## 2 Materials and Methods
### Motion Configurations Tested and Design of the Simulations  
#### 2.3.1 General Concepts  
In this section, the design of the simulations performed on the multi-body model is presented. While each of the following subsections refers to a certain set of simulations (S1, S2, S3, and S4), based on a specific combination of parameters that will be discussed, four common motion configurations were considered throughout the whole analysis. As stated in Section 2.1.2, the same knee flexion/extension pattern was chosen for all the simulations, and therefore the expression \"motion configuration\" refers only to the ankle joint. In particular, the configurations tested are the following:  
1. A: 1 DoF, flexion/extension about a fixed axis, parallel to the \(v\) axis;
2. B: 2 DoFs, flexion/extension about a moving axis, parallel to the \(v\) axis and translating horizontally in the antero-posterior direction (directed backward);
3. C: 2 DoFs, screw-like helical motion, realized by means of flexion/extension occurring about an axis parallel to the \(v\) axis, combined with a translation (directed laterally) along the same axis;
4. D: 6 DoFs, tri-planar motion, flexion/extension occurs about an axis that has 3 translational DoFs, same trends as the ones selected for B and C configurations, and has an average inclination in the frontal plane of 10 deg (the medial tip lower than the lateral one). To simulate the limited contributions of the two remaining rotational DoFs, a small oscillation of \(\pm 1\) deg about the average axis position was introduced, with the same curve trend of the flexion/extension DoF.  
Not all the simulations presented in the following were repeated for each motion configuration for the sake of clarity and conciseness. In particular, A e D configurations were always considered since they represent, in our belief, the two most significant motion conditions among the ones tested. The performance of the algorithms for the other configurations (B and C) were discussed in the first set of simulations (S1), which is still the most extended among the ones presented in this paper, to allow some discussion about those specific patterns of motion. On the other hand, configuration A is the most simplistic one, and it allows for proving the performance of each algorithm in the most restrained environment. Configuration D, although with some approximations, allows for testing the algorithms in a more realistic environment, in which the joint under investigation has full tri-planar motion. Such configuration is the one depicted in Figure 1. Translational and rotational motion patterns for the other configurations (A, B, and C) were derived from the 6-DoFs tri-planar motion by \"turning off\" the unnecessary DoFs. In all the motion configurations tested, three full gait cycles were considered.  
The noisy marker coordinates generated by the model, for each motion configuration and simulation parameter selected, were then elaborated through the AoR estimation algorithms shown in Sections 2.2.1 and 2.2.2. In particular, the helical axis estimation was performed for the three different weighting factors shown in Equations (1)-(3) and summarized in Table 2. Therefore, a total of four estimation algorithms were run for each multi-body simulation considered. For each repetition, dispersion parameters (\(d_{\text{eff}}\), \(X_{\text{eff}}\)) and rms distance \(\Delta\) were evaluated, as detailed in Section 2.2.3.