Mechanical imperfections

The purpose of this simple test is twofold. First of all, it is important to know how the target’s center point is affected by the rotation particularly because it is oriented towards the scanner at each instrument station. Ideally, the position of the center should not change when the target is rotated in any of the two directions (horizontal or vertical). However, the mechanical imperfections distort this ideal case and information is needed on how the orientation has to be done to minimize these errors.

Concerning the second aspect of the test, the mounting offset between the target and the prism has to be examined to make sure it corresponds to the one obtained from the technical specifications of both reflectors. The results of this test are scanner independent.

To answer the questions an indoor measurement setup was designed, shown in Figure 10 (left ima-ge). Two pillars stationed approximately 3 m apart were used for fixing the measuring instrument

Figure 10: Testing target construction defects. The numbers in the right image represent the directions in which each target was rotated, i.e., first left-right, then up-down.

on one side and the target on the other. Both tribrachs were levelled with precise tubular levels beforehand. The instrument Leica TS30 with an angular precision of 0.5” and distance precision of 0.6 mm + 1 ppm was chosen for the task. In each direction 1, 2, 3, and 4 (Figure 10, right image) the target was rotated by an incremental step of 10^◦ up to the angle of 40^◦, resulting in 17 target positions including the position where both rotations were 0^◦. Furthermore, in every position the target center was calculated in two stages. In the first stage, the horizontal and the vertical angle of the center point were computed from the four readings of the telescope’s crosshair as shown in Figure 10. The left-right readings were used to calculate the horizontal angle and the up-down readings to determine the vertical angle. In the second stage, the distance to the center was measured by placing the crosshair to the center of the small (2 mm) silver spot in the middle of the target. This

two stage approach has proved to be freer of ambiguity and more accurate than the one where the silver dot is used to determine all three coordinate components. Even if the distance in the second stage is measured along the direction which does not perfectly coincide with the one computed from stage one, the difference in the distance is negligible because the latter direction never leaves the silver spot. The presented measurement sequence was repeated for all four targets. Finally, the Leica reflector GHP1P with an additive constant bellow 0.1 mm was measured to be able to determine the mounting offset. All measurements to determine the horizontal and the vertical angle of the target centers in the first stage were performed with no redundancy for they were carried out manually.

In the second stage, five measurements were registered each time to estimate the distance to the target center. Concerning prism measurements, the ATR functionality was used again to perform five consecutive repetitions of all measured quantities, i.e., both angles and the distance.

The results of the test reveal that all distances were estimated with the precision which was always below 0.2 mm. These slope distances were finally transformed to horizontal distances which were used for comparison. As for the angular measurements to the prism, these were estimated with a standard deviation of 0.1” and 0.3”, which represents an insignificant level of dispersion at the distance of 3 m. In Table 2 the results of this test are shown with respect to the targets’ sensitivity to rotation. On average the positions of the target centers are the most prone to errors in the

forward-Table 2: Results of the targets’ sensitivity to rotation. ∆1, ∆2, ∆3 are the spans in which the centers are moving with ∆1 representing the left-right, ∆2 the forward-backward and the ∆3

the up-down direction as seen from the observation point.

Target ∆₁ [mm] ∆₂[mm] ∆₃[mm]

T1 1.2 0.8 0.2

T2 0.2 0.6 0.2

T3 0.7 0.9 0.2

T4 0.3 1.3 0.1

backward direction and the least in the up-down direction. With the exception of T1 (∆₁) and T4 (∆2) all spans are confined to a 1 mm range but in the submillimeter domain the targets’ sensitivity obviously is slightly different. A closer inspection indicates no strong correlation exists between the incidence angle and the size of the error in any direction ∆i at this small deviation level. On the other hand, the results reveal that the rotation of the target around its vertical axis contributes more to the error in the forward-backward direction than the rotation around its horizontal axis.

Consequently, the values of ∆₂in Table 2 are basically the outcome of rotation around the first axis and are even lower with respect to the horizontal axis. At this point a final conclusion is that it is hard to say what the spans would be like if the observations were carried out by rotating each target around its vertical axis in a full circle. Based on the results from Table 2, some values of ∆₁ and

∆2 would probably go beyond 1 mm compared to ∆3, which tends to be 3 to 4 times lower and quite stable within the observed 80^◦x 80^◦spherical window. However, such a measurement scheme cannot be done from one instrument station only and was not considered within the thesis.

Focusing now on the analysis of the mounting offset, the technical specifications of both reflectors show that the target center should be 114.5 mm (±0.7 mm) above the prism center, hence the offset should be limited to a vertical direction only. However, the results presented in Table 2 already indicated that particularly in the submillimeter domain the offset could not be constrained to this direction only. This fact can eventually be seen in Figure 11 where the positions of the target centers are illustrated with respect to the position of the prism center whereby taking into account the vertical offset of 114.5 mm. Based on Figure 11, the confirmation of the first impression about the

Figure 11: Mounting offset results. The views show the positions of the centers as seen from the observation point. The direction to the observation point is indicated by the arrows in the middle and right image for more clarity. The highlighted brackets represent the positions where both rotations are zero. All values on the axes are in [mm].

targets’ construction inequalities becomes firmly noticeable. Centers are clustered in a systematic pattern around the origin. In terms of the mounting offset, the centers for T1, T2 and T3 are about 0.5 mm lower but within the specified tolerance threshold. The same cannot be said about T4 with its centers arranged approximately around the value of 113.5 mm, i.e., 1 mm lower than specified.

At least for T4 this vertical eccentricity error should be considered every time the target is used especially because it seems to change very little. Therefore, the offset for T4 is the least accurate in the vertical direction, whereas all the others are nearly the same.

Concerning the eccentricity errors in the other two spatial directions, it is important how the spans

∆₁and ∆₂of each target are positioned with respect to the origin. For example, a 2 mm span would be split in half if positioned centrally. This fact can be observed for T1 with its ∆₁ being confined to a ±1 mm boundary. The graphs reveal that the most precise is T2 but almost the least accurate.

T1 and T3 are similar in precision but T3 tends to be more accurate. As for T4, its centers are the most disperse with some falling more than 1 mm from the origin. These largest deviations are obtained when the target is rotated around its vertical axis to the right side (direction 2 in Figure 10). Such an outcome for T4 supports the assumption that if the centers were observed in a full

circle around the vertical axis, the deviations for this target would probably be at least of this level if not a little more. On the other hand, the results for T1, T2 and T3 within this 80^◦ x 80^◦ spherical window suggest more strongly that the full circle observations would still produce centers fenced by the ±1 mm boundary. Finally, when examining the results, it also becomes clear that applying any kind of modelling routines to eliminate such horizontal eccentricity errors would be hard for no firm patterns to appear at this deviation level. Therefore, no reason exists that extending observations to a full circle would ultimately contribute to the modelling efficiency of this type of errors. It seems some of their effect can only be minimized through the transformation process if at all (see equation 3). Due to the manual observation process and the small deviation level, the increase in the reliability of the results could eventually be obtained by repeating the test at the same or perhaps different instrument-target distances.