Toward Task Autonomy in Robotic Cardiac Ablation: Learning-Based Kinematic Control of Soft Tendon-Driven Catheters

Abstract

The goal of this study was to propose and validate a control framework with level-2 autonomy (task autonomy) for the control of flexible ablation catheters. To this end, a kinematic model for the flexible portion of typical ablation catheters was developed and a 40-mm-long spring-loaded flexible catheter was fabricated. The feasible space of the catheter was obtained experimentally. Furthermore, a robotic catheter intervention system was prototyped for controlling the length of the catheter tendons. The proposed control framework used a support vector machine classifier to determine the tendons to be driven, and a fully connected neural network regressor to determine the length of the tendons. The classifier and regressors were trained with the data from the feasible space. The control system was implemented in parallel at user-interface and firmware and exhibited a 0.4-s lag in following the input. The validation studies were four trajectory tracking and four target reaching experiments. The system was capable of tracking trajectories with an error of 0.49 ± 0.32 and 0.62 ± 0.36 mm in slow and fast trajectories, respectively. Also, it exhibited submillimeter accuracy in reaching three preplanned targets and ruling out one nonfeasible target autonomously. The results showed improved accuracy and repeatability of the position control compared with the recent literature. The proposed learning-based approach could be used in enabling task autonomy for catheter-based ablation therapies.

Introduction

Background

Catheterization has become the standard-of-care for the treatment of cardiac atrial fibrillation (AFib).¹ AFib causes cardiac arrhythmia and is the most common cardiology reason for hospitalizations.¹ Inactivating the cardiac cells that cause the arrhythmia by heating, that is, radiofrequency ablation (RFA), or freezing, cryoablation, is the well-established option intervention with catheter-based techniques to treat AFib. Figure 1 depicts the catheter-based intervention for cardiac ablation schematically. For this treatment, a long, thin, and flexible catheter, on the tip of which a burner or freezer unit is embedded, is used. The catheter is surgically inserted into the patient's femoral vein and is steered toward the right or left atrium. Once the tip of the catheter is inside the atrium and in contact with the desired location on the atrial wall (usually medial and posterior walls), the surgeon turns on the radiofrequency pulse generator for burning (ablating) the cells.

FIG. 1.

The schematic shape of steerable ablation catheters in the right atrium during ablation interventions.

The pulse generating cells are mainly scattered on the posterior and medial wall of the atria. Therefore, the surgeon needs to manipulate the tip of catheter from its handle outside the body to reach multiple spots inside the atria for burning enough cells to manage the AFib favorably.

Maintaining stable contact between the tip of the catheter and the atrial wall is challenging due to the motion of the heart.² Also, the blood flow in the atrium may displace the tip of catheter, unless it is in secure contact with the wall. Studies have shown that to obtain optimal treatment, maintaining the contact force in the range of 0.2 ± 0.1 N is crucial.^2,3 Whereas the excessive force would puncture the cardiac wall and insufficient force would lead to an ineffective treatment.²

Another factor challenging the success of RFA is that the surgeon merely relies on the two-dimensional visual feedback of the catheter shape obtained from the X-ray fluoroscopy. Whereas in open surgery, the surgeon has both visual and tactile feedback on the surgical site. In this regard, a recent study has shown that the surgeon's tactile perception happens three times faster than the visual perception in cardiovascular interventions.⁴ Therefore, the surgeon's situational awareness is limited during the RFA procedure due to the compromised sense of touch and variability of the contact between the catheter and the cardiac wall.³

For more robust and dexterous manipulation of the catheters, soft robots have a favorable choice,⁵ as they offer flexibility and dexterity. Tendon-driven catheters are a widely adopted mode of actuation.⁶

A steerable catheter comprises a shape-controllable tip portion (4–10 cm), a non-steerable body (80–150 cm), and a control handle. The handle typically has a knob mechanism to wind/unwind the tendons that are internally connected to the tip portion. Usually, the steerable catheters are single or double curved, depending on the number of curves their tip portion could possess.⁶ Such catheters may also be embedded with sensors for direct measurement of contact force.^7,8 The steerable catheters have enhanced the surgeon's ability to establish and maintain contact between the catheter and atrial wall. Also, thanks to the high maneuverability of catheters, accessing the inferior and posterior wall of the atria has become possible.⁶

Despite the advantages of the steerable catheters, precise manipulation and targeting inside the dynamically moving atria are still a challenge. As an option, the robotic catheter intervention (RCI) systems have added unprecedented dexterity and precision to the electrophysiologic intervention.³ The currently available RCI systems, such as Sensei and Magellan Robotic System (Auris Health, CA), provide robotic catheter manipulation with level-0 (no autonomy) and level-1 (robot assistance) autonomy. A detailed definition of the levels of autonomy is provided in Yang et al.⁹

