Path Tracking of Highway Tunnel Inspection Robots: A Robust Enhanced Extended Sliding Mode Predictive Control Approach

Abstract

The irregular geometry of highway tunnel linings, combined with uneven terrain and external disturbances, often causes inspection robots to deviate from their predefined paths. Due to the strong coupling inherent in robotic systems, these deviations propagate to the end-effector, significantly compromising automated inspection accuracy and effectiveness. To tackle these issues, this study introduces an Enhanced Extended Sliding Mode Predictive Control (EESMPC) method, which integrates an adaptive Extended State Observer (ESO). The algorithm is derived from the robot chassis model and a desired trajectory error model, enabling precise contour profile tracking. Crucially, the integrated ESO actively estimates and compensates for unmodeled disturbances and system uncertainties within the state feedback, thereby enhancing both path tracking stability and precision. Comparative MATLAB simulations and experimental path tracking tests evaluated the performance against three other controllers. The results demonstrate that the EESMPC algorithm achieves superior tunnel lining tracking performance, exhibiting marked improvements in both tracking accuracy and system robustness. Consequently, this approach significantly enhances the automated inspection accuracy and operational efficiency of highway tunnel inspection robots.

1. Introduction

Highway tunnels, serving as critical infrastructure within urban transportation networks, are susceptible to structural deterioration—including lining cracks, water leakage, and spalling—during long-term service. This deterioration arises from factors such as construction defects and geological hazards [1,2,3], including seismic disasters [4], landslides, debris flows, etc. The progression of these defects significantly compromises tunnel serviceability and poses serious threats to transportation safety. Therefore, standardized tunnel defect inspection, essential for accurate damage diagnosis and evidence-based remediation strategies, has emerged as a critical challenge requiring urgent industry attention in tunnel operations and maintenance.

To address the demand for standardized highway tunnel inspection, tunnel inspection robot technology has advanced rapidly in recent years. Compared to traditional manual methods—characterized by high operational risks and low efficiency—robots equipped with autonomous navigation capabilities effectively replace human labor in lining inspection, significantly mitigating personnel safety hazards [5]. Modern inspection robots typically integrate high-precision sensors with advanced image processing techniques [6], substantially improving defect identification accuracy through multi-source data fusion. Current mainstream systems incorporate core technological features including: (1) robotic arms, (2) high-performance computer vision systems, (3) high-resolution 3D laser scanners, and (4) ultrasonic sensor arrays [7]. Victores et al. [8] developed an automated system using a mobile platform equipped with a robot for accurate geometric measurement of crack parameters. Mashimo’s robot [9] employed integrated CCD cameras and laser sensors, enabling automated acquisition of magnified crack imagery along with identification of concrete lining spalling and loosening defects. Nakamura [10] proposed a comprehensive full-cross-section inspection system achieving precise defect localization and dimensional quantification on tunnel surfaces.

However, substantial technical limitations remain in current highway tunnel inspection robots, primarily related to path-tracking accuracy and inspection stability. Irregular terrain conditions stemming from tunnel-construction-induced ground settlement and complex operational environments [11,12] frequently prevent robots from precisely adhering to predefined inspection paths [13]. More critically, due to kinematic chain effects, pose deviations of the mobile chassis are significantly amplified throughout the long-reach robotic arm, resulting in a pronounced amplification at the end-effector [14]. This severely degrades the positioning accuracy of mounted inspection equipment, directly compromising defect inspection reliability. Consequently, the measurement accuracy and operational efficiency of inspection robots are fundamentally constrained by the path-tracking control performance of their mobile chassis [15]. Optimizing path-tracking control strategies enables real-time compensation for pose deviations, effective suppression of environmental disturbances [16], and significant enhancement of inspection data spatial consistency and timeliness [17,18]. Therefore, developing high-precision chassis path-tracking control algorithms constitutes a fundamental requirement for improving the overall operational performance of tunnel inspection robots.

Tunnel inspection equipment is typically mounted on a mobile chassis platform. Consequently, the path-tracking control performance of this chassis directly dictates inspection accuracy and operational efficiency. As the inspection system’s core control element, high-precision path tracking ensures strict adherence to the predefined inspection trajectory, which is crucial for acquiring spatially consistent inspection data. Prevalent path tracking control algorithms primarily comprise five categories: PID Control [19,20], Pure Pursuit Control [21,22], Fuzzy Control [23,24], Model Predictive Control (MPC) [25,26], and Sliding Mode Control (SMC) [27,28,29]. However, significant limitations exist in these controllers. PID controllers lack environmental adaptability, exhibiting degraded control performance under complex tunnel disturbances. Pure Pursuit Control, being geometry-based, demonstrates increased tracking deviation in dynamic tunnel environments characterized by abrupt structural changes and non-uniform terrain. Fuzzy Control requires real-time processing of multi-source sensor information; its expanding rule base compromises real-time responsiveness due to computational load. MPC performance heavily relies on system model accuracy. Modeling errors—arising from the robot’s coupled multi-body dynamics and complex environment—lead to degraded control performance. SMC offers strong robustness and reduced model dependence, yet its inherent high-frequency chattering introduces tracking errors. Comparatively, SMC demonstrates distinct advantages for highly nonlinear tunnel inspection robots. However, effective chattering suppression remains a critical challenge. Chattering not only reduces trajectory tracking accuracy but also causes harmful vibrations in the end-mounted inspection equipment, severely compromising reliability. Therefore, designing enhanced SMC algorithms incorporating chattering suppression presents a promising approach for improving tunnel inspection robot path tracking performance.

Significant innovative research advances path tracking control methodologies. Brown et al. [30] proposed an integrated control framework leveraging Model Predictive Control (MPC) to achieve synergistic optimization of local path planning and tracking. Robust sliding mode controllers [31,32,33,34,35,36,37] were proposed for path tracking of wheeled mobile robots (WMRs) in uncertainty. The control algorithms presented above improve the performance of WMRs for path tracking, ensuring stability and precision during operation. However, these methods neglect the estimation of unknown disturbances.

To address problems of reduced precision caused by unknown disturbances encountered during path tracking, observer-based controllers to estimated disturbances were introduced for trajectory tracking control of robots. To address path tracking for rice seeding robots in paddy fields, Li [38] developed a fast terminal sliding mode controller integrated with a nonlinear observer. Meanwhile, Miranda [39] introduced a novel finite-time tracking controller for wheeled mobile robots (WMRs) under kinematic disturbances, utilizing an observer to estimate and compensate for these effects. To address trajectory tracking under disturbances, Rodríguez [40] proposed a robust control scheme that combines a disturbance observer with a proportional-retarded controller for WMRs. Ramírez [41] introduced a Linear Active Disturbance Rejection control scheme, incorporating a saturation-input strategy in the ESO design to suppress potential peaking phenomena. Meanwhile, Zhao [42] formulated a method utilizing integral sliding mode control alongside ESO to estimate and compensate for both external perturbations and unmodeled dynamics in WMR path-following tasks. Numerical simulations and experimental results confirm the efficacy and superior performance of these observer-based controllers. However, these methods have not been applied in the field of tunnel inspection. The tunnel environment poses significant challenges for sensors, and the uncertainty in measurements directly impacts controller performance. Irregularities in the tunnel linings, along with conditions such as water stains, dust, and uneven lighting, can hinder feature extraction or cause visual matching failures, leading to jumps or drift in absolute position and orientation estimates. Multi-sensor fusion serves as an essential strategy to enhance system robustness and address single-sensor failures. To efficiently estimate system performance, Yin [43] proposed a measurement precision model based on error analysis theory, to guide both hardware selection and spatial arrangement. To effectively monitor bridge cable health and bridge tower tops, Shi [44,45] used monitoring data based on an improved multi-rate data fusion method to analyze the bridge’s thermal field distribution and the time-dependent variation of tower displacements, achieving high fused displacement measurements precision. To achieve more reliable health evaluation and fault diagnosis for tunnel linings, research on health monitoring with intelligent applications of artificial intelligence utilized by inspection robots has been significantly advanced. Guo [46] introduced a dynamically constrained digital twin framework for fault diagnosis, leveraging insights from the digital twin paradigm and the Runge–Kutta method for dynamical simulation, supporting fault diagnosis under conditions of limited or missing fault data. Zhao [47] proposed a multi-feature health indicator based on joint feature distribution modeling, along with a prediction architecture named the Temporal–Self-Attention-based Dual-branch Transfer Adversarial Network. The framework is designed to enhance the generalizability of cross-working conditions.

