## 3 Results and Discussion
### Automation of the Scanning Procedure
Figure 6:0  
Figure 6:1  
Equations (8) and (9) correspond to the distances FC and HC in terms of the length \(l_{3}\) and angle \(\alpha\). Figure 6:2  
Figure 6:3  
Equations (10) and (11) express the coordinates of point A in terms of the coordinates of point B and the distances DB and AD. Figure 6:4  
Figure 6:5  
Equations (12) and (13) provide the coordinates of point A in terms of the length \(l_{1}\), the angles \(\theta_{1}\) and \(\theta_{2}\), and the length \(l_{2}\). Figure 6:6  
Figure 6:7  
Equations (14) and (15) give the coordinates of point C in terms of the coordinates of point B and the distances DB, FC, AD, and HC. Figure 6:8  
Figure 6:9  
Equations (1) to (17) represent the direct kinematic calculation of the positions for the end effector in which the 3D handle scanner is positioned to move on the established trajectory provided that the orientation of the structured light is perpendicular to the scanned surface. The forward kinematic model is useful for modeling and simulation purposes in order to check the operation of the device. Figure 9 shows the inverse kinematic scheme used to automate the scanning process using a handheld 3D scanner. In the inverse kinematic model, the parameters \(y_{\rm C}\), \(z_{\rm C}\), \(l_{1}\), \(l_{2}\), \(l_{3}\), and \(\alpha\) are known, and the parameters of the controlled joints (represented by points O, B, and A in Figure 9), \(\theta_{1}\), \(\theta_{2}\), and \(\theta_{3}\) must be obtained. In the following stage we present the deduction of the joint control parameters \(\theta_{1}\), \(\theta_{2}\), and \(\theta_{3}\) for point C with orientation \(\alpha\) and as a result of all the points along the scanning path. Equations (18) to (32) represent the inverse kinematic calculation of the positions for the end effector in which the 3D handle scanner is positioned to move on the established trajectory provided that the orientation of the structured light is perpendicular to the scanned surface. Equation (18) relates the y-coordinate of point A to the y-coordinate of point C (after considering the length FC). Figure 7:0  
Equation (19) relates the z-coordinate of point A to the z-coordinate of point C (after considering the length AF). Figure 7:1  
Equation (20) gives the y-coordinate of point A in terms of the y-coordinate of point C and the length \(l_{3}\) and the cosine of angle \(\alpha\). Figure 7:2  
Equation (21) gives the z-coordinate of point A in terms of the z-coordinate of point C and the length \(l_{3}\) and the sine of angle \(\alpha\). Figure 7:3  
Equation (22) establishes the relationship between the length l and the coordinates of point A using the Pythagorean theorem. Figure 7:4  
Equation (23) determines the length of OK (one of the sides of the triangle) in terms of length l\({}_{1}\), length l\({}_{2}\), and the cosine of angle \(\theta_{2}\). Figure 7:5  
Figure 7:6 Inverse kinematic schematics for automated scanning with a 3D handle scanner. Equation (24) determines the length of AK (another side of the triangle) in terms of length \(\mathrm{l}_{2}\) and the sine of angle \(\mathrm{\theta}_{2}\). Figure 7:7  
Equation (25) relates the length \(\mathrm{l}\) to the lengths of OK and AK using the Pythagorean theorem. Figure 7:8  
Equation (26) expands Equation (25) to express the length \(\mathrm{l}\) in terms of \(\mathrm{l}_{1}\), \(\mathrm{l}_{2}\), and the trigonometric components of angle \(\mathrm{\theta}_{2}\). Figure 7:9  
Equation (27) relates the sum of the squares of the y-coordinate and z-coordinate of point A to the sum of the squares of \(\mathrm{l}_{1}\) and \(\mathrm{l}_{2}\), and the trigonometric components of angle \(\mathrm{\theta}_{2}\). Figure 8.0  
Equation (28) simplifies Equation (27) by combining the trigonometric components. Figure 8.1  
Equation (29) solves the equation for angle \(\mathrm{\theta}_{2}\) based on the expressions derived in Equation (28).