Design of an Adaptive Fuzzy Sliding Mode Control with Neuro-Fuzzy system for control of a differential drive wheeled mobile robot

Abstract This paper presents the design of a novel trajectory tracking control strategy and the development of a mathematical model for a non-holonomic differential-drive wheeled mobile robot. The proposed control system utilizes a dual-loop approach, where the inner loop controls the dynamics by employing Adaptive Fuzzy Sliding Mode Control (AFSMC), and the outer loop, handles kinematics by utilizing an Adaptive Neuro-Fuzzy Inference System ;(ANFIS). The ANFIS is employed to minimize the error between the actual and desired velocities, providing a desired input for the inner loop. Meanwhile, the AFSMC is used to effectively control the system dynamics. The use of these dual-loop controllers considerably improves the system’s overall efficiency. The inner controller compensates for dynamic disturbances, while the outer controller manages velocity errors. We integrate the actuator dynamics and the chopper effect of the wheels in the dynamics modeling, which helps to increase the models accuracy. MATLAB was used to implement the controller, while circular and eight-shaped trajectories were generated to assess the performance of the proposed controller. In addition, a comparative analysis of different controllers such as PID, SMC, AFSMC, and AFSMC with ANFIS was presented. The simulations were conducted under uncertainties, and the proposed controller is better than other controllers at tracking desired trajectories. The Lyapunov stability analysis is employed to verify the stability of the proposed controller. This paper shows that the proposed dual-loop controller is stable and more robust to internal parameter variation and external disturbance for the examined system. In general, the AFSMC with ANFIS is superior in trajectory tracking for the examined system compared to other controllers.


PUBLIC INTEREST STATEMENT
Most human activities have been replaced by robots due to their versatility in various environments and their ability to perform tasks with quality, accuracy, and efficiency.In existing literature, much of the focus is on robotic system modeling and controller design, mainly centered on the simpler kinematics model.However, in scenarios involving high-speed movements or heavy load transportation, relying solely on the kinematics model may not provide optimal velocity tracking.To address this, we tackle the challenge by considering both kinematic and dynamic models in our approach.We designed an Adaptive Neuro-Fuzzy Inference System (ANFIS) with an AFSMC kinematic controller to reduce trajectory tracking errors in mobile robots to further enhance their efficiency.Unlike many previous studies, we also take into account the significance of actuator and chopper dynamics, as neglecting them can lead to imperfections in the control law, especially in high-speed torque and variable load scenarios.

Introduction
Most human activities have been replaced by robots due to their versatility in various environments and their ability to perform tasks with quality, accuracy, and efficiency.Robots have particularly played a crucial role in liberating humans from hazardous and dangerous tasks such as nuclear waste cleaning, mining, military combat operations, fire-fighting operations, sea exploration, and space exploration.A wheeled mobile robot (WMR) is a vehicle equipped with motors that enable it to move autonomously in a specified direction.Among the different types of robots, the popularity of the differential drive mobile robot (DDMR) has grown significantly due to its simplicity, ease of control, and flexibility (Leena & Saju, 2016).
Extensive research has been conducted on mobile robots, but most of the literature concerning DDMRs focus solely on their kinematics models.While this assumption works well for low-speed, low-acceleration, and light-load applications, it is essential to consider dynamic modeling for mobile robots designed to operate at high speeds and perform tasks with heavy loads (Albagul & Wahyudi, 2004).The low-level velocity control loops (kinematic control loop) implemented on mobile robots do not guarantee perfect velocity tracking.Therefore, accounting for robot dynamics becomes necessary to minimize tracking errors.In the outer loop, a desired velocity command is generated based on the robot's current position and the desired position.On the other hand, the inner loop compensates for the robot's dynamics to minimize the error between the robot's actual velocity and the desired velocity generated by the kinematic controller.
The major contribution of this research paper is to improve the proposed models accuracy by taking into account the actuator and chopper dynamic impacts on the system.The cascaded controller, which includes the inner loop (to manage the dynamic system) and the outer loop (to modify the velocities of the mobile robot), is intended to enhance the overall systems efficiency.The suggested controller's performance is investigated using various trajectories, including eightshaped and circular trajectories.In the presence of unknown disturbances, the cascaded controller demonstrates good trajectory tracking performance for the DDMR system.

Statement of the problem
Nowadays, robots are used for various industrial applications.However, most robots don't have the most appropriate controller which leads to unsatisfactory results.As a result, numerous scholars have proposed various control mechanisms to achieve the desired objectives.However, most of them worked on controlling a DDWMR by considering its kinematic model only, but such type of model limits the robots tasks leading to slow speed and light-load application.
This research paper considers both kinematics and dynamics.Moreover, in the dynamic model, the actuator and chopper effects are considered, which improves the accuracy of the model.The inner loop and outer loop control techniques are developed, and the former is used to regulate the system dynamics using Adaptive Fuzzy Sliding Mode Control (AFSMC), while the latter is utilized to manage kinematics by using an Adaptive Neuro-Fuzzy Inference System (ANFIS).
The sliding surface slope (λ) and switching gain k ð Þ are two tuning control parameters for the SMC.Most articles use trial and error approach to find these parameters, however these methods are ineffective for time-varying systems such as mobile robots.Fuzzy logic is employed in this study to change these control settings based on the current system situation.As a result, the proposed controller can tolerate parameter variations and external disturbances.Adaptive Fuzzy SMC (AFSMC) uses fuzzy logic to tune SMC parameters such as sliding surface slop (λ) and switching gain k ð Þ.The AFSMC detects speed errors and generates an input voltage to the actuator.
The fuzzy logic controller (FLC) has the ability to imitate human knowledge representation and explanations, and the ANN controller is capable of performing parallel computations of learning.The ANFIS is a type of supervised machine learning which uses label data as inputs and outputs.In this paper, the training and testing data is obtained from the pose error (input) and the angular and linear velocities (outputs).The input data obtained from pose errors and output data obtained from velocities are used to train the ANFIS model to estimate the velocities of the robot which are used as references inputs to the inner loop.
The main contribution of this research is to improve the models accuracy by taking into account the system's actuator and chopper dynamic effects.The cascaded controller, which consists of the inner loop (AFSMC) and outer loop (ANFIS), is intended to improve the overall efficiency of the system.The goal is to use the suggested controller to obtain a good trajectory tracking performance of the DDMR in the presence of unknown disturbances.To examine the performance of the suggested controller, eight shape and circular trajectories were examined.
The remaining sections of the paper are organized as follows.Section 1 presents prior work in the domain of mobile robots.Section 2 presents the derivation of the nonlinear mathematical modeling of DDWMR and the evaluation and simulation the system's open loop model, and section 3 presents the design control techniques for nonlinear WMR system.Section 4 discusses the simulation findings, and Section 5 presents the conclusion.

