' respective longitudinal component velocities, and   and   indicate the front and rear wheels' respective lateral component velocities (Figure 2).
The simplified model is shown in Figure 2, and the following relationship can be derived in Equation ( 4): where   represents the virtual wheel spacing.
If the virtual equivalent model is used to express the kinematic model, the simplified model of VT-UMbot is expressed as follows: Based on the speed of the fictitious left and right driving wheels, the simplified forward kinematics model determines the speed of the geometric COM, which is denoted by the following Equation ( 5): Based on the geometric COM speed to divide the speed of the left and right driving wheels, the simple inverse kinematics model is as the following Equation ( 6):

Road cracks image-based visual servo system
The camera on VT-UMbot is facing down and parallel to the ground.When feasible, a 2D Cartesian coordinate system is adopted for path tracking applications to keep things as simple as possible rather than a 3D coordinate system.In Figure 3, the following coordinate systems are defined: the inertial system     ; VT-UMbot coordinate system      is firmly fixed on VT-UMbot, at a height h above the land surface; the camera frame     , the center of which is situated at the camera's optical center and connected to the VT-UMbot frame     .The centers are coincident; the moving reference frame      defines the reference route C, with the   axis toward forward direction.In this paper, the front viewpoint  is regarded as VT-UMbot guidance point.

2.2.1
VT-UMbot camera system model VT-UMbot is considered to be a rigid body with wheels on the ground plane, which is represented by the generalized coordinate vector  ∈ ℝ 3 , defined as  = [, , ]  , where (, ) ∈ ℝ 2 is the Cartesian position on the plane, and  ∈ ℝ 3 is the direction of the x-axis of the VT-UMbot coordinate system      relative to the x-axis of the inertial system     .In this case, VT-UMbot satisfies the following Equation ( 7): A pinhole camera moving at body velocity (, ) ∈ ℝ 6 is considered, and any point  is observed in the road cracks scenario with the camera frame coordinate   = (  ,   ,   ) ∈ ℝ 3 .The speed of the point relative to the VT-UMbot coordinate system      is given by the following Equation ( 8): where  = (  ,   , 0) ∈ ℝ 3 and  = (0, 0,   ) ∈ ℝ 3 are the linear velocity and angular velocity of the camera, respectively, in the inertial system     .Point  can be represented in the image space using the following Equation ( 9): where (  ,   ) ∈ ℝ 2 is the coordinate of the point  represented in the image frame     ;  > 0 is the focal length of the camera lens;   ,   > 0 is the camera scaling factor; and ( 0 ,  0 ) ∈ ℝ 2 is the coordinate of the principal point.
Without losing generality, it is assumed that the mobile robot operates in the xy-plane and only rotates around the z-axis of the inertial system     ; In this case, by substituting Equation ( 8) into the time derivative of Equation ( 9), and only considering the allowed motion, it is obtained that (  , −  , −  ) = ( ẋ, ẏ, θ), and thus Equation ( 10) can be obtained: where   =   −  0 and   =   −  0 .Note that (5) associates the speed of point  in the image frame      with the speed of the camera in the inertial system     .