Addressing the insufficient path tracking accuracy of highway tunnel inspection robots operating in complex environments, this study introduces an EESMPC framework designed for chassis path tracking in tunnel environments. The method combines a SMPC with an adaptive ESO to achieve high-precision trajectory tracking under the irregular terrain conditions characteristic of tunnels. The proposed method innovatively integrates SMPC with adaptive ESO. The SMPC incorporates a sliding mode function within a receding-horizon optimization framework, thereby preserving the predictive optimization and constraint-handling capabilities of MPC while introducing the robustness of sliding mode control. The ESO unifies model uncertainties and external disturbances into a lumped disturbance, which is estimated and compensated for in a feedforward manner. This two-layer architecture fundamentally enhances the system’s accuracy and robustness under complex disturbances, such as those arising from irregular tunnel road surfaces, significantly enhancing trajectory tracking accuracy and anti-disturbance capabilities. This methodology provides a robust technical foundation for advancing intelligent inspection systems in highway tunnels, as validated in Figure 1.

The main contributions of this paper are as follows: A Robust EESMPC scheme for accurate path tracking during tunnel inspection tasks is proposed in specific application scenarios with multiple sources of uncertainty, such as road unevenness, model parameter perturbations, and sensor measurement noise, which mitigates the problem of tracking errors in robot inspection systems when faced with various disturbance scenarios. Aimed at enhancing the accuracy of path tracking in varying disturbance highway tunnel environments, an improved ESO with adaptive gain adjustment based on observation error is proposed to enhance the adaptability of the EESMPC strategy. The performance of the system is evaluated both in MATLAB(v2020a) simulation and in a real tunnel environment. The results indicate that EESMPC can withstand various highway tunnel scenarios, demonstrating higher robustness and accuracy than the other methods.

The article is arranged as follows: Section 2 introduces the disturbance model and system uncertainty assumption including analysis of tunnel lining profile disturbances and analysis of tunnel pavement disturbances. Section 3 introduces the kinematic model, dynamic model and path tracking pose error model of tunnel lining inspection robots. Section 4 provides the design of the enhanced extended sliding mode predictive controller with detailed derivation. Section 5 provides experiments and Section 6 provides the discussion, where the performance of the proposed controller is compared with three other controllers. Finally, Section 7 concludes this article.

2. Disturbance Model and System Uncertainty Assumption

During tunnel lining inspection in complex working conditions, unpredictable exogenous disturbances may affect the tunnel lining inspection robot. These pronounced vibrations compromise the inspection accuracy of the robot. Exogenous disturbances primarily include tunnel lining profile variations and tunnel pavement irregularities.

2.1. Disturbance Model of Tunnel Lining Profile

The tunnel lining is typically constructed in a curved shape along the direction of traffic, and structural discontinuities along the tunnel inspection line including abrupt concavities, convexities, or installed objects may introduce disturbances. As illustrated in Figure 2, these unanticipated profile variations constitute disturbance sources during inspection. Tunnel lining disturbances primarily manifest as constant or step-like disturbances.

$$d_{\mathrm{step}}\left(t\right)=A·H(t-t_0)$$

(1)

where A is the disturbance amplitude (e.g., the dimension or height difference of an abrupt lining change). $H(t-t_0)$ is the unit step function, indicating that the disturbance initiates and persists continuously starting at time $t_0$.

2.2. Disturbance Model of Tunnel Pavement

In highway tunnels with prolonged service and heavy traffic loads, road surface settlement is inevitable. This deformation can exhibit considerable variation, with unknown rocks or depressions potentially protruding from the pavement, as illustrated in Figure 3. As the robot inspects the lining, vibration excitation from the pavement surface transmitted to the mobile platform. Therefore, mitigating the vibration impact of the tunnel pavement on the robot is essential. Unknown rocks or dent disturbances on the tunnel pavement primarily manifest as constant or step-like disturbances.

$$d_{\mathrm{pm}}\left(t\right)=B·H(t-t_1)$$

(2)

where B is the disturbance amplitude, $H(t-t_1)$ is the unit step function, indicating that the disturbance initiates and persists continuously starting at time $t_1$.

3. Mathematics and Model

3.1. Kinematic Model

The robot chassis utilizes a dual-steerable-wheel configuration, closely aligning with bicycle model assumptions. Given the constant velocity maintained during inspection, developing the kinematic model based on the bicycle model is well-justified for this structure. The chassis motion is described as rotation about an instantaneous center of rotation $C_w$. As shown in Figure 4, the origin of the robot chassis coordinate frame is located at the center of mass $C_w$, and the velocity at $C_w$ is denoted $V_b$. The parameters are presented in

Table 1. Parameter symbols.

Symbol	Parameter
$O_b$	Center of rotation
$V_b$	Chassis center speed
$R_f$	Front steering wheel rotation radius
$L_f$	Distance from front wheel to center point
$v_f$	Front wheel speed
${\theta }_f$	Front steering wheel travel angle
${\theta }_{ft}$	Front steering wheel steering angle
$C_w$	Robot chassis center point
${\phi }_w$	Heading angle
$R_b$	Rotation radius
$R_r$	Rear steering wheel rotation radius
$L_r$	Distance from the rear wheel to the center point
$v_r$	Rear wheel speed
${\theta }_r$	Rear steering wheel travel angle
${\theta }_{rt}$	Rear steering wheel steering angle
r	Steering wheel radius

.

The Ackermann steering principle dictates that the steering axes of the front and rear wheels converge at the instantaneous center of rotation $C_w$. This geometric constraint, illustrated in Figure 4, governs the steering angles of all wheels.

