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Magnetic based In-situ Self 3D Pose Estimation for a Modular Soft Tendon-Driven Continuum Robot via IMU-Fusion

Zheng Cao, Guo Ning Sue, Xiangyun Bu, David Quinn, Junzhe Hu, Carmel Majidi

arXiv:2609.39950v1cs.ROcs.SYeess.SPeess.SY

TL;DR

Continuum-robot deformation makes accurate configuration estimation challenging, and vision-based feedback depends on unobstructed visibility. The paper combines embedded attitude and magnetic sensing in a modular system; across four scenarios and 20 trials, it reports 10.0 mm backbone RMSE and 14.9 mm tip RMSE, with closed-loop control demonstrated under disturbance.

Abstract

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Continuum robots are well suited for gentle manipulation because of their inherent compliance and ability to adapt to complex environments. However, their continuously deformable structure makes accurate configuration estimation challenging, particularly when external vision systems are unavailable or obstructed. In this work, we present an embedded pose sensing framework that combines inertial measurement units (IMUs) and active magnetic fields to estimate the robot configuration without relying on external cameras. The angular measurements from the IMU and magnetic-field references are fused to improve local orientation estimation and reduce accumulated orientation error during operation. This pose sensing scheme achieves an update rate of 16.7~Hz, allowing real-time feedback. The proposed system is experimentally validated through closed-loop control, where the estimated robot configuration is used to maintain the end-effector at a desired position while interacting with an object. These results demonstrate the potential of distributed magnetic--inertial sensing for real-time pose estimation and closed-loop control of continuum robots.

I. INTRODUCTION

Continuum robots’ distributed, continuous deformation complicates configuration estimation, while model-based estimates can be sensitive to material behavior and external loading. The paper motivates embedded magnetic–inertial sensing and presents a modular system for real-time reconstruction and feedback.

I. INTRODUCTION

  • Distributed deformation makes continuum-robot configuration harder to estimate than rigid-link configuration.Accurate feedback matters for closed-loop control.Source: §I · p. 1
  • The proposed modular magnetic–inertial architecture combines onboard attitude estimates and magnetic measurements for backbone reconstruction at 16.7 Hz.Continuous attitude polling and intermittent magnetic corrections support configuration updates without external visual tracking during deployment.Source: §I · p. 1
  • The work validates shape estimation under varied deformation and contact conditions and closed-loop tip-height control under an external disturbance.The stated contributions include modular embedded 3D shape reconstruction and a sensing and communication scheme for 16.7 Hz updates.Source: §I · p. 1

A. Pose and Configuration Estimation

  • Model-based estimates can degrade under nonlinearities, hysteresis, friction, and external loading, while vision requires unobstructed visibility.Strain sensing measures local deformation but may need complex models to reconstruct global shape; accelerometers cannot resolve full orientation and are affected by dynamic acceleration.Source: A. Pose and Configuration Estimation · pp. 1–2, §I · p. 1

B. Embedded Inertial and Magnetic Proprioception

Prior embedded inertial and magnetic methods offer proprioceptive information, but distributed magnetic–inertial systems face update-rate and scalability limits. The paper presents a modular platform integrating sensing and communication hardware in each robot segment.

B. Embedded Inertial and Magnetic Proprioception

  • Distributed inertial measurements have been used for shape and end-effector estimation, while magnetic methods infer local bending or learn shape mappings, and.Magnetic sensing supplies a directional reference but may require dedicated sources, robot-specific calibration, or learned mappings.Source: B. Embedded Inertial and Magnetic… · p. 2
  • Prior magnetic–inertial orientation sensing [13] remains limited in update rate and scalability across long modular robots.The proposed architecture combines local orientation estimates from multiple segments to reconstruct 3D configuration for real-time feedback.Source: B. Embedded Inertial and Magnetic… · p. 2

A. Design

  • The robot uses nine serial TPU 90A modules with integrated joints, designed for assembly, disassembly, replacement, and adjustable length.Each module carries a PCB with an IMU, microcontroller, planar coil, and switching MOSFET; boards share an I2C bus and stream measurements to a Raspberry Pi 5 running ROS 2.Source: A. Design · p. 2
  • The IMU tracks attitude without magnetic input during coil activation, while magnetic measurements correct arbitrary initial heading and subsequent drift.The PCB coil’s asymmetric outline breaks axial field symmetry; a 40 ms pulse at 3.3 V produces approximately 100 µT at the neighboring segment, against 0.6 µT noise per axis.Source: A. Design · p. 2

B. Sensing

Each joint is modeled as a fixed-radius spherical pivot without axial twist, and a learned mapping relates magnetic measurements to joint direction. Training samples cover the reachable spherical workspace and use inertial measurements as labels.

