A New Type-3 Fuzzy Predictive Approach for Mobile Robots

Abstract

Wheeled mobile robots are widely used for their excellent mobility and high maneuverability. The intelligentization of mobile robots to perform autonomous movement is one of the favorite topics in the robotic field. This paper proposes a new intelligent path-following scheme for mobile robots. A secure path is designed by the chaotic systems and an unknown switching mechanism. The proposed controller is constructed by the type-3 (T3) fuzzy logic systems (FLSs) and a predictive compensator. The T3-FLSs are taught to obtain good accuracy. In addition, the stability is mathematically investigated and guaranteed by the designed compensator. The simulations show that the case-study robot follows the planned secure path well and resists uncertainties.

1. Introduction

In today’s world, humans need robots that can do hard, repetitive tasks and specialized tasks that may danger human life, such as neutralizing mines and working in chemical factories that cause irreparable damage to humans. To build intelligent robots, it is essential to have a modern control system, suitable path-following methods, and algorithms [1].

To design an accurate path-following scheme, various controllers have been proposed. For example, in [2] a predictive controller is designed using steering constraints, and by simulations, the feasibility of the designed scheme is verified. In [3,4], a linear observer is designed to estimate the uncertainties, and then a sliding-mode controller (SMC) is designed. The fixed-time control strategy is developed in [5,6], and the convergence time is analyzed. In [7] an adaptive controller is suggested, and the barrier function technique is developed to cope with disturbances. The Euler–Lagrange motion is analyzed in [8], an optimal controller is designed, and boundedness of trajectories is ensured. In [9], a PID controller is developed, and an image processing approach tunes the parameters of PID. The nonsingular SMC is suggested in [10], and a recursive approach is designed to optimize the structure. In [11], by using the Udwadia–Kalaba theorem, some motion constraints are considered, and by a numerical approach, the tracking accuracy is studied. In addition to SMC, the $H_{\infty }$-based method is also applied to ensure the robustness of robots in complicated paths [12].

The dynamics of mobile robots are complicated, and this problem has many uncertainties and disturbances. Some intelligent controllers have been introduced to design modern control systems that are more resistant to perturbations. For instance, in [13] a feedback linearization control method is designed using FLSs, and the effect of geometrical constraints is analyzed. In [14], the conventional SMC is improved by FLSs, and the effect of nonlinearities on motion is studied. The PID controller using FLSs is suggested in [15], and it is applied on an eight-wheeled robot. In [16], FLS-based motion tracking is designed, and the bee colony algorithm optimizes it. In [17], an SMC is developed by FLSs, and the effect of friction torques is examined. In [18], a motion controller is designed, and the FLSs are used to construct the kinematic model. In [19], a fuzzy predictive controller is suggested to improve the motion on the slope. In [20], a terminal SMC is designed for path-following, and the FLSs are combined with wavelet neural networks to overcome uncertainties. The FLS-based control systems combined with the predictive scheme are developed in [21,22,23], and the efficacy of FLSs is examined in the path-following of mobile robots.

Recently, high-order FLSs and learning methods have been developed for mobile robots in hard practical situations [24,25,26]. In [27] a type-2 (T2) FLS is suggested to estimate the uncertainties of a mobile robot and based on the FLS-based dynamic, a backstepping controller is established. In [28], a T2-FLS-based controller is optimized by the shark smell optimization technique, and by comparison with type-1 counterparts, the navigation accuracy is analyzed. In [29], a T2 fuzzy controller is designed for Omni-Directional robots, and by comparison with type-1 counterparts, it is shown that T2 fuzzy controller results in better efficiency in the face of obstacles. The shark smell optimization method is developed in [30] to design a T2 fuzzy controller, and it is applied on a mobile robot. In [31] the navigation accuracy of various types of FLS-based controllers is compared, and it is demonstrated that by the T2-FLS-based controller the robot reached the reference trajectory in a shorter time. The T2-FLS-based PID controller is designed in [32], and by adding torque load the superiority of T2-FLSs is evaluated. In [33], a navigation scheme is developed using the bee colony algorithm and T2-FLSs, and the better adaptation of T2-FLSs is shown. Another approach to improve the accuracy of FLS-based controllers is the use of deep-learning methods [34,35,36]. Furthermore, in [37] an approach is developed for modeling problems that can be extended for the estimation of uncertainties.

More recently, type-3 FLSs have been used for various engineering problems, with better efficiency. For example, in [38], T3-FLSs are used for control systems, and in various examinations, the better identification accuracy of T3-FLSs is examined. In [39] T3-FLSs are used in a predictive controller, and it is confirmed that T3-FLSs give desirable control results. The efficiency of T3-FLSs is examined in [40] in real-world situations, and the superiority of T3-FLSs is shown. The forecasting ability of T3-FLSs is studied in [41], and by the use of the real-world data set of COVID-19, it is shown that T3-FLS shows better forecasting capability in uncertain problems.

To the best knowledge of the authors, T3-FLS-based controllers have not been studied for mobile robots. In this study, a new T3-FLS-based control technique is designed, and its robustness and stability are guaranteed. A predictive controller is designed to enhance tracking accuracy and better robustness. The suggested approach is applied for a secure path-following problem for patrol mobile robots. The suggested path is generated by chaotic systems under an unknown switching mechanism. The designed strong controller enables the robot to track a chaotic complicated path.

2. Kinematic Model

Considering the two-wheeled model (see Figure 1), the linear velocity is obtained as:

$$\begin{array}{c}\vartheta =\mathcal{R}\left({\dot{\varphi }}_{\mathcal{R}}+{\dot{\varphi }}_{\mathcal{L}}\right)/2\hfill \\ w=\mathcal{R}\left({\dot{\varphi }}_{\mathcal{R}}-{\dot{\varphi }}_{\mathcal{L}}\right)/2\hfill \end{array}$$

(1)

where, $\mathcal{R}$/$\mathcal{L}$ denote radius/dsitance. Then it can written:

$$\left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{P}\end{array}\right]=\left[\begin{array}{cc}\frac{\mathcal{R}}{2}& \frac{\mathcal{R}}{2}\\ 0& 0\\ \frac{\mathcal{R}}{2\mathcal{L}}& -\frac{\mathcal{R}}{2\mathcal{L}}\end{array}\right]\left[\begin{array}{c}{\dot{\varphi }}_{\mathcal{R}}\\ {\dot{\varphi }}_{\mathcal{L}}\end{array}\right]$$