Motivation and contributions

With the current RCI technology, maintaining the tip of the catheter in constant contact with the atrial wall is a surgical task that needs repeated correction of the tip position by the surgeon through the master module of the RCI. This has been reported to be cumbersome and increasing the cognitive load of the surgeon.^3,10 Therefore, an RCI system with level-2 autonomy (task autonomy) capable of autonomous tip position control of the catheter for establishing and maintaining contact with the atrial wall is favorable. With the task autonomy, the surgeon would determine the position of the catheter tip (e.g., from a previously three-dimensional (3D) map of the heart) and the robot would control the shape of the catheter to reach the target position. Also, the surgeon would be monitoring the maneuver of the robot and interfere if the need would be. A similar control schema has been proposed in Sharifi et al.¹¹ for extracardiac needle insertion.

In the present study, an inverse kinematic-based schema for the position control of the tip of a tendon-driven catheter was proposed and validated. The catheter was custom designed and fabricated with a four-tendon actuation mechanism. The main contribution of the proposed schema was that it was based on a nonlinear learning-based (artificial neural network [NN]) inverse kinematics. Such a learning-based approach would allow for implementing the control schema with a low computational surplus, which is a well-known limitation of the mechanistic models, for example, continuum mechanics-based models.³ Also, it would allow for intrinsic compensation of internal friction and backlash, as the learning-based model would be trained by the actuation data from the catheter before the task performance. In practice, since the catheters are single-use, such a model training (a.k.a calibration) can be automatized as a part of the system setup.

Another challenge was to resolve the redundancy of the catheter actuation. The catheter tip had two degrees-of-freedom (DoF) in the task space, while it had four actuation DoFs (one DoF for each tendon). Such a configuration made the catheter overactuated and actuation redundant. Utilizing a learning-based classifier, and by limiting the number of tendons to be simultaneously driven to only two, the redundancy was resolved and the catheter model was incrementally treated as a fully actuated system.

Based on the validation results, the proposed learning-based kinematic model and feedforward position control would provide a low-maintenance, fast, and accurate alternative to the currently available model-based and sensor-based control schema. To model and control the steerable catheters, researchers have adopted continuum mechanics,¹² differential geometry,¹³ particle-based,^14–16 and multibody dynamics^17,18 approaches. In the following, a review of the pertinent studies is presented.

Related studies

The previous research on modeling the dynamics and kinematics of flexible catheters can be categorized as mechanistic and heuristic (statistical) models. As for the mechanistic models, researchers have modeled the deformation of flexible catheters under external load with¹⁹ and without¹² internal tendon actuation. Theoretically, a flexible catheter has infinite DoFs. However, for simplification, the catheter shape of the catheter has been modeled as a curve^13,18,20,21 with a finite number of DoFs, or a set of small rigid segments with flexible joints.^17,22–25 In the curve-based models, a piecewise constant curvature (PCC) model has been widely adopted.¹⁷ The validity pf of the PCC assumption depends on the structure of the catheter. Since the mechanistic models often involve simplifying assumption on the catheter shape or the external loading, intrinsic nonlinearities such as dead-zones, friction, slack of the tendons, and material nonlinearities (e.g., hyperelasticity) are usually neglected. Also, as tendons can support merely tension but not compression, tendon slacking is a common phenomenon observed. To compensate for the slack, researchers have used symmetric agonist/antagonist tendon configurations.^6,26 The backlash caused by the tendon slack is a source of nonlinearity and must be compensated. Also, high tension in tendons results in non-negligible friction^27,28 and tendon elongation. Considering such nonlinearities in the mechanistic models, such as the Cosserat rod model,^13,20,29,30 would complicate the model and incur high computational cost, especially for inverse kinematics.^31–34

On the contrary, heuristic models such as Gaussian mixture models (GMMs) and NNs have shown to be a favorable option for real-time applications as they offer a low computational cost and the ability to capture the nonlinear effects such as dead-zone and friction.³ In an early study, Rafii-Tari et al.³⁵ utilized a GMM to obtain the kinematics of catheter maneuvers in an RCI setup. In another effort, Khoshnam and Patel³⁶ proposed a GMM to estimate the external force on RFA catheters using curvature analysis. Recently, Chi et al.³⁷ encoded the catheter kinematics in a GMM and used it for kinematic control of catheters. Although GMMs offer the inclusion of nonlinearities, such models need rigorous data preparation. NNs, as another heuristic option, were first used to acquire the kinematics of soft robots for compensating for the material nonlinearities.³⁸ Giorelli et al. utilized an NN for obtaining the kinematic Jacobians³⁹ of a soft robot in real time. NNs have also been used as a lookup table for obtaining the kinematics of soft robots obtained from other methods, for example, the finite element method^40–42 for real-time applications. A comprehensive methodological review of the methods on the taxonomy of steerable catheters and the modeling techniques can be found in Jelínek et al.⁶ and Hooshiar et al.,³ respectively.