Density outlier removal filter-based image processing
For road crack identification and tracking, the information that needs to be extracted from the red, green, blue (RGB) image captured by the camera is the position of the pixel point  of the road crack when the VT-UMbot moves.Based on this information, VT-UMbot can perform alignment operations when moving on the road.One way to obtain this information is to use image segmentation techniques and extract the centroid of the region of interest (ROI).Park et al. (2021) performed quantitative visual detection of ROI on image surface textures.ROI are delineated based on hue-saturation-value (HSV) images for further image processing, and this area is the focus of our analysis of the crack characteristics of the acquired images.The ROI is to focus on the key target of road cracks pre-drawn with chalk, which reduces processing time and increases accuracy.
A robust segmentation algorithm combining RGB color space to HSV color space conversion and image segmentation optimization algorithm based on density filtering is proposed to deal with the changing illumination conditions of road scenes.For color photographs, the HSV space is frequently preferable (Aqthobilrobbany et al., 2020).The brightness and color information are separated in the HSV model, and the hue and saturation components are closely connected to how people perceive color.To convert the RGB model to the HSV model, first, the values of the R, G, and B components are scaled to a range between 0 and 1, that is, divide by 255.After the conversion, V and S are both between 0 and 1, and H is between 0 • and 360 • (the result of the calculation may be less than 0; if it is less than 0, add 360).
In order to improve the segmentation process, the density outlier removal filter (Le et al., 2022) for smoothing and noise reduction is applied to the HSV channel.To determine the number of points that fall in the circle's center, the density filter generates a circle with a pixel in its center.The point is kept when the number exceeds the value supplied; if the number is less than the value given, the point is discarded as noise.The circle's radius and the number of points in the circle must be manually selected, but this technique runs rapidly and the points left by the sequential iteration are the densest.
1.For each point   in the ROI, determine a neighborhood of radius  (i.e., a circle with   as the center and  as the radius).2. If the number of points in the neighborhood is  <  ℎℎ , the point   is a noise point and is eliminated.
The arbitrary point  is given by the centroid coordinates (  ,   ) of the ROI in the actual road crack image as shown in Figure 4. On this basis, the displacement error relative to the center of the image is calculated.

2.2.3
Visual servoing for error correction without losing generality, it is assumed that during the VT-UMbot's motion through road rack, point  does not move along the y-axis of the image frame     , and its depth coordinate   remains constant.Under these assumptions, the control systems in Equations ( 11) and ( 12) take the following Equation ( 13): The coordinates of the feature point  (Figure 4) in VT-UMbot reference system       , that is, a part of the reference line (road cracks) can be calculated from the pixel coordinates in the picture plane since the intrinsic camera parameters and the camera's height h above the ground are known.
The problem of visual servoing is to ensure that the VT-UMbot visually follows the image target and keeps its direction aligned with the road crack trajectory.In this case, the control strategy can be simply defined as Equation ( 14): where   is the image error between the point  and the point  (Figure 3) on the x-axis of the image frame     , and  0 is assumed to be constant or zero.The actual road crack image error is shown in Figure 4.Then, in order to ensure that the image error   is stable to zero, the following control law is proposed as shown in Equation ( 15): where   > 0 is the proportional gain.Note that since   is measured by the camera and the vehicle's direction  can be updated by the odometer, all the signals required to implement the law of control as shown in Equation ( 15) are available.

Proposed CEAFC for tracking road cracks
A CEAFC for tracking road cracks is proposed.The CE optimization algorithm is introduced to adjust the PID parameters, and the membership function is designed to perform fuzzy reasoning on the input and output of the controller for tracking of road cracks.