Assuming the robot chassis undergoes planar motion, its pose in the world coordinate frame is defined by the vector $P={[x,y,{\phi }_w]}^T$, where $(x,y)$ specifies the central position and ${\phi }_w$ denotes the heading angle. Owing to the robot’s low inspection speed and smooth turning maneuvers, the time derivative of the heading angle ${\phi }_w$ is approximated by the angular velocity $\omega$, which is approximated by the angular velocity $C_w$. Under this assumption, the kinematic model for the chassis center point is expressed as:

$$\begin{array}{c}\dot{P}=\left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{{\phi }_w}\end{array}\right]=\left[\begin{array}{ccc}cos{\phi }_w& -sin{\phi }_w& 0\\ sin{\phi }_w& cos{\phi }_w& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{v}_x\\ v_y\\ \omega \end{array}\right]\hfill \end{array}$$

(3)

Since the robot chassis is subject to nonholonomic constraints and its motion range is confined to the pavement of the tunnel, it satisfies the following nonholonomic constraint equation:

$$\dot{y}·cos{\phi }_w-\dot{x}·sin{\phi }_w=0$$

(4)

To establish the generalized kinematic equation of the mobile robot, generalized coordinates are adopted to represent its pose coordinates $\mathit{q}={\left[xy{\phi }_w{\theta }_f{\theta }_r{\theta }_{ft}{\theta }_{rt}\right]}^T$. Assuming that the wheels satisfy the pure rolling without slipping condition, the nonholonomic constraints of the mobile platform can be formulated as:

$$\left\{\begin{array}{c}\dot{y}cos{\phi }_w-\dot{x}sin{\phi }_w=0\hfill \\ \dot{x}cos{\phi }_w+\dot{y}sin{\phi }_w=r·{\dot{\theta }}_f·cos{\theta }_{ft}\hfill \\ \dot{x}cos{\phi }_w+\dot{y}sin{\phi }_w=r·{\dot{\theta }}_r·cos{\theta }_{rt}\hfill \end{array}\right.$$

(5)

From the geometric relationships in Figure 4, it can be derived that

$$\left\{\begin{array}{c}cos{\theta }_{ft}={\displaystyle \frac{R_b}{R_f}}\hfill \\ sin{\theta }_{ft}={\displaystyle \frac{L_f}{R_f}}\hfill \\ cos{\theta }_{rt}={\displaystyle \frac{R_b}{R_r}}\hfill \\ sin{\theta }_{rt}={\displaystyle \frac{L_r}{R_f}}\hfill \end{array}\right.$$

(6)

$$\left\{\begin{array}{c}\dot{x}=r·{\dot{\theta }}_{f}cos{\theta }_{ft}cos{\phi }_w\hfill \\ \dot{y}=r·{\dot{\theta }}_{f}cos{\theta }_{ft}sin{\phi }_w\hfill \end{array}\right.$$

(7)

The above kinematic constraints can be expressed as a pose transition equation in matrix form, which can be written as:

$$\mathit{A}\left(\mathit{q}\right)\dot{\mathit{q}}=\mathbf{0}$$

(8)

$$\mathit{A}=\left[\begin{array}{ccccccc}-sin{\phi }_w& cos{\phi }_w& 0& 0& 0& 0& 0\\ cos{\phi }_w& sin{\phi }_w& 0& -r{\displaystyle \frac{R_b}{R_f}}& 0& 0& 0\\ cos{\phi }_w& sin{\phi }_w& 0& 0& -r{\displaystyle \frac{R_b}{R_r}}& 0& 0\end{array}\right]$$

(9)

Thus, through the coordinate transformation matrix, the generalized coordinates in the base coordinate system of the mobile chassis can be mapped to the actuation dynamics of the driving motor within the inertial coordinate system, and its matrix form is as follows:

$$\dot{\mathit{q}}=\mathit{S}\left(\mathit{q}\right)\dot{\mathit{\beta }}$$

(10)

where $\mathit{S}\left(\mathit{q}\right)=\left[\begin{array}{cccc}r{\displaystyle \frac{R_b}{R_f}}cos{\phi }_w& 0& 0& 0\\ r{\displaystyle \frac{R_b}{R_f}}sin{\phi }_w& 0& 0& 0\\ 0& 0& 1& 0\\ 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$, $\dot{\mathit{\beta }}=\left[\begin{array}{c}{\dot{\theta }}_f\\ {\dot{\theta }}_r\\ {\dot{\theta }}_{ft}\\ {\dot{\theta }}_{rt}\end{array}\right]$.

3.2. Dynamic Model

Using the Lagrange dynamic approach, the inspection robot is described as:

$$\mathit{M}\left(\mathit{q}\right)\ddot{\mathit{q}}+\mathit{V}(\mathit{q},\dot{\mathit{q}})=\mathit{B}{\mathit{\tau }}_b+{\mathit{A}}^T\left(\mathit{q}\right){\mathit{\lambda }}_b$$

(11)

where M denotes the mass, $V(q,\dot{q})$ encompasses the centripetal and Coriolis force, B maps the input transformation matrix, ${\tau }_b$ is the input vector comprising the wheel torques, and ${\lambda }_b$ is the Lagrange multiplier.

By rearranging Lagrange’s equations, it can be derived that:

$$\mathit{M}\left(\mathit{q}\right)=\left[\begin{array}{ccccccc}m& 0& 2m_{w}sin{\phi }_w& 0& 0& 0& 0\\ 0& m& -2m_{w}cos{\phi }_w& 0& 0& 0& 0\\ 2m_{w}sin{\phi }_w& -2m_{w}cos{\phi }_w& I& 0& 0& 0& 0\\ 0& 0& 0& I_{w1}& 0& 0& 0\\ 0& 0& 0& 0& I_{w1}& 0& 0\\ 0& 0& 0& 0& 0& I_{w2}& 0\\ 0& 0& 0& 0& 0& 0& I_{w2}\end{array}\right]$$

$\mathit{V}(\mathit{q},\dot{\mathit{q}})=\left[\begin{array}{c}2m_w{\dot{{\phi }_w}}^{2}cos{\phi }_w\\ 2m_w{\dot{{\phi }_w}}^{2}sin{\phi }_w\\ 2m_w\left(\dot{x}cos{\phi }_w+\dot{y}sin{\phi }_w\right)\dot{{\phi }_w}\\ 0\\ 0\\ 0\\ 0\end{array}\right]$,

$\mathit{B}=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$,${\mathit{\tau }}_b=\left[\begin{array}{c}{\tau }_f\\ {\tau }_r\\ {\tau }_{ft}\\ {\tau }_{rt}\end{array}\right]$ where $m=m_b+2m_w$, $I=I_b+2m_w(L_f^2+L_r^2)+2I_m$. m and I are the total mass and moment of inertia of the chassis, respectively, excluding the steering wheels. The steering wheel mass is $m_w$ and $I_b$ is the platform’s moment of inertia excluding the steering wheels. The moments of inertia related to the steering wheels are defined as follows: $I_m$ (about the chassis center), $I_{w1}$ (drive wheels about their axes), and $I_{w2}$ (steering wheels themselves).

