DentiBot: System Design and 6-DoF Hybrid Position/Force Control for Robot-Assisted Endodontic Treatment

RCT or endodontic treatment is a dental procedure performed to save a severely damaged or infected tooth [34]. Teeth consist of layers including enamel, dentin, and a pulp chamber containing dental pulp with blood vessels, nerves, and connective tissues. When the dental pulp becomes infected or damaged due to factors like decay, cracks, or trauma, it can lead to pain and abscess formation. RCT aims to remove the infected pulp, clean and shape the root canals, and fill them to prevent reinfection. The expected outcome is pain relief, elimination of infection, and preservation of the natural tooth, allowing it to function normally with proper care and potentially lasting a lifetime.

The procedure of endodontic treatment involves several key surgical steps, as illustrated in Fig. 3. Following the diagnosis of the need for RCT, local anesthesia is administered to provide numbing of the tooth and its surrounding area, ensuring a painless experience for the patient. Subsequently, a dental bur is employed to create an access opening in the tooth, typically through the crown [35]. This access point allows the dentist to reach the pulp chamber and root canals for further treatment.

After completing the initial steps, the cleaning and shaping stage, crucial in endodontic treatment, ensues to remove the damaged pulp tissue and infected dentin from the root canal systems. Dentists employ specialized endodontic files during this step, often utilizing rotary files with sharp cutting edges. By employing a careful in-and-out motion, the endodontic files are manipulated until they reach the apex of the root canal. To determine the precise location of the apex, an apex locator or radiographs may be used. Given the limited visibility, dentists heavily rely on their tactile sense to ensure accurate adjustment of the file insertion path. Gradually, the root canal is enlarged by using files of increasing sizes, which are determined by the dentist’s judgment based on the unique characteristics of each root canal.

Instrument fracture is a significant concern during endodontic treatment [26]. Removing a separate file in the root canal is technically sensitive. There is another failure circumstance called ledge, which means that the file drills in the wrong path and turn out to be stuck in the root canal. File misalignment and ledging can result in undesired remains of infected tissue in the root canal system and overpreparation of the root dentin. Hence, it is essential to align the file with the root canal and reduce the probability of instrument fracture and ledging.

After cleaning and shaping step, the dentist fills the root canals with biocompatible materials, usually gutta-percha and sealers, to ensure complete sealing of the root canals. Adequate coronal restoration is also necessary after obturation. Obturation and coronal restoration prevent microorganisms’ recontamination and byproducts from entering the root canal system. Finally, the dentist will fabricate a crown or other restoration to protect the tooth.

Overall, endodontic treatment, particularly the cleaning and shaping step, is a complex and delicate procedure that requires precise execution to ensure optimal outcomes for patients. However, the patients would be only locally anesthetized around their oral tissues. They still can arbitrarily move their head and jaw. Conventionally, the dentist would hold the patient’s jaw and compensate for any movement through their well-trained experience.

In this study, we focus on the automation of the cleaning and shaping step in RCT, helping the dentist with real-time force sensing and delicate movements by the robot. The following clinical requirements have been identified:

1. (R1)

   Achieving six degree-of-freedom (DoF) motions of the flexible endodontic file within the pulp chamber and root canals.

2. (R2)

   Implementing force guidance to address the absence of visual feedback and the risk of file fracture inside the curved root canal.

3. (R3)

   Tracking patient’s jaw motion during the procedure to maintain a fixed relative pose between the robot and the patient.

In order to perform robot-assisted endodontic treatment, several system specifications must be determined. Firstly, the required workspace is defined as a cylinder (see Fig. 3) to accommodate the size and shape of teeth. Considering the average diameter of molar teeth, approximately $10.7$ mm [36], a cylinder diameter of $13$ mm is chosen to account for outlier cases. The cylinder height is determined based on the length of root canals. Canine teeth have the longest root canal, approximately $24$ mm [37]. Therefore, the cylinder height is set to $28$ mm to provide sufficient surgical depth.

Considering the range of attitude in which the robot operates is also crucial. Before inserting the endodontic file, the robotic manipulator can be manually aligned with the target root canal. Therefore, the critical range of angles pertains to the movement of the file within a single root canal. Research indicates that there is a deviation of $3.8-5.0$ degrees in the attitude of the endodontic file during RCT [38]. This deviation is primarily caused by the conical shape of root canals, which have an entrance width of up to $1.5$ mm and gradually narrow down to $0.2-0.6$ mm at the apex [39]. To maintain precision and minimize the risk of file fracture, it is advisable to allow for the manipulation angle at least $10$ degrees for the roll and pitch axis. On the other hand, the yaw axis, or file rotation, shall have a full range of $360$ degrees.

In this study, NiTi (Nickel-Titanium) rotary files (ProTaper Universal SX–F3, Dentsply Sirona, Ballaigues, Switzerland), with the length of $21$ mm and the apical diameter of $0.18-0.30$ mm, are used in autonomous cleaning and shaping step. NiTi files are valued for their flexibility [40]. In general, NiTi files have a maximum bending angle more than $45$ degrees [41]. They can navigate curved root canal pathways with ease, reducing the risk of ledging [42]. ProTaper files are compatible with motorized handpieces or automated systems. The rotation speed is recommended at $150-350$ rpm. In addition, the mean apical force and axial torque need to be limited to $3.9$ N and $12$ mN$\cdot$m, respectively, to ensure the safety of the treatment [43].