PID parameter optimization with CE
To conduct the tracking control, the PID parameter tuning formula of the controller is as follows in Equation ( 16): where Δ  , Δ  , and Δ  are the tuning values produced by the adaptive fuzzy controller during the online working of the VT-UMbot.The CE method is used to determine the initial parameters   ,   , and   of PID gain.
The PID parameter optimization algorithm based on the CE method for the road crack trajectory tracking is designed.The CE method is iterative and based on a specific random mechanism.A reasonable choice is to use a probability density function (PDF), such as a normal distribution.Let ℎ(, ) be a family of PDFs in  parameterized by a real value vector : ℎ(, ).The purpose of the CE method is to find the minimum  over  (let  be a real function on ).The corresponding state  * satisfies the minimum:  * =  ( * ) = min ∈ |().In each iteration, the CE method generates a ( 1 ,  2 , …   ) and ( 1 ,  2 , …   ) sequence, such that  converges to  * and  converges to  * .The probability () of the estimated event   = { ∈ |() ≥ }.
A set of N fuzzy PID controllers  = (  ,   ,   ) with ℎ(, ) = (ℎ(  , ), ℎ(  , ), ℎ(  , )) are generated by the CE algorithm, and the cost function for each controller is calculated.The parameters of two fuzzy PID controllers ( 1 ,  1 ,  1 ) and ( 2 ,  2 ,  2 ) are designed by the CE optimization method.Then ℎ(, ) is updated using a set of best controllers.This set of controllers is defined by the  parameter.When the minimum value of the cost function or the maximum number of iterations is reached, the process ends.A large number of different simulation controllers are generated to obtain the optimal parameters of the fuzzy PID controller, some of which are fixed, such as the initial parameters of the VT-UMbot and the time of each simulation cycle.The integral time absolute error (ITAE) criterion is used to evaluate the performance of the control system in each iteration.To generate the controller for optimization in this work, the normal distribution is selected.The controller parameters (  ,   ,   ) for each iteration (ℎ = 1, 2, 3, …) are estimated by the mean and the variance is defined as μℎ = , where 4 ≤  lite ≤ 20.Based on the last update of the PDF of each PID controller, the CE method creates  = 30 controllers in each iteration.Five controllers with the lowest ITAE (  lite = 5) are selected to update the next value of the PDF parameter.In order to obtain the optimal PID parameters, 300 controllers are generated by numerical simulation, which correspond to 15 update iterations of the PID controller parameters.The initial values of all scaling factors in PDF are  0 = 0.5,  0 = 0.5.The determined values of the PDF for the parameters of the controllers are shown in Table 1.The course of PID parameters during 300 experiments is presented in Figure 5.

Adaptive fuzzy control for road crack trajectory
An adaptive fuzzy controller based on the CE method is designed and implemented.One (true) and zero (false) are the only two values used in conventional logic.This is insufficient to accurately simulate how people make decisions.Fuzzy logic can imitate human reasoning since it employs the complete range of time between 0 and 1.This controller's primary function is to maintain the VT-UMbot's tracking of road cracks.Figure 6 demonstrates a block diagram of the proposed CEAFC controller, where the fuzzy controller is constructed as a feedback loop.
The control signal is , while the error is   , between the image center coordinate and the image ROI center point; and   is the change of the current error and the previous error;  is the direction of the x-axis of the VT-UMbot coordinate system      relative to the x-axis of the inertial system     ;  is the change of the current  and the previous .Δ  , Δ  , and Δ  are adjusted by fuzzy logic.Fuzzification, knowledge base, fuzzy reasoning, and defuzzification are required phases in the fuzzy logic controller.The inputs are initially translated into a fuzzification module, where the numerically controlled variables that have been quantified are changed into qualitative values, such as negative (N), zero (Z), and positive (P).These linguistic variables are described by triangular and trapezoidal membership functions.The input is then processed by the fuzzy inference module using heuristic judgments before being sent into the defuzzification module.The program is defuzzified, and the   ,   , and   are adjusted and output.
In this article, a two-input three-output system is proposed.The fuzzy logic controller has two subcontrollers for distance and angle.The input parameters are   ,   and , .Its output parameters are Δ 1 , Δ 1 , Δ 1 and Δ 2 , Δ 2 , Δ 2 .The input data are given to the fuzzification system, where the fuzzy inference system makes recommendations and the output values of Δ 1 , Δ 1 , Δ 1 and Δ 2 , Δ 2 , Δ 2 are adjusted online by the defuzzification system.The membership functions of the input and output of the fuzzy controller are expressed as negative (N), zero (Z), and positive (P) as shown in Figures 7 and 8.The purpose of these rules is to correct the orientation, moving VT-UMbot to track road cracks.Finally, the defuzzification process is performed using the centroid method, and the PID parameters are output.The pseudocode of the tracking method is as in Algorithm 1.

CASE STUDY
To test and verify the effectiveness of the proposed method and VT-UMbot designed in this article, it was conducted on the road of the Chang'an University campus Overall structure of VT-UMbot.

A l g o r i t h m 1
Visual tracking control method.
Input: The tracking error,   , ; Output: Linear and angular velocity of the robot, Velocity.linear.x and Velocity.angular.z; in Xi'an, Shaanxi Province, for experimental surveys.Three different road cracks were selected as experimental scenarios.

