Manipulator Control of the Robotized TMS System with Incurved TMS Coil Case

Abstract

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The force/torque control, considering the incurved shape of the TMS coil shape, improved the adherence between the coil and the subject’s head. It also enhanced the accuracy of the coil.

Abstract

This paper proposes the force/torque control strategy for the robotized transcranial magnetic stimulation (TMS) system, considering the shape of the TMS coil case. Hybrid position/force control is used to compensate for the error between the current and target position of the coil and to maintain the contact between the coil and the subject’s head. The desired force magnitude of the force control part of the hybrid controller is scheduled by the error between the current and target position of the TMS coil for fast error reduction and the comfort of the subject. Additionally, the torque proportional to the torque acting on the coil’s center is generated to stabilize the contact. Compliance control, which makes the robot adaptive to the environment, stabilizes the coil and head interaction during force/torque control. The experimental results showed that the force controller made the coil generate a relatively large force for a short time (less than 10 s) for the fast error reduction, and a relatively small interaction force was maintained for the contact. They showed that the torque controller made the contact area inside the coil. The experiment also showed that the proposed strategy could be used for tracking a new target point estimated by the neuronavigation system when the head moved slightly.

1. Introduction

Transcranial magnetic stimulation (TMS) is a non-invasive and safe technique that induces a current in the brain by passing a brief and high-intensity magnetic field generated by a magnetic coil [1,2,3]. This technique is used in various areas such as brain mapping [4,5,6,7], studying brain–behavior relationships [6,8,9], and medical therapeutic applications [3,10,11].

A professional operator usually manipulates the TMS coil manually. However, adjusting the TMS coil by hand is limited in for reducing the error between the coil position and the target stimulation point. This error affects significantly TMS responses like the motor-evoked potential (MEP) and the TMS-evoked potential (TEP). In other words, the subjects may receive weaker stimulation to their brain by a small error [12]. Meanwhile, because each TMS session progresses for tens of minutes [13,14], it is difficult to compensate for the head movement of the subject by continuously holding the coil using a hand. Additional devices can be used [15,16] to solve this problem. However, because these devices make subjects uncomfortable owing to their poor wearing sensation, they cause stress. Since this stress makes subjects secrete cortisol, which affects neuronal activity [17], it can cause variability in the efficacy of the TMS [18,19].

To overcome these shortcomings, the robotic TMS system in which a robot manipulator controls the TMS coil has been proposed [18,20,21,22,23,24,25,26,27]. As robot manipulators have a high positioning accuracy and repeatability, they can improve the coil positioning accuracy. Furthermore, because the neuronavigation of the robot’s TMS system can estimate the subject’s head position and the target stimulation site, the operator does not have to continuously adjust the coil manually for a long time.

Some studies used the force/torque control strategies of a robot manipulator, which are methods to regulate the contact force between the robot manipulator’s end-effector and the environment, to improve robotized TMS systems. The first case is that of Zakaria [26]. Using adaptive force control and a neuro-fuzzy algorithm, he composed a system that can track slight head movement and improve safety. In the same year, Ritcher et al. [22] suggested a method for controlling the interaction force between the coil and the subject’s head by moving the coil in the direction of the end-effector of the robot manipulator or in the opposite direction. Jaroonsorn et al. [27] used a hybrid position/force control to maintain a constant interaction force between the head and the coil while compensating to ensure the position error. Noccaro et al. [21] used impedance control for the safe interaction between the coil and head.

This study used hybrid position/force control and proportional torque control, which generates the torque proportional to the torque error between the current and desired torque, taking into account the shape of the TMS coil. The coil we used in this study has an incurved shape, as shown in Figure 1, and we will utilize that characteristic. As the coil is curved and the human head is round, the contact area between them increased. This increases friction and reduces the instability caused by oscillations generated from sensor measurement noise or control signal disturbances. It also causes greater surface friction between them, physically prohibiting the slight head movement. And, because the surface contact reduces the distance between the coil and head, the magnetic field for stimulation is less affected by the slight head motion. However, the larger surface contact area also makes it difficult for the coil to compensate for the error between its current and target treatments with small force because of the larger friction while changing the force direction commanded by the robot controller. This problem was solved by scheduling the magnitude of the force. Proportional torque control is also applied to create a contact area between the coil and the head near the center of the coil. This stabilizes the contact. Compliance control, a force/torque control strategy to make the robot adaptive to the environment, also stabilizes the coil and head interaction during force/torque control.