Through the previous kinematic model, it can be derived that:

$${\mathit{S}}^T\left(\mathit{q}\right)\mathit{A}\left(\mathit{q}\right)=\mathbf{0}$$

(12)

Differentiating both sides of the kinematic equation $\dot{\mathit{q}}=\mathit{S}\left(\mathit{q}\right)\dot{\mathit{\beta }}$ with respect to time yields:

$$\ddot{\mathit{q}}=\mathit{S}\left(\mathit{q}\right)\ddot{\mathit{\beta }}+\dot{\mathit{S}}\left(\mathit{q}\right)\dot{\mathit{\beta }}$$

(13)

Substituting the above equation into the dynamic equation of the mobile platform, and premultiplying both sides by ${\mathit{S}}^T\left(\mathit{q}\right)$, it can be derived that:

$${\mathit{S}}^T\left(\mathit{q}\right)\mathit{M}\left(\mathit{q}\right)\left[\mathit{S}\left(\mathit{q}\right)\ddot{\mathit{\beta }}+\dot{\mathit{S}}\left(\mathit{q}\right)\dot{\mathit{\beta }}\right]={\mathit{S}}^T\left(\mathit{q}\right)\mathit{B}{\mathit{\tau }}_b$$

(14)

Equivalently,

$${\mathit{S}}^T\left(\mathit{M}\mathit{S}\ddot{\mathit{\beta }}+\mathit{M}\dot{\mathit{S}}\dot{\mathit{\beta }}+\mathit{V}\right)={\mathit{S}}^T\mathit{B}{\mathit{\tau }}_b$$

(15)

The generalized dynamic equation of the mobile platform has been derived.

3.3. Path Tracking Pose Error Model and Analysis

Achieving precise path tracking for the highway tunnel inspection robot necessitates a robust kinematic modelling framework. During inspection, the control system first generates a continuously differentiable reference path based on the fitted geometric features of the tunnel lining. To evaluate tracking performance, a robot pose error model is formulated. This model is solved in real-time to compute optimal velocity and steering commands, ensuring the inspection equipment maintains precise path tracking along the predefined trajectory.

Accounting for variations in tunnel diameters and operational standoff distances, the world coordinate frame for pose error calculation is established at the initial chassis center position (origin O), with the robot’s heading direction defining the positive x-axis. The reference pose is $P_{\mathrm{r}}={\left[\begin{array}{c}{x}_r,y_r,{\phi }_r\end{array}\right]}^{\mathrm{T}}$. Figure 5 defines the resulting pose error configuration.

Given the slowly varying tunnel geometry, the preview control objective is set as a point previewed at a certain distance ahead along the reference path. The tracking error relative to this preview point, expressed in the robot’s body frame, is:

$$P_e=\left[\begin{array}{c}{x}_e\\ y_e\\ {\phi }_e\end{array}\right]=\left[\begin{array}{ccc}cos{\phi }_w& -sin{\phi }_w& 0\\ sin{\phi }_w& cos{\phi }_w& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{x}_r-x\\ y_r-y\\ {\phi }_r-\phi \end{array}\right]$$

(16)

where $(x_r,y_r,{\phi }_r)$ are the coordinates of the preview point.

Expanding Equation (16) and differentiating each error component yields the differential expression of the pose error:

$$\left\{\begin{array}{c}{\dot{x}}_e=v_{rx}cos{\phi }_e-v_{ry}sin{\phi }_e-v_x+\omega y_e\hfill \\ {\dot{y}}_e=v_{rx}cos{\phi }_e+v_{ry}sin{\phi }_e-v_y+\omega x_e\hfill \\ {\dot{\phi }}_e={\omega }_r-\omega \hfill \end{array}\right.$$

(17)

Given the small curvature of highway tunnel geometries, heading angle errors remain minimal during operation. This justifies linearizing Equation (17) for controller implementation:

$${\dot{P}}_e=A·P_e+u_r-u$$

(18)

where $P_e$ represents the system state vector; $A=\left[\begin{array}{ccc}0& \omega & v_{ry}\\ -\omega & 0& v_{rx}\\ 0& 0& 0\end{array}\right]$ represents system gain; $u_r$ represents the target control input vector; u represents the actual control input vector.

The target control input $u_r$ is derived from the preview point coordinates:

$${\mathit{u}}_r=\left[\begin{array}{c}{v}_r·cos{\phi }_d\\ v_r·sin{\phi }_d\\ k_{\omega }·({\phi }_d-\phi )/T\end{array}\right]$$

(19)

where $v_r=min(max(v_{base},{\displaystyle \frac{d_r}{T}}),v_{max})$, $v_{base}$ is the allowable speed during inspection; $d_r$ is distance between the chassis center and preview point. When $v_{base}>{\displaystyle \frac{d_r}{T}}$, $v_r=v_{base}$, otherwise, $v_r={\displaystyle \frac{d_r}{T}}$. $v_{max}$ is the allowable maximum speed during inspection. T is the controller sampling period; $k_{\omega }$ is the angular velocity smoothing factor regulating rate of change.

With the pose error model established, the state vector $P_e$ governs the control system. The motion control input u regulates the steerable wheels’ dynamics to actively suppress, ensuring the robot chassis converges to and tracks the reference path during tunnel inspection.

4. Enhanced Extended Sliding Mode Predictive Controller Design

During highway tunnel inspection, the Enhanced Extended Sliding Mode Predictive Control (EESMPC) framework utilizes the generated reference path to achieve high-precision tracking. As shown in Figure 6, the controller processes position/orientation error inputs through three integrated stages: (1) State Expansion: The Extended State Observer (ESO) augments the error state vector by estimating and compensating unmodeled disturbances. (2) Sliding Mode Optimization: The SMPC core solves the constrained optimization problem. (3) Rolling Horizon Execution: The optimal control input is applied while shifting the prediction window forward each cycle. This architecture enables real-time disturbance suppression while maintaining the robot chassis within prescribed tracking bounds throughout tunnel inspections.

4.1. Design of SMPC

1.

Discretization for SMPC

To transform the continuous-time model into a form suitable for digital controllers and considering the subsequent need for the online solution of quadratic programming (QP) problems, the forward Euler method is used to discretize the continuous-time model. For sampling time T, the discrete state-space representation becomes:

$${\dot{P}}_e={\displaystyle \frac{P_e(k+1)-P_e\left(k\right)}{T}}=A·P_e+U$$

(20)

where $U=u_r-u$.

Here is the precise mathematical formulation of the robot pose error model state equation, incorporating academic rigor and consistent notation:

$$\left\{\begin{array}{c}{P}_e(k+1)=A_m{P}_e\left(k\right)+BU\left(k\right)\hfill \\ y\left(k\right)=C·P_e\left(k\right)\hfill \end{array}\right.$$

(21)

where $A_m=\left[\begin{array}{ccc}1& \omega T& V_{ry}T\\ -\omega T& 1& V_{rx}T\\ 0& 0& 1\end{array}\right]$, $B=\left[\begin{array}{ccc}T& 0& 0\\ 0& T& 0\\ 0& 0& T\end{array}\right]$,$C=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$.

2.

Sliding Surface Design

Based on the discretized pose error model, the following sliding surface is selected to enforce tracking convergence:

$$s\left(k\right)=K·P_e\left(k\right)=0$$

(22)

where the matrix gains $K=[k_1,k_2,k_3]$ is selected to satisfy both the stability/dynamic performance requirements of the quasi-sliding mode and the Hurwitz condition, consequently ensuring stability on the sliding surface.

The global sliding mode surface of the prediction model can be formulated as:

$$s_p\left(k\right)=K·P_e\left(k\right)-p\left(k\right)$$

(23)

where $p\left(k\right)$ is a designed function satisfying three conditions: $p\left(0\right)=K·P_e\left(0\right)$; when $t\to \infty$, $p\left(t\right)\to \infty$; $p\left(t\right)$ is first-order differentiable.