Generally, in tendon-driven catheters, the tip is connected to a series of parallel tendons that are aligned with the catheter's body and, at the other end, are connected to motors. The motors change the length of the tendons that consequently change the position and orientation of the tip. For quasistatic applications, that is, where the inertial effects are negligible due to slow maneuvers or low ratio of external forces to the weight of the catheters, the kinematic modeling is favorable. In kinematic models, the aim is to map the change of the length of the tendons (joint-space) to the changes of the tip pose (task space).^43–45

In the following, first, a kinematic model of a tendon-driven catheter is proposed and validated. Afterward, the overview of the RCI robotic system developed for the experiments is described. Furthermore, the details of the tendon selection classification, NN-based inverse kinematics, and the proposed catheter position controller are presented. In the end, the results of the validation studies on assessing the performance of the hypothesized level-2 autonomy in an ex vivo setup are discussed, followed by the concluding remarks.

Materials and Methods

In this section, an overview of the catheter fabrication and the RCI mechanism used in this study is provided. Afterward, the theoretical workspace of the catheter based on the constant bending radius assumption is obtained and its validity is investigated by experimental comparison. Furthermore, the proposed learning-based forward and inverse kinematic model of the catheter is described and the accuracy of those is investigated. In the end, the experimental test procedures and setups for studying the performance of the proposed trajectory tracking are described.

Catheter fabrication and assembly

The prototyped catheter in this study, named as MiFlex, is a tendon-driven catheter with four inextensible tendons. Figure 2 shows the fabrication steps and finally the size and assembly of the catheter. The selected dimensions for the catheter prototype were 6 mm in diameter and 40 mm in length. The selected diameter was to replicate an 18-Fr (1Fr = 1/3 mm) catheter, and a 40 mm length was according to Refs.^46,47 for the average transversal diameter of the right atrium in adults diagnosed with AFib.

FIG. 2.

Molding, degassing, curing, and assembly steps for prototyping the flexible catheter MiFlex.

For the fabrication, a cylindrical mold was rapid prototyped with a 3D printer (Replicator+; MakerBot, NY). Also, a square platform (16 × 16 × 8 mm), housing four through holes, was 3D printed to provide a platform for the fixed end of the catheter. The through holes were used to accommodate anchorage M2 screws for fixing the mold to the platform and later were used as guides for the four tendons.

The catheter comprised a steel compression coil spring with a nominal outer diameter of 5 mm and compressive stiffness of 0.35 N/mm. The spring was installed at the center of the cylindrical mold, while the silicone rubber material for the body of the catheter (Ecoflex™ 00-20; Smooth-on, Inc., PA) was filled in the mold. The use of coil spring would enhance the ability of the catheter to recover to its original shape by compensating for the viscous energy damping in the rubber material. After filling the mold, it was rested still in a vacuum chamber (Best Value Vacs, IL) under 29 mm Hg vacuum pressure for discarding the air bubbles (degassing). Furthermore, the degassed mold was rested for 24 h at 24°C for final curing. After curing, the platform was secured in a 3D-printed base. The reason for the base was to make the assembly modular and facilitate the replacement of the catheter and base. In practical cases, the catheter would be a single-patient use disposable, and thus, such a modular design could help in replacing the catheter.

RCI system overview

At the system level, the RCI system in this study was designed with three modules, that is, mechanical, electrical, and software modules. Figure 3a shows the components of the prototyped RCI system. Four independent stepper motors, identified as $M_{1 - 4}$ , were used to control the length of the tendons, that is, $L_{1 - 4}$ . The motors were controlled using two independent stepper motor drivers stacked on two ATmega microprocessors (Uno, Arduino Co., MA). Each microprocessor was also connected to two potentiometers (as rotary encoders) to measure the angle of the shaft of the motors.

FIG. 3.

(a) Components of the mechanical and electrical modules in the prototyped RCI device, and (b) software architecture of the UI and FW. FW, firmware; RCI, robotic catheter intervention; UI, user interface.

