## Appendix A Exclusion of the Ankle ROM Data  
As discussed in the main body of the work, the range of motion of the ankle (both from the left and right limb) is not used in our analysis. This choice is related to the fact that we identified this gait parameter as biased. This section provides insights into this matter. Figure 11 shows the gait parameters of all the subjects grouped by their class (i.e., HC, PC, CA, or HSP) to qualitatively distinguish the variability of the parameters for different neurodegenerative diseases and healthy subjects. To better represent the extent of the data distributions, we z-scored them before grouping the data by their class. As expected, the different classes revealed different distributions. A visible example is represented by most of the distributions of the gait parameters of patients affected by hereditary spastic paraplegia, which are significantly broader than the values collected from healthy controls. Additionally, one of the parameters showing a significant difference in the distributions of controls and neurodegenerative disease is the ROM of the ankle. On the one hand, this is expected, since many studies indicate a decrease in the ankle ranges of motion in patients affected by neurodegenerative diseases [35; 36; 37; 38]. On the other hand, it is also possible to notice a significant difference between the behaviors of HC distribution and the rest of the data when considering the left and right values of ankle ROM: when the distribution of right-ankle ROM (i.e., RANkleROM) has a higher average but also presents a partial overlap with the distribution of other classes, as in the case of the left-ankle ROM (i.e., LANkleROM), the HC distribution shows a different extent, with no overlap with other distributions of the same parameter. To better visualize and estimate the difference in the ankle range of motion across the different classes, Figure 12 shows a scatter plot of the values collected for each patient and the relative distribution for every class. The figure above clearly shows a different behavior of healthy controls and neurodegenerative diseases. Considering the distributions, when compared to the diseases, healthy controls showed higher values, and the difference in the distributions was more pronounced for the LANkleROM than for the RANkleROM. Additionally, subjects affected by neurodegenerative diseases showed similar values for the right and left limbs: Indeed, most of the points of the scatter plot were distributed along the bisector, with deviations in agreementwith the expected degree of fluctuations between the left and right limbs. Nevertheless, healthy subjects showed values predominantly higher for the left limb, with in-subject differences that reached up to \(20^{\circ}\), in contrast with what is reported in the literature for healthy subjects [43]. To statistically estimate the difference between left and right values of ankle ranges of motion between healthy controls and the subjects affected by neurodegenerative diseases, we performed the [PERSON]'s \(t\)-test, which was supported by effect size estimations such as the [PERSON]'s \(d\)[44] and [PERSON] \(g\)[45] that are used to estimate the effect size in the case of small samples. The results are summarized in Table 1.  
It can be noticed that the effect size was large both for the left and right limbs; however, the largest differences were consistently on the left ankle distributions. Similar conclusions can be driven considering the \(t\)-test results, which were statistically significant in all cases, but the left-limb comparisons produced p-values lower than the right-limb ones by several orders of magnitude. Additionally, a paired-samples \(t\)-test comparing the left- and right-ankle ROM distributions of healthy controls revealed a statistically significant difference, with \(t(64)=7.56\), where \(p=1.9\times 10^{-10}\). Indeed, these parameters, when included in the analysis, led to an increase in classification performance by around 10% across all the classification tasks.