## 4 Discussion
### AoR Estimation by Means of Helical Axis Is Significantly Affected by Time Discretization  
As reported in Section 3.1, by varying the range and speed of motion patterns, a significant dependency between estimation accuracy and the number of time frames available was observed for the IHA algorithm. In particular, this phenomenon was highlighted when high speed and range of motion are considered. Indeed, given the constant sampling rate of the simulated motion capture data, those conditions correspond to the worst time discretization of the trajectories used to reconstruct joint motion. As confirmed by the simulations presented in Section 3.3, the improvement in IHA estimation accuracy, when higher than  
Figure 10: The absolute value of the angular velocity signal, as imposed by the model (\(\omega_{\text{Ankle}}\), dashed line) in motion configuration A, and as estimated by the IHA algorithm (\(\omega_{\text{IHA}}\)). The plots were obtained by considering SNR\({}_{\text{md}}\) = 100 dB, and different levels of common noise signal: (**a**) SNR\({}_{\text{cmm}}\) = 100 dB; (**b**) SNR\({}_{\text{cmm}}\) = 90 dB; (**c**) SNR\({}_{\text{cmm}}\) = 80 dB; (**d**) SNR\({}_{\text{cmm}}\) = 70 dB. 100 Hz sampling frequencies are selected, is obvious, as signaled by the low values of linear dispersion \(d_{\text{eff}}\) and rms distance \(\Delta\) achieved (Table A3). Of course, high sampling frequency values such as 1000 Hz are not desirable during real experimentation due to the limitations of the instrumentation and data acquisition systems, though significant improvements in estimation accuracy are shown even for more feasible solutions, such as 250 and 500 Hz. Moreover, high discretization would result in a large number of time frames that can affect the efficiency of the algorithms in applications where the computational time is relevant. As shown in Section 3.3, the SARA approach does not seem to be significantly influenced by the number of time frames, regardless of the speed selected. This result is quite interesting and widens the field of application of this methodology to fast and wide motion patterns, even for relatively low sampling frequency. The dependency between AoR estimation accuracy, in terms of linear dispersion, and time discretization of the motion patterns is highlighted by the results shown in Figures 11a and 12a for representative conditions. The results presented in Section 3.1 show that, when the linear dispersion of the IHA set is high, mainly due to large ROMs and speed, the rms distance is also increased (see Figures 11b and 12b). This result signals that, in general, an increase in the linear dispersion has an impact on the estimation accuracy of the MHA position, regardless of the weighting factor chosen. As far as the angular dispersion \(\chi_{\text{eff}}\) is concerned, it seems to be independent of the several ROM/speed combinations tested, but it is affected far more significantly by the weight selected for MHA estimation. In particular, the corre  
Figure 11: Linear dispersion (**a**) and rms distance (**b**) in A and D motion configurations, for different number of frames, obtained by increasing the joint motion speed. Results are shown for IHA (w.1) and SARA algorithms. Figure 12: Linear dispersion (**a**) and rms distance (**b**) in A and D motion configurations, for different sampling rate. Results are shown for IHA (w.1) and SARA algorithms. et al. [34] shows the most limited angular dispersion among the several solutions, but it has an impact on the accuracy of AoR estimation in terms of its positioning (see Table 1, IHA (w.3) columns). The corresponding increase in rms distance was highlighted especially for motion configurations A (1 DoF, flexion/extension) and C (2 DoFs, helical motion), whereas it was quite negligible for the other configurations that share linear translation DoF of the AoR along the antero-posterior direction. The presence of translational motion occurring in the sagittal plane might mix up the analysis and should be investigated in the future.