Design of Adaptive Finite-Time Backstepping Control for Shield Tunneling Systems with Constraints

Abstract

This paper focuses on the finite-time tracking control problem of shield tunneling systems in the presence of constraints on the states and control input. By modeling the system based on the LuGre friction model, an effective method of tracking control in finite time is designed to overcome these actual constraints at the same time. First, the constraint on the system state is transformed into a symmetric constraint on the tracking error, and the constraint on control input is handled by designing an auxiliary differential equation. Then, radial basis function (RBF) neural networks are introduced to approximate the uncertainties. Next, using an adaptive finite-time backstepping method and choosing a logarithmic barrier Lyapunov function (BLF), a finite-time controller is designed to realize the finite-time stability of the closed-loop system. Finally, a simulation example is given to verify the correctness and validity of the theoretical results.

1. Introduction

A shield tunneling machine is a kind of construction machinery used for tunnel excavation, which has a complex structure and harsh working environment [1,2]. To reduce the risk of construction accidents caused by improper selection of tunneling parameters, it is necessary to use modern control technology to achieve more accurate control and provide useful guidance for operators.

Presently, many techniques are used in the construction of underground engineering [3,4,5,6]. During the tunneling process, the shield tunneling machine provides forward propulsive force through its thrust jacks (hydraulic cylinders), which will generate friction with the surrounding soil. Therefore, friction needs to be carefully considered in the force analysis. Presently, scholars have developed various models to study the friction force, and they have been widely applied [7,8,9,10]. The LuGre friction model proposed in [10] integrates the bristle deformation model and the Stribeck model, which can be used to establish the dynamic model of low-speed friction. For example, based on the LuGre friction model, refs. [11,12,13] investigated the non-uniform force friction compensation problem for the photoelectric tracking system, the friction compensation problem for a class of uncertain mechanical systems, and the adverse effect of nonlinear friction disturbance on tracking performance in permanent magnet synchronous motor servo systems, respectively. The propulsive velocity of the shield tunneling machine is relatively low during the tunneling process. LuGre friction model was adopted in [14,15] to describe the friction between the shield body and the surrounding soil during the tunneling process of the shield tunneling machine.

In practical engineering, the displacement, propulsive velocity, and propulsive force of the shield tunneling machine need to satisfy certain constraint conditions. For example, the tunneling machine could only advance forward, and the propulsive force should be positive and limited. These constraints were ignored in [14,15], which may affect construction safety and even lead to some unexpected situations. In the dynamic model of a shield tunneling machine based on the LuGre friction model, displacement and propulsive velocity are considered to be system states, while propulsive force is regarded as the actual control input. As such, these constraints are imposed on the system’s state and control inputs. By introducing new auxiliary variables to augment the system state, the control input constraint problem was converted into a state constraint problem in [16], therefore achieving consistent global finite-time stability. To convert the constrained output variable into an unconstrained output variable, an output transformation technique was presented in [17]. For a given output constraint or state constraint, the barrier Lyapunov function [18,19], the integral barrier Lyapunov function [20], the symmetric barrier Lyapunov function [21], or the asymmetric barrier Lyapunov function [22] were available to prevent the output or state from violating the constraints.

It should be noted that the shield tunneling machine could only advance forward, which determines that its displacement is non-negative and increasing monotonically. Meanwhile, the propulsive velocity and propulsive force are non-negative and finite. In the design of the control strategy, one needs to ensure that the full state and control input of the system always remain in a non-negative range. Moreover, as emphasized in [23,24], many tasks in practice are time-constrained and need to be accomplished within a finite time. Finite-time control ensures better convergence speed and accuracy for closed-loop control systems [25]. A class of uncertain all-state constrained strict feedback was investigated, and an adaptive finite-time command filtered backstepping control was designed in [26]. Using the finite-time Lyapunov stability theorem and saturated function, a control law to ensure both relative distance and circular velocity convergences to the prescribed values in a finite time was developed in [27], and the finite-time stability of networked control systems under denial-of-service attacks was considered in [28]. A group of heterogeneous autonomous underwater vehicle (AUV) systems with intermittent communication links was studied, and a finite-time trajectory-tracking control strategy was proposed in [29]. To investigate the problem of tracking control for a marine surface vehicle with output constraints, a finite-time control law with the virtue of backstepping technique was designed in [30]. An adaptive finite-time backstepping integral sliding mode control based on the inverse system method was presented for attitude tracking of the sail in [31]. The dynamic layout performance metrics were explored in [32], and both established and contemporary methods in trajectory planning, feedback, and feedforward control adaptive control were synthesized. The issue of trajectory tracking of helicopters under disturbances and input constraints was addressed in [33] via a fast finite-time backstepping framework.

In this paper, we investigate the finite-time tracking control problem of shield tunneling systems with constraints on the state and control input. First, the non-negative constraint on the state is transformed into a symmetric one, and the constraint on control input is handled by designing an auxiliary differential equation. Then, the finite-time convergence of the tracking error is realized by combining the barrier Lyapunov function (BLF) and the adaptive finite-time backstepping method. Meanwhile, radial basis function (RBF) neural networks are introduced to approximate some unknown functions. The simulation results show that the proposed adaptive finite-time backstepping control strategy for shield tunneling systems is effective.

2. Preliminaries and Problem Formulation

As shown in Figure 1, the shield tunneling machine cuts the soil through a rotating cutter head, while the pushing jacks (cylinders) provide forward thrust by pedaling on the built tunnel segments. The cut soil enters the soil bin and is then discharged through the screw conveyor. Whenever a shield tunneling machine excavates about $1.8$ m, it stops digging and then installs tunnel segments. After the installation of the pipe segments is completed, the shield tunneling machine starts a new excavation process. To facilitate control and assembly of tunnel segments, it is necessary to group the thrust jacks of the shield tunneling machine into four groups in four directions: up, down, left, and right. For example, in Figure 2, group A consists of the cylinders labeled 15, 16, and 1, while group C consists of the ones labeled 6, 7, 8, 9, and 10. Each group of cylinders serves as a control unit.

Consider the dynamic model of a shield tunneling system based on the LuGre friction model [34]

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{x}}_1=x_2,\hfill \\ {\dot{x}}_2={\displaystyle \frac{1}{M}}(F_u-F-J_1\left(x_2\right)z-J_2{x}_2),\hfill \\ \dot{z}=x_2-{\sigma }_0{\displaystyle \frac{|x_2|}{g\left(x_2\right)}}z,\hfill \end{array}\right.\end{array}$$

(1)

where $x_1$ and $x_2$ represent the displacement and propulsive velocity of the shield head, respectively, forming the state vector $x={[x_1,x_2]}^T$ of system (1); M is the mass of the shield head, and it is time-varying; $F_u$ denotes the total thrust of the cylinders, serving as the control input for system (1); function $g\left(x_2\right)\in {\mathbb{R}}_{>0}$ describes the Stribeck phenomenon, ${\sigma }_0$ is the stiffness; z is the internal friction state; F represents the earth pressure force exerted on cutter head and it is unknown and bounded; functions $g\left(x_2\right)$, $J_1$ and $J_2$ are defined as follows

$$\begin{array}{cc}\hfill & g\left(x_2\right)=F_c+(F_s-F_c)e^{-{(x_2/v_s)}^2},\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & J_1={\sigma }_0-{\sigma }_0{\sigma }_1{\displaystyle \frac{|x_2|}{g\left(x_2\right)}},J_2={\sigma }_1+{\sigma }_2,\hfill \end{array}$$

(3)

and the physical meanings of some symbols are briefly explained in

Table 1. Some symbols in LuGre model and system.

Symbols	Physical Meanings
$x_1$	Displacement
$x_2$	Propulsive velocity
M	Mass of shield head
F	Earth pressure force exerted on cutter head
z	Average deflection of the bristles
${\sigma }_0$	Stiffness of the bristles
${\sigma }_1$	Damping coefficient
${\sigma }_2$	Viscous coefficient
$F_c$	Coulomb friction force
$F_s$	Stiction force
$v_s$	Stribeck velocity
$F_u$	Propulsive force
$x_{1d}$	Desired displacement
$x_{2d}$	Desired propulsive velocity

.

In the field of motion control, the limitations of the actuator result in restricted control input for the shield tunneling system, making it difficult to achieve excessive control law values. In actual realization, the displacement and propulsive velocity are non-negative. To consider the complex motion form of the system, the following assumptions are needed

Assumption 1.

