Auto-Inspection System Using Optimized Fuzzy Sliding Mode Control Strategy for Tunnel Inspection

Abstract

Cities composed of many mountainous areas necessitate the use of many tunnels for roads and highways with a potential safety hazard. To determine the safety of tunnels, periodic tunnel inspections mainly rely on manual work, which is dangerous and slow. Therefore, this paper proposes an auto-inspection system for tunnel inspection consisting of a robotic arm, laser sensors, and an inspection radar to free inspectors from hazardous environments and high-intensity work. Based on the mathematical model in the inspection process, a sliding mode controller is designed and optimized with fuzzy control and hyperbolic tangent functions, and is used in a tunnel inspection robot system for the first time. The simulation results show that optimized fuzzy sliding mode control can improve the tracking accuracy and stability during the tracking process compared with the traditional algorithm. Curved line inspection and arch line inspection experiment tests demonstrate that our system can automatically inspect the tunnel, and that the optimized fuzzy sliding mode control provides a superior performance in terms of the tracking process, with the average error decreasing by 37.8% when compared to traditional algorithms. Therefore, the proposed system is of great significance for high-precision and high-stability unmanned automated tunnel inspection.

1. Introduction

With the development of the economy and continuous maturity of tunnel construction technology, tunnel construction is continuously and rapidly developing. During the service period of a tunnel, the internal lining is prone to defects such as deformation, corrosion, and voids, which can seriously threaten the normal use of the tunnel and even cause serious consequences. According to relevant data statistics, the disease rate of tunnels has exceeded 60%. It is necessary to regularly inspect and maintain tunnels to ensure that tunnels can be put into service for a longer period of time. Tunnel defect inspection mainly relies on manual work, but inspectors will be exposed to dangerous tunnel environments with polluted air and continuous traffic flow. At the same time, manual inspection is time-consuming, laborious, and inefficient, making it difficult to meet the needs of tunnel inspection.

Tunnel inspection systems based on robots and intelligent sensors can greatly reduce the work intensity of the personnel and the maintenance cost of tunnels. They also improve safety by performing inspections in dangerous environments instead of inspectors. Tunnel inspection systems consisting of cameras and various sensors were proposed to capture tunnel surface images in [1,2,3,4]. In their systems, they designed a computer vision system to collect sensor data to assure precise inspection with a high resolution. A robotic arm with various sensors mounted on a moving vehicle in a tunnel inspection process was presented in [5,6,7,8,9]. Their system was capable of acquiring and extracting cracks or other defects efficiently. Elisabeth Menendez [10] presented the ROBO-SPECT robotic system, composed of a mobile vehicle, an extended crane, and a high-precision robotic arm, using the resolved motion rate control (RMRC) as a kinematic control and iterative rotation algorithm as a guided approach.

To achieve precise control of the inspection system, it is necessary to design a well-performing control strategy. The robotic arm is a complex mechanism system with multiple inputs, multiple outputs, and strong nonlinearity. The influence of various uncertain factors such as external disturbances during the motion process makes it difficult to establish an accurate mathematical model. Sliding mode control (SMC) is a real-time robust closed-loop control strategy. It has the advantages of a strong real-time fast response and high control accuracy. The application of sliding mode control was introduced in robotic arm control with an uncertain nonlinear model [11,12], but it easily leads to chattering. Therefore, numerous studies have been conducted to advance SMC. A fractional integral sliding mode control was introduced on the sliding surface, further improving the system’s performance [13,14]. Adaptive-controller-based sliding mode control was proposed to estimate the uncertainties [15,16,17]. A non-singular fast terminal sliding mode controller (SONFTSMC) was designed to solve the trajectory tracking problem for robotic manipulators with dynamic uncertainty, external disturbance, and input saturation [18,19,20,21]. Fuzzy control has a strong learning ability, does not depend on precise mathematical models, and is particularly suitable for a nonlinear time-variant system, such as robotic arm systems. An improved fuzzy compensation sliding mode control strategy was proposed to control the angular velocity of the servo system in a robotic arm system with uncertainties to eliminate chattering [22].

For automatically inspecting tunnels, there are a few situations that affect the inspection accuracy and stability of robots. Firstly, the moving vehicle may deviate, and, secondly, the measuring line of the tunnel may be a curve. So, the irregular shape of the tunnel lining should be taken into consideration in research. The robot system should follow and adjust the changes in the tunnel line, and internal defects in the tunnel should be inspected so that the system is in contact with the tunnel wall or is maintained within a fixed range. A well-performing closed-loop control strategy is crucial for the trace following performance instead of open-loop control and human-based methods.