Patient movement is a critical consideration in robot-assisted RCT. On average, the patient’s linear displacement speed during the procedure is approximately $1$ mm/s, with a maximum speed of $2.5$ mm/s [14]. The angular displacement speed is around $0.5$ deg/s, with a maximum speed of $1$ deg/s. Precise tracking of the patient’s motion within these velocity ranges is crucial to ensure safety. In this study, we aim to reduce the tracking error of the DentiBot to $2$ mm, a significant improvement from its previous level of up to $5$ mm [24]. In the event that the patient exceeds the tolerable movement speed, the robot shall be programmed to immediately halt the surgical procedure, allowing the dentist to regain full control and ensure the patient’s safety.

In summary, the following specifications have been defined for the DentiBot:

1. (S1)

   Workspace: a cylinder working volume with the height of $28$ mm and diameter of $13$ mm. The minimum allowable roll and pitch angles are $10$ degrees.

2. (S2)

   Maximum apical force and axial torque exerted to the endodontic file: $3.9$ N and $12$ mN$\cdot$m.

3. (S3)

   Error tolerance for patient motion tracking: $2$ mm with the maximum velocity of $2.5$ mm/s.

The components of the DentiBot include a 6-DoF robotic manipulator, an endodontic handpiece, a 6-axis force/torque sensor, and a string-based PTM (as shown in Fig. 1). Each component has its own coordinate system, as defined in Fig. 4. The robotic manipulator performs the surgical operations like a dentist. The ground link of the manipulator, represented as $\{G\}$, serves as a fixed frame for the entire kinematic chain. The sixth link of the manipulator is denoted as $\{R\}$ and is fixedly connected to the base frame of the PTM, $\{B\}$. Below the base of the PTM, the force/torque sensor is attached to measure the contact between the endodontic file (denoted as $\{F\}$, installed on the dental handpiece) and the root canal (denoted as $\{P\}$). The force and torque measurements are output in the sensor’s frame $\{S\}$. While the dental handpiece drives the rotational motion of the endodontic file, the robotic manipulator maneuvers the file orientation and actuates the in-and-out motion to perform the cleaning and shaping of root canal systems.

On the patient side of the PTM, a teeth brace is mounted on the patient’s jaw and represented as $\{A\}$. The PTM estimates the coordinate transformation between $\{A\}$ and $\{B\}$ in real time. In practice, the transformation between $\{R\}$ and $\{B\}$ is already known in the mechanical design. The transformation between $\{A\}$ and $\{P\}$ can be acquired through 3D medical imaging. As a result, the relative pose between the robot and the target tooth of the patient, i.e., the coordinate transformation between $\{R\}$ and $\{P\}$, is intraoperatively obtained for patient tracking. In the following subsections, we will provide an explanation of the kinematics and calibration procedures for both the robotic manipulator and the force/torque sensor. The detailed design and analysis of the string-based PTM are given in Section IV.

A 6-DoF robotic manipulator (Meca500, MecaDemic, Montreal, Canada) carrying a motorized dental handpiece is employed to fulfill both the clinical requirement (R1) and the workspace specification (S1). Equipped with a motorized dental handpiece, the robotic manipulator positions the endodontic file within the designated cylindrical workspace with an accuracy of $0.1$ mm. The endodontic file can be easily replaced with various lengths and diameters to accommodate different treatment demands.

To establish the coordinate transformation between the robot frame $\{R\}$ and endodontic file frame $\{F\}$, the tool center point (TCP) calibration [44, 45] is applied. Note that $\{F\}$ is placed at the top of the screw-shape working part of the file, rather than the file tip. Thus, two endodontic files with different lengths are required in this calibration process.

The longer endodontic file is first installed on the dental handpiece. By manually commanding the robotic manipulator, the file tip is positioned to a fixed point with four different orientations. The relative poses between $\{G\}$ and $\{R\}$ at different orientations are obtained using the Denavit-Hartenberg (D-H) Method [46] and recorded. Next, the translational vector between $\{R\}$ and the file tip, denoted as ${}^{\mathrm{R}}\mathbf{t}_{\mathrm{F1}}\in\mathbb{R}^{3}$, is solved in the following least squares problem:

where ${}_{\mathrm{R}}^{\mathrm{G}}\mathbf{R}(j)$ and ${}^{\mathrm{G}}\mathbf{t}_{\mathrm{R}}(j)$, $j\in\{1,2,3,4\}$, represent the rotational matrix and translational vector, respectively, obtained from four different orientations in the TCP calibration process.

By repeating the same TCP calibration procedure on another shorter endodontic file, ${}^{\mathrm{R}}\mathbf{t}_{\mathrm{F2}}$ is obtained. Once completed, the rotational matrix ${}^{\mathrm{R}}_{\mathrm{F}}\mathbf{R}$ is calculated based on the directional vector ${{}^{\mathrm{R}}\mathbf{t}_{\mathrm{F1}}}-{{}^{\mathrm{R}}\mathbf{t}_{\mathrm{F2}}}$. The translational vector ${}^{\mathrm{R}}\mathbf{t}_{\mathrm{F}}$ is then derived by

where $\left\|{}^{\mathrm{F}}\mathbf{t}_{\mathrm{F1}}\right\|_{2}$ denotes the length of the first file’s working part.

