## 2 Experiment Methods
### Data Preprocessing  
Considering the differentiation in the balance control for the participants while walking, the rotation vector data are the most effective [33; 34; 35]. Therefore, the rotation vectors were extrapolated from the rotation matrix data recorded with the smartphone and the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Characteristic**} & **Overall** & **BMI 1** & **BMI 2** & **BMI 3** & **BMI 4** \\  & **(n = 170)** & **(\(<\)=5\%)** & **(\(>\)5\% \& \(<\)=85\%)** & **(\(>\)85\% \& \(<\)95\%)** \\ \hline \hline \multicolumn{2}{c}{**Gender**} & & & & \\ \hline Female & 86 (50.59\%) & 2 (2.33\%) & 55 (63.95\%) & 21 (24.42\%) & 8 (9.30\%) \\ \hline Male & 84 (49.41\%) & 2 (2.38\%) & 51 (60.71\%) & 14 (16.67\%) & 17 (20.24\%) \\ \hline \multicolumn{2}{c}{**Race**} & & & & \\ \hline Hispanic & 18 (10.59\%) & 0 & 13 (72.22\%) & 4 (22.22\%) & 1 (5.56\%) \\ \hline White &
\begin{tabular}{c} 133 \\ (78.24\%) \\ \end{tabular} & 4 (3.01\%) & 83 (62.41\%) & 26 (19.55\%) & 20 (15.04\%) \\ \hline Black & 3 (1.76\%) & 0 (0.00\%) & 0 (0.00\%) & 3 (100.00\%) & 0 (0.00\%) \\ \hline Asian & 4 (2.35\%) & 0 (0.00\%) & 2 (50.00\%) & 1 (25.00\%) & 1 (25.00\%) \\ \hline Other & 12 (7.06\%) & 0 (0.00\%) & 5 (41.67\%) & 4 (33.33\%) & 3 (25.00\%) \\ \hline Total & 170 & 4 (2.35\%) & 106 (62.35\%) & 35 (20.59\%) & 25 (14.71\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Demographic information of the student participants. Figure 2: X-Y, and Z-axis orientations of the smartphone during the walking balance test. X-axis is the right (\(-\)) and the left (\(+\)) side of the participant. Y-axis is the upward (\(-\)) and the downward (\(+\)) of the participant. Z-axis is the forward (\(+\)) and the backward (\(-\)) of the participant. The participants wearing the smartphone with the mHealth application walk about a total of 157 feet. (**a**) A participant wearing a smartphone; (**b**) Indoor physical education (PE) room. mHealth application. Using the mHealth application, the X-, Y-, Z-axis rotation vectors of the middle of the waistline were identified as the center of gravity (COG) of each participant. The X-axis represents the body motion angle between the right and the left side of a participant. The Y-axis represents the body motion angle between the upward and the downward movement of a participant. The Z-axis represents the body motion angle between the forward and the backward movements of a participant (see Figure 2). The X-, Y-, and Z-axis rotation vectors came from the rotation matrix. The rotation matrix data was collected using the rotation sensor in Android Open-Source Project (AOSP). Using the rotation sensor, the mHealth application determined the rotation matrix as:  
\[\text{A}=\left[ \begin{array}{ccc}\cos\theta\cos\psi&-\cos\phi\sin\psi+\sin \phi\sin\theta\cos\psi&\sin\phi\sin\psi+\cos\phi\sin\theta\cos\psi\\ \cos\theta\sin\psi&\cos\phi\,\cos\psi+\sin\phi\sin\theta\sin\psi&-\sin\phi\cos \psi+\cos\phi\sin\theta\sin\psi\\ -\sin\theta&\sin\phi\cos\theta&\cos\phi\cos\theta\end{array} \right]\]  
Let the rotation matrix be as:  
\[\text{A}=\left[ \begin{array}{ccc}R_{00}&R_{01}&R_{02}\\ R_{10}&R_{11}&R_{12}\\ R_{20}&R_{21}&R_{22}\end{array} \right] \tag{1}\]  
From the rotation matrix (\(3\times 3\)), we extracted the rotation vector X-axis (\(R_{\text{X}}\)), Y-axis (\(R_{\text{Y}}\)), and Z-axis (\(R_{\text{Z}}\)) using the following formula [36]:  
\[R_{x}=\frac{(R_{21}-R_{12})}{\sqrt{(R_{21}-R_{12})^{2}+(R_{02}-R_{20})^{2}+(R_ {10}-R_{01})^{2}}} \tag{2}\]  
\[R_{y}=\frac{(R_{02}-R_{20})}{\sqrt{(R_{21}-R_{12})^{2}+(R_{02}-R_{20})^{2}+(R _{10}-R_{01})^{2}}} \tag{3}\]  
\[R_{z}=\frac{(R_{10}-R_{01})}{\sqrt{(R_{21}-R_{12})^{2}+(R_{02}-R_{20})^{2}+(R _{10}-R_{01})^{2}}} \tag{4}\]  
Although each participant wore the smartphone on the same location of his/her body, the sensors in the smartphone appeared to be located with slightly different slopes.