Adaptive Fuzzy Command Filtered Tracking Control for Flexible Robotic Arm with Input Dead-Zone

Abstract

In this paper, an adaptive fuzzy tracking control method is proposed to address the issues of dead-zone and unobservable states in a flexible robotic arm system. The control design process begins with the utilization of a fuzzy logic system to approximate the nonlinear functions present in the flexible robotic arm system. To estimate the unobservable states of the system, a state observer is then designed. To alleviate the computational complexity during controller design, the command filtering technique is introduced. Additionally, the Nussbaum function is incorporated to address the unknown control gain problem. The stability of the system can be verified through the design of a Lyapunov function. This study’s simulation results demonstrate that the designed control system can closely track the specified reference signals. The closed-loop system effectively controls the flexible robotic arm, as verified through experimentation.

1. Introduction

Flexible robotic arms are gaining prominence in various fields, such as the manufacturing industry, the medical sector, and food processing. As a pivotal branch of robotics technology, they offer exceptional control precision, response speed, and load capacity [1,2]. In the pursuit of enhanced flexibility, precise adaptability, and heightened safety, flexible robotic arms are emerging as substitutes for rigid counterparts. Their distinctive design and material characteristics reveal significant potential, making them a preferred choice. They not only inherit the advantages of traditional robotic arms, such as outstanding control performance and high load-bearing capacity, but also excel in terms of their adaptability and multifunctionality. Their pliable structures and elastic properties enable them to effortlessly handle various workpieces with complex shapes, even within confined and hazardous environments. This flexibility allows them to overcome the limitations of traditional robotic arms in these scenarios [3,4]. However, it is important to note that flexible robotic arms also face certain challenges. Their flexible structures and complex motion characteristics introduce complexities such as nonlinearity, strong coupling, and temporal variations. Additionally, uncertainties in parameters, load disturbances, input saturation, and other issues significantly impact the accuracy and stability of the system. Addressing these factors and achieving higher levels of accuracy in control effectiveness remain pressing challenges that need to be resolved [5,6].

In this context, this paper aims to investigate the control challenges faced by flexible robotic arms when encountering these issues. In this paper, we will introduce an innovative control approach that combines techniques such as adaptive control, fuzzy control, and observer design to achieve trajectory tracking control for flexible robotic arms. Additionally, we will also discuss how to address common dead-zone input problems in flexible robotic arm systems and how to handle systems with unknown parameters. Through this research study, we aim to provide valuable insights for addressing the complex challenges in the control of flexible robotic arms and inspire future research directions.

In recent years, the control issues related to flexible robotic arms have garnered significant attention from both domestic scholars and scholars from elsewhere. To date, numerous effective control methods have been developed, including adaptive control [7,8], robust control [9,10], and sliding mode control [11], among others. For strictly feedback-form nonlinear systems, the backstepping design method is considered one of the most effective control approaches [12]. However, backstepping can only be applied to control systems with precisely known dynamics. In practical engineering, uncertainties and nonlinearities are commonly present, making it challenging to obtain precise mathematical models for these systems. Therefore, methods like adaptive fuzzy control [13] and neural network control [14], which do not rely on exact mathematical models, are better suited for designing controllers for unknown nonlinear systems. Combining these methods with backstepping has resulted in numerous research achievements. Furthermore, with the advancements in adaptive fuzzy control and backstepping design techniques, the backstepping method [15] has been extensively used to address control problems in uncertain nonlinear feedback systems. While the proposed adaptive control methods ensure control system stability and favorable control performance, traditional backstepping approaches face issues of computational explosion during controller construction due to the repeated differentiation of virtual control variables [16,17]. To address these problems, researchers have proposed dynamic surface control techniques [18,19,20]. These techniques introduce first-order filters in the process of backstepping recursion to handle computational complexity. However, the use of these filters leads to filtering errors, affecting control precision [21]. In order to mitigate these errors, researchers have suggested the adoption of command filtering techniques and the introduction of compensation signals. The authors of [21,22] successfully utilized command filtering techniques to address computational explosion problems and eliminate filtering errors, resulting in improved control precision.

In the context of addressing robotic arm control challenges, various innovative approaches have been proposed. For example, Yao et al. [23] introduced a novel adaptive fuzzy neural network control method to address the trajectory tracking control of robotic arms with output constraints and nonlinear inputs. Meanwhile, Wang et al. [24] explored an adaptive fuzzy sliding mode robust control algorithm to address trajectory tracking control for robotic arms in the presence of uncertain parameters and external disturbances. On a different note, Han et al. [25] presented an adaptive sliding mode control method to resolve robotic arm tracking control issues involving unknown external disturbances. Li et al. [26] introduced an adaptive fault-tolerant control strategy to handle actuator failures and uncertain boundary disturbances, achieving the control of robotic arms. Liu et al. [27] investigated a fault-tolerant control design to effectively control robotic arms in the presence of actuator failures, input saturation, and external disturbances. Zhao et al. [28] proposed an adaptive sliding mode control approach to address the issue of oscillations in control torque and verified the effectiveness of this control method through simulations. However, these methods require the measurement of velocity and angular displacement within the robotic arm system. Yet, in practical engineering scenarios, considering factors like the cost and size of the robotic arm, velocity sensors are often omitted, resulting in certain states being unmeasurable. When aiming to achieve more precise robotic arm control, it is necessary to obtain various states of the system during its operation. Therefore, designing an effective observer is of significant importance for the state estimation of a robotic arm with unobservable states.

Furthermore, the various random disturbances present during the operation of robotic arm systems also constitute a significant influencing factor on system stability [29]. Researchers have conducted in-depth studies on issues such as actuator saturation, friction constraints, and external disturbances [30,31,32]. It is worth noting that dead-zone inputs, as a type of nonlinear non-smooth problem [33,34,35], frequently arise in industrial processes.

However, the effects stemming from dead-zones are often overlooked for the convenience of controller design. If the dead-zone issue is not addressed, it can lead to discontinuities in the system’s response, resulting in instability and performance degradation near the dead-zone region. In practical operations, the inadequate resolution of the dead-zone problem can prevent the system from achieving expected performance and may even induce instability and accidents. Therefore, it is important to consider dead-zone issues in the design of control systems. However, there is limited research on flexible robotic arm systems with dead-zone inputs, and there are scarce outcomes concerning adaptive fuzzy command filter control for such systems. This motivated us to conduct this study. Additionally, we also conducted research on systems with unobservable states based on the above discussions.