Motivated by the above observations, this paper investigates a nonlinear control system for automatic tunnel inspection, and the mathematical model in the inspection process is established. The sliding mode controller is designed and optimized with fuzzy control and hyperbolic tangent functions to improve its control ability. MATLAB simulation and experiment tests are fabricated to prove that the system can automatically inspect the tunnel lines in different situations.

In

Table 1. A summary of key notations.

Symbol	Meaning	Symbol	Meaning
${\theta }_e$	Relative angle between the radar and the tunnel wall	$x_4$	Distance between sensor 4 and the tunnel wall
$x_1$	Distance between sensor 1 and the tunnel wall	$L$	Distance between sensor 1 and sensor 4
$d_p$	Distance between the radar surface and the tunnel wall	$\Delta a$	Deviation angle
$\Delta d$	Maintained distance between the radar surface and the tunnel wall	$P_{re}$	Expected position of the radar and the actual pose
${[{\theta }_1,{\theta }_2,{\theta }_3,{\theta }_4,{\theta }_5,{\theta }_6]}^T$	Joint angle corresponding to the current position	$P_s$	The pose of the robotic arm
${\alpha }_s,{\beta }_s,{\gamma }_s$	Euler angle of robotic arm pose	$\begin{array}{l}{n}_x,n_y,n_z\\ o_x,o_y,o_z\\ a_x,a_y,a_z\end{array}$	Projection of the unit direction vector corresponding to the coordinate axis on the x, y, and z axes
$x_s,y_s,z_s$	Position of the coordinate origin in the reference frame	$p_{x5},p_{y5},p_{z5}$	Position of the end of the robotic arm have changed
$\begin{array}{l}{n}_{x5},n_{y5},n_{z5}\\ o_{x5},o_{y5},o_{z5}\\ a_{x5},a_{y5},a_{z5}\end{array}$	Attitudes of the end of the robotic arm have changed	$P_e$	Actual and expected pose of the robot end system
$P_M$	Actual pose	$J(q)$	Velocity Jacobian matrix
$v$	Control variables	$\dot{q}$	Angular velocity of each joint
$u$	Output of the system	$e$	Input of the system is an error
$J^{-1}(q)$	Inverse of the velocity Jacobian matrix	$f\_SMC(e)$	Output of the SMC system
$x$	Actual pose	$x_d$	Expected pose
$s$	Sliding mode functions	$k_i$	Integral sliding mode constant coefficient
$\dot{s}$	Convergence law	$\tau$	Integral variables
$\epsilon ,k$	Coefficient of the convergence law	$c$	Sliding mode constant coefficient
$v_d$	Expected variables	$v_x,v_y,v_z$	Linear velocity component
$w_x,w_y,w_z$	Angular velocity component	$V$	Lyapunov function
$\lambda$	Adding coefficient	$sat(s)$	Switching function

, a summary of key notations is presented. The remainder of the paper is organized as follows. In Section 2, the inspection system will be studied and the corresponding mathematical model will be presented. The SMC law will be applied in Section 3. A fully functional experimental test will be introduced in Section 4, followed by the discussion of test results in Section 5. Finally, the conclusion will be offered in Section 6.

2. Inspection System and Mathematical Model

Our inspection system is composed of a robotic arm, a connecting bracket, six laser sensors, and an inspection radar, shown in Figure 1. The robotic arm serves as the carrier and actuator, lifting tunnel inspection sensors. The connecting bracket is equipped on the end of the robotic arm, connecting an inspection radar and laser sensors. The inspection radar is a PLT600 series geological radar that can penetrate the concrete of tunnels and inspect internal defects.

The laser sensors are LM-Q800T high-precision displacement sensors developed by senpum, divided into three groups: pitch ranging group, horizontal ranging group, and obstacle ranging group. The pitch ranging group can be used to inspect different tunnel lines, ensuring that the radar is parallel to the inner wall of the tunnel. The distance data in the horizontal direction along the tunnel line, obtained from the horizontal ranging group, are used to calculate the deviation between the actual pose and expected pose of the radar. The obstacle ranging group is installed on both ends of the connecting bracket to detect minor obstacles ahead.

The robotic arm used is AUBO-i16, and its Denavit–Hartenberg (D–H) linkage parameters are easily obtained from the model, shown in

Table 2. D–H linkage parameters of AUBO-i16 robotic arm: linkage of the robotic arm (Linkage), angle of each joint of the robotic arm (${\theta }_i$), link offset ($d_i$), link twist (${\alpha }_{i-1}$), link length ($a_{i-1}$), initial joint angle (offset).

