Fuzzy-Based Fixed-Time Nonsingular Tracker of Exoskeleton Robots for Disabilities Using Sliding Mode State Observer

Abstract

In this article, the position tracking control of the wheelchair upper-limb exoskeleton robotic system is investigated with the aim of rehabilitation of disabled people. Hence, the fuzzy nonsingular terminal sliding mode control method by using the state observer with a fixed-time convergence rate is designed in three main parts. In the first part, the fixed-time state observer is proposed for estimation of the states of the system. Secondly, the fixed-time convergence of position tracking error of the upper-limb exoskeleton robot system is examined by using the nonsingular terminal sliding mode control approach. In the third part, with the target of the improvement of the controller performance for removal of the chattering phenomenon which diminishes the controller performance, the fuzzy control method is used. Finally, the efficiency and proficiency of the proposed control method on the upper limb exoskeleton robotic system are demonstrated via the simulation results which are provided by MATLAB/Simulink software. In this part, simulation results are obtained based on different initial conditions in two examples using various desired values. Thus, it can be demonstrated that the proposed method applied to the upper-limb exoskeleton robot system is robust under various initial conditions and desired values.

1. Introduction

The wheelchair, which has a structure of two front support wheels and two rear driving wheels, is one of the best tools for elder or young people with disabilities. To put in other words, disabled people use this instrument to not only go out but also participate in outdoor and indoor activities [1,2,3,4,5]. Unfortunately, when disabled persons use a wheelchair, they encounter some problems including climbing steps, opening a door, and picking up an object which is placed on the floor or high shelf [6]. Therefore, some recent technologies, such as robotic technology, should be used to enhance the performance of the wheelchair system [7]. Exoskeleton is a robotic system that is accompanied by a wheelchair system in order to provide additional power and recover movement performance [8]. Wheelchair upper-lime exoskeleton provides considerable performance among different kinds of the exoskeleton robot in which it mixes the boons of rehabilitation and motion assistance of disabled or elder persons. Hence, more applicable controllers for the upper-limb exoskeleton are still needed to enhance the efficiency of the power assist [9].

Position tracking is the main function in the control of the wheelchair upper-limb exoskeleton system which can be accomplished by various control techniques. For the aim of the position tracking control of the wheelchair upper-limb exoskeleton robot, both proportional-derivative (PD) and proportional-integral-derivative (PID) controllers have been broadly recommended because of their independent and easy-to-tune features [10,11,12,13]. However, the usage of the integrator in the PID controller leads to a reduction in the closed-loop system’s bandwidth and removal of the steady-state error produced by uncertainty and external disturbance, and a high gain of the integrator can decrease the transient performance and system’s stability. As a result, the PID controller used in many wheelchairs’ upper-limb exoskeleton robotic systems is tuned via small integral gain. Moreover, both PD and PID control methods recommend the asymptotic convergence rate [14,15]. Hence, some control techniques have been used to improve the performance of the PD and PID control scheme for exoskeleton robot systems [16]. In order to provide the fast convergence of position tracking error of the upper-limb exoskeleton system and singularity removal in the control strategy, the non-singular sliding mode control (NSSMC) scheme has been proposed [17,18]. Whereas unknown states always exist in the upper-limb exoskeleton robot system, the performance level of the system will be decreased. Hence, by design of the state observer, the states are observed and denied properly which leads to the high performance of the system [19]. In order to boost tracking performance and remove the chattering in the controller of the system, the fuzzy logic control method has been recommended [20,21].

