## 2 Materials and Methods
### Lidar Projection Analysis
#### 2.3.4. Heel Tracking from Laser Scans and Step Length Calculation  
Figure 7 shows the recognition of the right and left heel from the raw point cloud data of the LiDAR device during walking. The frames represents the time frames of walking duration and the distance (m) represents the horizontal x-axis value obtained from the LiDAR device. Values less than 0 m determined the left leg while the right leg is identified by values more than 0 m. Hence, when the heel values change from negative to positive, then the position of the left leg changes to right leg, and vice versa. As a result, the distance between the point of initial contact of one foot and the point of initial contact of the opposite foot can be computed as the step length.  
During walking, the laser projection from the LiDAR device onto the lower leg of the subjects generates a round shape. These round shapes are the contours of the back of the subject's heel, as shown in Figure 8. As the subject walks towards the end of the walking distance, the clarity of the cloud points decreased. Therefore, in order to increase the accuracy of the heel estimation, we fitted a 4 th-order polynomial for each LiDAR scanning frame, as shown in Figure 9a,b.  
Figure 6. Scatter plot of the cloud points after frame extraction.  
Figure 8: Scatter plot of all the cloud points during walking: (**a**) the identified single-support leg of the first step; (**b**) the identified single-support leg of the second step.  
Figure 7: Heel estimation from both the right and left legs.  
As shown in Figure 9, the reported values of \(\mathrm{R}^{2}\) of the cubic polynomial is 0.9133 and 0.5908 for Frames 10 and 17, respectively. Meanwhile, the \(\mathrm{R}^{2}\) value for the 4 th-degree polynomial is 0.9916 and 0.8451 for Frames 10 and 17, respectively. The \(\mathrm{R}^{2}\) values of the lower-order polynomials were unacceptable. For the higher-order polynomial, despite the better \(\mathrm{R}^{2}\) values, the u shape that represented the heel was distorted. Hence, the 4 th-degree polynomial was selected as it retains the u shape significantly. Equation (1) represents the 4 th-degree polynomial, where \(n=4\).  
\[p(x)=p_{1}x^{n}+p_{2}x^{n-1}+\ldots+p_{n}x+p_{n+1} \tag{1}\]  
As shown from Figure 7, the frame that represents the single-support leg (\(SSL\)) phase was identified. From these identified frames \((Frame_{SSL_{i}})\) of \(i=[1,\ 10,\ 17,\ 23,\ 31,\ 37,\ 45]\), we calculated the lowest local minima, which represents the heel \((Heel_{SSL_{i}})\), as shown in Table 1.  
As shown in Figure 10, the average Step Length \((SL)\) can be obtained by calculating the differences between the adjacent rows of \((Heel_{SSL_{i}})\) in \((Frame_{SSL_{i}})\) with an \(SSL\) array dimension of \(N\). Equation (2) represents this operation.  
\[SL=\frac{1}{N}\sum_{i=1}^{N}(\|Heel_{SSL_{i}}-Heel_{SSL_{i+1}}\|) \tag{2}\]  
\begin{table}
\begin{tabular}{c c} \hline \multicolumn{2}{c}{**SSL**} \\ \hline _Frame_ & _Heel_ **(m)** \\ \hline
1 & 1.0174 \\
10 & 1.6750 \\
17 & 2.3338 \\
23 & 2.9590 \\
31 & 3.5844 \\
37 & 4.1045 \\
45 & 4.4287 \\ \hline \end{tabular}
\end{table}
Table 1: Tabulated 2-dimensional sample array of \(SSL\).  
Figure 9: Scatter plot of the extracted cloud points, fitted with a quadratic polynomial of the 3 rd degree and 4 th degree, and the estimated heel position: (**a**) extracted calculated data from Frame 10; (**b**) extracted calculated data from Frame 17.