# Open Loop Position Control of Soft Hydraulic Actuators for Minimally Invasive Surgery
## 4 Results
### Positioning
#### 4.1.3 Three DOF Hybrid Parallel Mechanism  
Figure 8 shows the path following results for the rigid three-DOF system. The performance was again satisfactory despite the presence of some offset and triangular skew as described in the previous section. Due to the integration of the prismatic joint, some  
\begin{table}
\begin{tabular}{l l l l l l l} \hline \hline \multirow{2}{*}{**Configuration**} & \multirow{2}{*}{**Test**} & \multirow{2}{*}{**RMSE/mm**} & \multicolumn{2}{c}{**Hysteresis**} & \multicolumn{2}{c}{**\% Error**} \\ \cline{5-8}  & & & **/mm** & **+ve** & **\(-\)ve** \\ \hline Single & Min. to Max. & 0.4713 & 0.7113 & +6.25\% & \(-\)4.12\% \\ Single with pulleys & Min. to Max. & 1.5062 & 0.9580 & +7.48\% & \(-\)1.34\% \\ Antagonistic & Min. to Max. & 0.3496 & 0.5145 & +3.67\% & \(-\)4.08\% \\ Single & Random & 0.3711 & - & +4.23\% & \(-\)4.75\% \\ Antagonistic & Random & 0.2632 & - & +2.80\% & \(-\)3.10\% \\ \hline \hline \end{tabular}
\end{table}
Table 2: Summary results table for each of the one-DOF tests.  
\begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline  & & & \multicolumn{2}{c}{**RMSE/mm**} & \multicolumn{2}{c}{**Max. Error Y/mm**} & \multicolumn{2}{c}{**Max. Error Z/mm**} \\ \cline{3-8}  & & **Y** & **Z** & **+ve** & **\(-\)ve** & **+ve** & **\(-\)ve** \\ \hline \multirow{4}{*}{No pulley} & Circle 2.5 mm \(\varnothing\) & 0.2818 & 0.4788 & 0.6360 & \(-\)0.3016 & 0.2670 & \(-\)0.8580 \\  & Circle 5 mm \(\varnothing\) & 0.6705 & 0.5458 & 1.4873 & \(-\)0.8085 & 0.9783 & \(-\)1.1430 \\  & Circle 7.5 mm \(\varnothing\) & 1.8396 & 1.5948 & 3.9157 & \(-\)4.3145 & 3.6205 & \(-\)3.9453 \\  & Raster scan & 3.3391 & 1.7345 & 10.0754 & \(-\)8.8439 & 4.3930 & \(-\)5.8534 \\ \hline \multirow{4}{*}{Pulley} & Circle 5 mm \(\varnothing\) & 0.3956 & 0.8915 & 1.0458 & \(-\)1.0229 & 1.6253 & \(-\)0.0897 \\  & Circle 10 mm \(\varnothing\) & 0.5805 & 1.2137 & 0.7261 & \(-\)1.5084 & 2.3462 & \(-\)0.2917 \\ \cline{1-1}  & Circle 15 mm \(\varnothing\) & 0.5131 & 0.5003 & 1.4162 & \(-\)1.2809 & 1.2756 & \(-\)0.2438 \\ \cline{1-1}  & Circle 20 mm \(\varnothing\) & 0.8735 & 0.9833 & 2.6011 & \(-\)2.2634 & 3.1025 & \(-\)1.6103 \\ \cline{1-1}  & Raster scan & 4.9296 & 0.9606 & 6.0424 & \(-\)6.1034 & 3.5801 & \(-\)3.5761 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Summary table for results of each of the two-DOF tests.  
Figure 7: Results of path following tests for the two-DOF setup. (**A**) Circle diameter of 2.5 mm, no pulleys. (**B**) 5 mm diameter, no pulleys. (**C**) 7.5 diameter, no pulleys. (**D**) Raster scan pattern, no pulleys. (**E**) Circle diameter 5 mm, with pulleys. (**F**) 10 mm diameter, with pulleys. (**G**) 15 mm diameter, with pulleys. (**H**) 20 mm diameter, with pulleys. (**I**) Raster scan pattern, with pulleys.  
hysteresis was present in the X direction because of the friction experienced between the Bowden tube and the flexible rod. It is worth noting that avoiding the effects of friction was one of the motivations for developing the soft actuators.  
The observed position in the helix path shows a deviation from the desired path when the path started again from the beginning, denoted by the straight line in the desired path aligned with the X axis. This occurred because there was no pause between the end and beginning of the repetitions, and as such, the prismatic joint position lagged the desired value significantly.  
Table 4 contains the RMSE in X, Y, and Z for the five different tests that were carried out, as well as the minimum and maximum errors. Note that the large RMSE and minimum error in X for the helix test was due to the transition from the maximum extension to the minimum, as shown in Figure 8C.