There exist positive scalars $V_{max}$ and $P_{max}$ such that the state $x_1$, $x_2$ and the control input $F_u$ of system (1) satisfy $0\le x_1$, $0\le x_2\le V_{max}$ and $0<F_u\le P_{max}$, respectively.

Assumption 2.

The mass function $M\left(t\right)$ and its derivative $M^{\prime }\left(t\right)$ are bounded, i.e., there exist positive scalars $M_{min}$, $M_{max}$ and $M_d$ such that $0<M_{min}\le M\left(t\right)\le M_{max}$, $0\le |M^{\prime }\left(t\right)|\le M_d$.

Remark 1.

In practical engineering, the shield tunneling machine can only move forward and not backward. Its displacement is non-negative and monotonically increasing. Similarly, the propulsive velocity is non-negative. At the same time, due to the limitations of the physical properties of the actuator, the propulsive force can only be non-negative and has an upper bound. In addition, during the tunneling process of the shield machine, the cutter head cuts the soil, and the cut soil then enters the soil bin, which increases the mass of the shield head. At the same time, the shield machine uses screw conveyors and other devices to continuously discharge the soil from the bin and then reduce the mass of the shield head accordingly. Therefore, the mass of the shield head fluctuates within a limited range.

2.1. Problem Statement

Suppose that the shield tunneling system with full state constraints satisfies Assumptions 1 and 2, our goal is to design the control input $F_u$ such that the displacement $x_1$ and propulsive velocity $x_2$ of the shield tunneling system can track up to the desired signals $x_{1d}$ and $x_{2d}$, where $x_{1d}$, $x_{2d}$ are known and satisfy ${\dot{x}}_{1d}=x_{2d}$.

Denote $e_1:=x_1-x_{1d}$ as the tracking error, then it obtains ${\dot{e}}_1={\dot{x}}_1-{\dot{x}}_{1d}=x_2-{\dot{x}}_{1d}$. Let $e_2:={\dot{e}}_1$, then ${\dot{e}}_2={\dot{x}}_2-{\ddot{x}}_{1d}$. Let $u=F_u-P_{max}/2$, and from $0<F_u\le P_{max}$, then it obtains $\left|u\right|\le P_{max}/2$, which transforms a non-negative constraint on the actual control into a symmetric constraint on the composite control. Among Assumption 1, $V_{max}$ is regarded as an upper bound on $x_{2d}$. Due to $0\le x_2\le V_{max}$ and $0\le x_{2d}\le V_{max}$, and from the definition of error, it follows $-V_{max}\le e_2=x_2-x_{2d}\le V_{max}$. In this way, the non-negative constraints of the state of the original system $x_2$ are transformed into the symmetric constraints of the error state $e_2$. In the case of $x_2$ being required to be non-negative, $x_1$ is monotonically increasing since ${\dot{x}}_1=x_2$. This shows that the state $x_1$ is non-negative if its initial value $x_1\left(0\right)\ge 0$. Next, we will take an adaptive control approach to estimate the unknown parameters online. Denote $\widehat{d}$, ${\widehat{\sigma }}_0$, ${\widehat{\sigma }}_1$, ${\widehat{J}}_2$ as the estimates of the unknown parameters $d:=-F+P_{max}/2$, ${\sigma }_0$, ${\sigma }_1$ and $J_2$, respectively, and denote $\tilde{d}:=d-\widehat{d}$, ${\tilde{\sigma }}_0:={\sigma }_0-{\widehat{\sigma }}_0$, ${\tilde{\sigma }}_1:={\sigma }_1-{\widehat{\sigma }}_1$, ${\tilde{J}}_2:=J_2-{\widehat{J}}_2$ as the corresponding estimation errors.

Thus, one can obtain

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{e}}_1=e_2,\hfill \\ {\dot{e}}_2={\displaystyle \frac{1}{M}}(u+d-J_1\left(x_2\right)z-J_2{x}_2)-{\ddot{x}}_{1d},\hfill \\ \dot{u}=-cu+v,\hfill \end{array}\right.\end{array}$$

(4)

where c is a positive constant, v is the new control to be designed (refer to (33)) so that the tracking errors $e_1$ and $e_2$ can enter a small neighborhood near the origin in a finite time. Here, the purpose of introducing the third equation in (4) is to eliminate the constraint on u.

2.2. Preliminaries

To derive the finite-time stability of system (4), some preliminaries are needed and listed below.

Lemma 1

([35]). For any vectors $x,y\in {\mathbb{R}}^n$ and positive scalar $\epsilon >0$, it holds

$$\begin{array}{c}\hfill xy\le {\displaystyle \frac{{\epsilon }^p}{p}}{\left|x\right|}^p+{\displaystyle \frac{1}{q{\epsilon }^q}}{\left|y\right|}^q,\end{array}$$

(5)

where positive constants $p>1$ and $q>1$ satisfy $(p-1)(q-1)=1$.

Lemma 2

([36]). For any positive real scalars m and n, vectors ϕ and ψ, and real function $\omega (\varphi ,\psi )>0$, it holds

$$\begin{array}{c}\hfill {\left|\varphi \right|}^m{\left|\psi \right|}^n\le {\displaystyle \frac{m}{m+n}}{\omega (\varphi ,\psi )\left|\varphi \right|}^{m+n}+{\displaystyle \frac{n}{m+n}}\omega {(\varphi ,\psi )}^{-{\displaystyle \frac{m}{n}}}{\left|\psi \right|}^{m+n}.\end{array}$$

(6)

Lemma 3

([37]). For any $x_i\in \mathbb{R}(i=1,2,\cdots ,n)$ and positive scalar $\gamma \in (0,1)$, it holds

$$\begin{array}{c}\hfill \left(\right|x_1|+\cdots +|x_n{\left|\right)}^{\gamma }\le |x_1{|}^{\gamma }+\cdots +{\left|x_n\right|}^{\gamma }.\end{array}$$

(7)

Lemma 4

([38]). For any scalar $k>0$, if $x\in (-k,k)$, then

$$\begin{array}{c}\hfill log\left({\displaystyle \frac{k^2}{k^2-x^2}}\right)\le {\displaystyle \frac{x^2}{k^2-x^2}}.\end{array}$$

(8)

Lemma 5

([26]). For nonlinear system $\dot{x}=f\left(x\right)$, if there exists a positive definite function $V\left(x\right)$ satisfying $\dot{V}\left(x\right)\le -aV\left(x\right)-b{V}^{\gamma }\left(x\right)+c$ with constants $a>0,b>0,0<\gamma <1$ and $0<c<\infty$, then the trajectory of system $\dot{x}=f\left(x\right)$ is practical finite-time stable. Moreover, there exists a scalar $0<\lambda <1$ such that

$$V\left(x\right)\le min\{{\displaystyle \frac{c}{a(1-\lambda )}},{\left({\displaystyle \frac{c}{b(1-\lambda )}}\right)}^{{\displaystyle \frac{1}{\gamma }}}\},$$

(9)

where $t\ge T_r$, the setting time is given by

$$T_r\le max\{t_0+{\displaystyle \frac{1}{a\lambda (1-\gamma )}}ln{\displaystyle \frac{a\lambda V^{1-\gamma }\left(t_0\right)+b}{b}},t_0+{\displaystyle \frac{1}{a(1-\gamma )}}ln{\displaystyle \frac{a{V}^{1-\gamma }\left(t_0\right)+b\lambda }{b\lambda }}\},$$

(10)

where $t_0$ denotes the initial time.

Remark 2.

Lemmas 1–4 are used to derive the design process of control strategy, and Lemma 5 is invoked in the proof of Theorem 1 to analyze the stability of shield tunneling system. From Lemma 5, one can provide a rigorous framework for designing controllers that guarantee fast and robust convergence and specifically give a small domain of convergence and a maximum value of convergence time. In this study, we will achieve the finite-time stability of the shield tunneling system using these Lemmas.