Figure 3b depicts the software architecture used for the feedback control of the motors, trajectory error estimation, and data storage. The software was composed of two components: the user interface (UI), running on the PC, and the firmware (FW), and loaded on the microprocessors. The UI was used to acquire the user inputs, that is, the desired trajectory and the target within the workspace. The FW role was to receive the desired tendon lengths from UI and relaying the current tendon lengths back to UI. The control framework, described in the Control Framework section, was implemented in the FW (low-level implementation), and the trajectory generation and task-space to joint-space mapping (inverse kinematics) were implemented in the UI (high-level implementation). Also, for validation purposes, the trajectory of the tip marker (red sphere at the tip of the catheter) was tracked in real-time using the two USB cameras (800 × 600 pixel resolution, model C920; Logitech, Lausanne, Switzerland) with a stereo-calibration adopted from Refs.^48,49 The stereo-vision verification on a chessboard template (depicted in Fig. 3b) showed an error of ±0.26 mm in detecting the corners of the squares in the template. It is noteworthy that the position feedback from the camera tracking served merely as a reference and was not used in the control framework.

Control framework

To enable the RCI system to exhibit task autonomy for trajectory following tasks, the presented control system in Figure 4 was proposed. The control goal was to attain and hold a given desired position, $P^{*}$ , in the task space (global Cartesian) through changing the length of tendons. For the user's intuitive inference, the choice of the Cartesian task space was favorable. However, the kinematics of the catheter was less complicated in spherical coordinates, and thus, a mapping $ℳ : R^{3} \to R^{2}$ from Cartesian to spherical coordinates was used. Afterward, the control system would determine the desired length of a set of tendons (control state) by which the tip of the catheter would reach $P^{*}$ .

FIG. 4.

The proposed learning-based feedforward control system for position control of the tip of the catheter.

The proposed control framework is a feedforward system. The utilization of an internal motor shaft position feedback in the motor drivers $M_{1 - 4}$ ensures that the desired tendon lengths are attained. The same technique has been successfully implemented in Yu et al.³² Such a control framework necessitated a mapping between the desired kinematic state in task-space $(θ^{*}, φ^{*})$ and the joint-space variables $L_{1 - 4}^{*}$ . Such a mapping constituted an inverse kinematic problem. Mechanistic modeling of the problem involves nonlinearities such as material nonlinearity, internal friction between the tendons and the body of the catheter, and tendon slacking. As an alternative to the mechanistic approach, the inverse kinematic problem was divided into learning-based classification and regression problems. For a given desired position $P^{*}$ , the classifier, that is, $C (θ^{*}, φ^{*})$ would determine tendon(s) to be pulled, while the regressor would determine the length of the said tendon(s) (desired lengths). Therefore, the objective of the inverse kinematics would be to determine the catheter configuration, that is, $(C_{k} : i - j, L_{i}, L_{j})$ in joint space, for a given desired position $P^{*}$ in the task space.

Inverse kinematics

Degrees of freedom

Figure 5 depicts a representative deformation of the catheter, where the Cartesian coordination system ${(\begin{matrix} X & Y & Z \end{matrix})}^{T}$ represents the fixed global frame, P is the position of the tip of the catheter, that is, the center of the red tip marker marked by the bright $+$ , r is the bending radius, $Γ$ is the bending plane, and O_b is the center of the bending arc $\hat{O P}$ .

FIG. 5.

Representative deformed shape of the catheter with constant bending radius.

Due to the relatively larger longitudinal stiffness compared with the bending stiffness, the compression of the catheter along its spine was neglected. Therefore, $|\hat{O P}| = 2 r θ = 40 m m c o n s t a n t .$ (1)

Also, $\vec{P} : = \vec{O} P$ was presented in the global Cartesian coordinates and spherical coordinates as follows: $\vec{P} = {(\begin{matrix} x & y & z \end{matrix})}^{T} \in S^{+},$ (2)

and $\vec{P} = {(\begin{matrix} ρ & θ & φ \end{matrix})}^{T} \in S^{\circ},$ (3)

where $S^{+} \in R^{3}$ and $S^{\circ} \in R^{3}$ are the Cartesian and spherical representations of the working space (surface) of the catheter, and $ρ \in R$ , $θ \in [0, π]$ , and $φ \in (- π, π]$ . The mapping from the Cartesian coordinates to the spherical and its versa were obtained as follows: $(\begin{matrix} ρ \\ θ \\ φ \end{matrix}) = (\begin{matrix} \sqrt{x^{2} + y^{2} + z^{2}} \\ arccos (\frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}}) \\ arctan 2 (y, x) \end{matrix}),$ (4) $(\begin{matrix} x \\ y \\ z \end{matrix}) = (\begin{matrix} ρ sin θ cos φ \\ ρ sin θ sin φ \\ ρ cos θ \end{matrix}),$ (5)