The robotic manipulator accepts joint velocity commands $\dot{\mathbf{q}}_{\mathrm{cmd}}\in\mathbb{R}^{6}$. The geometry Jacobian matrix $\mathbf{J}$ is applied to convert ${}^{\mathrm{F}}\mathbf{\dot{p}}_{\mathrm{cmd}}\in\mathbb{R}^{6}$, the velocity command along the linear ($x$,$y$,$z$) and rotary ($\phi$,$\psi$,$\theta$) axes with respect to the endodontic file frame $\{F\}$, to $\dot{\mathbf{q}}_{\mathrm{cmd}}$, the joint velocity command [47]:

Note that $q_{i}\in\mathbb{R}$, $i=\{1,2,...,6\}$, represents the angle of the $i$-th joint. The pose vector from the ground link to the six link, ${}^{\mathrm{G}}\mathbf{p}_{\mathrm{R}}\in\mathbb{R}^{6}$, is obtained from the forward kinematics of the 6-DoF robotic manipulator.

In order to facilitate intraoperative force guidance and fulfill the clinical requirement (R2) and force/torque limit specification (S2), we employ a 6-axis force/torque sensor (Mini40, ATI Industrial Automation, Apex, NC) that connects the sixth link of the robotic manipulator and the motorized dental handpiece to measure the force and torque applied to the endodontic file. Both force and torque measurements are originally obtained in the sensor’s frame $\{S\}$ with the resolution of $0.01$ N and $0.25$ mN$\cdot$m, respectively.

Before applying the force/torque measurements for force guidance, it is essential to conduct gravity compensation. As the orientation of the force/torque sensor and dental handpiece relative to the ground undergoes changes during the operation, compensation for the weight of the dental handpiece becomes necessary. We employ the compensation method introduced by Vougioukas et al. [48]:

where ${}^{\mathrm{S}}\boldsymbol{f}_{s}$ and ${}^{\mathrm{S}}\boldsymbol{f}_{0}$ denote the compensated and raw force measurements, respectively, in the force sensor’s coordinate frame $\{S\}$. ${}^{\mathrm{S}}\boldsymbol{\tau}_{s}$ and ${}^{\mathrm{S}}\boldsymbol{\tau}_{0}$ represent the compensated and raw torque measurements, respectively. $\boldsymbol{w}_{h}$ and $\boldsymbol{r}_{h}$ are the unknown weight and centroid position of the dental handpiece, respectively. ${}^{\mathrm{R}}_{G}\mathbf{R}$ can be obtained from forward kinematics of the robotic manipulator. ${}^{\mathrm{S}}_{R}\mathbf{R}$, however, includes an unknown mounting error along $\theta$-axis of the sensor.

To determine the unknown parameters like $\boldsymbol{w}_{h}$ and ${}^{\mathrm{S}}_{R}\mathbf{R}$, a least squares method is applied. First, we assign certain poses to the robotic manipulator and record both ${}^{\mathrm{R}}_{G}\mathbf{R}$ and ${}^{\mathrm{S}}\boldsymbol{f}_{0}$ at each pose. By assuming there is no applied force, i.e., ${}^{\mathrm{S}}\boldsymbol{f}_{s}$ equals zero, and reorganizing the right hand side of Eq. 5, the unknown variables are obtained. Similarly, $\boldsymbol{r}_{h}$ is identified from another least squares problem using Eq. 6.

Furthermore, as the sensor frame $\{S\}$ does not align with the endodontic file frame $\{F\}$, another coordinate transformation from $\{S\}$ to $\{F\}$ is required:

where ${}^{\mathrm{F}}_{\mathrm{R}}\mathbf{R}$ and ${}^{\mathrm{R}}_{\mathrm{S}}\mathbf{R}$ are obtained from the TCP calibration and gravity compensation processes, respectively. Note that the torque sensed in the file frame $\{F\}$ comprises both the torque sensed in frame $\{S\}$ and an extra torque resulting from the force ${{}^{\mathrm{F}}\boldsymbol{f}s}$ applied to the lever ${{}^{\mathrm{F}}\!\mathbf{t}_{\mathrm{S}}}$, which is determined from the CAD model.

To minimize the size of the passive pose measuring device required, we employ a string-based parallel kinematic mechanism. In essence, this mechanism calculates the relative pose between two rigid bodies by measuring the lengths of six strings connecting them. Previous research has introduced different configurations, specifically the 3-2-1 type and the 2-2-2 type. In the 3-2-1 type, three ball joints are affixed to the mobile object, with three, two, and one string(s) connected to each of these three joints, respectively [49, 50]. The relative pose can be determined analytically using trilateration, which has been applied in studying human walking patterns [51].

To address the issue of multiple solutions encountered in the 3-2-1 type measuring device, an alternative 2-2-2 type has been suggested [52]. In this configuration, the relative pose is determined through numerical methods like the Newton-Raphson method. However, when the device’s footprint is minimized, it also reduces the variation of the six string lengths with regard to the change in relative pose. Consequently, the estimation accuracy is sensitive to measurement noises.

In this study, we employ the “1-1-1-1-1-1” type, where each string connects to a separate ball joint, to develop a passive pose measuring device aimed at enhancing estimation accuracy. The relative pose is still determined using the Newton-Raphson method, which theoretically offers up to 40 potential solutions [53]. However, it is widely recognized that the majority of these solutions are complex numbers and can be eliminated by utilizing an initial guess in proximity to the true solution [54, 55].

