Disturbance Observer-Based Tracking Controller for n-Link Flexible-Joint Robots Subject to Time-Varying State Constraints

Abstract

This paper addresses the tracking control for an n-link flexible-joint robot system with full-state constraints and external disturbances. First, a nonlinear disturbance observer (NDO) is introduced to asymptotically estimate and suppress the influence of the related disturbances. Next, the constrained system under consideration is transformed into a new unconstrained system using state-dependent function (SDF) transformations. Subsequently, a NDO-based tracking controller that combines the backstepping method and filter technique is proposed in this work. Based on stability analysis, it can be proven that the tracking error converges to a predefined compact set, which can be arbitrarily small without violating the full-state constraints. Finally, simulation results are presented to demonstrate the validity of the suggested control algorithm.

1. Introduction

As a significant branch of robots, manipulators have evolved from rigid structures to flexible ones to meet the growing demands. Therefore, the studies on flexible-joint (FJ) robot systems have attracted wide attention [1,2,3,4,5,6]. Ref. [2] presented a comprehensive review of the mechanical design and synthesis of piezoelectric-actuated compliant micro positioning platforms based on the composition of flexible mechanisms. Ref. [3] proposed a dynamic modeling method for a multi-flexible-body robot system. Ref. [4] developed a nonlinear observer for an FJ manipulator to address the difficulty in measuring velocity signals. Ref. [5] analyzed the drive system and control strategy for a hyper-redundant continuous robot. In [6], an adaptive torque observer based on fuzzy inference was proposed for a permanent magnet synchronous machine (PMSM), and was further applied to an FJ manipulator.

The tracking control of flexible-joint (FJ) robots has indeed been a challenging and important area of research in control theory and engineering. The increasing order and nonlinear coupling characteristics of FJ robot systems have made the control particularly difficult [7,8,9,10,11]. Moreover, the introduction of an integral manifold approach has been significantly investigated in this field. By leveraging the accurate system modeling, the integral manifold approach facilitates the transformation of an FJ system into a rigid one, thereby reducing the design complexity [12,13]. Subsequently, a variety of advanced control methods, including backstepping control [14,15], PID control [16,17], sliding mode control [18,19], and others, have been introduced to address the control problems in FJ robots. In the presence of model uncertainty, adaptive control design plays a critical role in developing controllers that can effectively achieve the control objectives [20,21]. To enhance the system performance, adaptive control design often makes use of neural networks (NNs) and fuzzy systems, which possess universal approximation capabilities [22,23]. Ref. [24] proposed a unique adaptive neural network control technique for resolving the tracking control problem for flexible-joint robots with random noises. In [25], an adaptive fuzzy control scheme was proposed under an event-triggered mechanism.

While the aforementioned methods are effective for FJ robots’ tracking control, they may not adequately address the issue of tracking control in the presence of state constraints. For instance, when transporting goods, FJ manipulators must operate within the specified constraint zones to prevent crashes. To handle these constraints, robust control strategies for nonlinear systems, such as model predictive control [26,27] and set invariance notions [28,29] have been investigated. Subsequently, the use of barrier Lyapunov functions (BLFs) to handle the state constraints in nonlinear systems is becoming increasingly common [30,31,32]. An FJ robot’s adaptive fuzzy tracking control with full-state constraints was proposed in [33]. Ref. [34] proposed an event-triggered control approach to deal with output constraints and make the system finite-time convergent. In addition, a new control approach for an FJ robot manipulator with full-state constraints was provided in [35], where the investigated constrained system was transformed into a new unconstrained system by applying state-dependent function transformations (SDFs).

With the advancement of robot technology in recent years, enhancing robot performance to attain higher accuracy and faster speeds without increasing costs has gained more attention. The greatest obstacle to achieving this goal is the constraints that disturbances impose. In practical control scenarios, interference can readily degrade the control performance. Disturbance observer (DO) is a widely utilized technique in most of the trajectory tracking control literature for dealing with time-varying external disturbances [36]. In [37], an adaptive neural control based on a DO was presented for n-link FJ robots. A sliding mode disturbance observer was designed for a nonlinear system in [38]. Ref. [39] introduced a new DO-based neural network integral sliding mode controller with output constraints, which combined the advantages of neural networks, disturbance observer, and integral sliding mode.

Until now, there have been limited studies addressing the n-link FJ robot manipulator systems with full-state constraints and external disturbances. To address these challenges, this paper introduced a DO-based nonlinear tracking controller for the considered systems. Initially, the original dynamic equations are transformed into a chained system and the external disturbances are estimated by the introduced nonlinear disturbance observer. Subsequently, the system with full-state constraints is transformed into an unconstrained system by the SDF transformation, which can simplify the controller design. Using the backstepping method as well as the filter technique, the nonlinear anti-disturbance tracking controller is obtained. The key contributions of this paper can be outlined as follows:

(I)

The proposed method offers a wider range of applications. Compared with the single-link FJ manipulator in [35,40], the studied system in this paper belongs to a class of complex MIMO systems, which means that the proposed control method has a wider application range.

(II)

The proposed method is easier to implement. Compared with the BLF-based method [33], the proposed state-constrained method removes the feasibility conditions, which broadens the range of acceptable initial values and relaxes the requirements for control parameters.

(III)

To address the issue of external disturbances, an NDO is incorporated for the real-time estimation of disturbances. In the controller design phase, filter techniques are employed to efficiently mitigate the problem of differential explosion resulting from the repeated differentiation of virtual control inputs.

(IV)

Through simulation results, we demonstrate the effectiveness and robustness of the proposed method. The proposed method ensures the effective control performance, and enhances the system’s anti-disturbance capability. In addition, the system states do not violate the constraints under the design control strategy. And the tracking error can be made arbitrarily small by properly adjusting the design parameters.

The rest of this paper is organized as follows. Section 2 gives the problem formulation and preliminaries. DO-based control design and stability analysis are presented in Section 3. Section 4 provides the simulation results. Section 5 concludes this paper.

2. Problem Formulation and Preliminaries

Problem Formulation

The n-link flexible-joint (FJ) robotic manipulator dynamics can be expressed as follows:

$$\begin{array}{cc}\hfill M{\ddot{q}}_1+C(q_1,{\dot{q}}_1){\dot{q}}_1+G\left(q_1\right)+K(q_1-q_2)& =d\left(t\right),\hfill \\ \hfill J{\ddot{q}}_2+B{\dot{q}}_2-K(q_1-q_2)& =u\left(t\right)\hfill \end{array}$$

(1)

where $q_1,{\dot{q}}_1,{\ddot{q}}_1\in R^n$ denote the link position, velocity, and acceleration vectors, respectively; $M\left(q\right)\in R^{n\times n}$ is the inertia matrix; $C(q,\dot{q})\in R^{n\times n}$ is the centrifugal and Coriolis force matrix; $G\left(q\right)\in R^n$ is the gravity vector, $d\left(t\right)$ is the unknown disturbance vector; and $q_2,{\dot{q}}_2,{\ddot{q}}_2\in R^n$ denote the rotor angular position, velocity, and acceleration vectors, respectively. The positive definite diagonal matrices $K\in R^{n\times n},J\in R^{n\times n}$, and $B\in R^{n\times n}$ reflect the joint flexibility, actuator inertia, and natural damping term, respectively, and $u\left(t\right)\in R^n$ is the torque input. Here, we assume that ${lim}_{t\to \infty }\dot{d}\left(t\right)=0$.