The remainder of this paper is organized as follows: Section 2 describes the method of calibrating the coordinate system between the camera and the base of the robot manipulator. Section 3 explains the planning of the trajectory from the initial pose to the target pose until the TMS coil contacts the head. Section 4 describes the force and torque control strategies used to improve the accuracy and contact. Section 5 describes the manner in which the robot manipulator tracks the target point when the subject’s head moves. Section 6 describes the experimental setup and processes, and Section 7 presents the experimental result. A discussion of this study with the future work is presented in Section 8. Finally, the conclusions of this study are presented in Section 9.

2. Calibration Between the Robot Manipulator and Neuronavigation System

To control the robot manipulator using the coordinate data from the navigation system, we first express the poses measured by the camera to the coordinate system of the robot manipulator’s base. This section describes the method used to achieve the calibration.

Before explaining the calibration method, the notation of the transformation matrices is described as follows: ${}_{ }{}^{x}T_y$ expresses the pose of x described by the coordinate system of y. More clearly, ${}_{ }{}^{x}T_y$ is defined as follows:

(1)${}_{ }{}^{x}T_y=\left[\begin{array}{cc}{}_{ }{}^{x}R_y& {}_{ }{}^{x}t_y\\ 0& 1\end{array}\right],$

where ${}_{ }{}^{x}R_y$ is the rotation matrix that describes the orientation of x based on the coordinate system of y and ${}_{ }{}^{x}t_y$ is the position vector of x based on the coordinate system of y. The letters at the location of x and y are defined as follows: b is the base of the robot manipulator, c is the camera of the navigation system, and t is the tool, which is the TMS coil. These expressions are illustrated in Figure 2. Hereafter, the position of the TMS coil means the center of the coil’s floor, and the coordinate system of the TMS coil is the coordinate system whose axes are parallel to those of the end-effector of the robot manipulator and whose origin is the location of the TMS coil. Using these expressions, we can describe the relationship between the robot base coordinate and the camera coordinates using the following equation:

(2)${}_{ }{}^{b}T_c={}_{ }{}^{b}T_e{}_{ }{}^{e}T_t{}_{ }{}^{t}T_c.$

Theoretically, if the robot manipulator and camera do not move, ${}_{ }{}^{b}T_c$ is constant regardless of the location of the TMS coil. According to (2), ${}_{ }{}^{b}T_c$ can be obtained using three values according to (2): the pose of the end-effector of the robot manipulator expressed by the coordinate system of the base of the robot manipulator, the pose of the TMS coil expressed by the coordinate system of the navigation camera, and the CAD information of the TMS coil. Meanwhile, (2) can be rewritten as follows:

(3)${}_{ }{}^{b}T_c={}_{ }{}^{b}T_t{}_{ }{}^{t}T_c.$

Therefore,

(4)${}_{ }{}^{b}T_t={}_{ }{}^{b}T_c{}_{ }{}^{c}T_t.$

Using (4) after fixing ${}_{ }{}^{b}T_c$, the coordinates estimated by the camera can be converted to the robot base coordinate system.

3. Trajectory Planning

All trajectories in this study are planned using the trapezoidal velocity profile method [28] without the third phase. The trapezoidal velocity profile method is used to plan trajectories based on the trapezoidal velocity–time graph shown in Figure 3. The symbols in the graph are defined as follows: $a$ is the acceleration/deceleration during the accelerating/decelerating periods and $t_f$ is the total time required to execute the planned trajectory. If the trajectory length is fixed, these parameters are not independent; that is, some parameters can be determined if others are specified. For safety considerations, the values $a$ and $v_{\max }$ are specified. The trajectory length $s$ can be determined by calculating the area of the trapezoid surrounded by the velocity profile curve and $x$-axis in Figure 3, given by $s=v_{\max }\left(t_f-t_a\right)$. From this equation, we can derive

