**2. Dynamic model of one-wheel robot**

This section presents the fundamental parts that compound the Quarter-Car robot structure. **Figure 1** shows the platform used for the project simulations. The platform consists of a wheel–motor coupling that will emulate the behavior of a mobile robot

**Figure 1.** *One-wheel robot for control tests.*

axis when different surfaces interact. The measurements will be carried out: angular velocity, longitudinal velocity, current, and slip. Slippage will be controlled so that the tire does not slip and can correctly follow a trajectory.

### **2.1 DC motor**

Direct current (DC) motors are the most common actuators within control systems. It directly provides a rotational movement, and together with the wheels, rails, and cables, it can perform a translational action. The equivalent electrical circuit of the armature and the free body diagram of the rotor are shown in **Figure 2**. It is necessary to obtain a dynamic model that allows a correct analysis. The dynamic model of this servo system depends on the electrical and mechanical characteristics, such as the resistance *Ra*, the inductance *La*, the inertia *J* of the armature, the back electromotive force *vb*, and the friction *D*.

From **Figure 2**, it is possible to derive the following equations based on Newton's second law for rotational motion and Kirchhoff's second law, Eqs. (1) and (2), respectively.

$$\frac{d\alpha\_m}{dt} = \frac{K}{J}\dot{i}\_a - \frac{D}{J}\alpha\_m \tag{1}$$

$$\frac{di\_a}{dt} = \frac{e\_a}{L\_a} - \frac{K}{L\_a}\alpha\_m - \frac{R\_a}{L}i\_a \tag{2}$$

$$\frac{d\theta\_m}{dt} = \alpha\_m \tag{3}$$

#### **2.2 Tire**

Tire friction models are also indispensable for accurately reproducing friction forces for simulation purposes. A common assumption in most tire friction models is that the normalized tire friction, Eq. (4), is a nonlinear function of the normalized

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

relative velocity between the surface and the tire (slip coefficient *s*) with a different maximum.

$$\mu = \frac{F}{F\_n} = \frac{\text{Friction force}}{\text{Normal force}}\tag{4}$$

Furthermore, it is understood that *μ* also depends on vehicle speed and road surface conditions, among other factors. This work considers the simplified motion dynamics of a Quarter-Car model. The system is represented by Eqs. (5) and (6).

$$
\hbar \, m \dot{\psi} = F \tag{5}
$$

$$J\dot{\rho} = -rF + u\tag{6}$$

Where *m* is <sup>1</sup> <sup>4</sup> of the mass of the vehicle, and *J* and *r* are the inertia and radius of the wheel, respectively. *v* is the linear speed of the tire, and *ω* is the angular speed of the wheel, *u* is the acceleration or braking torque, and *F* is the friction force as shown in **Figure 3**.

The most common tire friction models used in the literature are those of algebraic slip/force relationships. They are defined as one-to-one maps (memory-less) between the friction *F* and the longitudinal slip rate *s*, which is defined in Eq. (7).

$$s = \begin{cases} s\_f = \frac{r\nu - v}{v} & \text{if } v > r\nu, \ v \neq 0 \text{ for breaking} \\ s\_m = \frac{r\nu - v}{r\nu} & \text{if } v < r\nu, \ a \neq 0 \text{ for movement} \end{cases} \tag{7}$$

Slippage results from the reduction in the effective circumference of the rim as a result of surface deformation due to the tire rubber's elasticity. This, in turn, implies that the longitudinal velocity *v* will not be equal to *rω*. The absolute value of the slippage is defined in the interval 0, 1 ½ �. When *s* ¼ 0, there is no slip (pure rotation), while ∣*s*∣ ¼ 1 indicates total slip/skid.

**Figure 3.** *One-wheel system with concentrated friction.*
