Magnetisch-inertiale In-situ-3D-Posenschätzung für modularen Soft-Sehnen-Kontinuumsroboter mittels IMU-Fusion
Die Autoren stellen ein modulares magnetisch-inertiales Sensorkonzept für einen sehnengetriebenen Soft-Kontinuumsroboter vor, das ohne externe Kameras auskommt. In 20 Versuchen über vier Szenarien erreicht die fusionierte Schätzung eine mittlere Backbone-RMSE von 10,0 mm und eine Spitzen-RMSE von 14,9 mm, entsprechend 5,5 Prozent der Roboterlänge. Der als Preprint veröffentlichte Ansatz liefert Echtzeit-Updates mit 16,7 Hz und demonstriert das Potenzial für geschlossene Regelung unter externer Last.
Wie kann die 3D-Konfiguration eines modularen, sehnengetriebenen Soft-Kontinuumsroboters ausschließlich mit eingebetteten IMU- und Magnetfeldmessungen in Echtzeit und ohne externe Kameras geschätzt werden?
Die kontinuierliche Verformung des Roboters macht die Bestimmung der Pose schwierig. Modellbasierte Schätzungen aus Aktuatorwinkeln sind anfällig für Materialnichtlinearitäten, Hysterese, Reibung und äußere Lasten. Externe Kameras benötigen freie Sicht und sind in verdeckten Umgebungen unbrauchbar. Eingebettete Inertialsensoren driften und liefern ohne Magnetkorrektur keine vollständige Orientierung; der bisherige magnetisch-inertiale Ansatz erreicht nur etwa 1,2 Hz und skaliert bei vielen Modulen schlecht.
Bisherige Verfahren setzen auf kinematische Modelle, externe Stereo- und Lern-basierte Bildsysteme, verteilte Dehnungs- oder faseroptische Sensoren sowie rein inertiale oder rein magnetische Messungen. Der verwandte magnetisch-inertiale Ansatz von Sue et al. kombiniert Spulen und Magnetometer, liefert aber nur etwa 1,2 Hz Aktualisierungsrate und ist für lange modulare Roboter unzureichend. Rein inertiale Schätzungen akkumulieren Drift und werden durch dynamische Beschleunigung gestört.
Der Roboter besteht aus neun seriellen TPU-90A-Modulen mit integriertem Hourglass-Gelenk; jedes Modul trägt eine Platine mit BNO086-IMU, ATtiny3226-Mikrocontroller, planarer PCB-Spule und Schalt-MOSFET. Ein ESP32-S3 fragt die Module über I2C ab und streamt die Daten an einen Raspberry Pi 5 mit ROS 2. Jedes Gelenk wird als festes sphärisches Drehgelenk ohne Torsion modelliert; ein gemeinsam trainiertes MLP bildet normalisierte Magnetfelddifferenzen auf die relative Richtung ab. Das Training nutzt rund 15.000 stationäre Posen über 10 Stunden, wobei die normalisierte Beschleunigung als Label dient. Die Fusion kombiniert quaternionenbasierte Lagewinkel mit periodischen Magnetkorrekturen über einen pro-Segment-Bias-Zustand und rekonstruiert die Backbone ohne Beschleunigungsintegration. Zusätzlich wird eine PID-Höhenregelung mit bedingter Integration für die Spitzenhöhe implementiert.
In 20 Versuchen über vier Szenarien erreicht die Fusion eine gepoolte Backbone-RMSE von 10,0±3,2 mm und eine Spitzen-RMSE von 14,9±6,5 mm (5,5 % der Roboterlänge). Für planares Curling liegen Backbone- und Spitzen-RMSE bei 7,6 und 8,8 mm, bei Hinderniskontakt bei 11,3 und 16,5 mm, bei freiem 3D-Curling bei 13,3 und 21,1 mm, bei manueller Manipulation bei 7,8 und 13,0 mm. Die rein magnetische Schätzung erreicht im Mittel 24,6 mm Backbone- und 42,3 mm Spitzen-RMSE; die inertialen Varianten liegen bei 114,8 bis 175,3 mm Backbone- und 176,2 bis 274,7 mm Spitzen-RMSE. Die Aktualisierungsrate beträgt 16,7 Hz; Magnetkorrekturen erfolgen alle 3 s pro Segment. In der Höhenregelung wird die Zielhöhe von 40 mm in 3,4 s erreicht, mit 15 % Überschwingen und Einregeln auf ±2 mm nach 5,5 s; nach Anhängen einer 500-g-Last erholt sich die Regelung innerhalb von 5,1 s auf ±2 mm um den Sollwert. Im zyklischen Betrieb beträgt die Drift der Fusion 4,1 mm/50 s gegenüber 51,1 mm/50 s der rein magnetischen Schätzung; im statischen Halten 1,2 mm/50 s gegenüber 3,3 mm.
