## 3 'Speed-ID-Accuracy' Model: Numerical Cases
Results are in Figure 4. Here, the accuracy of individuals is measured in relative terms, i.e., by the ratio (r\({}_{\text{SD}}\)) between the standard deviation of the endpoint position (SD(x)) and the target width (W): a high r\({}_{\text{SD}}\) value represents low relative accuracy of an individual in acquiring a given target. Curves in Figure 4 are obtained by setting the same range for A (20-45 cm) at a given W value. The ratio r\({}_{\text{SD}}\) linearly increases with ID for a given W: less accuracy performance is consequence of higher trajectory lengths as ID increases with A. Moreover, r\({}_{\text{SD}}\) decreases for higher W values at a given ID: higher accuracy can be obtained in meeting larger targets' width following a trajectory of a given length. Furthermore, r\({}_{\text{SD}}\) shows a higher sensitivity to ID for lower W values and same A range, as it can be observed in Figure 4.  
The 'Speed-ID-Accuracy' model (Equation (5)) has been applied also to a Circular Traveling Task (CTT) with constant target width along the trajectory. The ID of a circumferential trajectory (CTT) with constant target width can be easily obtained by setting \(\theta_{\text{f}}\ =\ 2\star\pi\) in Equation (11):  
\[\text{ID}_{\text{CTT}}\ =\ \frac{2\star\pi\ast\text{T}}{\text{W}} \tag{18}\]  
In [19], field data on movement time and success probability of reaching the target for different circle perimeters and constant target widths of CTTs are provided. By adopting the same approach \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Coefficient** & **a\({}^{\prime}\)** & **b\({}^{\prime}\)** & **c\({}^{\prime}\)** & **R\({}^{2}\) = 98.97\%** \\ \hline Value & 2.265 & 0.213 & 3.948 & _p_-value \textless{} 0.001 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Regression parameters of Equation (17) for the Tunnel Traveling Task. (Speed [cm/s]; A [cm]; W [cm]). Figure 4: Accuracy of the endpoint position in a Tunnel Traveling Task: (r\({}_{\text{SD}}\) = SD(x)/W) vs. ID for different W values. followed for the Tunnel Traveling Task, the regression parameters of the 'Speed-ID-Accuracy' model (Equation (5)) as well as the regression parameters of the task speed vs. the circle perimeter (A) and target width (W) (Equation (17)) have been obtained. Results of the regression analyses are in Tables 5 and 6, respectively. As shown in Table 5, the model (Equation (5)) well fits experimental data even if higher R\({}^{2}\) value could have been obtained in case of more available data. This is mainly due to the higher complexity of a CTT compared with TTT, causing a higher uncertainty in meeting the final target after a circular trajectory. By applying the classical 'Speed-Accuracy' model (Equation (4)) to the same data set in [19], the goodness of regression significantly declines, being R\({}^{2}\) = 19.85 and \(p\)-value = 0.147. As one can see, the usefulness of the proposed 'Speed-ID-Accuracy' model increases when reaching tasks have to follow more complex trajectories usually observed in actual cases. A very good regression of speed values on \(A\) and \(W\) values is obtained (see Table 6), thus confirming the linear dependency of the speed on the trajectory length (A) and the target width (W) (Equation (17)). By substituting ID\({}_{\text{CTT}}\) (Equation (18)) and v (Equation (17)) in Equation (5), the 'Speed-ID-Accuracy' model for the circular trajectory is obtained. In Figure 5, the r\({}_{\text{SD}}\) values obtained from experimental data [19] are plotted against the ID of circle perimeter (ID\({}_{\text{CTT}}\)) for different W values; in the same figure the corresponding r\({}_{\text{SD}}\) values obtained by the model are plotted. \begin{table}
\begin{tabular}{c c c c c} \hline
**Coefficient** & **a** & **b** & **c** & **\(\beta\)** & **R\({}^{2}\) = 80.66\%** \\ \hline Value & 0.010 & \(1.0\times 10^{-5}\) & 0.0154 & 2.673 & \(p\)-value = 0.001 \\ \hline \end{tabular}
\end{table} 
Table 6: Regression parameters of Equation (17) for the Circular Traveling Task (CTT).