# Simple Ultrasonic-Based Localization System for Mobile Robots
## 4 Evaluation and Discussion  
After applying the system to the ceiling of the experimental workplace (Figure 8), with an accuracy of \(+/-5\) mm, measurements were conducted at all points from A0 to E4 (25 points). At each point, 100 measurements of the tag length from each of the beacons were performed, meaning a total of 300 measurements were taken at each point. The beacons were synchronized and triggered exactly as they would be in the application, sequentially. Initially, a request for the first beacon (T1) came through via an HF signal. Subsequently, the beacon emitted an ultrasonic signal (40 kHz) lasting 20 periods. Using the Time of Flight (TOF) method, the processor in the tag evaluated the distance and sent a request to the next beacon (T2). This process was repeated 99 more times after measuring all three distances. Before the trilateration calculation, it was appropriate to verify the obtained results and evaluate the repeatability of distance measurements from the beacons at different points (Figure 9) using the standard deviation. Figure 8: (**a**) Beacons on the ceiling, (**b**) tag. The table clearly displays, for each point, the average value (row AVG) of measured lengths from the beacons (columns \(\text{dist}_{\text{T1}}\), \(\text{dist}_{\text{T2}}\), \(\text{dist}_{\text{T3}}\)) and the standard deviation from the average value (\(\pm\)), which expresses the repeatability of distance measurements at individual points. The display of points A0 to E4 correlates with Figure 3, which is a top view of the measured points. Green color indicates points that fall within the relevant range according to Figure 5. Figure 10 shows the standard deviation view of the distance \(\text{dist}_{\text{T1}}\). Figure 10: Standard deviations of measurements of distances from T1. Figure 9: Arithmetic averages of distances and their standard deviations measured at points A0 to E4 using our measurement system. The beacon was positioned at a height between points A0, A1, B0, and B1. In these points, which are closest to the ultrasonic transmission axis, the smallest standard deviations were measured (2.13 mm to 2.49 mm). The graph illustrates an increase in standard deviations as the measured points move farther from the ultrasonic transmission axis. The most unfavorable condition is observed at point E3 (16.92 mm). However, this point lies outside the area of good coverage by the signal from beacon T1. Among the points still within the signal coverage radius of this beacon, the highest standard deviation was recorded at point D2 (6.99 mm). Figure 11 displays the standard deviations of the measured distances dist\({}_{\text{T2}}\). In this case, beacon T2 was positioned between points D0, D1, E0, and E1. In these points, there are also minimal standard deviations (2.48 mm to 3.66 mm). The following Figure 12 illustrates the standard deviations of distance measurements from beacon T3. The graph is rotated 180\({}^{\circ}\) for better clarity. The beacon is located between points C3 and C4 (with standard deviations of 3.45 mm and 3.98 mm). The highest standard deviation was measured at point D0 (5.95 mm). In [32], a similar distance measurement system using the TOF method with a variable signal period and utilizing significantly more expensive hardware for measurement is employed. The standard deviation in this system was measured at 0.7 mm, which is 3 to 10 times more precise compared to (2.13 mm to 6.99 mm) in this article's system. However, the [32] system is not suitable for mobile robotics due to its dimensions. The distance measurement system using reflected ultrasonic signal measurement [33] achieves a distance measurement error of 13 mm to 234 mm. The system presented in this article exhibits a distance measurement error in valid measured points ranging from 6 mm to 21 mm. From the measured distances using trilateration calculations (Equations (6)-(23)), average coordinate values (row AVG) for x, y, and z measurements were computed for each point (Figure 13).