Employing the global sliding mode function method, $p\left(k\right)$ is designed as:

$$p\left(k\right)=s\left(0\right){\alpha }^k$$

(24)

where $\alpha$ is the global sliding mode control parameter, satisfying $0<\alpha <1$.

The switching function based on global sliding mode prediction control is then:

$$s_p\left(k\right)=K·P_e\left(k\right)-s\left(0\right){\alpha }^k$$

(25)

3.

Sliding mode prediction model

The sliding function value for the future time step $k+p$ is predicted by recursively applying the discretized state-space model and sliding surface definition:

$$s_p\left(k\right)=K\left[A_m{P}_e(k-1)+BU(k-1)\right]-s\left(0\right){\alpha }^k$$

(26)

The predicted sliding function state over the prediction horizon is:

$$\begin{array}{cc}{s}_p(k+p)\hfill & =K{A}_m^p{P}_e\left(k\right)+\sum _{i=1}^{p}K{A}_m^{i-1}BU(k+p-i)-s\left(0\right){\alpha }^{k+p}\hfill \end{array}$$

(27)

To mitigate the effects of uncertainties in prediction, feedback correction is applied to the predicted sliding function. Future predictions are corrected by utilizing the deviation between the present function value $s\left(k\right)$ and the prior prediction at $k-p$:

$$\begin{array}{cc}{\widehat{s}}_p(k+p)\hfill & =s(k+p)-h_p\left[s\left(k\right)-s_p(k\mid k-p)\right]\hfill \\ \hfill & =K{A}_m^p{P}_e\left(k\right)+\sum _{i=1}^{p}K{A}_m^{i-1}BU(k+p-i)-s\left(0\right){\alpha }^k+h_p\left[s\left(k\right)-s_p(k\mid k-p)\right]\hfill \end{array}$$

(28)

where $h_p$ is the correction coefficient, typically set as $h_p=1$, $s_p(k\mid k-p)=C{A}_m^p{P}_e(k-p)+{\sum }_{i=1}^{p}C{A}^{i-1}BU(k-i)-{\alpha }^k$.

4.

Determining the reaching law

For the SMPC design, the reference trajectory is defined by the widely used exponential reaching law:

$$\dot{s}=-\epsilon \mathrm{sgn}\left(s\right)-qs$$

(29)

where $ϵ$, q are reaching law parameters. Discretizing the above equation, we can get:

$$s(k+1)-s\left(k\right)=-\epsilon T\mathrm{sgn}\left(s\right(k\left)\right)-qTs\left(k\right)$$

(30)

Thus, the reference trajectory at time $k+p$ is:

$$s_r(k+p)=\mu s_r(k+p-1)-\eta \mathrm{sgn}\left[s_r(k+p-1)\right]$$

(31)

where $0<\mu =1-qT<1$, $\eta =\epsilon T>0$.

5.

Determining the control law

Within the SMPC framework, the control input depends not only on the current sliding surface but also on the predicted values of the sliding surface over a finite future horizon. This enhances performance while mitigating output chattering. The optimal control law is defined as the solution to a QP problem with the dual objective of minimizing trajectory tracking error and penalizing excessive control effort [48]. The cost function is designed as:

$$\begin{array}{cc}J=argmin\hfill & (\sum _{i=1}^N{\left[{\widehat{s}}_{\mathrm{p}}(k+i)-s_{\mathrm{r}}(k+i)\right]}^2+\sum _{j=0}^{M-1}{r}_j{\left[U(k+j)\right]}^2)\hfill \end{array}$$

(32)

where the prediction and control horizons are denoted by N and M, respectively. The weighting matrix $r_j$ regulates the trade-off between tracking accuracy and control energy. The corresponding cost function is expressed in matrix-vector form as:

$$\begin{array}{c}J={\left[{\widehat{S}}_p(k+1)-S_r(k+1)\right]}^T\left[{\widehat{S}}_p(k+1)-S_r(k+1)\right]+{\overline{U}}^T\left(k\right)R\overline{U}\left(k\right)\hfill \end{array}$$

(33)

where ${\widehat{S}}_p(k+1)={\left[{\widehat{s}}_p(k+1),{\widehat{s}}_p(k+2),\dots ,{\widehat{s}}_p(k+N)\right]}^T$.

Substituting Equation (28) into Equation (33) and rearranging yields:

$$\begin{array}{c}J={\left[\Phi P_e\left(k\right)+GU\left(k\right)+HE\left(k\right)-S_r(k+1)-E{\alpha }^{k}S\left(0\right)\right]}^T[\Phi P_e\left(k\right)+GU\left(k\right)+HE\left(k\right)\hfill \\ -S_r(k+1)-E{\alpha }^{k}S\left(0\right)]+{\overline{U}}^T\left(k\right)R\overline{U}\left(k\right)\hfill \end{array}$$

(34)

where $R=\mathrm{diag}(r_0,r_1,\dots ,r_{M-1})$,

E is an $N\times 3$ identity matrix,

$H=2({\Phi }^T\Phi +R)$ is an $N\times N$ symmetric positive definite matrix,

$S_r(k+1)={\left[s_r(k+1),s_r(k+2),\dots ,s_r(k+N)\right]}^T$,

$E\left(k\right)={\left[s\left(k\right)-s_p(k\mid k-1),s\left(k\right)-s_p(k\mid k-2),\dots ,s\left(k\right)-s_p(k\mid k-N)\right]}^T$,

$\overline{U}={\left[U\left(k\right),U(k+1),\dots ,U(k+M-1)\right]}^T$,

$\Phi =\left[\begin{array}{c}K{A}_m\\ K{A}_m^2\\ ⋮\\ K{A}_m^N\end{array}\right]$,$G=\left[\begin{array}{cccc}KB& 0& \dots & 0\\ K{A}_{m}B& KB& \dots & 0\\ ⋮& ⋮& & ⋮\\ K{A}_m^{N-1}B& K{A}_m^{N-2}B& \dots & B\end{array}\right]$.

Taking the partial derivative of Equation (34) and setting ${\displaystyle \frac{\partial J}{\partial \overline{U}\left(k\right)}}=0$, the SMPC control law is obtained:

$$\begin{array}{cc}\overline{U}\left(k\right)\hfill & ={(G^{T}G+R)}^{-1}{G}^T\left[S_r(k+1)-\Phi P_e\left(k\right)-HE\left(k\right)-{\alpha }^{k}Es\left(0\right)\right]\hfill \end{array}$$

(35)

Substituting the reference trajectory expression (Equation (31)) allows calculation of the control increments from time k to $k+M-1$.

The actual control input applied at time k is therefore:

$$u=u_r+U$$

(36)

After obtaining the actual control input, to further enhance control performance and reduce chattering caused by rapid switching, the following control input constraints are added:

$$\left\{\begin{array}{c}\left|u_i\right|=|u_{ri}+U_i|<u_{im}\hfill \\ |\Delta u_i|=|\Delta (u_{ri}+U_i)|<u_{idm}\hfill \end{array}\right.$$

(37)

where $u_i$ represent control inputs, $u_{ri}$ represent target control inputs, $U_i$ represent target control inputs from SMPC. $u_{im}$ are maximum allowable control inputs; $u_{idm}$ is the maximum allowable absolute control increment per step. These constraints are represented by matrices $U_m=\left[u_{1m},u_{2m},u_{3m},u_{1\mathrm{dm}},u_{2\mathrm{dm}},u_{3\mathrm{dm}}\right]$ according to system requirements.