Linkage	${\mathit{\theta }}_{\mathit{i}}$	${\mathit{d}}_{\mathit{i}}/(\mathbf{m})$	${\mathit{\alpha }}_{\mathit{i}}/(\mathbf{°})$	${\mathit{\alpha }}_{\mathit{i}-1}/(\mathbf{m})$	$\mathbf{o}\mathbf{f}\mathbf{f}\mathbf{s}\mathbf{e}\mathbf{t}/(\mathbf{°})$
1	${\theta }_1$	0.163	0	0	180
2	${\theta }_1$	0.191	−90	0	−90
3	${\theta }_3$	0	180	0.48	0
4	${\theta }_4$	0	180	0.37	−90
5	${\theta }_5$	0.1175	−90	0	0
6	${\theta }_6$	0.1035	90	0	0

.

Pose Deviation Conversion Model

A mathematical model is created to describe the behavior of the inspection system. The schematic diagram of the robotic arm during the detection process is shown in Figure 2. The X and Z coordinates represent the world coordinate system, and the movement direction of the moving vehicle is perpendicular to the paper surface, and $\phi$ represents the angle between the radar centerline and the horizontal direction. During this process, the system will continuously adjust its position and posture.

The pose deviation of the robot during the inspection process includes position deviation and pose deviation. During the process of inspection, position deviation may be caused by changes in tunnel shape or external interference, and pose deviation may be caused by bends in the tunnel, or the movement error of the robot.

For position deviation, deviations in the Z-axis and X-axis are important factors. The deviation in the Y-axis is ignored.

In Figure 3, the inspection situation of the system at $\phi$ angle is present. $x_2$ is the distance between sensor 2 and the tunnel wall, and $x_3$ is the distance between sensor 3 and the tunnel wall. During the inspection, the position and posture of the radar will be adjusted first to make the data of sensor 2 and sensor 3 be close, so that the radar is parallel to the inner wall of the tunnel. Then, the automatic correction task is carried out through the data of sensor 1 and sensor 4 placed on the left and right sides of the system.

In Figure 4, the relationship between laser sensor data and relative pose deviation is present. ${\theta }_e$ is a relative angle between the radar and the tunnel wall.

In Figure 4, $L$ is the distance between sensor 1 and sensor 4 in the horizontal direction. $x_1$ is the distance between sensor 1 and the tunnel wall, and $x_4$ is the distance between sensor 4 and the tunnel wall. The projection of the tunnel wall on the horizontal plane is a curve, so that the relative deviation angle ${\theta }_e$ can be described as

$${\theta }_e=\arctan (\frac{x_4-x_1}{L})$$

(1)

In Figure 4b, the dashed line represents the pose of the radar after correcting the deviation angle. $\Delta a$ is the distance between the radar surface and the sensor. The distance between the radar surface and the tunnel wall can be expressed as

$$d_p=\frac{x_1+x_4}{2}·\cos {\theta }_{re}-\Delta a$$

(2)

Assuming that $\Delta d$ is the distance between the radar surface and the tunnel wall that should be maintained, the relative distance deviation $d_{re}$ between them can be expressed as

$$d_{re}=d_P-\Delta d=\frac{x_1+x_4}{2}·\cos {\theta }_{re}-\Delta a-\Delta d$$

(3)

Based on the above, the relative pose error between the expected position of the radar and the actual pose can be calculated as

$$P_{re}=(\frac{x_1+x_4}{2}·\cos {\theta }_{re}-\Delta a-\Delta d,0,0,0,0,\arctan (\frac{x_4-x_1}{L}))$$

(4)

After obtaining the relative pose deviation, the relative pose deviation of the radar should be converted into the deviation between the expected pose and the actual pose of the robotic arm end.

Assuming that the joint angle corresponding to the current position is ${[{\theta }_1,{\theta }_2,{\theta }_3,{\theta }_4,{\theta }_5,{\theta }_6]}^T$, and the pose of the robotic arm is $P_s={[x_s,y_s,z_s,{\alpha }_s,{\beta }_s,{\gamma }_s]}^T$, then the corresponding homogeneous transformation matrix obtained by the robotic arm through forward Kinematics is

$$\left[\begin{array}{cccc}{n}_x& o_x& a_x& x_s\\ n_y& o_y& a_y& y_s\\ n_z& o_z& a_z& z_s\\ 0& 0& 0& 1\end{array}\right]$$

(5)