(2)

Considering the angular velocity, and orthogonal rotation, we have:

$$\left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{P}\end{array}\right]=\left[\begin{array}{cc}\frac{\mathcal{R}}{2}cos\left(P\right)& \frac{\mathcal{R}}{2}cos\left(P\right)\\ \frac{\mathcal{R}}{2}sin\left(P\right)& \frac{\mathcal{R}}{2}sin\left(P\right)\\ \frac{\mathcal{R}}{2\mathcal{L}}& -\frac{\mathcal{R}}{2\mathcal{L}}\end{array}\right]\left[\begin{array}{c}{\dot{\varphi }}_{\mathcal{R}}\\ {\dot{\varphi }}_{\mathcal{L}}\end{array}\right]$$

(3)

3. Dynamics

The dynamics are expressed as follows using the Lagrange method:

$$\begin{array}{c}\Psi \left(\eta \right)\ddot{\eta }+\upsilon \left(\eta ,\dot{\eta }\right)\dot{\eta }+G\left(\eta \right)+{\tau }_{\kappa }=F\left(\eta \right)\tau -A^T\gamma \hfill \end{array}$$

(4)

where, $\Psi$ is inertia matrix, $\upsilon$/G is Coriolis matrix/gravitational, ${\tau }_{\kappa }$/F is disturbance/input matrix, and $\tau$ is input vector. $\eta$ represents position. Considering $G=0$, we can write:

$$\Psi \left(\eta \right)=\left[\begin{array}{ccccc}m& 0& -mdsin\left(\delta \right)& 0& 0\\ 0& 0& mdcos\left(\delta \right)& 0& 0\\ -mdsin\left(\delta \right)& mdcos\left(\delta \right)& I& 0& 0\\ 0& 0& 0& I_w& 0\\ 0& 0& 0& 0& I_w\end{array}\right]$$

(5)

$$\upsilon \left(\eta ,\dot{\eta }\right)=\left[\begin{array}{ccccc}0& -mdcos\left(\delta \right)& 0& 0& 0\\ 0& -mdsin\left(\delta \right)& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]$$

(6)

$$A^T\gamma =\left[\begin{array}{ccc}-sin\left(\delta \right)& cos\left(\delta \right)& cos\left(\delta \right)\\ cos\left(\delta \right)& sin\left(\delta \right)& sin\left(\delta \right)\\ -\kappa & \mathcal{L}& -\mathcal{L}\\ 0& -\mathcal{R}& 0\\ 0& 0& -\mathcal{R}\end{array}\right]$$

(7)

Considering $\varsigma$ such that $\dot{\eta }=s\eta$, $\ddot{\eta }=\dot{s}\eta +s\dot{\eta }$, $\eta ={\left[\begin{array}{cc}{\dot{\varphi }}_{\mathcal{R}}& {\dot{\varphi }}_{\mathcal{L}}\end{array}\right]}^T$, ${\varsigma }^T{A}^T=0$, $A^T\gamma$ is eliminated and:

$$\left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{\delta }\\ {\dot{\varphi }}_{\mathcal{R}}\\ {\dot{\varphi }}_{\mathcal{L}}\end{array}\right]=\left[\begin{array}{cc}\frac{\mathcal{R}}{2}cos\left(\delta \right)& \frac{\mathcal{R}}{2}cos\left(\delta \right)\\ \frac{\mathcal{R}}{2}sin\left(\delta \right)& \frac{\mathcal{R}}{2}sin\left(\delta \right)\\ \frac{\mathcal{R}}{2\mathcal{L}}& -\frac{\mathcal{R}}{2\mathcal{L}}\\ 1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}{\dot{\varphi }}_{\mathcal{R}}\\ {\dot{\varphi }}_{\mathcal{L}}\end{array}\right]$$

(8)

$$\begin{array}{c}\Psi \left(\eta \right)\left(\dot{\varsigma }\eta +\varsigma \dot{\eta }\right)+\upsilon \left(\eta ,\dot{\eta }\right)\varsigma =F\left(\eta \right)\tau -A^T\gamma \hfill \end{array}$$

(9)

From (9), we have:

$$\overline{\Psi }\left(\eta \right)\dot{\eta }+\overline{\upsilon }\left(\eta ,\dot{\eta }\right)\eta =\overline{F}\left(\eta \right)\tau$$

(10)

where,

$$\begin{array}{c}\overline{\Psi }\left(\eta \right)=\hfill \\ \left[\begin{array}{cc}{I}_w+\frac{{\mathcal{R}}^2}{4{\mathcal{L}}^2}\left(m{\mathcal{L}}^2+I\right)& \frac{{\mathcal{R}}^2}{4{\mathcal{L}}^2}\left(m{\mathcal{L}}^2-I\right)\\ \frac{{\mathcal{R}}^2}{4{\mathcal{L}}^2}\left(m{\mathcal{L}}^2-I\right)& I_w+\frac{{\mathcal{R}}^2}{4{\mathcal{L}}^2}\left(m{\mathcal{L}}^2+I\right)\end{array}\right]\hfill \end{array}$$

(11)

$$\begin{array}{c}\overline{\upsilon }\left(\eta \right)=\hfill \\ \left[\begin{array}{cc}0& \frac{{\mathcal{R}}^2}{2{\mathcal{L}}^2}\left(m_c\kappa \dot{\delta }\right)\\ -\frac{{\mathcal{R}}^2}{2{\mathcal{L}}^2}\left(m_c\kappa \dot{\delta }\right)& 0\end{array}\right]\hfill \end{array}$$

(12)

$$\overline{F}\left(\eta \right)=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$

(13)

$$\begin{array}{c}m=m_c+2m_w\hfill \\ I=I_c+m_c{\kappa }^2+2m_w{\mathcal{L}}^2+2I_m\hfill \end{array}$$

(14)