Let $x_1={[q_{11},q_{12},\cdots ,q_{1n}]}^T$, $x_2={\dot{x}}_1={[{\dot{q}}_{11},{\dot{q}}_{12},\cdots ,{\dot{q}}_{1n}]}^T$, $x_3={[q_{21},q_{22},\cdots ,q_{2n}]}^T$ and $x_4={\dot{x}}_2={[{\dot{q}}_{21},{\dot{q}}_{22},\cdots ,{\dot{q}}_{2n}]}^T$. The dynamics of (1) then can be rewritten into the following form:

$$\left\{\begin{array}{cc}\hfill {\dot{x}}_1=& x_2\hfill \\ \hfill {\dot{x}}_2=& M^{-1}(-C(x_1,x_2)x_2-G\left(x_1\right)-K(x_1-x_3)+d\left(t\right))\hfill \\ \hfill {\dot{x}}_3=& x_4\hfill \\ \hfill {\dot{x}}_4=& J^{-1}(-B{x}_4+K(x_1-x_3)+u\left(t\right)).\hfill \end{array}\right.$$

(2)

The control purpose of this paper is to design an NDO-based tracking controller to realize the following objectives: (i) The desired trajectory $x_d={[x_{d1},x_{d2},\cdots ,x_{dn}]}^T$ can be tracked in the sense that $|z_{1i}\left(t\right)|=|x_{1i}\left(t\right)-x_{di}\left(t\right)|⩽\epsilon$ as $t\to \infty$ where $\epsilon >0$ is a given precision, and all signals are guaranteed to be bounded. (ii) All states are required to be maintained in the following constraints

$$\begin{array}{c}\hfill x_{di}\left(t\right)-F_{11}\left(t\right)<x_{1i}\left(t\right)<x_{di}\left(t\right)+F_{12}\left(t\right),\\ \hfill -F_{j1}\left(t\right)<x_{ji}\left(t\right)<F_{j2}\left(t\right)\end{array}$$

(3)

where $F_{ij}\left(t\right)$ are the prescribed constraints, with $i=1,2,\cdots ,n$, $j=2,3,4$ being the positive time-varying functions, and their first and second derivatives are also continuous and bounded.

An NDO developed in [36] is used in this research to estimate the external unknown bounded continuous disturbance. For the sake of description, we describe the above system (2) into the following compact form:

$$\begin{array}{ccc}\hfill \dot{x}& =& f\left(x\right)+g_1\left(x\right)u+g_2\left(x\right)d,\hfill \\ \hfill y& =& x_1\hfill \end{array}$$

(4)

where $x={[x_1,x_2,x_3,x_4]}^T$, and

$$f\left(x\right)=\left[\begin{array}{c}{x}_2\\ M^{-1}(-C(x_1,x_2)x_2-G\left(x_1\right)-K(x_1-x_3))\\ x_4\\ J^{-1}(-B{x}_4+K(x_1-x_3))\end{array}\right]$$

$$g_1\left(x\right)=\left[\begin{array}{c}{0}_{n\times n}\hfill \\ 0_{n\times n}\hfill \\ 0_{n\times n}\hfill \\ J^{-1}\hfill \end{array}\right],g_2\left(x\right)=\left[\begin{array}{c}{0}_{n\times n}\hfill \\ M^{-1}\hfill \\ 0_{n\times n}\hfill \\ 0_{n\times n}\hfill \end{array}\right].$$

The following NDO [36] is then employed to provide an online estimation of the disturbance $d\left(t\right)$:

$$\left\{\begin{array}{cc}\hfill & \dot{z}=-l\left(x\right)g_2\left(x\right)z-l\left(x\right)[g_2\left(x\right)p\left(x\right)+f\left(x\right)+g_1\left(x\right)u]\hfill \\ \hfill & \widehat{d}=z+p\left(x\right)\hfill \end{array}\right.$$

(5)

where l is the the gain matrix to be chosen. In this paper, the gain matrix l and p are chosen as $l\left(x\right)=[0_{n\times n},M_{n\times n},0_{n\times n},0_{n\times n}]$ and $p\left(x\right)=[0_{n\times n},M_{n\times n}{x}_2,$$0_{n\times n},0_{n\times n}]$ to satisfy that

$$\begin{array}{c}\hfill l\left(x\right)g_2\left(x\right)={\left(\begin{array}{ccc}1& \dots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & 1\end{array}\right)}_{n\times n}.\end{array}$$

Now, we define error $e=d\left(t\right)-\widehat{d}\left(t\right)$, then it can be shown that $\dot{e}=-l{g}_{2}e+\dot{d}$, which implies that ${lim}_{t\to \infty }e\left(t\right)=0$ under the condition ${lim}_{t\to \infty }\dot{d}\left(t\right)=0$.

3. Controller Design and Stability Analysis

The control design procedure in this section can be summarized as follows: First, the constrained system under consideration is transformed into an unconstrained system using the SDF transformation. Subsequently, the tracking controller is designed using backstepping and filter technique to guarantee that the system states of the resulting closed-loop system are all bounded and do not exceed the constraint boundaries. In addition, we can make $z_{1i}$ arbitrarily small by adjusting the design parameters. The control flow is shown in Figure 1.

Figure 1. The control flow.

3.1. System Transformation

The SDF transformation is introduced to convert the original constrained system (4) into a new unconstrained system. After creating the tracking errors $z_{1i}=x_{1i}-x_{di}$, the subsequent functions can be introduced as [41]:

$$\begin{array}{c}\hfill s_{1i}=\frac{z_{1i}}{(F_{11}+z_{1i})(F_{12}-z_{1i})},s_{ji}=\frac{x_{ji}}{(F_{j1}+x_{ji})(F_{j2}-x_{ji})}\end{array}$$

(6)

where $i=1,2,\cdots ,n$, $j=2,3,4.$ By introducing (6), the initial system (4) is changed into the new system described below:

$$\left\{\begin{array}{cc}\hfill {\dot{s}}_{1i}=& {\mu }_{1i}(h_{2i}{s}_{2i}-{\dot{x}}_{di})+{\nu }_{1i}\hfill \\ \hfill {\dot{s}}_{2i}=& {\mu }_{2i}{M}_i^{-1}(-C(x_1,x_2)x_2-G\left(x_1\right)-K(x_1-\delta +\widehat{d}+e)+{\nu }_{2i}\hfill \\ \hfill {\dot{s}}_{3i}=& {\mu }_{3i}{h}_{4i}{s}_{4i}+{\nu }_{3i}\hfill \\ \hfill {\dot{s}}_{4i}=& {\mu }_{4i}{J}_i^{-1}(-B{x}_4+K(x_1-x_3)+u)+{\nu }_{4i}\hfill \end{array}\right.$$

(7)

where

$$\begin{array}{ccc}\hfill {\mu }_{1i}& =& \frac{F_{11}{F}_{12}+z_{1i}^2}{{(F_{11}+z_{1i})}^2{(F_{12}-z_{1i})}^2},\hfill \\ \hfill {\nu }_{1i}& =& -\frac{[{\dot{F}}_{11}{F}_{12}+F_{11}{\dot{F}}_{12}+({\dot{F}}_{12}-{\dot{F}}_{11})z_{1i}]z_{1i}}{{(F_{11}+z_{1i})}^2{(F_{12}-z_{1i})}^2},\hfill \\ \hfill {\mu }_{ji}& =& \frac{F_{j1}{F}_{j2}+x_{ji}^2}{{\left(F_{j1}+x_{ji}\right)}^2{\left(F_{j2}-x_{ji}\right)}^2},\hfill \\ \hfill {\nu }_{ji}& =& -\frac{[{\dot{F}}_{j1}{F}_{j2}+F_{j1}{\dot{F}}_{j2}+({\dot{F}}_{j2}-{\dot{F}}_{j2})x_{ji}]x_{ji}}{{(F_{j1}+x_{ji})}^2{(F_{j2}-x_{ji})}^2},\hfill \\ \hfill h_{1i}& =& (F_{11}+z_{1i})(F_{12}-z_{1i}),\hfill \\ \hfill h_{ji}& =& (F_{j1}+x_{ji})(F_{j2}-x_{ji}),\hfill \end{array}$$

$\delta ={\left[h_{31}{s}_{31},h_{32}{s}_{32},\cdots ,h_{3n}{s}_{3n}\right]}^T$ and $i=1,2,\cdots ,n$, $j=2,3,4.$ The constraint of $z_{1i}\left(t\right)$ is automatically guaranteed for every initial condition meeting $x_{di}\left(0\right)-F_{11}\left(0\right)<x_{1i}\left(0\right)<x_{di}\left(0\right)+F_{12}\left(0\right)$ as long as $s_{1i}$ is constrained for $t\in [0,+\infty )$. It is crucial to guarantee the boundedness of $s_{1i}$ for any $t>0$ in order to satisfy the pre-specified constraint bound. If $\underset{t\to \infty }{lim}{s}_{1i}\left(t\right)=0$, then the asymptotic tracking control can be achieved.

