Comparison of Classical, Neural Network and Hybrid Models for Hysteretic Single-tendon Catheter Kinematics

Abstract

While robotic control of catheter motion can improve tip positioning accuracy, hysteresis arising from tendon friction and flexural deformation degrades kinematic modeling accuracy. In this paper, we compare the capabilities of three types of models for representing the forward and inverse kinematic maps of a clinical single-tendon cardiac catheter. Classical hysteresis models, neural networks and hybrid combinations of the two are included. Our results show that modeling accuracy is best when models are trained using motions corresponding to the anticipated clinical motions. For sinusoidal motions, recurrent neural network models provide the best performance. For point-to-point motions, however, a simple backlash model can provide comparable performance to a recurrent neural network.

I. Introduction

Tendon-actuated catheters are used extensively in percutaneous intracardiac procedures including for the repair and replacement of heart valves, the treatment of cardiac arrhythmias and the repair of occluded vessels. For example, pulmonary vein isolation is a therapeutic procedure used to treat atrial fibrillation, an irregular heart rhythm. In this procedure, a transseptal sheath is guided from the femoral vein in the groin into the right atrium of the heart, across the atrial septum and just into the left atrium (Fig.1).

Fig. 1.

Transcatheter pulmonary vein isolation. (a) Transseptal sheath extends from femoral vein in groin across atrial septum into left atrium. (b) Steerable catheter extending through transseptal sheath is used to ablate tissue at sequence of points encircling pairs of pulmonary veins.

A steerable ablation catheter is inserted through the transseptal sheath which is sequentially positioned at closely spaced sets of points around the pairs of pulmonary veins. At each point, the catheter contacts the tissue and, using radiofrequency energy, creates a scar. The rings of scars prevent abnormal electrical signals arising inside the pulmonary veins from propagating into the left atrium. The catheter pauses at each point for up to 60 seconds to create a scar.

Precise control of point-to-point motions, such as those used in pulmonary vein isolation, is challenging because the kinematic modeling of continuum robots is more difficult than for traditional robots comprised of rigid links and discrete joints. While mechanics-based models of continuum robot designs have been developed [1]–[3], they typically neglect hysteretic phenomena despite its known effect on modeling accuracy [4].

While feedback control can mitigate feedforward modeling inaccuracy, this too has its challenges. Few catheters incorporate tracking or shape sensors owing to the cost and complexity of including sensors in small cross sections. This forces the operator to close the feedback loop using real-time imaging to compensate for kinematic modeling error. This increases the level of attention that the operator must devote to catheter control and is challenging given the limitations of fluoroscopic and ultrasound imaging.

The alternative approach is to incorporate improved kinematic models into the robot controller that include accurate characterizations of hysteresis. The challenge for the robot designer, however, is that, while there are many possible techniques for modeling hysteresis, most investigations consider only one or two methods, do not model both forward and inverse kinematics and do not consider the effect of the anticipated clinical motions on model accuracy. Consequently, there are no definitive studies comparing the performance and characteristics of a broad range of models.

For example, both traditional hysteresis and deadband models as well as neural networks provide potential means of modeling kinematic hysteresis in catheters. Tendon-actuated hysteresis modeling has been considered in the context of mapping tendon tension to tube flexure or tip contact force [5], [6], examining how sheath curvature affects tendon hysteresis [7], and modeling the combined hysteretic behavior of a tendon-actuated catheter driven by a pneumatic artificial muscle [8].

These papers have considered a variety of hysteresis models including traditional models like the Preisach and Prandtl-Ishlinskii models [5], [8], as well as advanced approaches involving recurrent neural networks (RNNs) like Long short-term memory (LSTM) networks [6], [8], [9] and hybrid models combining RNNs and the Preisach model [7].

To provide a unified comparison of model types, this paper presents an experimental comparison of seven hysteretic models for the forward and inverse kinematics. An eighth model, comprised of a feedforward neural network (FNN), is used as a baseline non-hysteretic model for comparison since it can learn a nonlinear kinematic map but lacks the capability to model history dependence. FNNs represent the best performance achievable by a model that ignores hysteresis and so provide a baseline for quantifying the improvement achieved by other models when considering historical information.

Here, we study a single-tendon catheter that flexes in one direction. This design is commonly used clinically such as in the application shown in Fig. 1. Catheter motions are composed of a combination of axial translation, axial rotation and uni-directional flexure. While catheter translation and rotation can be performed smoothly, the tendon-actuated flexure of catheters fabricated from polymeric components exhibits a significant amount of hysteresis.