Li et al. [22] mixed the fuzzy estimation technique with the backstepping control procedure for position tracking control of the wheelchair upper-limb exoskeleton robot system. Riani et al. [23] proposed an integral terminal-SMC method using a model-based adaptive control procedure for the rehabilitation control of the wheelchair upper-limb exoskeleton system. Wu et al. [24] offered the fuzzy-based sliding mode admittance control for the exoskeleton robot system in order to provide an opportunity for patients to take part in actions actively. Rahmani and Rahman [25] recommended a fuzzy control scheme based on sliding mode control with the design of a new switching surface for the upper-limb exoskeleton system. Precup and Preitl [26] investigated the stability and sensitivity aspects of the fuzzy control systems. In [26], with the usage of Popov’s hyperstability theory accompanied by application on a mechatronic system, the stability of the fuzzy control system is demonstrated. Then, by using the mechatronic application on servo-systems, the sensitivity of the fuzzy control system is examined. Chen et al. [27] designed a machine learning-based artificial intelligence method using an evolved bat algorithm to investigate the stability of the TS fuzzy systems. Precup et al. [28] offer two data-driven-based sliding mode control and fuzzy control methods for the stability of the error dynamics of the nonlinear system. Moreover, the validity of this approach is proved by three-dimensional crane system laboratory equipment. Precup et al. [29] proposed the adaptive fuzzy control technique combined with model predictive control to address the problem related to multi-parametric quadratic programming and apply an electrohydraulic servo-system experimentally to validate the suggested method.

Table 1. Disadvantages of the existing methods.

Reference	Disadvantages
Li et al. [22]	The backstepping control technique is an iterative controller which leads to complexity in the control process, so it is not preferred for exoskeleton system owing complicated motion.
Riani et al. [23]	Whereas exoskeleton robot is located near the human body which its impression has to be considered in the design of the controller.
Wu et al. [24]	Although, this method relays on the exact dynamical model of the exoskeleton robot system which is not possible due to existence of modeling error.
Rahmani and Rahman [25]	The removal of the singularity problem which is realized by NSSMC method has not been investigated in this paper.

is provided to elaborate on the disadvantages of the existing methods which consider the control scheme of the upper-limb exoskeleton robot system compared with the proposed method.

As it can be found from the above discussion related to the design of the different control methods for position tracking of the wheelchair upper-limb exoskeleton robot system, no control method has been designed based on the fuzzy nonsingular terminal sliding mode control method using state observer for position tracking of the upper-limb exoskeleton robot system. Therefore, the main novelties of this article can be described briefly as follows:

−

Presentation of free and typical types of the dynamical robot system for describing a dynamical model of the upper-limb exoskeleton robot system;

−

Design of the fixed-time convergence rate state observer for compensation of the uncertainty in the states of the upper-limb exoskeleton robot system;

−

Proposition of the nonsingular terminal sliding mode control method with the target of the fixed-time convergence of position tracking error of the exoskeleton robot system;

−

Suggestion of fuzzy control procedure for improvement of the control input performance.

The rest of the paper is organized as follows: the dynamical model of the wheelchair upper-limb exoskeleton robot system is introduced in Section 2. In Section 3, the control procedure including the design of the state observer is provided. Presentation of the nonsingular terminal sliding mode control method and recommendation of the fuzzy control technique are explained in Section 4. Simulation results and some conclusions are provided in Section 5 and Section 6, respectively.

2. Description of Mechanical Model of Upper-Exoskeleton Robot

The wheelchair upper-limb exoskeleton robot system comprises two main parts as shown in Figure 1; a glove that is used for the user to accomplish the required capability and an upper-limb exoskeleton. In the upper limb exoskeleton part, there are four main joints with three degrees of freedom (3-DoF) such as joint 1 which is responsible for adduction and abduction function, joint 2 related to the internal and external rotation action, joint 3 for flexion and extension, and joint 4 is used for elbow flexion and extension [11].

As was mentioned in [11], the designed control method for position tracking of an exoskeleton robot system did not require the dynamical model of the system; hence, the following 2-DoF dynamical model is considered to represent a dynamical model of the typical robot system [30]:

$$M\left(q\right)\ddot{q}+C\left(q,\dot{q}\right)\dot{q}+{\tau }_g=\tau$$

(1)

where $q$, $\dot{q}$, and $\ddot{q}$ are the position, velocity, and accelerations of the robot system, respectively. The expression $M\left(q\right)$ is the inertia matrix by:

$$M\left(q\right)=\left[\begin{array}{cc}{m}_{11}& m_{12}\\ m_{21}& m_{22}\end{array}\right]$$