If there is a deflection angle ${\theta }_e$ between the inspection radar and the tunnel wall, it is very difficult to obtain the desired Euler angle of the end of the robotic arm, when its pitch angle $\phi$ is not zero. This paper proposes a more simple and effective method. Due to the fact that the axis of the fifth joint of the robotic arm is always parallel to the surface of the radar, the radar will move according to the fifth joint axis. Therefore, the fifth joint axis of the robot is used to correct the deviation in this paper. Firstly, to obtain the expected pose of the robot relative to the base, the fifth joint axis needs to move an angle $\Delta {\theta }_5$ to eliminate the corresponding deviation ${\theta }_e$, so that the expected angle after the robot is

$${[{\theta }_1,{\theta }_2,{\theta }_3,{\theta }_4,{\theta }_5+\arctan (\frac{x_4-x_1}{L}),{\theta }_6]}^T$$

(6)

The homogeneous transformation matrix of the robotic arm corresponding to Equation (6) is

$$\left[\begin{array}{cccc}{n}_{x5}& o_{x5}& a_{x5}& p_{x5}\\ n_{y5}& o_{y5}& a_{y5}& p_{y5}\\ n_{z5}& o_{z5}& a_{z5}& p_{z5}\\ 0& 0& 0& 1\end{array}\right]$$

(7)

From Equation (7), it can be seen that due to the rotation of the fifth joint axis of the robotic arm, the position and attitude of the end of the robotic arm have changed. The position change can be calculated by $p_{x5}$,$p_{y5}$,$p_{z5}$, and the attitude angle of the end of the robotic arm can be obtained as

$$\left\{\begin{array}{l}{\gamma }_5=-a\tan 2(n_{y5},n_{x5})\\ {\beta }_5=a\tan 2(-n_{z5},n_{x5}\cos \gamma +n_{y5}\sin \gamma )\\ {\alpha }_5=-a\tan 2(a_{x5}\sin \gamma -a_{x5}\cos \gamma ,o_{y5}\cos \gamma -o_{x5}\sin \gamma )\end{array}\right.$$

(8)

After the angle deviation is eliminated by the rotation of the fifth joint axis, the end position of the robotic arm will change. However, in practical applications, only the attitude angle of the robotic arm end is expected to change to eliminate the deviation angle, and its position is expected to remain consistent with the initial position. Therefore, its expected posture becomes

$$P_5=\left[\begin{array}{l}{x}_s\\ y_s\\ z_s\\ -a\tan 2(a_{x5}\sin \gamma -a_{x5}\cos \gamma ,o_{y5}\cos \gamma -o_{x5}\sin \gamma )\\ a\tan 2(-n_{z5},n_{x5}\cos \gamma +n_{y5}\sin \gamma )\\ -a\tan 2(n_{y5},n_{x5})\end{array}\right]$$

(9)

If both position and pose adjustments are existing during tunnel inspection, and the pitch angle $\phi$ of the robot during inspection is not zero, the relative deviation of its position will be projected in the robot base coordinate. Based on the above, the deviation between the actual and expected pose of the robot end system can be obtained as

$$P_e=\left[\begin{array}{l}\left(\frac{x_1+x_4}{2}-\Delta a-\Delta d\right)\cos {\theta }_e\cos \phi \\ 0\\ \left(\frac{x_1+x_4}{2}-\Delta a-\Delta d\right)\cos {\theta }_e\sin \phi \\ -a\tan 2(a_{x5}\sin \gamma -a_{x5}\cos \gamma ,o_{y5}\cos \gamma -o_{x5}\sin \gamma )-{\alpha }_s\\ a\tan 2(-n_{z5},n_{x5}\cos \gamma +n_{y5}\sin \gamma )-{\beta }_s\\ -a\tan 2(n_{y5},n_{x5})-{\gamma }_s\end{array}\right]$$

(10)

3. Inspection System Control Strategy

To achieve the purpose of adaptive inspection, a Lyapunov-based nonlinear control algorithm will be derived and applied to the system for the deviation adjustment control of the system during the inspection process.

3.1. Control Scheme Based on SMC (Sliding Mode Control) Algorithm

As shown in Figure 5, according to the pose deviation model mentioned above, the relative pose error $P_{re}$ between the actual and expected pose of the radar can be calculated based on the data received by the sensor displacement. After that, the error $P_e$ between the expected and actual pose of the robot end system is obtained by combining the relative pose error $P_{re}$ and the actual pose $P_M$.