This paper proposes an adaptive fuzzy state feedback control method to achieve trajectory tracking in flexible robotic arm systems, taking into account the presence of dead-zones and unobservable states. The method utilizes a fuzzy logic system to approximate nonlinear functions and a state observer to estimate unobservable states in nonlinear systems. To address computational explosion issues, filtering techniques are integrated into the controller design, and compensation signals are introduced to mitigate filtering errors. This paper concludes by demonstrating that the adaptive fuzzy control approach ensures the boundedness of all variables in the closed-loop system, enabling accurate tracking of reference signals. The effectiveness of the system can be verified through simulations, as we will show in this paper.

The remainder of this paper is structured as follows: Section 2 presents the preparatory work. Section 3 introduces the design of the state observer. Section 4 discusses the controller design and stability analysis. Section 5 provides the simulation and experimental results regarding the robotic arm. Section 6 presents our conclusions.

2. Preliminaries

2.1. System

The flexible robotic arm is a non-rigid single-link system with a DC motor, as shown in Figure 1. The system’s dynamic equations can be derived using the Euler–Lagrange equations, yielding the mechanical arm system’s equations of motion.

$$\left\{\begin{array}{l}{J}_1{\ddot{q}}_1+F_1{\dot{q}}_1+K(q_1-\frac{q_2}{N})+mgd\mathit{cos}{q}_1=0\\ J_2{\ddot{q}}_2+F_2{\dot{q}}_2-\frac{K}{N}(q_1-\frac{q_2}{N})=K_{t}i\\ LDi+Ri+K_b{\dot{q}}_2=u\end{array}\right.,$$

(1)

where $J_1$ and $J_2$ are the inertias, $F_1$ and $F_2$ are the viscous friction constants, $q_1$ is the angular positions of the link, $q_2$ is the motor shaft, $m$ is the link mass, $g$ is the acceleration of gravity, $d$ is the position of the link’s center of gravity, $K$ is the spring constant, $N$ is the gear ratio, $K_t$ is the torque constant, $R$ and $L$ are the armature resistance and inductance, $i$ is the armature current, $K_b$ is the back-emf constant, $u$ is the armature voltage.

If only $q_1$ is measurable and $K_{t}K=J_1{J}_{2}NL$, then the state variables can be defined as ${\xi }_1=q_1$, ${\xi }_2={\dot{q}}_1$, ${\xi }_3=q_2$, ${\xi }_4={\dot{q}}_2$, ${\xi }_5=i$. The mathematical model of the system (1) is as follows:

$$\left\{\begin{array}{l}{\dot{\xi }}_1={\xi }_2\\ {\dot{\xi }}_2=-\frac{mgd}{J_1}\mathit{cos}y-\frac{F_1}{J_1}{\xi }_2-\frac{K}{J_1}({\xi }_1-\frac{{\xi }_3}{N})\\ {\dot{\xi }}_3={\xi }_4\\ {\dot{\xi }}_4=\frac{K}{J_{2}N}({\xi }_1-\frac{{\xi }_3}{N})-\frac{F_2}{J_2}{\xi }_4-\frac{K_t}{J_2}{\xi }_5\\ {\dot{\xi }}_5=-\frac{R}{L}{\xi }_5-\frac{K_b}{L}{\xi }_4-\frac{1}{L}u\\ y={\xi }_1\end{array}\right.,$$

(2)

The pure feedback form of the system (2) is as follows:

$$\left\{\begin{array}{l}{\dot{\xi }}_1={\xi }_2\\ {\dot{\xi }}_2=f_2({\xi }_1,{\xi }_2,{\xi }_3)+{\xi }_3\\ {\dot{\xi }}_3={\xi }_4\\ {\dot{\xi }}_4=f_4({\xi }_1,{\xi }_2,{\xi }_3,{\xi }_4,{\xi }_5)+{\xi }_5\\ {\dot{\xi }}_5=f_5({\xi }_1,{\xi }_2,{\xi }_3,{\xi }_4,{\xi }_5,u)+u\\ y={\xi }_1\end{array}\right.,$$

(3)

where $f_2\left({\xi }_1,{\xi }_2,{\xi }_3\right)=-\frac{mgd}{J_1}\mathit{cos}y-\frac{F_1}{J_1}{\xi }_2-\frac{K}{J_1}\left({\xi }_1-\frac{{\xi }_3}{N}\right)-{\xi }_3$, $f_4\left({\xi }_1,{\xi }_2,{\xi }_3,{\xi }_4,{\xi }_5\right)=\frac{K}{J_{2}N}\left({\xi }_1-\frac{{\xi }_3}{N}\right)-\frac{F_2}{J_2}{\xi }_4-\frac{K_t}{J_2}{\xi }_5-{\xi }_5$, $f_5\left({\xi }_1,{\xi }_2,{\xi }_3,{\xi }_4,{\xi }_5,u\right)=-\frac{R}{L}{\xi }_5-\frac{K_b}{L}{\xi }_4-\frac{1}{L}u-u$.

According to the authors of [27,28], we can express the dead-zone in the following form:

$$h_i\left(t\right)=\left\{\begin{array}{l}{-m}_i{b}_i\hspace{1em} \hspace{1em}{v}_i>b_i\\ -{m_{i}v}_i -b_i\le v_i\le b_i\\ m_i{b}_i\hspace{1em}\hspace{1em}\hspace{1em} v_i<b_i\end{array}\right.,$$

(4)

From (4), we have $\left|h_i(t)\right|\le {h_i}^{\ast }$.