(5)$t_f={\displaystyle \frac{s}{v_{\max }}}+t_a.$

Additionally, $t_f$ must be at least $2t_a$. In other words,

(6)${\displaystyle \frac{s}{v_{\max }}}+t_a\ge 2t_a.$

By substituting $t_a=v_{\max }/a$ into Equtation (6), we obtain

(7)$v_{\max }\le \sqrt{sa}.$

To ensure all trajectory plans meet this condition, if the specified $v_{\max }$ is larger than $\sqrt{sa}$, it is automatically adjusted to $\sqrt{sa}$.

Based on this method, the trajectory plan for the TMS coil consists of four phases, which are proceeded sequentially. The first to third phases are related to locating the coil near the desired coil position while avoiding collisions with the subject’s head. Each phase is illustrated in Figure 4. The meanings of the symbols are as follows: $x_i$ represents the initial position of the TMS coil when the trajectory planning begins. $x_o$ is the center of the virtual sphere for the path planning, as detailed in Section 3.1. $x_{si}$ and $x_{sf}$ are the points on the virtual sphere’s surface closest to $x_i$ and $x_f$. Following the third phase, the fourth phase begins, which involves the zero calibration of the force sensor attached to the manipulator’s wrist. The detailed explanations are provided below.

3.1. First Phase: Approach from the Initial Position to the Spherical Surface

The virtual spherical surface is generated before the robot manipulator begins to operate. Its center is 25 mm above the subject’s head center, as estimated by the neuronavigation system. The robot manipulator is prohibited from approaching the inside of the sphere to avoid collisions with subject before the coil approaches the subject’s head (third phase). The reason why the center of the sphere is slightly above the head is discussed in Section 3.3. After generating a spherical surface, the coil is moved from $x_i$ to $x_{si}$. Upon reaching $x_{si}$, the coil adjusts its orientation to face the center of the sphere $x_o$. In other words, the coil’s orientation is gradually aligned with the direction of the vector $x_{\mathrm{o}}-x_i$. The processes of the positioning and orientation alignment are performed separately to avoid singularities in the generated path.

3.2. Second Phase: Trajectory on the Sphere Surface

As the shortest path between two points on a spherical surface is the shorter arc connecting those points [29], the trajectory between $x_{si}$ and $x_{sf}$ is decided as the shortest arc of the great circle of the virtual sphere between those two points, as shown in Figure 4b. In addition, while the coil moves on the spherical surface, it faces the center of the sphere.

3.3. Third Phase: Approaching the Subject’s Head from the Point of the Spherical Surface

The coil moves from $x_{sf}$ to $x_f$ at the same velocity. The coil orientation is maintained during this phase. Since the last pose of the next phase is the same as that of this phase, $x_f-x_o$ is the coil orientation before starting the force/torque control. The trajectory is graphically shown in Figure 4c.

If the center of the sphere is above the head, $x_f-x_o$ is more perpendicular to the head face, as shown in Figure 5. This helps the center of the coil approach the head surface. Trajectory planning is completed when the coil is in contact with the subject’s head. The system determines whether the coil is in contact with the force measured by a force sensor embedded in the wrist of the robot manipulator. If the measured force exceeds a specified threshold, the control system recognizes that the coil is in contact with the head of the subject.

3.4. Fourth Phase: Zero Adjustment of the Force/Torque Control

To remove the bias of the force/torque sensor, it is initialized to zero after moving the coil 5 mm away from the last trajectory point in the third phase. Subsequently, the coil is moved back to that point again to start the force/torque control.

4. Force and Torque Control for Adhering Between the TMS Coil and the Head

If the TMS coil is controlled only by its position, then it may not adhere to the head of the subject. In this case, the position of the brain targeted by TMS can vary significantly due to the head’s movement, even if it is small. In addition, it is not possible to compensate for a relatively small error without considering the interaction between the coil and the head. In this study, a force/torque control strategy is proposed to ensure the coil remains attached to the head while minimizing the error between the target and current coil positions. This is achieved after the trajectory planning described in the previous section, through the application of compliance control, hybrid position/force control, and proportional torque control.