The error $P_e$ is input to the SMC, which outputs the control variables $v$ and $w$, which represent the linear and angular velocities of the robot end system, respectively.

The relationship between the end velocity and angular velocity with the angular velocity of each joint can be stated as

$$v=J(q)\dot{q}$$

(11)

where $J(q)$ is the velocity Jacobian matrix. Then, the system will output the incremental angle of each joint angle of the robotic arm. In the meantime, the radar will be driven by the robot and the pose of the radar will be obtained.

3.2. Design of SMC

Assuming that the input of the system is error $e$ between the expected pose and the actual pose of the robot end system, the output of the system $u$ can be stated as

$$u=\dot{q}=J^{-1}(q)v=J^{-1}(q)·f\_SMC(e)$$

(12)

The sliding surface is designed as follows:

$$\left\{\begin{array}{l}e=x_d-x\\ s=ce+k_i{\displaystyle {\int }_{-\infty }^{t}e(\tau )}d\tau \end{array}\right.$$

(13)

For the exponential convergence law, the convergence speed gradually decreases from a large value to zero, which not only shortens the convergence time but also reaches the switching surface very smoothly. The exponential convergence law is selected as

$$\dot{s}=-\epsilon \mathrm{sgn}s-ks \epsilon >0, k>0$$

(14)

Based on the above Equations (13) and (14), the control variables $v$ can be developed as

$$\left\{\begin{array}{l}\dot{s}=c\dot{e}+k_{i}e=-\epsilon \mathrm{sgn}s-ks\\ \dot{e}={\dot{x}}_d-\dot{x}=-\frac{1}{c}(\epsilon \mathrm{sgn}s+ks+k_{i}e)\\ v=v_d+\frac{1}{c}(\epsilon \mathrm{sgn}s+ks+k_{i}e)\end{array}\right.$$

(15)

where $v$ is a matrix with six rows and one column:

$$v={\left[v_x v_y v_z {\omega }_x {\omega }_y {\omega }_z\right]}^T$$

(16)

Based on the above definitions and calculations, the SMC law can be developed as

$$u=\dot{q}=J^{-1}(q)v=J^{-1}\left[v_d+\frac{1}{c}(\epsilon \mathrm{sgn}s+ks+k_{i}e)\right]$$

(17)

To perform stability verification on the SMC, the Lyapunov function is defined as

$$V=\frac{1}{2}{s}^2$$

(18)

By simultaneously taking derivatives on both sides of Equation (18), it can be obtained that

$$\dot{V}=-\epsilon \left|s\right|-k{s}^2$$

(19)

where $\epsilon >0, k>0$, then, $-\epsilon \left|s\right|-k{s}^2\le 0$.

Based on the above calculations,

$$\dot{V}=s\dot{s}\le 0$$

(20)

According to Lyapunov theorem, the system is stable, the sliding surface will eventually converge to zero, and the rate of convergence depends on $k$.

3.3. Design of Optimized Fuzzy SMC

In this paper, fuzzy control algorithm is combined with sliding mode control algorithm, and the switching gain $k$ is optimized in real-time by introducing fuzzy logic and fuzzy rules. When the error is large, a larger gain is chosen, and when the error is small, a smaller gain is chosen in order to weaken the system’s chattering phenomenon.

The error $E$ and the change rate of $E_c$ are selected as the input variables of the fuzzy controller, and the switching gain $k$ is selected as the output.

According to

Table 3. Input variable fuzzification: input variables of the fuzzy controller (Input), switching gain ($k$), error ($E$), change rate of error ($E_c$).

Input	Physical Domain	Fuzzy Domain	Quantization Factor
$E$	[−3, 3]	[−6, 6]	0.5
$E_c$	[−6, 6]	[−6, 6]	1
$k$	[−0.3, 0.3]	[−6, 6]	0.05

, the input error $E(t)$ and the change rate of error $E_c(t)$ are fuzzified, as well as the output $k$. By determining quantization factors, the physical domain of input and output quantities is transformed into a fuzzy domain. In the interval of the fuzzy domain, the fuzzy subsets of $E$, $E_c$ and $k$ are defined as seven linguistic variables, namely (NB, NM, NS, ZO, PS, PM, PB), where seven linguistic variables represent negative large, negative medium, negative small, zero, positive small, positive medium, positive large. The corresponding fuzzy domain intervals are [–6, –4], [−4, −2], [−2, 0], [0, 2], [2, 4], and [4, 6].

Triangular membership function is selected as the input membership function. When the variables are input into the fuzzy controller and the variables output by the fuzzy controller are not at the endpoints of the fuzzy language, the degree of membership can be calculated using the triangular membership function, as shown in Figure 6.