Experiment setting
The designed VT-UMbot is composed of a hardware platform based on a differential independent drive and a software system based on ROS.Aiming at the subtle and complex geometric characteristics of pavement cracks, the differential drive chassis is selected to better meet the tracking task.To achieve miniaturization and unmanned construction operations, VT-UMbot is developed based on ROS.

TA B L E 2
Hardware configuration of visual tracking with an unmanned mobile robot.

Structural design
The proposed hardware platform is a four-wheeled differential drive robot equipped with multiple sensors for road cracks tracking (Figure 9).Its configuration is shown in Table 2.
The following primary elements make up the VT-UMbot hardware layer:

Parameter setting
This paper adopts the ROS (Bore et al., 2019;Okumuş & Kocamaz, 2019) based on the Linux environment.The most popular robot operating system in use today, ROS, includes machine learning, motion planning, object identification, navigation and location, and robotic arm motion control.Hardware abstraction, control of low-level devices, the implementation of common functions, inter-process messaging, and packet management are just a few of the common operating system features that ROS combines.ROS is a robotic middleware that provides services that can be used to control various families of robots and enhances the ability to interact with different environments based on specific tasks.The main modules in the software architecture are represented in a block diagram (Figure 10).ROS has good compatibility with Ubuntu18.04,so the ROS robot software environment based on the Linux system Ubuntu18.04 is installed and configured.The source codes of this research project were written in C++ and Python languages, compiled and debugged, and the tracking codes were tested on the ROS.

System integration
A Realsense D435i camera with a 135-degree field of view using an red, green, blue and depth (RGB-D) perception unit, front-mounted, is used to extract road cracks features in VT-UMbot's front view scene.The image format is RGB, with 8-bit channels per color.Starting from the camera frame, the proposed visual servo system is responsible for extracting the features of road cracks traced with chalk from the unstructured road environment and conducting error correction.The system can determine the center and direction of the road cracks trajectory and output the coordinate error and angular error in real time.
The VT-UMbot tracking control system is carried out on Nvidia Jetson AGX Xavier (32G).The ROS framework, which offers the foundation for in-the-moment communication, is used by the unmanned mobile robot.The channel for the ROS client is formed by the ROS master node, which is configured on the controller.The ROS nodes can simultaneously subscribe to and publish topics to other ROS nodes.The control node publishes the speed topic ∕_ to the processing node, and the robot chassis subscribes to the speed topic to drive the four brushless motors to control VT-UMbot to track road cracks.The control system and the robot four-wheel differential chassis use a control and navigate (CAN) to universal serial bus (USB) serial communication module, which is convenient for communication between the upper computer and the lower computer.The operator communicates with the control system using HUAWEI wireless routers.

Field testing of VT-UMbot on campus roads
The straight, curved, and continuous turning road cracks are considered as actual scenes for field testing.The effectiveness of tracking road cracks for different scenarios is checked and evaluated.

3.2.1
Checking the effectiveness of straight road crack tracking The proposed CEAFC method with VT-UMbot for road cracks developed in this article needs to be validated.First, a straight crack in the asphalt road was selected (Figure 11a), which is a single crack with a width of 2 mm, and VT-UMbot was used to carry out a straight road crack tracking experiment, and the proposed method was verified.
As shown in Figure 12, the tracking effect of our method is significantly improved, with certain robustness and convergence.The biggest crack tracking error is 12.19 mm, and the mean absolute error is 3.84 mm.The crack tracking response is fast and meets the requirements of straight crack tracking.

Checking the effectiveness of curved road crack tracking
To improve the experimental samples without loss of generality, a curved crack in the cement concrete road was also selected (Figure 11b), which is a single crack with a width of 3 mm, and the VT-UMbot was used to carry out a curved road crack tracking experiment.As shown in Figure 12, the tracking effect with our method is good, with the biggest crack tracking error being 16.25 mm and the mean abso-lute error being 4.20 mm.The crack tracking response is fast and meets the crack tracking requirements for curved road crack.

