## 5 Experiments and Results
### Regulation and Motion-Compensation Tests  
Since the end-effector inevitably contains errors due to both kinematic and non-kinematic factors, there should be errors in the final pose when the robot is controlled by open-loop control. Therefore, it is necessary to perform a process of optimizing a link parameter for a desired area through robot calibration. With this calibration procedure, it is also required to employ a feedback control using an external measuring device to further minimize the final errors.  
If the subject is stationary throughout the entire stimulation session, it is possible to reduce the error only by estimating the current pose of the robot. However, this is not the case because the subject cannot immobilize his/her head for a long time. The brain stimulation is usually performed repetitively in clinical situations, with each session lasting several minutes. Therefore, the subject is highly likely to move. We needed to track this movement. Thus, a DRF was attached to the head for tracking.  
Firstly, we performed regulation task. We gave three different target postures to the robot and estimated the error at each posture at steady state. Secondly, we performed a head motion compensation task. To verify the motion compensation performance, the subject intentionally generated small movements while the target pose was specified in the navigation program and the robot was in feedback control. For safety, control stopped when the DRF became invisible or the distance between the robot DRF and the head DRF exceeded 300 mm. In the experiment, the tracking device introduced in Section 4 reported the position and orientation of several DRFs with a sampling rate of 30 Hz. The control period of the whole system was set to 35 ms.  
Figure 9 shows the results: a is the result when three different static target postures were given, and b is the result for head motion compensation. The black lines are corrected target postures obtained through the DRF, and the blue dot-dash (-.) lines are the poses of the end-effector obtained through the tracking system. Moreover, the red lines represent error graphs for position and orientation. The position error (\(\mathbf{e}_{p}\)) is expressed by the XYZ-coordinate using Equation (4), and the orientation error (\(\mathbf{e}_{o}\)') is calculated by using the ZYZ Euler angle. We selected the ZYZ Euler angle because it is more intuitive than other representations.  
The regulation task shows that the position errors at steady state are 0.264 mm, 0.047 mm, and 0.072 mm, respectively. Orientation errors were less than 0.04 radian at all three postures (Figure 9a). This accuracy is thought to be sufficient for TMS brain stimulation because TMS targeting accuracy is more than a few millimeters. For motion compensation task, as shown in Figure 9b, the end-effector follows the subject's head, compensating the error, and finally, the error also becomes almost zero when the subject stops moving. The error seems to be high in particular when the head speed is high. This is because we set the velocity limitation of the robot for safety. The control goal is to remove the error when the target is stationary.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Point** & **Max.** & **Min.** & **Std.** \\ \hline
1 & 0.0350 & 0.0043 & 0.0097 \\
2 & **0.0504** & 0.0058 & 0.0162 \\
3 & 0.0467 & 0.0061 & 0.0115 \\
4 & 0.0260 & 0.0059 & 0.0061 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Result of repeatability experiment (unit: mm).