# Kinesthetic Learning Based on Fast Marching Square Method for Manipulation
## 4 Experimental Evaluation of FML Algorithm
### Experiments Based on Human Handwriting Paths  
Recorded motions in the LASA human handwriting library are two-dimensional, i.e., \(\xi=[x;y]\in\mathbb{R}^{2}\). For each generated pattern, seven demonstrations are provided, where the starting motion point is represented by different initial positions and the ending position is the same point in all cases. All the provided demonstrations have the same number of data points and may intersect with each other. Among the 30 handwriting motions in the library, 26 of them include one single pattern that represents one-shot  
Figure 18: Schematic of MDP. learning examples. Three of them are formed by two patterns and one by three patterns, representing multiple-shot-learning examples. For this experiment, it has been decided to make use of some predetermined and constant values for all the tests. As such, it is possible to see how the method works without being affected by a variable change. For this experiment, a \(sat=0.1\) and \(aoi=25\ px\) were taken as starting data. The experiment in two dimensions is divided into two different cases. The first case is used to compare the developed method (FML) against a group of methods from state of the art without taking into account the auto-learning process. This will allow verifying how well the generated paths fit learned data. The second case will compare the results, taking into account the auto-learning process, which allows one to evaluate the advantages of the algorithm in terms of trajectory optimization and speed. Figures 19 and 20 show the result of using FML in the LASA dataset. In Figure 19, the result of using the method without using the auto-learning process can be seen. In this case, there is no environment exploration so the method perfectly follows the data learned by demonstration. Figure 20 shows the method using the auto-learning process. As it is shown in this sub-figure, the method generates an optimization in path length and the time to solve the different cases. Overall, the figures show how reproductions (blue lines) closely match demonstrations (red points) when the auto-learning is not working, and when this process is working, the method has the ability to obtain correct paths in which the execution time is shorter. This promotes the developed algorithm having a balance between the exploration of the environment and the exploitation of learned data. In addition to the obtained values, several tests have been performed with variations in both the saturation and size of the area of influence, obtaining always satisfactory results for saturation values between \(sat=(0.05,1]\). Figure 19: Qualitative results of FML applied on the LARSA dataset only considering data exploitation. Fast Marching Learning is applied without including the auto-learning process. No exploration capabilities are available, only exploitation. 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. Once results have been presented qualitatively with the LASA dataset, a quantitative comparison has been made with some of the most widely used algorithms in the state of the art for the trajectory generation using learning methods (presented in Section 1). Those algorithms are SEDS, CLF-DM, FSM-DM, UMIC and Modified DS, whose source codes are all available online. Their characteristics are presented in Table 1.  
For this process, a quantitative comparison was made using two error functions: swept error area (SEA) [26] and velocity mean square error \(V_{rmsc}\)[41]. The SEA equation is defined as follows:  
\[SEA=\sum_{k=0}^{N}A(pt_{e}(k),pt_{e}(k+1),pt_{d}(k),pt_{d}(k+1)) \tag{8}\]  
where \(A(pt_{e}(k),pt_{e}(k+1),pt_{d}(k),pt_{d}(k+1))\) corresponds to the area of the tetragon generated by the four points represented as \(pt_{e}(k),pt_{e}(k+1),pt_{d}(k),pt_{d}(k+1)\), where \(pt_{e}\) and \(pt_{d}\) indicate the sampled points in the trajectory reproduced by the method and the demonstrated trajectories, respectively.