The hysteresis models compared include two classical models (backlash [10] and Prandt1-Ishlinskii [11]), three neural network models (LSTM, GRU and a feedforward neural network with an input buffer) and two hybrid models (combinations of backlash and LSTM). The paper also investigates two classes of catheter motions: sinusoidal and point-to-point.

The remainder of the paper is arranged as follows. Section II defines the kinematic inputs and outputs. It also introduces the classical models, neural network models, and hybrid models used for hysteretic kinematics modeling. Section III describes the sets of experiments performed to evaluate the models along with modeling results. Conclusions are presented in Section IV.

II. Modeling Hysteretic Kinematics

As shown in Fig. 2, when a tensile force or displacement is applied to a tendon attached to the tip of a flexible catheter, it causes the catheter to deflect. The kinematic input can be considered to be either tendon displacement or tension force. We assume that, while catheter flexure is hysteretic, there is a unique relationship between flexure angle, $\theta$, tip offset from the longitudinal axis, $y$, and tip displacement along the longitudinal axis, $x$. Thus, the output of the kinematic map can be taken as one of $\{\theta ,x,y\}$ and the remaining two variables can be computed from the first. As described below, we investigated the utility of a variety of models for representing the forward and inverse kinematic maps.

Fig. 2.

Clinical cardiac catheter. Kinematic input is tendon displacement, $d$, or tension, $t$. Kinematic output is $\theta ,x$ or $y$.

A. Classical hysteresis models

Numerous mathematical models have been developed to describe nonlinear hysteresis behavior exhibited by materials and systems [12], [13]. Here, we consider two models, a backlash model and a frequency-dependent Prandtl–Ishlinskii model.

The virtue of the backlash model is its simplicity. As shown in Fig. 3(a), it is described by four parameters and was proposed to describe and compensate for the dead-zone, hysteresis or backlash characteristics in control systems [10], [14]. The change in output, $v\left(t\right)$, verses input, $u\left(t\right)$, is characterized by two straight lines connected by horizontal line segments and is given by

$$v\left(t\right)=\left\{\begin{array}{ll}{m}_L\left[u\left(t\right)-c_L\right]& \text{if}u\left(t\right)\le z_L\\ m_R\left[u\left(t\right)-c_R\right]& \text{if}u\left(t\right)\ge z_R\\ v(t-1)& \text{if}{z}_L\le u\left(t\right)\le z_R\end{array}\right.$$ (1)

$$\begin{array}{rr}& \\ & z_L=\frac{v(t-1)}{m_L}+c_L,z_R=\frac{v(t-1)}{m_R}+c_R\end{array}$$ (2)

Fig. 3.

Hysteresis models. (a) Four-parameter backlash model. (b) Hysteresis loop of Prandtl-Ishlinskii model. (c) Hybrid LSTM-Backlash model (HB1). (d) Parallel hybrid LSTM-Backlash model (HB2).

Here, $m_L,m_R,c_L$, and $c_R$ are parameters defining the slopes and positions of the two straight lines. The inverse form of the backlash model to be used for inverse kinematic modeling is given by

$$u\left(t\right)=\left\{\begin{array}{ll}u(t-1)& \text{if}v\left(t\right)=v\left(t-1\right),\\ \frac{v\left(t\right)}{m_L}+c_L& \text{if}v\left(t\right)<v\left(t-1\right),\\ \frac{v\left(t\right)}{m_R}+c_R& \text{if}v\left(t\right)>v\left(t-1\right).\end{array}\right.$$ (3)

Among those models developed specifically for hysteresis, e.g., Preisach, Prandtl-Ishlinskii (PI), and Bou–Wen, the PI model has merit in that model identification and inversion are somewhat easier than other models [12], [13]. Although the classical PI model does not account for rate-dependent behavior, several modified versions incorporate this capability. We use here the version proposed in [11]. The rate-dependent PI model is formulated by superposing time-dependent play operators and is given by