Under imposed control input constraints, solely the initial control increment $U\left(k\right)$ of the optimized sequence is deployed. By continually repeating this optimization based on the receding horizon principle, the optimal actual control input for the system is attained.

6.

System stability verification

For stability verification, the SMPC design employs the addition of a terminal constraint. Based on the cost function expression (Equation (32)), assuming:

$$V\left(k\right)=\sum _{i=1}^N{\left[{\widehat{s}}_p(k+i)-s_r(k+i)\right]}^2+\sum _{j=0}^{M-1}{r}_j{\left[U(k+j)\right]}^2$$

(38)

Assuming that at time $k+N$, the predicted state reaches the reference trajectory, achieving ideal sliding mode dynamics.

The attained optimum of the cost function, which is $V^0\left(k\right)$ at each cycle, is chosen as the Lyapunov function candidate for proof of stability.

The cost function has a sum-of-squares form, ensuring $V^0\left(k\right)$ is positive definite. To simplify, the control and prediction horizons are both set to N:

$$\begin{array}{c}{V}^0(k+1)=min(\sum _{i=1}^N{\left[{\widehat{s}}_p(k+i+1)-s_r(k+i)\right]}^2+\sum _{j=0}^{M-1}{r}_j{\left[U(k+1+j)\right]}^2)\hfill \end{array}$$

(39)

Considering the optimization process and the terminal constraint, it can be shown:

$$\begin{array}{cc}{V}^0(k+1)\hfill & =min\{V\left(k\right)-{\left[{\widehat{s}}_p(k+1)-s_r(k+1)\right]}^2-r_0{U}^2\left(k\right)+[{\widehat{s}}_p(k+1+N)\hfill \\ & -s_r(k+1+N){]}^2+r_0{U}^2(k+N)\}\hfill \end{array}$$

(40)

$$\begin{array}{cc}{V}^0(k+1)\hfill & =V^0\left(k\right)-{\left[{\widehat{s}}_p(k+1)-s_r(k+1)\right]}^2+r_0{U}^2\left(k\right)\hfill \end{array}$$

(41)

Since $V^0(k+1)\le V^0\left(k\right)$, the derivative of $V^0\left(k\right)$ is negative definite. Therefore, the system is stable under the proposed SMPC law with terminal constraint.

4.2. Adaptive Extended State Observer

Highway tunnel environments introduce complex disturbances—including aerodynamic effects, uneven terrain, and sensor noise—that degrade robot path-tracking performance. As these disturbances are often unmeasurable, this paper incorporates an ESO. In the ESO approach, the collective unknown disturbance is modeled as an augmented state, which leads to the construction of an expanded state-space model:

$$Z(k+1)=A_{m}Z\left(k\right)+BU\left(k\right)-L\left(CZ\left(k\right)-y\left(k\right)\right)$$

(42)

where $Z(k+1)$ is the new extended state vector, $L\left(k\right)={\left[\begin{array}{ccc}{l}_1& l_2& l_3\end{array}\right]}^T$ is the gain vector of the observer.

Define the estimation error between the original state and its estimate as $E_z\left(k\right)=Z\left(k\right)-P_e\left(k\right)$. From Equations (21) and (42), the error dynamics are:

$$E_z(k+1)=(A_m-LC)E_z\left(k\right)$$

(43)

To ensure $E_z$ converges to 0, the observer gain L must be designed such that all eigenvalues of $[A_m-LC]$ have negative real parts.

The ESO for the original system is thus:

$$\left\{\begin{array}{c}{E}_z\left(k\right)=Z\left(k\right)-y\left(k\right)\hfill \\ Z(k+1)=A_{m}Z\left(k\right)-L{E}_z\left(k\right)+BU\left(k\right)\hfill \end{array}\right.$$

(44)

Since different highway tunnels present varying disturbance environments, an improved ESO with adaptive gain adjustment based on observation error is proposed to enhance the adaptability of the SMPC strategy and balance the convergence speed and noise robustness.The adaptive gain is designed with the principle that a larger error magnitude leads to a higher gain for faster convergence, while a smaller error results in a reduced gain to mitigate noise.

In this paper, an adaptive observer gain is designed as $L(k+1)=L\left(k\right)+\alpha ·E_z\left(k\right)$, where $\alpha =0.1$ is the constant adjustment coefficient.

After introducing the improved adaptive ESO, the state variables input to the system are modified as follows:

$$Z_e\left(k\right)=P_e\left(k\right)+E_z\left(k\right)$$

(45)

The improved ESO can adaptively adjust observer parameters based on the system’s actual performance, improving estimation accuracy and system responsiveness. It is suitable for systems in highway tunnel environments where parameters vary over time or exhibit uncertainties.

After incorporating the improved ESO, the input to the proposed controller becomes:

$$\begin{array}{c}U\left(k\right)={(G^{T}G+R)}^{-1}{G}^T\left[S_r(k+1)-\Phi Z_e\left(k\right)-HE\left(k\right)-{\alpha }^{k}Es\left(0\right)\right]\\ ={(G^{T}G+R)}^{-1}{G}^T\left[S_r(k+1)-\Phi (P_e\left(k\right)+e\left(k\right))-HE\left(k\right)-{\alpha }^{k}Es\left(0\right)\right]\hfill \end{array}$$

(46)

Solving the SMPC optimization problem yields the optimal control sequence $U_k^{\ast }$, and at each step, only the first element of the optimized sequence is implemented as the control action.

$$u\left(k\right)=u^{\ast }\left(k\right|k)$$

(47)

Through the construction of the improved ESO, unknown disturbances within the state variables can be effectively estimated and compensated, thereby improving the stability and tracking accuracy of the path-following system.

4.3. Numerical Simulation

MATLAB simulations were performed to assess the control performance of the robust EESMPC. The path tracking performance of EESMPC was compared with that of the SMC, MPC, and PID controllers. Simulation settings included an initial pose of $P_o={\left[\begin{array}{ccc}0,& 0,& 0\end{array}\right]}^T$, and a velocity parameter $v_0=0.5$ m/s. EESMPC parameters are listed in

Table 2. Parameter symbols of EESMPC.

Parameter	Value
K	${\left[\begin{array}{ccc}0.85,& 0.85,& 0.25\end{array}\right]}^T$
$r_i$	$0.2$
N	5
M	1
$\mu$	$0.6$
$\eta$	$0.01$
$\alpha$	$0.6$
$U_m$	$\left[\begin{array}{cccccc}0.55,& 0.55,& 0.345,& 0.15,& 0.15,& 0.174\end{array}\right]$
k	$0.2$

, while the PID controller used $k_p=100$, $k_i=0$, $k_d=30$. The $k_p$, $k_i$, $k_d$ gains of the PID controller were preliminarily determined using the Ziegler–Nichols tuning method. Based on this initial tuning, manual fine-tuning was performed according to step-response performance to achieve a balance between response and stability. SMC parameters are listed in

Table 3. Parameter symbols of SMC.

