**4. Simulations and results**

**Figure 10** shows the diagram of the model, which outputs the wheel axis's slip, current, and longitudinal speeds calculated from the wheel's radius and angular velocity. **Tables 4** and **5** display the values of the parameters used for the simulation.

**Figure 10.**

*Diagram model for simulation.*


**Table 4.** *Parameters of the DC motor.*

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


#### **Table 5.**

*Parameters of the robot's structure.*

**Figure 11.** *Response of the Quarter-Car robot to surfaces (a) dry asphalt, (b) wet asphalt, and (c) ice.*

The motor values correspond to a DC motor. **Table 5** shows the values that were used for the mechanical structure of the robot.

Assuming that the motor that is coupled to the robot's wheel is fed at 12 V and only the speeds and the slip are measured, we can say that the system is an open-loop representation, whose behavior is shown in **Figure 11** for each of the surfaces presented in **Table 6**. In **Figure 11a**, it can be seen that when the tire crosses a dry asphalt surface, the longitudinal velocity increases considerably and tries to equalize the angular velocity. On the other hand, when the tire crosses a wet asphalt surface, the longitudinal velocity takes longer to equal the angular velocity, as shown in **Figure 11b**. Finally, the angular velocity tends to its maximum permissible value when


#### **Table 6.**

*Parameters for computing the coefficient of friction depending on the surface.*

the tire makes contact with an ice-covered surface. In contrast, the longitudinal rate grows negligibly as a function of time; see **Figure 11c**. This is because the coefficient of friction is very close to zero, which means that the friction is zero, causing the tire to tend to skid without moving considerably longitudinally.

The Algorithm 1 presents the way to compute the friction coefficient used for simulation.

#### **Algorithm 1:**


The behavior of the robot as a function of longitudinal and radial speed is presented in **Figure 12**, which contains the behavior over time of the difference in rates, better known as sliding. As shown in **Figure 11**, the speeds of the wheel have different magnitudes. This means that the angular speed is much more significant in the first moments than the longitudinal speed. This behavior is without any controller, so there is no device that regulates the speeds from the beginning of the operation. However, in **Figure 12**, it can be seen that the slip tends to zero when the speeds reach the same magnitude on dry and wet asphalt surfaces, **Figure 11**, while for the icy surface, there is always a slip value, and the tire remains rotating without moving longitudinally.

#### **4.1 PID-like fuzzy controller results**

For the design of the PID-like fuzzy controller, the methodology presented in the section 3 was used; see **Figure 13**.

In order to solve the problem of different speeds, a slip controller will be designed whose reference input is from a seven segments motion profile. The wheel–motor

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

**Figure 12.** *Slip response to a 12 V input.*

**Figure 13.** *PID-like fuzzy controllers and the Quarter-Car robot structure.*

system must ensure the same angular and longitudinal speed by following the change in reference speed at any time. In **Figure 14**, the velocity profile is shown as input. It goes through two integrators because the jerk profile is being generated, so when integrating once, you get the reference acceleration, and the second time you get the input velocity. The longitudinal velocity is compared concerning the motion profile. The idea is that the rim moves according to the reference and can work with the surfaces presented in **Table 6**. The difference between the reference speed and the longitudinal speed produces the error, which is the input to the master controller. The

**Figure 14.** *Fuzzy controller.*

output of this controller is compared to the slip to generate an internal control loop to prevent slippage.

The Quarter-Car and the control system used for the simulation are shown in **Figure 14**. The diagram shows a cascade controller with a longitudinal speed master loop and a current slave loop. Both control structures are based on fuzzy logic. For the master loop, a self-tuning PID controller is used. This structure is shown in detail in **Figure 15**. The same rules were used and the same number of linguistic values. The difference lies in the range of operation of the linguistic variables of gains *kp*, *kd*, and *ki*. These ranges were proposed from the PID controller answer from the previous section. For the slave loop, a PI structure with self-tuning was proposed, and the same strategy as the one mentioned above was used. Only the range of values of the input to the internal *kp* and *ki* profit parts has been modified.

The ranges of values are smaller and are intended to compare the response of the master controller with the slip produced by the tire and the selected surface to evaluate.

The simulation result is shown in **Figure 15**, where each surface shows a similar behavior when following the reference and where the angular and longitudinal velocities do not offer a significant displacement. This is to the slide controller function. For example, in **Figure 15**, a dry asphalt surface is used, and it can be seen that the speeds adhere to the reference. In 15–*b*, it can be seen that the longitudinal velocity tends to

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

decrease in magnitude, although the change in it is not significant. Finally, **Figure 15c** shows the behavior of the angular and longitudinal velocities on an ice surface. It is shown that the controller is reacting well, the longitudinal speed reduces its magnitude, and the angular speed follows the profile. The latter makes sense since the slip functions the angular and longitudinal velocities. Therefore, the magnitude of the velocities must be similar depending on the surface over which the plant moves.

The slip measured by the simulation is shown in **Figure 16**. As can be seen, there is less slip for dry asphalt. This was to be expected since this type of surface provides a better grip on the tires to the surface. On the other hand, we see that slip tends to increase in magnitude when the robot's wheel moves on a wet surface, similar to a larger vehicle when it rains. Finally, when the surface contains ice, the landslide tends to increase in magnitude since surfaces with ice have a minimal coefficient of friction, causing the landslide to increase in magnitude if it is not controlled.

The current measured in the simulation is shown in **Figure 17**. The dry asphalt surface presents a greater magnitude of current mainly due to the coefficient of friction since there is always a grip; that is, there is a considerable friction force. When the tire runs on a wet surface, it tends to decrease the coefficient of friction, making this one turn more efficiently, thus reducing the current supplied to the motor. Finally, when the surface is ice, the current tends to decrease since the wheel will tend to slip, which means that the amount of current that is going to be injected into the motor is less, and in order to compensate for the speed in the wheel or on the motor shaft.

The final position or target position is deduced from the 7-segment profile. **Figure 18** shows the position of the system with its respective reference. **Table 7** displays the values of the desired and measured positions of the different surfaces.

**Table 8** shows the performance of the controllers against their respective surfaces using the root mean square error (RMSE). As can be seen, the PID controller shows a better performance than the fuzzy controller on a dry asphalt surface. The adaptive

**Figure 16.** *Slip behaviour.*

**Figure 17.** *Current consumption.*

**Figure 18.** *Perfil de 7 segmentos.*


**Table 7.** *Final position values.*

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


**Table 8.**

*Performance values.*

PID-like fuzzy controller starts to increase its performance when there is a change of surface, for example, for the wet surface. Finally, better performance is appreciated when the surface is icy since the traction controller allows longitudinal speed to be controlled by controlling the wheel's speed, making the latter follow the reference.