Next, let us define $\beta \left(t\right)=Htanh\left(\frac{1+\lambda t}{H}\right)$, which is called a scaling function with $H>1$ and $\lambda >0$. The following characteristics of $\beta \left(t\right)$ are crucial for the subsequent work: (1) $\beta \left(t\right)$ strictly increases and $\beta \left(t\right)⩾\frac{1}{2}$ for all t; (2) $\dot{\beta }\left(t\right)\le \lambda \beta \left(t\right)$; and (3) $\underset{t\to \infty }{lim}\beta \left(t\right)=H$.

Remark 1.

The following characteristics of $s\left(x\right)$ are noteworthy: (i) $F_{11}\left(t\right)$ and $F_{12}\left(t\right)$ are smooth, time-varying, strictly positive functions. As a result, $s_{1i}\left(x\right)$ is invertible and strictly growing. (ii) ${\mathrm{s}}_{1i}\left(0\right)=0$, $s_{1i}\left(x\right)\to \infty$ as $x_{1i}\left(t\right)\to -F_{11}\left(t\right)$ or $x_{1i}\left(t\right)\to F_{12}\left(t\right)$.

3.2. Controller Design

To facilitate the controller design using the backstepping method, the following state transformations are given as:

$$\begin{array}{c}\hfill w_{1i}=s_{1i},w_{ji}=s_{ji}-{\gamma }_{ji}\end{array}$$

(8)

in which $i=1,2,\cdots ,n$, $j=2,3,4$ and ${\gamma }_{ji}$ is the output of the first-order filter which is described as

$$\begin{array}{c}\hfill {\varpi }_{ji}{\dot{\gamma }}_{ji}=[\beta {(\frac{{\alpha }_{j-1,i}}{h_{ji}}-{\gamma }_{ji}]}^3\end{array}$$

(9)

where $h_{ji}$ has been given above, ${\alpha }_{j-1,i}$ is the virtual control input and ${\varpi }_{ji}>0$ is a design parameter.

Then, in order to further achieve the aforementioned practical tracking control target, an error transformation is achieved by using the scaling function:

$$\begin{array}{c}\hfill {\zeta }_{1i}=\beta w_{1i},{\zeta }_{ji}=\beta w_{ji}.\end{array}$$

(10)

In addition, we define

$$\begin{array}{c}\hfill y_{ji}={\gamma }_{ji}-\frac{{\alpha }_{j-1,i}}{h_{ij}},Y_{ji}=\beta y_{ji}\end{array}$$

(11)

which can ensure the stability of the first-order filter for subsequent stability analysis.

To achieve the desired control objective, we present the controller design step by step as follows:

Step 1: In this step, we choose a Lyapunov function as

$$\begin{array}{c}\hfill V_1=\sum _{i=1}^n{V}_{1i},V_{1i}=\frac{1}{2}{\zeta }_{1i}^2+\frac{1}{2}{Y}_{2i}^2.\end{array}$$

(12)

Differentiating $V_{1i}$, one obtains

$$\begin{array}{c}\hfill {\dot{V}}_{1i}={\zeta }_{1i}{\dot{\zeta }}_{1i}+Y_{2i}{\dot{Y}}_{2i}\end{array}$$

(13)

in which

$$\begin{array}{c}\hfill {\zeta }_{1i}{\dot{\zeta }}_{1i}=\beta {\zeta }_{1i}({\mu }_{1i}{h}_{2i}{w}_{2i}+{\mu }_{i1}{h}_{2i}{y}_{2i}-{\mu }_{1i}{\dot{x}}_{di}+{\nu }_{1i}+{\beta }^{-1}\dot{\beta }{w}_{1i})+\beta {\zeta }_{1i}{\mu }_{1i}{\alpha }_{1i},\end{array}$$

(14)

and

$$\begin{array}{ccc}\hfill Y_{2i}{\dot{Y}}_{2i}& ⩽& Y_{2i}(\beta {\dot{y}}_{2i}+\lambda \beta y_{2i})\hfill \\ & =& Y_{2i}[\beta ({\dot{\gamma }}_{2i}-{\left(\frac{{\alpha }_{1i}}{h_{2i}}\right)}^{\prime })+\lambda \beta y_{2i}]\hfill \\ & =& -\frac{\beta Y_{2i}^4}{{\varpi }_{2i}}+\beta {\xi }_{2i}{Y}_{2i}+\lambda Y_{2i}^2\hfill \end{array}$$

(15)

where ${\xi }_{2i}=-\frac{{\dot{\alpha }}_{1i}}{h_{2i}}+\frac{{\alpha }_{1i}{\dot{h}}_{2i}}{h_{2i}^2}$.

Combining (13) with (14) and (15) yields that

$$\begin{array}{ccc}\hfill {\dot{V}}_{1i}& ⩽& \beta {\zeta }_{1i}({\mu }_{1i}{h}_{2i}{w}_{2i}+{\mu }_{i1}{h}_{2i}{y}_{2i}-{\mu }_{1i}{\dot{x}}_{di}\hfill \\ & & +{\nu }_{1i}+{\beta }^{-1}\dot{\beta }{w}_{1i})+\beta {\zeta }_{1i}{\mu }_{1i}{\alpha }_{1i}-\frac{\beta Y_{2i}^4}{{\varpi }_{2i}}+\beta {\xi }_{2i}{Y}_{2i}+\lambda Y_{2i}^2.\hfill \end{array}$$

(16)

Using Young’s inequality to calculate the terms on the right hand of (16), the following inequalities are obtained as:

$$\begin{array}{ccc}\hfill \beta {\zeta }_{1i}{\mu }_{1i}{h}_{2i}{w}_{2i}& ⩽& \beta {\mu }_{1i}^2{\zeta }_{1i}^2{h}_{2i}^2{w}_{2i}^2+\frac{\beta }{4},\hfill \\ \hfill \beta {\zeta }_{1i}{\mu }_{1i}{h}_{2i}{y}_{2i}& ⩽& \beta r_{1i}{\zeta }_{1i}^2{\mu }_{1i}^2+\frac{\beta }{4r_{1i}}{h}_{2i}^2{Y}_{2i}^2\hfill \\ & ⩽& 2r_{1i}\beta {\zeta }_{1i}^2{\mu }_{1i}^2+\frac{\beta }{2r_{1i}}{h}_{2i}^4{Y}_{2i}^4+\frac{\beta }{8r_{1i}},\hfill \\ \hfill -\beta {\zeta }_{1i}{\mu }_{1i}{\dot{x}}_{di}& ⩽& r_{1i}\beta {\mu }_{1i}^2{\zeta }_{1i}^2{\dot{x}}_{di}^2+\frac{\beta }{4r_{1i}},\hfill \\ \hfill \beta {\zeta }_{1i}{\nu }_{1i}& ⩽& \beta {\nu }_{1i}\left|{\zeta }_{1i}\right|⩽l_{1i}\beta {\zeta }_{1i}^2{\nu }_{1i}^2+\frac{\beta }{4l_{1i}},\hfill \\ \hfill \beta {\zeta }_{1i}{\beta }^{-1}\dot{\beta }{w}_{1i}& ⩽& l_{1i}\beta {\lambda }^2{\zeta }_{1i}^2{w}_{1i}^2+\frac{\beta }{4l_{1i}},\hfill \\ \hfill \beta {\xi }_{2i}{Y}_{2i}& ⩽& \frac{\beta {\xi }_{2i}^2}{2}+\frac{\beta Y_{2i}^4}{4}+\frac{\beta }{4},\hfill \\ \hfill \lambda Y_{2i}^2& ⩽& {\lambda }^2{Y}_{2i}^4+\frac{1}{4}.\hfill \end{array}$$