Literature reviews
Numerous researchers have explored different papers related to the control of Differential Drive Wheeled Mobile Robots (DDMR) using various control techniques.For this literature review, we have carefully selected recent and relevant papers to discuss.The reviewed papers are as follows: In (Zangina et al., 2020), the authors proposed a nonlinear PID Controller for Trajectory Tracking Control of a Differential Drive Mobile Robot (DDMR).They designed a nonlinear PID controller for both kinematic and dynamic models, utilizing a trial and error approach with six tuning parameters to enable the Robot to track a desired trajectory.While the simulation results show that the proposed nonlinear PID controller achieves stability and trajectory tracking control, the accuracy falls short.It should be noted that the nonlinear PID Controller is widely used in industries due to its simplicity in design, ease of implementation, and high efficiency for linear systems.However, satisfactory performance may be challenging to attain in nonlinear systems.
In (Lee et al., 2018), the authors applied a Fuzzy-PID controller for path tracking of a DDMR and conducted a comparative study with PID controllers.The simulation results indicated that the fuzzy-PID controller outperformed the classical PID controller with arbitrary initial states.The authors concluded that the fuzzy-PID controller is a suitable choice for path tracking control of DDMRs when compared to the PID controller.Nevertheless, it should be noted that fuzzy-PID is heavily dependent on expert knowledge, and its utilization of fuzzy techniques doesn't always guarantee system stability and robustness.
In (Thay et al., 2018), the authors proposed Sliding Mode Control (SMC) as a dynamic controller for trajectory tracking control of an autonomous mobile robot system.They compared the performance of the proposed SMC-based dynamic controller to a PID-based dynamic controller, both with and without uncertainties.The simulation results showed that the proposed dynamic controller based on SMC achieved better performance than the dynamic controller based on PID.However, the chattering in traditional SMC limited its practical applicability due to its undesirable control input behavior.
A number of researchers have proposed various variants of Sliding Mode Control (SMC) techniques to address chattering while enhancing performance and robustness.Factors such as reaching law selection, sliding surface, and gain tuning significantly impact the effectiveness of SMC in minimizing chattering and improving trajectory tracking for mobile robot.Some approaches include boundary layer SMC, Quasi-SMC with smoothing functions (Benaziza et al., 2017), Terminal SMC (Benaziza et al., 2017;Moudoud et al., 2023), and adaptive SMC (Phuc et al., 2021).These methods offer viable solutions to overcome the challenges associated with chattering in SMC.
In (Moudoud et al., 2023), the authors proposed an Adaptive Terminal Integral Sliding Mode Control (ATISMC) for trajectory tracking in a Wheeled Mobile Robot (WMR).It exhibits robustness, fast convergence, and chattering avoidance.However, incorporating terminal sliding mode may increase complexity, especially for complex dynamic systems like mobile robot.
In (Benaziza et al., 2017), the authors presented two control laws (Quasi-SMC and global terminal sliding mode (GTSM) control) for the trajectory tracking control of non-holonomic Mobile Robots.The quasi sliding mode control was proposed for the angular velocity to converge the angle error to zero quickly with asymptotic stability and to avoid actuator chattering effects.The GTSM control was proposed for linear velocity to bring the position error to zero and ensure asymptotic stability using the Lyapunov theory.The simulation results demonstrated that this proposed controller achieved faster convergence of tracking errors for given circular and sinusoidal reference trajectories.However, it's important to note that the authors did not consider the system dynamics in this paper.
In (Phuc et al., 2021), the authors presented an Adaptive Fuzzy Sliding Mode Control (AFSMC) for trajectory tracking control of a non-holonomic Mobile Robot system.They designed SMC, Fuzzy SMC (FSMC), and AFSMC controllers and compared their performance in converging the Mobile Robot on desired position, velocity, and orientation trajectories.The Simulink results demonstrated the efficiency of the proposed AFSMC, which showed strong resistance and could handle problems such as parameter variation and system disturbance while also eliminating chattering.However, it is worth mentioning that this paper focused on designing only a dynamic controller, which is not ideal for trajectory tracking control of DDMRs.Additionally, actuator dynamics were not considered, which may lead to reduced system performance.
In existing literature, much of the focus is on system modeling and controller design, mainly centered on the simpler kinematics model.However, in scenarios involving high-speed movements or heavy load transportation, relying solely on the kinematics model may not provide optimal velocity tracking.To address this, we tackle the challenge by considering both kinematic and dynamic models in our approach.We design an Adaptive Neuro-Fuzzy Inference System (ANFIS) with an AFSMC kinematic controller to reduce trajectory tracking errors in mobile robots.Unlike many previous studies, we also take into account the significance of actuator and chopper dynamics, as neglecting them can lead to imperfections in the control law, especially in high-speed torque and variable load scenarios.Our study involves using an electrical actuator to drive the robot's wheels, and we carefully consider the supplied force or torque in the dynamic model of the system.By meticulously considering both kinematic and dynamic aspects and accounting for actuator dynamics and chopper effect, we aim to improve the accuracy of the model for dynamic differential mobile robots (DDMRs).To further enhance the system's efficiency, we design a cascaded controller with an inner loop (AFSMC) and an outer loop (ANFIS).This comprehensive approach seeks to achieve superior trajectory tracking performance in DDMRs.

