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Shaft-Configuration-Adaptive Catheter Tip Position Estimation via Motor-History Conditioned Residual Learning

Peihan Zhang, Michael C. Yip, Ankur Kapoor, Young-Ho Kim

arXiv:2609.26304v1cs.ROeess.SY

TL;DR

Task-dependent shaft configurations and actuation history complicate catheter tip estimation. The paper uses an initial sweep to calibrate a geometric backbone and encode shaft context for a motor-history-conditioned GRU residual estimator. Across evaluated bent-shaft recordings, it reports lower RMSE than both geometric baselines.

  • Problem

    Unknown task-dependent shaft configurations alter tendon routing and distal response, while friction, slack, and compliance make tip position depend on preceding actuation.

    Source: §I · p. 1

  • Method

    An initial multidirectional sweep calibrates a geometric backbone and encodes shaft context, which conditions a GRU residual that uses motor history for subsequent tip estimation.

    Source: §I · p. 2

  • Results

    3.3 mm mean per-run 3-D tip-position RMSE was reported, a 53% reduction relative to PDL and 59% relative to PCC.

    Source: §V · p. 7

  • Takeaways & Limitations

    The initial-sweep shaft-configuration context is the dominant contributor to accuracy in the reported ablations, while residual-learner choice and torque add smaller gains.

    Source: D. Discussion · p. 6, §V · p. 7

    Initial calibration requires reference tip positions, and shaft configuration is assumed fixed afterward; repositioning invalidates the context and geometric calibration.

    Source: D. Discussion · p. 6

Abstract

from arXiv · show

Tendon-driven continuum manipulators are widely used in medical applications, where accurate tip-position estimation is essential for precise navigation and instrument positioning. However, patient anatomy and procedural setup impose task-dependent unknown shaft configurations, while friction, slack, and compliance introduce hysteresis, making tip estimation challenging. This paper presents a motor-history-conditioned gated recurrent unit (GRU) residual estimator for three-dimensional catheter tip estimation without direct shaft-configuration sensing. First, an initial multidirectional sweep strategy is applied to calibrate a geometric catheter model backbone, and encode the motor-angle and drive-torque response into a shaft-configuration context vector. During subsequent motion, the context conditions a GRU that predicts a task-space residual correcting this backbone, relying on motor measurements alone. The context remains fixed for the current shaft configuration, while the recurrent state captures the evolving actuation history. Across four disposable intra-cardiac echocardiography catheters and 16 bent shaft configurations, the method achieves 3.3mm open-loop tip RMSE, a 59% reduction relative to the constant-curvature baseline.

I. INTRODUCTION

Task-dependent shaft bends and actuation hysteresis make catheter tip estimation difficult, while direct shaft sensing may require unavailable sensors or usable images. The paper combines a calibrated geometric model with a motor-history-conditioned GRU residual and adapts it to shaft configurations through an initial sweep.

I. INTRODUCTION

  • Unknown shaft bends change tendon routing and distal response, while friction, slack, and compliance make tip position depend on preceding actuation.Direct shape sensing can require added sensors or usable image observations that are not always accessible on the device.Source: §I · p. 1
  • The estimator combines a geometric nominal prediction with a GRU residual conditioned on shaft context and motor-angle and drive-torque history.After the initial sweep, estimation uses motor measurements alone, without shaft-configuration sensing.Source: §I · p. 2
  • The same recurrent-network weights apply across configurations and catheters of the same device family because unit- and configuration-specific behavior is absorbed by the context and geometric parameters.Source: §I · p. 2
  • A multidirectional sweep both calibrates the geometric model and encodes shaft context, so adapting to a new configuration requires a repeated sweep rather than retraining.Source: §I · p. 2
  • 3.3 mm open-loop tip-position RMSE, with a 59% reduction relative to baseline, was achieved in validation across catheter units and shaft configurations.The validation used four disposable catheter units and configurations with unknown bending directions and magnitudes.Source: §I · p. 2

II. RELATED WORK

Prior approaches address shaft configuration and actuation-history dependence through sensing, proximal-signal compensation, or temporal models. The paper combines both adaptation needs in a motor-measurement-driven estimator that uses a sweep rather than retraining for a new configuration.

II. RELATED WORK

  • Shape-sensing approaches reconstruct or observe shaft geometry, [9], while routing sensors can inform transmission compensation,.Source: §II · p. 2
  • Many clinical catheters lack integrated fibers or a usable shaft view [26], and temporal models are typically trained for one configuration [28].The proposed approach instead encodes an initial multidirectional sweep as shaft context and calibrates the geometric model from that same sweep; a new configuration requires a repeated sweep rather than retraining.Source: §II · p. 2
  • Motor-side methods infer transmission effects from proximal signals [3], [4], while analytical and data-driven temporal models represent actuation-history dependence,,,, [17],, [18].Source: §II · p. 2

III. METHOD

The method estimates catheter-tip position relative to a frame near the bending section, focusing on motor-driven bending response. For a fixed shaft configuration, it combines a geometric backbone with a residual predicted from motor history and initial-sweep context.