4.1. Force Control

For the force control of the proposed system, we selected the hybrid position/force control strategy, which controls the robot position and the interaction force at the same time by confining each control orientation perpendicularly. It is selected not only to maintain the contact between the coil and head, but also considering the position control to compensate the error. The following paragraphs describe how to realize this strategy.

The notations described in Figure 6 is defined as follows: $x_t$ represents the location of the TMS coil and the unit vector $u_1$ represents the orientation of the coil. In addition, $u_2^{\prime }≔\left(x_t-x_f\right)/\parallel x_t-x_f\parallel$. For the hybrid position–force control, the unit vector $u_2$ is defined as perpendicular to $u_2^{\prime }$ and $u_1$, or

(8)$u_2={\displaystyle \frac{u_2^{\prime }-\left(u_1\cdot u_2^{\prime }\right)u_1}{\parallel u_2^{\prime }-\left(u_1\cdot u_2^{\prime }\right)u_1\parallel }}.$

The process for calculating $u_2$ described above corresponds to the blue block of Figure 7. Suppose that $\theta$ is the angle between $u_1$ and $u_2^{\prime }$. Then $\theta$ is the same as ${\cos }^{-1}\left(u_1\cdot u_2^{\prime }\right)$, described in the yellow block of Figure 7. The desired force generated by the coil is expressed as:

(9)$F=F(u_1\left|\cos \theta \right|+u_2\left|\sin \theta \right|)$

where F denotes the magnitude of what is generated by the coil. How the weight between the force and position controllers was decided is described in the orange and green block in Figure 7. In this control scheme (9), the role of the term including $u_1$ is the force controller, which maintains contact between the coil and the head. The role of the term including $u_2$ is the position controller which compensates the error between the coil’s current and target positions. Even if $u_2$ does not consider the component of the error parallel to the orientation of the coil, it does not matter because $u_1$ will compensate for that by adjusting the coefficients $\left|\cos \theta \right|$ and $\left|\sin \theta \right|$ from (9). Meanwhile, if the target point estimated by the navigation system is located above the head, the coefficient $\left|\cos \theta \right|$ makes the error larger. This is because while the force control is trying to reduce the magnitude of the error, which is parallel to $u_1$, the error component parallel to $u_2$ becomes larger owing to both orientations being opposite in this situation. In other words, $\theta$ can be near $180°$. However, adhering the coil to the head rather than compensating for the error is more important if the coil is not too far from the estimated target point. The force control described above is shown in Figure 7.

SInce the incurved coil case contacts the head with a larger area than the flat one, a larger force is required to rapidly reduce the coil’s position error. However, transferring a larger force to the head for a longer time makes the subject uncomfortable. To solve this problem, $F$ is scheduled based on the coile position error, as shown in Figure 8. The maximum force $F_{\max }$ is set for the safety and hardware limitations. The minimum force $F_{\min }$ is set to maintain contact between the coil and head. $e_o$ is the coil error when the manipulator begins the force control. e is the coil’s current error. $r_1$ and $r_2$ are the scaling factors for determining the error interval where F varies linearly with e.

4.2. Torque Control

The floor shape of the coil used in this study is curved and symmetrical, as shown in Figure 1. Therefore, the contact area between the head of the subject and the TMS coil can be increased by locating the contact area closer to the center of the coil. By increasing the contact area, the coil can contact stably and better resist any slight movement of the head by increasing the surface friction. To make the contacting surface near the center of the coil, proportional torque control is applied as described below.