Mathematical model
In order to design a controller for any physical system, we must first develop a precise mathematical model.The dynamics and kinematics of a differential drive robot are derived in this section, as well as the limitations (non-holonomic constraint) of the kinematic model.Robot's local mobility is reduced due to the pure rolling nature of the wheels.This constraint is referred to as a nonholonomic constraint, and it is discussed.

Coordinate systems
In order to describe the position of the DDWMR in this environment, first need to be defined Inertial Coordinate (global frame) and robot Coordinate.The Inertial frame is considered as the reference frame and is denoted as X I ; Y I f g.And the robot coordinate system is denoted as X r ; Y r f g.The origin of the robot frame is the mid-point A on the axis between the wheels, as shown in Figure 1 (Dhaouadi & Hatab, 2013).The robot's center of mass C is assumed to be on the symmetry axis, at a distance d from the origin A. In the inertial reference frame, the robot's pose can be mathematically described as: The mapping among these two frames is an essential issue that needs to be addressed at this point.
Let X r ¼ where R θ ð Þ is the rotation matrix.
This transformation will enable also the handling of motion between frames.
The relationship between the velocities in the inertial frame and the robot frame is described by Equation 3. As will be seen in the next section, it is critical in the development of the DDMR kinematic and dynamic models.

Non-holonomic constraint
Two non-holonomic constraint equations characterize the motion of a differential-drive mobile robot, which are derived from two main assumptions: no lateral slip motion and pure rolling constraint.The robot can only move in a curved motion (forward and backward) but not sideways, according to the first constraint.This condition, as written in Equation 4, means that the velocity of the center point A along the lateral axis is zero in the robot frame.The second constraint states that each wheel maintains a single point of contact with the ground.There is no movement of the wheel along its X-axis and no slipping of the wheel along its Y-axis.In Equation 6, the speed of the contact points in the robot frame are related to the speed of the wheels.
Using the orthogonal rotation matrixR θ ð Þ, the velocity in the inertial frame gives The robot's velocities can be calculated as a function of the velocities of the robot's center point, A, in the inertial frame of reference.

Figure 1. Differential drive wheeled mobile robot (DDWMR).
The rolling constraint equations can be written as follows, using the rotation matrix: Figure 2 depicts the rolling motion constraint (Dhaouadi & Hatab, 2013).
Using the contact points velocities from equation x; y ð Þ and substituting in x; y ð Þ the three constraint equations can be written in the following matrix form: The above constraints matrix Λ q ð Þ will be used for the DDWMR dynamic modeling.

Kinematic model
The fundamental goal of kinematic modeling for the DDWMR is to represent the robot velocities in relation to the wheel speed as well as the robot's geometric parameters.The average linear speed of the two wheels in the robot frame is the linear speed of each wheel in the robot frame.

And the angular velocity of the DDWMR is
The DDMRs velocities in the robot frame can now be expressed in terms of the inertial frame's center-point A velocities as follows: The DDMR velocities can be represented in terms of the linear and angular velocities of DDMR in the inertial frame to obtain an alternative form for the above kinematic model.The final kinematic model looks like this: For implementation and control reasons, it is often necessary to control the robot by wheels angular velocities ω R ; ω L the transformation have straightforward form where L; R, ω R andω L are distance between wheels, radius of wheel, angular velocities of right and left wheel.

Dynamic modeling of the DDMR
The dynamic model of the DDMR is vital for simulation study of the DDMR motion and for designing different motion control algorithms.A non-holonomic DDMR with n generalized coordinates q 1 ; q 2 ; q 3 ; . . .q n ð Þ and subject to m constraints can be described using the following equations of motion: where: , τ, and Λ T q ð Þ are an nxn systematic positive definite inertia matrix, centripetal and coriolis matrix, the surface friction matrix, gravitational vector, vector of bounded unknown disturbances, input matrix, input vector, matrix involved with the kinematic constraint, and the Lagrange multipliers vector, respectively (Dhaouadi & Hatab, 2013).By considering the kinetic and potential energies of a system, the Lagrange method is used to derive the equations of motion for that system.The Lagrange equation is written as follows: where L ¼ K T À P T is the Lagrangian function, K T ; P T are the total kinetic and potential energy of the system, respectively.q i are the generalized coordinates, F is the generalized force vector, Λ is the constraints matrix, and λ 0 is the vector of Lagrange multipliers associated with the constraints.Since the DDMR is moving in X I ; Y I f g plane, therefore the potential energy of the DDMR is considered to be zero.
For the DDMR, the generalized coordinates are selected as The DDMR's kinetic energies are equal to the sum of the kinetic energy of the robot platform without wheels plus the kinetic energy of the wheels and actuators.
The kinetic energy of the robot platform is While the kinetic energy of the right and left wheel is Using (Equation 20-Equation 22) along with Equation 7,Equation 8), the total kinetic energy of the DDMR is where m is the total mass of the robot, I is the total equivalent inertia, I w is the moment of inertia of each driving wheel with a motor about the wheel axis,I m is the moment of inertia of each driving wheel with motor about the wheel diameter.I c is the moment of inertia of the WMR about the vertical axis through the center of mass, m w is the mass of each driving wheel (with actuator), m C is the mass of WMR without the driving wheels and actuators (PMDC motor).
The final Lagrangian function, L ¼ K T the equations of motion of the DDMR are given by: where, Equation 24 is then transformed into a more convenient form for the purposes of control and simulation.The main objective is that it eliminate the constraint term Λ T q ð Þλ 0 in equation since the Lagrange multipliers λ i 0 are unknown.This is accomplished by first determining the reduced vector.
Finally, the vector is reduced and transformed into an alternative form, which is represented by DWMR's linear and angular velocity.It is simple to show that the model in Equation 24 can be rearranged in the following compact form using the kinematic model in Equation 11,Equation 12), which is more fully explained in (Dhaouadi & Hatab, 2013).
where, τ R ; τ Lv are the torque in the right and left wheel.

