## 5 Validation
### Algorithmic Accuracy  
With the high mechanical accuracy determined in the pose repeatability experiment, the focus will now be set on evaluating potential error magnitudes and sources resulting from the control input side of the projection system. Iterative algorithms in combination with measurement errors are not able to deliver a perfect solution, but they can satisfy the given metrics as adequately as possible (if possible, and if convergence was reached). The metrics of the presented algorithm are Equations (16), (19), (23) and (24). As described previously, these equations should be as close as possible to 0, and all of these equations express angles. The errors are assumed to be small, enabling the neglect of the non-linear characteristics of the errors and the observing of the mean values for possible systematic errors and the standard deviation for potential outliers. Since the mean value and standard deviation of the position estimation in Equations (16) and (19) originates from the angle difference between two vectors, it is impossible to directly correlate them to the accuracy of the position estimation. However, it is possible to assess the quality of the result in terms of moving head steps. Ideally, the error should be smaller than one step, as this would limit the algorithmic accuracy to below the mechanical limits. For the analysis of the transformation estimation error, it is assumed that the position error is small and that the light beam contacts a flat surface perpendicularly, eliminating spot distortion. Considering that the exact surface will be neglected, and no distance information can be derived from the angle error, it is not possible to reconstruct meaningful Cartesian coordinates. Therefore, it is expedient to combine the angle errors of the \(pan^{\circ}\) and \(tilt^{\circ}\) angles of each point to a single angle error, as shown in Equation (25). \[\Delta combined^{\circ}=arccos(cos(\Delta pan^{\circ})*cos(\Delta tilt^{\circ})) \tag{25}\]  
With this combined angle error \(\Delta combined^{\circ}\), the point position error \(e\) can be calculated (Equation (26)) as a circle around the true center of a point \(P_{n}\), as seen in Figure 12. The distance \(d\) is calculated from the estimated projector position and the Cartesian coordinatesof the reference point \(P_{n}\). From the individual point position errors, a mean value and standard deviation can be calculated to evaluate the quality of the final result. \[e=d\cdot sin(\Delta combined^{\circ}) \tag{26}\]  
The mean values, standard deviations, and equivalent step counts are listed in Table 1 for the position and transformation estimation as well as the point position error. The values of the position estimation suggest an adequate result, with the \(pan\) angle being more accurate than the \(tilt\) angle. After the transformation estimation, a large error in the mean value of the \(tilt\) angle can be observed, while the \(pan\) angle error remains in the same magnitude. This deviation propagates to the point position error, where the mean value is much larger than that of the standard deviation. The low error at the position estimation stage suggests that a result with an adequate fit for the given input values was found, which, however, deviates from the true projector position and therefore led to the mean value deviation in the \(tilt\) angle. After manually tuning the result of the estimated moving head position, it was found that the \(z\) component was the cause of the inaccuracy. A hypothesis for the source of this inaccuracy is the much wider \(pan\) angle range of the reference points compared to the \(tilt\) angle range (see Table 10), since the \(x\) and \(y\) components are accurate and have a stronger correlation to the \(pan\) angle. This hypothesis is supported by the fact that the accuracy of the result increases when selecting reference points with an overall smaller \(pan\) range, even though the overall number of points is decreased.