Suppose that the force between the subject’s head and the floor of the coil is consistent and that the force’s distribution on the contact area is uniform. Then, the closer the torque acting on the center of the coil’s floor is to zero, the closer the centroid of the contact area is to the center of the coil’s floor. Therefore, the torque is commanded to the controller as follows:

(10)$\tau \left({\tau }_c\right)=k_p{\left[{\tau }_{cx},{\tau }_{cy},0\right]}^T$

where ${\tau }_c={\left[{\tau }_{\mathrm{cx}},{\tau }_{cy},{\tau }_{cz}\right]}^T$ is the measured torque acting on the center of the coil’s floor, and $k_p$ is the proportional gain of the torque control. Since it is similar to P (proportional) control, where the desired torques along the x- and y-axes are zero, it is referred to as “proportional torque control” in this study. The coordinate system of $\tau$ and ${\tau }_c$ are the same as the tool’s. The z value of τ is zero because it cannot tilt the coil, hence it rarely affects the adhesion. This overall control is described visually in Figure 9.

4.3. Compliance Control

Compliance control is implemented using the URScript function provided by the manufacturer for the adaptive characteristics between the robot and the environment [30]. It stabilizes the interactions between the TMS coil and the head. As the amount of torque generated along the z-axis of the tool coordinate is minimal, compliance control is applied for all orientations except the z-rotation axis of the tool coordinates.

5. Tracking the Target Point with TMS Coil

The subject’s head may move during the TMS sessions. If the head movement is not slight, the target point of the TMS coil must be changed. If the head does not move out quickly from the coil, it adheres to the head almost continuously while the head is moving because of the force controller of the hybrid position/force controller (9). In this case, the target point can be tracked by varying $x_f$ in the force controller. The experimental results demonstrate the effectiveness of the proposed method.

6. Experiment

6.1. Experimental Setup

A Polaris VEGA ST camera was used for our navigation system in our experiment, together with an NDI passive marker and NDI stylus (Figure 10). One marker was attached to the coil to calibrate the camera and robot manipulator (Figure 11). The other one was attached to the headband to estimate the center of the dummy head and the target point of the TMS point. The manipulator used in this research was UR5e from Universal Robot. The robot control program was programmed using Visual Studio C++ 2019 with the ur_rtde library [31], developed by SDU Robotics. A force/torque sensor embedded in the wrist of the robot manipulator was used to measure the force and torque. As the sensor does not have high accuracy or precision [32], it can show that the control strategy in this study can be used with inexpensive force/torque sensors. The TMS coil used in this study was developed by AT&C. The distance between the robot manipulator and the camera was approximately 2.3~2.5 m, and the dummy was located based on the measurement volume of Polaris VEGA ST [33] and the manipulability of the robot manipulator. Moreover, the dummy was moved by adjusting the micrometer of the dummy stage (Figure 12) to test the target point tracking algorithm.

6.2. Experimental Protocol

First, the camera estimated the position of the marker attached to the coil. The robot controller read the TMS location information from the camera and calculated the matrix ${}_{ }{}^{b}T_c$. The navigation system used the location information of both ears, the nose, and the middle of the forehead to estimate the location of the center of the head and the target point of the TMS coil. In this study, we set the left dorsolateral prefrontal cortex as the target point.

After preparing the robotized TMS task, the robot manipulator was instructed to locate the TMS coil at the target point. The force/torque control is applied by using the built-in force control function of the URScript programming language [30]. First, it is shown that the position controller of the hybrid position/force controller compensates for the error between the current TMS coil position and the target TMS coil point by comparing two controllers with the same forces: the hybrid position/force controller, that we suggested in (9), and the force controller without the position controller, which is $F=F{u}_1$. Second, we applied the suggested force controller with the desired forces of 5, 10, 20, 30, and 40 N without torque control. This was to explore the relationship between the magnitude of the desired force and the error of the TMS coil to the desired point and decide how to set the desired force. After selecting, the force controller was applied with a proportional torque controller with gains $k_p=0.0, 1.0, 2.0,4.0,$ and 4.5 to optimize. Until parameter $F$ and $k_p$ were chosen, the dummy was fixed during the experiment. After determining all control parameters for the next experiment, we moved the dummy 7 mm to the x and y axes, as shown in Figure 11, after the coil arrived at the target point. We estimated a new target point from the navigation system and made the robot-manipulated coil track that. From all the experiments, we obtained the error data between the current and estimated target positions of the TMS coil, the torque applied to the center of the coil, and the force applied to the coil.