In order to cope with the unknown control gain sign ${\beta }_j(j=1,\cdots ,m)$, we introduce the Nussbaum gain function; the Nussbaum gain function, $N\left(\zeta \right)$, has the following properties:

$$\underset{s\to \mathrm{\infty }}{lim}\mathit{sup}\frac{1}{s}{\int }_0^{s}N\left(\zeta \right)d\zeta =\mathrm{\infty },$$

(5)

$$\underset{s\to \mathrm{\infty }}{lim}\mathit{inf}\frac{1}{s}{\int }_0^{s}N\left(\zeta \right)d\zeta =-\mathrm{\infty },$$

(6)

For this paper, the Nussbaum function was chosen as $N\left(\zeta \right)=\mathit{exp}({\zeta }^2)\mathit{cos}(\frac{\pi }{2}\zeta )$. The following lemma regarding the property of Nussbaum function gain is used in the controller design.

Lemma 1 ([36,37,38]).

Consider a special Nussbaum function $N\left(\zeta \right)=\mathit{exp}({\zeta }^2)\mathit{cos}(\frac{\pi }{2}\zeta )$ and let $V\left(t\right)$ and ${\zeta }_j\left(t\right)$ be smooth functions defined on $\left[0,t_f\right]$ with $V\left(t\right)\ge 0\forall t\in \left[0,t_f\right]$. If the following inequality holds, then $V\left(\cdot \right),{\zeta }_j\left(\cdot \right)$ and ${\sum }_{j=1}^m{\int }_0^t{d}_j({\beta }_j{N}^{\prime }\left({\zeta }_j\right)+1){\dot{\zeta }}_j{e}^{\alpha \tau }d\tau$ must be bounded on $\left[0,t_f\right]$.

$$0\le V\left(t\right)\le V\left(0\right)+e^{-\alpha t}{\sum }_{j=1}^m{\int }_0^t{d}_j({\beta }_j{N}^{\prime }\left({\zeta }_j\right)+1){\dot{\zeta }}_j{e}^{\alpha \tau }d\tau +\lambda ,$$

(7)

where $\alpha >0$ and $\lambda >0$ are constants, ${\beta }_j$ is a nonzero constant, and  $d_j$ is a suitable constant.

Lemma 2 ([39]).

Define command filtering:

$${\dot{\kappa }}_1={\omega }_n{\kappa }_2,$$

(8)

$${\dot{\kappa }}_2=-2\varsigma {\omega }_n{\kappa }_2-{\omega }_n{\kappa }_1-{\alpha }_1,$$

(9)

If input ${\alpha }_1$ holds for all $t\ge 0$, satisfying $\left|{\dot{\alpha }}_1\right|\le p_1$ and $\left|{\ddot{\alpha }}_1\right|\le p_2$, and $p_1>0$ and $p_2>0$ are constants, ${\kappa }_1\left(0\right)={\alpha }_1\left(0\right),{\kappa }_2\left(0\right)=0$. For any $\delta >0$, there exists ${\omega }_n>0$ in $\varsigma \in \left(\mathrm{0,1}\right)$ so that $\left|{\kappa }_1-{\alpha }_1\right|\le \delta ,{\dot{\kappa }}_1,{\ddot{\kappa }}_1,{\stackrel{⃛}{\kappa }}_1$ is bounded.

The purpose of this paper is to design an adaptive fuzzy controller to enable the system’s output angular position $y$ to track the reference signal $y_r$ as closely as possible. Additionally, in the flexible robotic arm system, all signals are bounded.

2.2. Fuzzy Logic System

A fuzzy logic system (FLS) [40] consists of four parts: the knowledge base, the fuzzifier, the fuzzy inference engine working on fuzzy rules, and the defuzzifier. The knowledge base for the FLS comprises a collection of fuzzy if–then rules of the following form:

$R^l$: If $x_1$ is $F_1^l$ and $x_2$ is $F_2^l$ and … and $x_n$ is $F_n^l$,

then, $y$ is $G^l$, $l=\mathrm{1,2},\dots ,N$.

Here, $x=x_1,\dots ,{x_n}^T$ and $y$ are the input and output of the fuzzy logic system. $F_i^l$ and $G^l$ are fuzzy sets. Associated membership functions ${\mu }_{F_i^l}\left(x_i\right)$ and ${\mu }_{G^l}\left(y\right)$ are defined. $N$ represents the number of rules. The fuzzy logic system can be expressed as follows:

$$y\left(x\right)=\frac{{\sum }_{l=1}^N\overline{y_l}{\prod }_{i=1}^n{\mu }_{F_i^l}\left(x_i\right)}{{\sum }_{l=1}^N\left[{\prod }_{i=1}^n{\mu }_{F_i^l}\left(x_i\right)\right]},$$

(10)

where $\overline{y_l}=\underset{y\in R}{max}{\mu }_{G^l}\left(y\right)$.

Define the fuzzy basis function:

$${\varphi }_l=\frac{{\prod }_{i=1}^n{\mu }_{F_i^l}\left(x_i\right)}{{\sum }_{l=1}^N({\prod }_{i=1}^n{\mu }_{F_i^l}\left(x_i\right))},$$

(11)

where $\theta {=[{\overline{y}}_1,{\overline{y}}_2,\dots ,{\overline{y}}_N]}^T{=[{\theta }_1,{\theta }_2,\dots ,{\theta }_N]}^T$ and ${\varphi }^T\left(x\right)=\left[{\varphi }_1\left(x\right),\cdots ,{\varphi }_N\left(x\right)\right]$.

The general form of a fuzzy logic system is as follows:

$$y\left(x\right)={\theta }^T\varphi \left(x\right),$$

(12)

Lemma 3 ([41]).

The function $f\left(x\right)$ is smooth on the compact set $\Omega \subseteq R^N$. For a given constant $\epsilon >0$, there exists $y\left(x\right)={\theta }^T\varphi \left(x\right)$ so that:

$$\underset{x\in \Omega }{sup}\left|f\left(x\right)-{\theta }^T\varphi \left(x\right)\right|\le \epsilon ,$$

(13)

According to [42,43], we define:

$$\theta =arg\underset{\theta \in R^N}{min}\left\{\underset{x\in \Omega }{sup}\left|f\left(x\right)-{\theta }^T\varphi \left(x\right)\right|\right\},$$

(14)

Then,

$$f\left(x\right)={\widehat{\theta }}^T\varphi \left(x\right)+\epsilon ,$$

(15)

where $\left|\epsilon \right|\le {\epsilon }^{\ast }$, ${\epsilon }^{\ast }$ is an unknown constant.