$$\begin{gathered} v\left(t\right):=a_{0}u\left(t\right)+\sum _{i=1}^k{a}_i{\Psi }_{r_i}\left(t\right)\left[u\right], \\ {\Psi }_{r_i}\left(t\right)\left[u\right]=\text{max}\left(u\left(t\right)-r_i\left(t\right),\text{min}\left(u\left(t\right)+r_i\left(t\right), \\ {\Psi }_{r_i}\left(t_{j-1}\right)\left[u\right]\right)\right)\text{for}t\in \left[t_{j-1},t_j\right],r_i\left(t\right)=\xi i+\beta \left|\dot{u}\left(t\right)\right|\text{for}i=1,\dots ,k, \end{gathered}$$ (4)

where ${\Psi }_{r_i}\left(t\right)\left[u\right]$ are the time dependent play operators and $r_i$ is a time dependent threshold function of these play operators. In this paper, $k$ is set to 4 and an example loop is depicted in Fig. 3(b) for parameters $\left\{a_0,a_1,a_2,a_3,a_4,\xi ,\beta \right\}=\left\{\mathrm{0.1,0.2,0.3,0.4,0.5,0.1,2}\right\}$.

For modeling the inverse kinematics, we use the inverse form of the rate-dependent PI model given by [15]. The parameters to be optimized are $\left\{b_0,b_1,b_2,b_3,b_4,\xi ,\beta \right\}$ and the model is given by

$$\begin{gathered} u(t):=c_{0}v(t)+\sum _{i=1}^k{c}_i{\Psi }_{s_i}(t)[v], \\ {\Psi }_{s_i}\left(t\right)\left[v\right]=\text{max}\left(v\left(t\right)-s_i\left(t\right),\text{min}\left(v\left(t\right)+s_i\left(t\right),{\Psi }_{s_i}\left(t_{j-1}\right)\left[v\right]\right)\right)\text{for}t\in \left[t_{j-1},t_j\right], \\ {r}_i\left(t\right)=\xi i+\beta \left|\dot{v}\left(t\right)\right|\text{for}i=1,\dots ,k, \\ {s}_i\left(t\right)=s_{i-1}\left(t\right)+\left(r_i\left(t\right)-r_{i-1}\left(t\right)\right)\sum _{j=0}^{i-1}{b}_j\text{for}i=2,\dots ,k, \\ {c}_0=\frac{1}{b_0},c_i=\frac{1}{{\sum }_{j=0}^i{b}_j}-\frac{1}{{\sum }_{j=0}^{i-1}{b}_j}\text{for}i=1,\dots ,k. \end{gathered}$$ (5)

B. Neural network models

Neural networks are effective tools for identifying complex patterns in data. Feedforward Neural Networks (FNNs) utilize a single-direction architecture to learn input-output mappings and so lack the capability to model any dependence on the history of motion. We use FNNs in this paper to represent the best that a non-hysteretic model can do in reproducing the forward and inverse kinematics.

Recurrent Neural Networks (RNNs) are specifically designed to handle sequences so their structures allow them to model hysteresis. Long short-term memory (LSTM) networks incorporate memory cells and gating mechanisms which make them more effective at dealing with long sequences [16]. Gated Recurrent Unit (GRU) networks have a simplified architecture compared to LSTMs, which reduces their computational complexity while maintaining their performance in many tasks [17].

In addition to RNNs and GRUs, we also consider a modified FNN which incorporates a history input buffer [18], [19]. In this modification, the input is comprised of the current input plus a sequence of preceding inputs. This FNN with History Input Buffer (FNN-HIB) is able to reproduce hysteretic behavior because it creates a map between a sequence of prior inputs and the current output. Its structure is simpler than that of an RNN, but its implementation requires the use of the input buffer.

In this paper, all of the neural network models incorporate an input layer, two hidden layers of dimension 64 and an output layer. All of the input layers are comprised of a single node except for the FNN-HIB, which is comprised of 50 nodes, serving as the buffer storing a sequence of 49 prior inputs plus the current input value.

C. Hybrid models

Since enhanced modeling performance can often be achieved by combining neural networks with phenomenological models, we also consider two hybrid models integrating neural networks with classical hysteresis models. Fig. 3(c) and (d) shows the two models considered which each combine an LSTM network and a backlash model. In the first model (Fig. 3(c)), the LSTM is used to calculate instantaneous values of the four backlash parameters, $\left\{m_L,m_R,c_L,c_R\right\}$.

$$\begin{gathered} \left(m_L,m_R,c_L,c_R\right)=f_{\text{lstm}}\left(u\right), \\ {v}_{HB1}=f_b\left(\left(m_L,m_R,c_L,c_R\right),u\right) \end{gathered}$$ (6)