Considering (4) and (10), we obtain:

$$\begin{array}{c}\left[\begin{array}{cc}0& \frac{{\mathcal{R}}^2}{2{\mathcal{L}}^2}\left(m_c\kappa \dot{\delta }\right)\\ -\frac{{\mathcal{R}}^2}{2{\mathcal{L}}^2}\left(m_c\kappa \dot{\delta }\right)& 0\end{array}\right]\left(\begin{array}{c}\dot{\vartheta }\\ \dot{w}\end{array}\right)+\hfill \\ \left[\begin{array}{cc}0& -m_{c}dw\\ m_{c}dw& 0\end{array}\right]\left(\begin{array}{c}\vartheta \\ w\end{array}\right)=\left[\begin{array}{cc}\frac{1}{\mathcal{R}}& 0\\ 0& \frac{1}{\mathcal{R}}\end{array}\right]\left(\begin{array}{c}{u}_1\\ u_2\end{array}\right)\hfill \end{array}$$

(15)

4. General View on the Suggested Approach

An overview view of the designed approach is depicted in Figure 2. The issue is that the following tracking errors are approached to zero by controllers $u_1$ and $u_2$.

$${\chi }_i\left(t\right)={\eta }_i\left(t\right)-{\eta }_{d_i}\left(t\right),i=1,2$$

(16)

where, ${\eta }_i$ is robot position. The dynamics (15) are estimated as:

$$\begin{array}{c}{\dot{\widehat{\eta }}}_1={\Xi }_1\left(\eta |P_1\right)+{\mu }_1\hfill \\ {\dot{\widehat{\eta }}}_2={\Xi }_2\left(\eta |P_2\right)+{\mu }_2\hfill \end{array}$$

(17)

where, ${\Xi }_1\left(\eta |P_1\right)$, and ${\Xi }_2\left(\eta |P_2\right)$ are the suggested FLSs and $P_1$, and $P_2$, are learnable parameters. Unknown dynamics and input nonlinearities should be compensated by the controller.

5. Type-3 FLS

The dynamics of the under control system is perturbed by various factors. Then an strong estimator on basis of T3-FLSs is suggested to tackle the perturbations. The main structure is shown in Figure 3.

(1)

The inputs of FLS are considered as ${\zeta }_1,{\zeta }_2,\frac{d}{dt}{\zeta }_1,\frac{d}{dt}{\zeta }_2$.

(2)

The memberships of fuzzy set (FS) should be computed in this step. For ${\tilde{\Lambda }}_i^j$ (j-th FS for input ${\zeta }_i$), the memberships are [42]:

$${\overline{\xi }}_{{\tilde{\Lambda }}_{i|{\overline{\sigma }}_i}^j}=\left\{\begin{array}{c}1-{\left(\frac{\left|{\zeta }_i-{˩}_{{\tilde{\Lambda }}_i^j}\right|}{{\underline{{\rm Y}}}_{{\tilde{\Lambda }}_i^j}}\right)}^{{\overline{\sigma }}_i}if{˩}_{{\tilde{\Lambda }}_i^j}-{\underline{{\rm Y}}}_{{\tilde{\Lambda }}_i^j}<{\zeta }_i\le {˩}_{{\tilde{\Lambda }}_i^j}\\ 1-{\left(\frac{\left|{\zeta }_i-{˩}_{{\tilde{\Lambda }}_i^j}\right|}{{\overline{{\rm Y}}}_{{\tilde{\Lambda }}_i^j}}\right)}^{{\overline{\sigma }}_i}if{˩}_{{\tilde{\Lambda }}_i^j}<{\zeta }_i\le {˩}_{{\tilde{\Lambda }}_i^j}+{\overline{{\rm Y}}}_{{\tilde{\Lambda }}_i^j}\\ 0if{\zeta }_i>{˩}_{{\tilde{\Lambda }}_i^j}+{\overline{{\rm Y}}}_{{\tilde{\Lambda }}_i^j}or{\zeta }_i\le {˩}_{{\tilde{\Lambda }}_i^j}-{\underline{{\rm Y}}}_{{\tilde{\Lambda }}_i^j}\end{array}\right.$$

(18)

$${\overline{\xi }}_{{\tilde{\Lambda }}_{i|{\underline{\sigma }}_i}^j}=\left\{\begin{array}{c}1-{\left(\frac{\left|{\zeta }_i-{˩}_{{\tilde{\Lambda }}_i^j}\right|}{{\underline{{\rm Y}}}_{{\tilde{\Lambda }}_i^j}}\right)}^{{\underline{\sigma }}_i}if{˩}_{{\tilde{\Lambda }}_i^j}-{\underline{{\rm Y}}}_{{\tilde{\Lambda }}_i^j}<{\zeta }_i\le {˩}_{{\tilde{\Lambda }}_i^j}\\ 1-{\left(\frac{\left|{\zeta }_i-{˩}_{{\tilde{\Lambda }}_i^j}\right|}{{\overline{{\rm Y}}}_{{\tilde{\Lambda }}_i^j}}\right)}^{{\underline{\sigma }}_i}if{˩}_{{\tilde{\Lambda }}_i^j}<{\zeta }_i\le {˩}_{{\tilde{\Lambda }}_i^j}+{\overline{{\rm Y}}}_{{\tilde{\Lambda }}_i^j}\\ 0if{\zeta }_i>{˩}_{{\tilde{\Lambda }}_i^j}+{\overline{{\rm Y}}}_{{\tilde{\Lambda }}_i^j}or{\zeta }_i\le {˩}_{{\tilde{\Lambda }}_i^j}-{\underline{{\rm Y}}}_{{\tilde{\Lambda }}_i^j}\end{array}\right.$$

(19)

$${\underline{\xi }}_{{\tilde{\Lambda }}_{i|{\overline{\sigma }}_i}^j}=\left\{\begin{array}{c}1-{\left(\frac{\left|{\zeta }_i-{˩}_{{\tilde{\Lambda }}_i^j}\right|}{{\underline{{\rm Y}}}_{{\tilde{\Lambda }}_i^j}}\right)}^{\frac{1}{{\overline{\sigma }}_i}}if{˩}_{{\tilde{\Lambda }}_i^j}-{\underline{{\rm Y}}}_{{\tilde{\Lambda }}_i^j}<{\zeta }_i\le {˩}_{{\tilde{\Lambda }}_i^j}\\ 1-{\left(\frac{\left|{\zeta }_i-{˩}_{{\tilde{\Lambda }}_i^j}\right|}{{\overline{{\rm Y}}}_{{\tilde{\Lambda }}_i^j}}\right)}^{\frac{1}{{\overline{\sigma }}_i}}if{˩}_{{\tilde{\Lambda }}_i^j}<{\zeta }_i\le {˩}_{{\tilde{\Lambda }}_i^j}+{\overline{{\rm Y}}}_{{\tilde{\Lambda }}_i^j}\\ 0if{\zeta }_i>{˩}_{{\tilde{\Lambda }}_i^j}+{\overline{{\rm Y}}}_{{\tilde{\Lambda }}_i^j}or{\zeta }_i\le {˩}_{{\tilde{\Lambda }}_i^j}-{\underline{{\rm Y}}}_{{\tilde{\Lambda }}_i^j}\end{array}\right.$$