Lemma 4 ([44]).

The Young’s inequality is defined as follows: for any variable $x,y\in R$, the following inequality must hold:

$$\mathrm{x}\mathrm{y}\le \frac{o^p}{p}{\left|x\right|}^p+\frac{1}{{qo}^q}{\left|y\right|}^q,$$

(16)

where $o>0$, $p>1$, $q>1$, $\left(p-1\right)\left(q-1\right)=1$.

Before proceeding with the fuzzy state observer design, we first express the dead-zone [36] as a slowing time-varying input dependent function with a bounded disturbance:

$$u=mv+h,$$

(17)

The dead zone function is shown in Figure 2, where $b_i>0$, $b_j<0$. $m_i(v)$ is the unknown smooth function.

According to Equations (4) and (17), we can express the dead zone as follows:

$$u=m(v-sat\left(t\right)),$$

(18)

$$u=\left\{\begin{array}{c}{m}_i\left(v_i-b_i\right) v_i>b_i\\ 0 -b_i\le v_i\le b_i\\ m_i\left({v_i+b}_i\right) v_i<b_i\end{array}\right.,$$

(19)

where the asymmetric saturation function is defined as follows:

$$sat\left(t\right)=\left\{\begin{array}{c}{b}_i v_i>b_i\\ v_i -b_i\le v_i\le b_i\\ b_i v_i<b_i\end{array}\right.,$$

(20)

3. Fuzzy Adaptive Observer

The state ${\xi }_i(i=\mathrm{2,3},\mathrm{4,5})$ of the flexible robotic arm system are unobservable. Therefore, it is necessary to establish a suitable state observer to monitor the system’s state. The primary purpose of the observer is to provide a real-time estimate $\widehat{\xi }(t)$ for the state $\xi (t)$ in the aforementioned model (3).

To start with, a fuzzy logic system is employed to approximate the unknown nonlinear functions within the robotic arm system. The nonlinear functions are represented as follows: $-\frac{mgd}{J_1}\mathit{cos}y-\frac{F_1}{J_1}{\xi }_2-\frac{K}{J_1}\left({\xi }_1-\frac{{\xi }_3}{N}\right)-{\xi }_3$, $\frac{K}{J_{2}N}\left({\xi }_1-\frac{{\xi }_3}{N}\right)-\frac{F_2}{J_2}{\xi }_4-\frac{K_t}{J_2}{\xi }_5-{\xi }_5$, $-\frac{R}{L}{\xi }_5-\frac{K_b}{L}{\xi }_4-\frac{1}{L}u-u$.

The model of the fuzzy adaptive observer is depicted as follows:

$$\left\{\begin{array}{l}{\stackrel{.}{\widehat{\xi }}}_1={\widehat{\xi }}_2+k_1(y-{\widehat{\xi }}_1)\\ {\stackrel{.}{\widehat{\xi }}}_2={\theta }_2^T{\varphi }_2({\xi }_1,{\widehat{\xi }}_2,{\widehat{\xi }}_{3,f})+{\widehat{\xi }}_3+k_2(y-{\widehat{\xi }}_1)\\ {\stackrel{.}{\widehat{\xi }}}_3={\widehat{\xi }}_4+k_3(y-{\widehat{\xi }}_1)\\ {\stackrel{.}{\widehat{\xi }}}_4={\theta }_4^T{\varphi }_4({\xi }_1,{\widehat{\xi }}_2,{\widehat{\xi }}_3,{\xi }_4,{\xi }_{5,f})+{\widehat{\xi }}_5+k_4(y-{\widehat{\xi }}_1)\\ {\stackrel{.}{\widehat{\xi }}}_5={\theta }_5^T{\varphi }_5({\xi }_1,{\widehat{\xi }}_2,{\widehat{\xi }}_3,{\xi }_4,{\xi }_5,u_f)+u+k_5(y-{\widehat{\xi }}_1)\\ y={\xi }_1\end{array}\right.,$$

(21)

Formula (21) can be rewritten as follows:

$$\left\{\begin{array}{l}\stackrel{.}{\widehat{\xi }}=A\widehat{\xi }+Ky+\overline{F}+C_{1}u\\ y=C_2\widehat{\xi }\end{array}\right.,$$

(22)

where $\overline{F}=[0,{\theta }_2^T{\varphi }_2\left({\xi }_1,{\widehat{\xi }}_2,{\widehat{\xi }}_{3,f}\right), 0, {\theta }_4^T{\varphi }_4\left({\xi }_1,{\widehat{\xi }}_2,{\widehat{\xi }}_3,{\xi }_4,{\xi }_{5,f}\right), {\theta }_5^T{\varphi }_5\left({\xi }_1,{\widehat{\xi }}_2,{\widehat{\xi }}_3,{\xi }_4,{\xi }_5,u_f\right){]}^T$, $\widehat{\xi }=[{\widehat{\xi }}_1,{\widehat{\xi }}_2\cdots ,{\widehat{\xi }}_5{]}^T$, $A=\left[\begin{array}{c}-k_1\\ ⋮\\ -k_5\end{array}\begin{array}{c}\\ I_4\\ \cdots \end{array}\begin{array}{c}\\ \\ 0\end{array}\right]$, $K=[k_1,\cdots ,k_5{]}^T$, $C_1^T=\left[\mathrm{0,0},\mathrm{0,0},1\right]$, $C_2=\left[\mathrm{1,0},\mathrm{0,0},0\right]$.

By selecting appropriate parameters, $k_i(i=1,\cdots ,5)$, ensure that the polynomial $p\left(s\right)=s^5+k_1{s}^4+\cdots +k_5$ satisfies the Hurwitz matrix condition. In other words, given a matrix of $Q^T=Q>0$, there exists a positive definite symmetric matrix, $P^T=P>0,$ meaning that

$$A^{T}P+PA=-Q,$$

(23)

Define the observer error as $e=\xi -\widehat{\xi }=[e_1,\cdots ,e_5{]}^T$; then,

$$\dot{e}=Ae+\epsilon +\tilde{\theta },$$

(24)

where $\epsilon =[0,{\epsilon }_2,0,{\epsilon }_4,{\epsilon }_5{]}^T$, ${\tilde{\theta }}_i={\widehat{\theta }}_i-{\theta }_i$, $i=2,4,5$.