(2)

with $m_{11}=a_1+2a_{3}cos\left(q_2\right)-2a_{4}sin\left(q_2\right)$, $m_{12}=m_{21}=a_2+a_{3}cos\left(q_2\right)+a_{4}sin\left(q_2\right)$, $m_{22}=a_2$. The expression $C\left(q,\dot{q}\right)$ is the Coriolis and centrifugal forces matrix as:

$$C\left(q,\dot{q}\right)=\left[\begin{array}{cc}-h{\dot{q}}_2\hfill & -h\left({\dot{q}}_1-{\dot{q}}_2\right)\\ h{\dot{q}}_1\hfill & 0\end{array}\right]+\left[\begin{array}{cc}{v}_1& 0\\ 0& v_2\end{array}\right]$$

(3)

with $h=a_{3}sin\left(q_2\right)-a_{4}cos\left(q_2\right)$. The expression ${\tau }_g$ is the gravitational forces matric as:

$${\tau }_g=\left[\begin{array}{cc}{F}_{c1}& 0\\ 0& F_{c2}\end{array}\right]\left[\begin{array}{c}sgn\left({\dot{q}}_1\right)\\ sgn\left({\dot{q}}_2\right)\end{array}\right]$$

(4)

In addition, $\tau$ is the control input. Moreover, the required parameters and their unit and quantity are given in

Table 2. Required parameters.

Parameter	Value	Parameter	Value
$m_1$	$1 \mathrm{kg}$	$v_2$	$2.7 {\mathrm{kgm}}^2/\mathrm{s}$
$m_e$	$2 \mathrm{kg}$	$I_1$	$0.12 {\mathrm{kgm}}^2$
$l_1$	$1 \mathrm{m}$	$I_2$	$0.25 {\mathrm{kgm}}^2$
$l_{c1}$	$0.5 \mathrm{m}$	$a_1$	$2.745 {\mathrm{kgm}}^2$
$l_{ce}$	$0.25 \mathrm{m}$	$a_2$	$0.375 {\mathrm{kgm}}^2$
$F_{c1}$	$5 \mathrm{Nm}$	$a_3$	$0.433 {\mathrm{kgm}}^2$
$F_{c2}$	$5 \mathrm{Nm}$	$a_4$	$0.25 {\mathrm{kgm}}^2$
$v_1$	$5.5 {\mathrm{kgm}}^2/\mathrm{s}$

. With the simplification of Equation (1) we have:

$$\ddot{q}=-M^{-1}\left(q\right)C\left(q,\dot{q}\right)\dot{q}-M^{-1}\left(q\right){\tau }_g+M^{-1}\left(q\right)\tau$$

(5)

3. Preliminaries

In this part, some essential assumptions, remarks, and Lemmas are presented which are used in the design of the control strategy.

Definition 1

([31]).For any nonlinear system as:

$$\dot{x}=f\left(x,t\right), x\left(0\right)=x_0$$

(6)

where $x_0\in {ℝ}^n$,$f: {ℝ}^{+}\times {ℝ}^n\to {ℝ}^n$, the origin is the equilibrium point. Hence, if the origin is stable based on Lyapunov stability theory and the convergence time $T$ is bounded and holds the condition of $T<T_{max}$ where $T_{max}>0$; therefore, the nonlinear system is globally fixed-time stable.

Lemma 1 (

[32]).For any nonlinear system as:

$$\dot{x}=-\alpha si{g}^{{\gamma }_1}\left(x\right)-\beta si{g}^{{\gamma }_2}\left(x\right), x\left(0\right)=x_0$$

(7)

where ${\gamma }_1>1$, $0<{\gamma }_2<1$ and $\alpha ,\beta >0$, the origin of the system can be fixed-time stable and the convergence time is calculated as:

$$T_0\le \frac{1}{\alpha \left({\gamma }_1-1\right)}+\frac{1}{\beta \left(1-{\gamma }_2\right)}$$

(8)

Lemma 2 (

[32]).For any$k\in {ℝ}^{+}$, it realizes that:

$$\left\{\begin{array}{l}{\displaystyle {\displaystyle \sum }_{i=1}^n}{\left|x_i\right|}^k\ge n^{1-k}{\left({\displaystyle {\displaystyle \sum }_{i=1}^n}\left|x_i\right|\right)}^k, if k>0\\ {\displaystyle {\displaystyle \sum }_{i=1}^n}{\left|x_i\right|}^{1+k}\ge {\left({\displaystyle {\displaystyle \sum }_{i=1}^n}{\left|x_i\right|}^2\right)}^{\frac{1+k}{2}}, if 0<k<1\end{array}\right.$$