Checking the effectiveness of continuous turning road crack tracking
To improve the experimental samples without loss of generality, a continuous turning crack of the cement concrete road was selected (Figure 11c), which is a single crack with a width of 3 mm, and the VT-UMbot was used for road cracks tracking experiments.
As shown in Figure 12, the tracking effect with our method is good, the biggest crack tracking error is 18.95 mm, and the mean absolute error is 4.80 mm.The crack tracking response is fast and meets the requirements for continuous turning road crack tracking.

RESULTS AND DISCUSSION
The proposed CEAFC method is coded in C++ and Python based on ROS Melodic environment with Linux system Ubuntu18.04and conducted on VT-UMbot.

Performance evaluation of the proposed CEAFC method
Experiments were carried out on campus roads to verify road cracks tracking performance.Mean absolute error and standard deviation of error were used as performance evaluation metrics.The road cracks tracking errors of VT-UMbot with our proposed method during the tracking process are shown in Figure 12.A straight crack in the asphalt road is considered as crack #1, and for the case of crack #1, the mean absolute error and the standard deviation of the crack tracking for our method are 3.84 and 1.86 mm, respectively.A curved crack in the cement concrete road is considered as crack #2, and for the case of crack #2, the mean absolute error and the standard deviation of the crack tracking for our method are 4.20 and 2.60 mm, respectively.A continuous turning crack of the cement concrete road is considered as crack #3, and for the case of crack #3, the mean absolute error and the standard deviation of the crack tracking for our method are 4.80 and 3.15 mm, respectively.
Figures 13-15 show the frequency distribution of   .For the case of crack #1, the tracking errors of our method are [−12.19 -12.18][mm] and 75.39% of the measured tracking errors are less than 4 mm.For the case of crack #2, the tracking errors of our method oscillate in the range of [−16 -16.25][mm] and 70.49% of the measured tracking errors are less than 4 mm.For the case of crack #3, the tracking errors of our method oscillate in the range of [−18.95 -17.60](mm), and 67.07% of the measured track-ing errors are less than 4 mm.Our method shows the significant efficiency of the road cracks tracking with VT-UMbot.The proportion of tracking errors less than 4 mm is more than 65%.It is clearly obvious that our method demonstrates high accuracy performance and robust control.

Performance comparison with other methods
To further demonstrate the effectiveness of our proposed method, it is compared with other four control methods.The Quantitative control is considered as Method 1.In this method, the linear velocity and angular velocity of the robot are quantitatively controlled.The wheels drive must precisely control the motor to move forward or backward at a certain linear velocity and a certain angular velocity.ROS velocity messages consist of two sets of 3D vectors, linear.x,linear.y,linear.zand angular.x,angular.y,F I G U R E 1 3 Frequency distributions of tracking error   with our proposed method and other four methods for crack #1.