(20)

$${\underline{\xi }}_{{\tilde{\Lambda }}_{i|{\underline{\sigma }}_i}^j}=\left\{\begin{array}{c}1-{\left(\frac{\left|{\zeta }_i-{˩}_{{\tilde{\Lambda }}_i^j}\right|}{{\underline{{\rm Y}}}_{{\tilde{\Lambda }}_i^j}}\right)}^{\frac{1}{{\underline{\sigma }}_i}}if{˩}_{{\tilde{\Lambda }}_i^j}-{\underline{{\rm Y}}}_{{\tilde{\Lambda }}_i^j}<{\zeta }_i\le {˩}_{{\tilde{\Lambda }}_i^j}\\ 1-{\left(\frac{\left|{\zeta }_i-{˩}_{{\tilde{\Lambda }}_i^j}\right|}{{\overline{{\rm Y}}}_{{\tilde{\Lambda }}_i^j}}\right)}^{\frac{1}{{\underline{\sigma }}_i}}if{˩}_{{\tilde{\Lambda }}_i^j}<{\zeta }_i\le {˩}_{{\tilde{\Lambda }}_i^j}+{\overline{{\rm Y}}}_{{\tilde{\Lambda }}_i^j}\\ 0if{\zeta }_i>{˩}_{{\tilde{\Lambda }}_i^j}+{\overline{{\rm Y}}}_{{\tilde{\Lambda }}_i^j}or{\zeta }_i\le {˩}_{{\tilde{\Lambda }}_i^j}-{\underline{{\rm Y}}}_{{\tilde{\Lambda }}_i^j}\end{array}\right.$$

(21)

where, the upper/lower memberships for ${\tilde{\Lambda }}_i^j$ denoted by ${\overline{\xi }}_{{\tilde{\Lambda }}_{i|{\overline{\sigma }}_i}^j}$/${\overline{\xi }}_{{\tilde{\Lambda }}_{i|{\underline{\sigma }}_i}^j}$ and ${\underline{\xi }}_{{\tilde{\Lambda }}_{i|{\overline{\sigma }}_i}^j}$/${\underline{\xi }}_{{\tilde{\Lambda }}_{i|{\underline{\sigma }}_i}^j}$, respectively. ${˩}_{{\tilde{\Lambda }}_i^j}$, denotes the center of ${\tilde{\Lambda }}_i^j$. ${\underline{{\rm Y}}}_{{\tilde{\Lambda }}_i^j}$, and ${\overline{{\rm Y}}}_{{\tilde{\Lambda }}_i^j}$ denote the distances of ${˩}_{{\tilde{\Lambda }}_i^j}$ to the left/right points of ${\tilde{\Lambda }}_i^j$ (see Figure 4).

(3)

The rule firing degrees are:

$${\overline{\zeta }}_{{\overline{\sigma }}_i}^l={\overline{\xi }}_{{\tilde{\Lambda }}_{1|{\overline{\sigma }}_i}^{p_1}}·{\overline{\xi }}_{{\tilde{\Lambda }}_{1|{\overline{\sigma }}_i}^{p_2}}\cdots {\overline{\xi }}_{{\tilde{\Lambda }}_{1|{\overline{\sigma }}_i}^{p_n}}$$

(22)

$${\overline{\zeta }}_{{\underline{\sigma }}_i}^l={\overline{\xi }}_{{\tilde{\Lambda }}_{1|{\underline{\sigma }}_i}^{p_1}}·{\overline{\xi }}_{{\tilde{\Lambda }}_{1|{\underline{\sigma }}_i}^{p_2}}\cdots {\overline{\xi }}_{{\tilde{\Lambda }}_{1|{\underline{\sigma }}_i}^{p_n}}$$

(23)

$${\underline{\zeta }}_{{\overline{\sigma }}_i}^l={\underline{\xi }}_{{\tilde{\Lambda }}_{1|{\overline{\sigma }}_i}^{p_1}}·{\underline{\xi }}_{{\tilde{\Lambda }}_{1|{\overline{\sigma }}_i}^{p_2}}\cdots {\underline{\xi }}_{{\tilde{\Lambda }}_{1|{\overline{\sigma }}_i}^{p_n}}$$

(24)

$${\underline{\zeta }}_{{\underline{\sigma }}_i}^l={\underline{\xi }}_{{\tilde{\Lambda }}_{1|{\underline{\sigma }}_i}^{p_1}}·{\underline{\xi }}_{{\tilde{\Lambda }}_{1|{\underline{\sigma }}_i}^{p_2}}\cdots {\underline{\xi }}_{{\tilde{\Lambda }}_{1|{\underline{\sigma }}_i}^{p_n}}$$

(25)

(4)

The output of FLS is written as:

$$h=\frac{{\displaystyle \sum _{i=1}^{n_{\sigma }}}\left({\underline{\sigma }}_i{\underline{h}}_i+{\overline{\sigma }}_i{\overline{h}}_i\right)}{{\displaystyle \sum _{i=1}^{n_{\sigma }}}\left({\underline{\sigma }}_i+{\overline{\sigma }}_i\right)}$$

(26)

where,

$${\overline{h}}_i=\frac{{\displaystyle \sum _{l=1}^{n_r}}\left({\overline{\zeta }}_{{\overline{\sigma }}_i}^l{\overline{\upsilon }}_l+{\underline{\zeta }}_{{\overline{\sigma }}_i}^l{\underline{\upsilon }}_l\right)}{{\displaystyle \sum _{l=1}^{n_r}}\left({\overline{\zeta }}_{{\overline{\sigma }}_i}^l+{\underline{\zeta }}_{{\overline{\sigma }}_i}^l\right)}$$