Overall system model including actuator dynamics
In most literature on designing a Mobile Robot controller (Bosera et al., 2023;Hu & Woo, 2006), the actuator dynamics are often overlooked, despite being crucial for overall Robot dynamics.Electrical actuators are preferred for driving Mobile Robots due to their high controllability (Mija et al., 2014;Yulin, 2010).In this study, we opted for a PMDC motor to drive the Robot's wheels.To adjust the motor's armature voltage, we employed a chopper, which effectively converts a constant voltage source into a variable one, significantly enhancing DC motor performance.This chopper, a power electronics device, rapidly connects and disconnects the load from the source, generating chopped DC voltage (V O ) at the load terminals.Among the power semiconductor devices used in the chopper circuit, MOSFETs are the most common and suitable for highfrequency switching, benefiting from zero storage time for minority carriers (Eya et al., 2022).Hence, we selected MOSFETs for their advantageous characteristics.
The operation frequency of chopper ( and its duty cycle is: D ¼ t on =T To calculate the duty cycle D ð Þ, assume that the converter is in a steady-state condition.The switches are treated as an ideal, and the losses in the inductive and the capacitive elements are ignored for the sake of simplicity.Besides this, assume that the converter only operates in continuous conduction mode.Therefore, the average output is given by Equation 27.Thus, we observed that the average output voltage of a chopper can be controlled by varying the duty cycle of the circuit and this paper uses a constant switching frequency.In this manner, varying the on-time of the system, while the chopping frequency is kept fixed.Generally this method is achieving the appropriate variable voltage by manipulating the width of the pulse (PWM).
The schematic diagram of the PMDC motor with converter, gear, and its load is shown in Figure 3 (Mihailo, 2012).The DC-DC converter (chopper) which is given by Figure 3's output gives the variable output dc voltage V O ¼ V a ð Þ needed to drive the motor and the DC motor is represented by its equivalent circuit which consists of an inductor L a ð Þ and resistor R a ð Þ in series with the back EMF E a ð Þ.To reduce the speed of the wheel (increased torque), DC motors are coupled with high reduction gears as well as the gears are directly connected with wheel.
The armature voltage V a is used as a control input while the field circuit conditions remain constant.In fact, the armature circuit equations for a DC motor are given by Equation 28.
where,i a ; K b ; K t ; τ; N; τ m andω m are the armatures current, back EMF constant, and torque constant, output torque applied to the wheel, gear ratio, motor torque, and rotor velocity.
By integrating the equations for each motor with the mechanical dynamics of the DDMR, we obtain the complete dynamic equations for the DDMR with actuators.Any additional torque disturbances affecting the wheels can be included as additive terms to the motor torque.Figure 4 presents a block diagram representing the system.The forward kinematic model, when combined with the dynamic model, forms a comprehensive model for simulating and analyzing the DDMR.Mobile Robots are mechanically connected to the robot wheels through gears.
The motion equations for the Motors are directly related to the dynamics of the Robot's mechanical system.This rearrangement of the equation gives us: It is important to note that the input to the driving mechanism of the robot is the armature voltage of the DC motors.Therefore, by ignoring inductance, La, and using Equation 28-Equation 30), the following equation is obtained.Finally, the complete nonlinear dynamics model representation of the wheeled mobile robot with actuator dynamics becomes:

Open loop model verification simulation
The model verification of the open loop model is achieved using MATLAB Simulink.The DC-motor and mobile robot parameters from (Yousfi Allagui et al., 2021) were obtained and a constant input voltage V R ; V L ð Þ was given to the dynamic model.As we have seen from the simulation response depicted in Figures 5-10, the behavior of the Simulink model of Mobile Robot is similar to the behavior of the real-time physical system.As a result, we can generalize that our developed model has properly symbolized the real-time system.
In this section, we validate the mobile robot's dynamic model by applying diverse input voltages and comparing its kinematic response to the real system's behavior.In the simulation, four openloop scenarios were tested for the mobile robot.The model's responses closely matched the expected behavior, demonstrating its accuracy in representing the system.While not identical due to some unmeasured parameters, the model closely approximates the physical robot's behavior, making it suitable for controller design and future applications.The validated dynamic model serves as a strong foundation for improving the robot's control and overall performance.

Controller design
In this paper, the trajectory tracking controller design and organization are divided into two phases.First, the kinematic (inner loop) controller is used to generate the desired linear and angular velocities based on differences between the desired position x d ; y d ð Þ and the actual robot position x; y ð Þ.The second part is the dynamic controller (outer loop) design, which is used to compensate for the dynamic effects of the mobile robot.The complete structure of the trajectory tracking controller is shown at Figure 11.

AFSMC for trajectory tracking of mobile robot
The trajectory tracking control of the wheeled mobile robot based on an AFSMC is discussed in this section.This controller allows the mobile robot to move towards the desired pose and track any reference Trajectories.Adaptation in Fuzzy Slide Mode Controller can offer a system's ability of online-adjustment as well as enhancement of the system tracking performance.The Supervisory Controller is an adaptive controller that can search for the current performance of the system and manipulate the controller's parameters to enhance the system's performance (Ashagrie et al., 2021).The boundary layer/saturation function is used in this work at the discontinuous part of SMC instead of using the Signum function; due to this replacement the chattering effect in the control   input is eliminated.Furthermore, the adaptive Fuzzy Inference scheme is utilized to minimize the effect of perturbation.Since the Lambda (λ) andk are factors that have a vital impact on the discontinuous component of the control system, the Adaptive part of the proposed controller is to manipulate these two significant gains to maximize the controller performance.Therefore, if these factors are properly adjusted, the SMC can be more convenient for disturbance rejection from the proposed system.Consequently, AFSMC has a strong resistance to uncertainty and can often solve problems.The block diagram of the AFSMC is presented in Figure 12.

