Fuzzy adaptive back stepping control of wheeled mobile robot

2.1. Kinematic model

The kinematic model of a WMR defines its motion based on velocity inputs and system constraints. Understanding the kinematics is essential for developing accurate trajectory tracking and motion control strategies. This section establishes the fundamental equations governing the motion of a WMR, providing a basis for designing effective control algorithms. By analyzing the kinematic equations, it is ensured that the control inputs result in smooth and precise navigation while maintaining stability. A wheeled mobile robot is considered with two actuator wheels.

Fig. 1WMR on Cartesian coordination

The geometric center of the robot is located at point C, while the midpoint is also between the two actuator wheels. The robot’s center of mass is at point M, and the distance from point C to M is denoted as $d$. The coordinates of the center of mass M in the global coordinate system {$O,X,Y\}$ are $\left(x,y\right)$. The robot’s angle of rotation $\theta$ is positive for counterclockwise rotation. The distance from the geometric center to the center of the operator wheel is $R$. The radius of the moving wheels is denoted by $r$. It is assumed that the robot’s geometric center and center of mass do not coincide, although the likelihood of alignment is high. It is also assumed that the wheels rotate easily and do not slip. The non-holonomic structure can be expressed as follows:

Therefore, the kinematic description of the robot can be expressed as follows. The straight-line speed and the angular speed of the robot. It is presented Eq. (2) as a matrix in the following format:

Considering the transfer as follows:

where, $Z_1$ and $Z_2$ are reference coordination of $x$ and $y$ axes, respectively, in circle – shaped.

Therefore, the kinematic description of the robot model is shown in the following transformation:

Because Eq. (5) has a vertical transfer, there is no change in the unit’s values, and it is concluded that the tracking error of the original model converges to zero. Assume that the robot's reference kinematic model is given as follows:

And $Z_{1r}$ and $Z_{2r}$ are real coordination motion of robot in $x$ and $y$ axes, respectively. Where $v_r$ the ideal straight is linear speed and $w_r$ is the ideal angular speed of the robot. The tracking error of the system is given as follows:

The kinematic model provides a foundation for controlling WMRs by describing their motion in terms of linear and angular velocities. The transformation of reference coordinates enables precise trajectory tracking, minimizing errors in position and orientation. These equations serve as a crucial step in designing adaptive controllers that can handle uncertainties and external disturbances. The insights gained from this model contribute to the development of robust control strategies, which will be further enhanced through the integration of dynamic modeling and adaptive fuzzy logic.

2.2. Dynamic model

When the parameter is known, a common problem in tracking is the design of velocity and input controllers to make Eq. (8) asymptotically stable. However, in reality, it is very difficult to accurately identify the parameter. With the uncertainty of the parameter in this model, it will have an adaptive tracking controller. In engineering practice, formulation of non-holonomic control problems can be realized in dynamic levels, while force and torque are considered as control inputs, the dynamic description of the robot will be as follows [36]:

The matrices of the robot system in Eq. (9) are expressed as follows:

The general coordinate is represented by $q$, while $M$ is a symmetric, definite positive inertial matrix. It serves as the Coriolis matrix, and the side-to-side indication shows the extent of surface wear. $G$ is the gravity vector, ${\tau }_d$ represents limited unknown disturbances, including unstructured dynamics, $B$ is the input transfer matrix $\tau =\left({\tau }_1-{\tau }_2\right)$ and are the applied torque to the left and right wheels. $G$ represents the vector of constraint forces, while $A$ is the matrix associated with constraints. When considering the mobile robot under non-holonomic constraints, the matrix $A$ will look like this:

According to the Eq. (11):

By deriving from the sides of the equation $\dot{q}=S\left(q\right)V$ and inserting it into the dynamic Eq. (9) and multiplying $S^T\left(q\right)$ by its sides follow as:

Having:

The Eq. (9) will be transferred as follows:

The matrices of the dynamic are presented in Eq. (14), [37]:

where $I$, that is the moment of inertia of the robot, $m$ is the mass of the robot, $R$ is the distance between the geometric center and the center of the wheels of the robot operator, and $r$ is the radius of the wheels of the operator. If $V_c={\left[v_c,w_c\right]}^T$ is the virtual speed control of the system kinematics from Eq. (11), It is assumed that the tracking error rate is as follows:

By inserting Eq. (16) into Eq. (14) and using the linear property of the robot’s inertial parameters:

While:

The $\phi$ is the vector of the robot’s inertial parameters, such as the moment of inertia and mass, while $Y$ represents a specific matrix that is unrelated to the robot’s inertial characteristics. In real motion, the surface friction vector and the irregularity vector are limited by a certain function. Therefore, $\stackrel{-}{F}\left(\dot{q}\right)$, and ${\stackrel{-}{\tau }}_d$ are both bound (limited) by the same definite function [37]:

where $N\left(q,\dot{q}\right)$ is defined a specific value. The tracking problem for robot dynamics involves designing a control force or torque$\tau$to create a closed-loop system, asymptotically stable, using Eq. (8) and Eq. (17).

3.1. Design of backward adaptive controller

To ensure the tracking error of Eq. (11) converges to zero, that assumes that the error estimation of $d$ is $\tilde{d}=\widehat{d}-d$. Therefore, the virtual path tracking control law from the robot’s kinematic model is given as follows: Let’s assume $\forall t\in \left[0,+\infty \right]$ that both $Z_{1r}$ and $Z_{2r}$ are bounded, with the lower limit being zero. It compensates for the following conditions:

Then, if it applies the speed control law according to Eq. (26) and the adaptive control law according to Eq. (24) to Eq. (8), the kinematic tracking error described by Eq. (11) will be asymptotically stable with Eq. (20):

That is, the tracking error tends to zero:

The adaptive control law $\widehat{d}$ is as follows:

While $k_1>$0 and $k_2<$0 and a is a positive constant [37].

3.2. Dynamic adaptive control

In Eq. (21), tracking control only kinematic speed is considered, although in reality, it is very difficult to have an ideal speed control. It should also be stated that the error between the operator speed and the ideal speed control will not be zero, which means $\eta \ne 0$ that Eq. (23) is the only ideal kinematic speed control. To achieve torque control, the tracking error speed in Eq. (14) must converge to zero. Also, in this article, the value of $\phi$ is unknown and $\widehat{\phi }$ is an estimate of the $\phi$ therefore $\tilde{\phi }=\widehat{\phi }-\phi$. This article assumes an unknown value for the tracking error, which serves as an estimate of its own error. Robot is assumed that $\forall t\in \left[0,+\infty \right)$, $Z_{1r}$ and $Z_{2r}$, that are bounded, and then using the speed control law in Eq. (26) and the dynamic adaptive control law in Eq. (27) and Eq. (28) are mentioned [2]:

Tracking error position $e_p\to 0$, $\left(p=\mathrm{1,2},3\right)$:

The adaptive control law $\widehat{\phi }$ is as follows [2]:

While $k_3>$ 0 and $k_4>$ 0, $\mathrm{\Gamma }$ is a positive matrix and all design parameters [36].

3.3. Fuzzy set

A fuzzy logic system consists of the following 4 general parts:

– Fuzzy rule base.

– Fuzzifier.

– Non-phase generator.

– Inference engine.