Die Arbeit ist ein Preprint und nicht begutachtet; ein veröffentlichter Code oder Datensatz wird im Paper nicht berichtet. Die Langzeitaufnahmen ohne unabhängige Ground Truth können Schleichverformung nicht von Schätzerdrift trennen; die Genauigkeit der Höhenregelung bleibt unbestätigt, da die Regler-Schätzung nicht aufgezeichnet wurde. Die RMSE enthält Referenz-, Rekonstruktions- und Ausrichtungsunsicherheit; bei Hinderniskontakt werden 10 bis 14 % der Frames rekonstruiert, und Kompression verletzt die Annahme fester Segmentlängen. Enge Krümmungen überschreiten den Trainingsbereich von 25 bis 30 Grad, manuelle Kräfte können die No-Twist- und Fixed-Length-Annahmen verletzen. Magnetkorrekturen bringen in kurzen 13 bis 30 Sekunden dauernden Versuchen nur geringen Nutzen, und die Fusionsparameter wurden nicht weiter abgestimmt. Die Auswertung erfolgt auf einem einzigen Roboteraufbau; Hardware- und Kalibrierungsunterschiede zwischen Sitzungen bestehen.
Kurzfristig, innerhalb von 1 bis 3 Jahren, könnten Hersteller von Soft-Greifern und Inspektionsrobotern die eingebettete Propriozeption für kamerafreie Formrückmeldung in verdeckten Umgebungen nutzen, sofern Robustheit gegenüber Fertigungstoleranzen und größere Trainingsdatensätze nachgewiesen werden. Mittelfristig, in mehr als 3 Jahren, ist der Einsatz in minimal-invasiven Instrumenten und in der Lebensmittelhandhabung denkbar, wenn Driftkorrektur, Miniaturisierung, Sterilität und Zulassung gelöst sind. Voraussetzung ist eine unabhängige Validierung der Regelgenauigkeit und der Langzeitdrift auf Zielplattformen.
Vollständige Studie
Magnetic based In-situ Self 3D Pose Estimation for a Modular Soft Tendon-Driven Continuum Robot via IMU-Fusion
Veröffentlicht unter CC BY-NC-SA 4.0. Wiedergabe mit Namensnennung, Original unter arXiv:2609.39950 (PDF).
Abstract
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
Soft continuum robots deform continuously to conform to complex environments and handle delicate objects, enabling gentle manipulation [1], food harvesting [2], and minimally invasive procedures [3, 4, 5]. However, their distributed deformation makes configuration estimation more challenging than for rigid-link robots.
Reliable configuration feedback is essential for closed-loop control. Estimates derived from actuation inputs through kinematic or mechanical models [6, 7] are sensitive to material nonlinearities, hysteresis, friction, and external loading [8, 9]. External vision provides accurate measurements but requires clear visibility [10], motivating embedded proprioception. Strain sensors measure local deformation but may require complex models for global shape reconstruction [11]. Accelerometers provide inclination relative to gravity but cannot resolve full orientation and are affected by dynamic acceleration [12]. Magnetic measurements provide complementary directional information for distributed shape estimation [13].
Distributing magnetic–inertial sensing across modular robots requires efficient acquisition and communication. The approximately 1.2 Hz estimation rate of [13] limits dynamic tracking and feedback control. We address this limitation with embedded sensing modules that combine onboard attitude estimates and magnetic measurements to reconstruct the three-dimensional backbone. Continuous attitude polling and intermittent magnetic corrections enable configuration updates at 16.7 Hz. The architecture supports distributed sensing and closed-loop control under external loading without external visual tracking during deployment.
The main contributions are:
- A modular magnetic–inertial sensing architecture for embedded three-dimensional shape reconstruction.
- A sensing and communication scheme providing configuration updates at 16.7 Hz.
- Experimental validation of shape estimation under varied deformation and contact conditions, and closed-loop tip-height control under an external disturbance.
II RELATED WORKS
II-A Pose and Configuration Estimation
Accurate configuration estimation is essential for continuum robots because their compliant, continuously deformable structures make the relationship between actuation and robot shape difficult to predict. Model-based approaches estimate robot configuration from actuation inputs using kinematic or mechanical models [6, 7], but their accuracy can degrade in the presence of material nonlinearities, hysteresis, friction, and external loading.
External sensing has therefore been widely used to directly observe the robot configuration. Vision-based methods have employed stereo cameras and learning-based approaches for continuum-robot pose estimation [14], while other systems combine visual tracking with additional displacement measurements for real-time configuration estimation [15]. Although these approaches can provide accurate measurements, they depend on external sensing infrastructure and an unobstructed line of sight.