The second model of Fig. 3(d) is a parallel connection of the models. Here, the backlash model has fixed parameters $\left\{m_L,m_R,c_L,c_R\right\}$ which are learned in an initial training session. Subsequently, the LSTM is trained to model the error of the backlash model. The equations of this hybrid model are

$$v_{HB2}=f_{\text{lstm}}\left(u\right)+f_b\left(\left(m_L,m_R,c_L,c_R\right),u\right)$$ (7)

III. Experiments

We performed experiments on the clinical single-tendon catheter of Fig. 2 using the custom drive system of Fig.4. The catheter is made up of a polymer tube with laser-cut tubing embedded inside. It is comprised of a stiff proximal section and a flexible, steerable distal section. The non-steerable proximal section was constrained using a table-mounted stabilizer block as shown. The steerable distal segment, which allows for bending, measures 6 cm in length. A stainless-steel tendon wire runs through a channel near the outer polymer layer of the catheter, enabling controlled movement of the flexible section.

Fig. 4.

Catheter drive system.

The proximal end of the tendon was wrapped around a pulley integrated with a cartridge to which the sheath is attached. The pulley was driven thorugh a bevel gear by a DC motor (Faulhaber 2342012CR) via servo drive (Accelnet ACJ-055–09). A PC running ROS 2 was used to control the tendon with the servo drive reporting tendon displacement and tension. An electromagnetic sensor (Ascension trakSTAR) was used to track tip position and orientation with accuracy of ±1.4mm and ±0.35°.

When initializing the robot, tendon slack can cause significant trial-to-trial variation. Tendon pretension, however, creates a precurvature in the sheath. To balance these effects, we conducted preliminary experiments to determine the minimum pretension producing repeatable results.

Experimental trials were conducted using varying levels of pretension. To quantify the repeatability of the catheter’s performance, the average standard deviation across five trials was calculated for each pretension level. The minimum pretension was chosen as the lowest pretension value that yielded a standard deviation less than 0.5°. This minimum pretension value was 4.92N corresponding to an initial bending angle of 20°. The tension $\left(F\right)$ was calculated using the equation $F=I{K}_t/r$, where $I$ represents the measured current, $K_t$ denotes the torque constant, and $r$ is the radius of the motor.

We also observed that, with further flexure of the sheath, it would never return to this initial bending angle, but instead would relax to a larger angle, 30°. The initial flexure, occurring only once during the catheter’s initialization, is not modeled since we focus on the catheter’s repeatable behavior flexing over a range of angles greater than or equal to this minimum angle of 30°. To avoid including data from this nonrepeatable initial flexure in the modeling data, a preliminary commanded motion was performed corresponding to a 0.1 Hz sinusoidal bending to a maximum of 90° and then back to the relaxed bending angle of 30°.

Following these initialization steps, custom motions were imposed as tendon displacements to gather data for modeling purposes. The collected data was sampled at 25 Hz to ensure uniformity across the modeling datasets.

Training for all the models included the use of a mean square error (MSE) loss function and the Adam optimizer in the PyTorch framework, with a learning rate set to 0.001. Each model was trained with a data sequence length of 50 and a batch size of 16. The early stopping strategy was adopted to select the best model based on validation performance within 100 epochs.

Four sets of experiments were performed. The first set investigated hysteresis properties, the second set compared alternative kinematic mappings, and the third set evaluated the effect of training motions on model performance. The final set comprehensively compared all models based on what was learned in the preceding experiments. Each set is detailed below.

A. Hysteresis Properties

We first investigated the hysteresis properties for the catheter of Fig. 2. These characteristics include frequency dependence, the selection of kinematic input variable (tendon displacement versus tendon tension), and repeatability.

We examined frequency dependence by comparing the catheter’s bending angle response to tendon displacement and tension across five frequencies of decaying sinusoidal signals. Figs. 5(a) and (b) show that the output amplitudes decrease to a small degree with increasing frequency, indicating a slight frequency dependence.

Fig. 5.

Properties of hysteresis for the clinical cardiac catheter. (a) Bending angle versus tendon displacement for 5 decaying sinusoidal input frequences. (b) Bending angle versus tendon tension for 5 decaying sinusoidal input frequences. (c) Bending angle versus tendon displacement for 5 trials with an input frequency of 0.5 Hz.