Lemma 6

([39]). For any nonlinear function $f\left(Z\right):{\mathbb{R}}^n\to \mathbb{R}$, Z is an input vector on a compact set ${\mathrm{\Omega }}_Z\in {\mathbb{R}}^n$, which can be approximated by an RBF neural network as

$$\begin{array}{c}\hfill f\left(Z\right)={W^{\ast }}^{T}S\left(Z\right)+\delta \left(Z\right),\forall Z\in {\mathrm{\Omega }}_Z\subset {\mathbb{R}}^n\end{array}$$

(11)

where $\delta \left(Z\right)$ is the approximation error satisfies $\left|\delta \right(Z\left)\right|\le \epsilon$, scalar $\epsilon >0$, vector $S\left(Z\right)={[s_1\left(Z\right),s_2\left(Z\right),\cdots ,s_{\ell }\left(Z\right)]}^T\in {\mathbb{R}}^{\ell }$ represents the basis function vector, positive integer $\ell >1$ denotes the number of neural network nodes and the Gaussian function $s_i\left(Z\right)(i\in \{1,2,\cdots ,\ell \})$ is given as

$$\begin{array}{c}\hfill s_i\left(Z\right)=exp\left(-{\displaystyle \frac{{(Z-{\mu }_i)}^T(Z-{\mu }_i)}{{\nu }_i^2}}\right),\end{array}$$

(12)

where ${\mu }_i$ and ${\nu }_i$ are the center and the width of $s_i\left(Z\right)$, respectively. The ideal weight vector $W^{\ast }$ is

$$\begin{array}{c}\hfill W^{\ast }=arg\underset{W\in {\mathbb{R}}^{\ell }}{min}\left\{\underset{Z\in {\mathrm{\Omega }}_Z}{sup}|f\left(Z\right)-W^{T}S\left(Z\right)|\right\}\end{array}$$

(13)

where $W={[W_1,W_2,\cdots ,W_{\ell }]}^T\in {\mathbb{R}}^{\ell }$ is the weight vector of neural network.

Remark 3.

It is known that the RBF neural network poses powerful nonlinear approximation ability. With the RBF neural network, we can efficiently handle complex nonlinear problems by mapping the input vector directly to the hidden space and utilizing radial basis functions (e.g., Gaussian functions) as activation functions. This nonlinear mapping enables RBF neural networks to approximate any complex nonlinear function. RBF neural networks, as a special case of BP neural networks, have a simpler structure, a fast learning convergence rate, and a better generalization ability. At present, RBF neural networks have been applied to various fields, such as function approximation, pattern recognition and classification, fault diagnosis, and image processing. In this paper, RBF neural networks are selected to approximate the unknown nonlinear functions in the process of control strategy design.

3. Controller Design

Before designing the controller, we list the needed parameters in

Table 2. The representation of the parameters.

Parameters	Representation
$e_1$	Tracking error of displacement
$e_2$	Tracking error of propulsive velocity
$V_{max}$	Maximum value of the propulsive velocity
$P_{max}$	Maximum value of the propulsive force
$M_{min}$	Minimum value of the mass function
$M_{max}$	Maximum value of the mass function
$M_d$	Maximum value of the derivative of the mass function
u	$F_u-P_{max}/2$, composite control
v	New control to be designed
$\widehat{•}$	Estimated value of •
$\widetilde{•}$	$•-\widehat{•}$, error between estimated value $\widehat{•}$ and true value •
$V_i$	Lyapunov function
${\eta }_i$	Coordinate transformation
${\alpha }_i$	Virtual control
$c_i$	Feedback gain
$r_i$	Adaptation gain

.

First, we make a coordinate transformation as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\eta }_1=e_1,\hfill \\ {\eta }_2=e_2-{\alpha }_1,\hfill \\ {\eta }_3=u-{\alpha }_2,\hfill \end{array}\right.\end{array}$$

(14)

where ${\alpha }_1$ and ${\alpha }_2$ are the virtual controls to be designed (refer to (19) and (26)).

We regard system (4) as three subsystems. For each subsystem, we design a virtual control law to ensure its stability. Then, we can obtain the control law v for the entire system (4). Therefore, we need to adopt the following three steps.

Step 1: Let $f_1\left(Z_1\right)={\displaystyle \frac{{\eta }_1}{2}}+c_1{\eta }_1^{2\lambda -1}$ with $Z_1={[x_1,x_{1d}]}^T$, scalars $\lambda \in (0,1)$ and $c_1\in {\mathbb{R}}_{>0}$. From Lemma 6, $f_1\left(Z_1\right)$ can be approximated by an RBF neural network. Thus, we have

$$\begin{array}{c}\hfill f_1\left(Z_1\right)=W_1^{\ast T}{S}_1\left(Z_1\right)+{\delta }_1\left(Z_1\right),\left|{\delta }_1\left(Z_1\right)\right|\le {\epsilon }_1.\end{array}$$

(15)

where $W_1^{\ast }$ is the ideal weight coefficient, $S_1\left(Z_1\right)$ is the basis function vector, ${\delta }_1\left(Z_1\right)$ is the approximation error and ${\epsilon }_1$ is a positive number. Define ${\theta }_1={\parallel W_1^{\ast }\parallel }^2$, ${\widehat{\theta }}_1$ as the estimate of ${\theta }_1$, and ${\tilde{\theta }}_1={\theta }_1-{\widehat{\theta }}_1$ as the estimation error.

The Lyapunov function is chosen as

$$\begin{array}{c}\hfill V_1={\displaystyle \frac{1}{2}}{\eta }_1^2+{\displaystyle \frac{1}{2r_1}}{\tilde{\theta }}_1^2,\end{array}$$

(16)

where $r_1$ is a positive constant. Thus, it obtains

$$\begin{array}{cc}\hfill {\dot{V}}_1& ={\eta }_1({\eta }_2+{\alpha }_1)-{\displaystyle \frac{1}{r_1}}{\tilde{\theta }}_1{\dot{\widehat{\theta }}}_1\hfill \\ \hfill & \le -c_1{\eta }_1^{2\lambda }+{\eta }_1(c_1{\eta }_1^{2\lambda -1}+{\displaystyle \frac{{\eta }_1}{2}}+{\alpha }_1)+{\displaystyle \frac{{\eta }_2^2}{2}}-{\displaystyle \frac{1}{r_1}}{\tilde{\theta }}_1{\dot{\widehat{\theta }}}_1.\hfill \end{array}$$

(17)

According to Lemma 2 and the definition of ${\theta }_1$, it follows

$$\begin{array}{cc}\hfill {\eta }_1{f}_1\left(Z_1\right)=& {\eta }_1(W_1^{\ast T}{S}_1\left(Z_1\right)+{\delta }_1\left(Z_1\right))\hfill \\ \hfill \le & {\displaystyle \frac{{\eta }_1^2{\parallel W_1^{\ast }\parallel }^2{S}_1^T{S}_1}{2a^2}}+{\displaystyle \frac{a^2}{2}}+{\displaystyle \frac{{\eta }_1^2}{2b^2}}+{\displaystyle \frac{b^2{\delta }_1^2}{2}}\hfill \\ \hfill \le & {\displaystyle \frac{{\eta }_1^2{\theta }_1{S}_1^T{S}_1}{2a^2}}+{\displaystyle \frac{a^2}{2}}+{\displaystyle \frac{{\eta }_1^2}{2b^2}}+{\displaystyle \frac{b^2{\epsilon }_1^2}{2}},\hfill \end{array}$$

(18)

where a and b are positive constants. The virtual control ${\alpha }_1$ is designed as

$$\begin{array}{c}\hfill {\alpha }_1=-c_1{\eta }_1-{\displaystyle \frac{{\eta }_1{\widehat{\theta }}_1{S}_1^T{S}_1}{2a^2}}-{\displaystyle \frac{{\eta }_1}{2b^2}}\end{array}$$

(19)

and ${\widehat{\theta }}_1$ is updated according to

$$\begin{array}{c}\hfill {\dot{\widehat{\theta }}}_1=r_1{\displaystyle \frac{{\eta }_1^2{S}_1^T{S}_1}{2a^2}}-2{\widehat{\theta }}_1.\end{array}$$

(20)

Substituting the virtual control (19) and the update rule (20) of ${\widehat{\theta }}_1$ into (17), we obtain

$$\begin{array}{c}\hfill {\dot{V}}_1\le -c_1{\eta }_1^{2\lambda }-c_1{\eta }_1^2+{\displaystyle \frac{{\eta }_2^2}{2}}+{\displaystyle \frac{2}{r_1}}{\tilde{\theta }}_1{\widehat{\theta }}_1+{\displaystyle \frac{a^2}{2}}+{\displaystyle \frac{b^2{\epsilon }_1^2}{2}}.\end{array}$$