Parameter	Value
C	$diag(0.18,0.015,0.023)$
$ϵ$	$0.008$
$\rho$	$0.13$
$\Delta$	$0.05$

, and The design and derivation of SMC are presented in Appendix A. The switching surface coefficients of the SMC were selected based on the desired closed-loop pole placement principle to ensure the dynamic characteristics of the sliding mode. The switching gain was determined according to the reaching condition for the existence of the sliding mode. The thickness of the boundary layer was chosen by trading off tracking accuracy against control smoothness. The simulation parameters of the MPC controller are detailed in

Table 4. Parameter symbols of MPC.

Parameter	Value
$N_p$	10
$N_c$	5
Q	$diag(30,30,15)$
R	$diag(1,1,2)$
$R_f$	$diag(0.5,0.5,1)$

. The weighting matrices of the MPC controller were determined through systematic trial-and-error combined with engineering experience. The target was to balance the squared tracking error and the squared control input over the prediction horizon. $N_p$ and $N_c$ were selected considering both the system dynamics and the available realtime computational capability. For a fair comparison, a uniform sampling period of $T=10$ ms is used.

The desired trajectory of the system is given as $P_{d1}={\left[\begin{array}{ccc}0.2,& 0,& 0\end{array}\right]}^T$. The corresponding tracking performance, obtained from simulations, is presented in Figure 7. The results indicate that the proposed method achieves convergence in 0.3 s, and the MPC method’s convergence time is 0.4 s, and the SMC method’s convergence time is 0.7 s and the convergence time of PID is 1.5 s. By comparing, it can be observed that the proposed EESMPC can make the tracking errors converge to a smaller error than the other controllers.

The sine trajectory of the system is given as $P_{d2}=0.2sin\left({\displaystyle \frac{2\pi }{5}}t-5\right)$. The data of position tracking are shown in Figure 8. The results indicate that the proposed method’s convergence time is 0.3 s, and the MPC method’s convergence time is 0.4 s, and the SMC method’s convergence time is 0.7 s and the convergence time of PID is 1.5 s. From the comparison, it can be observed that the proposed EESMPC enables the tracking errors to converge to a smaller value than the other controllers.

From the analysis of disturbances in Section 2, step position disturbances will be encountered in the desired trajectory. To verify the disturbance effect of step disturbances in the tunnel lining on the tracking trajectory, based on the simulation results of the sine curve, external step disturbances are introduced. The tracking trajectories are set to −0.14 m between 3 s and 3.4 s and −0.12 m between 7 s and 7.3 s. The simulation results of the tracking of reference trajectories with step disturbances are shown in Figure 9. It can be observed that after step disturbances are applied all the algorithms are quickly adjusted to adapt to the step disturbances. In contrast, the convergence speed to the sinusoidal trajectory is faster.

The performance and precision of the proposed algorithm were evaluated by employing a set of quantitative indicators to benchmark it against the four candidate controllers, as shown in

Table 5. Evaluation indicators of distance tracking errors.

Controller	Converge Time/s	Max Error/m	Overshoot/%	DRT/m	MAE/m	RMSE/m
SMC	$0.68$	$0.010$	$13.5$	$1.08$	$0.0040$	$0.0181$
MPC	$0.45$	$0.005$	$5.5$	$0.94$	$0.0041$	$0.0193$
EESMPC	$0.34$	$0.002$	$2.0$	$0.84$	$0.0023$	$0.0148$
PID	$1.28$	$0.030$	$2.5$	$1.41$	$0.0095$	$0.0271$

. The max error refers to the maximum deviation of the controller when encountering disturbances after convergence. In the table, DRT refers to the disturbance rejection time, and MAE and RMSE are the abbreviations for mean absolute error and root mean square error, respectively.

From the results of evaluation indicators, compared with the other controllers, the proposed method achieves a markedly shorter convergence time, a smaller max error, a smaller overshoot and a shorter disturbance rejection time. In contrast with the other controllers, the MAE and RMSE of the proposed method are smaller.

From the simulation results, for time-varying position tracking with disturbances, the proposed EESMPC maintains close alignment with the ideal curve’s trajectory, and achieves rapid regulation with minimal overshoot.

The numerical simulation results confirm that the proposed method exhibits superior dynamic response and steady-state accuracy. It delivers quantitatively controllable transient responses, precisely adjustable convergence time, and high steady-state precision in tunnel inspection trajectory tracking.

5. Experiments

In order to further assess the applicability and reliability of the proposed approach within actual highway tunnel settings, field experiments were conducted in a highway tunnel under construction.

Path tracking error during field trials was derived from multi-source sensors. To intuitively visualize tracking deviation relative to the tunnel lining contour, experimental error data was quantified as the relative deviation between the robot chassis position and the fitted reference curve. The robot features a controller built around an NVIDIA Xavier NX processor, which delivers 21 TOPs of computing power, thereby guaranteeing both high computational efficiency for the algorithm and real-time performance of the controller.

5.1. Target Path Point Acquisition

The target path for the tunnel inspection robot’s path tracking in this study is not pre-defined but acquired in real-time by four laser sensors mounted laterally on the robot. The method for obtaining target path points is illustrated in Figure 10.

Let the current values of the four sensors be $L_1$, $L_2$, $L_3$, $L_4$. These path point data are added to the fitting dataset. At the next time step, new sensor values $L_1^{\prime }$, $L_2^{\prime }$, $L_3^{\prime }$, $L_4^{\prime }$ are similarly added to the dataset. This process repeats. Subsequent fitting of the path point data within the dataset yields the target path function for tracking.

Due to sensor limitations and external disturbance, anomalous sensor data can affect curve fitting and ultimately robot tracking performance. Therefore, the acquired sensor data requires filtering. Considering highway tunnel applications, a limiting moving average filter was employed. This method averages only a finite queue of recent data, improving real-time performance while eliminating fluctuations and rejecting abrupt outliers.

5.2. Path Fitting Method Based on Least Squares

Based on the acquired target path points, data processing is required to derive the target path function that the robot chassis will track. Owing to external disturbances, sensor data is inherently contaminated with noise and errors. Moreover, the target path function needs to predict data trends to facilitate timely comparison with the current pose for motion control purposes.

The Least Squares Method (LSM) [48] identifies the optimal function fit for given data by minimizing the sum of squared errors. In this study, the LSM was employed to fit a polynomial to the sensor data, thereby obtaining the target function for the curve fitting stage. Considering the surface characteristics of tunnel walls, a polynomial model was selected. To ensure smoothness and continuity, a cubic polynomial was utilized, expressed as follows:

$$y=a{x}^3+b{x}^2+cx+d$$

(48)

Given a set of sample data points $P(x,y)$, the sum of squared errors s is:

$$s=\sum _{i=1}^m{\left[a{x}_i^3+b{x}_i^2+c{x}_i+d-y_i\right]}^2$$

(49)

Expressing s in matrix form:

$$X_v=\left[\begin{array}{cccc}1& x_1& x_1^2& x_1^3\\ 1& x_2& x_2^2& x_2^3\\ ⋮& ⋮& ⋮& ⋮\\ 1& x_m& x_m^2& x_m^3\end{array}\right],\theta =\left[\begin{array}{c}d\\ c\\ b\\ a\end{array}\right],y_r=\left[\begin{array}{c}{y}_1\\ y_2\\ y_3\\ ⋮\\ y_m\end{array}\right]$$

(50)

where $X_v$ is a Vandermonde matrix, $\theta$ is the coefficient vector, and $y_r$ is the output vector.