Furthermore, comparing the hysteresis loops of Figs. 5(a) and (b) reveals that using tendon displacement as the input provides a smoother, more-uniform loop than tendon tension. Anticipating that this loop will be easier to model, we use tendon displacement as the input variable for kinematic modeling.

As shown in Fig. 5(c), we also examined the repeatability of hysteresis by performing multiple trials with the same tendon motion. These results confirm that the kinematic behavior is repeatable and so is amenable to the proposed modeling approach.

B. Model Accuracy versus Choice of Output Variable

For the three output variables shown in Fig. 2, it is expected that they are functionally dependent on each other. For example, if the assumption of constant curvature holds, then the tangent angle $\theta$ and tip position coordinates $x$ and $y$ are related by

$$\begin{gathered} x=\frac{l}{\theta }\text{sin}\left(\theta \right), \\ y=\frac{l}{\theta }(1-\text{cos}(\theta \left)\right) \end{gathered}$$ (8)

where $l$ is the length of the bending section of the catheter. Assuming these variables are related to each other, we should be able to train a hysteretic model for one output variable and then compute the other two using a non-hysteretic functional model.

We used preliminary experiments to model the relationship between $\{x,y,\theta \}$ and to compare the relative accuracy of the hysteretic models in mapping directly to the three output variables. These experiments showed that, while $\{x,y,\theta \}$ are not related by the constant curvature assumption, their relationship can be accurately modeled using a quadratic equation as shown in the block of Fig. 6(b).

Fig. 6.

Direct and Composite Mappings to tip coordinates, $x$ and $y$. (a) Direct mapping. (b) Composite mapping. (c) Example motion comparison of the direct and composite mappings using the backlash model.

Furthermore, we determined that the hysteretic mapping from tendon displacement to $x$ and $y$ is more nonlinear than the map from tendon displacement to $\theta$. While the backlash and PI models proved adept at reproducing the mapping from tendon displacement, $d$, to $\theta$ (Fig. 6(a)), they struggled to directly predict $x$ and $y$. For these models, we found that more accurate maps to $x$ and $y$ could be produced by the composite map of Fig. 6(b). The hysteretic model maps first to $\theta$ and then a quadratic function is used to map $\theta$ to $x$ or $y$. An example motion prediction comparing the two mappings for the backlash model is shown in Fig. 6(c).

Since neural networks are particularly adept at capturing nonlinear relationships, it was observed as expected that the mappings of Fig. 6(a) and (b) produced equivalent accuracy for all neural network and hybrid models tested.

C. Modeling Clinical Motions

In this paper, two types of motions are considered for hysteretic kinematic modeling: sinusoidal and point-to-point motions. Sinusoidal motions of varying amplitude and frequency are commonly employed when assessing hysteresis [7], [8], [20]. In clinical settings, such periodic motions are used to create oscillating catheter flexion for scanning motions of an imaging probe, e.g., as discussed in [8].

The most common catheter-based procedures, however, such as the example of pulmonary vein isolation in Fig. 1, require the catheter to move between a sequence of tip positions with long pauses at each location. These point-to-point motions can be modeled as stationary time periods interspersed with segments of constant velocity.

We collected motion data of both types for model training, validation and testing as described below.

1). Sinusoidal Motion Data:

Tendon displacements, $d\left(t\right)$, given by decaying sinusoids were used to represent periodic motions. Data was collected for seven frequencies and three offsets as given below.

$$d\left(t\right)=d_{\text{max}}{e}^{-\tau t}\left(\text{sin}\left(2\pi ft-\frac{\pi }{2}\right)+c\right)+d_{\text{offset}},$$ (9)

$$\begin{array}{lll}t\hfill & \in \hfill & \left\{0,\dots ,t_{\max }\right\}\hfill \\ f\hfill & \in \hfill & \left\{0.1,0.15,0.2,0.3,0.4,0.45,0.5\right\}\text{Hz}\hfill \\ \tau \hfill & =\hfill & f\log \left(7/6\right)\text{Hz}\hfill \\ d_{\text{max}}\hfill & =\hfill & 3\text{mm}\hfill \\ \left\{c,d_{\text{offset}}\right\}\hfill & \in \hfill & \left(\left\{1,0\text{mm}\right\},\left\{0,3\text{mm}\right\},\left\{-1,6\text{mm}\right\}\right)\hfill \end{array}$$ (10)

The maximum time, $t_{\text{max}}$, and decay rate, $\tau$, were selected so that data for seven full cycles could be collected. The maximum tendon displacement, $d_{\text{max}}$, was selected to correspond to a tip bending angle of ${90}^{\circ }$. The three pairs of $\left\{c,d_{\text{offset}}\right\}$ produce oscillations in which the simusoidal maxima are all at the maximum tendon displacement, the sinusoidal means are all at the mean tendon displacement, and the sinusoidal minima are all at the pretension tendon displacement, respectively.