(27)

$${\underline{h}}_i=\frac{{\displaystyle \sum _{l=1}^{n_r}}\left({\overline{\zeta }}_{{\underline{\sigma }}_i}^l{\overline{\upsilon }}_l+{\underline{\zeta }}_{{\underline{\sigma }}_i}^l{\underline{\upsilon }}_l\right)}{{\displaystyle \sum _{l=1}^{n_r}}\left({\overline{\zeta }}_{{\underline{\sigma }}_i}^l+{\underline{\zeta }}_{{\underline{\sigma }}_i}^l\right)}$$

(28)

The output can be rewritten as:

$$\widehat{O}\left(\zeta |P\right)=P^{T}C$$

(29)

where,

$$C^T=\left[{\underline{C}}_1,...,{\underline{C}}_{n_r},{\overline{C}}_1,...,{\overline{C}}_{n_r}\right]$$

(30)

$$P^T=\left[{\underline{\upsilon }}_1,...,{\underline{\upsilon }}_{n_r},{\overline{\upsilon }}_1,...,{\overline{\upsilon }}_{n_r}\right]$$

(31)

$${\underline{C}}_l=\frac{{\displaystyle \sum _{i=1}^{n_{\sigma }}}{\underline{\sigma }}_i{\underline{\zeta }}_{{\underline{\sigma }}_i}^l}{{\displaystyle \sum _{i=1}^{n_{\sigma }}}\left({\underline{\sigma }}_i+{\overline{\sigma }}_i\right){\displaystyle \sum _{l=1}^{n_r}}\left({\overline{\zeta }}_{{\underline{\sigma }}_i}^l+{\underline{\zeta }}_{{\underline{\sigma }}_i}^l\right)}+\frac{{\displaystyle \sum _{i=1}^{n_{\sigma }}}{\overline{\sigma }}_i{\underline{\zeta }}_{{\overline{\sigma }}_i}^l}{{\displaystyle \sum _{i=1}^{n_{\sigma }}}\left({\underline{\sigma }}_i+{\overline{\sigma }}_i\right){\displaystyle \sum _{l=1}^{n_r}}\left({\overline{\zeta }}_{{\overline{\sigma }}_i}^l+{\underline{\zeta }}_{{\overline{\sigma }}_i}^l\right)}$$

(32)

$${\overline{C}}_l=\frac{{\displaystyle \sum _{i=1}^{n_{\sigma }}}{\underline{\sigma }}_i{\overline{\zeta }}_{{\underline{\sigma }}_i}^l}{{\displaystyle \sum _{i=1}^{n_{\sigma }}}\left({\underline{\sigma }}_i+{\overline{\sigma }}_i\right){\displaystyle \sum _{l=1}^{n_r}}\left({\overline{\zeta }}_{{\underline{\sigma }}_i}^l+{\underline{\zeta }}_{{\underline{\sigma }}_i}^l\right)}+\frac{{\displaystyle \sum _{i=1}^{n_{\sigma }}}{\overline{\sigma }}_i{\overline{\zeta }}_{{\overline{\sigma }}_i}^l}{{\displaystyle \sum _{i=1}^{n_{\sigma }}}\left({\underline{\sigma }}_i+{\overline{\sigma }}_i\right){\displaystyle \sum _{l=1}^{n_r}}\left({\overline{\zeta }}_{{\overline{\sigma }}_i}^l+{\underline{\zeta }}_{{\overline{\sigma }}_i}^l\right)}$$

(33)

6. Predictive Controller

The predictive control scheme is described as:

$$\underset{{\mu }_{p_z}\left(h\right),...,{\mu }_{p_z}(h+n_C)}{min}J={\displaystyle \sum _{h=i}^{i+n_P}}\iota ·{\widehat{e}}_z^2\left(h\right)+q·\Delta {\mu }_{p_z}^2\left(h\right),$$

(34)

where,

$${\widehat{e}}_z^{}\left(i\right)=y_{\mathrm{T}3-\mathrm{FLS}\left(2\right)}\left({\underline{\widehat{e}}}_z|P_z\right),$$

(35)

where, $\iota$ and q denote constants, and

$${\underline{\widehat{e}}}_z={\left[{\widehat{e}}_z^{}\left(i-1\right),...,{\widehat{e}}_z^{}\left(i-\tau \right),{\mu }_{p_z}\left(i-1\right)\right]}^T.$$

(36)

$P_z$ is tuned as:

$$P_z\left(i\right)=P_z\left(i-1\right)+\kappa \left({\underline{e}}_z-{\underline{\widehat{e}}}_z\right)C_z^{},$$

(37)

where, $\kappa \in \left[0,1\right]$, and $C_z^{}$ is:

$$\begin{array}{c}{C}_z^T=\frac{0.5}{{\displaystyle \sum _{j=1}^K}{\overline{\sigma }}_j}\left[\frac{{\underline{\sigma }}_1\left({\overline{C}}_{{\underline{\sigma }}_1}^1+{\underline{C}}_{{\underline{\sigma }}_1}^1\right)}{{\displaystyle \sum _{l=1}^R}\left({\overline{C}}_{{\underline{\sigma }}_j}^1+{\underline{C}}_{{\underline{\sigma }}_j}^1\right)},...,\frac{{\underline{\sigma }}_{K_z}\left({\overline{C}}_{{\underline{\sigma }}_{K_z}}^1+{\underline{C}}_{{\underline{\sigma }}_{K_z}}^1\right)}{{\displaystyle \sum _{l=1}^R}\left({\overline{C}}_{{\underline{\sigma }}_j}^1+{\underline{C}}_{{\underline{\sigma }}_j}^1\right)}\right.,\hfill \\ \frac{{\underline{\sigma }}_1\left({\overline{C}}_{{\underline{\sigma }}_1}^{R_z}+{\underline{C}}_{{\underline{\sigma }}_1}^{R_z}\right)}{{\displaystyle \sum _{l=1}^R}\left({\overline{C}}_{{\underline{\sigma }}_j}^1+{\underline{C}}_{{\underline{\sigma }}_j}^1\right)},...,\frac{{\underline{\sigma }}_K\left({\overline{C}}_{{\underline{\sigma }}_{K_z}}^{R_z}+{\underline{C}}_{{\underline{\sigma }}_{K_z}}^{R_z}\right)}{{\displaystyle \sum _{l=1}^R}\left({\overline{C}}_{{\underline{\sigma }}_j}^1+{\underline{C}}_{{\underline{\sigma }}_j}^1\right)},\hfill \\ {\overline{\sigma }}_1\frac{\left({\overline{C}}_{{\overline{\sigma }}_1}^1+{\underline{C}}_{{\overline{\sigma }}_1}^1\right)}{{\displaystyle \sum _{l=1}^R}\left({\overline{C}}_{{\overline{\sigma }}_j}^l+{\underline{C}}_{{\overline{\sigma }}_j}^l\right)},...,{\overline{\sigma }}_K\frac{\left({\overline{C}}_{{\overline{\sigma }}_{K_z}}^1+{\underline{C}}_{{\overline{\sigma }}_{K_z}}^1\right)}{{\displaystyle \sum _{l=1}^R}\left({\overline{C}}_{{\overline{\sigma }}_j}^l+{\underline{C}}_{{\overline{\sigma }}_j}^l\right)},\hfill \\ \left.{\overline{\sigma }}_1\frac{\left({\overline{C}}_{{\overline{\sigma }}_1}^{R_z}+{\underline{C}}_{{\overline{\sigma }}_1}^{R_z}\right)}{{\displaystyle \sum _{l=1}^R}\left({\overline{C}}_{{\overline{\sigma }}_j}^l+{\underline{C}}_{{\overline{\sigma }}_j}^l\right)},...,{\overline{\sigma }}_K\frac{\left({\overline{C}}_{{\overline{\sigma }}_{K_z}}^{R_z}+{\underline{C}}_{{\overline{\sigma }}_{K_z}}^{R_z}\right)}{{\displaystyle \sum _{l=1}^R}\left({\overline{C}}_{{\overline{\sigma }}_j}^l+{\underline{C}}_{{\overline{\sigma }}_j}^l\right)}\right],\hfill \end{array}$$

