## 6 Results and Discussions
### The Outer Loops
#### 6.2.3 Global Results  
Table 6 contains the results of force control with velocity inner loop and with various types of outer loops. It should be noted that \(\mathbf{FF}\), \(\mathbf{K_{pf}}\), \(\mathbf{K_{if}}\), and \(\mathbf{K_{cf}}\) are the gains of the outer force loops. The table shows the mean, standard deviation, and the error respect to the reference value of 5 N. Also, it incorporates the maximum and minimum values reached by the force control. Finally, the quantity of picks outside the range of \(\pm 1\) N are shown.  
Table 6 represents the value of the best experiments for every regulator (outer loop). It may be appreciated that the error is very small for all the outer loops.  
According to the theory, PV should be more damped than a simple proportional, albeit it is slower to converge. According to the experiments, PV control has a slightly worse average error and standard deviation. However, it has fewer peaks, and they are lower. In the same way, a PD algorithm is applied. The results show as the derivative action in the force is worse than the feedback velocity action.  
The smallest average error is obtained with P + FF control. However, PI and PIV have better standard deviation and fewer peaks outside the range of \(\pm 1\) N. It should be emphasized that the feedforward term was constant in these experiments. The results can be improved by adjusting the feedforward gain based on the stiffness matrix.  
Due to the number of criteria that determine the performance of force control, a prioritization matrix has been made. The aim of this matrix is determining the most suitable algorithm for the outer loop. The criteria for evaluating the algorithms are  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Outer** & \multirow{2}{*}{**FF**} & \multirow{2}{*}{\(\mathbf{K_{pf}}\)} & \multirow{2}{*}{\(\mathbf{K_{f}}\)} & \multirow{2}{*}{\(\mathbf{K_{if}}\)} & \multirow{2}{*}{**Mean**} & \multicolumn{2}{c}{**Standard**} & \multirow{2}{*}{**Error**} & \multirow{2}{*}{**Max**} & \multirow{2}{*}{**Min**} & \multicolumn{2}{c}{**N\({}^{\circ}\)****Picks**} & \multicolumn{2}{c}{**N\({}^{\circ}\)****Picks**} \\
**Loop** & & & & & & & & & & \(<\)**4 N** & \(>\)**6 N** \\ \hline P & & 0.001 & & & 4.9932 & 0.4445 & 0.13\% & 6.9728 & 3.2728 & 324 & 392 \\ PV & & 0.001 & & 0.03 & 4.9897 & 0.4504 & 0.21\% & 6.8789 & 3.1789 & 231 & 420 \\ PD & & 0.001 & & 0.03 & 4.9855 & 0.6097 & 0.29\% & 6.9084 & 2.9084 & 297 & 456 \\ PI & & 0.001 & 0.0001 & & 4.9949 & 0.4484 & 0.10\% & 6.8634 & 3.5634 & 320 & 324 \\ PIV & & 0.001 & 0.0001 & & 0.03 & 4.9945 & 0.4340 & 0.11\% & 7.1285 & 3.4285 & 204 & 355 \\ P + FF & \(1\times 10^{-6}\) & 0.001 & & & 4.9990 & 0.4531 & 0.02\% & 6.8982 & 3.2982 & 389 & 341 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparison of outer loops.  
Figure 10: Integral action (PI) vs. feedforward action (P + FF) comparison.  
* A mean close to the reference force;
* A minimum standard deviation;
* Fewer peaks above and below the reference force;
* Lower value of maximum and minimum force.  
Furthermore, it is preferable to use algorithms where the adjustment of the gains of the force control is not dependent on the knowledge of the features of the inner loop.  
Table 7 presents a prioritization matrix where the different algorithms are evaluated according to the criteria above. The weights of the criteria were previously determined, these weights should be decided according to the final application (in this case, a polishing task). The options were evaluated with a score from 1 to 5, with 5 being the best score.  
The total scores indicate that the use of a PIV algorithm is the most suitable option for this example. However, the P + FF algorithm is also a good option if the inner loop structure is known.