The design of the proposed string-based PTM is illustrated in Fig. 5. This module incorporates six string potentiometers (M150, TE Connectivity, Schaffhausen, Switzerland), each with a high resolution of $0.2$ mm. These potentiometers are designed to handle a maximum stroke of $38$ mm and weigh $14$ g. To maintain appropriate tension, each potentiometer employs a torque spring, exerting a force of approximately $1.1$ N. These potentiometers are mounted on the base adapter, which is affixed to the end effector of the 6-DoF robotic manipulator, as shown in Fig. 5a. The ends of the strings are securely anchored at six distinct positions on a custom-made teeth brace, as drawn in Fig. 5b. The coordinate transformation between the base and the anchor, denoted as ${}_{A}^{B}\textbf{T}$, is determined during the surgical procedure by measuring the lengths of the six strings ($l_{i},i\in\{1,2,\cdots,6\}$) and solving for the forward kinematics in real time.

To accommodate the restricted maximum stroke of the string potentiometers, which is only 38 mm, an extension string has been integrated. This extension guarantees that when the endodontic file, held by the dental handpiece, is inserted into the target root canal, the string potentiometer can operate throughout its entire stroke range. It’s worth noting that before employing the string potentiometers, a calibration procedure is conducted to establish the conversion between the output voltage and the actual string length.

To enhance string length sensitivity and prevent any collisions between the strings and the dental handpiece, a deliberate arrangement of string potentiometers and anchor positions has been implemented. The goal is to maximize the distance between each pair of base or anchor joints while minimizing the overall footprint of the module. For the base adapter, three string potentiometers are positioned on each side of the dental handpiece to evenly distribute the tension forces generated by the potentiometers, as illustrated in Fig. 5a. On each side, the sensors are situated to maximize the area spanned by the base positions of the strings ($\mathbf{b}_{i},i\in\{1,2,\cdots,6\}$), but meanwhile maintain a safe distance between the strings and the dental handpiece. A similar rationale is followed in determining the anchor positions ($\mathbf{a}_{i},i\in\{1,2,\cdots,6\}$). An example of the teeth brace can be seen in Fig. 5b. Further details regarding the joint positions are provided in Tab. I. It is important to note that the order of anchor positions along the $y$-axis closely matches the order of the string bases. This arrangement helps minimize the risk of strings crossing over each other.

The PTM estimates the relative pose between the robot and the patient, i.e., the pose vector $\mathbf{p_{s}}\in\mathbb{R}^{6}$ that represents the linear and angular displacements of the homogeneous transformation matrix between the anchor frame $\{A\}$ and the base frame $\{B\}$. This estimation is achieved by solving the forward kinematics problem of the proposed string-based parallel kinematic mechanism. In this work, the Jacobian-based Newton-Raphson method [54] is applied.

To start with, the method solving for the inverse kinematics problem of the PTM is introduced. For any relative pose, the corresponding string length $l_{i}$ can be obtained by

where $\mathbf{a}_{i}$ and $\mathbf{b}_{i}$ denote the anchor and base joint position, respectively. The calculation of inverse kinematics at the relative pose $\mathbf{p_{s}}$ is denoted as $InvKine$($\mathbf{p_{s}}$), which returns the six string lengths, denoted as $\mathbf{l}_{\mathbf{p}}\in\mathbb{R}^{6}$, $\mathbf{l}_{\mathbf{p}}=\begin{bmatrix}l_{1}&l_{2}&\cdots&l_{6}\end{bmatrix}^{T}$.

Given the measured string lengths $\mathbf{l}\in\mathbb{R}^{6}$, the forward kinematics is solved iteratively. In each step, the estimated string lengths $\mathbf{l}_{\mathbf{p}}$ are calculated from the inverse kinematics at the current estimated pose. Then, the string length error $\Delta\mathbf{l}$ between the measurement $\mathbf{l}$ and the estimate $\mathbf{l}_{\mathbf{p}}$ are computed and used to update the estimated pose. The iterative solver continues until $||\Delta\mathbf{l}||_{2}$ is less than the predefined error tolerance, $tol$, which is set as $1\times 10^{-10}$ mm in this work. This pose estimation algorithm is summarized as Algorithm 1. Note that $\mathbf{p}_{s0}$ denotes the initial guess, which is set as the relative pose estimated at the previous sampling time. $J(\mathbf{p}_{s})$ returns the Jacobian matrix, which characterizes how the relative pose changes at $\mathbf{p}_{s}$ in response to variations in string lengths. This matrix is carried out using a numerical sensitivity analysis.

Last but not least, the estimated pose vector $\mathbf{p}_{s}$ in the PTM base frame $\{B\}$ is transformed to the file frame $\{F\}$ for patient tracking control, which will be elaborated in Section V.

The kinematic performance of the proposed string-based PTM is analyzed in this subsection. First, the error sensitivity of PTM estimation using the proposed configuration is analyzed and compared with that of the other two configurations. Second, we examine the workspace of the proposed PTM. Last, we investigate the presence of any singularities, mechanical interference, or string-crossing issues during the movement of the endodontic file within the designated cylinder workspace. All the simulations are done with the kinematic model of the DentiBot, including the PTM joint positions listed in Tab. I.