where $arctan 2 (y, x)$ is the two-parameter nonsingular tangent inverse function defined as follows: $arctan 2 (y, x) = (\begin{matrix} 2 arctan (\frac{y}{\sqrt{x^{2} + y^{2}} + x}) & x > 0 \\ 2 arctan (\frac{\sqrt{x^{2} + y^{2}} - x}{y}) & x \leq 0, y \neq 0 \\ π & x < 0, y = 0 \end{matrix}$ (6)

Since the single-plane bending condition necessitates $O_{b} \in Γ$ and given that $\bar{O_{b} O} ⊥ \hat{O P}$ , O_b would unconditionally remain on the XY-plane throughout the bending. Also, in , $ρ = 2 r sin θ .$ (7)

Substituting r from Equation (1) in Equation 7 yields the following kinematic constraint between ρ and $θ$ : $ρ \frac{θ}{sin θ} = 40 m m c o n s t a n t .$ (8)

In fact, Equation (8) describes the locus of the theoretical workspace of the tip of the catheter in terms of the spherical coordinates ρ and θ. Moreover, it shows that the workspace is independent of $φ$ . Given that O_b coincides on the z = 0 plane, the workspace would necessarily be axisymmetric with respect to Z-axis.

Figure 6a depicts the theoretical workspace of the catheter at various $θ$ s. The shape of the constructed workspace is in accordance with the experimental findings in other studies.^21,50 Another deduction of Equation (8) was that the minimum number of independent coordinates to fully describe the catheter tip position was $(ρ, φ)$ or $(θ, φ)$ . Therefore, the catheter had two DoFs. Figure 6a depicts the feasible workspace of the catheter overlaid on the theoretical workspace. The feasible workspace of the catheter was obtained by sequentially pulling the tendons in all the possible dual-tendon configurations, labeled as tendon classes. $C_{1} : 1 - 2, C_{2} : 2 - 3, C_{3} : 3 - 4, C_{4} : 4 - 1, C_{5} : 2 - 4, C_{6} : 3 - 1$ . To obtain a complete feasible space, for each $C_{k} : i - j$ , tendon i would increment for 1 mm (up to 10 mm), while tendon j would complete a 10 mm sweep. Meanwhile, the position of the tip of the catheter was tracked using the stereo-cameras.

FIG. 6.

(a) Comparison of the theoretical workspace and the feasible workspace of the catheter, and (b) the distribution of the minimum distance between the experimental tip positions and the theoretical workspace. Color images are available online.

Tendon selection: support vector machine classification

The feasible space data showed that for a given point in the feasible space, the tendon configuration might not be unique. Whereas for a subdomain in the feasible space, multiple combinations of tendons would pose the catheter tip at a similar position. Such a circumstance defined a redundant control space that needed to be resolved. The redundancy resolution was performed first by selecting the tendon classes $C_{1 - 4}$ , which exhibited the least overlap in the $φ - θ$ plane (Fig. 7a). Afterward, a support vector machine (SVM) classifier with a linear kernel was trained with $θ$ and $φ$ as the features and $C_{1 - 4}$ as the categories. The data set of the feasible space, containing the spherical coordinates of the feasible space (n = 16,100), was divided into training and validation data with a 70:30 ratio. The model training and validation were performed using the MATLAB Classification Toolbox (MATLAB R2019b; MathWorks, MA). Figure 7 depicts the feature space of the training data, the contour of the classified subspaces on XY-plane, and the confusion matrix of the SVM classification. With adopting the accuracy metric defined as

FIG. 7.

(a) Feature space constructed by θ and φ. The figure shows that at relatively large θ, φ determines the choice of tendon class, (b) the contour of the classified feasible space on XY-plane, and (c) confusion matrix for tendon class prediction resulting from the validation data ( $n = 4830$ ). Color images are available online.

a c c u r a c y = (1 - \frac{\sum o f f - d i a g o n a l c o m p o n e n t s}{\sum a l l c o m p o n e n t s}) \times 100,

(9)

the accuracy of the classifier for tendon class prediction was estimated as 97.3%.