7. Result

For convenience, the symbols used in this section are defined as follows: e is the positional error magnitude between $x_t$ and $x_f$. In other words,

(11)$e=\parallel x_f-x_t\parallel .$

$e_{\mathrm{n}}$ and $e_{\mathrm{p}}$ are the magnitudes of the error components that are normal and parallel to the orientation of the TMS coil, respectively. That is,

(12)$e_{\mathrm{n}}=\left(x_f-x_t\right)\cdot u_1$

and

(13)$e_{\mathrm{p}}=\left(x_f-x_t\right)\cdot u_2.$

7.1. Calibration Error

With ${}_{ }{}^{b}T_c$ obtained from the calibration, we measured the calibration errors between the estimated and actual robot positions using 80 random poses. The points for each pose were seleted from a hemisphere centered within the range where the points could serve as the center of a virtual sphere used for path planning. The orientations corresponding to each point were set such that the coil faced the center of the hemisphere. The maximum, minimum, average, and standard deviation of the errors are presented at Table 1.

The calibration error may cause the navigation-estimated target point above or inside the head. This results in a problem in that the robot cannot compensate for the error along the coil’s orientation. This is discussed in detail in Section 7.3.

7.2. Error Compensation

While experimenting with the proposed hybrid position/force controller and one without the position controller component, the data for e, $e_n$, and $e_p$ were collected during the force control after path planning, as shown in Figure 13. Torque control was not applied, and the dummy was fixed during the experiment. As shown in Figure 13a,b, e is smaller with the hybrid position/force controller than without the position controller because without the position controller, the force controller only deals with the interaction between the robot and the environment. Particularly from Figure 13c–f, the hybrid position/force controller compensated $e_n$ more than $e_p$ because of the position controller component. The force control component was considered to be an adhesion between the TMS coil and the head instead of compensating for errors parallel to $u_1$.

7.3. 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, $e_{\mathrm{n}}$ is much smaller than $e_{\mathrm{p}}$. There are two reasons for this: the value of $e_{\mathrm{n}}$ is theoretically zero when the force control is initialized because when the trajectory planning ends, the coil is on the segment $\overline{x_{sf}{x}_f}$, and its orientation is the same as $x_{sf}-x_f$, from Figure 4c. And, $e_{\mathrm{n}}$ 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. Each case is illustrated in Figure 15. If $e_{\mathrm{n}}$ is so small with these cases, $\theta$ can be near $180°$. This can be verified by Figure 14c,d, where if $e_{\mathrm{p}}$ was increased during the experiment, then $\theta$ is near $180°$.

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:

(14)$F=F\left(e\right)$

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 $e, e_{\mathrm{n}}, e_{\mathrm{p}}$, 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.

As shown in Figure 16, the error was reduced rapidly, as intended, and the converged e value was less than 3 mm. The large force over 20 N was applied to the head briefly—less than 10 s. From these results, (14) was used for the desired force of the force controller.

7.4. Setting $k_p$ for the Proportional Torque Control

After determining F, the same experiments were performed with the force and torque control and the e and ${\tau }_c/F_c$ values were obtained, as shown in Figure 18 for each $k_p$ during the force control. The values are described in Figure 18. As the applied torque on the object was proportional to the applied force on it, we can reject the effect of the control force from ${\tau }_c$ by dividing it by $F_c$. Thus, the value ${\tau }_c/F_c$ can be compared by the cases that applied different forces to check how close the contact area between the dummy and the coil was to the center of the coil.

From Figure 18a, it can be seen that a larger $k_p$ reduces e more slowly during the initial period of the experiment and increases the steady-state e value. However, this is not very meaningful: the difference between the maximum and minimum times to the first time the value of e decreases below 5 mm is 1.244 s. The maximum error difference between each $k_p$ after 10 s from the start of the experiment was less than 1.1 mm. Thus, $k_p$ was decided by the ${\tau }_c/F_c$ values, as shown in Figure 18b. It is shown that the larger $k_p$, the lower ${\tau }_c/F_c$ value after around 20 s from the start of the experiment. Meanwhile, using a $k_p$ larger than 4.0 made a large oscillation to that value, so it makes the coil pose unstable. It is natural that if the proportional control gain of the PID control system is very large, it makes the system unstable because of the large overshoot. From these results, $k_p=4.0$ is chosen.

