## 6 Positioning Accuracy
### Discussion  
All 100 readings collected at each check point during each of the three experiments were treated as one sample. For each sample the Most Probable Value (MPV) was calculated. The difference between MPV and true position of the check point as well as its standard deviation can be found in Table 2. \begin{table}
\begin{tabular}{c c c c} \hline
**Check Point** & **Experiment 1 Error \(\pm\)** & **Experiment 2 Error \(\pm\)** & **Experiment 3 Error \(\pm\)** \\
**Number** & **Standard Deviation (m)** & **Standard Deviation (m)** & **Standard Deviation (m)** \\ \hline
1 & 0.072 \(\pm\) 0.051 & 0.172 \(\pm\) 0.047 & 0.070 \(\pm\) 0.043 \\
2 & 0.219 \(\pm\) 0.097 & 0.236 \(\pm\) 0.063 & 0.081 \(\pm\) 0.035 \\
3 & 0.184 \(\pm\) 0.076 & 0.257 \(\pm\) 0.048 & 0.083 \(\pm\) 0.046 \\
4 & 0.121 \(\pm\) 0.053 & 0.122 \(\pm\) 0.049 & 0.086 \(\pm\) 0.039 \\
5 & 0.140 \(\pm\) 0.057 & 0.144 \(\pm\) 0.057 & 0.133 \(\pm\) 0.028 \\
6 & 0.064 \(\pm\) 0.056 & 0.166 \(\pm\) 0.048 & 0.080 \(\pm\) 0.030 \\
7 & 0.151 \(\pm\) 0.042 & 0.089 \(\pm\) 0.044 & 0.097 \(\pm\) 0.033 \\
8 & 0.187 \(\pm\) 0.068 & 0.288 \(\pm\) 0.058 & 0.061 \(\pm\) 0.039 \\
9 & 0.175 \(\pm\) 0.048 & 0.182 \(\pm\) 0.078 & 0.033 \(\pm\) 0.099 \\
10 & 0.277 \(\pm\) 0.082 & 0.153 \(\pm\) 0.142 & 0.099 \(\pm\) 0.041 \\
**Average** & **0.159 \(\pm\) 0.063** & **0.181 \(\pm\) 0.063** & **0.082 \(\pm\) 0.043** \\ \hline \end{tabular}
\end{table} 
Table 2: Difference between Most Probable Value (MPV) and true position \(\pm\) standard deviation. First column contains the check point number, which are used to identify each of the ten known test positions. Columns 2, 3, and 4 contain error and standard deviation for each test in each of the three experiments. On average, Experiment 3 (3D trilateration with room calibration factor) showed the best accuracy and Experiment 2 (3D trilateration without room calibration) the worst. Average difference in accuracy between Experiments 1 (2D trilateration) and 2 is only around 14%, whereas Experiment 3 has twice the accuracy of 1 and more than twice the accuracy of 2. In addition, Experiment 3 on average has a smaller standard deviation. Interestingly, test point 5, located in the middle of the room, gave approximately the same error in all three experiments. Overall, the three experiments produced stable sub-meter accuracy. Only in Experiment 2, the worst reading (test point 10) was calculated to be more than half a meter away from its true position, although when averaged with the other readings at this point produced an error of only 15.3 cm. Experiment 3, on average, had sub-decimeter accuracy (<10 cm). In order to analyze whether there was a systematic direction/pattern in which trilateration calculations (on average) were displaced in relation to true position, arrow vectors drawn to scale indicating distance, and direction of error displacement were placed on the floor plan for each of the three experiments (Figure 3). In Experiment 1 there is no clear trend in which the error vector occurs, so it is unlikely to have been caused by systematic error. When compared to Experiment 3, the error vector is relatively large. Most likely this is the direct result of the mobile phone and microphones being positioned in the same plane. The speaker on the phone was pointing directly upward, so being on the same plane meant the source (mobile speaker) was rotated roughly 90 degrees away from each microphone. Given the highly directional nature of ultrasound, this resulted in weaker, less stable signal detection even though the average distance to each microphone was shorter. In the other two experiments, the phone was roughly 1.6 m lower than the microphones, resulting in the source speaker being positioned at an angle substantially less than 90 degrees to the microphones. For Experiment 2, it can be observed that arrows mostly point inward, towards the center of the room--a predictable outcome.