(9)

4. Main Results

In this part, firstly, a state observer with a fixed-time convergence rate is designed with the aim of estimating the uncertain states. In the following, the nonsingular terminal sliding mode control technique based on a state observer is used with the aim of fixed-time convergence of the position tracking of the upper-limb exoskeleton system. Finally, fuzzy-based nonsingular terminal sliding mode control using a state observer is designed to improve the performance of the controller which suffers from the chattering phenomenon.

4.1. Design of the State Observer

In this part, the fixed-time state observer is proposed as:

$$\dot{\psi }=-M^{-1}\left(q\right)C\left(q,\dot{q}\right)\dot{q}-M^{-1}\left(q\right){\tau }_g+M^{-1}\left(q\right)\tau +{\kappa }_{1}si{g}^{{\eta }_1}+{\kappa }_{2}si{g}^{{\eta }_2}$$

(10)

where $\psi$ is the estimator of the value $\dot{q}$, $\epsilon =\dot{q}-\psi$ is the estimator error, ${\kappa }_1,{\kappa }_2>0$, $0<{\eta }_1<1$ and ${\eta }_2>1$.

Theorem 1.

Presume the dynamical system of the upper-limb exoskeleton system as (5). Using the state observer (10), the observation error converges to zero in the fixed time$T_0\le \frac{1}{2^{\frac{{\eta }_{1+1}}{2}}{\kappa }_1\left(1-\frac{{\eta }_{1+1}}{2}\right)}+\frac{1}{2^{\frac{{\eta }_{2+1}}{2}}{\kappa }_2\left(\frac{{\eta }_{1+1}}{2}-1\right)}$.

Proof. 

Construct the Lyapunov function as:

$$V_0=0.5{\epsilon }^2$$

(11)

By taking the time derivative of the considered candidate Lyapunov function, one attains that:

$${\dot{V}}_0=\epsilon \dot{\epsilon }$$

(12)

Considering the estimator error, one attains:

$${\dot{V}}_0=\epsilon \left(\ddot{q}-\dot{\psi }\right)$$

(13)

and substituting (5) and (10), we have:

$$\begin{gathered} {\dot{V}}_0=\epsilon -M^{-1}\left(q\right)C\left(q,\dot{q}\right)\dot{q}-M^{-1}\left(q\right){\tau }_g+M^{-1}\left(q\right)\tau +M^{-1}\left(q\right)C\left(q,\dot{q}\right)\dot{q} \\ +M^{-1}\left(q\right){\tau }_g-M^{-1}\left(q\right)\tau -{\kappa }_{1}si{g}^{{\eta }_1}-{\kappa }_{2}si{g}^{{\eta }_2} \end{gathered}$$

(14)

Performing some simplification and removal of the same expressions, results in:

$${\dot{V}}_0=\epsilon \left(-{\kappa }_{1}si{g}^{{\eta }_1}-{\kappa }_{2}si{g}^{{\eta }_2}\right)$$

(15)

With loss of generality, we have:

$${\dot{V}}_0\le -{\kappa }_1{\left|\epsilon \right|}^{{\eta }_1+1}-{\kappa }_2{\left|\epsilon \right|}^{{\eta }_2+1}$$

(16)

Based on Lemma 2, one can gain:

$${\dot{V}}_0\le -2^{\frac{{\eta }_{1+1}}{2}}{\kappa }_1{\left(0.5{\epsilon }^2\right)}^{\frac{{\eta }_1+1}{2}}-2^{\frac{{\eta }_{2+1}}{2}}{\kappa }_2{\left(0.5{\epsilon }^2\right)}^{\frac{{\eta }_2+1}{2}}$$

(17)

As a result, based on Definition 1 and Lemma 1, the observer estimator error is the globally fixed-time stable with the convergence time $T_0\le \frac{1}{2^{\frac{{\eta }_{1+1}}{2}}{\kappa }_1\left(1-\frac{{\eta }_{1+1}}{2}\right)}+\frac{1}{2^{\frac{{\eta }_{2+1}}{2}}{\kappa }_2\left(\frac{{\eta }_{1+1}}{2}-1\right)}$. □