Next, after a straightforward substitution, (16) can be rewritten as

$$\begin{array}{ccc}\hfill {\dot{V}}_{1i}& ⩽& \beta {\zeta }_{1i}({\mu }_{1i}{\alpha }_{1i}+r_{1i}{\Phi }_{1i}{\zeta }_{1i}+l_{1i}{\Psi }_{1i}{\zeta }_{1i})+\beta {\Delta }_{1i}\hfill \\ & & +\beta {\zeta }_{1i}^2{\mu }_{1i}^2{h}_{2i}^2{w}_{2i}^2+\frac{\beta }{2r_{1i}}{h}_{2i}^4{Y}_{2i}^4-\frac{\beta Y_{2i}^4}{{\varpi }_{2i}}+\frac{\beta Y_{2i}^4}{4}+\frac{\beta {\xi }_{2i}^2}{2}+\frac{\beta }{4}+{\lambda }^2{Y}_{2i}^4+\frac{1}{4}\hfill \end{array}$$

(17)

in which ${\Phi }_{1i}={\mu }_{1i}^2({\dot{x}}_{di}^2+2)$, ${\Psi }_{1i}={\nu }_{1i}^2+{\lambda }^2{w}_{1i}^2$ and ${\Delta }_{1i}=\frac{1}{4r_{1i}}+\frac{1}{8r_{1i}}+\frac{1}{2l_{1i}}+\frac{1}{4}$. In view of the structure of (17), the virtual controller is chosen as ${\alpha }_1=A_1$ with

$$\begin{array}{ccc}\hfill A_1& =& {[A_{11},A_{12},\cdots ,A_{1n}]}^T,\hfill \\ \hfill A_{1i}& =& -\frac{1}{{\mu }_{1i}}(k_{1i}+r_{1i}{\Phi }_{1i}+l_{1i}{\Psi }_{1i}){\zeta }_{1i}.\hfill \end{array}$$

(18)

By selecting $\frac{1}{{\varpi }_{2i}}⩾\frac{h_{2i}^4}{2r_{1i}}+\frac{1}{4}+{\frac{\lambda }{\beta }}^2+{\varpi }_{2i}^{*}$ which represents a positive design parameter, and combining (17) and (18), we have

$$\begin{array}{c}\hfill {\dot{V}}_{1i}⩽-k_{1i}\beta {\zeta }_{1i}^2-\beta {\varpi }_{2i}^{*}{Y}_{2i}^4+\beta {\phi }_{1i}+\beta {\mu }_{1i}^2{\zeta }_{1i}^2{h}_{2i}^2{w}_{2i}^2+\frac{\beta {\xi }_{2i}^2}{2}\end{array}$$

(19)

where ${\phi }_{1i}={\Delta }_{1i}+\frac{3}{4}$, which further leads to

$$\begin{array}{c}\hfill {\dot{V}}_1⩽-\sum _{i=1}^n{k}_{1i}\beta {\zeta }_{1i}^2-\sum _{i=1}^n\beta {\varpi }_{2i}^{*}{Y}_{2i}^4+\sum _{i=1}^n\beta {\phi }_{1i}+\sum _{i=1}^n\beta {\mu }_{1i}^2{\zeta }_{1i}^2{h}_{2i}^2{w}_{2i}^2+\sum _{i=1}^n\frac{\beta {\xi }_{2i}^2}{2}\end{array}$$

(20)

where the item ${\displaystyle \sum _{i=1}^n}\beta {\mu }_{1i}^2{\zeta }_{1i}^2{h}_{2i}^2{w}_{2i}^2$ will be dealt with in the following stage.

Step 2: The second candidate Lyapunov function is picked as

$$\begin{array}{c}\hfill V_2=\sum _{i=1}^n{V}_{2i},V_{2i}=\frac{1}{2}{\zeta }_{2i}^2+\frac{1}{2}{Y}_{3i}^2.\end{array}$$

(21)

Then, it is simple to derive the time derivative of $V_{2i}$ as

$$\begin{array}{ccc}\hfill {\dot{V}}_{2i}& =& {\zeta }_{2i}{\dot{\zeta }}_{2i}+Y_{3i}{\dot{Y}}_{3i}\hfill \\ & ⩽& \beta {\zeta }_{2i}({\mu }_{2i}{\Gamma }_{1i}+{\mu }_{2i}{M_i}^{-1}(K\tau +K\sigma +e)+v_{2i}-{\dot{\gamma }}_{2i}+{\beta }^{-1}\dot{\beta }{w}_{2i})\hfill \\ & & +\beta {\zeta }_{2i}{\mu }_{2i}{M}_i^{-1}K{\alpha }_2-\frac{\beta Y_{3i}^4}{{\varpi }_{3i}}+\beta {\xi }_{3i}{Y}_{3i}+\lambda Y_{3i}^2\hfill \end{array}$$

(22)

where ${\Gamma }_{1i}={M_i}^{-1}(-C(x_1,x_2)x_2-G\left(x_1\right)-K{x}_1+\widehat{d})$, ${\xi }_{3i}=-\frac{{\dot{\alpha }}_{2i}}{h_{3i}}+\frac{{\alpha }_{2i}{\dot{h}}_{3i}}{h_{3i}^2}$, $\tau =[h_{31}{w}_{31},$$h_{32}{w}_{32},$ $\cdots ,h_{3n}{w}_{3n}{]}^T$, $\sigma ={\left[h_{31}{y}_{31},h_{32}{y}_{32},\cdots ,h_{3n}{y}_{3n}\right]}^T$. Then, using Young’s inequality, one has

$$\begin{array}{ccc}\hfill \beta {\zeta }_{2i}{\mu }_{2i}{\Gamma }_{1i}& ⩽& r_{2i}\beta {\zeta }_{2i}^2{\mu }_{2i}^2{\Gamma }_{1i}^2+\frac{\beta }{4r_{2i}},\hfill \\ \hfill \beta {\zeta }_{2i}{\mu }_{2i}{\Lambda }_{1i}\tau & ⩽& \beta {\zeta }_{2i}^2{\mu }_{2i}^2{\tau }^T\tau +\frac{\beta }{4}{\Lambda }_{1i}{\Lambda }_{1i}^T,\hfill \\ \hfill \beta {\zeta }_{2i}{\mu }_{2i}{\Lambda }_{1i}\sigma & ⩽& 2\beta r_{2i}{\zeta }_{2i}^2{\mu }_{2i}^2+\frac{\beta }{2r_{2i}}{\rho }^T\rho +\frac{\beta }{8r_{2i}}{\left({\Lambda }_{1i}{\Lambda }_{1i}^T\right)}^2,\hfill \\ \hfill \beta {\zeta }_{2i}{\mu }_{2i}{M}_i^{-1}e& ⩽& r_{2i}\beta {\zeta }_{2i}^2{\mu }_{2i}^2+\frac{\beta }{4r_{2i}}{∥M_i^{-1}∥}^2{e}^{T}e,\hfill \\ \hfill \beta {\zeta }_{2i}{v}_{2i}& ⩽& l_{2i}\beta {\zeta }_{2i}^2{v}_{2i}^2+\frac{\beta }{4l_{2i}},\hfill \\ \hfill -\beta {\zeta }_{2i}{\dot{\gamma }}_{2i}& ⩽& l_{2i}\beta {\zeta }_{2i}^2{\dot{\gamma }}_{2i}^2+\frac{\beta }{4l_{2i}},\hfill \\ \hfill \beta {\zeta }_{2i}{\beta }^{-1}\dot{\beta }{w}_{2i}& ⩽& l_{2i}\beta {\lambda }^2{\zeta }_{2i}^2{w}_{2i}^2+\frac{\beta }{4l_{2i}},\hfill \\ \hfill \beta {\xi }_{3i}{Y}_{3i}& ⩽& \frac{\beta {\xi }_{3i}^2}{2}+\frac{\beta Y_{3i}^2}{4}+\frac{\beta }{4}\hfill \end{array}$$

and

$$\begin{array}{c}\hfill \lambda Y_{3i}^2⩽{\lambda }^2{Y}_{3i}^2+\frac{1}{4}\end{array}$$

where ${\Lambda }_{1i}=M_i^{-1}K$, $\rho ={\left[h_{31}{Y}_{31},h_{32}{Y}_{32},\cdots ,h_{3n}{Y}_{3n}\right]}^T$. As a result, we can obtain the following result as

$$\begin{array}{ccc}\hfill {\dot{V}}_{2i}& ⩽& \beta {\zeta }_{2i}({\mu }_{2i}{\Lambda }_{1i}{\alpha }_2+r_{2i}{\Phi }_{2i}{\zeta }_{2i}+l_{2i}{\Psi }_{2i}{\zeta }_{2i})+\beta {\Delta }_{2i}+\beta {\zeta }_{2i}^2{\mu }_{2i}^2{\tau }^T\tau \hfill \\ & & +\frac{\beta }{4r_{2i}}{∥M_i^{-1}∥}^2{e}^{T}e+\frac{\beta }{2r_{2i}}{\rho }^T\rho -\frac{\beta Y_{3i}^4}{{\varpi }_{3i}}+\frac{\beta Y_{3i}^4}{4}+\frac{\beta {\xi }_{3i}^2}{2}+\frac{\beta }{4}+{\lambda }^2{Y}_{3i}^4+\frac{1}{4}\hfill \end{array}$$