Choose the Lyapunov function $V_0$ as

$$V_0=e^{T}Pe,$$

(25)

$${\dot{V}}_0=-e^{T}Qe+2e^{T}P\left(\epsilon +\tilde{\theta }\right),$$

(26)

By using the Young’s inequality, we can obtain the following:

$${\dot{V}}_0\le -e^{T}Qe+3e^2+P^2\left({\epsilon }_2^{{\ast }^2}+{\epsilon }_4^{{\ast }^2}+{\epsilon }_5^{{\ast }^2}\right)+P^2\left({\tilde{\theta }}_2^T{\tilde{\theta }}_2+{\tilde{\theta }}_4^T{\tilde{\theta }}_4+{\tilde{\theta }}_5^T{\tilde{\theta }}_5\right),$$

(27)

4. Design of Fuzzy Adaptive Controller and Stability Analysis

A block diagram of the control system, implementing tracking control of the flexible robotic arm using adaptive fuzzy control and employing an observer to estimate unobservable states, is shown in Figure 3.

Define the error variables $z_i(i=1,\cdots ,5)$, where $y_r$ is the system’s reference signal. To address the issue of computational explosion in backstepping control, command filtering is introduced at this stage.${\overline{\alpha }}_i(i=1,\cdots ,4)$ represents the filtered controller signals;${\alpha }_i(i=1,\cdots ,4)$ represents the virtual control functions. ${\xi }_i$ is the system’s state value (obtained as ${\widehat{\xi }}_i$ through a state observer), which serves as the state estimate.

$$\left\{\begin{array}{l}{z}_1=y-y_r\\ z_2={\widehat{\xi }}_2-{\overline{\alpha }}_1\\ z_3={\widehat{\xi }}_3-{\overline{\alpha }}_2\\ z_4={\widehat{\xi }}_4-{\overline{\alpha }}_3\\ z_5={\widehat{\xi }}_4-{\overline{\alpha }}_4\end{array}\right.,$$

(28)

The defined compensation error signal is $r_i=z_i-{\omega }_i(i=\mathrm{1,2},\cdots ,5)$.

The command filter is defined as:

$${\dot{\kappa }}_1=\omega {\kappa }_2,$$

(29)

$${\dot{\kappa }}_2=-2\varsigma \omega {\kappa }_2-\omega ({\kappa }_1-{\alpha }_i),$$

(30)

where $\omega >0$ and $\varsigma \in (\mathrm{0,1}]$ are the designed parameters, ${\overline{\alpha }}_i(t)={\kappa }_1(t)$ is the output of each filtering signal, and ${\alpha }_{i-1}$ is the input to the command filter. The initial condition are ${\kappa }_1(0)={\alpha }_i(0)$ and ${\kappa }_2(0)=0$.

Step 1: The derivative of $r_1$ is as follows:

$$\begin{array}{cc}{\dot{r}}_1& ={\dot{z}}_1-{\dot{\omega }}_1\hfill \\ & ={\widehat{\xi }}_2+k_1\left(y-{\widehat{\xi }}_1\right)-{\dot{y}}_r-{\dot{\omega }}_1\end{array},$$

(31)

Choose the Lyapunov function as follows:

$$V_1=V_0+\frac{1}{2}{r}_1^2,$$

(32)

The derivative of $V_1$ is as follows:

$$\begin{array}{cc}{\dot{V}}_1& ={\dot{V}}_0+r_1{\dot{r}}_1\hfill \\ & ={\dot{V}}_0+r_1\left(r_2+{\omega }_2+{\epsilon }_1+{\overline{\alpha }}_1-{\dot{y}}_r-{\dot{\omega }}_1\right)\end{array},$$

(33)

where ${\epsilon }_i=k_i\left(y-{\widehat{\xi }}_1\right)\left(i=1\cdots 5\right);{{\epsilon }_i}^{\ast }$ is an unknown constant.

As a result of the Young’s inequality,

$${\epsilon }_1{r}_1\le \frac{1}{2}{r}_1^2+\frac{1}{2}{\epsilon }_1^{{\ast }^2},$$

(34)

Through substituting Equation (31) into Equation (30), we can obtain the following:

$${\dot{V}}_1\le {\dot{V}}_0+r_1\left(r_2+{\omega }_2+\frac{1}{2}{r}_1+{\overline{\alpha }}_1-{\dot{y}}_r-{\dot{\omega }}_1\right)+\frac{1}{2}{\epsilon }_1^{{\ast }^2},$$

(35)

Choose the virtual control function ${\alpha }_1$ as:

$${\alpha }_1=-c_1{z}_1+{\dot{y}}_r-\frac{1}{2}{r}_1,$$

(36)

The compensatory signal ${\dot{\omega }}_1$ is given by the following:

$${\dot{\omega }}_1=-c_1{\omega }_1+{\omega }_2+\left({\overline{\alpha }}_1-{\alpha }_1\right),$$

(37)

where $c_1>0$ is a designed parameter.

Through substituting Equations (32) and (33) into Equation (31), we can obtain the following:

$${\dot{V}}_1\le {\dot{V}}_0-c_1{r}_1^2+\frac{1}{2}{\epsilon }_1^{{\ast }^2}+r_1{r}_2,$$

(38)

Step 2: Choose the virtual control function ${\alpha }_2$, the adaptive rate ${\dot{\theta }}_2$, and the compensatory signal ${\dot{\omega }}_2$ as follows:

$${\alpha }_2=-c_2{z}_2-{\theta }_2^T{\varphi }_2\left({\xi }_1,{\widehat{\xi }}_2,{\widehat{\xi }}_{3,f}\right)+{\dot{\overline{\alpha }}}_1-r_1-r_2,$$

(39)

$${\dot{\theta }}_2=r_2{\varphi }_2\left({\xi }_1,{\widehat{\xi }}_2,{\widehat{\xi }}_{3,f}\right)-{\sigma }_2{\theta }_2,$$

(40)

$${\dot{\omega }}_2=-c_2{\omega }_2+{\omega }_3+\left({\overline{\alpha }}_2-{\alpha }_2\right),$$

(41)

where ${\sigma }_2>0$ and $c_2>0$ are designed parameters.