Sliding mode control stability using Lyapunov stability
The sliding surface vector s The control signal is composed of two terms which are the equivalent U eq À � and the switching U Sw ð Þ control laws, represented by: where, Also, the design of the switching law is derived from the control algorithm and the control law is written as: Then, substituting the equation, we get This study investigates the stability of the dynamic control using the Lyapunov criteria.The Lyapunov method is applied to analyze the stability of the system with SMC.A quadratic Lyapunov candidate function, expressed in terms of the sliding surfaces (Moudoud et al., 2021(Moudoud et al., , 2022)), is utilized for this purpose.The Lyapunov function derivative is calculated as: Remark 1: The proposed modified SMC ensures guaranteed stability for the robot manipulator, with positive definite V and negative definite _ V for allt !0. k 1 andk 2 Must be positive.The sliding variables s 1 and s 2 converge to zero as t ! 1.By preserving this stability criteria, the approach leverages the advantages of classical SMC with Lyapunov stability analysis, ensuring a stable system.The sliding mode controller guarantees closed-loop system stability, demonstrating the effectiveness of the approach.

Fuzzy sliding mode controller
In this section, we present the Fuzzy Sliding Mode Controller (FSMC), a hybrid approach that combines fuzzy logic with SMC.By integrating a FLC in place of the equivalent dynamic part of SMC as shown in Figure 12, FSMC inherits robustness and stability while effectively addressing parameter variations and dynamic challenges.The FLC utilizes linguistic variables and a rule base to determine control actions based on real-time feedback, ensuring precise trajectory tracking for the mobile robot.FSMC incorporates three linguistic variables: "Error" (e) with linguistic terms NB, NM, NS, Z, PS, PM, PB; "Change of Error" ( _ e ̇) with the same linguistic terms; and "Equivalent input part of SMC" U eq À � with linguistic terms NB, NM, NS, Z, PS, PM, PB.By intelligently adapting control actions using fuzzy logic inference, FSMC achieves accurate trajectory tracking and handles uncertainties and dynamic variations.The rule base for FSMC with the linguistic variables is presented in Table 1, comprising of 49 rules to determine the corresponding output (Equivalent control part of SMC).

Self Tuning Fuzzy Sliding Mode Control (ST-FSMC)
The Self-Tuning Fuzzy Sliding Mode Control (ST-FSMC) is an advanced control method that combines adaptive techniques with SMC to enhance trajectory tracking of mobile robots.The FLC plays a vital role, using triangular membership functions to efficiently evaluate input and output variables for fuzzy inference.By employing fuzzy rules, the FLC dynamically adjusts the control parameter (α) based on the current input signal (s), optimizing the control process.Online tuning of gain parameters (λ i andk i ) for each robot ensures accurate trajectory tracking and stability.To achieve smoother control, a saturation function replaces the signum function in the discontinuous part of SMC, and an adaptive fuzzy inference scheme enhances robustness by effectively handling

ANFIS controller for mobile robot system
The trajectory tracking of the wheeled mobile robot based on an ANFIS control allows the mobile robot to move towards the desired pose and allows it to track any given trajectory.

Adaptive Neuro-Fuzzy Inference System
The Adaptive Neuro-Fuzzy Inference System (ANFIS) is a powerful combination of Artificial Neural Networks (ANNs) and Fuzzy Logic (FL).It offers fast learning and adaptive interpretation to model complex patterns and understands nonlinear relationships (Chopra et al., 2021).ANFIS is widely used for approximating highly intricate nonlinear systems, boasting accuracy and interpretability, which enhance its generalization ability.The ANFIS architecture automatically generates knowledge in the form of fuzzy "if-then" rules by determining the optimal membership functions and parameters for the rule base, thereby serving as a comprehensive and automated machine learning solution (Eya et al., 2023).Additionally, ANFIS provides a method to address nonholonomic problems (Mondada et al., 1994).

ANFIS controller for non-holonomic WMR
The purpose of a controller is to set the appropriate input to the wheel to keep the Robot on track in the given desired trajectory.There is a difficulty in compensating for the difference between the robot's track Þ to converge to zero, therefore, it's hard to get accurate parameters from real robots.Consequently, a new approach based Adaptive   Neuro-Fuzzy is taking into account kinematic models of the proposed system.Generally, the control of a wheeled mobile robot consists of following a reference trajectory and attempting to measure both the position and orientation with respect to a fixed frame.This approach is composed of two neuro-fuzzy controllers, both position and orientation control, allowing tracking of these desired trajectories.If we combine the position and orientation controls, we can control the robot's trajectory.Inputs e x ; e y and e θ are the differences in between the real position of the robot given by The linguistic variable "distance" is obtained by using Equation 45.
The linguistic variable "alpha" which is the angle of orientation that the MR forms between the straight line of actual and target position is calculated by using Equation 45.
where,v and ω are given by ANFIS position control, and w is given by ANFIS orientation control.Here, the angular velocity of two wheels ω R andω L are given by Equation 15.

