3.1 Phase I: One-Dimensional (1D) Control

To achieve a state of equilibrium with a degree of freedom, when the main platform shifts between 0 and 180 degrees, the mobile platform must gradually modify its position to always remain parallel to the horizontal reference base. The schematic designs for Phase I as well as the devices used and their operation principles are described below:

Figure 1 provides a top view of the diagrams developed for Phase I of the Project.

Fig. 1 One-dimensional prototype 

Figure 2 provides a side view of Fig. 1, wherein the angle formed between the main and mobile platforms can be observed.

Fig. 2 Platform design 

Figure 3 denotes the actual Phase I implementation.

Fig. 3 Working platform 

3.2 Phase II: Two-Dimensional (2D) Control

The Inventor tool was used to design the 2D prototype. This tool is a parameter-driven 3D solid modeling suite supported by the AutoDesk Company. The prototype components designed were then printed on a 3D printer. The prototype includes mobile joints that provide three degrees of freedom, one for each x, y, and z axis, denominated as roll, pitch, and yaw respectively, as well as a support for the LiDAR sensor [⁸].

To achieve a state of equilibrium with two degrees of freedom, when the main platform of the prototype (shown in purple) shifts between 0 and 180 degrees along the x or y axes, the mobile platform that supports the LiDAR sensor must gradually modify its position to always remain parallel to the horizontal reference base.

Figure 5 denotes the actual implementation of the 2D prototype, wherein the TF Mini Plus LiDAR sensor may be observed already installed in the support.

Fig. 4 CAD structure 

Fig. 5 Structure with LiDAR 

3.3 Phase III: Moving Platform Distance Measurements Using the LiDAR Sensor

We used the TF Mini Plus sensor to measure distances. This is a time of flight (ToF) sensor capable of measuring objects located at 10 cm to 12 m. It is powered with 5 V and can send information through the serial port at 115,200 bauds.

Figure 6 denotes the connection between the TF Mini Plus sensor and Arduino UNO module.

Fig. 6 Arduino connections with LiDAR 

The following table lists the specifications for the TF Mini Plus sensor used.

3.4 Components Used

3.4.1 Accelerometer

According to Newton's second law:

F→=m⋅a→.

From the above expression, a body with a mass of "m” subjected to acceleration will experience a force "F" that is proportional to the said acceleration. This force is used to activate the internal micro-mechanized structure of the accelerometer, through which the acceleration of the body may be measured independently in each of the three X, Y, and Z axes.

Moreover, when the acceleration values from each axis are combined, the angle of inclination at which the body is with respect to each plane can be trigonometrically calculated, as shown in Fig. 7. For the purposes hereof, the ADXL345 accelerometer has been used to determine the pitch and roll inclination angles of the main base. Figure 8 displays the connection diagram between the ADXL345 module and Arduino UNO platform.

Fig. 7 Angle of inclination 

Fig. 8 Arduino–accelerometer connection 

3.4.2 Servomotor

A servomotor (or servo) is a special type of direct current (DC) motor that provides controlled rotation movements and high torque. For these two features, servomotors require a pulse-width-modulation (PWM)-based control signal and gearbox.

Figure 9 illustrates the block diagram for a servomotor control system.

Fig. 9 Block chart 

To implement the 2D platform, we used two Tower Pro MG995 servomotors. Each motor provides a torque of 15 Kg-cm and the maximum rotation angle of 180 degrees at an operating voltage of 4.8–7.2 V.

Table 1 Sensor specifications 

Range	0.1–12 m
Accuracy	+5 cm (0.1–6 m); 1% (6–12 m)
Resolution	5 mm
Frequency	1–1000 Hz
Sensibility	70 k lux
Temperature	−20°C–60°C

3.5 Mathematical Motor Model

The servomotor system uses the DC motor to control the mechanical position of the rotor according to the voltage supplied by a magnetic field [⁵]. Figure 10 illustrates the motor used in the present work [¹⁰].

Fig. 10 DC motor 

Using Kirchhoff's law of voltage, the following equation is obtained:

The torque (T) of the motor for the armature load “i” with a (K) factor:

The Counter EMF (Eb) related to the angular velocity:

Regarding the model of a DC motor and Newton's law combined with Kirchhoff's law:

Using Laplace, Equations (1) and (2) can be restated as:

Then, from Equations (3) and (4), we obtain Equation (5):

The block diagram for the DC motor is displayed in Fig. 11.

Fig. 11 Block diagram 

In Equation (6), we obtain the transfer function with an output angle of θ as a function of voltage (V(s)):

Considering the DC motor diagram from Fig. 11, Equation (7) provides the transfer function for the angular velocity (ω) against the input (V(s)).

From Equation (7), we can obtain the transfer function Equation (8):

3.6 Control Strategies

In this work, we used three control strategies, and each one is assessed below:

3.6.1 Pulse-Width-Modulation-Based Serial Proportional Controller

PWM is one of the most efficient servomotor control techniques. It generates a periodic square wave wherein the wave portion that remains in the high state of the signal will be proportional to the distance that the motor will travel.

The timer contains the pulse width in a micro-controller; therefore, when the overflow occurs, the PWM signal will remain high until the moment when the comparator generates an interruption after the pulse has elapsed. The length of the pulse will indicate the angle of the rotation of the motor.

Figure 12 contains some examples of typical angular positions for a servomotor and the corresponding control pulses.

Fig. 12 PWM chart 

For the servomotor to be positioned and held in the desired location, regardless of disturbances or external forces to which it may be subjected, the proportional control must be serially applied and sustained over time.

3.6.2 Proportional Integrator Derivative (PID) Controller

The PID controller was implemented in the design using the Simulink/Matlab PID Controller block. This block optimizes the parameters of the PID control function [⁵].

To improve the performance of the stabilization response, we used a PID controller in the 3D prototype designed. Thereafter, it was replicated in Simulink (a Matlab modeling software) and Sim Mechanics, and the proportional, integral, and derivative control parameters were defined.

Figure 13 denotes the behavior of the Z axis before the optimization of the PID parameters. Therein, the overshoot and unwanted vibration of the platform can be observed.

Fig. 13 Overshoot and vibration 

Finally, Fig. 14 denotes the results from the stability analysis using the PID controls that were obtained from physical prototype tests.

Fig. 14 Block respond and tuned respond with Matlab 

Figure 15 denotes the results from the stability analysis after applying the PID controller to the physical prototype.

Fig. 15 PID-based stability 

As can be observed, the pitch and yaw exhibit a slight short and overshoot, which is then offset, achieving the system stabilization. Moreover, the roll control evidences one with the best results, with an overshoot close to zero. Figure 16 shows the roll results for the fuzzy logic controller.

Fig. 16 2D output signal 

3.7 Tests and Results

The distance measurement tests toward a fixed target located 2.40 m off the LiDAR sensor, with the platform in motion, are denoted.

As the pitch angle decreases, the platform slightly approaches the target, and the distance detected by the sensor decreases. As the pitch angle increases, the platform slightly moves away from the target, and the distance detected by the sensor increases. In both the cases, the distance variation is minimal mainly because the LiDAR sensor is always kept in a horizontal position pointing toward the target.

Figure 17 denotes the tests results when changing the roll angle.

Fig. 17 2D output signal 

At different roll angles, when changing the position of the platform toward the sides in parallel direction to the target, the distance detected by the LiDAR sensor remains constant.

These results are expected because the roll controller always maintains the sensor in the same horizontal position from the target.