Step 3: Choose the virtual control function ${\alpha }_3$ and the compensatory signal ${\dot{\omega }}_3$ as follows:

$${\alpha }_3=-c_3{z}_3+{\dot{\overline{\alpha }}}_2-r_2-\frac{1}{2}{r}_3,$$

(42)

$${\dot{\omega }}_3=-c_3{\omega }_3+{\omega }_4+\left({\overline{\alpha }}_3-{\alpha }_3\right),$$

(43)

where $c_3>0$ is a designed parameter.

Step 4: Choose the virtual control function ${\alpha }_4$, the adaptive rate ${\dot{\theta }}_4$, and the compensatory signal ${\dot{\omega }}_4$ as follows:

$${\alpha }_4=-c_4{z}_4-{\theta }_4^T{\varphi }_4\left({\xi }_1,{\widehat{\xi }}_2,{\widehat{\xi }}_3,{\xi }_4,{\xi }_{5,f}\right)+{\dot{\overline{\alpha }}}_3-r_3-r_4,$$

(44)

$${\dot{\theta }}_4=r_4{\varphi }_4\left({\xi }_1,{\widehat{\xi }}_2,{\widehat{\xi }}_3,{\xi }_4,{\xi }_{5,f}\right)-{\sigma }_4{\theta }_4,$$

(45)

$${\dot{\omega }}_4=-c_4{\omega }_4+{\omega }_5+\left({\overline{\alpha }}_4-{\alpha }_4\right),$$

(46)

where $c_4>0$ and ${\sigma }_4>0$ are designed parameters.

Step 5: Choose the Lyapunov function as follows:

$$V_5=V_4+\frac{1}{2}{r}_5^2+\frac{1}{2}{\tilde{\theta }}_5^T{\tilde{\theta }}_5,$$

(47)

The derivative of $V_5$ is as follows:

$${\dot{V}}_5={\dot{V}}_4+r_5\left({\theta }_5^T{\varphi }_5\left({\xi }_1,{\widehat{\xi }}_2,{\widehat{\xi }}_3,{\xi }_4,{\xi }_5,u_f\right)\right.+u+{\epsilon }_5-{\dot{\overline{\alpha }}}_4-{\dot{\omega }}_5)-{\tilde{\theta }}_5^T{\dot{\theta }}_5,$$

(48)

As a result of the Young’s inequality,

$${\epsilon }_5{r}_5\le \frac{1}{2}{r}_5^2+\frac{1}{2}{\epsilon }_5^{{\ast }^2},$$

(49)

$$-r_5{\tilde{\theta }}_5^T{\varphi }_5\left({\xi }_1,{\widehat{\xi }}_2,{\widehat{\xi }}_3,{\xi }_4,{\xi }_5,u_f\right)\le \frac{1}{2}{r}_5^2+\frac{1}{2}{\tilde{\theta }}_5^T{\tilde{\theta }}_5,$$

(50)

Through substituting Equations (45) and (46) into Equation (44), we can obtain the following:

$$\begin{array}{cc}{\dot{V}}_5\le & {\dot{V}}_4+r_5\left({\theta }_5^T{\varphi }_5\left({\xi }_1,{\widehat{\xi }}_2,{\widehat{\xi }}_3,{\xi }_4,{\xi }_5,u_f\right)\right.+\left.u+\frac{1}{2}{r}_5-{\dot{\overline{\alpha }}}_4-{\dot{\omega }}_5\right)\\ & +\frac{1}{2}{\epsilon }_5^{{\ast }^2}+{\tilde{\theta }}_5^T{\dot{\tilde{\theta }}}_5\hfill \end{array},$$

(51)

As a result of the Young’s inequality,

$$r_{5}h\le \frac{1}{2}{r}_5^2+\frac{1}{2}{h}^2,$$

(52)

where $h^{\ast }$ is a constant representing the maximum value of $h$.

Through substituting Equation (17) into Equation (48), we can obtain the controller $v$ as follows:

$$v=\dot{N}\left(\zeta \right)(-c_5{z}_5-{\theta }_5^T{\varphi }_5\left({\xi }_1,{\widehat{\xi }}_2,{\widehat{\xi }}_3,{\xi }_4,{\xi }_5,u_f\right)+{\dot{\overline{\alpha }}}_4-r_4-\frac{3}{2}{r}_5),$$

(53)

$$\dot{\zeta }=r_5(-c_5{z}_5-{\theta }_5^T{\varphi }_5\left({\xi }_1,{\widehat{\xi }}_2,{\widehat{\xi }}_3,{\xi }_4,{\xi }_5,u_f\right)+{\dot{\overline{\alpha }}}_4-r_4-\frac{3}{2}{r}_5),$$

(54)

The compensatory signal ${\dot{\omega }}_5$ is given by the following:

$${\dot{\omega }}_5=-c_5{\omega }_5,$$

(55)

The adaptive rate ${\dot{\theta }}_5$ is given by the following:

$${\dot{\theta }}_5=r_5{\phi }_5\left({\xi }_1,{\widehat{\xi }}_2,{\widehat{\xi }}_3,{\xi }_4,{\xi }_5,u_f\right)-{\sigma }_5{\theta }_5,$$

(56)

where $c_5>0$ and ${\sigma }_5>0$ are designed parameters.

Theorem 1.

For the flexible robotic arm system (1) with unobservable states and dead-zone inputs, the state observer (21); virtual control functions (36), (39), (42), (44); parameter adaptation rates (40), (45), (56); and the controller (53) ensure bounded signals within the closed-loop system. Moreover, by selecting appropriate design parameters, the tracking error converges to a small neighborhood.

Proof of Theorem 1.