(38)

The output of IT3-FLS(2) is written as:

$${\widehat{e}}_z(i+h)={\widehat{e}}_{z,forced}(i+h)+{\widehat{e}}_{z,free}(i+h),$$

(39)

where,

$${\widehat{e}}_{z,free}(i+h)=y_{\mathrm{IT}3-\mathrm{FNN}\left(2\right)}\left({\underline{\widehat{e}}}_z(i+h)|P_z\right),$$

(40)

$${\widehat{e}}_{z,forced}(i+h)={\displaystyle \sum _{\iota =0}^{h-1}}{\ell }_{\iota }\Delta {\mu }_{p_z}(i+h-\iota +1),$$

(41)

$$s(i-1)=\left[{\widehat{e}}_{z,step}(i+h)-{\widehat{e}}_{z,free}(i+h)\right]/d{\mu }_{p_z}\left(i\right),$$

(42)

$${\widehat{e}}_{z,step}(i+h)=y_{\mathrm{IT}3-\mathrm{FNN}\left(2\right)}\left({\underline{\widehat{e}}}_z(i+h)|P_z\right).$$

(43)

$${\underline{\widehat{e}}}_z(i+h)={\left[{\widehat{e}}_z(i+h-1)...,{\widehat{e}}_z^{}\left(i+h-\tau \right),{\mu }_{p_z}\left(i+h-1\right)\right]}^T.$$

(44)

$${\mu }_{p_z}\left(i\right)=\left\{\begin{array}{c}{\mu }_{p_z}\left(i-1\right)+d{\mu }_{p_z}\left(i\right)ifh>i-1,\hfill \\ {\mu }_{p_z}\left(i-1\right)else\hfill \end{array}\right..$$

(45)

The step size is denoted by $d{\mu }_{p_z}\left(i\right)$. The above equations can be rewritten as:

$$Q=S\Delta U_{p_z}+Q_{free},$$

(46)

where,

$$Q=\left[\begin{array}{c}{\widehat{e}}_z(i+1)\\ {\widehat{e}}_z(i+2)\\ ⋮\\ {\widehat{e}}_z(i+n_p)\end{array}\right],$$

(47)

$$S=\left[\begin{array}{cccc}{s}_0& 0& \cdots & 0\\ s_1& s_0& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ s_{n_p}-1& s_{n_p}-1& \cdots & s_0\end{array}\right],$$

(48)

$$\Delta U_{p_z}=\left[\begin{array}{c}\delta {\mu }_{p_z}\left(i\right)\\ \delta {\mu }_{p_z}(i+1)\\ ⋮\\ \delta {\mu }_{p_z}(i+n_p-1)\end{array}\right],$$

(49)

$$Q_{free}=\left[\begin{array}{c}{\widehat{e}}_{z,free}(i+1)\\ {\widehat{e}}_{z,free}(i+2)\\ ⋮\\ {\widehat{e}}_{z,free}(i+n_p)\end{array}\right],$$

(50)

$${\widehat{e}}_{z,free}(i+h)=y_{\mathrm{IT}3-\mathrm{FNN}\left(2\right)}\left({\underline{\widehat{e}}}_z(i+h)|P_z\right),$$

(51)

$${\underline{\widehat{e}}}_z(i+h)={\left[{\widehat{e}}_z(i+h-1)...,{\widehat{e}}_z^{}\left(i+h-\tau \right),{\mu }_{p_z}\left(i+h-1\right)\right]}^T,$$

(52)

$${\mu }_{p_z}\left(i\right)=\left\{\begin{array}{c}{\mu }_{p_z}\left(i-1\right)ifh>i-1,\hfill \\ {\mu }_{p_z}\left(i\right)else\hfill \end{array}\right..$$

(53)

From (34) and (46), one has:

$$\begin{array}{c}J=\iota Q^{T}Q+q\Delta U_{p_z}^T\Delta U_{p_z}\hfill \\ \begin{array}{c}=\iota {\left(S\Delta U_{p_z}+Q_{free}\right)}^T\left(S\Delta U_{p_z}+Q_{free}\right)\hfill \\ +q\Delta U_{p_z}^T\Delta U_{p_z}.\hfill \end{array}\hfill \end{array}$$

(54)

From (54), $\partial J/\partial \Delta U_{p_z}$ is obtained as:

$$\begin{array}{c}\partial J/\partial \Delta U_{p_z}=0\hfill \\ \Rightarrow \iota S^T{Q}_{free}+\Delta U_p\left(\iota S^{T}S+q\right)=0.\hfill \end{array}$$

(55)

From (55), $\Delta U_{p_z}$ is written as:

$$\Delta U_{p_z}=\iota {\left(\iota S^{T}S+q\right)}^{-1}{S}^T{Q}_{free}.$$

(56)

The control signal ${\mu }_{p_z}\left(i\right)$ is constructed as:

$${\mu }_{p_z}\left(i\right)={\mu }_{p_z}(i-1)+\Delta {\mu }_{p_z}\left(i\right),$$

(57)

where, $\Delta {\mu }_{p_z}\left(i\right)$ is the first term in vector $\Delta U_{p_z}$ in (56).