A. Problem formulation

  • The formulation considers four catheter degrees of freedom and estimates tip position in a frame attached near the proximal end of the bending section, focusing on the response to two motor-angle inputs.Source: A. Problem formulation · p. 3
  • For a fixed shaft configuration, the estimator combines a geometric backbone with a motor-history-conditioned residual, with the initial sweep supporting backbone calibration and context encoding.Source: §III · pp. 2–3, A. Problem formulation · p. 3

B. Geometric backbone and init sweep strategy

The geometric backbone maps motor angles to a nominal tip position and is calibrated for the current shaft configuration from an initial multidirectional sweep. The sweep also supplies context for the residual estimator.

B. Geometric backbone and init sweep strategy

  • The single-section piecewise-constant-curvature (PCC) and PDL models are calibrated for the current shaft configuration from the initial multidirectional sweep.Source: B. Geometric backbone and init sweep… · p. 3
  • The per-direction linear model (PDL) fits tip inclination, displacement, and azimuth along each bending ray, then interpolates between rays to represent direction-dependent responses.Its per-ray response functions are linear in their coefficients, while the motor-input dependence and resulting three-dimensional reconstruction are nonlinear.Source: B. Geometric backbone and init sweep… · p. 3
  • With the shaft configuration held fixed, outward-and-return sweeps pair motor angles with EM reference tip positions to calibrate the backbone and provide motor-angle and drive-torque history for context encoding.The EM measurements are used for calibration, training, and evaluation, but not as input to the context encoder or subsequent estimator; fitted parameters and context remain fixed during subsequent motion.Source: B. Geometric backbone and init sweep… · p. 3, Figure 2 · p. 3

C. Motor-history-conditioned GRU residual

The estimator uses a fixed shaft-configuration context with a recurrent state that evolves during motion. The GRU uses motor-derived features and the geometric model’s nominal prediction to estimate a residual.

C. Motor-history-conditioned GRU residual

  • The GRU predicts a tip-position residual from the fixed shaft-configuration context and evolving motor history.The geometric prediction supplies the nominal response that the residual corrects.Source: C. Motor-history-conditioned GRU… · pp. 3–4
  • A context encoder summarizes the normalized, subsampled initial-sweep sequence into a fixed-length vector.The sweep features collect motor angles and drive-reported torques.Source: C. Motor-history-conditioned GRU… · p. 4
  • Each recurrent step uses the nominal prediction and motor-derived channels, including motor angles, angular differences, and drive torques.The normalized feature and repeated context are supplied to the recurrent core.Source: C. Motor-history-conditioned GRU… · p. 4
  • The output head rescales the network output, and the hidden state is initialized at the start of each motion sequence.The context vector and geometric parameters remain fixed while the recurrent state evolves at each sample.Source: C. Motor-history-conditioned GRU… · p. 4

A. Implementation Details

The implementation uses motor-side measurements for estimation and electromagnetic tracking for reference positions and poses. Data span four catheters and 16 bent shaft configurations, with held-out motion segments used for testing.

A. Implementation Details

  • Two distal and proximal electromagnetic tracking sensors provide reference poses for calibration, training, and evaluation; estimation uses motor-side measurements alone.The catheter has two tip-bending motors with measured angles and drive-reported torque signals.Source: A. Implementation Details · p. 4
  • Each catheter contributes 48 runs across 16 bent configurations, with three repeated runs per setting; each run begins with an eight-ray sweep.The straight configuration is a reference, and the reported results use the bent configurations.Source: A. Implementation Details · p. 4, Figure 4 · p. 4, Figure 3 · p. 4
  • Within-catheter evaluation holds out all repeats of one shaft setting; only its initial sweeps support calibration, while the motion segment is reserved for testing.For leave-one-catheter-out evaluation, all recordings from the held-out catheter are excluded from model-development steps.Source: A. Implementation Details · p. 4
  • The training loss masks padded positions, and the reported materials include a tip-position RMSE table and a PCC, PDL, and PDL+GRU comparison plot.The plot shows medians, quartiles, and outliers.Source: A. Implementation Details · p. 4, Table I · p. 5, Figure 5 · p. 5
  • Tip accuracy is mean per-run 3-D position RMSE with equal run weights, and learned-model errors are averaged over three seeds.Position-error statistics are rounded to one decimal place and percentages to whole numbers.Source: A. Implementation Details · p. 5

B. Catheter-tip estimation results

PDL+GRU lowers tip-position RMSE against both geometric baselines, with lower errors across the reported distribution and motor-input magnitudes. Large motor excursions remain more difficult, and a random-motion example shows residual local deviations.