7.5. Tracking the Moved Target Point

In this experimental section, the head was moved slowly while the coil was adhered to the head. During the experiment, the measured reaction force $F_c$ was collected between the TMS coil and head and the desired force F; those data are shown in Figure 19a. While moving, $F_c$ did not decrease abruptly. This implied that the coil continuously interacted with the head. From approximately 12 s shown in the graph, shown in Figure 19a, F increased suddenly because of the variance of e. Subsequently, $F_c$ did not follow F well because much of the applied force was used to compensate for the error. However, this does not cause problems because the error information before the head position was stabilized is meaningless after moving the head and interacting with the coil. After the head movement, the coil tracked the target point using force control. As the adhesion between the coil and the head was maintained during movement, tracking the target point with the suggested force/torque controller was possible. The error information is shown in Figure 19b.

This experiment can be generalized to the real head of the subject, which may move during the TMS session. In this experiment, the target points of the coil were manually tracked after the dummy moved. This approach can be extended to real-time tracking by periodically applying the tracked target point from the neuronavigation system to $x_f$ in Figure 7.

8. Discussion

The adhesion between the subject’s head and TMS coil was improved by using the characteristics of the coil’s floor, which can help reduce the effect of small movements of the subject’s head on the robotized TMS task. Calibration and trajectory planning were performed to achieve this goal. The transformation matrix from the camera coordinate system to the robot manipulator’s base coordinate system was calculated from the calibration work. Using this matrix, the robot manipulator’s base coordinate system can express the coordinate data from the navigation system. While generating the trajectory, a sphere, including the subject’s head, is generated to prevent collision between the head and coil by preventing the coil from invading the sphere. After following the planned trajectory, several force/torque control strategies are applied for better adherence: compliance control for the interaction between the coil and head and proportional torque control to make the center of the contact area between the head and coil near the center of the coil’s floor. When using these strategies, the magnitude of the force used is scheduled by e to reduce e faster. Through this scheduling, a low e can be maintained with a lower force after applying a large force for a short time.

In the system used in our experiment, a force/torque sensor embedded inside the UR5e manipulator was used without additional external sensors. Therefore, this study can be utilized without external force/torque sensors. However, more precise sensors would be helpful for a low-desired force control of their robot manipulator because of their better precision.

The interaction force between the coil and head during the force/torque control in this study was larger than that in existing studies [21,22,26,27]. However, because the floor of the coil case used in previous studies was flat, the contact surface area during the interaction between the coil and the head was larger in this study than in previous studies. Therefore, the pressure transferred to the head during the force control was not significantly higher than in previous studies.

In future work, real-time tracking will be applied to the present system using the control strategy developed in this study. The real-time system will be realized by sending the head and target point of the coil periodically from neuronavigation system to the robot controller. By embodying this function, subjects of our system may feel less pressured to fix their head. Additionally, to mimic the moving of the human head, the moving dummy stage will be developed using servomotors and the additional structures. The clinical experiments with real patients will also be conducted.

9. Conclusions

This paper presented the force/torque controller for a robotized TMS system using an incurved coil. A hybrid position–force control was used to improve the adherence between the TMS coil and the subject’s head and to improve the accuracy between the current and the desired point of the TMS coil. The proportional torque control is used to reduce the torque applied on the center of the coil, which makes the contact area between the coil wider. This increases the stability of the contact.

From the experimental results, a larger force applied to the force control of the robot manipulator reduced e more rapidly. However, if a large force is applied to the subject, it can cause discomfort. Nevertheless, applying a low force to the force control of the robot manipulator causes the slow compensation of the error between the coil’s current and the target positions because of the large contact area between the coil and the head. Therefore, the applied force was scheduled using e. If e is reduced, the applied force is reduced until 5 N. The minimum force is set to maintain the adhesion between the coil and the head. The z-axis component of the error might have been larger during the experiment if the target point estimated by the navigator was above the head. However, this can be sacrificed to ensure firm adhesion between the coil and the head.