7. Stability Analysis

The main results concerning stability are given in Theorem 1.

Theorem 1.

The stability is guaranteed by controllers (58) and (59), compensators (60) and (61) and adaptation rules (62) and (63).

$${\mu }_1=-{\Xi }_1\left(\nu |P_{{\eta }_1}\right)-{\iota }_1{e}_1+D_t^{\beta }{\eta }_{d_1}+{\mu }_{c_1}+{\mu }_{p_1},$$

(58)

$${\mu }_2=-{\Xi }_2\left(\nu |P_{{\eta }_2}\right)-{\iota }_1{e}_2+D_t^{\beta }{\eta }_{d_2}+{\mu }_{c_2}+{\mu }_{p_2},$$

(59)

$${\mu }_{c_1}=-tanh\left(e_1^{}\right)\left({\overline{w}}_1+{\overline{\mu }}_{p_1}\right),$$

(60)

$${\mu }_{c_2}=-tanh\left(e_2^{}\right)\left({\overline{w}}_2+{\overline{\mu }}_{p_2}\right),$$

(61)

$$D_t^{\beta }{P}_{{\eta }_1}^{}=\gamma e_1^{}{C}_{{\eta }_1},$$

(62)

$$D_t^{\beta }{P}_{{\eta }_2}^{}=\gamma e_2^{}{C}_{{\eta }_2},$$

(63)

where, $0<\gamma \le 1$, ${\overline{\mu }}_{p_1}$ and ${\overline{\mu }}_{p_2}$ are the upper bound of ${\mu }_{p_1}$ and ${\mu }_{p_2}$, respectively.

Proof. 

Consider IT3-FLSs(1) with optimal parameters, then we have:

$$D_t^{\beta }{\eta }_1={\Xi }_1^{*}\left(\nu |P_{{\eta }_1}^{*}\right)+{\mu }_1+w_1,$$

(64)

$$D_t^{\beta }{\eta }_2={\Xi }_2^{*}\left(\nu |P_{{\eta }_2}^{*}\right)+{\mu }_2+w_2,$$

(65)

where, $w_1$ and $w_2$ are estimation errors. Considering control signals (58) and (59), one has:

$$\begin{array}{c}{D}_t^{\beta }{e}_1={\Xi }_1^{*}\left(\nu |P_{{\eta }_1}^{*}\right)-{\Xi }_1\left(\nu |P_{{\eta }_1}\right)-{\iota }_1{e}_1\hfill \\ +w_1+{\mu }_{c_1}+{\mu }_{p_1},\hfill \end{array}$$

(66)

$$\begin{array}{c}{D}_t^{\beta }{e}_2={\Xi }_2^{*}\left(\nu |P_{{\eta }_2}^{*}\right)-{\Xi }_2\left(\nu |P_{{\eta }_2}\right)-{\iota }_2{e}_2\hfill \\ +w_2+{\mu }_{c_2}+{\mu }_{p_2}.\hfill \end{array}$$

(67)

From (66) and (67), one has:

$${\Xi }_1^{*}\left(\nu |P_{{\eta }_1}^{*}\right)-{\Xi }_1\left(\nu |P_{{\eta }_1}\right)={\tilde{P}}_{{\eta }_1}^T{C}_{{\eta }_1},$$

(68)

$${\Xi }_2^{*}\left(\nu |P_{{\eta }_2}^{*}\right)-{\Xi }_2\left(\nu |P_{{\eta }_2}\right)={\tilde{P}}_{{\eta }_2}^T{C}_{{\eta }_2}.$$

(69)

From (68) and (69), the (66) and (67) are rewritten as:

$$\begin{array}{c}{D}_t^{\beta }{e}_1={\tilde{P}}_{{\eta }_1}^T{C}_{{\eta }_1}-{\iota }_1{e}_1\hfill \\ w_1+{\mu }_{c_1}+{\mu }_{p_1},\hfill \end{array}$$

(70)

$$\begin{array}{c}{D}_t^{\beta }{e}_2={\tilde{P}}_{{\eta }_2}^T{C}_{{\eta }_2}-{\iota }_2{e}_2\hfill \\ w_2+{\mu }_{c_2}+{\mu }_{p_2}.\hfill \end{array}$$

(71)

Now consider Lyapunov function as:

$$\begin{array}{c}V=\frac{1}{2}{e}_1^2+\frac{1}{2}{e}_2^2\hfill \\ +\frac{1}{2\gamma }{\tilde{P}}_{{\eta }_1}^T{\tilde{P}}_{{\eta }_1}^{}+\frac{1}{2\gamma }{\tilde{P}}_{{\eta }_2}^T{\tilde{P}}_{{\eta }_2}^{}.\hfill \end{array}$$

(72)

Then $D_t^{q}V$ is:

$$\begin{array}{c}{D}_t^{q}V\le e_1^{}{D}_t^q{e}_1^{}+e_2^{}{D}_t^q{e}_2^{}\hfill \\ -\frac{1}{\gamma }{\tilde{P}}_{{\eta }_1}^T{D}_t^{\beta }{P}_{{\eta }_1}^{}-\frac{1}{\gamma }{\tilde{P}}_{{\eta }_2}^T{D}_t^{\beta }{P}_{{\eta }_2}^{}.\hfill \end{array}$$

(73)

Substituting from (70) and (71) into (73), yields:

$$\begin{array}{c}{D}_t^{q}V\le -{\iota }_1{e}_1^2-{\iota }_2{e}_2^2\hfill \\ +w_1{e}_1^{}+e_1^{}{\mu }_{c_1}+e_1^{}{\mu }_{p_1}\hfill \\ +w_2{e}_2^{}+e_2^{}{\mu }_{c_2}+e_1^{}{\mu }_{p_2}\hfill \\ +{\tilde{P}}_{{\eta }_1}^T{C}_{{\eta }_1}{e}_1^{}-\frac{1}{\gamma }{\tilde{P}}_{{\eta }_1}^T{D}_t^{\beta }{P}_{{\eta }_1}^{}\hfill \\ +{\tilde{P}}_{{\eta }_2}^T{C}_{{\eta }_2}{e}_2^{}-\frac{1}{\gamma }{\tilde{P}}_{{\eta }_2}^T{D}_t^{\beta }{P}_{{\eta }_2}^{}.\hfill \end{array}$$

