## 3 Results and Discussion
### Automation of the Scanning Procedure
Figure 8.2  
Equation (30) calculates the tangent of angle \(\mathrm{\beta}\) using the z-coordinate and y-coordinate of point A.  
Figure 8.3  
Equation (31) calculates the tangent of angle \(\mathrm{\gamma}\) using the lengths AK and OK, and solves the equation for angle \(\mathrm{\gamma}\) based on those trigonometric functions. Figure 8.4  
Equation (32) describes the value of \(\mathrm{\theta}_{1}\) as a function of various variables. It starts with \(\mathrm{\theta}_{1}\) equaling the sum of \(\mathrm{\beta}\) and \(\mathrm{\gamma}\) expressed by relationships (30 and 31)  
Figure 8.5  
Introducing the coordinates of the set of points \(\mathrm{y}_{\mathrm{C}}\) and \(\mathrm{z}_{\mathrm{C}}\) and computing the joint angles with the above-presented relations, we can develop the control program of the device in order to follow the prescribed scanning path. The programming of low-cost controllers is much easier and the cost of the device is much lower than in the case of a robot. Figure 10a shows the modeling of the robot arm and the end effector, as we used V-RealmBuilder (Matlab-Simulink design program) with the help of which it was possible to simulate the minimum and maximum trajectories of the robot arm as a function of angles \(\mathrm{\theta}_{1}\), \(\mathrm{\theta}_{2}\), and \(\mathrm{\theta}_{3}\) shown in Figure 8.6  
Based on the diagram shown in Figure 10b we can analyze the feedback from the real environment where the scanner will be used. Simulations were performed to generate the trajectory of the proposed arm for automated scanning using a Solidworks design program. In order to simulate the trajectories of the arm's micrometer, the Motion Control function in Solidworks was used following the scanned path represented in Figures 8 and 9. Figure 11 shows the virtual simulation of the scanning of sneaker type A using a robotic arm. In this case, scanning it took 20 s to follow the proposed trajectory based on the mathematical model and inverse kinematics. Figure 12 shows the time required for automated 3D scanning using a robotic arm for footwear type A along the point path on scanning trajectory shown in Figure 8.7  
Figure 13 shows the virtual simulation of the scanning of sneaker type B using a robotic arm. In this case it took 17.8 s to follow the proposed trajectory based on the mathematical model and inverse kinematics. Figure 14 shows the time required for automated 3D scanning using a robotic arm for footwear type B along the point path on scanning trajectory shown in Figure 8.8  
Following the simulations performed, the scanning process and the times required for scanning the shoe types A and B are shown. In the simulation shown below, the paths are identical for both sneakers but the path times differ depending on the structure of the scanned sneaker. Figure 8.9 Virtual simulation for automated scanning with a 3D handle scanner for footwear type A.  
\[\alpha=\pi-\delta \tag{1}\]0 Processing the graphical interface for the robotic arm: (**a**) modeling of the robotic arm and the end effector, (**b**) diagram used to analyze the feedback from the real environment where the scanner will be used.