## 3 Results
### Results of S4 Set: Common and Random Sources of Additive Noise on Markers' Trajectories  
In Appendix A, Table A4, the dispersion parameters and the rms distance resulting from the analysis are collected for different combinations of common and random noise signals added to the markers' trajectories provided by the multi-body model. Figures 8 and 9 show a direct comparison between the MHAs obtained with the different weighting factors and the functional axes calculated with the SARA algorithm, respectively, for motion configurations A and D. The exact AoR is also shown to ease the interpretation of the results. All the plots refer to the shank LCS.  
The results highlight the significant impact of marker noise on the accuracy of AoR estimated by helical axis methodology. This effect seems to be particularly highlighted in the transition between SNR\({}_{\text{cmn}}\) = 100 dB and SNR\({}_{\text{cmn}}\) = 90 dB, and it increases at low SNR\({}_{\text{rnd}}\) values. For SNR\({}_{\text{cmn}}\) values lower than 90 dB, the addition of a random noise signal does not seem to increase the linear dispersion of the IHA sets in a significant way. The same behavior is highlighted both in A and D motion configurations. On the other hand, the angular dispersion is generally increased by lowering the SNRs of both source signals, and this effect is even more relevant for configuration A. Surprisingly, although linear and angular dispersion parameters show a significant increase even at relatively low levels of noise, the accuracy of AoR estimation by means of the MHA is generally not compromised for SNR\({}_{\text{cmn}}\) higher than 80 dB, regardless of the SNR\({}_{\text{rnd}}\) values selected. This effect is more evident for configuration A as compared to the tri-planar motion configuration (D).  
Figure 7: 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 two plots were obtained by considering the same sampling frequency (\(f_{\text{S}}\) = 100 Hz) and different motion speed: (**a**) lowest speed, V1; (**b**) highest speed, V3. Dispersion parameters are reported in each plot.  
Figure 8: AoRs calculated with MHA, by choosing different weighting factors, and SARA, for motion configuration A. Each row and column correspond to a particular level of common and random noise signals, in terms of signal-to-noise ratios. The exact AoR imposed by joint kinematics setup is highlighted in dashed black line. All axes units are in mm.  
Figure 9: AoRs calculated with MHA, by choosing different weighting factors, and SARA, for motion configuration D. Each row and column correspond to a particular level of common and random noise signals, in terms of signal-to-noise ratios. The exact AoR imposed by joint kinematics setup is highlighted in dashed black line. All axes units are in mm.  
By comparing the results presented so far with the outcomes of the SARA approach, the latter clearly outperforms the helical axis methodology for all the noise conditions tested (see Figures 8 and 9). In particular, the rms distance between the exact and estimated AoRs remains at very low values (below 5 mm) for almost all the trials, whereas it reaches about 20 mm (in the worst scenario) when the helical axis method is used. Figure 10 shows the estimation of the angular velocity performed by the IHA algorithm and its comparison with the angular velocity set as input of the multi-body model by choosing motion configuration A and different SNR\({}_{\text{cmm}}\) values. A progressively worse evaluation of the angular velocity is obtained by increasing the SNR of the common noise signal, affecting the accuracy of the estimation algorithm.