ANFIS controller based kinematic trajectory tracking
The ANFIS design comprises three models: ANFIS 1 (input: distance and alpha, output: linear speed "v"), ANFIS 2 (input: distance and alpha, output: angular speed "w1"), and ANFIS 3 (input: theta Error, output: angular speed "w2").The total angular speed is obtained by adding "w1" from ANFIS 2 and "w2" from ANFIS 3. The ANFIS design process involves four key areas performed by the Neuro-Fuzzy Designer: loading, plotting, and clearing data; generating or loading the initial FIS structure; training the FIS; and validating the trained FIS.In the loading phase, a Training data set with input/output data for modeling the system is loaded as column vectors, with the output data in the last column.The designer displays the data in a plot, with training, testing, and checking data annotated in blue, diamonds, and pluses, respectively.
Next, the initial FIS structure is specified by generating it using grid partitioning or by loading a previously saved Sugeno-type FIS structure from a file or MATLAB workspace.The graphical representation of the initial FIS structure can be viewed by clicking on "Structure." The training phase involves choosing an optimization method (hybrid or backpropagation) to adjust the membership function parameters based on the training data.Stopping criteria, such as the number of training epochs and error tolerance, are set.Clicking "Train Now" adjusts the membership function parameters and displays error plots.
After training, the FIS model is validated using separate testing or checking data sets.Validation data is loaded, and clicking "Test Now" plots the test data against the FIS output in red on the plot.The Neuro-Fuzzy design structure is shown in Figure 15. Figure 16 illustrates the structure of Anfis11, ANFIS 1's position controller, with "distance" and "alpha" as inputs and "v" as the output.Figure 17 displays Anfis12, ANFIS 2's position controller, with the same inputs and "w" as the output.Figure 18 depicts Anfis13, ANFIS 3's orientation controller, with "theta Error" as the input and "w" as the output.
The ANFIS kinematic controllers play a crucial role in adjusting robot motion direction and swiftly moving it toward the target pose.ANFIS learns and refines parameters in membership functions and fuzzy rules using training data, enabling the mobile robot to track a desired trajectory.The neuro-fuzzy controller proves to be effective in tracking predefined reference trajectories, making it a valuable tool in mobile robot control.
Ultimately, the mobile robot implements the dual controller, as depicted in Figure 11.This dual controller combines both kinematic and dynamic control approaches to effectively control the mobile robot system.

Results and discussion
This section evaluates the performance of the proposed control method under different conditions: with and without internal parameter change and system disturbance.The simulation is conducted

Simulation result of the tracking control for mobile robot
For the simulation, the given reference trajectory for the Robot Mobile is selected as a circular trajectory to test the tracking ability of the proposed controller and conventional controllers.In the case of a circular reference trajectory, assuming the robot starts from the origin (0, 0) and the reference trajectory starts from the point (1, 0), simulation time for trajectory tracking is 10 sec.Figures 20-24 show the circular reference trajectory tracking performance response of the system for PID, SMC, SMC with saturation (boundary layer), AFSMC, and AFSMC with ANFIS kinematic controller, respectively.From the figures, we observe that the actual trajectory follows or tracks the reference trajectory from the initial to the final within the given simulation time with different performance.The results of the control input right wheel motor is presented in Figure 30, while that of the left wheel is shown in Figure 31.The results of the right wheel motor using boundary layer SMC is shown in Figure 32, while that of the left is shown in Figure 33.
The performance index of IAE, ITSE, and ITAE are presented in Tables 3 and 4, respectively.3 show an average steady-state error of around 0.09 and a settling time of 3.8 seconds.From Table 4, we  obtain the average error performance index, including average ISE, average IAE, average ITSE, and average ITAE, which are measured at 0.174, 0.279, 0.028, and 0.201, respectively, indicating the PID controller's performance.
Figure 26 illustrates the position and orientation tracking errors for the SMC controller.The x-position tracking error e x starts at an initial error of 1 and converges to zero with a notably shorter settling time of 1.8 seconds.The tracking errors (e y ande θ Þreach maximum values of 0.04 and 0.06, respectively, and then diminish to nearly zero within the specified settling time.The results in Tables 3 and 4 indicate an average steady-state error of approximately 0.0015, a settling time of 1.8 seconds, and average error performance index parameters (average ISE, average IAE, average ITSE, and average ITAE) measuring 0.05, 0.12, 0.014, and 0.11, respectively.This suggests that the SMC performance provides higher accuracy for the given system when compared to PID.The steady-state error is decreased, settling time is reduced by 2 seconds, and all error  performance index parameters are lowered when using the Pure SMC controller.However, a notable drawback of SMC is the introduction of chattering into the control input, as evident in Figures 30 and 31, which can make it challenging to implement in real-time applications.
To address the chattering issue in pure SMC, the signum function is replaced with a saturation function.Figure 27 displays the position and orientation tracking errors when using boundary layer method in SMC using saturation function.As a result, the average steady-state error increases to 0.0032, and the settling time is prolonged by 1.1 second.Additionally, the average error performance index, which includes average ISE, average IAE, average ITSE, and average ITAE, experiences increments to 0.11, 0.23, 0.0235, and 0.19, respectively, as shown in the results.It is observed from the simulation results that replacing the signum function with the saturation function results in increased tracking error.However, the significant advantage is the elimination       Figure 28, it showcases the position and orientation tracking errors e x ; e y ande θ À � of the Mobile Robot with AFSMC.The x-position tracking error (e x ;) starts at an initial error of 1 and converges to zero with a remarkably short settling time of 2.1 seconds.Similarly, the tracking errors e y ande θ À � reach maximum values of 0.039 and 0.05, respectively, and then decay to nearly zero within the specified settling time.The simulation results summarized in Tables 3 and 4 reveal an average steady-state error of around 0.002 and a settling time of 1.8 seconds.From Table 4, we obtain the average error performance index, which includes average ISE, average IAE, average ITSE, and average ITAE, measuring 0.044, 0.1138, 0.0946, and 0.0918, respectively.In conclusion, the replacement of the signum function with a saturation function leads to increased tracking error in SMC with boundary layer.However, it successfully eliminates chattering.On the other hand, the AFSMC controller exhibits accurate and efficient tracking with minimal settling time and reduced error performance index values, making it a favorable choice for the Mobile Robot system.
Figure 29 illustrates the position and orientation tracking errors of the mobile robot using AFSMC with an ANFIS kinematic controller.The x-position tracking error starts at an initial error of 1 and asymptotically approaches zero with a remarkably short settling time of 1.6 seconds.Likewise, the tracking errors (e y ande θ Þ reach maximum values of 0.036 and 0.05, respectively, and then diminish to nearly zero within the specified settling time.The simulation results summarized in Tables 3 and  4 indicate an average steady-state error of around 0.0014 and a settling time of 1.8 seconds.From Table 4, we obtain the average error performance index, including average ISE, average IAE, average ITSE, and average ITAE, which are measured at 0.043, 0.107, 0.0077, and 0.076, respectively.This suggests that the mobile robot asymptotically follows the reference trajectory with minimal acceptable error when utilizing AFSMC with an ANFIS kinematic controller.Throughout the simulation and numerical results of Figures 25 to 29, the performance responses of all trajectory tracking controllers are used for comparison.From the results, it is evident that the proposed controllers (SMC, AFSMC, and AFSMC with ANFIS controller) outperform other controllers (PID, SMC, SMC with saturation) in terms of reducing settling time, minimizing steady-state error, and decreasing error performance index (ISE, IAE, ITSE, and ITAE).Notably, chattering in control effort is observed in the case of pure SMC.Hence, the performance responses of AFSMC and AFSMC with ANFIS controllers are found to be superior to that of pure SMC, boundary layer SMC with saturation, and PID for tracking the reference trajectory.Moreover, for practical implementation,         Based on these findings, the performance of pure SMC is nearly indistinguishable from that of the proposed controller.Figure 41 shows the pose tracking error plot for SMC with boundary layer.The results indicate an increased average steady-state error of 0.0413 and an extended settling time by 2.1 seconds.From Table 5, the average error performance index reveals the following increments: average ISE, average IAE, average ITSE, and average ITAE are raised to 0.00176, 0.0203, 0.00053, and 0.0192, respectively.Figure 42 demonstrates the plotted pose tracking error for AFSMC.The results show an increased average steady-state error of 0.0413 and a slightly  prolonged settling time of 0.1 seconds.As shown in Table 5, the average error performance index exhibits changes, with average ISE, average IAE, average ITSE, and average ITAE increasing to 0.00047, 0.01572, 0.00026, and 0.015, respectively.In summary, the proposed AFSMC with ANFIS kinematic controller exhibits superior performance compared to conventional controllers, with minimal steady-state error and shorter settling time.According to the results, the performance of AFSMC closely resembles that of the proposed controller.