Tendon length estimation: NN regression

In the control framework, the next step was to determine the desired length of each tendon through NN regression. To this end, four individual deep feedforward (DFF) NNs, denoted as $N N_{k}$ , $(k = 1 \dots 4)$ , were trained with the classified $(θ, φ) T$ as input and values of $(L_{i}, L j) T$ as output for the four tendon classes, that is, $C_{k} : i - j$ . In addition to the input and output layers, each $N N_{k}$ network had 10 hidden layers (with five neurons each) with fully connected architecture and rectified linear unit activation functions. The training was performed using the damped least-square method, a.k.a. Marquardt–Levenberg algorithm in MATLAB Neural Network Toolbox (MATLAB R2019b; MathWorks, Inc., MA). Similar to the classification, the data set was divided with a 70:30 ratio for training and validation, respectively. Table 1 summarizes the adjusted goodness-of-fit (adj-R²) and error of prediction for each $N N_{k}$ .

Table 1.

Goodness-of-Fit (adj-R²) and Average Percentage of Prediction Error ( $Ē_{%}$ ) for the Neural Networks $N N_{1 - 4}$

$C_{k} : i - j$	adj- $R^{2}$		$Ē_{%}$
$C_{k} : i - j$	$L_{i}$	$L_{j}$	$L_{i}$ (mean $\pm$ SD)	$L_{j}$ (mean $\pm$ SD)
$N N_{1} : 1 - 2$	0.97	0.97	3.1 ± 0.5	3.5 ± 0.4
$N N_{2} : 2 - 3$	0.99	0.94	2.8 ± 0.4	4.2 ± 0.3
$N N_{3} : 3 - 4$	0.94	0.98	4.4 ± 0.6	4.7 ± 0.6
$N N_{4} : 4 - 1$	0.97	0.98	3.6 ± 0.2	4.3 ± 0.3

Control implementation

A robotic system with level-2 autonomy should keep the surgeon in the control loop for supervisory privileges, that is, task initiation and termination, and trajectory selection. To meet this requirement, the proposed control framework was implemented in the UI software using object-oriented and multithread programming techniques. To increase the computational efficiency of the control loop, the SVM classifier $C (θ^{*}, φ^{*})$ , four NNs $N N_{1 - 4}$ , and the trajectory update loop were implemented in parallel. Thanks to the parallelization, the control loop in UI exhibited an average refresh-rate of 164 ± 12 Hz. Also, at the microprocessor level, the refresh-rate of the tendon length control loop was set to 1 kHz.

Experimental Validation

Experiment I: trajectory tracking

To study the performance of the proposed position control framework, the system was tested in tracking four desired trajectories. The trajectories were of circular, triangular, infinity sign, and spiral shapes and were denoted by $T_{\circ}, T_{Δ}, T_{\infty}, T_{\partial},$ respectively. Also, two time periods of 5 and 10 s per repetition were set to simulate slow and fast tasks, respectively. Each trajectory was repeated 10 times at each speed. The trajectories were defined in preprocessing with 50 intermediary points in the Cartesian task space such that the XY-projection of the intermediary points would be within the XY-projection of the feasible workspace (Fig. 6b).

Each trajectory was tested in an individual test session such that the test sessions would include ten repetitions of the slow tasks followed by ten repetitions of the fast task. During the tests, the UI would update the desired position $P^{*}$ on-line according to the temporal sequence of the intermediary points in each trajectory and the control loop would solve the inverse kinematics for each of the intermediary points in real-time. As a reference for comparison, the position of the tip of the catheter was tracked with the stereo-cameras during the tests.

Figure 8 depicts the desired and experimental trajectories demonstrated by the proposed RCI system, and Table 2 presents the performance of the control system in terms of root-mean-square (RMS)-error (average of five repetitions), error range, and time-lag in the fast and slow tasks for the four trajectories. Also, Figure 9 depicts the change of the length of tendons $L_{1 - 4}$ for one repetition of $T_{\circ}$ task with slow speed.

FIG. 8.

Desired and experimental trajectories for (a) circular, (b) spiral, (c) triangular, and (d) infinity-shape trajectories. Color images are available online.

FIG. 9.

The desired and attained change in the length of the tendons for T_o task. Tendon classes were automatically selected by the classifier so as to follow a full circle. Color images are available online.

Table 2.