To achieve the fuzzy inference process, a fuzzy rule table should be established based on $s$, $\dot{s}$ and symbols. Only when $s\dot{s}<0$, the system will move towards the sliding surface and stabilize near it. So, when $s\dot{s}<0$, the value $k$ should decrease to keep the system stable on the sliding surface. And, when $s\dot{s}>0$, $k$ should increase to enhance the regulatory effect of the approaching law, so that the system can approach the sliding surface faster. Therefore, the following fuzzy control rule in

Table 4. Input variable fuzzification. Error ($E$), and change rate of error ($E_c$). Negative large, negative medium, negative small, zero, positive small, positive medium, positive large (NB, NM, NS, ZO, PS, PM, PB).

	${\mathit{E}}_{\mathit{c}}$	NB	NM	NS	ZO	PS	PM	PB
$\mathit{E}$
NB		NB	NB	NB	NM	NM	NM	ZO
NM		NB	NB	NB	NM	NM	NS	PS
NS		NM	NB	NM	NS	ZO	PS	PS
ZO		NM	NM	NS	ZO	PS	PM	PM
PS		NM	NS	ZO	PS	PM	PB	PM
PM		NS	ZO	PS	PM	PB	PB	PB
PB		ZO	PS	PM	PB	PB	PB	PB

is established. The right gain value is obtained through optimization from the fuzzy rule, and the logical relationship between input and output can be seen from the fuzzy rule table. When actually writing a program to control a robotic arm, it is necessary to design a fuzzy rule table that can be queried by the program, and store the form in the memory of the industrial computer. After establishing fuzzy rules, the inference of fuzzy logic can be completed, and, finally, the fuzzy quantity needs to be transformed into an accurate output quantity through the operation of resolving the fuzzy. The defuzzification method selected in this paper is weighted average, and the output obtained is

$$z=\frac{{\displaystyle \sum _{i=0}^m(\eta (z_i)\cdot z_i)}}{{\displaystyle \sum _{i=0}^m\eta (z_i)}}$$

(21)

In Equation (21), $z$ is the exact value after defuzzification; $z_i$ is the value within the fuzzy domain; $\eta (z_i)$ is the degree of membership of $z_i$; $m$ is the number of membership degrees. Due to the fact that the value of switching gain should be greater than zero, the final output should be an absolute value.

In the above work, a fuzzy sliding mode controller is designed, but the output $\epsilon \mathrm{sgn}s+ks$ of the controller will cause chattering when the system approaches the sliding surface. Therefore, a new switching function instead of the traditional sign function is introduced to reduce chattering in this paper. By utilizing the boundedness of hyperbolic tangent functions, and adding coefficients $\lambda$, the new switching function is described as

$$sat(s)=\tanh (\lambda ·s)=\frac{e^{\lambda s}-e^{-\lambda s}}{e^{\lambda s}+e^{-\lambda s}}$$

(22)

To perform stability verification again, the Lyapunov function is defined as

$$V=\frac{1}{2}{s}^2$$

(23)

The new convergence law is obtained as

$$\dot{s}=-\epsilon \frac{e^{\lambda s}-e^{-\lambda s}}{e^{\lambda s}+e^{-\lambda s}}-ks$$

(24)

Based on the above definitions and Equations (22) and (23), it can be obtained that

$$\dot{V}=-\epsilon ·s·\tanh (\lambda ·s)-k{s}^2$$

(25)

where $\lambda >0, k>0, \epsilon >0$; in the meantime, $s·\tanh (\lambda ·s)$ is monotonically increasing when $s>0$. And, when $s=0$, $s·\tanh (\lambda ·s)=0$. Therefore, $-\epsilon ·s·\tanh (\lambda ·s)-k{s}^2<0$, when $s>0$. And $s·\tanh (\lambda ·s)$ is an even function, so $-\epsilon ·s·\tanh (\lambda ·s)-k{s}^2<0$ when $s<0$. According to Lyapunov theorem, the system is stable.

3.4. Performance Simulation and Comparison of SMC

A simulation program is written in MATLAB to verify the correction ability and tracking performance by comparing the simulation results of the SMC and the optimized fuzzy SMC. The system model parameters are shown in

Table 5. Summary of system model parameters. Step size of simulation (Step), total simulation time (Time), relative angle between the radar and the tunnel wall (${\theta }_{re}$), distance between the radar surface and the tunnel wall ($d_{re}$), and constant parameters for simulation ($\epsilon ,c,k,k_i$).