(21)

Step 2: Let $f_2\left(Z_2\right)={\displaystyle \frac{{\eta }_2(k_{b2}^2-{\eta }_2^2)}{2}}+{\displaystyle \frac{c_2{\eta }_2^{2\lambda -1}}{{(k_{b2}^2-{\eta }_2^2)}^{\lambda -1}}}+{\displaystyle \frac{{\eta }_2}{2(k_{b2}^2-{\eta }_2^2)}}-M{\ddot{x}}_{1d}-M\dot{{\alpha }_1}$ with $Z_2={[x_1,x_{1d},{\widehat{\theta }}_1,x_2,x_{2d}]}^T$ and positive constants $c_2$ and $k_{b2}$. Similarly, from Lemma 6, it is known that

$$\begin{array}{c}\hfill f_2\left(Z_2\right)=W_2^{\ast T}{S}_2\left(Z_2\right)+{\delta }_2\left(Z_2\right),\left|{\delta }_2\left(Z_2\right)\right|\le {\epsilon }_2\end{array}$$

(22)

where $W_2^{\ast }$ is the ideal weight coefficient, $S_2\left(Z_2\right)$ is the basis function vector, ${\delta }_2\left(Z_2\right)$ is the approximation error and ${\epsilon }_2$ is a positive number. Define ${\theta }_2={\parallel W_2^{\ast }\parallel }^2$, ${\widehat{\theta }}_2$ as the estimate of ${\theta }_2$, and ${\tilde{\theta }}_2={\theta }_2-{\widehat{\theta }}_2$ as the estimation error. The Lyapunov function is chosen as

$$\begin{array}{c}\hfill V_2=V_1+{\displaystyle \frac{M\left(t\right)}{2}}log({\displaystyle \frac{k_{b2}^2}{k_{b2}^2-{\eta }_2^2}})+{\displaystyle \frac{1}{2r_2}}{\tilde{\theta }}_2^2+{\displaystyle \frac{1}{2r_3}}{\tilde{d}}^2+{\displaystyle \frac{1}{2r_4}}{\tilde{\sigma }}_0^2+{\displaystyle \frac{{\sigma }_0}{2r_5}}{\tilde{\sigma }}_1^2+{\displaystyle \frac{1}{2r_6}}{\tilde{J}}_2^2,\end{array}$$

(23)

where $r_i(i=2,\cdots ,6)$ are positive constants. By taking the derivative of the second term, we obtain

$$\begin{array}{cc}\hfill {\left({\displaystyle \frac{M\left(t\right)}{2}}log\left({\displaystyle \frac{k_{b2}^2}{k_{b2}^2-{\eta }_2^2}}\right)\right)}^{\prime }=& {\displaystyle \frac{M^{\prime }\left(t\right)}{2}}log\left({\displaystyle \frac{k_{b2}^2}{k_{b2}^2-{\eta }_2^2}}\right)+{\displaystyle \frac{M\left(t\right)}{2}}·{\displaystyle \frac{k_{b2}^2-{\eta }_2^2}{k_{b2}^2}}·{\displaystyle \frac{2k_{b2}^2{\eta }_2{\dot{\eta }}_2}{{(k_{b2}^2-{\eta }_2^2)}^2}}\hfill \\ \hfill =& {\displaystyle \frac{M^{\prime }\left(t\right)}{2}}log\left({\displaystyle \frac{k_{b2}^2}{k_{b2}^2-{\eta }_2^2}}\right)+{\displaystyle \frac{{\eta }_2}{k_{b2}^2-{\eta }_2^2}}(u+d-J_1\left(x_2\right)z\hfill \\ \hfill & -J_2{x}_2-M{\ddot{x}}_{1d}-M{\dot{\alpha }}_1).\hfill \end{array}$$

In accordance with the coordinate change (14), $u={\eta }_3+{\alpha }_2$. Consequently,

$$\begin{array}{cc}\hfill {\dot{V}}_2=& {\dot{V}}_1+{\displaystyle \frac{{\eta }_2}{k_{b2}^2-{\eta }_2^2}}\left({\eta }_3+{\alpha }_2+d-{\sigma }_{0}z+{\sigma }_0{\sigma }_1{\displaystyle \frac{|x_2|}{g\left(x_2\right)}}z-J_2{x}_2-M{\ddot{x}}_{1d}-M\dot{{\alpha }_1}\right)\hfill \\ \hfill & +{\displaystyle \frac{M^{\prime }\left(t\right)}{2}}log\left({\displaystyle \frac{k_{b2}^2}{k_{b2}^2-{\eta }_2^2}}\right)-{\displaystyle \frac{1}{r_2}}{\tilde{\theta }}_2{\dot{\widehat{\theta }}}_2-{\displaystyle \frac{1}{r_3}}\tilde{d}\dot{\widehat{d}}-{\displaystyle \frac{1}{r_4}}{\tilde{\sigma }}_0{\dot{\widehat{\sigma }}}_0-{\displaystyle \frac{{\sigma }_0}{r_5}}{\tilde{\sigma }}_1{\dot{\widehat{\sigma }}}_1-{\displaystyle \frac{1}{r_6}}{\tilde{J}}_2{\dot{\widehat{J}}}_2.\hfill \end{array}$$

From Lemma 4 and Assumption 2, we have

$$\begin{array}{c}\hfill {\displaystyle \frac{M^{\prime }\left(t\right)}{2}}log({\displaystyle \frac{k_{b2}^2}{k_{b2}^2-{\eta }_2^2}})\le {\displaystyle \frac{M_d}{2}}{\displaystyle \frac{{\eta }_2^2}{k_{b2}^2-{\eta }_2^2}}.\end{array}$$

Notice that ${\displaystyle \frac{{\eta }_2}{k_{b2}^2-{\eta }_2^2}}{\eta }_3\le {\displaystyle \frac{{\eta }_3^2}{2}}+{\displaystyle \frac{{\eta }_2^2}{2(k_{b2}^2-{\eta }_2^2)}}$, ${\displaystyle \frac{{\eta }_2^2}{2}}={\displaystyle \frac{{\eta }_2}{k_{b2}^2-{\eta }_2^2}}·{\displaystyle \frac{{\eta }_2(k_{b2}^2-{\eta }_2^2)}{2}}$, it obtains

$$\begin{array}{cc}\hfill {\dot{V}}_2\le & -c_1{\eta }_1^{2\lambda }-c_1{\eta }_1^2+{\displaystyle \frac{2}{r_1}}{\tilde{\theta }}_1{\widehat{\theta }}_1+{\displaystyle \frac{a^2}{2}}+{\displaystyle \frac{b^2{\epsilon }_1^2}{2}}+{\displaystyle \frac{{\eta }_3^2}{2}}+{\displaystyle \frac{{\eta }_2}{k_{b2}^2-{\eta }_2^2}}({\displaystyle \frac{M_d{\eta }_2}{2}}+f_2\left(Z_2\right)\hfill \\ \hfill & +{\alpha }_2+d-{\sigma }_{0}z+{\sigma }_0{\sigma }_1{\displaystyle \frac{|x_2|}{g\left(x_2\right)}}z-J_2{x}_2)-{\displaystyle \frac{c_2{\eta }_2^{2\lambda }}{{(k_{b2}^2-{\eta }_2^2)}^{\lambda }}}-{\displaystyle \frac{1}{r_2}}{\tilde{\theta }}_2{\dot{\widehat{\theta }}}_2-{\displaystyle \frac{1}{r_3}}\tilde{d}\dot{\widehat{d}}\hfill \\ \hfill & -{\displaystyle \frac{1}{r_4}}{\tilde{\sigma }}_0{\dot{\widehat{\sigma }}}_0-{\displaystyle \frac{{\sigma }_0}{r_5}}{\tilde{\sigma }}_1{\dot{\widehat{\sigma }}}_1-{\displaystyle \frac{1}{r_6}}{\tilde{J}}_2{\dot{\widehat{J}}}_2.\hfill \end{array}$$

(24)