The frequencies $f\in \left\{\mathrm{0.1,0.2,0.4,0.5}\right\}\text{Hz}$ were used for training while $f=0.3\text{Hz}$ was used for validation and $f\in \left\{\mathrm{0.15,0.45}\right\}\text{Hz}$ were used for testing.

2). Point-to-Point Motion Data:

Point-to-point motions were composed of 20 segments which alternated between constant nonzero velocity and zero velocity. The nonzero velocity segments performed tendon displacements connecting 10 positions randomly selected from the range, $d\in \left\{\mathrm{0,6}\right\}\text{mm}$. Each displacement was performed at a constant velocity randomly selected in the range 0.5–6mm/sec and followed by a random-duration zero-velocity segment. The tendon displacements $d\left(t\right)$ are formulated as below:

$$d\left(t\right)=\left\{\begin{array}{ll}{v}_i\left(t-t_{2(i-1)}\right)+d_{i-1},& t_{2(i-1)}\le t<t_{2(i-1)+1}\\ d_i,& t_{2(i-1)+1}\le t<t_{2i}\end{array}\right.$$ (11)

Here, $i=\mathrm{1,2},\dots ,10,v_i\in \left[\mathrm{0.5,6}\right]\text{mm}/\text{sec}$ are constant velocities, $d_0=0$, and $d_i\in \left[\mathrm{0,6}\right]\text{mm}$ are the displacements and $t_{2(i-1)+1}$ are the time points at which the motion changes from constant velocity to zero velocity. The duration of nonzero velocity segments $t_{2(i-1)+1}-t_{2(i-1)}=\frac{d_i-d_{i-1}}{v_i}$. The duration of stationary segments $t_{2i}-t_{2(i-1)+1}\in \left[\mathrm{1,5}\right]\text{sec}$. Also, to ensure that the data was sufficiently exciting, each tendon displacement, $\Delta d$, was forced to satisy $0.5mm<\Delta d<3mm$.

Twenty trials were performed with fifteen of these trials used for training and the remaining five used for validation. Twenty additional trials were performed for testing in which stationary segments had random durations of 1–10 seconds and tendon displacements were not constrained.

During initial training experiments, it was observed that minimizing mean square error is not sufficient for producing well-behaved kinematic models. It is also necessary that the model produce a constant output for a constant input. Otherwise, the model could predict that the catheter tip could continue moving in a neighborhood around the actual configuration. The structure of the backlash model (and the hybrid models constructed from it) is such that it inherently incorporates the constant-input constant-output property. In contrast, neural network models can produce fluctuating outputs under static input conditions due to their tendency to overfit. To overcome this problem, we introduced a new term in the loss function to supplement the mean square error (MSE) term. The new term penalized the ratio of the standard deviation of model output, $\sigma \left(\hat{v}\right)$, to standard deviation of model input, $\sigma \left(u\right)$, computed over the data sequence length of 50 samples. In the loss function, $L$, below, $w_1$ and $w_2$ are scalar weighting functions and $ϵ>0$ is a small constant intended to prevent a zero denominator.

$$L=w_1\text{MSE}(v,\hat{v})+w_2\frac{\sigma \left(\hat{v}\right)}{\sigma \left(u\right)+ϵ}$$ (12)

The weighting factors were selected as $w_1=1$ and $w_2\in \left\{0,{10}^{-4}\right\}$, based on empirical evaluation. This new term penalized changes in the output when the input was constant and resulted in much improved performance during stationary portions of point-to-point motions. During model training, both values of $w_2$ were compared and the model with lowest error is reported.

3). Training versus Testing Motion Type:

To investigate if the models would be sensitive to the type of motion that they were trained on, we compared how models trained on one type of motion were able to predict the other type of motion.