Applying the LSM, the optimal coefficient vector is:

$$\theta ={\left(X_v^T{X}_v\right)}^{-1}{X}_v^T{y}_r$$

(51)

The solved cubic polynomial coefficients are substituted into Equation (49), yielding the optimal curve function that minimizes the sum of squared errors.

5.3. The Actual Tunnel Environment

Experimental trials were primarily executed within the left vehicular lane. The tunnel features a concrete pavement surface with intentionally preserved unevenness, closely replicating real-world operational scenarios. The test configured within the tunnel environment is illustrated in Figure 11. For a comprehensive evaluation, two different tunnel lining scenarios are adopted to test the performance of EESMPC.

Multiple automated inspection experiments were conducted in a tunnel environment to assess the precision, stability, and overall performance of the designed path tracking control algorithm. The pavement within the tunnel is uneven, and there exist unknown protrusions and dents in the tunnel, leading to numerous unknown disturbance factors. Disturbances encountered in the path tracking are unknown dents and uncertain protusions. The maximum disturbance amplitude is 0.2 m. All four controllers were sequentially implemented in the robot’s main control unit, with autonomous inspection tasks executed separately for each algorithm. To ensure a fair comparison, five experiments were conducted for each controller. Through extensive on-site debugging, key parameters were determined, including inspection distance baseline $d_b=6000$ mm.

Two distinct tunnel lining scenarios were tested. In scenario 1, the robot operated at 1.0 m/s for 30 s. In scenario 2 (a different tunnel), the speed was increased to 1.5 m/s with the same 30 s duration. Both tests started from predefined initial chassis poses. The experimental setup mirrors that used in the preceding numerical simulations.

6. Results and Discussion

The integrated laser sensor array was employed to acquire distance measurements, which provided real-time pose data for the system. The measurements from four displacement lasers subsequently served as the central focus of the experimental analysis.

Figure 12 presents the experimental results acquired from the tunnel scenario 1. In Figure 12, the light-yellow areas indicate error bars. The values of error bars are computed as follows:

$$V_{eb}=6000\pm (1.96\ast {\displaystyle \frac{SD}{n}})$$

(52)

where $V_{eb}$ is the value of error bars. $SD=\sqrt{{\displaystyle \frac{{\sum }_{i=1}^n{(x_i-\overline{x})}^2}{n-1}}}$, 6000 mm is the zero error line, and 1.96 is the commonly used z-value for a 95% confidence interval.

It can be seen from the figure that the laser data obtained by the controller proposed falls within the error bars, while the data from the other three controllers partially lie outside the error bars. As evidenced by the experimental data, the proposed controller exhibits a reduced error in laser measurements relative to the other three controllers. In contrast, the EESMPC algorithm maintains deviations consistently near the zero-baseline, characterized by shorter settling time, reduced maximum deviation, lower absolute error, and enhanced stability. The MAE and RMSE values of the laser during path tracking are present in

Table 6. MAE values of laser data in scenario 1.

Controller	Laser 1/mm	Laser 2/mm	Laser 3/mm	Laser 4/mm
SMC	$34.2133$	$26.6147$	$34.7240$	$50.6507$
MPC	$34.5040$	$53.4320$	$21.7080$	$34.4400$
EESMPC	$17.5857$	$17.1493$	$17.5640$	$29.3960$
PID	$52.2907$	$49.2653$	$44.0773$	$63.2053$

and

Table 7. RMSE values of laser data in scenario 1.

Controller	Laser 1/mm	Laser 2/mm	Laser 3/mm	Laser 4/mm
SMC	$28.9407$	$28.2914$	$28.2322$	$43.2642$
MPC	$32.5603$	$25.7905$	$18.5917$	$41.1369$
EESMPC	$20.7921$	$17.6897$	$16.6565$	$34.8105$
PID	$35.4748$	$38.3295$	$36.5904$	$61.4735$

.

As summarized in Table 6 and Table 7, the MAE values for the proposed controllers are $17.5857$ mm, $17.1493$ mm, $17.5640$ mm and $29.3960$ mm, and the RMSE values for the proposed controllers are $20.7921$ mm, $17.6897$ mm, $16.6565$ mm and $34.8105$ mm. The proposed method outperforms the other controllers across key metrics, achieving high-precision control alongside improved energy efficiency.

Figure 13 presents the experimental results acquired from the tunnel scenario 2. In Figure 13, the values of the error bars are the same as Figure 12.

As evidenced by the experimental data, the proposed controller exhibits a reduced error in laser measurements relative to the other three controllers, characterized by shorter settling time, reduced maximum deviation, and lower absolute error. The RMSE values of the laser during path tracking are present in

Table 8. RMSE values of laser data in scenario 2.

Controller	Laser 1/mm	Laser 2/mm	Laser 3/mm	Laser 4/mm
SMC	$37.5335$	$30.8116$	$43.7759$	$65.6551$
MPC	$43.2911$	$57.3810$	$25.2770$	$45.3589$
EESMPC	$22.6997$	$19.6680$	$22.4903$	$37.5399$
PID	$59.8191$	$61.9208$	$54.8949$	$75.8679$

.

As summarized in Table 8, the RMSE values for the proposed controllers are 22.6997 mm, 19.6680 mm, 22.4903 mm and 37.5399 mm. The proposed method outperforms the other controllers across key metrics, achieving high-precision control alongside improved energy efficiency. This leads to the higher tracking accuracy of the EESMPC controller, validating its comprehensive performance advantage for tunnel lining inspection. In scenario 2, the larger RMSE values are primarily attributed to the combined effects of higher testing speeds and more complex scenario dynamics.

In conclusion, for the path tracking control of highway tunnel inspection robots, all evaluation metrics of the proposed EESMPC algorithm outperform those of the other three controllers. This verifies the accuracy and stability of the designed path tracking controller, thereby confirming its effectiveness in path tracking control for the robot chassis during autonomous inspection tasks.

7. Conclusions

External disturbances and irregular terrain within highway tunnels often cause inspection robots to deviate from their predefined paths. Critically, pose deviations of the robot chassis result in significant positioning errors, which severely compromises inspection accuracy and further degrades inspection effectiveness. To address this challenge, a robust EESMPC algorithm is proposed in this study. The proposed EESMPC incorporates an adaptive ESO to estimate and compensate for unmeasured system state parameters and disturbances, thereby enhancing the control precision and stability of the path tracking system. The validation through comparative MATLAB simulations and experiments confirms that the proposed EESMPC algorithm achieves superior tracking performance and robustness in following target curves within demanding tunnel environments. Overall, the proposed EESMPC effectively enhances the automated inspection accuracy and operational efficiency of highway tunnel inspection robots. In this study the disturbance models were simplified. To facilitate theoretical derivation and controller design, external disturbances and model uncertainties were simplified as bounded step signals. However, actual disturbances in tunnel environments may exhibit more complex time-varying or correlated characteristics. Tunnel-lining unevenness and pavement irregularities are often coupled. This simplified model may not fully capture all dynamic features of real-world disturbances, which could affect control performance optimality under extreme or unmodeled dynamic scenarios. Controller performance may also vary with the accuracy of the sensors used as well as the response speed of the chassis wheels. Future research may consider more refined disturbance models, such as those based on stochastic processes or data-driven approaches.