## 5 Validation
### Theoretical and Mechanical Limits  
From the derivation of the \(pan^{\circ}\) and \(tilt^{\circ}\) axis control in Appendix B.1, the limits of the control input accuracy can be derived. For the moving head used in this paper, it is \(pan_{step}=0.008272\frac{\gamma}{dig}\) and \(tilt_{step}=0.002747\frac{\gamma}{dig}\). The testing showed that the moving head still performs movement for single-digit (\(dig\)) increments and decrements; however, the resulting accuracy of the small movements was not further evaluated. Comparable to industrial robot systems, the moving head can be characterized in terms of pose accuracy and pose repeatability [41]. The Pose Accuracy determines how precise an arbitrary given point will be hit by the light spot. The Pose Repeatability determines how precise the same point will be hit from different starting points and different movement velocities between the start and end point. The Pose Accuracy is generally harder to determine for light projection systems as the light beam does not have an end point until it contacts a surface. The pose repeatability, however, can be assessed without complication for isolated cases. From those results, a general idea about the systems accuracy can be derived. To gain an initial estimation of the pose repeatability, a simple experiment was conceptualized as shown in Figure 11. A target, as shown in Figure 10, is aligned to an arbitrary center point. The moving head is moved to a point on a concentric circle around this center point and then back to the center. The deviation from the actual center point is measured and saved. Since the movement speed depends on the distance, and since the movement path chosen by the moving head is not always the shortest or direct path, different circle diameters and positions on the circle must be tested. It is posited that the initial orientation of the axis, especially the \(tilt\) axis, has a significant influence on the pose repeatability due to the backlash within the moving head mechanism. Figure 11: Experimental setup for determining the Pose Repeatability. Figure 10: Light spot on the target (light grey circles: 1 mm, black circles: 5 mm); red circle marks the initial position of the light spot. Inaccuracies result from an uneven background. The described pose repeatability experiment was carried out by mounting the target shown in Figure 10 onto the workpiece. This way, realistic and relevant measurements can be generated, as the actual distance and orientation are used. A variety of starting points was selected, from far apart (>1 m) to achieve a high movement velocity, to short distances (5 mm) to achieve a very slow movement velocity. The starting points included single-axis movements and combined movements. However, none of the starting points could offer a measurable deviation from the marked center point to be observed. The achieved pose repeatability of the described experiment exceeded the expectations. When manually agitating the moving head to forcefully alter its \(pan\) and \(tilt\) angles, it moves back precisely to the original position, indicating that the axes are servo controlled or have position feedback, which is a possible explanation for the high level of accuracy observed. In order to gain more than a qualitative understanding of the pose repeatability, a laser tracker was used to directly measure the position at the light exit window. A retro reflector was mounted on the moving head, and the same experiment as described previously was conducted. After each return to the center of the target, the position of the retro reflector was measured with the laser tracker. The measurements and results are listed in Table 11, and the resulting standard deviations are \(\sigma_{x}\) = 79.79 \(\upmu\)m, \(\sigma_{y}\) = 19.30 \(\upmu\)m and \(\sigma_{z}\) = 107.88 \(\upmu\)m, while the reported standard deviation from the laser tracker for its position measurements is a magnitude lower, indicating that the measurements are sufficiently precise.