Summary of the Performance of the Control System for the Tip Position of the Catheter in Four Trajectories with Slow and Fast Speeds

Trajectory	RMS-error (mm)	Error range (mm)	Time-lag (s)
$T_{\circ}$
Slow	$0.56 \pm 0.41$	(−0.84,1.07)	$0.41$
Fast	$0.73 \pm 0.43$	(−1.24,1.4)	0.38
$T_{Δ}$
Slow	$0.31 \pm 0.24$	(−0.35,0.47)	$0.37$
Fast	$0.40 \pm 0.20$	(−0.54,0.81)	$0.43$
$T_{\infty}$
Slow	$0.62 \pm 0.54$	(−1.05,1.14)	$0.44$
Fast	$0.84 \pm 0.44$	(−2.31,1.24)	$0.41$
$T_{\partial}$
Slow	$0.46 \pm 0.39$	(−0.89,1.26)	$0.34$
Fast	$0.52 \pm 0.31$	(−0.76,1.21)	$0.40$
Average
Slow	$0.49 \pm 0.48$	(−0.78,0.99)	$0.39$
Fast	$0.62 \pm 0.35$	(−1.17,1.16)	$0.41$

RMS, root-mean-square.

The results showed that the system was fairly accurate in following the desired trajectories as the average RMS-error for the four trajectories was 0.49 ± 0.32 and 0.62 ± 0.36 mm for the slow and fast speeds, respectively. Also, the time-lag between the input and output was consistent among the trajectories with an average of 0.4 s for all trajectories. A limiting factor in estimating the time-lags was the frame-rate of the cameras, that is, 33 ± 5 Hz, as the image frame time-stamps were used as the synchronization benchmark. Therefore, the computed time-lag might have been smaller than the reported values. Another finding in this experiment was that the system was more accurate at slow-speed than fast-speed trajectories. However, for both the speeds, the accuracy was within the acceptable practical precision of ±1 mm.³ Moreover, small spikes in the temporal variation of the tip position of the catheter, for example, Figure 9 at t = 4 s and t = 8 s, were due to the change of tendon classes. At these time instances, the controller switched the classes, and thus, the tendon lengths before and after these instances were estimated by different NNs. Nevertheless, the spikes are relatively small and decreased with the trajectory progressing farther from the tendon class boundaries. Authors sought such switches between the tendon classes as a source of deviations from the planned trajectory observed in the experiments.

Experiment II: target reaching

The second experiment was performed on a freshly excised bovine myocardial tissue. Four arbitrary target points denoted as $P_{1 - 4}$ in Figure 10a were selected and marked manually. Afterward, each point was selected on the top- and side-view live images in the UI software and registered as a target. The test procedure was that after selecting each specific point, the UI would determine if the point is within the catheter's reach and if so, it would plan a direct path. Upon the user's approval, the system would initiate the task to hit the point, that is, $P_{1 - 4}$ , maintain the target position for three seconds, return to its resting position, that is, $P_{\circ} = {(0, 0, 40)}^{T}$ , and repeat the task for four repetitions.

FIG. 10.

(a) Marked positions on the bovine myocardial tissue for reaching tasks, $P_{1 - 4}$ , and (b) Cartesian tip position of the catheter in four repetitions to reach P₃.

This experiment was to replicate a robot-assisted ablation intervention with level-2 autonomy, where the surgeon would only specify the preplanned location of the target point(s) on intraoperative images for ablation, based on which the robot would reach the position, hold the position for the period on ablation (typically less than 30 s), and move to a resting position.

Figure 10b shows the variation of the tip position of the catheter in Cartesian task space for a representative point P₃, and Table 3 summarizes the position of $P_{1 - 4}$ , temporal average maintained positions in repetitions, RMS-error of in reaching tasks, and the repeatability of reached positions (standard deviation of the accumulative average reached positions for all repetitions).

Table 3.

Performance of the Robotic Catheter Intervention in Autonomous Reaching to the Targets $P_{1 - 4}$ Selected by the User in Live Images

Target	Position (mm)	Reached position (mm)	RMS-error (mm)	Repeatability (mm)
P ₁	${(\begin{matrix} - 6.9 & - 9.7 & 37.4 \end{matrix})}^{T}$	${(\begin{matrix} - 7.1 & - 9.0 & 37.0 \end{matrix})}^{T}$	${(\begin{matrix} 0.39 & 0.47 & 0.42 \end{matrix})}^{T}$	$\pm (\begin{matrix} 0.3 & 0.3 & 0.3 \end{matrix})$
P ₂	${(\begin{matrix} 5.2 & - 8.4 & - 35.7 \end{matrix})}^{T}$	${(\begin{matrix} 5.9 & - 8.1 & - 34.9 \end{matrix})}^{T}$	${(\begin{matrix} 0.37 & 0.52 & 0.44 \end{matrix})}^{T}$	$\pm (\begin{matrix} 0.3 & 0.3 & 0.3 \end{matrix})$
P ₃	${(\begin{matrix} 9.5 & - 10.4 & - 36.1 \end{matrix})}^{T}$	${(\begin{matrix} 9.1 & - 10.7 & - 36.7 \end{matrix})}^{T}$	${(\begin{matrix} 0.43 & 0.36 & 0.51 \end{matrix})}^{T}$	$\pm (\begin{matrix} 0.3 & 0.3 & 0.3 \end{matrix})$
$P_{4}^{a}$	${(\begin{matrix} 18.7 & - 9.5 & - 37.2 \end{matrix})}^{T}$	—	—	—
Average	—	—	${(\begin{matrix} 0.4 & 0.45 & 0.46 \end{matrix})}^{T}$	$\pm (\begin{matrix} 0.3 & 0.3 & 0.3 \end{matrix})$