According to Lemma 1 and the definition of ${\theta }_2$, it has

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\eta }_2}{k_{b2}^2-{\eta }_2^2}}{f}_2\left(Z_2\right)& ={\displaystyle \frac{{\eta }_2}{k_{b2}^2-{\eta }_2^2}}(W_2^{\ast T}{S}_2\left(Z_2\right)+{\delta }_2\left(Z_2\right))\hfill \\ \hfill & \le {\displaystyle \frac{{\eta }_2^2{\theta }_2{S}_2^T{S}_2}{2a^2{(k_{b2}^2-{\eta }_2^2)}^2}}+{\displaystyle \frac{a^2}{2}}+{\displaystyle \frac{{\eta }_2^2}{2b^2{(k_{b2}^2-{\eta }_2^2)}^2}}+{\displaystyle \frac{b^2{\epsilon }_2^2}{2}}.\hfill \end{array}$$

(25)

The virtual control ${\alpha }_2$ is designed as

$$\begin{array}{cc}\hfill {\alpha }_2=& -c_2{\eta }_2-{\displaystyle \frac{{\eta }_2{\widehat{\theta }}_2{S}_2^T{S}_2}{2a^2(k_{b2}^2-{\eta }_2^2)}}-{\displaystyle \frac{{\eta }_2}{2b^2(k_{b2}^2-{\eta }_2^2)}}-{\displaystyle \frac{M_d{\eta }_2}{2}}-\widehat{d}+{\widehat{\sigma }}_{0}z\hfill \\ \hfill & -{\widehat{\sigma }}_0{\widehat{\sigma }}_1{\displaystyle \frac{|x_2|}{g\left(x_2\right)}}z+{\widehat{J}}_2{x}_2.\hfill \end{array}$$

(26)

Moreover, the update rules for ${\widehat{\theta }}_2$, $\widehat{d}$, ${\widehat{\sigma }}_0$, ${\widehat{\sigma }}_1$ and ${\widehat{J}}_2$ are defined as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\dot{\widehat{\theta }}}_2=r_2{\displaystyle \frac{{\eta }_2^2{S}_2^T{S}_2}{2a^2{(k_{b2}^2-{\eta }_2^2)}^2}}-2{\widehat{\theta }}_2,\hfill \\ \hfill & \dot{\widehat{d}}=r_3{\displaystyle \frac{{\eta }_2}{k_{b2}^2-{\eta }_2^2}}-2\widehat{d},\hfill \\ \hfill & {\dot{\widehat{\sigma }}}_0=-r_4{\eta }_2{\displaystyle \frac{z-{\widehat{\sigma }}_1\frac{|x_2|}{g\left(x_2\right)}z}{k_{b2}^2-{\eta }_2^2}}-2{\widehat{\sigma }}_0,\hfill \\ \hfill & {\dot{\widehat{\sigma }}}_1=r_5{\displaystyle \frac{{\eta }_2\frac{|x_2|}{g\left(x_2\right)}z}{k_{b2}^2-{\eta }_2^2}}-2{\widehat{\sigma }}_1,\hfill \\ \hfill & {\dot{\widehat{J}}}_2=-r_6{\displaystyle \frac{{\eta }_2{x}_2}{k_{b2}^2-{\eta }_2^2}}-2{\widehat{J}}_2.\hfill \end{array}\end{array}$$

(27)

Substituting the virtual control (26) and the update rule (27) into (24), we obtain

$$\begin{array}{cc}\hfill {\dot{V}}_2\le & -c_1{\eta }_1^{2\lambda }-c_1{\eta }_1^2+{\displaystyle \frac{2}{r_1}}{\tilde{\theta }}_1{\widehat{\theta }}_1+a^2+{\displaystyle \frac{b^2({\epsilon }_1^2+{\epsilon }_2^2)}{2}}+{\displaystyle \frac{{\eta }_3^2}{2}}-{\displaystyle \frac{c_2{\eta }_2^2}{k_{b2}^2-{\eta }_2^2}}-{\displaystyle \frac{c_2{\eta }_2^{2\lambda }}{{(k_{b2}^2-{\eta }_2^2)}^{\lambda }}}\hfill \\ \hfill & +{\displaystyle \frac{2}{r_2}}{\tilde{\theta }}_2{\widehat{\theta }}_2+{\displaystyle \frac{2}{r_3}}\tilde{d}\widehat{d}+{\displaystyle \frac{2}{r_4}}{\tilde{\sigma }}_0{\widehat{\sigma }}_0+{\displaystyle \frac{2{\sigma }_0}{r_5}}{\tilde{\sigma }}_1{\widehat{\sigma }}_1+{\displaystyle \frac{2}{r_6}}{\tilde{J}}_2{\widehat{J}}_2\hfill \end{array}$$

(28)

due to the fact that ${\sigma }_0{\sigma }_1-{\widehat{\sigma }}_0{\widehat{\sigma }}_1={\sigma }_0{\tilde{\sigma }}_1+{\tilde{\sigma }}_0{\widehat{\sigma }}_1$.

Step 3: Let $f_3\left(Z_3\right)={\displaystyle \frac{{\eta }_3(k_{b3}^2-{\eta }_3^2)}{2}}+{\displaystyle \frac{c_3{\eta }_3^{2\lambda -1}}{{(k_{b3}^2-{\eta }_3^2)}^{\lambda -1}}}-{\dot{\alpha }}_2$ with $Z_3=[x_1,x_2,u,x_{1d},x_{2d},{\widehat{\theta }}_1$, ${\widehat{\theta }}_2,\widehat{d},{\widehat{\sigma }}_0,{\widehat{\sigma }}_1,{\widehat{J}}_2{]}^T$, and positive constants $c_3$ and $k_{b3}$. Furthermore, from Lemma 6, it follows

$$\begin{array}{c}\hfill f_3\left(Z_3\right)=W_3^{\ast T}{S}_3\left(Z_3\right)+{\delta }_3\left(Z_3\right),\left|{\delta }_3\left(Z_3\right)\right|\le {\epsilon }_3\end{array}$$

(29)

where $W_3^{\ast }$ is the ideal weight coefficient, $S_3\left(Z_3\right)$ is the basis function vector, ${\delta }_3\left(Z_3\right)$ is the approximation error and ${\epsilon }_3$ is a positive number. Define ${\theta }_3={\parallel W_3^{\ast }\parallel }^2$, ${\widehat{\theta }}_3$ as the estimate of ${\theta }_3$, and ${\tilde{\theta }}_3={\theta }_3-{\widehat{\theta }}_3$ as the estimation error. The Lyapunov function is chosen as

$$\begin{array}{c}\hfill V=V_2+{\displaystyle \frac{1}{2}}log\left({\displaystyle \frac{k_{b3}^2}{k_{b3}^2-{\eta }_3^2}}\right)+{\displaystyle \frac{1}{2r_7}}{\tilde{\theta }}_3^2,\end{array}$$

(30)

where $r_7$ is a positive constant. Notice that ${\displaystyle \frac{{\eta }_3^2}{2}}={\displaystyle \frac{{\eta }_3}{k_{b3}^2-{\eta }_3^2}}·{\displaystyle \frac{{\eta }_3(k_{b3}^2-{\eta }_3^2)}{2}}$, it follows

$$\begin{array}{cc}\hfill \dot{V}=& \dot{V_2}+{\displaystyle \frac{{\eta }_3}{(k_{b3}^2-{\eta }_3^2)}}(-cu+v-{\dot{\alpha }}_2)-{\displaystyle \frac{1}{r_7}}{\tilde{\theta }}_3{\dot{\widehat{\theta }}}_3\hfill \\ \hfill \le & -c_1{\eta }_1^2-c_1{\eta }_1^{2\lambda }-{\displaystyle \frac{c_2{\eta }_2^2}{k_{b2}^2-{\eta }_2^2}}-{\displaystyle \frac{c_2{\eta }_2^{2\lambda }}{{(k_{b2}^2-{\eta }_2^2)}^{\lambda }}}+a^2+{\displaystyle \frac{b^2({\epsilon }_1^2+{\epsilon }_2^2)}{2}}-{\displaystyle \frac{c_3{\eta }_3^{2\lambda }}{{(k_{b3}^2-{\eta }_3^2)}^{\lambda }}}\hfill \\ \hfill & +{\displaystyle \frac{{\eta }_3}{(k_{b3}^2-{\eta }_3^2)}}\left(f_3\left(Z_3\right)-cu+v\right)+{\displaystyle \frac{2}{r_1}}{\tilde{\theta }}_1{\widehat{\theta }}_1+{\displaystyle \frac{2}{r_2}}{\tilde{\theta }}_2{\widehat{\theta }}_2\hfill \\ \hfill & +{\displaystyle \frac{2}{r_3}}\tilde{d}\widehat{d}+{\displaystyle \frac{2}{r_4}}{\tilde{\sigma }}_0{\widehat{\sigma }}_0+{\displaystyle \frac{2{\sigma }_0}{r_5}}{\tilde{\sigma }}_1{\widehat{\sigma }}_1+{\displaystyle \frac{2}{r_6}}{\tilde{J}}_2{\widehat{J}}_2-{\displaystyle \frac{1}{r_7}}{\tilde{\theta }}_3{\dot{\widehat{\theta }}}_3.\hfill \end{array}$$

