# Crouch Gait Analysis and Visualization Based on Gait Forward and Inverse Kinematics
## 3 Results  
### Gait Analysis on the Workspace  
A visualization of the Cartesian coordinates of the reference frames of the model described in Section 2.1.2 are shown in Figure 9. The coordinates have been calculated using the joint angles of the dataset of the 2392-\(OpenSim\) model [49] as input in Equations (12)-(19). This model is well-known and widely used in biomechanical modeling settings; the experimental design used to obtain the gait data is presented in [50]. The number of samples (\(M\)) of the normal, mild crouch, and severe crouch gait signals are (51), (119), and Af (101), respectively. These are the values used in this work to present the results using the gait cycle (%) as the reference. While the length of the eight rigid-body segments \(l_{1R}\), \(l_{2R}\), \(l_{3R}\), \(l_{4R}\), \(l_{1L}\), \(l_{2L}\), \(l_{3L}\), and \(l_{4L}\) was adopted from [51], in this work, we aim at numerical validation for the proposed methods for forward and inverse kinematics using the widely used gait data. For this reason, we use only a single trial for each gait (normal, mild crouch, and severe crouch). However, it is possible to take difference gait trials as input. The 4D plot shows a sagittal view of the right and left frame reference coordinates for gaits during a gait cycle: normal (N) in rows 1 and 2, mild crouch (MC) in rows 3 and 4, and severe crouch (SC) in rows 5 and 6 (Figure 9). In this work, we considered the global frame \(\Sigma_{0}\) to be fixed, that is, gait analysis is similarly assumed on a treadmill. The color bar on the left side of the plots is related to the percentage of the elapsed gait cycle, and corresponds with one sample of the cycle in the plots. Figure 8: 3D virtual gait visualization framework. In Figure 9, two main comparative approaches in the sagittal plane are shown. The comparison between lower limbs and the comparison of abnormal gaits were both performed by taking the normal gait pattern as the reference. In a normal gait, there is a correspondence between the coordinates of the right and left references; the difference is a phase shift of half a cycle from one to the other. For mild and severe crouch gaits, the same relationship as in the pattern does not occur. This allows for the establishment of the degree of severity between lower limbs. The same figure presents the workspace performance of the normal gait as the pattern. The pattern can be used to compare abnormal gaits based on both the joint workspace range and the shape factor. As mentioned in Section 1, most gait analysis works have reported only on the sagittal plane, because the analysis is focused on the hip, knee, and ankle joints. However, certain abnormalities related to pelvic movements, as well as global analysis, cannot be performed in the sagittal plane alone. For this reason, in this work we have extended the analysis to frontal plane (Figure 10) and transverse plane (Figure 11). Figure 9: Sagittal 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), rows 1 and 2; for mild crouch gait (MC), rows 3 and 4; and for severe crouch gait (SC), rows 5 and 6. Columns 1, 2, 3, and 4 plot the hip, knee, ankle, and toe, respectively. In the same way as the sagittal plane, the frontal plane allows us to visualize and compare the gait workspace performance between both lower limbs and different types of gaits together with the influence of the pelvic list, which is related to one of the lower limbs (in this case, the right one). In Figure 12, the Euclidean distances between the frame references (knees, ankles, and toosh left and right limbs are shown. Equations (20)-(22) were used to calculate each sample of the gait cycle of each gait (N, MC, and SC). The ranges of the Euclidian distance in a normal gait are 0.29 m \(\leq d_{\Sigma_{2R}\Sigma_{2L}}\leq\) 0.43 m, 0.30 m \(\leq d_{\Sigma_{2R}\Sigma_{2L}}\leq\) 0.68 m, and 0.29 m \(\leq d_{\Sigma_{4R}\Sigma_{4L}}\leq\) 0.76 m for knees, ankles, and toes, respectively.