F I G U R E 1 4
Frequency distributions of tracking error   with our proposed method and other four methods for crack #2.angular.z,where linear.x is the velocity in m/s representing the forward direction.The angular.z represents the angular velocity of the robot's rotation around the center, in radians per second.This article concerns the relationship between these two speeds.The remaining four members are negligible and have no practical significance for a four-wheeled robot on the ground plane.
The quantitative control of linear velocity and P control of linear velocity is considered as Method 2. In this method, the linear velocity is quantitatively controlled, and the angular velocity is proportionally controlled.By tuning the parameter   , the error is multiplied by the gain of the P controller to output a command signal.
The P control is considered as Method 3 (Iwendi et al., 2019;Ünker, 2022).In this method, the P controller's input signal reacts proportionally to the output signal.The controller is expressed mathematically,  =   ().In the formula,   is the proportional coefficient.The P controller simply modifies the signal's strength during signal transformation, not its phase.
The PID control is considered as Method 4 (Aqeel-Ur-Rehman & Cai, 2020; MohandSaidi & Mellah, 2022).The fundamental idea behind PID is to modify the feedback signal, particularly the error between the desired and measured states.The proportional gain   can quickly respond to errors, thereby reducing steady-state errors and calculating the control action.The integral gain   can make the system error zero and eliminate the steady-state error.However, too strong an integral action will increase the overshoot of the system and even cause the system to oscillate.The differential gain   can reduce the overshoot, overcome the oscillation, and improve the stability of the system.
The road cracks tracking errors of VT-UMbot with the other four methods during the tracking process are shown in Figure 12.For the case of crack #1, the tracking errors of Method 1, Method 2, Method 3, and Method 4 oscillate in the range of [−40 -39.27][mm], [−38.68 -36.56][mm], [−34.24 -33.34][mm], and [−20.31 -17.60][mm], respectively.For the case of crack #2, the tracking errors of Method 1, Method 2, Method 3, and Method 4 oscillate in the range of [−74.87 -91.44][mm], [−37.92 -37.91][mm], [−37.92 -35.22][mm], and [−20.31 -18.95][mm], respectively.For the case of crack #3, the tracking errors of Method 1, Method 2, Method 3, and Method 4 oscillate in the range of [−105.62 -90.72The road cracks tracking curves and errors obtained based on our proposed method and the other four methods are compared as shown in Table 3.For the case of crack #1, it can be seen that the tracking performance of our proposed method is the best, the peak value of the tracking error curve is the lowest, only 12.19 mm (i.e., the maximum tracking error), and the mean absolute error is only 3.84 mm.Compared with the trajectory tracking curve obtained by Method 4, our proposed method can improve the tracking accuracy of the real road cracks scene by 24.41% (calculated by mean absolute error) and the control robustness for the real road cracks scene by 48.62% (calculated by standard deviation).For the case of crack #2, it can be seen that the tracking performance of our proposed method is the best, the peak value of the tracking error curve is the lowest, only 16.25 mm (i.e., the maximum tracking error), and the mean absolute error is only 4.20 mm.Compared with the trajectory tracking curve obtained by Method 4, our proposed method can improve the tracking accuracy of the real road cracks scene by 37.13% (calculated by mean absolute error) and the control robustness for the real road cracks scene by 41.96% (calculated by standard deviation).For the case of crack #3, it can be seen that the tracking performance of our proposed method is the best, the peak value of the tracking error curve is the lowest, only 18.95 mm (i.e., the maximum tracking error), and the mean absolute error is only 4.80 mm.Compared with the trajectory tracking curve obtained by Method 4, our proposed method can improve the tracking accuracy of the real road cracks scene by 37.17% (calculated by mean absolute error) and the control robustness for the real road cracks scene by 46.79% (calculated by standard deviation).
The real tracking trajectories of VT-UMbot during three road cracks scenarios are shown in Figures 16-18.These crack tracking trajectory images are synthesized from road crack pictures collected by the camera at a frame rate of 15 Hz using an image stitching algorithm based on scaleinvariant feature transformation.It is clearly obvious that our proposed method has a better performance with minimizing the tracking error and improving the robustness.In addition, the tracking error of crack #1 is slightly higher than crack #2 and higher than crack #3 because there is a gradual increase in bending and irregularity degree from crack #1 to crack #3, and it is probably due to inertia during the steering of VT-UMbot.It can be seen from the tracking evaluation metrics that VT-UMbot can finish the road cracks tracking task well, which verifies the feasibility and effectiveness of the VT-UMbot platform and proposed CEAFC method, and lays a solid foundation for the intelligent repair of road cracks.