User interface software overruled P₄ as it was out of the feasible space of the catheter.

The results of experiment II showed that the RCI system was successful in autonomous reaching to the preplanned targets with a spatial error of 0.75 mm, that is, the norm of the average RMS-error for $P_{1 - 3}$ . Also, the system was capable of ignoring P₄ as being out of the feasible zone of the catheter. Moreover, the system showed fairly repeatable target hits with a spatial repeatability of ±0.52 mm, that is, the norm of repeatability of x, y, and z.

Conclusions

The goal of this study was to propose and validate a control framework with level-2 autonomy (task autonomy) for kinematic control of flexible ablation catheters. Also, it was hypothesized that through learning-based classification and regression, the inverse kinematics of the soft catheter could be captured within the practical precision, that is, ±1 mm. Also, with a one-time preoperative calibration (training), the material and geometric nonlinearities involved in the deformation of the catheter, for example, friction, large deformation, were captured and compensated, thus, simplifying the catheter model.

All the modules of the RCI system in this study were developed in-house and thus allowed for maximal software/hardware integration. Authors believe that such integration was a key in the observed accuracy of the system. The validation study for the trajectory tracking and target reaching also showed fair accuracy and repeatability for position control of the catheter. More specifically, this study exhibited better spatial accuracy and repeatability compared with Yu et al.,⁵⁰ Li et al.,^51,52 Yip et al.,⁵³ Tan et al.,³⁴ and Slawinski et al.,⁵⁴ in which the reported accuracies and repeatabilities were not submillimeter.

Another promising finding was that the proposed control system did not exhibit a dead-zone at the proximity of the resting position. Such a dead-zone has been reported in other studies, for example, Yu et al.,⁵⁰ and has been sought related to the slack of the tendons. Moreover, the distributed implementation of the control system, that is, the inverse kinematics in the UI module and tendon length control in the FW, allowed for multithread computation parallelization of the control system. Such a multithread parallelization was crucial in reaching the small time-lag.

For further improvement of this study, researchers are encouraged to replace the stepper motors with servo-motors, which would facilitate dynamic torque control on the catheter. Also, releasing the two-tendon drive condition and allowing for more tendons to be pulled simultaneously would increase the feasible space of the catheter. Nevertheless, the latter may incur considerable compression in the catheter's spine and violate the constant bending radius assumption.

Furthermore, increasing the number of tendons would have a double effect worth the investigation. On one hand, more tendons would potentially lead to finer control of the pose of the catheter. Nevertheless, more tendons would worsen the redundancy of the catheter. Consequently, it would increase the tendon driving classes and the occurrence of the switching between the classes factorially. The latter was sought as a source of deviation from the planned trajectories in this study.

Another extension of this work would be adding a linear DoF to the base of the catheter, for example, as proposed by Yu et al.,⁵⁰ or adding a second set of tendons terminated midspan of the catheter length. Such extensions would expand the feasible space of the catheter spatially, however, would increase the DoFs and might change the kinematics of the catheter.

Furthermore, a combination of the precise position control with a displacement-based tissue contact model, for example, as proposed in Refs.,^55,56 for controlling the catheter-tissue contact forces would be a possible extension of this work. Such an extension would also possibly facilitate the provision of intraoperative haptic feedback, as proposed in Refs.^3,57

Footnotes

Acknowledgments

Authors express their gratitude to the Natural Science and Engineering Council (NSERC) of Canada, Fond du Recherche Nature et Technologie (FRQNT) of Quebec, and Concordia University for their support. The materials and methods disclosed in this manuscript are protected by the US Provisional Patent Application No. 62/962522 filed on January, 17, 2020.

Author Disclosure Statement

No competing financial interests exist.

Funding Information

This research was supported by NSERC Vanier CGS Scholarship and Concordia Public Scholar Award of A.H., NSERC CREATE Innovation-at-the-Cutting-Edge (ICE) Grant of J.D., and Concordia University Research Chair and NSERC grant of M.P.

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