4.2. Fast Non-Singular Terminal Sliding Mode Control

In this part, nonsingular terminal sliding surface is proposed as:

$$s=\dot{e}+c_{1}e+c_{2}si{g}^{\vartheta }\left(e\right)$$

(18)

where $c_1,c_2>0$, $\vartheta >0$ and we have:

$$\begin{gathered} e=q-q_d \\ \dot{e}=\dot{q}-{\dot{q}}_d \end{gathered}$$

(19)

By taking the time derivative from (18), we have:

$$\dot{s}=\ddot{e}+c_1\dot{e}+c_2\vartheta {\left|e\right|}^{\vartheta -1}\dot{e}$$

(20)

Substitution from (5), it can obtain:

$$\dot{s}=-M^{-1}\left(q\right)C\left(q,\dot{q}\right)\dot{q}-M^{-1}\left(q\right){\tau }_g+M^{-1}\left(q\right)\tau +\left(c_1+c_2\vartheta {\left|e\right|}^{\vartheta -1}\right)\dot{e}-{\ddot{q}}_d$$

(21)

Therefore, the nonsingular terminal sliding mode controller based on a fixed-time state observer is designed as:

$$\begin{gathered} \tau ={\tau }_{eq}+{\tau }_{fn} \\ {\tau }_{eq}=-M\left(q\right)\left(-M^{-1}\left(q\right)C\left(q,\dot{q}\right)\dot{q}-M^{-1}\left(q\right){\tau }_g+\left(c_1+c_2\vartheta {\left|e\right|}^{\vartheta -1}\right)\dot{e}-{\ddot{q}}_d\right) \\ {\tau }_{fn}=-M\left(q\right)\left({\kappa }_{3}si{g}^{{\eta }_3}+{\kappa }_{4}si{g}^{{\eta }_4}\right) \end{gathered}$$

(22)

where ${\kappa }_3,{\kappa }_4>0$, ${\eta }_3>1$ and $0<{\eta }_4<1$.

Theorem 2.

Consider the dynamical equation of the upper-limb exoskeleton robot system as (5), tracking error as (19) and non-singular terminal sliding surface as (20). Moreover, the state observer is designed as (10). In this case, the position tracking error of the upper-limb exoskeleton robot system is converged to zero with a fixed-time convergence rate under the designed controller (22).

Proof. 

The candidate Lyapunov function is formed as:

$$V_1=0.5s^2$$

(23)

Taking the time derivative of the formed Lyapunov function yields:

$${\dot{V}}_1=s\dot{s}$$

(24)

Substitution of from Equation (21), we have:

$${\dot{V}}_1=s\left(-M^{-1}\left(q\right)C\left(q,\dot{q}\right)\dot{q}-M^{-1}\left(q\right){\tau }_g+M^{-1}\left(q\right)\tau +\left(c_1+c_2\vartheta {\left|e\right|}^{\vartheta -1}\right)\dot{e}-{\ddot{q}}_d\right)$$

(25)

With substitution of the control input (22) and consideration of Assumption 1, it can result:

$$\begin{gathered} {\dot{V}}_1=s\left(-{\kappa }_{3}si{g}^{{\eta }_3}-{\kappa }_{4}si{g}^{{\eta }_4}\right) \\ \le -{\kappa }_3{\left|s\right|}^{{\eta }_3+1}-{\kappa }_4{\left|s\right|}^{{\eta }_4+1} \end{gathered}$$

(26)

Based on the same strategy used in Theorem 1, the position tracking error of the upper-limb exoskeleton robot system is converged to zero with a fixed-time convergence time $T_0\le \frac{1}{2^{\frac{{\eta }_{3+1}}{2}}{\kappa }_3\left(1-\frac{{\mathsf{\eta }}_{3+1}}{2}\right)}+\frac{1}{2^{\frac{{\eta }_{4+1}}{2}}{\kappa }_4\left(\frac{{\eta }_{4+1}}{2}-1\right)}$. □