(31)

According to Lemma 1 and the definition of ${\theta }_3$, it obtains

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\eta }_3}{k_{b3}^2-{\eta }_3^2}}{f}_3\left(Z_3\right)& ={\displaystyle \frac{{\eta }_3}{k_{b3}^2-{\eta }_3^2}}(W_3^{\ast T}{S}_3\left(Z_3\right)+{\delta }_3\left(Z_3\right))\hfill \\ \hfill & \le {\displaystyle \frac{{\eta }_3^2{\theta }_3{S}_3^T{S}_3}{2a^2{(k_{b3}^2-{\eta }_3^2)}^2}}+{\displaystyle \frac{a^2}{2}}+{\displaystyle \frac{{\eta }_3^2}{2b^2{(k_{b3}^2-{\eta }_3^2)}^2}}+{\displaystyle \frac{b^2{\epsilon }_3^2}{2}}.\hfill \end{array}$$

(32)

Therefore, the new control v is designed as

$$\begin{array}{c}\hfill v=-c_3{\eta }_3+cu-{\displaystyle \frac{{\eta }_3{\widehat{\theta }}_3{S}_3^T{S}_3}{2a^2(k_{b3}^2-{\eta }_3^2)}}-{\displaystyle \frac{{\eta }_3}{2b^2(k_{b3}^2-{\eta }_3^2)}},\end{array}$$

(33)

where the updating rule for ${\widehat{\theta }}_3$ is defined as

$$\begin{array}{c}\hfill {\dot{\widehat{\theta }}}_3=r_7{\displaystyle \frac{{\eta }_3^2{S}_3^T{S}_3}{2a^2{(k_{b3}^2-{\eta }_3^2)}^2}}-2{\widehat{\theta }}_3.\end{array}$$

(34)

Substituting the new control (33) and the update rule (34) of ${\widehat{\theta }}_3$ into (31), we obtain

$$\begin{array}{cc}\hfill \dot{V}\le & -c_1{\eta }_1^2-{\displaystyle \frac{c_2{\eta }_2^2}{k_{b2}^2-{\eta }_2^2}}-{\displaystyle \frac{c_3{\eta }_3^2}{k_{b3}^2-{\eta }_3^2}}-c_1{\eta }_1^{2\lambda }-{\displaystyle \frac{c_2{\eta }_2^{2\lambda }}{{(k_{b2}^2-{\eta }_2^2)}^{\lambda }}}-{\displaystyle \frac{c_3{\eta }_3^{2\lambda }}{{(k_{b3}^2-{\eta }_3^2)}^{\lambda }}}\hfill \\ \hfill & +{\displaystyle \frac{2}{r_1}}{\tilde{\theta }}_1{\widehat{\theta }}_1+{\displaystyle \frac{2}{r_2}}{\tilde{\theta }}_2{\widehat{\theta }}_2+{\displaystyle \frac{2}{r_3}}\tilde{d}\widehat{d}+{\displaystyle \frac{2}{r_4}}{\tilde{\sigma }}_0{\widehat{\sigma }}_0+{\displaystyle \frac{2{\sigma }_0}{r_5}}{\tilde{\sigma }}_1{\widehat{\sigma }}_1+{\displaystyle \frac{2}{r_6}}{\tilde{J}}_2{\widehat{J}}_2+{\displaystyle \frac{2}{r_7}}{\tilde{\theta }}_3{\widehat{\theta }}_3\hfill \\ \hfill & +{\displaystyle \frac{3}{2}}{a}^2+{\displaystyle \frac{b^2({\epsilon }_1^2+{\epsilon }_2^2+{\epsilon }_3^2)}{2}}.\hfill \end{array}$$

(35)

Based on Lemma 1, the item ${\displaystyle \frac{2}{r_1}}{\tilde{\theta }}_1{\widehat{\theta }}_1$ reduces to

$$\begin{array}{cc}\hfill {\displaystyle \frac{2}{r_1}}{\tilde{\theta }}_1{\widehat{\theta }}_1=& {\displaystyle \frac{2}{r_1}}{\tilde{\theta }}_1{\theta }_1-{\displaystyle \frac{2}{r_1}}{\tilde{\theta }}_1^2\hfill \\ \hfill \le & -{\displaystyle \frac{1}{r_1}}{\tilde{\theta }}_1^2+{\displaystyle \frac{1}{r_1}}{\theta }_1^2\hfill \\ \hfill =& -{\displaystyle \frac{1}{r_1}}{\tilde{\theta }}_1^2+{\displaystyle \frac{1}{r_1}}{\theta }_1^2+{\left({\displaystyle \frac{{\tilde{\theta }}_1^2}{2r_1}}\right)}^{\lambda }-{\left({\displaystyle \frac{{\tilde{\theta }}_1^2}{2r_1}}\right)}^{\lambda }.\hfill \end{array}$$

(36)

Let $\varphi =1$, $\psi ={\displaystyle \frac{{\tilde{\theta }}_1^2}{2r_1}}$, $n=\lambda$, $m=1-\lambda$, $\omega ={\lambda }^{{\displaystyle \frac{\lambda }{1-\lambda }}}$, from Lemma 2, we have

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{{\tilde{\theta }}_1^2}{2r_1}}\right)}^{\lambda }\le (1-\lambda ){\lambda }^{{\displaystyle \frac{\lambda }{1-\lambda }}}+{\displaystyle \frac{{\tilde{\theta }}_1^2}{2r_1}}.\end{array}$$

(37)

Thus, it holds

$$\begin{array}{c}\hfill {\displaystyle \frac{2}{r_1}}{\tilde{\theta }}_1{\widehat{\theta }}_1\le -{\displaystyle \frac{1}{2r_1}}{\tilde{\theta }}_1^2+{\displaystyle \frac{1}{r_1}}{\theta }_1^2+(1-\lambda ){\lambda }^{{\displaystyle \frac{\lambda }{1-\lambda }}}-{\left({\displaystyle \frac{{\tilde{\theta }}_1^2}{2r_1}}\right)}^{\lambda }.\end{array}$$

(38)

Similarly, one can obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\displaystyle \frac{2}{r_2}}{\tilde{\theta }}_2{\widehat{\theta }}_2\le -{\displaystyle \frac{1}{2r_2}}{\tilde{\theta }}_2^2+{\displaystyle \frac{1}{r_2}}{\theta }_2^2+(1-\lambda ){\lambda }^{{\displaystyle \frac{\lambda }{1-\lambda }}}-{\left({\displaystyle \frac{{\tilde{\theta }}_2^2}{2r_2}}\right)}^{\lambda },\hfill \\ \hfill & {\displaystyle \frac{2}{r_3}}\tilde{d}\widehat{d}\le -{\displaystyle \frac{1}{2r_3}}{\tilde{d}}^2+{\displaystyle \frac{1}{r_3}}{d}^2+(1-\lambda ){\lambda }^{{\displaystyle \frac{\lambda }{1-\lambda }}}-{\left({\displaystyle \frac{{\tilde{d}}^2}{2r_3}}\right)}^{\lambda },\hfill \\ \hfill & {\displaystyle \frac{2}{r_4}}{\tilde{\sigma }}_0{\widehat{\sigma }}_0\le -{\displaystyle \frac{1}{2r_4}}{\tilde{\sigma }}_0^2+{\displaystyle \frac{1}{r_4}}{\sigma }_0^2+(1-\lambda ){\lambda }^{{\displaystyle \frac{\lambda }{1-\lambda }}}-{\left({\displaystyle \frac{{\tilde{\sigma }}_0^2}{2r_4}}\right)}^{\lambda },\hfill \\ \hfill & {\displaystyle \frac{2{\sigma }_0}{r_5}}{\tilde{\sigma }}_1{\widehat{\sigma }}_1\le -{\displaystyle \frac{{\sigma }_0}{2r_5}}{\tilde{\sigma }}_1^2+{\displaystyle \frac{{\sigma }_0}{r_5}}{\sigma }_1^2+(1-\lambda ){\lambda }^{{\displaystyle \frac{\lambda }{1-\lambda }}}-{\left({\displaystyle \frac{{\sigma }_0{\tilde{\sigma }}_1^2}{2r_5}}\right)}^{\lambda },\hfill \\ \hfill & {\displaystyle \frac{2}{r_6}}{\tilde{J}}_2{\widehat{J}}_2\le -{\displaystyle \frac{1}{2r_6}}{\tilde{J}}_2^2+{\displaystyle \frac{1}{r_6}}{J}_2^2+(1-\lambda ){\lambda }^{{\displaystyle \frac{\lambda }{1-\lambda }}}-{\left({\displaystyle \frac{{\tilde{J}}_2^2}{2r_6}}\right)}^{\lambda },\hfill \\ \hfill & {\displaystyle \frac{2}{r_7}}{\tilde{\theta }}_3{\widehat{\theta }}_3\le -{\displaystyle \frac{1}{2r_7}}{\tilde{\theta }}_3^2+{\displaystyle \frac{1}{r_7}}{\theta }_3^2+(1-\lambda ){\lambda }^{{\displaystyle \frac{\lambda }{1-\lambda }}}-{\left({\displaystyle \frac{{\tilde{\theta }}_3^2}{2r_7}}\right)}^{\lambda }.\hfill \end{array}\end{array}$$