A Monte Carlo simulation is conducted to assess the error sensitivity of pose estimation when employing different string-based parallel kinematic mechanisms. We compared three distinct types: the 3-2-1 type, the 2-2-2 type, and the proposed configuration. In each case, the maximum position estimation error is found across $N$=$10,000$ simulations, considering randomly perturbed string length measurements. Note that the actual pose was set at the origin, with an initial guess positioned 20 mm away from it. The disturbance $\mathbf{\epsilon_{l}}\in\mathbb{R}^{6}$ is generated as a uniformly distributed random number with the maximum ranging from $0$ to $0.5$ mm. The simulation results, as shown in Fig. 6, indicate that the proposed method has less estimation error than the other two types. In practice, the maximum measuring error of string lengths is around $0.2$ mm, which results in a maximum estimation error of $1.84$ mm and mean error of $0.57$ mm. This result satisfies the requirement for accurate patient tracking (S3).

The analysis of PTM workspace aims to ascertain its suitability for facilitating unrestricted maneuvering of the endodontic file within the prescribed cylindrical workspace. Given that the 6-DoF robotic manipulator’s workspace inherently exceeds the necessary cylindrical workspace, constraints are imposed on the PTM workspace based on the $38$ mm stroke limitation of the string potentiometers. It is assumed that the PTM is initialized with the endodontic file’s tip positioned at the central point of the cylindrical workspace. In this specified pose, each string length can be extended or shortened by a maximum of $19$ mm. The workspace analysis is conducted within a cubic volume with a $40$ mm width surrounding the cylindrical workspace, employing a $0.5$ mm resolution. Dexterity, defined as the ratio of valid orientations within a $\pm 5$-degree range along the roll and pitch angles, is calculated for each test point within the cube. Test points are deemed part of the PTM workspace only if they satisfy two conditions: (1) the stroke of the string potentiometers remains within the defined constraint and (2) full dexterity can be achieved at that specific point. The outcome of this workspace analysis is illustrated in Fig. 7a, where the green and yellow volumes represent the required cylindrical workspace and the PTM workspace, respectively. Importantly, the required cylindrical workspace is entirely encompassed within the region where the PTM achieves full dexterity without exceeding the maximum stroke of the string potentiometers.

Subsequently, the assessment of singularities or multiple solutions within the required cylindrical workspace for PTM estimation is conducted. In this analysis, the tip of the endodontic file undergoes random movements within the cylindrical workspace over a duration of $5000$ seconds, sampled at intervals of $0.01$ seconds. The maximum velocity of the DentiBot is constrained to $2.5$ mm/s, reflecting the maximum speed of patient movement. The test trajectory is depicted in Fig. 7a. Initially, both the starting point and the initial estimate $p_{s0}$ are positioned at the center of the cylindrical workspace. Then, the estimated pose from the preceding timestamp serves as the initial estimate for the subsequent timestamp. Consequently, no instances of singularity or multiple solutions are observed throughout the analysis. The maximum position and orientation error along the test trajectory are $1.77\times 10^{-12}$ mm and $1.11\times 10^{-12}$ degrees, respectively.

Last, an analysis is conducted to confirm there is no string-crossing and mechanical interference when applying the PTM along with the DentiBot. When analyzing the PTM estimation error along the test trajectory in the previous assessment, the distances between the six strings ($\mathfrak{s}_{i},i\in\{1,2,\cdots,6\}$) and the six side edges of the hexagonal prism enclosing the dental handpiece ($\mathfrak{h}_{i},i\in\{1,2,\cdots,6\}$) are also computed. Note that due to the symmetric design of the PTM, $\mathfrak{h}_{1}$, $\mathfrak{h}_{2}$, and $\mathfrak{h}_{3}$ are closer to $\mathfrak{s}_{1}$, $\mathfrak{s}_{2}$, and $\mathfrak{s}_{3}$, while $\mathfrak{h}_{4}$, $\mathfrak{h}_{5}$, and $\mathfrak{h}_{6}$ are closer to $\mathfrak{s}_{4}$, $\mathfrak{s}_{5}$, and $\mathfrak{s}_{6}$. Out of a potential $66$ interference pairs, we assess $24$, as some pairs, like $\mathfrak{s}_{1}$-$\mathfrak{h}_{4}$, are unlikely to experience interference. The results, depicted in Fig. 7b, demonstrate that the distances between strings consistently exceed $5$ mm. The narrowest separation between strings and the dental handpiece is $0.4$ mm, observed in the case of pair $\mathfrak{s}_{4}$-$\mathfrak{h}_{5}$ during an extreme roll angle of the handpiece. This analysis confirms the absence of string-crossing and mechanical interference when employing the PTM.

The distinctive feature of endodontic files lies in their flexibility, setting robot-assisted RCT apart from automated dental implant surgery. This flexibility, however, introduces an additional displacement when external forces are applied to the file, necessitating careful compensation. In this study, we compensate for the displacement resulting from file bending by implementing a virtual spring force proportional to the estimated amount of deflection. The method for estimating file deflection is introduced in this subsection.

The endodontic file is considered as a beam fixed at one end on the dental handpiece and subjected to external forces from multiple directions on the other end. Since the deflection along the axial axis is negligible, only the resultant radial force $f$ is considered, as illustrated in Fig. 8. Note that $f$ represents the summation of external forces and surface pressure inside the root canal as a single resultant force. The absolute value of $f$ is identical to the sensed force (by abuse of notation, denoted as $f_{s}$ here) to yield the net force equals zero. As a result, the effective leverage length $l_{a}$ can be derived as

where $\tau_{s}$ denotes the sensed torque.