4.3. Fuzzy Nonsingular Terminal Sliding Mode Control

As it is obvious that in the designed controller (22), the constant parameters ${\kappa }_3,{\kappa }_4$ impact the performance of the controller such that the small value of them can decrease the tracking performance of the controller while the big value of them leads to better tracking performance but results in the chattering phenomenon. For removal of the effects of the constant parameters ${\kappa }_3,{\kappa }_4$, the Takagi–Sugeno (T-S) fuzzy approach fuzzy control technique is used. Hence, the following procedure for the design of the controller is considered as [11]:

Condition (a): if $s>0$

$$u_1={\tau }_{eq}+{\kappa }_3+{\kappa }_4$$

(27)

Condition (b): if $s<0$

$$u_2={\tau }_{eq}-{\kappa }_3-{\kappa }_4$$

(28)

where s is the premise variable of the fuzzy rules with the following membership function:

$$\left\{\begin{array}{c}{\mu }_1\left(s\right)=\frac{1}{1+e^{-\lambda s}}, s>0\\ {\mu }_2\left(s\right)=\frac{1}{1+e^{\lambda s}}, s<0\end{array}\right.$$

(29)

where $\lambda >0$ and following condition holds ${\mu }_1\left(s\right)+{\mu }_2\left(s\right)=1$ such that ${\mu }_1\left(s\right),{\mu }_2\left(s\right)>0$.

Based on Equation (29), one can attain that the various values of $\lambda$ give rise to different membership functions as shown in Figure 2.

Therefore, fuzzy nonsingular terminal sliding mode controller based on state observer is as follows [11]:

$$\tau ={\mu }_1\left(s\right)u_1+{\mu }_2\left(s\right)u_2$$

(30)

The block diagram of the proposed method is depicted in Figure 3 to elaborate the procedure of the control process in detail as:

5. Simulation Results

In this part, the Simulink environment in MATLAB software is selected to provide some simulation results. Afterward, simulation results based on the fuzzy nonsingular terminal sliding mode control approach using the fixed-time state observer are provided. Later, two examples with different initial conditions and various desired values are examined. Thus, proficiency, productivity, and robustness of the recommended method are demonstrated based on these two examples. In each example, simulation results are obtained according to three initial conditions in

Table 3. Considered initial condition.

Number	Initial Condition
$1$	$q\left(0\right)={\left[0.5,0.5\right]}^T$
$2$	$q\left(0\right)={\left[1,1\right]}^T$
$3$	$q\left(0\right)={\left[-0.5,-0.5\right]}^T$

and a specific desired value in

Table 4. Desired values.

Number	Initial Condition
$\mathrm{Example} 1$	$q_d={\left[sin\left(t\right),sin\left(t\right)\right]}^T$
$\mathrm{Example} 2$	$q_d={\left[0.5cos\left(\pi t\right),0.5cos\left(\pi t\right)\right]}^T$

. Moreover, it is worth noting that, in each example, the parameters of the control process are tuned based on a trial-and-error approach, the selected parameters are the same in both examples and are considered in

Table 5. Control parameters.

Parameter	Value	Parameter	Value
${\kappa }_1$	${\left[100,100\right]}^T$	${\eta }_4$	$2$
${\kappa }_2={\kappa }_4$	${\left[5,5\right]}^T$	$c_1$	${\left[10,10\right]}^T$
${\kappa }_3$	${\left[10,10\right]}^T$	$c_2$	${\left[5,5\right]}^T$
${\eta }_1={\eta }_3$	$0.2$	$\vartheta$	$1.4$
${\eta }_2$	$1.2$	$\lambda$	$1$

.

A. 

Example 1

Simulation results are obtained according to the initial conditions, desired values, and control parameters which are specified in Table 3, Table 4 and Table 5. As one can obtain from Figure 4, the joints of the upper-limb exoskeleton robot system track the specified desired values with a fixed-time convergence rate. Moreover, the robustness of the proposed method in tracking the desired values is observed by using different initial conditions. It can be seen that under different initial conditions, the desired tracking performance is accomplished very well. According to Figure 5, the tracking error signals are converged to zero with a fixed-time convergence rate. Moreover, the robustness in the convergence of the tracking error is perceived by using different initial conditions. From Figure 6, the trajectories of the nonsingular terminal sliding surface converge to zero with a fixed-time convergence rate. Under different initial conditions, it can be attested that sliding surfaces have robust performance.