(39)

Substituting (38) and (39) into (35), it has

$$\begin{array}{cc}\hfill \dot{V}\le & -c_1{\eta }_1^2-{\displaystyle \frac{c_2{\eta }_2^2}{k_{b2}^2-{\eta }_2^2}}-{\displaystyle \frac{c_3{\eta }_3^2}{k_{b3}^2-{\eta }_3^2}}-c_1{\eta }_1^{2\lambda }-{\displaystyle \frac{c_2{\eta }_2^{2\lambda }}{{(k_{b2}^2-{\eta }_2^2)}^{\lambda }}}-{\displaystyle \frac{c_3{\eta }_3^{2\lambda }}{{(k_{b3}^2-{\eta }_3^2)}^{\lambda }}}-{\displaystyle \frac{1}{2r_1}}{\tilde{\theta }}_1^2\hfill \\ \hfill & -{\displaystyle \frac{1}{2r_2}}{\tilde{\theta }}_2^2-{\displaystyle \frac{{\tilde{d}}^2}{2r_3}}-{\displaystyle \frac{{\tilde{\sigma }}_0^2}{2r_4}}-{\displaystyle \frac{{\sigma }_0{\tilde{\sigma }}_1^2}{2r_5}}-{\displaystyle \frac{{\tilde{J}}_2^2}{2r_6}}-{\displaystyle \frac{1}{2r_7}}{\tilde{\theta }}_3^2-{\left({\displaystyle \frac{{\tilde{\theta }}_1^2}{2r_1}}\right)}^{\lambda }-{\left({\displaystyle \frac{{\tilde{\theta }}_2^2}{2r_2}}\right)}^{\lambda }\hfill \\ \hfill & -{\left({\displaystyle \frac{{\tilde{d}}^2}{2r_3}}\right)}^{\lambda }-{\left({\displaystyle \frac{{\tilde{\sigma }}_0^2}{2r_4}}\right)}^{\lambda }-{\left({\displaystyle \frac{{\sigma }_0{\tilde{\sigma }}_1^2}{2r_5}}\right)}^{\lambda }-{\left({\displaystyle \frac{{\tilde{J}}_2^2}{2r_6}}\right)}^{\lambda }-{\left({\displaystyle \frac{{\tilde{\theta }}_3^2}{2r_7}}\right)}^{\lambda }\hfill \\ \hfill & +{\displaystyle \frac{{\theta }_1^2}{r_1}}+{\displaystyle \frac{{\theta }_2^2}{r_2}}+{\displaystyle \frac{d^2}{r_3}}+{\displaystyle \frac{{\sigma }_0^2}{r_4}}+{\displaystyle \frac{{\sigma }_0{\sigma }_1^2}{r_5}}+{\displaystyle \frac{J_2^2}{r_6}}+{\displaystyle \frac{{\theta }_3^2}{r_7}}+7(1-\lambda ){\lambda }^{{\displaystyle \frac{\lambda }{1-\lambda }}}+{\displaystyle \frac{3}{2}}{a}^2\hfill \\ \hfill & +{\displaystyle \frac{b^2({\epsilon }_1^2+{\epsilon }_2^2+{\epsilon }_3^2)}{2}}.\hfill \end{array}$$

By following the aforementioned backstepping procedure, and substituting v in (33) into the third equation of system (4), we can obtain actual control $F_u$. The sketched procedure is summarized in

Table 3. The sketched procedure of design controller.

Stage 1	(1) Establishing system model (1)
Preparatory work	(2) Transforming constraints
	(3) Introducing auxiliary system $\dot{u}=-cu+v$
Stage 2	(4) Making a coordinate transformation (14)
Design controller	(5) Designing the virtual control ${\alpha }_1$ in (19) and the update rule of
	${\widehat{\theta }}_0$ in (20)
	(6) Designing the virtual control ${\alpha }_2$ in (26) and update rules of
	${\widehat{\theta }}_2$, $\widehat{d}$, ${\widehat{\sigma }}_0$, ${\widehat{\sigma }}_1$ and ${\widehat{J}}_2$ in (27)
	(7) Designing the control v in (33) and the update rule of ${\widehat{\theta }}_3$ in (34)
	(8) Obtaining the actual control $F_u=u+P_{max}/2$

.

Now, we discuss the finite-time stability of the error system (4) as follows

Theorem 1.

For the error system (4) that satisfies Assumptions 1 and 2, the trajectory of the system (4) is practical finite-time stability if the virtual controls ${\alpha }_1$ and ${\alpha }_2$, the new control v, and the parameters update rules satisfy (19), (20), (26), (27), (33) and (34). Then, the tracking errors $e_1$ and $e_2$ can converge to a small neighborhood near the origin in a finite time, i.e., the displacement and velocity of the shield tunneling machine can track the desired signal in a finite time.

Proof. 

Let $c_0=min\{c_1,c_2,c_3\}$, from Lemmas 3 and 4, we have

$$\begin{array}{cc}\hfill & -c_1{\eta }_1^2-{\displaystyle \frac{c_2{\eta }_2^2}{k_{b2}^2-{\eta }_2^2}}-{\displaystyle \frac{c_3{\eta }_3^2}{k_{b3}^2-{\eta }_3^2}}-c_1{\eta }_1^{2\lambda }-{\displaystyle \frac{c_2{\eta }_2^{2\lambda }}{{(k_{b2}^2-{\eta }_2^2)}^{\lambda }}}-{\displaystyle \frac{c_3{\eta }_3^{2\lambda }}{{(k_{b3}^2-{\eta }_3^2)}^{\lambda }}}\hfill \\ \hfill & \le -c_0\left({\eta }_1^2+{\displaystyle \frac{{\eta }_2^2}{k_{b2}^2-{\eta }_2^2}}+{\displaystyle \frac{{\eta }_3^2}{k_{b3}^2-{\eta }_3^2}}+{\eta }_1^{2\lambda }+{\displaystyle \frac{{\eta }_2^{2\lambda }}{{(k_{b2}^2-{\eta }_2^2)}^{\lambda }}}+{\displaystyle \frac{{\eta }_3^{2\lambda }}{{(k_{b3}^2-{\eta }_3^2)}^{\lambda }}}\right)\hfill \\ \hfill & \le -c_{min}({\eta }_1^2+log\left({\displaystyle \frac{k_{b2}^2}{k_{b2}^2-{\eta }_2^2}}\right)+log\left({\displaystyle \frac{k_{b3}^2}{k_{b3}^2-{\eta }_3^2}}\right)+{\eta }_1^{2\lambda }+log{\left({\displaystyle \frac{k_{b2}^2}{k_{b2}^2-{\eta }_2^2}}\right)}^{\lambda }\hfill \\ \hfill & +log{\left({\displaystyle \frac{k_{b3}^2}{k_{b3}^2-{\eta }_3^2}}\right)}^{\lambda }),\hfill \end{array}$$