 

Choose the Lyapunov function as follows:

$$V_5=V_0+{\sum }_{i=1}^5\frac{1}{2}{r}_i^2+{\sum }_{i=\mathrm{2,4},5}\frac{1}{2}{\tilde{\theta }}_i^T{\tilde{\theta }}_i,$$

(57)

As a result of the Young’s inequality,

$$-r_i{\tilde{\theta }}_i^T{\varphi }_i\left({\xi }_1,{\widehat{\xi }}_2,\cdots ,{\widehat{\xi }}_{i,f}\right)\le \frac{1}{2}{r}_i^2+\frac{1}{2}{\tilde{\theta }}_i^T{\tilde{\theta }}_i,$$

(58)

$$\begin{array}{cc}-{\sigma }_i{\tilde{\theta }}_i^T{\theta }_i& \le -{\sigma }_i{\tilde{\theta }}_i^T\left({\tilde{\theta }}_i+{\widehat{\theta }}_i\right)\hfill \\ & \le -{\sigma }_i{\tilde{\theta }}_i^T{\tilde{\theta }}_i+{\sigma }_i{{\tilde{\theta }}_i}^2+{\sigma }_i{\widehat{\theta }}_i\hfill \\ & \le -{\sigma }_i{\tilde{\theta }}_i^T{\tilde{\theta }}_i+{\sigma }_i{{\widehat{\theta }}_i}^2\hfill \end{array},$$

(59)

Then,

$$\begin{array}{cc}{\dot{V}}_5\le & {\dot{V}}_0-{\sum }_{i=1}^5{c}_i{r}_i^2+{\sum }_{i=1}^5\frac{1}{2}{\epsilon }_i^{{\ast }^2}-{\sum }_{i=\mathrm{2,4},5}{\sigma }_i{\tilde{\theta }}_i^T{\theta }_i\hfill \\ & +{\sum }_{i=\mathrm{2,4},5}\frac{1}{2}{\tilde{\theta }}_i^T{\tilde{\theta }}_i+\frac{1}{2}{h^{\ast }}^2\hfill \end{array},$$

(60)

$$\begin{array}{cc}{\dot{V}}_5\le & -e^{T}Qe+3e^2+P^2\left({\epsilon }_2^{{\ast }^2}+{\epsilon }_4^{{\ast }^2}+{\epsilon }_5^{{\ast }^2}\right)+P^2\left({\tilde{\theta }}_2^T{\tilde{\theta }}_2+{\tilde{\theta }}_4^T{\tilde{\theta }}_4+{\tilde{\theta }}_5^T{\tilde{\theta }}_5\right)\hfill \\ & -{\sum }_{i=1}^5{c}_i{r}_i^2+{\sum }_{i=1}^5\frac{1}{2}{\epsilon }_i^{{\ast }^2}+{\sum }_{i=\mathrm{2,4},5}\frac{1}{2}{\tilde{\theta }}_i^T{\tilde{\theta }}_i-{\sum }_{i=\mathrm{2,4},5}{\sigma }_i{\tilde{\theta }}_i^T{\tilde{\theta }}_i\hfill \\ & +{\sum }_{i=\mathrm{2,4},5}{\sigma }_i{{\widehat{\theta }}_i}^2+\frac{1}{2}{h^{\ast }}^2\hfill \end{array},$$

(61)

Formula (61) can be transformed into the following:

$${\dot{V}}_5\le -C{V}_5+D,$$

(62)

where $C=\mathit{min}\left(\frac{\left({\lambda }_{min}\left(Q\right)-3\right)}{{\lambda }_{min}\left(P\right)},2c_i,2\left({\sigma }_k-\frac{1}{2}-P^2\right)\right)i=\mathrm{1,2},\cdots ,5,k=\mathrm{2,4},5$, $D={\sum }_{i=\mathrm{2,4},5}{\sigma }_i{{\widehat{\theta }}_i}^2+{\sum }_{i=1}^5\frac{1}{2}{\epsilon }_i^{{\ast }^2}+\frac{1}{2}{h^{\ast }}^2+P^2\left({\epsilon }_2^{{\ast }^2}+{\epsilon }_4^{{\ast }^2}+{\epsilon }_5^{{\ast }^2}\right)$.

By integrating Equation (59) over the interval $[0,t]$, we can obtain the following inequality:

$$0\le V_5\left(t\right)\le V_5\left(0\right)e^{-ct}+\frac{D}{c}(1-e^{-ct}),$$

(63)

According to Equation (60), it can be demonstrated that all signals within the closed-loop system are bounded. Thus, we have the following:

$$\left|z_1\left(t\right)\right|=\sqrt{2(V_5\left(0\right)\exp \left(-ct\right)+D/c)},$$

(64)

As $t\to \mathrm{\infty }$ and $\underset{t\to \mathrm{\infty }}{\lim }\exp \left(-ct\right)=0$, it follows that $\underset{t\to \mathrm{\infty }}{\lim }\left|z_1\left(t\right)\right|\le \sqrt{2D/c}$. By modifying the value of variable ${\sigma }_i$ and $c_i$, the tracking error is minimized as much as possible. □

5. Simulation

In this section, the effectiveness of the method is demonstrated through simulations on a flexible robotic arm. The parameters of the flexible robotic arm were selected as shown in

Table 1. Parameters of the flexible robotic arm.

Parameters	Value
$J_1,J_2$	$50 \mathrm{K}\mathrm{g}{\mathrm{m}}^2$
$K_t$	$10 \mathrm{N}\mathrm{m}/\mathrm{A}$
$K_b$	$0.976 \mathrm{N}\mathrm{m}/\mathrm{A}$
$m$	$0.1 \mathrm{K}\mathrm{g}$
$g$	$9.8 \mathrm{N}/\mathrm{K}\mathrm{g}$
$F_1,F_2$	$0.01 \mathrm{N}\mathrm{m}\mathrm{s}/\mathrm{r}\mathrm{a}\mathrm{d}$
$R$	$2 \mathsf{\Omega }$
$K$	$20$
$L$	$100 \mathrm{H}$
$N$	$1$

.

