# Manipulator Control of the Robotized TMS System with Incurved TMS Coil Case
## 7 Result
### Setting the Desired Force of the Force Controller  
The error data were collected when the different forces were applied to the force controller as shown in Figure 14a-c. There was no torque control and dummy head movement in this experiment. Figure 14a is similar to Figure 14c because, during the experiment, \(e_{\text{n}}\) is much smaller than \(e_{\text{p}}\). There are two reasons for this: the value of \(e_{\text{n}}\) is theoretically zero when the force control is initialized because when the trajectory planning ends, the coil is on the segment \(\overline{\mathbf{x}_{sf}\mathbf{x}_{f}}\), and its orientation is the same as \(\mathbf{x}_{sf}-\mathbf{x}_{f}\), from Figure 4c. And, \(e_{\text{n}}\) is compensated by the position controller component of the proposed force controller. And from Figure 14c, \(e_{p}\) is not fully compensated or even increased during the force control. The reason for this is one of the following: the target point of the TMS coil was above the head because of the navigation system error or the scalp deformation, or the calculated coil's position was inside the head because of the deformation of the head. Figure 13: Comparison of \(e,\ e_{\text{n}}\), and \(e_{\text{p}}\) when the hybrid/position force is used or not: (**a**) \(e\) when the input force is 5 N; (**b**) \(e\) when the input force is 10 N; (**c**) \(e_{\text{n}}\) when the input force is 5 N; (**d**) \(e_{\text{n}}\) when the input force is 10 N; (**e**) \(e_{\text{p}}\) when the input force is 5 N; and (**f**) \(e\) when the input force is 10 N.  
Each case is illustrated in Figure 15. If \(e_{\text{n}}\) is so small with these cases, \(\theta\) can be near 180\({}^{\circ}\). This can be verified by Figure 14c,d, where if \(e_{\text{p}}\) was increased during the experiment, then \(\theta\) is near 180\({}^{\circ}\). Meanwhile, as shown in Figure 14a, \(e\) reduced faster during the initial stage of the experiment when the desired force was larger, as shown in Table 2, whereas continuously applying a large force to the head could make the subject uncomfortable. However, because of the large contact surface between the coil and head from the shape of the curved floor of the coil, it is slow to use a low force for force control to compensate \(e\). For this reason, the desired force is set as in the following equation for faster error reduction and selecting a suitable force for interacting with the head and coil:  
\[F=F(e) \tag{14}\]  
Figure 15: Two cases where \(\mathbf{u}_{2}\) is opposite to \(\mathbf{u}_{1}\): (**a**) When the navigation-estimated target point is above the head; (**b**) When the calculated TMS coil position is inside the head. where \(e_{o}\) is the initial value of the \(e\) after the force control is initiated. If the \(e\) value was larger than 20% of \(e_{o}\), \(F\) was fixed at 40 N to quickly reduce \(e\). If the \(e\) value was between 10% and 20% of \(e_{o}\), \(F\) was reduced linearly by \(e\) to reduce the interacting force between the coil and the head. The minimum value of \(F\) was set to 5 N to maintain the coil and head contact. Figure 16 shows the \(e\), \(e_{n}\), \(e_{p}\), and the interaction force \(F_{c}\) between the head and the coil, measured by the force/torque sensor embedded in the robot manipulator, during the experiment when the desired force followed (14). From Figure 16b, the desired force \(F\) was much larger than \(F_{c}\) during the initial period of the experiment because there were many vacancies between the coil and the head when the force control was started from the state, like in Figure 17a. In this period, much of the force commanded for the force controller was used for moving the robot manipulator rather than an interaction between the coil and the head.