(23)

where ${\Delta }_{2i}=\frac{1}{4r_{2i}}+\frac{1}{4}{\Lambda }_{1i}{\Lambda }_{1i}^T+\frac{1}{8r_{2i}}{\left({\Lambda }_{1i}{\Lambda }_{1i}^T\right)}^2+\frac{3}{4l_{2i}}$, ${\Phi }_{2i}=3{\mu }_{2i}^2+{\Gamma }_{1i}^2{\mu }_{2i}^2$ and ${\Psi }_{2i}=v_{2i}^2+{\dot{\gamma }}_{2i}^2+{\lambda }^2{w}_{2i}^2$.

The virtual controller in (23) is chosen as

$$\begin{array}{c}\hfill {\alpha }_2=K^{-1}M{A}_2\end{array}$$

(24)

where $A_2={[A_{21},A_{22},\cdots ,A_{2n}]}^T$, and a detailed description of $A_{2i}$ is given as

$$\begin{array}{c}\hfill A_{2i}=-\frac{1}{{\mu }_{2i}}(k_{2i}+r_{2i}{\Phi }_{2i}+l_{2i}{\Psi }_{2i}+{\mu }_{1i}^2{w}_{1i}^2{h}_{2i}^2){\zeta }_{2i}-\beta {\mu }_{2i}{\zeta }_{2i}{\tau }^T\tau .\end{array}$$

(25)

Then, by selecting $\frac{1}{{\varpi }_{3i}}⩾\frac{1}{4}+{\frac{\lambda }{\beta }}^2+{\varpi }_{3i}^{*}$, (23) can be rewritten as

$$\begin{array}{ccc}\hfill {\dot{V}}_{2i}& ⩽& -k_{2i}\beta {\zeta }_{2i}^2-\beta {\varpi }_{3i}^{*}{Y}_{3i}^4+\beta {\phi }_{2i}-\beta {\mu }_{1i}^2{\zeta }_{1i}^2{h}_{2i}^2{w}_{2i}^2\hfill \\ & & +\frac{\beta }{4r_{2i}}{∥M_i^{-1}∥}^2{e}^{T}e+\frac{\beta }{2r_{2i}}{\rho }^T\rho +\frac{\beta {\xi }_{3i}^2}{2}\hfill \end{array}$$

(26)

where ${\phi }_{2i}={\Delta }_{2i}+\frac{3}{4}$. This ultimately results in

$$\begin{array}{ccc}\hfill {\dot{V}}_2& ⩽& -\sum _{i=1}^n{k}_{2i}\beta {\zeta }_{2i}^2-\sum _{i=1}^n\beta {\varpi }_{3i}^{*}{Y}_{3i}^4+\sum _{i=1}^n\beta {\phi }_{2i}+\sum _{i=1}^n\beta {\mu }_{1i}^2{\zeta }_{1i}^2{h}_{2i}^2{w}_{2i}^2\hfill \\ & & +\sum _{i=1}^n\frac{\beta {\xi }_{3i}^2}{2}+\sum _{i=1}^n\frac{\beta }{4r_{2i}}{∥M_i^{-1}∥}^2{e}^{T}e+\sum _{i=1}^n\frac{\beta }{2r_{2i}}{\rho }^T\rho .\hfill \end{array}$$

(27)

Step 3: The candidate Lyapunov function is defined as:

$$\begin{array}{c}\hfill V_3=\sum _{i=1}^n{V}_{3i},V_{3i}=\frac{1}{2}{\zeta }_{3i}^2+\frac{1}{2}{Y}_{4i}^2.\end{array}$$

(28)

The time derivative of $V_{3i}$ is computed as

$$\begin{array}{ccc}\hfill {\dot{V}}_{3i}& =& {\zeta }_{3i}{\dot{\zeta }}_{3i}+Y_{4i}{\dot{Y}}_{4i}\hfill \\ & ⩽& \beta {\zeta }_{2i}({\mu }_{3i}{h}_{4i}{w}_{4i}+{\mu }_{3i}{h}_{4i}{y}_{4i}+v_{3i}-{\dot{\gamma }}_{3i}\hfill \\ & & +{\beta }^{-1}\dot{\beta }{w}_{3i})+\beta {\zeta }_{3i}{\mu }_{3i}{\alpha }_{3i}-\frac{\beta Y_{4i}^4}{{\varpi }_{4i}}+\beta {\xi }_{4i}{Y}_{4i}+\lambda Y_{4i}^2\hfill \end{array}$$

(29)

where ${\xi }_{4i}=-\frac{{\dot{\alpha }}_{3i}}{h_{4i}}+\frac{{\alpha }_{3i}{\dot{h}}_{4i}}{h_{4i}^2}$. Similar to Step 1 and Step 2, direct calculation produces

$$\begin{array}{ccc}\hfill {\dot{V}}_{3i}& ⩽& \beta {\zeta }_{3i}({\mu }_{3i}{\alpha }_{3i}+r_{3i}{\Phi }_{3i}{\zeta }_{3i}+l_{3i}{\Psi }_{3i}{\zeta }_{3i})+\beta {\Delta }_{3i}+\beta {\zeta }_{3i}^2{\mu }_{3i}^2{h}_{4i}^2{w}_{4i}^2\hfill \\ & & +\frac{\beta }{2r_{3i}}{h}_{4i}^4{Y}_{4i}^4-\frac{\beta Y_{4i}^4}{{\varpi }_{4i}}+\frac{\beta Y_{4i}^4}{4}+\frac{\beta {\xi }_{4i}^2}{2}+\frac{\beta }{4}+{\lambda }^2{Y}_{4i}^4+\frac{1}{4}\hfill \end{array}$$

(30)

where ${\Phi }_{3i}=2{\mu }_{3i}^2$, ${\Psi }_{3i}=v_{3i}^2+{\dot{\gamma }}_{3i}^2+{\lambda }^2{w}_{3i}^2$ and ${\Delta }_{3i}=\frac{1}{8r_{3i}}+\frac{3}{4l_{3i}}+\frac{1}{4}$.

The third virtual control law ${\alpha }_3$ can be written as ${\alpha }_3=A_3$ with

$$\begin{array}{ccc}\hfill A_3& =& {[A_{31},A_{32},\cdots ,A_{3n}]}^T,\hfill \\ \hfill A_{3i}& =& -\frac{1}{{\mu }_{3i}}(k_{3i}+r_{3i}{\Phi }_{3i}+l_{3i}{\Psi }_{3i}){\zeta }_{3i}.\hfill \end{array}$$

(31)

By choosing $\frac{1}{{\varpi }_{4i}}⩾\frac{h_{4i}^4}{2r_{3i}}+\frac{1}{4}+{\frac{\lambda }{\beta }}^2+{\varpi }_{4i}^{*}$, one can obtain

$$\begin{array}{c}\hfill {\dot{V}}_{3i}⩽-k_{3i}\beta {\zeta }_{3i}^2-\beta {\varpi }_{4i}^{*}{Y}_{4i}^4+\beta {\phi }_{3i}+\frac{\beta {\xi }_{4i}^2}{2}+\beta {\mu }_{3i}^2{\zeta }_{3i}^2{h}_{4i}^2{w}_{4i}^2\end{array}$$