Fig. 7 displays the results of our evaluation, comparing the performance of LSTM models trained on these two different datasets. When tested on a sinusoidal motion, as shown in Fig. 7(a), the tip angle prediction of the model trained on sinusoidal data matches well with experimental data, whereas the model trained on point-to-point motions fails to capture the sinusoidal patterns accurately. Conversely, when applied to a point-to-point motion, as shown in Fig. 7(b), the model trained on sinusoidal data struggles to handle the stationary periods, while the model trained on point-to-point data successfully maintains a constant output during these periods.

Fig. 7.

Effect of training motion data on model performance. (a) Comparison of LSTM models trained on sinusoidal and point-to-point motions when tested on a sinusoidal motion. (b) Comparison of LSTM models trained on sinusoidal and point-to-point motions when tested on a point-to-point motion.

We also trained models on both types of motions, but found that the resulting cross-trained models were not as accurate as models trained and tested on a single motion type. Inverse kinematic modeling produced similar results.

D. Model Comparison

The seven hysteresis models of Section II were compared with the memoryless FNN model with respect to reproducing the catheter’s forward and inverse kinematics using the test data. As noted above, modeling errors are minimized if training and testing are performed on the same motion type (sinusoidal or point-to-point) so only those results are reported here. Forward and inverse maps for the three output variables, $\{\theta ,x,y\}$ were modeled. Since modeling errors were comparable for the three outputs, only results for $\theta$ are reported here.

Fig. 8 presents a comparison of forward and inverse kinematic modeling errors for all the models. The figure includes the mean, standard deviation and maximum error for each model computed over the test trials. One-sided Wilcoxon signed-rank tests were conducted to determine whether the errors produced by the models were statistically greater or less than each other. Table I summarizes the results of these tests.

Fig. 8.

Kinematic modeling error for tip angle, $\theta$. (a) Forward kinematic map - mean, standard deviation and maximum RMS error. (b) Inverse kinematic map - mean, standard deviation and maximum RMS error. Maximum errors are noted above standard deviation bars.

TABLE I.

Statistical Equivalence of Models.

		FNN	Backlash	PI	LSTM	GRU	FNN-HIB	HB1
Backlash	FS / IS	< / <
	FP / IP	< / <
PI	FS / IS	< / <	> / <
	FP / IP	< / <	> / >
LSTM	FS / IS	< / <	< / <	< / <
	FP / IP	< / <	< / ~	< / <
GRU	FS / IS	< / <	< / <	< / <	~ / ~
	FP / IP	< / <	~ / ~	< / ~	> / ~
FNN-HIB	FS / IS	< / <	< / <	< / ~	> / >	> / >
	FP / IP	< / <	> / >	~ / >	> / >	> / >
HB1	FS / IS	< / <	< / <	< / ~	~ / >	~ / >	< / ~
	FP / IP	< / <	< / ~	< / <	~ / ~	< / ~	< / <
HB2	FS / IS	< / <	< / <	< / <	~ / ~	~ / ~	< / <	~ / <
	FP / IP	< / <	< / <	< / <	~ / <	< / <	< / <	~ / <

A “~” indicates equivalent modeling error, while “<” or “>” indicate that the model in the first column on the left has significantly less or greater error than the model in the first row, with a significance level of $p<0.05$. FS = Forward model for sinusoidal motions.IS = Inverse model for Sinusoidal motions. FP = Forward model for Point-to-point motions. IP = Inverse model for Point-to-point motions.

In Fig. 8, the leftmost model with the highest error is the FNN. Since it cannot reproduce hysteresis, its prediction represents an average of the hysteresis loop. For the forward kinematic mapping, this results in it overestimating an increasing bending angle by an average of 2.5° for sinusoidal motions and 2.25° for point-to-point motions. For decreasing bending angles, it underestimates the bending angle by the same amounts. The inverse FNN models similarly produce an averaged prediction of tendon displacement.

For sinusoidal motions, all of the neural network models except FNN-HIB and HB1 provide excellent and statistically equivalent performance. Of these, the LSTM and GRU models provide the smallest mean and maximum errors considering both forward and inverse kinematics. For example, the LSTM model reduced forward mean and maximum prediction errors to 17% and 23% of the nonhysteretic FNN model predictions. Inverse mean and maximum prediction errors were reduced to 23% and 66% of the nonhysteretic model predictions.