Disturbance rejection and sensitivity to internal parameter variation
The results of a proposed controller and convectional controllers (AFSMC, FSMC, SMC, and PID) are compared in this section under internal parameter change and system disturbance.Keep all of the parameters the same as presented in [38], with the exception of the robot's mass, which is changed by 0.03 kg.This implies taking into account the value of the plant parameter's uncertainty and also considering τ d ¼ 2 þ sin 0:1t 1 þ cos 0:1t ½ � T as an external system disturbance.Based on the simulation results in Table 6, we can compare the performance of the designed controller with that of another conventional controller in a presence of disturbance and internal parameter change.
Based on the results, the proposed controller exhibits consistent performance with and without system disturbances and internal parameter variations.The average error performance index, for ISA, IAE, ITSE, and ITAE, only increases by small margins (0.01, 0.06, 0.005, and 0.013, respectively), and the settling time experiences a minor increase of 0.3 seconds.This indicates that the designed controller (AFSMC with ANFIS kinematic controller) demonstrates excellent robustness against fluctuations in internal parameters and performs even better in the presence of system disturbances, achieving minimal tracking error compared to PID, SMC with saturation, and AFSMC controllers.
On the other hand, PID, SMC with the saturation function, and AFSMC controllers are considerably affected by disturbances and changes in internal parameters in contrast to AFSMC with ANFIS Kinematics Controller.Table 6 provides a summarized comparison of the proposed controller's performance with another conventional controller under the presence of disturbance and internal parameter variations.The simulation results indicate that the proposed controller consistently performs well, whether encountering external disturbances or internal parameter changes.The average error performance index (ISA, IAE, ITSE, and ITAE) only experiences marginal increments (0.01, 0.06, 0.005, and 0.013, respectively), and the settling time increases slightly by 0.3 seconds.This highlights the robustness of the designed controller (AFSMC with ANFIS kinematic controller) in handling variations in internal parameters and its ability to perform admirably even in the presence of external disturbances, achieving minimal tracking error when compared to PID, SMC with saturation, and AFSMC controllers.In contrast, PID, SMC with the boundary layer, and AFSMC controllers are highly affected by disturbances and internal parameter changes when compared to the proposed controller (AFSMC with ANFIS Kinematics Controller).
The PID controller exhibits increased average error performance index, comprising average ISE, average IAE, average ITSE, and average ITAE, by 0.14, 0.34, 0.1, and 0.47, respectively, and the settling time is prolonged by 0.8 seconds.As for SMC with boundary layer, the average error performance index experiences increments of 0.138, 0.14, 0.06, and 0.088 for average ISE, average IAE, average ITSE, and average ITAE, respectively, and the settling time extends by 0.7 seconds.In the case of AFSMC, the average error performance index undergoes changes of 0.04, 0.016, 0.048, and 0.027 for average ISE, average IAE, average ITSE, and average ITAE, respectively, and the settling time increases by 0.5 seconds.
The results suggest that AFSMC performs comparably well with the proposed controller when compared to PID and SMC.However, under higher system perturbation magnitudes, relying solely on controlling the system dynamics proves inadequate.Consequently, the proposed AFSMC with an ANFIS kinematic controller emerges as a superior solution, effectively addressing the challenges posed by internal parameter variations and external disturbances.The designed controller (AFSMC with ANFIS kinematic controllers) demonstrates heightened robustness and insensitivity to parametric variations in comparison to other controllers (AFSMC, SMC, and PID).This makes it highly suitable for application in systems with variable conditions, ensuring stable and reliable performance.

