# Bearing Estimation for Indoor Localization Systems Using Planar Circular Photodiode Arrays
## 4 Measurements and Evaluation
### Light Intensity
#### 4.2.2 Effects of PCPA Tilt  
In principle, the PCPA was designed with the assumption that both the beacons and the utilized photodiodes are aligned on a plane, i.e., the normal vector of the PCPA is vertical. However, in real world applications (e.g., localization of a mobile robot platform) the PCPA may be tilted. Thus it is important to analyze the deviations caused by a small amount of tilt.  
In our experiments we utilized the setup and rotation pad depicted on Figure 6. The PCPA was assembled with multiple mounts, which provided a tilt of \(0^{\circ}\), \(1^{\circ}\), \(3^{\circ}\), \(5^{\circ}\), and \(10^{\circ}\).  
The results of the measurements can be seen on Figure 11, where the maximum absolute amplitude errors are plotted relative to the \(0^{\circ}\) tilt measurement values. The exact utilized error function was derived as follows. The measured not-normalized sensitivity characteristics of channel \(m\), measured at bearing \(\varphi\), using tilt angle of \(\alpha\), are denoted by \(S_{m}(\varphi,\alpha)\). The reference measurement is \(S_{m}(\varphi,0)\), which is shown in Figure 8. The measurement error \(\Delta S_{tilt}(\varphi,\alpha)\) at bearing angle \(\varphi\), caused by the tilting of the sensor, is defined as follows:  
\[\Delta S_{tilt}(\varphi,\alpha)=\underset{m}{\text{max}}\left(abs\left( \frac{S_{m}(\varphi,0)}{\text{max}(S_{m}(\varphi,0))}-\frac{S_{m}(\varphi, \alpha)}{\text{max}(S_{m}(\varphi,0))}\right)\right) \tag{22}\]  
Figure 11: Maximum absolute error of the relative amplitude measurements with different amount of sensor tilt.  
Figure 10: Comparison of light intensity measurements with different reflective materials at 0.5 m: (**a**) black paper, (**b**) white paper, (**c**) aluminum foil, (**d**) mirror. Blue line: measured intensity, red dot: angular placement of the beacon.  
Notice that in (22) the first term is the normalized reference sensitivity characteristics of channel \(m\), while the second term is measured with the tilted sensor using the same (reference) normalization factor. Thus (22) provides a metric to characterize the measurement error, due to tilting. Figure 11 shows the measured \(\Delta S_{tilt}\) values for \(\alpha=1,3,\ 5,\ 10\).  
As Figure 11 shows, the largest error is expected in directions toward the tilt, i.e., in our case towards directions \(0^{\circ}\) and \(180^{\circ}\). The error is small in directions perpendicular to the tilting direction, in this case \(90^{\circ}\) and \(270^{\circ}\). It should be noted that the error never reaches 0 and a small asymmetry is also present, due to the measurement and processing noise and a minimal reflection from the environment. Additionally, it is notable that the \(\alpha=10\) tilt for the PCPA is an exaggerated case: most of the intended applications for the device provide less deviation (e.g., in case of industrial mobile robots the tilt is limited by the maximum inclination of ramps).  
Based on the results shown in Figure 11, it is evident that the effect of the sensor tilt is negligible to the measured amplitude values (mostly below 5%) compared to the deviations caused by reflective surfaces (e.g., Figures 9 and 10, where the deviation can be as much as 10%-660%).  
It is important to note, that the relatively low amplitude variations caused by the tilt of the PCPA is due to the shape of the photodiodes. The selected type has a cylindrical front side, in contrast to the most common half sphere shape, providing smaller vertical sensitivity.