**2.3 Slip-force models**

Slip/force models aim to describe the slip movement through its force/surface dependency mapping *F s*ð Þ : *s*↦*F*. They can also depend on the vehicle speed *v*, that is, *F s*ð Þ , *v* , and vary when the characteristics of the road change.

One of the best-known models of this type is the Pacejka model [23], also known as the "Magic Formula". It has been shown that this model agrees adequately with the experimental data obtained under particular conditions of constant linear and angular velocity. Pacejka's model is represented by Eq. (8).

$$F(\mathfrak{s}) = D \sin(\operatorname{Corcttan}(B\mathfrak{s} - E(B\mathfrak{s} - \operatorname{arctan}(B\mathfrak{s}))))\tag{8}$$

It can be seen that the Eq. (8) contains a set of parameters: *B*, *C*, *D*, and *E*. These parameters depend on the tire's physical properties and the vehicle's dynamic state. In the Eq. (8), *D* represents the maximum coefficient, *C* represents the shape coefficient and influences the shape of the curve, *B* is the stiffness coefficient, and *E* is the coefficient of curvature [24].

#### **2.4 Quarter-car robot model**

For the design of the traction controller, the Eqs. (1)–(3)represents the engine dynamics, and the Eqs. (5) and (6) represent the dynamics of the wheel of a differential robot.

**Figure 4** shows the interaction of the dynamic equations of the motor with the wheel of the robot. It is observed that the relationship between the equations is the angular velocity of the motor shaft and the longitudinal movement of the robot, as expressed by the Eq. from (9) to the Eq. (13).

$$\frac{di\_a}{dt} = \frac{e\_a}{L\_a} - \frac{K}{L\_a}\alpha\_m - \frac{R\_a}{L}i\_a \tag{9}$$

**Figure 4.** *Block diagram of the motor-wheel system from Eqs. (9)–(13).*

*PID-like Fuzzy Controller Design for Anti-Slip System in Quarter-Car Robot DOI: http://dx.doi.org/10.5772/intechopen.110497*

$$\frac{d\phi\_m}{dt} = \frac{K}{J}\dot{i}\_d - \frac{D}{J}\phi\_m - \frac{r\_w \cdot Ft}{G\_R \cdot J} \tag{10}$$

$$\frac{d\theta\_m}{dt} = \alpha\_m \tag{11}$$

$$\frac{dv}{dt} = \frac{Ft}{m} \tag{12}$$

$$\frac{d\mathbf{x}\_p}{dt} = \nu \tag{13}$$

The slip ratio is presented in Eq. (14) and is a function of the angular velocity.

$$\mathbf{s} = \frac{a\_b - a\_f}{|a\_f| + tol}, \quad tol \simeq \mathbf{0} \tag{14}$$

On the other hand, rewriting the Eq. (14) as a function of the radius of the wheel (*rw*) and the longitudinal velocity (*v*).

$$\mathfrak{s} = \frac{\alpha r\_w - v}{|\alpha r\_w| + tol}, \quad tol \simeq \mathbf{0} \tag{15}$$

Where *ωrw* and *v* represent the longitudinal speed of the wheel computed according to the radial speed times the radius of the tire and the longitudinal velocity measured, respectively. To calculate the traction force, the Eq. (16) of Pacejka and Sharp [23] is used.

$$F(\mathfrak{s}) = D \sin(\operatorname{Corcttan}(B\mathfrak{s} - E(B\mathfrak{s} - \operatorname{arctan}(B\mathfrak{s}))))\tag{16}$$

where *B*, *C*, *D*, and *E* are constants, *s* represents the slip value. **Figure 5** shows the implementation of Eqs. (15) and (16) in block diagram form.

On the other hand, if an imbalance is detected between the wheels of the robot, it is necessary to adjust the speed of the wheel that rotates at a higher rate with the one that turns at a lower speed [25]. To carry out this criterion, it is necessary to propose

#### **Figure 5.**

*Block diagram of the motor-wheel system from Eqs. (9)–(13) for the calculation of the traction force from the coefficient of friction.*

behavioral variables that mathematically model the speed adjustment. In order to carry out the previous step, it is required to define a database that protects all the behavior information. This information is the linguistic variable. In addition, the operating range of the controller must be considered, which is translated into modeling the behavior in the decompensation of the wheels.