Conclusion
In this paper, we presented a model for a differential-drive wheeled mobile robot and enhanced the model accuracy by considering actuator and chopper dynamic effects and verified the model using simulations with real-time scenarios.An innovative dual loop control approach combining AFSMC for dynamic system control with ANFIS Kinematic Controller was presented.Extensive simulations and analysis compared the proposed controller with PID, SMC, and FSMC.The stability of the controller was verified using the Lyapunov criterion, and various evaluation criteria, including steady-state error and settling time, and error performance indices (ISE, IAE, ITSE, ITAE), were employed for performance comparison.
Based on the simulation results using circular and eight-shaped reference trajectories, it is evident that the proposed dual controller outperforms conventional controllers (PID, SMC, and AFSMC) in terms of tracking performance, even in the presence of uncertainty.This demonstrates the superior ability of the proposed controller to enhance trajectory tracking performance and robustness.Furthermore, the proposed controller exhibits adaptability to various reference trajectories, establishing it as a reliable choice for dynamic systems.Notably, the proposed controller effectively mitigates the chattering effect in the control input by using boundary layer method, leading to smoother control efforts.Generally, it demonstrates remarkable robustness in handling uncertainties, outperforming other controllers under uncertain conditions.These findings solidify the potential and applicability of the proposed controller in practical control systems.
the robot frame and inertial frame coordinates of the given point, respectively.The two coordinates are then related in the following way:

Figure 3 .
Figure 3. Chopper-fed DC motor with gear and load equivalent circuit.

Figure 5 .
Figure 5. Open loop result when the right wheel set at 24V and left wheel is 0V.

Figure 6 .
Figure 6.The result of robot motion in xy-plane when right wheel is 24V and left wheel is 0V.

Figure 7 .
Figure 7. Open loop result when the left wheel set at 24V and right wheel is 0V.

Figure 8 .
Figure 8.The result of robot motion in xy-plane when left wheel is 24V and right wheel is 0V.

Figure
Figure 9. Open-loop result when the right and left wheel input are equal to 24V.
a positive design parameter and e ¼ e v e w ½ � T is the error value between the desired and actual linear and angular velocity of mobile robot v d ; w d ð Þ and v; w ð Þ, respectively.

Figure
Figure 12.Block diagram of the Adaptive Fuzzy Sliding Mode Controller.

Figure
Figure 15.The Neuro-Fuzzy Designer structure.

Figure
Figure 19.Simulink model of mobile robot using AFSMC dynamic with ANFIS kinematic control.

Figure
Figure 22.Circular trajectory tracking using boundary layer SMC using saturation function.

Figure 25
Figure 25 displays the position and orientation tracking errors for the PID controller.The x-position tracking error e x starts at an initial error of 1 and gradually decreases to zero over a relatively long time of approximately 3.8 seconds.The y-position and orientation tracking errors (e y ande θ Þ start from zero, reach maximum errors of 0.08 and 0.18, respectively, and then converge to almost zero within the specified settling time.The simulation results summarized in Table3show an average steady-state error of around 0.09 and a settling time of 3.8 seconds.From Table4, we

Figure
Figure 24.Circular trajectory tracking using AFSMC with ANFIS kinematic controller.

Figure 25 .
Figure 25.Position and orientation tracking error for PID controller.

Figure 26 .
Figure 26.Position and orientation tracking error for pure SMC.

Figure
Figure 27.Position and orientation tracking error for boundary layer SMC using saturation function.

Figure 28 .
Figure 28.Position and orientation tracking error for AFSMC.

Figure
Figure 29.Position and orientation tracking error for ANFIS with AFSMC.

Figure 30 .
Figure 30.Control input (V) for right wheel motor using SMC.

Figure 31 .
Figure 31.Control input (V) for left wheel motor using SMC.

Figure
Figure 32.Control input (V) for right wheel motor using boundary layer SMC.

Figure 33 .
Figure 33.Control input (V) for left wheel motor using boundary layer SMC.

Figure
Figure 34.Eight-shape type trajectory tracking using PID controller.

Figure
Figure 35.Eight-shape type trajectory tracking using pure SMC.

Figure
Figure 37. Eight-shape type trajectory tracking using AFSMC Controller.
Figure 39 (PID controller) shows increased average steady-state error of 0.0413 and an extended settling time of 3.1 seconds.The average error performance index parameters (average ISE, average IAE, average ITSE, and average ITAE) are increased to 0.00305, 0.0446, 0.0014, and 0.0365, respectively.

Figure 39 .
Figure 39.Position and orientation tracking error for PID controller.

Figure 40
Figure 40 depicts the pose (position and orientation) tracking errors for pure SMC.The results indicate that the average steady-state error increases to 0.0413, and the settling time is prolonged by 0.6 seconds.As presented in Table5, the average error performance index shows incremental values, with average ISE, average IAE, average ITSE, and average ITAE rising to 0.00086, 0.017, 0.00047, and 0.0164, respectively.

Figure 40 .
Figure 40.Position and orientation tracking error pure SMC.

Figure
Figure 41.Position and orientation tracking error for boundary layer SMC using saturation function.

Figure 42 .
Figure 42.Position and orientation tracking error for AFSMC.

Figure 43 .
Figure 43.Position and orientation tracking error for proposed controller (AFSMC with ANFIS).

Table 1 . Rule base for dynamic equivalent control part of SMC
perturbations.Figures 13 and Figure14depicts the membership functions, and Table2presents the Rule Base for the fuzzy logic supervisory.

Table 4 . Performance of AFSMC with ANFIS kinematic controller System Performance
as evident in Figures32 and 33.This approach offers promising properties for further enhancing SMC by eliminating chattering through the use of boundary layer techniques.