Now, the performance of the fuzzy control procedure is demonstrated based on Figure 7. As one can find, the control input signal owns very slight chattering which can be observed in Figure 7. Moreover, under different initial conditions, control input possesses boundedness and robustness performance. In the last part of example 1, the performance of the fixed-time state observer is investigated in Figure 8 and Figure 9. As one can observe, the states $q_1$ and $q_2$ can be estimated accurately based on the recommended state observer. Moreover, by using various estimators based on different initial conditions, the robustness of the fixed-time state-observer is proved. In addition, time responses of the estimation error are displayed in Figure 10, which attests to the robustness and accuracy of the suggested state observer.

B. 

Example 2

As mentioned before, for demonstration of the efficiency of the fuzzy-based nonsingular terminal sliding mode control approach, a second example considering a new desired trajectory is taken in this part. Hence, the simulation results including desired tracking, tracking error, sliding surface, control inputs, fixed-time estimator, and estimation error are obtained and depicted in Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17, respectively. As can be found from these figures, the fixed-time tracking control, performance of the fuzzy procedure over control input, and fixed-time state observer are accomplished suitably.

Therefore, according to both examples that have been presented above, desired tracking control of the upper-limb exoskeleton robot system based on the fuzzy nonsingular terminal sliding mode control technique is accomplished appropriately. Moreover, the high effect of the fuzzy control procedure over the performance of the control input of the upper-limb exoskeleton robot system is demonstrated. Finally, the high performance of the fixed-time state observer in the approximation of the unknown states of the upper-limb exoskeleton robot system is achieved.

Remark 1 [33]: At the end of the simulation part, the following table related to the integral of the absolute value of error (IAE) based on different initial conditions is provided to compare the IAE values of the nonsingular terminal sliding mode control (NTSMC) and fuzzy-based NTSMC methods. IAE values are obtained from the following formula as:

$$I\left(x\right)={{\displaystyle \int }}_0^t\left|x\right| dx$$

(31)

where $x$ is the signal.

As one can observe from the IAE values presented in

Table 6. Comparison of the IAE value based on NTSMC and fuzzy-NTSMC methods.

Example	Initial Condition	NTSMC	Fuzzy-NTSMC
1	$q\left(0\right)={\left[0.5,0.5\right]}^T$	0.27	0.18
	$q\left(0\right)={\left[1,1\right]}^T$	0.98	0.79
	$q\left(0\right)={\left[-0.5,-0.5\right]}^T$	0.37	0.25
2	$q\left(0\right)={\left[0.5,0.5\right]}^T$	0.15	0.1
	$q\left(0\right)={\left[1,1\right]}^T$	0.45	0.33
	$q\left(0\right)={\left[-0.5,-0.5\right]}^T$	1.21	0.92

, the IAE values of the fuzzy-based NTSMC method are smaller than those from the NTSMC method. Thus, the proficiency of the fuzzy method compared to the non-fuzzy method is demonstrated.

6. Conclusions

As one can observe, no one cannot neglect the helpful impacts of the wheelchair upper-limb exoskeleton robot system. So, in this project, with rehabilitation purposes for disabled people, fuzzy based on a nonsingular terminal sliding mode control scheme is presented for position tracking of the upper-limb exoskeleton robot system. Then, uncertain states which exist in the upper-limb exoskeleton robot system can be compensated with the usage of the fixed-time state observer. In addition, position tracking control of the upper limb exoskeleton robot system is realized by the nonsingular terminal sliding mode control technique. Moreover, the performance of the controller is enhanced based on the fuzzy rules procedure. Hence, the chattering phenomenon is removed from the control input, so, the performance of the controller is improved. At last, simulation results are prepared to show the productivity of the recommended control method on the upper-limb exoskeleton robot system using MATLAB/Simulink. As cases for future work, the experimental validation of the proposed method on a practical upper-limb exoskeleton robot and the design of the control method for the desired tracking of the upper-limb exoskeleton robot under model uncertainty, external disturbance, and input saturation can be considered.