Symbol	Value	Units
Step	0.1	s
Time	5	s
$d_{re}$	0.0050	m
${\theta }_{re}$	0.0040	rad
$\epsilon$	0.0002	/
$c$	0.5	/
$k$	0.15	/
$k_i$	0.01	/

. The switch function of the optimized fuzzy SMC is introduced as $sat(s)=\tanh (2s)$.

In the process of tracking constant expected values as shown in Figure 7, compared to traditional SMC, the optimized fuzzy SMC designed in this paper has a shorter convergence time, and its maximum deviation values are smaller. This means that the convergence speed and stability of the system have been optimized.

The optimized fuzzy SMC designed in this paper can effectively reduce the chattering of the joint angle of the robot. The effect is compared with traditional SMC. The initial pose of the robot is set as

$$P_s={[-0.0223,-0.1878,0.8876,1.5402,-0.0017,-1.6356]}^T$$

Then, the joint angle corresponding to the robot is

$$J_s={[-0.056,-0.799,-1.561,-0.793,-1.579,-0.000]}^T$$

The running cycle of the program is $T=0.1s$.

The target pose of the robot is set as

$$P_r={[-0.0723,-0.1958,0.8826,1.5752,0.0383,1.6316]}^T$$

The simulation result is shown in Figure 8. It can be seen that the optimized fuzzy SMC effectively reduces joint angle chattering and the value of the joint angle, and it improves the stability and accuracy of the system.

To simulate the constantly changing posture of the robot system during inspection, the result of tracking the changing posture using two algorithms is compared in Figure 9. The desired target for the y-axis is set as $y_1=0.025\sin (0.5t)$.

The attitude angle around the y-axis is set as $y_2=0.025\sqrt{t}$.

The running cycle of the program is 0.1 s.

As shown in Figure 9, compared with the traditional SMC, the tracking error using the optimized fuzzy SMC is lower, indicating that the designed algorithm is more stable and the accuracy is higher.

4. Experimental System

In order to verify the performance of the inspection system, the tunnel inspection robot test platform is constructed. Two different experimental environments were established to simulate the inspection environment of tunnels, and the inspection accuracy and control performance in different simulated experimental environments were tested.

4.1. The Tunnel Inspection Robot Test Platform

The tunnel inspection robot test platform consists of a mobile vehicle, a robotic arm, an inspection radar, and an electrical control system, as shown in Figure 10. The main controller in the electrical control system is a NVIDIA TX2 board, and the control algorithm can be compiled and downloaded to the main controller.

4.2. The Inspection Environment Simulation of Tunnels

Two scenarios for inspecting tunnels are simulated in this experiment, and the inspection distance between the laser sensor and the wall is set at 0.27 m. During the process of inspecting the curved line, the mobile vehicle moves along the preset path. The joint angle through the designed controller is input to the robot system to make the radar remain in the inspection range at every moment.

During the tunnel inspection process, the first scenario is to inspect the arch line. The simulation test environment is composed of two arc support frames and endurance plates as shown in Figure 11a, with a distance of 2 m from the top to the ground and a diameter of 2.4 m. In this inspection, the initial distance between the laser sensor and the wall is set at 0.305 m.

During the arch line inspection, the movement path of the mobile vehicle is a preset line with a deviation in the X direction. The robot system adaptively adjusts the joint angle according to the real-time output of the control algorithm.

As most of the tunnels are curved, the tunnel inspection robot automatically inspecting the turning area is essential. The second scenario is to inspect a curved line. A curved line inspection environment with 1.7 m height, 4 m length, and 0.3 m arc depth is constructed, as shown in Figure 11b.

During the curved line inspection, the movement path of the mobile vehicle is a preset curve similar to the curve model. In this scenario, the distance between the laser sensor and the wall is set at 0.27 m in order to ensure that the distance between the radar and the wall is 3 cm, which is optimal.

5. Test Results and Discussion

To verify the accuracy and stability of the designed control algorithm, multiple automatic inspection experiments were conducted on the basis of the experimental environment built in Section 4. During the experiment, distance data were obtained from sensors 1 and 4 in the control system, which were used to present the pose of the system in this paper.

In order to evaluate the performance of the two designed algorithms, the maximum absolute error, the inspection average absolute error, and the root mean square error were compared as evaluation indicators during the inspection.

