## 3 Results
### Results of Similarity through Dynamic Time Warping  
Figure 6a describes the GRFs and each stage in the stance phase with the normalized time. Figure 6b,c show the results of the accumulated cost matrices for patient A (b) and patient B (c) and the warping path, which is expressed by pink-colored curves for each plot. If there is no difference in the gait cycle between patients with hip OA and healthy participants, the warping path draws a diagonal line. However, if there is some difference during the gait, the warping path has curves at different stages. We applied the DTW algorithm to see the gait cycle similarity between the two groups. From the left for each person, three examples are shown along the x, y, and z axes of the GRF values of the foot of one patient. We can see the description of these directions in Figure 6a. Looking at the direction in which the subject is walking, the x-axis represents the medial-lateral direction, the y-axis represents the anterior-posterior direction, and the z-axis represents the vertical direction, which is the gravity direction. The horizontal axis of each matrix in Figure 6b,c is the average value of healthy people's data, and the vertical axis  
Figure 6: Results of accumulated cost matrices for patient A (**b**) and patient B (**c**) and the warping path, which is expressed as pink-colored curves for each plot. is the value of one patient. In the DTW matrix, a lower the cost is represented by a brighter yellowish color, and the higher the cost, the darker the indigo-based color. Furthermore, the pink line represents the minimum path in the DTW matrix. This path is a diagonal line as the two GRF data sets are similar. The stance phase is divided into three parts, as shown in Figure 5(a). Looking at the GRF_x plot of patient A in Figure 5(b), there is a slight difference from a healthy person in the medial-lateral direction in the initial contact stage and a little difference in the midstance phase as well. The GRF_y plot shows a difference in the anterior-posterior part in the initial contact and propulsion stages. Finally, the GRF_z plot shows a big difference in the midstance. This means that differences in the gait originated from unconscious avoidance motions to alleviate the produced pain. In the case of patient B in Figure 5(c), it can be seen that the GRF_y plot is similar to that of healthy people, and the GRF_x plot has much difference overall, with a significant difference present in the midstance stage. Furthermore, looking at the GRF_z plot, we can see a significant difference in the midstance stage. In the GRF_z plot of both patients, the midstance stage is the most different from the data of healthy people. Unlike the other phases, the midstance stage supported the body weight with only a single limb. Thus, the midstance, which felt much weight, differed significantly. All patients walked with reduced body weight in the midstance stage. Data with reduced body weight came from limping movements to avoid pain. Figure 7 shows the results of the accumulated cost matrices for all patients, and the warping path, which is expressed by dark-blue-colored curves: (a) represents the results of GRF_x, (b) represents that of GRF_y, and (c) represents that of GRF_z. Although a clear difference exists, it is not easy to find a clear pattern. This means that the pattern of relieving pain during the gait is expressed in many ways. Through the DTW algorithm, it was possible to determine which stage differed from healthy people. However, the most challenging thing in this study was the various abnormalities according to each patient. For example, despite the same hip OA, the various abnormalities from that difference in gait originated from unconscious avoidance motions to alleviate pain. Figure 7: Results of accumulated cost matrices for all patients and warping path, which is expressed by dark-blue-colored curves: (a) represents the results of the x-axis ground reaction force (GRF_x), (b) represents that of the y-axis GRF_y, and (c) represents that of the z-axis GRF_z.