Meanwhile, when the head moved during the TMS session, the coil continuously adhered to the head if it moved slowly. Therefore, the coil can track the moved target point using our force control strategy.

Author Contributions

Conceptualization, J.K. and G.-H.Y.; methodology, J.K.; software, J.K.; validation, J.K.; formal analysis, J.K.; investigation, J.K.; resources, G.-H.Y.; data curation, J.K.; writing—original draft preparation, J.K.; writing—review and editing, G.-H.Y.; visualization, J.K.; supervision, G.-H.Y.; project administration, G.-H.Y.; funding acquisition, G.-H.Y. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The experimental data are available by request to authors.

Conflicts of Interest

The authors declare no conflicts of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Table

Figure 1. The shape of the coil.

Figure 2. Transformation matrices of the coordinate. Each matrix describes the coordinates of the arrival point of the arrow using the coordinate system of the starting point of the arrow.

Figure 3. Velocity–time graph for the trapezoidal velocity profile method.

Figure 4. Trajectories for three phases: (a) First phase; (b) Second phase; (c) Third phase.

Figure 5. The last coil pose of the third phase of the sphere’s center [Forumla omitted. See PDF.] on and above the head’s center.

Figure 6. Vectors [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.], which affect the orientation of the force control.

Figure 7. Block diagram describing the force control.

Figure 8. Schedule of the force gain [Forumla omitted. See PDF.].

Figure 9. Block diagram of the proportional torque control.

Figure 10. (a) Stylus and the marker-attached headband; (b) Dummy stage.

Figure 11. Experimental environment.

Figure 12. The coil adhered to the left dorsolateral prefrontal cortex of the dummy.

Figure 13. Comparison of e, [Forumla omitted. See PDF.], and [Forumla omitted. See PDF.] when the hybrid/position force is used or not: (a) [Forumla omitted. See PDF.] when the input force is 5 N; (b) [Forumla omitted. See PDF.] when the input force is 10 N; (c) [Forumla omitted. See PDF.] when the input force is 5 N; (d) [Forumla omitted. See PDF.] when the input force is 10 N; (e) [Forumla omitted. See PDF.] when the input force is 5 N; and (f) [Forumla omitted. See PDF.] when the input force is 10 N.

Figure 14. Comparison of e, [Forumla omitted. See PDF.], [Forumla omitted. See PDF.], and [Forumla omitted. See PDF.] for each desired force: (a) [Forumla omitted. See PDF.]; (b) [Forumla omitted. See PDF.]; (c) [Forumla omitted. See PDF.]; and (d) [Forumla omitted. See PDF.].

Figure 15. Two cases where [Forumla omitted. See PDF.] is opposite to [Forumla omitted. See PDF.]: (a) When the navigation-estimated target point is above the head; (b) When the calculated TMS coil position is inside the head.

Figure 16. Values during the experiment when the variable desired force is applied: (a) [Forumla omitted. See PDF.], [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.]; (b) F and [Forumla omitted. See PDF.].

Figure 17. TMS coil and the head: (a) When the force control is started; (b) When the coil adhered to the head.

Figure 18. The experimental result for each [Forumla omitted. See PDF.]: (a) e; (b) [Forumla omitted. See PDF.].

Figure 19. Results of the tracking experiment: (a) Measured reaction force between the TMS coil and the dummy head while the head is moving; (b) The error between the current coil position and new target position after moving the dummy.

Table 1

Calibration Error Data.

	Error (mm)
Maximum	12.1604
Minimum	3.1598
Average	6.3677
Standard deviation	2.7342

Table 2

Ratio of Error Over Time Relative to the Initial Error for Each Desired Force.

Time Elapsed (s)	5 N	10 N	20 N	30 N	40 N
1	95.63%	91.03%	81.72%	72.44%	64.30%
3	89.14%	78.90%	59.25%	41.38%	26.90%
5	83.81%	68.30%	42.50%	19.85%	3.15%

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