Next, the displacement of the file tip caused by the resultant external force $f$, denoted as the maximum deflection $\delta$, can be obtained from the beam deflection formula [56]:

Note that $E$ represents Young’s modulus, which is around $80$ Gpa for NiTi files. $I$ denotes the cross-sectional area moment of inertia of endodontic files. Given that the diameter of endodontic files varies from $0.3$ mm at the tip to $1.2$ mm at the base, we consider half of the maximum diameter, i.e., $0.6$ mm, for inertia calculations. This model (Eq. 11) can be applied separately to the x- and y-axes of the endodontic file frame $\{F\}$.

The compensation for the estimated file deflection involves applying a virtual spring force. This virtual spring force will be integrated with the 6-DoF hybrid position/force control, which will be further explained in the subsequent subsection. It is worth noting that the estimated leverage length $l_{a}$ may exceed the length of the file due to measurement noises, particularly when only the file tip is in contact or when the applied external force is relatively small. To tackle this issue, we establish an upper limit for $l_{a}$, which is set equal to $l$, the known length of each endodontic file. Additionally, the file flexibility compensator is activated only when the measured force along the x- or y-axis exceeds a predefined threshold, which we have set at $0.03$ N in our system.

In order to minimize the contact force between the endodontic file and the root canal while simultaneously maintaining the relative pose between the robot and the patient, a hybrid position/force control strategy is proposed for the DentiBot. The position control utilizes the pose estimation $\mathbf{p}_{s}$ from the PTM to track the patient’s motion. The force control applies the force/torque measurement $\boldsymbol{\xi}_{s}\in\mathbb{R}^{6}$,

to adjust the file insertion path and prevent file failure or ledging. By combining these two control strategies, the hybrid position/force controller maintains a delicate balance between tracking accuracy and minimizing contact forces in real-time. This ensures that the DentiBot remains precise and effective throughout the cleaning and shaping process.

The control block diagram for the DentiBot is shown in Fig. 9. The controller comprises of an inner and an outer loop, which work together to track patient movement and adjust file insertion path, respectively. The inner loop, running at the sampling rate of $100$ Hz, implements a closed-loop position control to regulate the relative pose between the robot and the patient, i.e., $\mathbf{p}_{s}$ measured from the PTM, to the initial pose $\mathbf{p}_{d}$. The outer loop, running at the sampling rate of $20$ Hz, generates the force-guided path correction $\mathbf{p}_{\mathrm{adm}}$ based on the force/torque measurement $\boldsymbol{\xi}_{s}$, which then modifies the initial pose $\mathbf{p}_{d}$. This hybrid configuration is also called “implicit force control” in literature [57]. Note that all the signals are represented in the file frame $\{F\}$.

In the inner control loop, the difference between $\mathbf{p}_{d}$ (or $\mathbf{p}_{d}-\mathbf{p}_{adm}$ if $\mathbf{p}_{adm}$ is a non-zero vector) and $\mathbf{p}_{s}$, denoted as $\mathbf{p}_{\mathrm{err}}$, is fed into a proportional-derivative (PD) controller, which generates the velocity command $\mathbf{\dot{p}}_{\mathrm{cmd}}$ to the 6-DoF robotic manipulator:

where

Note that $k$ is the discrete time index. $C_{\mathrm{PD}}(z)$ is the six-dimensional PD position controller with the proportional gain $k_{p}$ and the derivative gain $k_{d}$ for each linear or angular axis. When the patient moves, i.e., $\Delta\mathbf{p}\neq\mathbf{0}$, the PTM estimates the $\Delta\mathbf{p}$ along with the current robot pose $\mathbf{p}$, resulting in none-zero $\mathbf{p}_{\mathrm{err}}$. The use of a PD position controller is advantageous as it provides closed-loop stability, i.e., asymptotically convergence of $\mathbf{p}_{\mathrm{err}}$, and ease of parameter tuning, while also preventing overshoot or oscillations within the narrow space of the root canal.

As for the outer control loop, the measured contact force and torque $\boldsymbol{\xi_{s}}$ are sent to the file flexibility compensator, which estimates the file deflection along the x- and y-axis, denoted as $\delta_{x}$ and $\delta_{y}$, respectively, using Eq. 11. Next, the compensation for file deflection is achieved by introducing virtual spring forces applied to the original force/torque measurements:

where

Note that $k_{f}$ denotes the virtual spring constant, which is a design parameter in the file flexibility compensator. The inclusion of extra forces during file flexibility compensation allows the admittance controller, the next function block, to generate a larger path adjustment, thereby diminishing file deflection.

The DentiBot utilizes an admittance controller to ensure appropriate response to contact force and torque [24]. Admittance control has been widely adopted to improve the safety during human-robot collaboration [58]. Generally, admittance control enables a robot to mimic the behavior of a second-order mass-damper system [59]. In the context of autonomous cleaning and shaping in RCT, the desired force/torque, denoted as $\boldsymbol{\xi}_{\mathrm{d}}$, consists of zero values for all elements except the z-axis and $\theta$-axis (file rotation around the z-axis). The endodontic file’s insertion motion is driven by a positive desired contact force along the z-axis. Conversely, no control is exerted on the $\theta$-axis to prevent the interactive torque generated by file rotation and drilling from affecting the DentiBot’s movement. Apart from these two axes, the contact force/torque should remain at zero.