Embedded deformation sensing provides an alternative for self-contained proprioception. Strain and fiber-optic sensing methods measure local deformation along the robot body and reconstruct the global configuration from distributed measurements [16, 17]. However, these approaches often require calibration or a model relating local sensor measurements to the global robot shape.
II-B Embedded Inertial and Magnetic Proprioception
Inertial sensing has been explored as a compact approach for embedded configuration estimation. [18] combined inertial sensing with actuator measurements to estimate continuum-robot pose, while [12] used distributed inertial measurement units (IMUs) to reconstruct robot shape and estimate the end-effector position. These studies demonstrate that distributed orientation measurements can provide proprioceptive information without external tracking.
Magnetic sensing has similarly been used for embedded shape estimation. [19] used magnetic measurements to infer local bending and reconstruct continuum-robot shape, while [20] and [21] demonstrated learning-based reconstruction from embedded magnetic-field measurements. Magnetic sensing provides an additional directional reference but can require dedicated magnetic sources, robot-specific calibration, or learned mappings.
More closely related to this work, magnetic and inertial measurements have been combined for embedded orientation estimation [13]. However, existing implementations remain limited in update rate and scalability when distributed across long, modular continuum robots. In this work, we develop a modular magnetic–inertial proprioceptive sensing architecture that combines local orientation estimates from multiple robot segments to reconstruct the three-dimensional configuration at a rate suitable for real-time feedback control.
III METHODOLOGY
III-A Design
The robot comprises nine serial TPU 90A modules, each 20 mm high, with two disks (50 mm in diameter, 2 mm thick) connected by an hourglass joint (16 mm high, 5 mm minimum diameter). Compared to existing designs [13] [22], our architecture features a soft mechanical module with integrated joints that simplifies the mechanical structure and reduces the number of separate components (Fig. 2 (a)). This approach facilitates assembly and disassembly while preserving modularity, allowing individual modules to be replaced and the robot length to be adjusted. In addition, each mechanical module has one magnetic coil instead of a single magnetic placed on the base, allowing infinite number of modules to be stacked. Each module carries a PCB integrating a BNO086 IMU, ATtiny3226 microcontroller, planar coil, and switching MOSFET (Fig. 2 (b)). Following [13], each microcontroller reads its IMU over SPI and exposes registers on a shared I2C bus. An ESP32-S3 polls the boards at 400 kHz and streams measurements over USB to a Raspberry Pi 5 running ROS 2. Separate logic and coil rails isolate the 3.3 V electronics from coil switching.
The BNO086 Game Rotation Vector fuses accelerometer and gyroscope measurements without magnetic input, allowing attitude tracking during coil activation. Magnetic measurements correct its arbitrary initial heading and subsequent drift. Raw magnetometer readings arrive at approximately 90 Hz.
The wound coil in [13] is replaced by additively wound traces on both sides of a 0.8 mm FR4 PCB with 2 oz copper. The coil trace is defined by the following equation:
r(\phi)=R_{0}\!\left[1+\sum_{k=1}^{3}\alpha_{k}\cos(k\phi+\beta_{k})\right],where \boldsymbol{\alpha}=(0.085,0.100,0.050) and \boldsymbol{\beta}=(0.5,0.2,-0.4) rad; successive turns use constant pitch. Its asymmetric outline breaks axial field symmetry (Fig. 2 (b)) and thus provide more features in the magnetic field to be captured and process down stream in the sensing pipelin. Approximately 19 turns per layer of 10 mil traces give a resistance near 1\,\Omega. A 40 ms pulse at 3.3 V produces approximately 100\,\muT at the neighboring segment, compared with 0.6\,\muT noise per axis.
III-B Sensing
III-B1 Joint Model and Training
Following [13], each joint is modeled as a fixed-radius spherical pivot without axial twist (Fig. 3 (a)). Its relative pose is therefore represented by a unit direction \mathbf{u}\in S^{2}. Unlike piecewise constant curvature [23], these local assumptions do not prescribe bending along the full backbone. The pivot radius is 10 mm; a 9 mm platform offset (height of sensing electronics), and a 10 mm before the pivot of the next module, giving a total pivot spacing of L=29 mm.
Training used a fixed proximal coil and a distal sensor actuated by four antagonistic tendons. Camera feedback from two webcams and AprilTags guided the joint toward targets sampled uniformly over a spherical cap (Fig. 3 (b) (c)): \cos\lambda\sim\mathcal{U}[\cos\lambda_{\max},1] and \theta\sim\mathcal{U}[-\pi,\pi), with \lambda_{\max} slightly below the 30^{\circ} mechanical limit. \lambda and \theta are colatitude and azimuth respectively. Equal-solid-angle Fibonacci cells and sampling quotas improved coverage.
