# Kinesthetic Learning Based on Fast Marching Square Method for Manipulation
## 4 Experimental Evaluation of FML Algorithm
### Experiments Based on Human Handwriting Paths
To visually understand the equation, an explanation using schematics is shown in Figure 21. \(V_{rmsc}\) can be computed following the next equation: \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Global** & **Multiple** & **Single-Step** & **Direct Application to** \\  & **Stability** & **Demonstrations** & **Learning** & **Higher Dimension** \\ \hline SEDS & Yes & Yes & No & No \\ CLF-DM & Yes & Yes & No & No \\ FSM-DM & Yes & Yes & No & No \\ UMIC & Yes & Yes & Yes & No \\ Modified DS & Yes & Yes & Yes & Yes \\ Out method & & & & \\ (FML) & Yes & Yes & Yes & Yes \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Characteristics of the different algorithms compared. Figure 20: Qualitative results of FML applied on the LARSA dataset considering that exploration and exploitation are merged using the auto-learning process. Fast Marching Learning is applied considering the auto-learning process. Exploration capabilities are combined with exploitation, so new paths can be found. Arrows correspond to the velocity map, blue lines are FML results and red points are taught data. Paths start in black dots and end in black stars. \[V_{rmse}=\sqrt{\frac{1}{T}\sum_{t=0}^{T}\left\|\dot{x}_{d}^{t}-\dot{x}_{m}^{t} \right\|^{2}} \tag{9}\]  
where \(\dot{x}_{m}^{t}\) represents the velocity of the trajectory generated by the learned method and \(\dot{x}_{d}^{t}\) represents the velocity of the trajectories in demonstrations, where 0 means that the velocity of the method generates a response as fast as learned data. Therefore, the SEA score can be understood as the error in the reproduction of a learned path, comparing the shapes between learned demonstrations and the generated path for each method, while \(V_{rmse}\) measures the preservation of the velocities of the demonstrated motions. A comparison of between the results of the FML method and the other selected methods is presented in Table 2.  
As expressed in Table 2, FML has clear advantages over the other methods. The mean SEA has a lower value, which means that it has a better reproduction when keeping the same starting points. FML also has a lower \(V_{rmse}\) value, which means that reproductions are as smooth as demonstrations, which is one of the previously presented advantages. When FML used auto-learning, the SEA error cannot be used because, due to the path optimization of the FML, a shorter path length is obtained, which does not allow a point-to-point comparison with the kinesthetically acquired data. Therefore, only \(V_{rmse}\) can be used. FML with auto-learning achieves a lower velocity due to the resulting path length reduction, as previously represented in Figure 20. Thus, it can be observed that shorter paths generate faster responses. FML achieves higher accuracy in terms of similarity based on the two error functions. In the case of SEA, the proposed method improves by approximately 69.78% \begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & **Mean SEA (mm2)** & **Mean \(V_{rmse}\) (mm/s)** \\ \hline SEDS & \(8.18\times 10^{5}\) & 142.9 \\ CLF-DM & \(5.72\times 10^{5}\) & 71.8 \\ FSM-DM & \(8.37\times 10^{5}\) & 172.6 \\ UMIC & \(8.79\times 10^{5}\) & 143.2 \\ Modified DS & \(4.07\times 10^{5}\) & 56.9 \\ Out method (FML) & \(1.23\times 10^{5}\) & 12.48 \\ Out method (FML) & & \\ using auto-learning & – & 9.17 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Reproduction error of different methods on the LASA dataset. Figure 21: Visual explanation of the swept error area (SEA) [26]. and \(V_{rnse}\) promotes approximately 78.06% without the auto-learning process and 83.85% with the auto-learning process compared with the most restrictive method (modified DS).