(32)

where ${\phi }_{3i}={\Delta }_{3i}+\frac{3}{4}$. Under (30), the following result can be expressed as

$$\begin{array}{c}\hfill {\dot{V}}_3⩽-\sum _{i=1}^n{k}_{3i}\beta {\zeta }_{3i}^2-\sum _{i=1}^n\beta {\varpi }_{4i}^{*}{Y}_{4i}^4+\sum _{i=1}^n\beta {\phi }_{3i}+\sum _{i=1}^n\frac{\beta {\xi }_{4i}^2}{2}+\sum _{i=1}^n\beta {\mu }_{3i}^2{\zeta }_{3i}^2{h}_{4i}^2{w}_{4i}^2.\end{array}$$

(33)

Step 4: Taking into account the following Lyapunov function $V_4={\displaystyle \sum _{i=1}^n}{V}_{4i},V_{4i}=\frac{1}{2}{\zeta }_{4i}^2$, it is simple to derive the time derivative of $V_{4i}$ as

$$\begin{array}{ccc}\hfill {\dot{V}}_{4i}& =& {\zeta }_{4i}{\dot{\zeta }}_{4i}\hfill \\ & ⩽& \beta {\zeta }_{4i}({\mu }_{4i}{\Gamma }_{2i}+v_{4i}-{\dot{\gamma }}_{4i}+{\beta }^{-1}\dot{\beta }{w}_{4i})+\beta {\zeta }_{4i}{\mu }_{4i}{J}_i^{-1}u\hfill \end{array}$$

(34)

where ${\Gamma }_{2i}=J_i^{-1}(-B{x}_4+K(x_1-x_3))$. Using Young’s inequality and the analysis described in the previous steps, we have

$$\begin{array}{c}\hfill {\dot{V}}_{4i}⩽\beta {\zeta }_{4i}(r_{4i}{\Phi }_{4i}{\zeta }_{4i}+l_{4i}{\Psi }_{4i}{\zeta }_{4i})+\beta {\Delta }_{4i}+\beta {\zeta }_{4i}{\mu }_{4i}{J}_i^{-1}u\end{array}$$

(35)

where ${\Phi }_{4i}={\mu }_{4i}^2{\Gamma }_{2i}^2$, ${\Psi }_{4i}=v_{4i}^2+{\dot{\gamma }}_{4i}^2+{\lambda }^2{w}_{4i}^2$, and ${\Delta }_{4i}=\frac{1}{4r_{4i}}+\frac{3}{4l_{4i}}$.

In this final step, the actual control law u is configured as $u=J{A}_4$ where

$$\begin{array}{ccc}\hfill A_4& =& {[A_{41},A_{42},\cdots ,A_{4n}]}^T,\hfill \\ \hfill A_{4i}& =& -\frac{1}{{\mu }_{4i}}(k_{4i}+r_{4i}{\Phi }_{4i}+l_{4i}{\Psi }_{4i}+{\mu }_{3i}^2{w}_{3i}^2{h}_{4i}^2){\zeta }_{4i}.\hfill \end{array}$$

(36)

Substituting u into (35), we obtain that

$$\begin{array}{c}\hfill {\dot{V}}_{4i}⩽-k_{4i}\beta {\zeta }_{4i}^2-\beta {\mu }_{3i}^2{\zeta }_{3i}^2{h}_{4i}^2{w}_{4i}^2+\beta {\phi }_{4i}\end{array}$$

(37)

where ${\phi }_{4i}={\Delta }_{4i}$, which further leads to

$$\begin{array}{c}\hfill {\dot{V}}_4⩽-\sum _{i=1}^n{k}_{4i}\beta {\zeta }_{4i}^2-\sum _{i=1}^n\beta {\mu }_{3i}^2{\zeta }_{3i}^2{h}_{4i}^2{w}_{4i}^2+\sum _{i=1}^n\beta {\phi }_{4i}.\end{array}$$

(38)

Thus, the entire design process for the tracking controller is described in detail.

3.3. Stability Analysis

Using the previous analysis, the result of this section can be presented in the following theorem.

Theorem 1.

For the n-link FJ manipulator system formulated by (1), based on the introduced nonlinear disturbance observer (5), the tracking control law u defined by (36) can achieve the following objectives:

(i) 

The states of the resulting closed-loop system are bounded and do not violate the constraints for all t.

(ii) 

For any given $\epsilon >0$, there exist proper design parameters such that the tracking error $z_{1i}\left(t\right)$ satisfies $\left|z_{1i}\left(t\right)\right|⩽\epsilon$ as $t\to \infty$, and the error signal $z_{1i}\left(t\right)$ can be made arbitrarily small.

Proof. 

Combining Formulas (20), (27), (33), and (38), we can have

$$\begin{array}{ccc}\hfill \dot{V}& =& {\dot{V}}_1+{\dot{V}}_2+{\dot{V}}_3+{\dot{V}}_4\hfill \\ & ⩽& -\sum _{j=1}^4\sum _{i=1}^n{k}_{ji}\beta {\zeta }_{ji}^2-\sum _{j=2}^4\sum _{i=1}^n\beta {\varpi }_{ji}^{*}{Y}_{ji}^4+\sum _{j=1}^4\sum _{i=1}^n\beta {\phi }_{ji}\hfill \\ & & +\sum _{j=2}^4\sum _{i=1}^n\frac{\beta {\xi }_{ji}^2}{2}+\sum _{i=1}^n\frac{\beta }{4r_{2i}}{∥M_i^{-1}∥}^2{e}^{T}e+\sum _{i=1}^n\frac{\beta }{2r_{2i}}{\rho }^T\rho .\hfill \end{array}$$

(39)

Then, the following compact sets are defined as ${\Omega }_V=\{{\displaystyle \sum _{j=1}^4}{\displaystyle \sum _{i=1}^n}{\zeta }_{ji}^2+{\displaystyle \sum _{j=2}^4}{\displaystyle \sum _{i=1}^n}{Y}_{ji}^2⩽2\delta \}\subset {\mathbb{R}}^{8n-1}$, ${\Omega }_x=\{{[x_{di},{\dot{x}}_{di},{\ddot{x}}_{di}]}^T:x_{di}^2+{\dot{x}}_{di}^2+{\ddot{x}}_{di}^2⩽\gamma \}\subset {\mathbb{R}}^{3n}$ with $\delta$ and $\gamma$ being the positive constants. In the compact set ${\Omega }_V\times {\Omega }_x$, there are positive constants ${\theta }_{ji}$ and ${\overline{h}}_3$ such that $\left|{\xi }_{ji}\right|⩽{\theta }_{ji}$ and $h_{3i}^2\le {\overline{h}}_3$.

Thus, by using Young’s inequality, we have $y_i^2\le y_i^4+\frac{1}{4}$. Now, (39) can be rewritten as

$$\begin{array}{ccc}\hfill \dot{V}& ⩽& -\sum _{j=1}^4\sum _{i=1}^n{k}_{ji}\beta {\zeta }_{ji}^2-\sum _{j=2}^4\sum _{i=1}^n\beta {\varpi }_{ji}^{*}{Y}_{ji}^2+\beta {\sigma }_1+\beta {\tau }_1\hfill \\ & ⩽& -\beta {\eta }_{1}V+\beta {\sigma }_1+\beta {\tau }_1\hfill \end{array}$$

(40)

where ${\varpi }_{3i}^{*}$ has been updated as ${\varpi }_{3i}^{*}-{\overline{h}}_3/\left(2r_{2i}\right)>0$, ${\eta }_1=min\left\{\left.2k_{ji},2{\varpi }_{ji}^{*}\right\}\right.$, ${\sigma }_1={\displaystyle \sum _{j=1}^4}{\displaystyle \sum _{i=1}^n}{\phi }_{ji}+\frac{1}{2}{\displaystyle \sum _{j=1}^4}{\displaystyle \sum _{i=1}^n}{\theta }_{ji}^2+\frac{1}{4}{\displaystyle \sum _{j=2}^4}{\displaystyle \sum _{i=1}^n}{\varpi }_{ji}^{*}$, ${\tau }_1={\displaystyle \sum _{i=1}^n}\frac{\beta }{4r_{2i}}{e}^{T}e$ $·{∥M_i^{-1}∥}^2$. By selecting $o=\frac{{\sigma }_1+{\tau }_1}{{\eta }_1}$, as a result, we can obtain that $V=o$ for $\dot{V}=0$. So, if $V\left(0\right)⩽o$, then $V\left(t\right)⩽o$ for $t⩾0$. Then, by multiplying (40) by $e^{{\eta }_1{\int }_0^t\beta \left(s\right)ds}$, one readily has

$$\begin{array}{c}\hfill \frac{d}{dt}\left(e^{{\eta }_1{\int }_0^t\beta \left(s\right)ds}V\left(t\right)\right)⩽e^{{\eta }_1{\int }_0^t\beta \left(s\right)ds}\beta ({\sigma }_1+{\tau }_1).\end{array}$$

