# Crouch Gait Analysis and Visualization Based on Gait Forward and Inverse Kinematics
## 3 Results
### Gait Analysis on the Workspace
Furthermore, Figure 12 shows the values of the triangular areas between the pelvis (base frame) and the joint reference frames (knees, ankles, and toes) (Section 2.1.3) of each gait trial, calculated by applying Equations (23)-(25). For the three triangles (Figure 3), the area for normal gait for each gait sample is greater than the areas in crouch gaits. For  
Figure 11: Transverse plane view of the Cartesian coordinates of hip, knee, ankle, and toe of both right and left lower limbs during a gait cycle. For normal gait (N), row 1; for mild crouch gait (MC), row 2; and for severe crouch gait (SC), row 3. Columns 1, 2, 3, and 4 are used to plot hip, knee, ankle, and toe, respectively. Figure 10: Frontal plane view of the Cartesian coordinates of hip, knee, ankle, and toe of both right and left lower limbs during a gait cycle. For normal gait (N), row 1; for mild crouch gait (MC), row 2; and for severe crouch gait (SC), row 3. Columns 1, 2, 3, and 4 are used to plot the hip, knee, ankle, and toe, respectively. the three triangles in a normal gait, the ranges of the areas are 0.066 m\({}^{2}\leq A_{P-K}\leq\) 0.092 m\({}^{2}\), 0.118 m\({}^{2}\leq A_{P-A}\leq\) 0.271 m\({}^{2}\), and 0.113 m\({}^{2}\leq A_{P-T}\leq\) 0.332 m\({}^{2}\). The maximum values of the areas in normal and mild crouch gait occur in the initial contact and the opposite contact events, while a severe crouch gait does not occur in the same events. This could be useful for relating the problem to the stage of the gait cycle. Table 1 presents a summary of the arithmetic mean, standard deviation, and root mean square level for both the distance and area metrics. In Table 1, the value that best reflects the difference between the gaits for the six metric values is the arithmetic mean, closely followed by the RMS value, with the less significant for this purpose being the standard deviation. In a gait data analytics approach, it is possible to choose the most useful metric during the feature selection stage. In addition, all the results presented in this work are based only on one gait cycle, which is enough for the analysis, as the gait cycle can be considered a quasiperiodic signal. Although the assessment of the Euclidean distances and triangular areas are useful only for local analysis, the centroid of each of the areas and the total centroid used to obtain a metric of the location of the geometric center of the kinematic chain are useful in performing a global analysis. In Figure 13, frontal and transverse views of the global centroid for the three triangles of the normal gait adn mild and severe crouch gaits are shown. The plots in the same figure establish the range of workspace of a normal gait that can be used to assess abnormal gaits. \begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Normal (N)**} & \multicolumn{3}{c}{**Mild Crouch (MC)**} & \multicolumn{3}{c}{**Severe Crouch (SC)**} \\ \hline
**METRIC** & **MEAN** & **STD** & **RMS** & **MEAN** & **STD** & **RMS** & **MEAN** & **STD** & **RMS** \\ \hline \(d_{\Sigma_{\text{L}}\Sigma_{\text{L}}}\)(m) & 0.373 & 0.049 & 0.376 & 0.350 & 0.038 & 0.352 & 0.346 & 0.038 & 0.348 \\ \(d_{\Sigma_{\text{L}}\Sigma_{\text{L}}}\)(m) & 0.520 & 0.138 & 0.538 & 0.432 & 0.092 & 0.441 & 0.381 & 0.058 & 0.386 \\ \(d_{\Sigma_{\text{L}}\Sigma_{\text{L}}}\)(m) & 0.563 & 0.157 & 0.584 & 0.459 & 0.107 & 0.471 & 0.405 & 0.066 & 0.410 \\ \(A_{P-K}\)(m\({}^{2}\)) & 0.081 & 0.008 & 0.08 & 0.077 & 0.007 & 0.077 & 0.076 & 0.007 & 0.076 \\ \(A_{P-A}\)(m\({}^{2}\)) & 0.203 & 0.053 & 0.209 & 0.147 & 0.030 & 0.150 & 0.136 & 0.025 & 0.138 \\ \(A_{P-T}\)(m\({}^{2}\)) & 0.239 & 0.065 & 0.248 & 0.178 & 0.041 & 0.183 & 0.161 & 0.031 & 0.164 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Statistical measures for Euclidean distances and triangular areas of joint reference frames: Arithmetic Mean (MEAN), Standard Deviation (STD), and Root Mean Square value (RMS) for normal (N), mild crouch (MC), and severe crouch (SC) gaits.