# Crouch Gait Analysis and Visualization Based on Gait Forward and Inverse Kinematics
## 3 Results
### Gait Analysis on the Joint Space  
Using the Cartesian coordinates calculated for each gait sample of normal gait and mild and severe crouch gaits with the proposed forward kinematics method (see Figure 1) as input in Equations (4)-(19), the joint angles of the eight degrees of freedom were calculated. However, Cartesian coordinates could be obtained from a Motion Capture (MoCap) system as well. Figure 14 shows the joint angle signals for the movements of pelvic rotation, pelvic list, hip and knee flexoextension, and ankle dorsi/ plantar flexion of both the left and right lower limbs. For the normal gait, considering the right lower limb as the reference [2], the joint angles of the left lower limb are out of phase by half a cycle  
Table 2 shows the minimum (Min) and maximum (Max) values of the range of motion (ROM) for each joint angle (Figure 14). From these values, the ROM of the normal gait is established as the gait pattern of reference. Then, gait analysis in the joint space is  
Figure 14: Joint angles for pelvic rotation (\(q_{1}\)), pelvic list (\(q_{2}\)), hip flexoextension (\(q_{3}\)), knee flexoextension (\(q_{4}\)), and ankle dorsi/plantar flexion of the lower limbs for normal gait (black), mild crouch gait (green), and severe crouch gait (blue). The vertical line (magenta) represents the end of the stance phase. Figure 13: Sagittal, frontal, and transverse views of the global centroids of the triangles between joint references for a cycle of normal, mild crouch, and severe crouch gait cycles. supported systematically and numerically by this pattern to determine abnormal gaits such as crouch gaits. To objectively compare the normal gait and mild and severe crouch gaits in the anatomical space, statistical metrics were calculated. The arithmetic mean \(\overline{q}\), standard deviation \(q_{STD}\), root mean square \((q_{RMS})\), and shape factor \((q_{SF})\) values of the joint angles are shown in Table 3.  
Table 3 provides quantitative information on the gait performance over the anatomical space, which can be used for local and global analysis. Considering the arithmetic mean as the metric for analysis, the statistically significant difference between the three gaits is established, which allows abnormal behaviors regarding the gait pattern to be classified and the degree of abnormality to be determined. In addition, in a normal gait, the arithmetic mean of the joint angles of both lower limbs are equal, qw the difference between the two is only offset by half a cycle. However, mild and severe crouch gaits are not the same. From the mean values, it is observed that the lower limbs do not have similar performance between them with respect to the gait pattern. Thus, it is possible to determine the degree of abnormality for each lower limb with respect to both each other and the overall pattern. The standard deviation for the pelvis rotation angle \(q_{1}\) is greater in severe crouch gait than in normal and mild crouch gaits. In both crouch gaits, the \(q_{STD}\) value of hip flexoextension is higher in the right joint than in the left one, which indicates a greater abnormality in the left hip. In both knee flexoextension and ankle dorsi/plantar flexion, the abnormalities occur in both joints, being more severe in mild crouch gaits. Regarding the RMS value in Table 3, both \(q_{1}\) and \(q_{2}\) have higher values for mild and severe crouch gaits than the gait pattern. Although the arithmetic mean, standard deviation, and RMS level allow for the establishment of objective metrics to assess abnormal gaits regarding gait patterns, the signal