## 4 Results
### SCARA Robot  
As the second test case a 4-dof SCARA robot is used. The kinematic diagram scheme and the geometrical dimensions of the robot are shown in Figure 5(a), while its main electric and mechanical parameters are reported in Table 3. The system performs a linear rest to rest motion of the end-effector between the Cartesian positions \(\mathbf{P}_{0}=[-0.175,\,0.400,\,0.300]\) m and \(\mathbf{P}_{1}=[0.175,\,0.350,\,0.200]\) m with initial and final orientation angles about the z-axis of \(\pi/3\) rad and \(\pi\) rad, respectively (see Figure 5(b)), and returns back to \(\mathbf{P}_{0}\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{3}{*}{**Motor**} & \multicolumn{3}{c}{**Vertical Plane**} & \multicolumn{3}{c}{**Horizontal Plane**} \\  & **Energy (J)** & \multicolumn{2}{c}{**Energy**} & \multicolumn{2}{c}{**Energy (J)**} & \multicolumn{2}{c}{**Energy (J)**} \\  & \(\circ\) & \(\bullet\) & **Reduction (\%)** & \(\circ\) & \(\bullet\) & **Reduction (\%)** \\ \hline
1 & 10.15 & 2.06 & 79.70 & 5.62 & 1.88 & 66.59 \\
2 & 9.43 & 2.29 & 75.69 & 7.31 & 2.14 & 70.63 \\ Total & 19.58 & 4.35 & 77.78 & 12.93 & 4.02 & 68.90 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Consumed energy per cycle with (\(\bullet\)) or without (\(\circ\)) energy reducing devices. Figure 5: Electric power without (top) and with (bottom) springs of motor 1 (**a**) and motor 2 (**b**) when the mechanism lies on the horizontal plane. Fifth-degree polynomial profiles have been chosen for the motion planning in the operative space as shown in Figure 7. The same figure also shows the corresponding position, velocity, and acceleration profiles of actuated joints. For this second test case, it has been assumed to provide all the four motors with regenerative drives. Conversely, the elastic elements are added only to the first two joints, that are the ones undergoing the largest motion. A lower bound of 5 Nm has been set for both the stiffnesses (\(k_{1}^{L}\), \(k_{2}^{L}\)) which is inside their admissible ranges:  
\[ \begin{array}{l}0<k_{1}^{L}<84.5\,\mathrm{Nm/rad}\\ 0<k_{2}^{L}<180\,\mathrm{Nm/rad}\end{array}  \tag{61}\]  
For the analyzed trajectory the optimum spring design is \(k_{1}^{*}=42\) Nm/rad, \(\theta_{1}^{*}=0.7418\) rad, \(k_{2}^{*}=59\) Nm/rad, and \(\theta_{2}^{*}=1.88\) rad. Such a design leads to a reduction of the electric energy consumption as it can be seen by comparing the areas delimited by the electric power curves in Figure 8 when torsional springs in parallel with the motors are used (second row) or not (first row). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameters** & **Joint 1** & **Joint 2** & **Joint 3** & **Joint 4** \\ \hline M [kg] & 12 & 15 & \(-\) & 2 \\ K\({}_{r}\)\([-\), m/rad\(]\) & \(1/60\) & \(1/100\) & \(1/210\) & \(1/30\) \\ I [kgm\({}^{2}\)] & \(2.08\cdot 10^{-5}\) & \(2.08\cdot 10^{-5}\) & \(1.70\cdot 10^{-6}\) & \(1.70\cdot 10^{-6}\) \\ F\({}_{p}\) [Nms/rad, Ns/m] & 0.03 & 0.03 & 0.2 & 0.001 \\ F\({}_{s}\) [Nm, Ns] & 0.4 & 0.4 & 0.1 & 0.02 \\ R [\(\Omega\)] & 39.4 & 39.4 & 18.2 & 18.2 \\ K\({}_{e}\) [Vs/rad] & 0.588 & 0.588 & 0.16 & 0.16 \\ K\({}_{t}\) [Nm/A] & 0.588 & 0.588 & 0.16 & 0.16 \\ \(\eta\) [\(-\)] & 1 & 1 & 1 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: SCARA parameters. Figure 6: SCARA robot: (**a**) kinematic schematic and (**b**) end-effector trajectory path in Cartesian space. In order to quantify the benefits of the concurrent addition of regenerative drives and compliant elements to the system, Table 4 states the values of the consumed energy per cycle employing or not such devices. Figure 8: Electric power without (top) and with (bottom) springs of motor 1 (**a**) and motor 2 (**b**). Figure 7: SCARA trajectory planning: (**a**) TCP coordinates vs. time and (**b**) joint coordinates vs time. By comparing the consumed energy reduction of motors 3 and 4 with that of motors 1 and 2, it appears evident the positive contribution of the compliant elements in the minimization of the energy consumption.