As shown in Figure 12, the arch line inspection test process is divided into three stages, namely the approaching stage, the waiting stage, and the inspection stage. At the end of the inspection stage, the distance between the two sensors and the wall is 0.27 m, indicating that the robot can automatically correct the pose error of the radar. After the approaching stage, the robot system enters the waiting stage, and the waiting time is 2 s. And then, during the inspection stage, the mobile vehicle is set to move along the preset path. In the middle of the inspection stage, it can be seen that there is a clear fluctuation area due to a deviation in the path. Additionally, the optimized fuzzy SMC has a faster rate of convergence and smaller error than the traditional SMC.

In order to verify the repeatability of the control system, multiple experiments were conducted. Taking the mean value of absolute error as the final average absolute error, the average results from multiple experiments are shown in

Table 6. Evaluation index of sensor data for arch line inspection test: laser sensor 1 (1), laser sensor 4 (4).

Sensor	Algorithm Type	Max Absolute Error (m)	Average Absolute Error (m)	Root Mean Square Error (m)
1	Traditional SMC	0.0108	0.0004	0.0051
	Optimized fuzzy SMC	0.0067	0.0002	0.0032
4	Traditional SMC	0.0109	0.0003	0.0023
	Optimized fuzzy SMC	0.0056	0.0002	0.0012

.

The maximum absolute error of sensor 1 using the traditional SMC is 10.8 mm, and the average absolute error is 0.4 mm. Correspondingly, the maximum absolute error of the tests using the optimized fuzzy SMC is 6.7 mm, and the average absolute error is 0.2 mm.

Similarly, the maximum absolute error of sensor 4 using the traditional SMC is 10.9 mm, and the average absolute error is 0.3 mm. The maximum absolute error and average absolute error of sensor 4 using the optimized fuzzy SMC are 5.6 mm and 0.2 mm, respectively.

From the comparison results of root mean square error, the experimental results show that when detecting the simulated arch line, the maximum absolute error between the radar and the inspection line decreases by 40.1%, and the average absolute error decreases by 27.7%.

The sensor data for the curved line inspection test are shown in Figure 13. For the tests, only the data from the inspection stage are analyzed. During the inspection, the mobile vehicle is set with the same preset path for autonomous movement. It can be seen that there has been a continuous change in the movement of the mobile vehicle in this experiment. However, the error from optimized fuzzy SMC is smaller than traditional SMC.

Similarly, in order to verify the repeatability of the control system, multiple experiments were conducted. The average results from multiple experiments are shown in

Table 7. Evaluation index of sensor data for curved line inspection test: laser sensor 1 (1), laser sensor 4 (4).

Sensor	Algorithm Type	Max Absolute Error (m)	Average Absolute Error (m)	Root Mean Square Error (m)
1	Traditional SMC	0.0193	0.0079	0.0086
	Optimized fuzzy SMC	0.0129	0.0053	0.0055
4	Traditional SMC	0.0185	0.0066	0.0092
	Optimized fuzzy SMC	0.0126	0.0040	0.0060

.

The maximum absolute error of sensor 1 using the traditional SMC is 19.3 mm, and the average absolute error is 7.9 mm. Correspondingly, the maximum absolute error and average absolute error of sensor 1 using the optimized fuzzy SMC are 12.9 mm and 5.3 mm, respectively.

Similarly, the maximum absolute error of sensor 4 using the traditional SMC is 18.5 mm, and the average absolute error is 6.6 mm. The maximum absolute error and average absolute error of the tests using the optimized fuzzy SMC are 12.6 mm and 4.0 mm, respectively.

From the comparison results of root mean square error, when detecting the simulated curved line, the maximum absolute error between the radar and the inspection reference line decreases by 41.6%, and the average absolute error decreases by 37.8%.

From the above discussions, it clearly shows that compared with traditional SMC, optimized fuzzy SMC converges faster and the error is smaller, indicating that the optimized fuzzy SMC has a better performance in the process of the inspection test.

6. Conclusions

The advanced control system of a tunnel robot can positively improve tunnel inspection accuracy and stability. To inspect tunnels automatically, an auto-inspection system for tunnel inspection using an optimized fuzzy SMC strategy has been proposed. The control algorithm has been studied using both numerical and experimental tests. The simulation and experimental results demonstrate that our system can automatically inspect the tunnel, and the optimized fuzzy sliding mode controller provides superior performance in terms of tracking process with the average error decreasing by 37.8% and compensating for the unknown interfere in the inspection. The optimized algorithm offers smaller error, better stability, stronger anti-interference ability when compared to traditional SMC. Therefore, the proposed system is of great significance for high-precision and high-stability tunnel inspection to release inspectors from high-intensity labor and dangerous environments.