1) Tip estimation accuracy:

  • 3.3 mm pooled mean per-run RMSE for PDL+GRU compares with 8.1 mm for PCC and 7.0 mm for PDL across all recordings.The reductions are 4.7 mm (59%) relative to PCC and 3.7 mm (53%) relative to PDL.Source: 1) Tip estimation accuracy · p. 5

2) Error distribution and motor-input magnitude:

3) Random motion example:

Over the displayed 0–170 s of motion, PDL+GRU maintains lower RMSE than PCC and PDL, with smaller between-run dispersion and no pronounced drift.

3) Random motion example:

  • Over 0–170 s, PDL+GRU mean run-bin RMSE increases from 2.3 mm in the first 10 s bin to 3.5 mm in the last.The corresponding ranges are 6.9–8.1 mm for PCC and 5.2–7.2 mm for PDL.Source: 4) Error over motion time · p. 5, Figure 8 · p. 5
  • PDL+GRU has a smaller between-run inter-quartile range than either geometric baseline at every displayed box group.Fig. 8 shows mean 10 s run-bin RMSE curves and between-run quartiles and medians.Source: 4) Error over motion time · p. 5, Figure 8 · p. 5
  • The reported time-resolved results show no pronounced drift over the displayed motion.The section reports lower typical error over time alongside the RMSE and between-run dispersion comparisons.Source: 4) Error over motion time · p. 5

C. Ablation studies

The ablations compare residual learners, motor-torque and sweep-context inputs, and transfer to held-out catheters. The reported results favor GRU and especially sweep context, while showing modest-penalty transfer within the device family.

C. Ablation studies

1) Residual-estimator comparison:

  • PDL+GRU achieves 3.3 mm pooled RMSE versus 3.6 mm for PDL+MLP, a paired reduction of 0.3 mm (7%; 95% interval 0.2–0.3 mm).PDL+GRU lowers mean error for each catheter and improves 81% of recordings; both learners use the same backbone and history/calibration inputs.Source: 1) Residual-estimator comparison · p. 6

2) Motor torque and initial-sweep context:

  • Adding sweep context reduces pooled RMSE by 2.1 mm with torque and 2.2 mm without torque; torque reduces it by less than 0.1 mm with context.Pooled RMSE is 5.5 mm without torque or context, 5.4 mm with torque alone, 3.4 mm with context alone, and 3.3 mm with both.Source: 2) Motor torque and initial-sweep… · p. 6, Figure 10 · p. 6

3) Model adaptation to new catheters:

  • 3.7 mm pooled RMSE under leave-one-catheter-out testing compares with 3.3 mm under within-catheter setting-held-out evaluation.Held-out catheter sweeps refit the geometric model and compute context only; they do not train or fine-tune the GRU.Source: 3) Model adaptation to new catheters · p. 6, Figure 11 · p. 6

D. Discussion

The discussion attributes the estimator’s gains chiefly to configuration-specific context, while noting calibration and fixed-shaft assumptions. The conclusion reports 3.3 mm mean per-run 3-D RMSE and identifies larger-scale and prospective evaluation as future needs.

D. Discussion

  • 7.0 mm with PDL fell to 3.3 mm with PDL+GRU, while PDL+GRU’s advantage over PDL+MLP was 0.3 mm.Source: D. Discussion · p. 6
  • Context with torque gave a 2.1 mm gain, whereas torque added below 0.1 mm when context was available, making configuration-specific calibration more influential in this setup.Source: D. Discussion · p. 6
  • The shaft configuration is assumed fixed after the sweep; repositioning invalidates the context and geometric parameters, requires a new sweep, and is not detected online.Source: D. Discussion · pp. 6–7
  • Larger motor-input magnitudes were associated with higher run-bin RMSE, and future work should address reduced calibration budgets and prospective evaluation on additional catheters.Source: D. Discussion · p. 7, §V · p. 7

V. CONCLUSION

  • 3.3 mm mean per-run 3-D tip-position RMSE represents a 53% reduction relative to PDL and 59% relative to PCC.Source: §V · p. 7
  • Leave-one-catheter-out evaluation gave 3.7 mm pooled RMSE; transfer was evaluated retrospectively on four units from one device family.Source: §V · p. 7, D. Discussion · p. 7
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