The fuzzy rule base for the fuzzy system consists of a set of if-then rules as follows:

${\mu }_G^l\left(y\right)$ and ${\mu }_{F_i}^l\left(x_i\right)$ that are fuzzy membership functions, ${\mu }_{F_i}^l\left(x_i\right)$ is called the degree of membership of $x_i$ in $F_i$. Basic fuzzy functions are defined as follows [38], [36]:

A fuzzy logic system can be written as follows:

4.1. Robot observation without noise

The reference model of the robot is shown as a red circle, and the tracker model can also be seen as blue. In order to evaluate the tracker models, these two graphs can be seen next to each other in the third graph. As illustrated in Fig. 4, the kinematic back stepping adaptive control law operates in the absence of noise, as previously discussed in the chapter. The control input or effort in the adaptive observer mode, as per Eq. (8), reflects the errors of the robot under noise-free conditions. In the feedback controller, the control torque representing the control effort without noise is described by Eq. (26). The tracking error, defined as the difference between the control speed and the mobile robot’s speed, is formulated in Eq. (16). Based on Eq. (27), which defines the dynamic adaptive control law for the mobile robot, this control law vector evolves and is depicted in Fig. 4.

4.2. Robot observation with noise

In the presence of noise in the mobile robot system model, the relevant diagrams are depicted as follows: Based on Eq. (20), which incorporates external friction and friction force, as shown below, over the simulated time period of the mobile robot. Fig. 5 illustrates the presence of noise in the backward adaptive observer mode, depicting the reference and real models. The accuracy of the adaptive observer can be fully understood in Fig. 5 illustrates the adaptive observer control law for estimating kinematic behavior in the presence of noise. It details the adaptation of linear and angular velocities using the kinematic back stepping approach under noisy conditions, emphasizing the system’s state error due to external noise. The control torque, determined by the adaptive dynamic back stepping control approach, is designed for the first and second system inputs. Additionally, the system’s tracking speed is derived from Eq. (16) and Eq. (27).

Fig. 4Result of robot observation without noise

4.3. Fuzzy control with noise

This section introduces the incorporation of fuzzy logic into the backward comparative perspective. The purpose of applying the fuzzy approach is to introduce fuzzy parameters, known as control gains, into the system. These parameters, $k_1$, $k_2$, and $k_3$ in Eq. (24) and Eq. (28), are fuzzified to enhance system control. Designing fuzzy parameters involves fundamental concepts detailed in the section on fuzzified parameter. Fig. 6 illustrates the real model and fuzzy reference of the robot. Essentially, the integration of fuzzy logic with the adaptive feedback controller aims to automate the tuning of control gains instead of relying on manual or trial-and-error methods. As described by Eq. (26), fuzzy logic adjusts these control functions within a range defined by membership functions, thereby optimizing the system's ability to track its path. Based on the fuzzy design for adaptive back stepping control of the system, the kinematic back stepping adaptive control parameters, considering both noise and fuzzification, are described as follows: angular velocity and linear velocity (see Eq. (23)). The adaptive control error for kinematic fuzzy back stepping in the presence of noise is given by Eq. (26), which expresses the control torque as the control effort designed using backward adaptive fuzzy control and applied to the system input. The tracking speed in both linear and angular modes in the presence of noise is demonstrated through the design of the backward adaptive fuzzy controller. Fig. 6 illustrates the parameters defined in Eq. (27), representing the adaptive control law utilizing the fuzzy approach in the presence of noise.

Fig. 5Result of robot observation with noise

5.1. The first element

Based on Table 1, the inputs and outputs of the mobile robot should be determined, distinguishing between fuzzified inputs and non-fuzzified outputs for the system (see Fig. 7).

Fig. 6Result of fuzzy robot control with noise

5.2. The second element

Selection of rules for input and output membership functions (see Fig. 7).

Fig. 7Adaptive values of the K1, K2, and K3

5.3. The third element

Selecting the type of membership functions for input and output (triangular, Gaussian, trapezoidal, etc.). To perform the fuzzification of the parameters for the backward adaptive controller, the graphs related to these three elements can be fully reviewed and analyzed. Initially, the dependence of the inputs and outputs of the system and the controller is illustrated in Fig. 8. The selected membership functions are triangular, indicating the degree of connection between the inputs and outputs. Finally, this provides information on the magnitude of the control parameters fed to the system in each time frame to enable proper system control.