## 3 Results
### Results and Evaluation of Attitude Estimation
As can be seen from Figure 7 (Right), the estimated value is not stable even after the sensor becomes stationary when \(\beta\) was increased. Therefore, the parameter for the correction of \(Yaw\) is set to \(\beta=0.01\). Next, the parameters \(\beta_{0},c_{1}\) for coping with magnetic disturbance are obtained. The value of Equation (37) takes a value from 0 to 1, and the final maximum value of Equation (35) must be \(\beta=0.01\); thus, we set \(\beta_{0}=0.01\). The value of \(c_{1}\) should be determined by the magnitude of the difference between the measured geomagnetic field under normal conditions and that under disturbance. In the following Table 7, we show the measured values when an iPhone and an AppleWatch are placed close to the measured device. The measured values are averaged over 10 s. The measured geomagnetic field under normal conditions was averaged for 100 s, and the result of the measurement was 45.349 uT. \begin{table}
\begin{tabular}{c c} \hline \(\alpha_{0}\) & \(RP_{norm}\) **[s]** \\ \hline
0.1 & 0.072 \\
0.2 & 0.103 \\
0.3 & 0.166 \\
0.4 & 0.179 \\ \hline \end{tabular}
\end{table} 
Table 6: Average of \(RP_{norm}\) when \(a_{0}\) of the complementary filter is varied. \begin{table}
\begin{tabular}{c c} \hline \(e_{1}\) & \(RP_{norm}\) **[rad]** \\ \hline
0.01 & 0.282 \\
0.001 & 0.222 \\
0.0001 & 0.212 \\ \hline \end{tabular}
\end{table} 
Table 4: Average of \(RP_{norm}\) when \(e_{1}\) of the complementary filter is varied. \begin{table}
\begin{tabular}{c c} \hline \(e_{1}\) & \(RP_{norm}\) **[rad]** \\ \hline
0.1 & 0.212 \\
0.01 & 0.179 \\
0.001 & 0.073 \\ \hline \end{tabular}
\end{table} 
Table 5: Average of \(RP_{norm}\) when \(e_{2}\) of the complementary filter is varied. From Table 7, the error is found to occur at a distance as close as 20 cm, and becomes very large when the distance is closer than 10 cm. In practical applications, smartwatches and similar devices may be closer than 10 cm; thus, errors that occur in such cases must be dealt with. In order to avoid large errors in the estimates, we do not consider the geomagnetic values when the absolute error is about 3 \(\mu\)T. From the shape of the graph of \(e^{-ax^{2}}\), the value of \(c_{1}\) is set to about 0.8. From the above results, the parameters of the correction in the complementary filter are determined as follows: \(\alpha_{0}=0.2,\ (e_{1},e_{2})=(0.0001,0.01),\ \beta_{0}=0.01,\ c_{1}=0.8\). To evaluate how the geomagnetic sensor copes with disturbances, we compare the results between the cases with and without a geomagnetic sensor value as well as the cases with a fixed value of \(\beta\) and a variable value of \(\beta\). Because the only error that can occur in each case is the _Yaw_ value, only the _Yaw_ value is considered for the error assessment. Each estimated value is the error from the initial state when the sensor is rotated in various directions for 10 s and then stopped. The following Table 8 shows the error values of _Yaw_ axis estimation without and with the geomagnetic sensor in addition to the angular velocity and acceleration sensors. From Table 8, the errors are within 1 degree in each case when the geomagnetic sensor is used, whereas the errors are more than about 10 degrees in the case when the geomagnetic sensor is not used. This indicates that the geomagnetic sensor is important for attitude estimation. The following Table 9 shows the error of the _Yaw_ axis estimation with the AppleWatch in close contact both when the value of \(\beta\) is fixed and when it is changed dynamically to cope with the disturbance.