Applying admittance control, the force-guided path correction $\mathbf{p}_{\mathrm{adm}}$ is calculated by

where

Note that the mass coefficient $m_{a}$, damper coefficient $b_{a}$, and constant gain $k_{a}$ are design parameters in the admittance controller. These parameters can vary for each linear axis ($x$, $y$, $z$) and rotary axis ($\phi$, $\psi$, $\theta$). The sampling period of the outer control loop is denoted as $T$, which is set as $0.05$ s in our system.

The hybrid position/force control strategy is utilized specifically during the cleaning and shaping phase of RCT. Prior to this step or in risky situations, only admittance control is employed, allowing the dentist to manually handle and maneuver the dental handpiece. When operating under pure admittance control, the PD position controller is bypassed, meaning that the admittance controller directly generates the velocity command $\mathbf{\dot{p}}_{\mathrm{cmd}}$ for the 6-DoF robotic manipulator. The transition between each of these automated steps will be further elaborated in the subsequent subsection.

The typical workflow of robot-assisted root canal cleaning and shaping, utilizing the DentiBot system, is outlined as follows. After the dentist creates an access open to the pulp chamber and root canal, the autonomous cleaning and shaping procedure commences. The admittance control is activated, enabling the dentist to manipulate the dental handpiece, attached to the DentiBot, and align the endodontic file with the target root canal. Subsequently, the PTM establishes a connection between the robot and the patient, registering the initial relative pose ${\mathbf{p}}_{d}$. The endodontic file is then automatically inserted into the root canal using the hybrid position/force control. Once inserted, the cleaning and shaping of the root canal is initiated.

In the cleaning and shaping stage, the endodontic file remains engaged until its tip reaches the apex of the root canal, indicating the completion of the instrumentation. To prevent fracturing, the file automatically reverses if the axial torque exceeds the upper limit. Once the file reaches the apex, it is withdrawn from the root canal. At this point, the dentist can choose to either replace the endodontic file, typically with a larger diameter, and repeat the procedure, or decide to conclude the treatment. This workflow is modeled as a finite state machine to facilitate surgical automation. As illustrated in the statechart diagram (Fig. 10), there are totally five states: Idle, Insertion, Cleaning and Shaping, Reverse, and Disengage. The specific actions and transitions associated with each state are described below.

In the cleaning and shaping step of RCT, it is essential for dentists to measure the length of the root canal and monitor the depth of file insertion. Conventional manual surgery lacks intraoperative visual feedback for this process. To overcome this limitation, we propose using preoperative CT scans to create a 3D model of the target root canal. The pathway of the root canal is also detected. Our GUI displays the preoperative model along with real-time visualization of the file being inserted into the root canal. This intraoperative real-time feedback allows dentists to monitor the surgical progress during the robot-assisted procedure.

The process of reconstructing the root canal model and detecting its central path is illustrated in Fig. 11. In this study, we utilize a micro-CT scanner (U-CT, MILabs, Houten, Netherlands) to capture high-resolution CT cross-sectional images of the teeth models at a resolution of 10 $\mu$m. A sample of the root canal model and corresponding CT images is presented in Fig. 11. To segment the root canal, we convert the grayscale CT images into binary format and apply the Canny edge detection algorithm [60] to each cross-sectional image. By stacking the contours segmented from CT cross-sectional images, the root canal 3D model is reconstructed.

After retrieving the root canal 3D model, the central path of the root canal is detected by fitting a sixth-order polynomial function on the x-z plane and y-z plane, respectively:

where a set of $(x_{rc},y_{rc},z_{rc})$ satisfying the above equation denotes the central path. $g_{j}$ and $h_{j}$, $j\in\{1,2,\cdots,7\}$, represent the coefficients of the polynomial with respect to the x-axis and y-axis, respectively. The length of the central path can then be calculated, providing valuable information for dentists in selecting appropriate endodontic files for the cleaning and shaping step. It also allows them to determine if the file tip has reached the apex of the root canal.

The GUI of the DentiBot system is depicted in Fig. 12b. It provides a visualization of the root canal 3D model and the DentiBot, which includes the robotic manipulator and dental handpiece. The virtual manipulator is capable of synchronous 6-DoF motions by receiving 6-axis joint data through the UDP protocol. Preoperative calibration establishes fixed homogeneous transformation matrices, such as ${}_{R}^{B}\textbf{T}$, ${}_{P}^{A}\textbf{T}$, and ${}_{F}^{R}\textbf{T}$. ${}_{A}^{B}\textbf{T}$ is derived from real-time PTM estimation. The relative pose between the endodontic file and the root canal model, represented as ${}_{P}^{F}\textbf{T}$, is thereby calculated and plotted based on

During the autonomous cleaning and shaping process, the GUI provides dentists with visual feedback, displaying the current operation state and real-time file insertion depth. This functionality enables dentists to closely monitor the surgical progress and intervene if necessary. Fig. 12b exemplifies the GUI interface, depicting the insertion of the endodontic file into the preoperatively reconstructed root canal. The GUI dynamically calculates and presents the real-time insertion depth, supplying dentists with crucial information throughout the procedure.