(40)

where $c_{min}=min\{{\displaystyle \frac{2c_0}{M_{min}}},2c_0,{\left({\displaystyle \frac{2}{M_{min}}}\right)}^{\lambda }{c}_0,2^{\lambda }{c}_0\}$. Thus, it holds

$$\begin{array}{cc}\hfill \dot{V}\le & -{\phi }_{1}V-{\phi }_1{V}^{\lambda }+\beta ,\hfill \end{array}$$

(41)

where

$$\begin{array}{cc}\hfill {\phi }_1:=& min\{c_{min},1\},\hfill \\ \hfill \beta :=& {\displaystyle \frac{{\theta }_1^2}{r_1}}+{\displaystyle \frac{{\theta }_2^2}{r_2}}+{\displaystyle \frac{d^2}{r_3}}+{\displaystyle \frac{{\sigma }_0^2}{r_4}}+{\displaystyle \frac{{\sigma }_0{\sigma }_1^2}{r_5}}+{\displaystyle \frac{J_2^2}{r_6}}+{\displaystyle \frac{{\theta }_3^2}{r_7}}+7(1-\lambda ){\lambda }^{{\displaystyle \frac{\lambda }{1-\lambda }}}+{\displaystyle \frac{3}{2}}{a}^2\hfill \\ \hfill & +{\displaystyle \frac{b^2({\epsilon }_1^2+{\epsilon }_2^2+{\epsilon }_3^2)}{2}}.\hfill \end{array}$$

From the Lemma 5, it is known that the trajectory of the system (4) is practical finite-time stability, the displacement and propulsive velocity of the shield tunneling system track to the desired signals in a finite time.

$$T\le max\{t_0+{\displaystyle \frac{1}{{\phi }_1\gamma (1-\lambda )}}ln{\displaystyle \frac{{\phi }_1\gamma V^{1-\lambda }\left(t_0\right)+{\phi }_1}{{\phi }_1}},t_0+{\displaystyle \frac{1}{{\phi }_1(1-\lambda )}}ln{\displaystyle \frac{{\phi }_1{V}^{1-\lambda }\left(t_0\right)+{\phi }_1\gamma }{{\phi }_1\gamma }}\},$$

(42)

where $0<\gamma \le 1$ and $t_0$ denotes the initial instant. □

Remark 4.

Theorem 1 provides a finite-time tracking control method for shield tunneling systems with constraints on states and input. For handling the constraint on control input u, the third equation in (4) is introduced, and the new control v is designed as (33), which can ensure that the control input u satisfies the constraint condition. If the constraint on control input u is not considered, the $u={\alpha }_2$ and ${\eta }_3=0$ hold, which may lead to intense oscillations of u. In contrast, due to the inhibitory effect of integration on oscillations, it can be observed from the third equation in (4) and (33) that the control input u with constrain designed in this paper is smoother. In contrast to the ones in [14,15], the method proposed in this paper can achieve finite-time stability of shield tunneling systems with some constraints.

4. Simulation Study

To validate the accuracy of the aforementioned findings, we analyzed the excavation construction data from ring 355 of Beijing Metro Line 2 within the first 1000 s. The actual displacements of four groups of cylinders were depicted in Figure 3. Correspondingly, the actual propulsive force of the shield tunneling machine is shown in Figure 4.

From Figure 3, one can find that the propulsive velocity is approximately 1 mm/s. Therefore, the desired signals are selected as $x_{1d}=0.001t$ and $x_{2d}=0.001$ m/s. The ring is in the pebble layer of $C_1$ level. The unloaded mass of the shield head is about 340 tons, and the friction coefficient between the shell of the shield tunneling machine and the soil is measured as $0.21$, so the Coulomb friction is about $F_c=6.99\times {10}^5$ N. In the shutdown state, the thrust of the shield tunneling machine is kept at $7.77\times {10}^6$ N to maintain balance, i.e., the static friction force is $F_s=7.77\times {10}^6$ N. The diameter of the cutter head is $6.18$ m. The depth of the tunnel is $12.8$ m, the diameter and the length of the body of the shield tunneling machine are $6.14$ m and $8.28$ m, respectively. The specific gravity of the soil is about $1.28\times {10}^5$ Kg/m³, and the coefficient of the lateral pressure caused by soil and water is about $0.47$. Form the empirical formula, the friction force is calculated as $f\approx 4.8\times {10}^6$ N. For the ideal case of the propulsive velocity $x_2$ being constant, the value of $J_1\left(x_2\right)z+J_2{x}_2$ with $\dot{z}=x_2-{\sigma }_0{\displaystyle \frac{|x_2|}{g\left(x_2\right)}}z$ is mainly determined by the values of $F_c$ and ${\sigma }_2$, so we can deduce ${\sigma }_2\approx 4.1\times {10}^9$ Ns/m. For clarity, the values of some parameters are listed in

Table 4. The values of some parameters in simulation.

Parameters	Value	Unit
M	$4\times {10}^5+50,000\times (1-cos\left(t\right))$	Kg
$F_c$	$6.99\times {10}^5$	N
$F_s$	$7.7\times {10}^6$	N
${\sigma }_0$	$5\times {10}^8$	N/m
${\sigma }_1$	200	Ns/m
${\sigma }_2$	$4.1\times {10}^9$	Ns/m
$v_s$	0.3	mm/s

.

Let the upper bound of the control input $P_{max}=1.28\times {10}^4$ kN, initial values $x_1\left(0\right)=0$, $x_2\left(0\right)=0$ and $F_u\left(0\right)=7.77\times {10}^4$ kN, parameters $k_{b2}=0.0012$, $k_{b3}=10,000$ and $a=0.1$, $b=0.3$, $c=50$, $c_1=c_2=100$, $c_3=500$, $r_i=0.05(i=1,2,\cdots ,7)$. Moreover, the RBF neural network is chosen as Gaussian function centers uniformly distributed between $[-1.5,1.5]$ with a width of 2 (i.e., ${\mathsf{\mu }}_i=-2+0.5i$ and ${\nu }_i=2(i=1,2,\cdots ,7)$).

For the error system (4), the update rules of the virtual controls ${\alpha }_1$ and ${\alpha }_2$, the signals v and ${\widehat{\theta }}_1$, ${\widehat{\theta }}_2$, ${\widehat{\theta }}_3$, $\widehat{d}$, ${\widehat{\sigma }}_0$, ${\widehat{\sigma }}_1$ and ${\widehat{J}}_2$ are designed according to Theorem 1. Based on the above design, we run a simulation for 300 s using MATLAB R2023b (MathWorks, Inc., Natick, MA, USA). The simulation results are shown in Figure 5, Figure 6 and Figure 7, which show the response curves of the system states, desired signals, and control input, respectively. From Figure 5, one can see that the curves of the system state $x_1$ and the desired displacement are almost identical. Similarly, the system state $x_2$ can track the desired velocity in Figure 6. Correspondingly, the propulsive force of cylinders is depicted in Figure 7. Obviously, after starting for 10 s, the value of the propulsive force $F_u$ is almost kept as $1.26\times {10}^7$ N and meets the constraint on the input.

When the constrain on the control input is not considered, then $u={\alpha }_2$ and $F_u=u+P_{max}/2$. At this time, the control input $F_u$ without constrain is shown in Figure 8. Obviously, the control input $F_u$ without constraint oscillates sharply, and its maximum is approximately $1.5\times {10}^7$ N and larger than the desired upper bound $1.28\times {10}^7$ N. Compared with Figure 4 and Figure 8, $F_u$ with constraint in Figure 7 is smoother and then easier to implement in shield machine operation, which shows that the adaptive finite-time backstepping control strategy proposed in this paper is effective.

5. Conclusions

In this paper, the finite-time tracking control problem of shield tunneling systems is studied based on the LuGre friction model. The constraint on the state is handled by combining the use of transformation and a logarithmic barrier Lyapunov function, while the constraint on the control input is handled by designing an auxiliary differential equation. Meanwhile, the unknown functions are approximated by radial basis function neural networks. Furthermore, using an adaptive finite-time backstepping method, a finite-time tracking control strategy is proposed, and the finite-time stability of the closed-loop system is achieved. Finally, the theoretical results are verified via the actual data of practical engineering, which shows the effectiveness of the proposed finite-time control method for shield tunneling systems with constrains. In the future, we will try to utilize the technique of sliding mode control (SMC) by combining sampled data with physical models, and then apply the obtained results to practical engineering.