For point-to-point motions, hybrid model HB2 provided the best error reduction, which was not as substantial as the reduction observed for sinusoidal motions. Specifically, the HB2 model reduced forward mean and maximum prediction errors to 43% and 55% of the nonhysteretic FNN model predictions. Inverse mean and maximum prediction errors were reduced to 39% and 68% of the nonhysteretic model predictions. The LSTM and hybrid HB1 models were second best and provided statistically equivalent error reduction. Of note, while the backlash model produced slightly larger errors than the LSTM and hybrid models, the difference may be acceptable in practice.

Fig. 9 compares the HB2 and FNN-HIB models with the non-hysteretic FNN model for a point-to-point motion trial. The effect of neglecting hysteresis with the FNN model is most obvious during the stationary periods. For the forward kinematics of Fig. 9(a), the non-hysteretic FNN model tends to overestimate tip angle when angle is increasing and underestimate it when tip angle is decreasing. For the inverse kinematics of Fig. 9(b), this relationship of over- and under-estimating tendon displacement is reversed. The FNN-HIB model adds an input buffer to the FNN model enabling it to reduce mean and maximum errors to 66% and 65% in the forward model. While mean error of the inverse model is also reduced to64% of the FNN model, the maximum error of the FNN-HIB model exceeds that of the FNN model.

Fig. 9.

Forward and inverse model predictions for a point-to-point motion for FNN, HB2 and FNN-HIB models. (a) Forward kinematic model output versus time. (b) Inverse kinematic model output versus time. (c) Forward kinematic model error versus time. (d) Inverse kinematic model error versus time.

IV. CONCLUSIONS

This paper investigated the hysteretic kinematic modeling of a single-tendon catheter, offering a comparative analysis of eight models, including two classical hysteresis models, four neural networks, and two hybrid models. Among the neural network models, the feedforward model is used as the non-hysteretic baseline since it can model complex nonlinear functions, but not time-dependent behavior.

In contrast to the neural network models, it was observed that the classical models were not as effective at modeling nonlinear behavior. This meant that while they could be used to directly produce the map from tendon displacement to bending angle, a two part mapping was needed to accurately produce tip positional coordinates. This additional complexity can be be viewed as a disadvantage of the classical models.

On the other hand, the classical models inherently exhibited the property that a constant input produced a constant output. In contrast, RNNs are not constrained in this way and the loss function had to be modified to explicitly penalize output motion when the input was constant.

The impact of anticipated motion type on model training and performance was examined. While a standard approach in modeling hysteresis is to perform training and testing with sinusoidal motions, the clinical applications that employ such motions are limited. Many applications consist of positioning the catheter tip at a sequence of positions. At each position, the catheter is held stationary.

It was observed that smaller modeling errors were obtained for models tested on the same type of motion they were trained on. In addition, errors were lower for sinusoidal motions than for point-to-point motions. This is to be expected since if one were to approximate point-to-point motions as sums of sinusoids, the approximations would require a much larger ranges of frequencies than those used in clinical sinusoidal motions. It is not unreasonable that models trained on a small range of frequencies can reproduce motions with those frequencies better than motions containing a much greater range of frequencies. Similarly, models trained on a larger range of frequencies will produce a good overall fit to motions with many frequencies, but not as good a fit for motions with a few frequencies when compared with models trained specifically on those frequencies.

For those applications that do involve sinusoidal motions, the LSTM and GRU models provide excellent forward and inverse kinematic models. The hybrid models also perform comparably, but since they are more complex, there is no advantage to using them. In contrast, the FNN-HIB model provided slightly worse behavior while also being more complex since its implementation requires an input buffer. The backlash and PI models were inferior to all of the hysteretic neural network models.

For applications using point-to-point motions, the HB2 model edged out the LSTM and HB1 models in reducing error. Interestingly, the backlash model performs almost as well. Its simplicity, using only four parameters, suggests its potential appropriateness for some point-to-point applications. If, however, the output variable is selected to be tip coordinate $x$ or $y$, instead of $\theta$, then the backlash model does require the composite mapping of Fig. 6 to be used. The classical PI model and the FNN-HIB model provide inferior performance to all of the other models for point-to-point motions.

One limitation of this study was that the proximal portion of the catheter was constrained to be straight. While for transfemoral procedures, much of the proximal length is approximately straight, the distal portion inside the heart is often deflected by an outer sheath. While this deflection will change the hysteresis properties of the catheter, it is unlikely to change the relative performance of competing hysteresis models and so was not considered here. In practical use, however, proximal curvature should be considered (see, e.g., [7]).