1) Joint Model and Training:

  • Each joint is modeled as a fixed-radius spherical pivot without axial twist, representing relative pose by a unit direction rather than prescribing whole-backbone curvature.The pivot radius is 10 mm, and total pivot spacing is 29 mm.Source: 1) Joint Model and Training · pp. 2–3
  • The sensing model assumes the upper-segment IMU and magnetometer move over a hemispherical surface around a virtual pivot; a fitted sphere has 0.19 mm RMSE.Two cameras track world and joint tags while randomly targeted orientations guide sampling across the workspace.Source: Figure 3 · p. 3
  • Training samples target orientations across a spherical cap just below the 30° mechanical limit, with Fibonacci cells and quotas used to improve coverage.Colatitude is sampled uniformly in cosine and azimuth uniformly over [−π, π).Source: 1) Joint Model and Training · p. 3
  • Approximately 15,000 poses collected over 10 h train an MLP to map normalized magnetic-field differences to a unit vector using cosine loss.Stationary normalized acceleration provides training labels; cameras guide coverage and check consistency rather than supply labels.Source: 1) Joint Model and Training · p. 3

2) Acquisition:

Attitude polling is interleaved with magnetic bursts to provide 16.7 Hz updates, with parent-coil corrections every approximately three seconds. Calibration corrects magnetic measurements and fits sensor alignment, while reconstruction uses the no-twist assumption.

2) Acquisition:

  • A ten-segment sweep takes approximately 52 ms and achieves a 16.7 Hz attitude update rate, interleaved with a 40 ms coil pulse every 300 ms.Each receiver’s coil field is computed from fresh coil-on samples minus its latest confirmed coil-off baseline; parent-coil correction arrives about every three seconds while attitude tracking continues.Source: 2) Acquisition · p. 3

3) Calibration and Reconstruction:

  • The robot is recorded hanging straight to define the reference direction, and reconstruction uses the no-twist mapping between parent and child directions.The parent rotation is evaluated at coil activation.Source: 3) Calibration and Reconstruction · p. 3
  • Calibration subtracts ambient offsets, fits ellipsoid-based gain and soft-iron correction, then estimates remaining alignment over a 45 s initialization window.Stationary data identify tilt alignment, while bending motion additionally identifies axial alignment.Source: 3) Calibration and Reconstruction · p. 3

4) Fusion:

The fusion method estimates and corrects slowly varying directional error for each segment, then reconstructs the backbone at the attitude update rate without integrating acceleration. The section also describes tip-height feedback and its safeguards.

4) Fusion:

  • Quaternion heading drift is approximately 0.65% of angular travel, and fusion estimates its slowly varying directional error with a per-segment rotation state.The unobservable component about the segment direction is damped toward zero.Source: 4) Fusion · pp. 3–4
  • The backbone is reconstructed at the attitude update rate without integrating acceleration.Log denotes the minimum rotation between directions.Source: 4) Fusion · p. 4

C. Closed-Loop Tip Height Control

  • Tip height is controlled independently of heading by selecting a tendon and commanding spool velocity from height error, its integral, and its derivative.Conditional integration is limited to errors under 10 mm; the measured height derivative is low-pass filtered.Source: C. Closed-Loop Tip Height Control · p. 4
  • Trials establish height and spool-position references after a 2 s settling period and reject starts more than 5 mm above the straight configuration.Source: C. Closed-Loop Tip Height Control · p. 4

IV. EXPERIMENTAL SETUP

The experiments use a nine-segment tendon-driven robot and motion-capture ground truth to assess pose estimation across four conditions, including contact and externally imposed deformation.

A. Platform and Ground Truth

  • The robot has nine 29 mm segments, four antagonistic tendons, and velocity-controlled servomotors; Optitrack Motive provides 100 Hz ground truth.Estimators process identical recordings at 16.7 Hz; accuracy is reported as position RMSE across segments and frames and as tip RMSE.Source: A. Platform and Ground Truth · p. 4

B. Pose Estimation Scenarios

  • The four tests cover slow planar curling, obstacle contact with unknown obstacle location, free 3D curling, and manual 3D manipulation with slack tendons.Manual manipulation tests shapes beyond the actuated workspace without actuator or tendon-length inputs.Source: B. Pose Estimation Scenarios · p. 4

C. Demonstrations

The demonstrations cover sensing during visual occlusion, fused-feedback tip-height holding under load, and estimator behavior during cyclic actuation and a prolonged static hold.