Selection of Fuzzy Membership Functions:

$$\begin{gathered} \begin{array}{c}{\mu }_{F_2^l}\left({\xi }_1,{\widehat{\xi }}_2,{\widehat{\xi }}_{3,f}\right)=\mathit{exp}\left[-\frac{{\left({\xi }_1-l\right)}^2}{2}\right]\times \mathit{exp}\left[-\frac{{\left({\widehat{\xi }}_2-l\right)}^2}{2}\right]\times \mathit{exp}\left[-\frac{{\left({\widehat{\xi }}_{3,f}-l\right)}^2}{2}\right]\end{array} \\ \begin{array}{cc}{\mu }_{F_4^l}\left({\xi }_1,{\widehat{\xi }}_2,{\widehat{\xi }}_3,{\widehat{\xi }}_4,{\widehat{\xi }}_{5,f}\right)=& \mathit{exp}\left[-\frac{{\left({\xi }_1-l\right)}^2}{2}\right]\times \mathit{exp}\left[-\frac{{\left({\widehat{\xi }}_2-l\right)}^2}{2}\right]\times \mathit{exp}\left[-\frac{{\left({\widehat{\xi }}_3-l\right)}^2}{2}\right]\\ & \times \mathit{exp}\left[-\frac{{\left({\widehat{\xi }}_4-l\right)}^2}{2}\right]\times \mathit{exp}\left[-\frac{{\left({\widehat{\xi }}_{5,f}-l\right)}^2}{2}\right]\hfill \end{array} \\ \begin{array}{cc}{\mu }_{F_5^l}\left({\xi }_1,{\widehat{\xi }}_2,{\widehat{\xi }}_3,{\widehat{\xi }}_4,{\widehat{\xi }}_5,u_f\right)=& \mathit{exp}\left[-\frac{{\left({\xi }_1-l\right)}^2}{2}\right]\times \mathit{exp}\left[-\frac{{\left({\widehat{\xi }}_2-l\right)}^2}{2}\right]\times \mathit{exp}\left[-\frac{{\left({\widehat{\xi }}_3-l\right)}^2}{2}\right]\\ & \times \mathit{exp}\left[-\frac{{\left({\widehat{\xi }}_4-l\right)}^2}{2}\right]\times \mathit{exp}\left[-\frac{{\left({\widehat{\xi }}_5-l\right)}^2}{2}\right]\times \mathit{exp}\left[-\frac{{\left(u_f-l\right)}^2}{2}\right]\hfill \end{array} \end{gathered}$$

where $l=-2,\cdots ,2$.

The reference signal is given as $y_r=\mathit{sin}\left(t-1\right)$. Select virtual control functions ${\alpha }_1$(36) ${\alpha }_2($39), ${\alpha }_3($42), ${\alpha }_4($44), $v($53); the adaptation rate ${\theta }_2($40), ${\theta }_4($45), ${\theta }_5($56); and initial selection $x_1\left(0\right)=0.03$, $x_2\left(0\right)=0.01$, $x_3\left(0\right)=0$, $x_4\left(0\right)=0$, $x_5\left(0\right)=0$, with parameter selections of $k_1=10$, $k_2=10$, $k_3=2$, $k_4=2$, $k_5=0.01$, $c_1=60$, $c_2=15$, $c_3=60$, $c_4=10$, $c_5=10$, ${\omega }_n=900$,$\varsigma =0.9$, ${\sigma }_2=20$, ${\sigma }_4=20$, ${\sigma }_5=20$.

Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10 display the simulation results. Figure 4 represents the trajectory between the output signal $y$ and the reference signal $y_r$. Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9 exhibit the trajectories of the robotic arm’s state values and their corresponding estimates. Figure 10 illustrates the trajectory signals $v$ of the controller under the influence of dead-zone inputs.

As can be seen from Figure 4, output signal $y$ effectively tracks the given reference signal $y_r$. From Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9, it is evident that the observer designed in this study is capable of accurately estimating the actual state. Figure 10 illustrates that the controller exhibits rapid convergence and good stability.

Figure 11 represents the trajectories of $e$ with different values of $c_i$, where $e=y-y_r$. The parameter e1 is chosen as follows: $c_1=60$, $c_2=15$, $c_3=60$, $c_4=10$, $c_5=10$. The parameter e2 is chosen as follows: $c_1=5$, $c_2=5$, $c_3=5$, $c_4=5$, $c_5=5$. The parameter e3 is chosen as follows: $c_1=15$, $c_2=15$, $c_3=15$, $c_4=15$, $c_5=15$. The parameter e4 is chosen as follows: $c_1=40$, $c_2=40$, $c_3=40$, $c_4=40$, $c_5=40$. As can be seen from Figure 9, by increasing the values of $c_1$, $c_2$, $c_3$, $c_4$, $c_5$, the trajectory tracking error e decreases accordingly. The experimental results correspond to the conclusion mentioned in Section 4 which states that $\underset{t\to \mathrm{\infty }}{\lim }\left|z_1\left(t\right)\right|\le \sqrt{2D/c}$.

Figure 12 represents the trajectories of $v$ with different values of $c_i$, where v1 is parameterized with $c_1=5$, $c_2=5$, $c_3=5$, $c_4=5$, $c_5=5$. v2 is parameterized with $c_1=15$, $c_2=15$, $c_3=15$, $c_4=15$, $c_5=15$. v3 is parameterized with $c_1=40$, $c_2=40$, $c_3=40$, $c_4=40$, $c_5=40$. As $c_i$ increases, better control performance can be achieved.

The simulation results indicate that the observer designed in this study has a good estimation performance and that the control method employed exhibits strong robustness.

6. Conclusions

In this paper, an adaptive fuzzy output feedback control method was proposed to address the issues of dead-zones and unknown model parameters in joint flexible robotic arm systems. The method integrates command filtering and fuzzy observer techniques to achieve the tracking control of desired trajectories for the flexible robotic arm system. The approach firstly involves employing a fuzzy logic system to approximate the nonlinear functions within the flexible robotic arm system. It subsequently involves designing a state observer to estimate the unobservable states of the system. The application of command filtering helps to address computational complexity concerns during controller design. Moreover, the Nussbaum function is introduced to tackle unknown control gain problems. The stability of the system can be verified by designing a Lyapunov function. Our simulation results confirm the effectiveness of the closed-loop system in addressing the challenges posed by dead-zones and unknown parameters in flexible robotic arm systems, thereby achieving effective control of the flexible robotic arm. The future extension of this study will focus on collaboration issues among multiple robotic arms in unstructured and collaborative environments. It will explore control strategies capable of achieving collaborative operations, task allocation, and coordination to accomplish complex objectives.