CONCLUSION AND FUTURE WORK
In this article, an overall framework of VT-UMbot for visual tracking is proposed to accomplish the mission of road cracks tracking, in which VT-UMbot is designed by sliding steering kinematics modeling.A visual servo system is constructed to ensure strong feature extraction ability for crack trajectory while maintaining a small computing power.A CEAFC method is proposed to guarantee the accuracy and robustness of tracking control.
Extensive experiments are conducted on three road cracks scenarios (straight road crack, curved road crack, and continuous turning road crack).Experimental results of our proposed method and the other four methods demonstrate the feasibility and effectiveness of our proposed method with strengths such as high accuracy tracking control, fast convergence, and robustness.
In the future, the collection of road crack depth images and infrared images will be considered, and research on lightweight of deep learning models for embedded devices will be carried out for road cracks (not only single cracks but also for alligator cracks) without tracing with chalk.Moreover, other control methods will be considered to further optimize the accuracy and robustness of the tracking control algorithm.The repair equipment may be added to VT-UMbot for achieving the miniaturization and unmanned road cracks repair equipment.

F I G U R E 2
Simplified equivalent schematic diagram of visual tracking with an unmanned mobile robot (VT-UMbot) model.COM, center of mass; ICR, instantaneous center of rotation.

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Top view of VT-UMbot and its camera.ICR, instantaneous center of rotation.

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Course of the proportional integral differential (PID) parameters optimization: (a) proportional gain, (b) integral gain, and (c)derivative gain.

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Proposed cross-entropy-based adaptive fuzzy control (CEAFC) controller block diagram of VT-UMbot.F I G U R E 7 Membership functions of the input and output of the proposed CEAFC controller for distance.F I G U R E 8 Membership functions of the input and output of the proposed CEAFC controller for angle.
1. Mechanical design: VT-UMbot is a four-wheel mobile robot driven by differential slip.Four 200 W brushless servo motors are used to drive the robot.The motor is F I G U R E 1 0 Software architecture of VT-UMbot.ROS, robot operating system.small, light in weight, large in output, maintenancefree, and highly efficient.2. Controller: VT-UMbot configuration uses an embedded computer Nvidia Jetson AGX Xavier as the main processor, which includes: 512 CUDA cores and 64 tensor cores Nvidia Volta GPU, v8.2 8 cores ARM CPU and 32 GB DDR4 memory.3. Sensors: The camera Intel Realsense D435i feeds raw data and depth pictures to the robot, helping it detect and classify things in the work area.VT-UMbot employs a variety of sensors to gather information from the surroundings.The encoder converts the change of displacement and angle into a photoelectric pulse or digital signal through photoelectric conversion, which is applied to the measurement of displacement, velocity, and acceleration.

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Tracking error   with our proposed method and other four methods for road cracks.

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I G U R E 1 5 Frequency distributions of tracking error   with our proposed method and other four methods for crack #3.TA B L E 3Results of tracking accuracy measurements conducted on road cracks.

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Tracking trajectory comparison of our proposed method and other four methods for straight road crack.

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Tracking trajectory comparison of our proposed method and other four methods for curved road crack.

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Tracking trajectory comparison of our proposed method and other four methods for continuous turning road crack.
The study presented in the article was partially supported by the National Key Research and Development Program of China (No. 2021YFB2601000), National Natural Science Foundation of China (No. 52078049), Fundamental Research Funds for the Central Universities, CHD (No. 300102210302, No. 300102210118), the 111 Project of Sustainable Transportation for Urban Agglomeration in Western China (No. B20035), and Natural Science Foundation of Shaanxi Province of China (S2022-JC-YB-0169).The authors also appreciate the help of graduate student Han Hong for experimental assistance.Open access publishing facilitated by Monash University, as part of the Wiley -Monash University agreement via the Council of Australian University Librarians.R E F E R E N C E SAqeel-Ur-Rehman, & Cai, C. (2020).Autonomous mobile robot obstacle avoidance using fuzzy-PID controller in robot's varying dynamics.2020 39th Chinese Control Conference (CCC), Shenyang, China (pp.2182-2186).Aqthobilrobbany, A., Handayani, A. N., Lestari, D., Muladi, Asmara, R. A., & Fukuda, O. (2020).HSV based robot boat navigation system.2020 International Conference on Computer Engineer- Error calculation from a road crack image.
, the following vision-based control system is obtained as presented in Equation (11) and Equation (12): )  (11) F I G U R E 4