1) Cloth Grasping:

  • During shirt lifting and curling around the shirt, onboard backbone estimates continue despite occlusion by the cloth and robot arm.Source: 1) Cloth Grasping · p. 4

2) Closed-Loop Height Hold:

  • Fused feedback maintains a 40 mm tip-height target and is tested after a 500 g tip load is attached.Height is independent of heading error; this demonstration uses eight modules rather than nine in the other experiments.Source: 2) Closed-Loop Height Hold · p. 4

1) Cyclic Actuation:

  • Cyclic actuation compares pulling and releasing estimates at matched spool positions over 50 pull–release cycles lasting 500 s.The commanded half-turn strokes and return positions are tracked by encoder feedback at 50 Hz.Source: 1) Cyclic Actuation · p. 4

2) Static Hold:

  • A fixed curled configuration is held for 30 min while joint-angle and tip-position changes quantify estimator drift.The assessment particularly tracks the quaternion chain and fused estimate.Source: 2) Static Hold · p. 4

V. RESULTS

Fusion produces lower reported position errors than the magnetic-only and inertial variants across the evaluated scenarios. Errors vary with deformation and contact, while model-variant changes have little effect compared with receiver alignment.

A. Pose Estimation Against Motion Capture

  • Across four scenarios and 20 trials, fusion achieves 10.0 mm backbone RMSE and 14.9 mm tip RMSE, with 9.3° link angular RMSE.Tip error is 5.5% of the 270 mm robot length.Source: A. Pose Estimation Against Motion… · p. 4
  • Magnetic-only estimation averages 24.6 mm backbone and 42.3 mm tip RMSE, while constrained and unconstrained inertial variants saturate in every trial.Their backbone RMSEs are 114.8 and 175.3 mm, respectively; position errors generally accumulate toward the tip.Source: A. Pose Estimation Against Motion… · p. 4

1) Scenario Comparison:

  • Free 3D curling has the largest fused errors, at 13.3 mm backbone and 21.1 mm tip RMSE; obstacle contact yields 11.3 and 16.5 mm, versus 7.6 and 8.8 mm for planar curling.Contact compression violates the fixed-length approximation, and marker reconstruction contributes to an 8.8 mm error floor versus 2.0 mm without contact.Source: 1) Scenario Comparison · p. 4, Figure 4 · p. 5
  • Manual manipulation achieves 7.8 mm backbone and 13.0 mm tip RMSE despite occlusion, twisting, and compression, without actuator inputs.The best trial reaches 3.5 mm backbone RMSE against a 1.4 mm metric floor.Source: 1) Scenario Comparison · p. 6

2) Model Variants and Alignment:

  • Five training and architecture variants differ by less than 0.4 mm in fused backbone RMSE and 0.2° in magnetic angular error, while omitting receiver alignment increases error.Source: 2) Model Variants and Alignment · p. 6

B. Demonstrations

The demonstrations show stable estimation during cloth grasping, closed-loop height regulation with disturbance recovery, and measurable differences between fused and magnetic-only estimates during cyclic actuation.

1) Cloth Grasping:

  • Across stationary intervals of 7, 20, and 14 s during cloth grasping, fused tip estimates varied by at most 6 mm, versus 40 mm for magnetic-only and 50–280 mm for inertial variants.Estimation continued during occlusion by the cloth and arm; these measurements characterize stability rather than absolute accuracy.Source: 1) Cloth Grasping · p. 6

2) Closed-Loop Height Hold:

  • The 40 mm target was reached in 3.4 s, with 15% overshoot and settling within ±2 mm after 5.5 s; height was then held at 40.6±0.04 mm for 10 s.Fused feedback ran at 18.6 Hz using 8 modules.Source: 2) Closed-Loop Height Hold · p. 6
  • The 500 g load lowered the tip; height recovered to within 2 mm of target 5.1 s after its minimum and subsequently held 39.0 ± 0.3 mm.Bending redistributed from distal to proximal joints after load application.Source: 2) Closed-Loop Height Hold · p. 6

1) Cyclic Actuation:

  • Across 49 analyzed cycles over 500 s, the final phase-matched tip difference reached 29.1 mm for fusion, while magnetic-only differences ranged from 29 to 44 mm without a clear trend.Baseline change and cycle-to-cycle variation were 16.1 and 2.3 mm for fusion, versus 4.4 and 7.9 mm for magnetic-only estimation; without ground truth, these changes cannot distinguish sensor drift from creep or imperfect motion repetition.Source: 1) Cyclic Actuation · p. 6

2) Static Hold:

Long-duration and fusion analyses show bounded measured displacement but do not establish absolute drift; fusion supports shape reconstruction and height control, with limits on short-trial corrections and comparative accuracy.