(41)

Integrating (41) from 0 to t yields that

$$\begin{array}{cc}\hfill V\left(t\right)\le & e^{-{\eta }_1{\int }_0^t\beta \left(s\right)ds}V\left(0\right)+{\sigma }_1{e}^{-{\eta }_1{\int }_0^t\beta \left(s\right)ds}{\int }_0^t{e}^{{\eta }_1{\int }_0^q\beta \left(s\right)ds}\beta \left(q\right)dq\hfill \\ \hfill & +e^{-{\eta }_1{\int }_0^t\beta \left(s\right)ds}{\int }_0^t{e}^{{\eta }_1{\int }_0^q\beta \left(s\right)ds}{\tau }_1\left(q\right)\beta \left(q\right)dq.\hfill \end{array}$$

(42)

On the right-hand side of (42), we compute the second term to obtain

$$\begin{array}{ccc}\hfill {\sigma }_1{e}^{-{\eta }_1{\int }_0^t\beta \left(s\right)ds}{\int }_0^t{e}^{{\eta }_1{\int }_0^q\beta \left(s\right)ds}\beta \left(q\right)dq& =& {\sigma }_1{e}^{-{\eta }_1{\int }_0^t\beta \left(s\right)ds}{\int }_0^t{e}^{{\eta }_1{\int }_0^q\beta \left(s\right)ds}d({\int }_0^q\beta \left(s\right)ds)\hfill \\ & =& \frac{{\sigma }_1}{{\eta }_1}{e}^{-{\eta }_1{\int }_0^t\beta \left(s\right)ds}(e^{{\eta }_1{\int }_0^q\beta \left(s\right)ds}\left|\begin{array}{c}t\\ 0\end{array}\right.)⩽\frac{{\sigma }_1}{{\eta }_1}.\hfill \end{array}$$

(43)

Since $\underset{t\to \infty }{lim}{\tau }_1\left(t\right)=0$, it can be deduced that there is a constant $c>0$ such that $\left|{\tau }_1\left(t\right)\right|<\epsilon$ as $t>c$. For the third term on the right-hand side of (42), we have

$$\begin{array}{ccc}\hfill & & e^{-{\eta }_1{\int }_0^t\beta \left(s\right)ds}{\int }_0^t{e}^{{\eta }_1{\int }_0^q\beta \left(s\right)ds}{\tau }_1\left(q\right)\beta \left(q\right)dq\hfill \\ & =& e^{-{\eta }_1{\int }_0^t\beta \left(s\right)ds}{\int }_0^c{e}^{{\eta }_1{\int }_0^c\beta \left(s\right)ds}{\tau }_1\left(q\right)\beta \left(q\right)dq\hfill \\ & +& e^{-{\eta }_1{\int }_0^t\beta \left(s\right)ds}{\int }_c^t{e}^{{\eta }_1{\int }_c^q\beta \left(s\right)ds}{\tau }_1\left(q\right)\beta \left(q\right)dq.\hfill \end{array}$$

(44)

The term $e^{-{\eta }_1{\int }_0^t\beta \left(s\right)ds}{\int }_0^c{e}^{{\eta }_1{\int }_0^c\beta \left(s\right)ds}{\tau }_1\left(q\right)\beta \left(q\right)dq\to 0$ as $t\to \infty$, and the term $e^{-{\eta }_1{\int }_0^t\beta \left(s\right)ds}$${\int }_c^q{e}^{{\eta }_1{\int }_c^q\beta \left(s\right)ds}{\tau }_1\left(q\right)\beta \left(q\right)dq$ satisfies

$$\begin{array}{ccc}\hfill & & e^{-{\eta }_1{\int }_0^t\beta \left(s\right)ds}{\int }_c^t{e}^{{\eta }_1{\int }_c^q\beta \left(s\right)ds}{\tau }_1\left(q\right)\beta \left(q\right)dq\hfill \\ & ⩽& \epsilon e^{-{\eta }_1{\int }_0^t\beta \left(s\right)ds}{\int }_c^t{e}^{{\eta }_1{\int }_c^q\beta \left(s\right)ds}\beta \left(q\right)dq\hfill \\ & ⩽& \frac{\epsilon }{{\eta }_1}{e}^{-{\eta }_1{\int }_0^t\beta \left(s\right)ds}(e^{{\eta }_1{\int }_c^q\beta \left(s\right)ds}\left|\begin{array}{c}t\\ c\end{array}\right.)⩽\frac{\epsilon }{{\eta }_1}.\hfill \end{array}$$

(45)

Finally, combining (44) and (45) yields that

$$\begin{array}{ccc}\hfill V\left(t\right)& ⩽& e^{-{\eta }_1{\int }_0^t\beta \left(s\right)ds}V\left(0\right)+\frac{{\sigma }_1}{{\eta }_1}+\frac{\epsilon }{{\eta }_1}\hfill \\ & ⩽& V\left(0\right)+\frac{{\sigma }_1}{{\eta }_1}+\frac{\epsilon }{{\eta }_1}.\hfill \end{array}$$

(46)

For any initial condition $x_{ji}\left(0\right)$ satisfying the desired constraints, in view of (46), we have $V\in L_{\infty }$, so ${\zeta }_{ji}\in L_{\infty }$, $Y_{ji}\in L_{\infty }$. Considering (10) and (11), one can get $w_{ji}={\beta }^{-1}{\zeta }_{ji}\in L_{\infty }$, $y_{ji}={\beta }^{-1}{Y}_{ji}\in L_{\infty }$. Further, it is easy to acquire that according to (7) and (8), $x_{1i}$ is constrained since the desired trajectory $x_{di}$ is constrained. Nevertheless, as $w_{ji}\in L_{\infty }$, it follows that if $-F_{11}\left(0\right)<z_{1i}\left(0\right)<F_{12}\left(0\right)$, then $-F_{11}\left(t\right)<z_{1i}\left(t\right)<F_{12}\left(t\right)$. When the inequality $(F_{11}\left(t\right)+z_{1i}\left(t\right))(F_{12}\left(t\right)-z_{1i}\left(t\right))<{(F_{11}\left(t\right)+F_{12}\left(t\right))}^2$ is applied, one has $\left|z_{1i}\left(t\right)\right|⩽\frac{F}{H}\sqrt{2(V\left(0\right)+\frac{{\sigma }_1}{{\eta }_1}+\frac{\epsilon }{{\eta }_1})}$ as $t\to \infty$, where $F={max}_{t⩾0}{(F_{11}\left(t\right)+F_{12}\left(t\right))}^2$. Thus, $\forall \epsilon >0$, one chooses $H>\frac{F}{\epsilon }\sqrt{2(V\left(0\right)+\frac{{\sigma }_1}{{\eta }_1}+\frac{\epsilon }{{\eta }_1})}$ to ensure that the practical tracking is realized since the tracking error $z_{1i}\left(t\right)$ satisfies $\left|z_{1i}\left(t\right)\right|⩽\epsilon$ as $t\to \infty$.

Now, it is easy to conclude that ${\gamma }_{ji}\in L_{\infty }$, $s_{1i}\in L_{\infty }$, $s_{ji}\in L_{\infty }$ with $(j=2,3,4)$, which further suggests that ${\alpha }_{j-1,i}$, u are bounded. As a result, all signals of the resulting closed-loop system are bounded and remain within the time-varying constrained regions for all time. The proof is now complete. ☐

Remark 2.