The DentiBot system, including the 6-DoF robotic manipulator and dental handpiece, was controlled by an NI LabVIEW Real-Time Target (Intel Core i7-3770 Processor) via the EtherCAT protocol. The Real-Time Target also received measurements from the force/torque sensor and PTM. On the other hand, the GUI was implemented on another personal computer (Intel Core i7-8700 Processor). It established communication with the Real-Time Target using the UDP protocol.

The experimental setups for different evaluations are summarized in Fig. 13. For the selection of control parameters, our approach involved designing the parameters for the admittance controller, which responds to external contact forces, followed by determining the gain in the file flexibility compensator. Since the endodontic file exhibits symmetry along its x- and y-axes, we utilized a linear platform to generate one-dimensional contact with the endodontic file (Fig. 13a). Initially, the linear platform was aligned with the file, after which it was moved to exert external force on the file. By employing the admittance controller and file flexibility compensator, the endodontic file effectively tracked the motion of the linear platform. The positions of the linear platform and the endodontic file were recorded using a motion capture system (Impulse X2E, PhaseSpace Inc., San Leandro, CA) with a precision of $0.1$ mm [61]. An active marker was affixed to the top of the dental handpiece, while three additional markers were attached to the linear platform. These three measured positions established a coordinate system $\{P\}$ for the linear platform, where the x-axis represents the primary travel axis. To evaluate the tracking performance, we calculated the tracking error by comparing the distance between the linear platform and the dental handpiece. Our objective was to identify a set of control parameters that minimizes the tracking error.

For the evaluation of the PTM, our objective was to verify its accuracy in measuring both translational and rotational motions. In the experiment, the base of the PTM was securely fixed on the table, while the anchor was attached to the end effector of the 6-DoF robotic manipulator (Fig. 13b). The robotic manipulator, with a path accuracy of $0.1$ mm, emulated the patient’s motion and controlled the movement of the PTM’s anchor. With this setup, the kinematic chain can be expressed as:

This equation signifies that the PTM’s measurements, ${}_{A}^{B}\textbf{T}$, can be mapped to the motion of the robot arm, ${}_{R}^{G}\textbf{T}$, by multiplying two precalibrated transformations, ${}_{R}^{A}\textbf{T}$ and ${}_{B}^{G}\textbf{T}$. Therefore, in this evaluation, we compared the measurements of ${}_{R}^{G}\textbf{T}$, one is obtained from the PTM, the other is obtained from the readings of the robot’s joint encoders.

To assess the performance of robot-patient alignment during robot-assisted endodontic treatment, we employed a 6-DoF Stewart platform installed with a root canal model as the patient motion simulator (Fig. 13c). The same setup was also applied to the preclinical evaluation on root canal models. Here, the patient motion simulator was controlled independently by an NI myRIO-1990 controller. To ensure data integrity, there was no communication between the DentiBot and the patient motion simulator. The DentiBot autonomously executed the procedure and achieved real-time alignment under 6-DoF patient motions. The motion of both the DentiBot and the patient motion simulator was captured using the motion capture system. For coordinate system establishment, seven active markers were mounted on the robotic manipulator and the patient motion simulator, with each device having three markers. An additional marker was placed on the base of the endodontic file to obtain more accurate file position data. The position data were then transformed into the patient’s frame $\{P\}$ to calculate the alignment error.

This subsection presents an evaluation of the performance of autonomous cleaning and shaping when using the DentiBot in RCT. The surgical workflow depicted in Fig. 10 was employed. During the pre-clinical evaluation, the patient motion simulator generated 6-DoF motions as the one applied in the engineering evaluation (Fig. 13c). To ensure precise file insertion while maintaining the relative pose between the robot and the patient, the proposed 6-DoF hybrid position/force control method was activated.

The evaluation involved testing two types of root canal models, as depicted in Fig. 18. Model A was an acrylic root canal model, which is commonly used for dental training. Model B was AA temp resin teeth, which were created by 3D printing denture materials based on CT scans of human root canals. For each type, five models with subtle difference are tested. All the models underwent micro-CT scanning before and after the robot-assisted procedure for further quantitative analysis. Prior to the autonomous procedure, the dentist selected the appropriate endodontic file type. In the case of the acrylic model, the procedure involved using three files (SX, S1, and S2) with increasing diameters. For the resin teeth, three additional files (F1, F2, and F3) were used to further enlarge the root canal cavity.

The results of the pre-clinical evaluation are summarized in Tab. VI. Overall, all models successfully underwent the complete procedure without any instances of file fracturing or ledging. The acrylic models exhibited a larger enlargement ratio, which can be attributed to their narrower root canal pathway. Importantly, there was no disparity in the root canal length after treatment, indicating the efficacy of intraoperative monitoring of file insertion depth. However, in some models, a slight reduction in root canal length was observed following the autonomous procedure, likely due to the presence of debris deposited near the apex.

Fig. 19 illustrates the CT scan results of the representative root canal models examined in the pre-clinical evaluation. Comparing the acrylic models to the AA temp resin models, it is evident that the former exhibited a greater degree of curvature. Nevertheless, the DentiBot successfully completed the cleaning and shaping process autonomously, reaching the apex with the endodontic file. As illustrated in Fig. 19, the root canals were effectively enlarged and well-shaped following the robot-assisted procedure. Notably, Model B2 displayed a smoother wall of the root canal after treatment. These results have proven the DentiBot’s capability to autonomously perform the cleaning and shaping step in RCT.