2) Static Hold:

  • During a 31.9 min static hold, fused and magnetic-only tip displacements reached 11.1 and 12.7 mm, with RMS scatter of 0.6 and 1.3 mm, respectively.The shared downward trend was consistent with curl relaxation, but without independent ground truth it does not establish an absolute drift bound.Source: 2) Static Hold · p. 6

A. Effect of Magnetic Fusion

  • Magnetic corrections offered little benefit in 13–30 s trials; magnetic estimates updated every 3 s, compared with attitude updates at 16.7 Hz and heading drift of approximately 0.65% of angular travel.Greater magnetic weighting may degrade short-term accuracy while potentially helping after larger accumulated motion.Source: A. Effect of Magnetic Fusion · p. 6
  • Across 49 cycles, fusion showed approximately 16 mm baseline change and 2.3 mm cycle-to-cycle variation, versus 4 mm and 7.9 mm for magnetic-only estimation.Retuning or joint-specific bias modeling may improve correction at the cost of greater variability; both estimators showed similar 8–9 mm trends during the 32 min static hold, consistent with creep but not excluding shared estimation error.Source: A. Effect of Magnetic Fusion · p. 6
  • Fusion reconstructed the backbone without acceleration integration, avoiding the 100–300 mm errors developed by inertial variants within 1 s.Height regulation settled to ±2 mm in 5.5 s, but height is heading-insensitive and the response was reconstructed from logged attitudes because the controller’s estimate was not recorded.Source: A. Effect of Magnetic Fusion · pp. 6–7
  • Overall tip RMSE was 14.5 mm (5.4% of length), below the approximately 7–10% reported for longer IMU-only arms but above the 2.9% reported for bend-sensor–IMU fusion.The comparison is limited because platforms and metrics differ.Source: A. Effect of Magnetic Fusion · p. 7

B. Measurement and Evaluation Limitations

Evaluation uncertainty arises from reference measurements, contact reconstruction, and fitted alignment; without independent ground truth, small method differences and height-control accuracy remain unresolved.

B. Measurement and Evaluation Limitations

  • RMSE combines reference, reconstruction, and alignment uncertainty; planar references used single markers, while 3D references inferred missing rotation from chain pitch when only two markers were visible.Planar marker-center offsets may also affect the reference.Source: B. Measurement and Evaluation… · p. 7
  • Obstacle contact compressed joints, requiring interior-segment reconstruction in 10–14% of frames; restricting evaluation to measured positions reduced fused RMSE from 11.3 to 10.5 mm.Tight curls exceeded the training range, and manual forces could violate no-twist and fixed-length assumptions; friction, speed, occlusion, manual placement, and session-to-session hardware and calibration also varied.Source: B. Measurement and Evaluation… · p. 7
  • Independent magnetic alignment reduced magnetic RMSE from 25.2 to 15.1 mm, while fused RMSE changed by only 0.1 mm, demonstrating sensitivity to the comparison frame.Receiver alignment used attitudes from other trials in the same scenario, so sensing channels were not fully independent; one world heading cannot correct relative segment drift.Source: B. Measurement and Evaluation… · p. 7
  • Without independent ground truth, long recordings cannot separate creep from estimator drift, and height-control accuracy remains unverified.These limitations constrain conclusions about differences of a few millimetres between methods but are insufficient to explain the much larger inertial errors.Source: B. Measurement and Evaluation… · p. 8

VII. CONCLUSIONS

  • The conclusion reports pooled backbone and tip RMSEs of 10.0±3.2 and 14.9±6.5 mm across 20 trials in four scenarios, while noting that control accuracy requires independent ground-truth validation.Future work includes improving drift correction, expanding the training workspace, extending feedback from tip height to backbone shape, and adding force sensing.Source: §VII · p. 8
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