Different from the BLF method, which is commonly used to solve the constraint problem such as [33], this paper adopts the SDF-based control method, which does not need to consider the influence of the feasibility conditions on the virtual controller, and simplifies the control design. Compared with the single-link FJ manipulator in [35,40], the studied system in this paper belongs to a class of complex MIMO systems, which means that the proposed control method has a wider application range. In addition, we considered the presence of the mismatched disturbance.

Remark 3.

In this paper, the control design of the DO-based tracking controller relies on the known system parameters M, C, G, B, and K, which is one of the limitations of the proposed approach. On the other hand, the backlash and flexibility are not considered temporarily in this paper. Therefore, our future works will focus on the compensation problem of the backlash and hysteresis for uncertain FJ manipulators such as [42].

4. Simulation Results

In this section, a two-link FJ manipulator system is used as an example to verify the feasibility of the proposed tracking controller based on the introduced NDO (5), whose dynamic equation is expressed as

$$\begin{array}{cc}\hfill M{\ddot{q}}_1+C(q_1,{\dot{q}}_1){\dot{q}}_1+G\left(q_1\right)+K(q_1-q_2)& =d\left(t\right),\\ \hfill J{\ddot{q}}_2+B{\dot{q}}_2-K(q_1-q_2)& =u\left(t\right)\end{array}$$

(47)

where $x_1={[x_{11},x_{12}]}^T=q_1={[q_{11},q_{12}]}^T$, $x_2={[x_{21},x_{22}]}^T={\dot{q}}_1={[{\dot{q}}_{11},{\dot{q}}_{12}]}^T$, $x_3={[x_{31},x_{32}]}^T=q_2={[q_{21},q_{22}]}^T$, $x_4={[x_{41},x_{42}]}^T={\dot{q}}_2={[{\dot{q}}_{21},{\dot{q}}_{22}]}^T$, $B=diag[B_1,B_2]$, $J=diag[J_1,J_2]$, and $K=diag[K_1,K_2]$. Combining system (2), we have $d={[d_1,d_2]}^T$, $u={[u_1,u_2]}^T$. The detailed expressions of M, C and G can be founded in [43]. It should be noted that M, C, and G can be calculated as long as some physical parameters of the FJ manipulator are known. In simulation, the parameters of the two-link FJ manipulator system are given in Table 1.

Table 1. Parameters of the two-link FJ manipulator system.

Variables/Parameters	Meaning	Value	Unit
$m_1$	mass of rod 1	1	kg
$m_2$	mass of rod 2	1	kg
$l_1$	length of rod 1	1	m
$l_2$	length of rod 2	1	m
$J_1$	joint flexibility of rod 1	1	$\mathrm{m}/{\mathrm{s}}^2$
$J_2$	joint flexibility of rod 2	1	$\mathrm{m}/{\mathrm{s}}^2$
$B_1$	damping factor of rod 1	0.1	none
$B_2$	damping factor of rod 2	0.1	none
$K_1$	joint stiffness of rod 1	100	$\mathrm{N}·\mathrm{m}·/\mathrm{rad}$
$K_2$	joint stiffness of rod 2	100	$\mathrm{N}·\mathrm{m}·/\mathrm{rad}$
g	gravitational acceleration	9.8	$\mathrm{m}/{\mathrm{s}}^2$

The external disturbances are selected as $d_1\left(t\right)=\frac{1}{1+t}+1,d_2\left(t\right)=\frac{1}{1+t}+1$. The reference trajectories are defined as $x_{d1}=0.5sint,x_{d2}=0.5sint$. Now, the control objectives can be described as (1) output variables $x_{11},x_{12}$ can track $x_{d1}=0.5sint,x_{d2}=0.5sint$, and (2) the following full-state constraints are realized: $x_{d1}\left(t\right)-F_{11}\left(t\right)<x_{11}\left(t\right)<x_{d1}\left(t\right)+F_{12}\left(t\right)$, $x_{d2}\left(t\right)-F_{11}\left(t\right)<x_{12}\left(t\right)<x_{d2}\left(t\right)+F_{12}\left(t\right)$, $-F_{21}\left(t\right)<x_{21}\left(t\right)<F_{22}\left(t\right)$, $-F_{21}\left(t\right)<x_{22}\left(t\right)<F_{22}\left(t\right)$, $-F_{31}\left(t\right)<{\dot{x}}_{11}\left(t\right)<F_{32}\left(t\right)$, $-F_{31}\left(t\right)<{\dot{x}}_{12}\left(t\right)<F_{32}\left(t\right)$, $-F_{41}\left(t\right)<{\dot{x}}_{21}\left(t\right)<F_{42}\left(t\right)$, $-F_{41}\left(t\right)<{\dot{x}}_{22}\left(t\right)<F_{42}\left(t\right)$, where $F_{11}\left(t\right)=F_{12}\left(t\right)=F_{21}\left(t\right)=F_{22}\left(t\right)=F_{31}\left(t\right)=F_{32}\left(t\right)=F_{41}\left(t\right)=F_{42}\left(t\right)=5$. The initial conditions are set as $x_1\left(0\right)={[0,0]}^T$, $x_2\left(0\right)={[0,0]}^T$, $x_3\left(0\right)={[0,0]}^T$, $x_4\left(0\right)={[0,0]}^T$, ${\gamma }_2\left(0\right)=\left[{\gamma }_{21}\left(0\right),{\gamma }_{22}\left(0\right)\right]$, ${\gamma }_3\left(0\right)=\left[{\gamma }_{31}\left(0\right),{\gamma }_{32}\left(0\right)\right]$, ${\gamma }_4\left(0\right)=\left[{\gamma }_{41}\left(0\right),{\gamma }_{42}\left(0\right)\right]$. The design parameters are chosen as $k_{11}=k_{12}=2.5$, $k_{21}=k_{22}=4.5$, $k_{31}=0.1$, $k_{32}=0.15$, $k_{41}=5$, $k_{42}=25$, $r_{11}=r_{12}=r_{21}=r_{22}=r_{31}=r_{32}=r_{41}=r_{42}=l_{11}=l_{12}=l_{21}=l_{22}=l_{31}=l_{32}=l_{41}=l_{42}=0.001$, ${\omega }_{21}={\omega }_{22}=0.001$, ${\omega }_{31}={\omega }_{32}=0.0015$, ${\omega }_{41}={\omega }_{42}=0.0001$, $\lambda =0.8$.

The simulation results are shown in Figure 2, Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7. From Figure 2 and Figure 3, it can be seen that the outputs $x_{11}$ and $x_{12}$ of the two-link FJ manipulator can effectively track the reference trajectories $x_{d1}$ and $x_{d2}$, and all system states are within the constraint boundaries. As depicted in Figure 4 and Figure 5, the virtual control signals and outputs of the first-order filter are bounded, which also indicate the reliability of the filter’s performance. Figure 6 shows that the disturbance estimation $\widehat{d}\left(t\right)$ generated by the NDO can converge asymptotically to the true disturbances $d\left(t\right)$, which emphasizes the effectiveness of NDO. Figure 7 illustrates that the control inputs are also bounded. These results clearly demonstrate that the suggested approach effectively achieves the desirable control performance, successfully meeting the specified control objectives.

Figure 2. The responses of states $x_{11}$, $x_{12}$, $x_{21}$, and $x_{22}$.

Figure 3. The responses of states $x_{31}$, $x_{32}$, $x_{41}$, and $x_{42}$.

Figure 4. The responses of virtual control signals.

Figure 5. The responses of first-order filters’ states.

Figure 6. Disturbance estimation results.

Figure 7. The responses of control inputs.

5. Conclusions

This paper focuses on addressing the issue of tracking control for n-link flexible-joint robot systems, considering both external disturbances and full-state constraints. To tackle this problem, we employ a transformation technique to convert the original dynamic system into a chained system. Additionally, a nonlinear disturbance observer is introduced to mitigate the impact of disturbances. Subsequently, the constrained system is transformed into an unconstrained system through SDF transformation, enabling the design of a tracking controller using the backstepping and filter technique. By appropriately selecting design parameters, we can ensure the stability of the resulting closed-loop system, adherence to the desired constraints, and a significant reduction in tracking errors. The proposed method is validated through simulations performed on a two-link FJ manipulator system. Our future works will study the anti-disturbance control for other underactuated systems, such as underactuated crane systems [44,45,46].