## 3 Results
### Results of S1 Set: Range and Speed of Motion Patterns  
Figure 4 shows an example of IHA sets resulting from the analyses, along with the resulting dispersion parameters and MHA, when a constant weighting factor is selected. The same calculation was performed for all the ROM/speed combinations and weighting factors, and then the results were compared to the outcomes of the SARA approach. Figure 5 shows a direct comparison between the MHAs obtained with the different weighting factors and the functional axes calculated with the SARA algorithm for motion configuration A. The exact AoR is also shown to ease the interpretation of the results. The same comparison, performed for configuration D, is shown in Figure 6, in two different views. All the plots refer to the shank LCS. The full set of the dispersion parameters and the rms distance resulting from the analysis is reported in Appendix A, Table A1. The number of frames (N\({}_{\text{I}}\)) considered by each algorithm is also presented. Figure 4: IHA (yellow lines) and MHA (black dashed line) estimation for different motion configurations: **(a)** 1 DoF, flexion/extension; **(b)** 2 DoFs, flexion/extension + horizontal displacement; **(c)** 2 DoFs, helical motion; **(d)** 6 DoFs, tri-planar motion. Linear (in mm) and angular (in deg) dispersion parameters are reported in each plot. Lateral and medial tips of the exact AoR are represented in red and blue colors, green points are the intersection of the IHAs with two quasi-sagittal planes orthogonal to \(v\) axis and located at \(\pm\)50 mm from the origin of the shank LCS. Given the constant sampling rate equal to 100 Hz, an increase in motion speed corresponds to a reduction in the number of frames available to represent each gait cycle. When performing on the lowest number of samples, the IHA algorithm exhibits a relevant decrease in the general performance, even in configuration A, as signaled by the increase in the linear dispersion. Especially for configurations A and B, this effect seems even clearer when the largest ROMs are considered, with the linear dispersion increasing up to 16.7 mm (conf. A) and 27.8 mm (conf. B). Curiously, the linear dispersion of the IHA set observed for configuration B is orthogonal to the horizontal displacement of the exact AoR imposed by the model, as shown in Figure 4b. Regarding the helical motion configuration (C), the IHA algorithm performs similarly for any of the ROMs considered; however, the linear dispersion is still increased and exceeds 10 mm at the highest speed. For the tri-planar motion configuration (D), the IHA algorithm shows similar results in terms of linear dispersion when the narrower ROMs are selected, regardless of the speed selected. However, for the widest ROM, a similar increase in linear dispersion is obtained at highest speed. As far as the rms distance \(\Delta\) is concerned, the same dependency with the ROM/speed combinations shown for the linear dispersion parameters is observed. Figure 5: AoRs calculated with MHA, by choosing different weighting factors, and SARA, for motion configuration A. Each row and column correspond to a particular range of motion (ROM) and speed (V) selected, using the same nomenclature detailed in Section 2.3.2. The exact AoR imposed by joint kinematics setup is highlighted in dashed black line. All axes units are in mm. As regards the angular dispersion of the IHA sets, the results do not show a significant correlation with the ROM/speed combination selected. In particular, the angular dispersion is always lower than 1.4 deg for configuration A, 0.8 deg for configuration B, 1.1 deg for configuration C, and 7 deg for configuration D. The increase in angular dispersion in tri-planar motion is expected due to the increased variability of the exact AoR in that configuration.