## 3 Results
### Analysis of Locally Optimal Selected Solutions  
Six techniques (Figures 3a and 4d) were selected and further analyzed (Table 2). In what follows, a lowering of one arm occurring at the same time as a raising of the other arm is referred to as \"arm inversion\" (e.g., see Figure 3b). An adduction (resp. abduction) immediately followed by an abduction (resp. adduction) is referred to as add-abduction (resp. abd-adduction). Finally,  
Figure 2: Number of twists generated by optimal techniques as a function of the hand path length. The color code represents the mean angle from the arm movement to the best tilting plane (BTP). The size of each circle represents the time ratio wherein the arms are moving. Magenta points represent the selected solutions. Black boxes delimit the most dense areas of solutions. The magenta box delimits a cluster of solutions which seem to be greatly affected by the use of the BTP; an enlargement of this area is presented above the graph. The black dashed line presents the linear regression over all four types of problem. some motions obtained with the 3D model are combinations of complex non-planar arm motions, hard to accurately describe with words. We refer the reader to the Supplementary Multimedia Material available online at [[http://www.mdpi.com/2076-3417/10/23/8363/s1](http://www.mdpi.com/2076-3417/10/23/8363/s1)]([http://www.mdpi.com/2076-3417/10/23/8363/s1](http://www.mdpi.com/2076-3417/10/23/8363/s1)) to get deeper insights into these 3D arm techniques. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**\#** & **Type** & **Twists** & **\(\mathbf{\phi}\)** & **Complexity** & **Effort** & **\(\mathbf{\Delta_{perf}\pm\sigma_{perf}}\)** & **\(\mathbf{\Delta_{kin}\pm\sigma_{kin}}\)** & **Convergence** \\  & & & [\({}^{\circ}\)] & **[m]** & **[Nm]** & [\({}^{\circ}\)] & [\({}^{\circ}\)] & **[\({}^{\circ}\)]** \\ \hline I & 2D-PHP & 1.01 & 56 & 6.92 & 4.5 & 2.37 \(\pm\) 0.06 & 10 \(\pm\) 6 & 87 \\ II & 2D-UHP & 2.94 & 41 & 18.2 & 151.3 & 3.11 \(\pm\) 0.04 & 12 \(\pm\) 7 & 27 \\ \hline III & 3D-PHP & 3.02 & 42 & 16.4 & 89.3 & 4.31 \(\pm\) 0.26 & 19 \(\pm\) 5 & 91 \\ IV & 3D-PHP & 3.12 & 45 & 15.0 & 64.4 & 3.77 \(\pm\) 0.34 & 17 \(\pm\) 9 & 82 \\ V & 3D-PHP & 3.28 & 25 & 10.8 & 140.9 & 5.17 \(\pm\) 0.34 & 25 \(\pm\) 17 & 78 \\ VI & 3D-UHP & 4.90 & 52 & 51.0 & 512.6 & 4.85 \(\pm\) 0.31 & 37 \(\pm\) 25 & 8 \\ \hline \hline \end{tabular} Note: effort needed by the 2D model and effort needed by the 3D model are not comparable. As explained in [7], 2D model does not include torques in shoulder change in plane of elevation, which are effective in the 3D model.  
\end{table} 
Table 2: Characteristics of selected locally optimal arm techniques, including the number of twists, the mean angle between the BTP and the plane of motion of the arms (column \(\phi\)), the length of the hand path (column Complexity), the sum of the joint torques (column Effort), the robustness (column \(\Delta_{perf}\pm\sigma_{perf}\) ), the reliability of the robustness (column \(\Delta_{kin}\pm\sigma_{kin}\)) and the convergence rate of the reoptimization (column Convergence). Figure 3: Animations of 2D techniques presented in a frame attached to the root segment in order to emphasize the motions of the arms. All arm motions occur in the frontal plane. Figure 4: Animations of 3D techniques presented in a frame attached to the root segment in order to emphasize the motion of the arms. Technique I (2D-PHP) is a 2D technique close to the one widely used by athletes to execute a backward somersault with one twist. Technique II (2D-UHP) is the 2D technique generating the most twists without a penalty for the hand path. Techniques III, IV and V (3D-PHP) are different 3D techniques which appear to yield more than three twists, while preserving a reasonable complexity. Techniques IV and V begin similarly. Even though technique IV is more complex than V at the end of the skill, it terminates in a more usual way with the arm raising in the sagittal plane just before landing on the trampoline bed.