(74)

From (62) and (63), the inequality (74), becomes:

$$\begin{array}{c}{D}_t^{q}V\le -{\iota }_1{e}_1^2-{\iota }_2{e}_2^2\hfill \\ +w_1{e}_1^{}+e_1^{}{\mu }_{c_1}+e_1^{}{\mu }_{p_1}\hfill \\ +w_2{e}_2^{}+e_2^{}{\mu }_{c_2}+e_1^{}{\mu }_{p_2}.\hfill \end{array}$$

(75)

Considering the upper bounds of $w_1$, $w_2$, ${\mu }_{p_1}$ and ${\mu }_{p_2}$ as ${\overline{w}}_1$, ${\overline{w}}_2$, ${\overline{\mu }}_{p_1}$ and ${\overline{\mu }}_{p_2}$, one has:

$$\begin{array}{c}{D}_t^{q}V\le -{\iota }_1{e}_1^2-{\iota }_2{e}_2^2\hfill \\ +{\overline{w}}_1\left|e_1^{}\right|+e_1^{}{\mu }_{c_1}+\left|e_1^{}\right|{\overline{\mu }}_{p_1}\hfill \\ +{\overline{w}}_2\left|e_2^{}\right|+e_2^{}{\mu }_{c_2}+\left|e_2^{}\right|{\overline{\mu }}_{p_2}.\hfill \end{array}$$

(76)

Substituting the compensators from (60) and (61), results in:

$$D_t^{q}V\le -{\iota }_1{e}_1^2-{\iota }_2{e}_2^2.$$

(77)

Form (77), it is concluded that $e_1\in {\ell }^2$ and $e_2\in {\ell }^2$, then through the Barbalat’s lemma, the stability (in the sense of asymptotic) is ensured. □

Remark 1.

In this paper, a T3-FLS is suggested for uncertainty estimation. Compared with type-2 FLSs, the T3-FLSs result in better accuracy in noisy practical conditions. In addition, the complexity of T3-FLSs is not much more than the type-2 counterpart to be a problem.

Remark 2.

It the designed scheme, the dynamics of the robot are considered to be unknown, and are estimated by the suggested T3-FLS. Then, the designed controller can be applied for other systems with same dimensionality.

8. Simulations

The dynamics of the reference plant is considered as:

$$\left\{\begin{array}{c}{D}_t^{0.98}{\chi }_{11}=35\left({\chi }_{12}-{\chi }_{11}\right)+35{\chi }_{12}{\chi }_{13}\hfill \\ D_t^{0.98}{\chi }_{12}=25{\chi }_{11}-5{\chi }_{11}{\chi }_{13}+{\chi }_{12}+{\chi }_{14}\hfill \\ D_t^{0.98}{\chi }_{13}={\chi }_{11}{\chi }_{12}-4{\chi }_{13}\hfill \\ D_t^{0.98}{\chi }_{14}=-100{\chi }_{12}\hfill \end{array}\right.$$

(78)

The simulation parameters are given in

Table 1. Simulation conditions.

ℜ	m	0.10	-
L	m	0.50	-
w	rad/s	−1	1
d	m	0.01	0.30
$I_c$	Kg m${}^2$	6	12
$m_c$	kg	87	181
$m_w$	kg	1.50	-
$I_m$	Kg m${}^2$	0.003	-
u	m/s	0	1
$I_w$	Kg m${}^2$	0.008	-

. The four switching modes are depicted in Figure 5. The output signals $\zeta 1,{\zeta }_2$ along with the reference signals are depicted in Figure 6. A good tacking efficiency is seen from Figure 6. The corresponding control signals are shown in Figure 7. The estimated signals and path portraits in all modes are given in Figure 8 and Figure 9, in which good tracking is seen. The designed chaotic path with suddenly switching criteria results in a strong secure path for patrol robots.

For a comparison between various typed of FLSs, the dynamics estimation of (15) without a controller, is considered as:

$$\left\{\begin{array}{c}{\dot{\widehat{\zeta }}}_1={\widehat{H}}_1\left(\zeta |{\theta }_1\right)\hfill \\ {\dot{\widehat{\zeta }}}_2={\widehat{H}}_2\left(\zeta |{\theta }_2\right)\hfill \end{array}\right.$$

(79)

where, $c_{{\tilde{A}}_{\iota |\alpha }^l},l\in \left[-70,70\right]$, $\overline{\alpha }\in \left\{0.8,1\right\}$, $\overline{\alpha }\in \left\{0.9,1\right\}$ and $\underline{\alpha }\in \left\{0.5,0.6\right\}$. Moreover, a noise is added as follows: the variance is equal to 0.04 and 0.15 and the mean is equal to zero. The obtained RMSEs are given in

Table 2. Comparison of RMSE.

Variance	FLS	${\widehat{\mathit{\zeta }}}_1-{\mathit{\zeta }}_1$	${\widehat{\mathit{\zeta }}}_2-{\mathit{\zeta }}_2$
0	IT2-FLS	4.2414	5.1804
	T3-FLS	1.8174	2.1140
0.05	IT2-FLS	2.2527	3.2128
	T3-FLS	1.7624	1.2441
0.1	IT2-FLS	5.3274	6.2230
	T3-FLS	1.79984	1.7779

. The RMSEs in Table 2 reveal the better efficiency of the suggested approach.

9. Conclusions

In this paper, a control system is proposed for mobile robots. The dynamics of robots are online estimated by an optimized T3-FLS. A predictive compensator is designed to deal with unpredicted disturbances. The stability is analyzed, and it is shown that the suggested approach is very effective. A chaotic path following is proposed for patrol robots. The suggested path is strongly secure, and it is switched between various chaotic paths. In simulations, four modes are considered for path following. The robot’s path is suddenly switched from one path to another. Each path is generated by a chaotic signal that is complicated and unpredictable. To better verify the effectiveness of the suggested T3-FLS, a comparison with other types of FLS is presented, and it is seen that T3-FLSs are more suitable for high-uncertain environments. For future studies, the suggested controller can be applied to a real prototype mobile robot.