At each stationary target, two to three pulses, each at most 40 ms, provided six coil-on samples. The coil field was \Delta\mathbf{B}=\mathbf{m}_{\mathrm{on}}-\mathbf{b}(t), where \mathbf{b}(t) was fitted linearly to surrounding coil-off measurements (ambient). \mathbf{m}_{\mathrm{on}} denotes the measured magnetic reading when the coil was on. Approximately 15,000 unique poses were collected over 10 h.
With the base leveled, normalized stationary acceleration \hat{\mathbf{a}}=\mathbf{a}/\|\mathbf{a}\| provides the parent axis in the child sensor frame. These training labels avoid camera and pivot-fitting errors; cameras guide coverage and check consistency. An MLP maps normalized \Delta\mathbf{B} to a unit vector, using cosine loss \ell=1-f(\Delta\mathbf{B})^{\top}\hat{\mathbf{a}}. Hereafter, f the function approximated by the MLP, includes input normalization, which removes multiplicative field-amplitude variation.
III-B2 Acquisition
Continuous attitude polling is interleaved with magnetic bursts. Reading each 96-byte sensor block takes about 3.4 ms; a ten-segment (1 base module and 9 sensing module on TPU) sweep takes approximately 52 ms, with an achieved attitude update rate of 16.7 Hz. Every T_{\mathrm{pair}}=300 ms, one coil fires for 40 ms while the master polls only the other segments’ 12-byte magnetometer blocks, taking approximately 3 ms per round.
For each receiver, the coil field is the mean of fresh coil-on samples minus its latest confirmed coil-off baseline. Samples are accepted only after settling and a bitwise change indicating a new conversion. Each segment receives a parent-coil correction every NT_{\mathrm{pair}}\approx 3 s, while attitude tracking continues between bursts.
III-B3 Calibration and Reconstruction
Let R_{i}=R(\mathbf{q}_{i})R(\mathbf{q}_{i}^{0})^{\top}, where \mathbf{q}_{i}^{0} is recorded with the robot hanging straight, and let \mathbf{e}=-\hat{\mathbf{z}}. Quaternion and magnetic directions are
\displaystyle\mathbf{u}_{i}^{q} \\ \displaystyle=R_{i}\mathbf{e}, \\ \displaystyle\mathbf{u}_{i}^{m} \\ \displaystyle=R_{i-1}M_{\mathbf{e}}\,f(A_{i}W_{i}\Delta\mathbf{B}_{i}), \\ \displaystyle M_{\mathbf{e}} \\ \displaystyle=2\mathbf{e}\mathbf{e}^{\top}-I.Under the no-twist assumption, M_{\mathbf{e}} converts the predicted parent axis in the child frame into the child direction in the parent frame. The parent’s rotation is evaluated at coil activation.
Ambient subtraction removes constant offsets. An ellipsoid fit to ambient-field measurements during sensor rotation determines the gain and soft-iron correction W_{i}. The remaining alignment A_{i} is fitted over a 45 s initialization window by minimizing the angle between M_{\mathbf{e}}f(A_{i}W_{i}\Delta\mathbf{B}_{i}) and R_{i-1}^{\top}\mathbf{u}_{i}^{q}. Stationary data identify tilt alignment; bending motion additionally identifies axial alignment.
III-B4 Fusion
Quaternion heading drift measured approximately 0.65% of angular travel. We estimate its slowly varying directional error using a per-segment rotation state \mathbf{b}_{i}:
\hat{\mathbf{u}}_{i}=\exp([\mathbf{b}_{i}]_{\times})\mathbf{u}_{i}^{q},\qquad P_{i}\leftarrow P_{i}+\sigma_{b}^{2}\Delta t\,I.At each magnetic update,
\displaystyle\mathbf{y}_{i} \\ \displaystyle=\operatorname{Log}(\mathbf{u}_{i}^{q}\!\to\!\mathbf{u}_{i}^{m})-\mathbf{b}_{i}, \\ \displaystyle K_{i} \\ \displaystyle=P_{i}(P_{i}+\sigma_{m}^{2}I)^{-1}, \\ \displaystyle\mathbf{b}_{i} \\ \displaystyle\leftarrow\mathbf{b}_{i}+K_{i}\mathbf{y}_{i},\qquad P_{i}\leftarrow(I-K_{i})P_{i}.Here \operatorname{Log} denotes the minimum rotation between directions. The unobservable component about \mathbf{u}_{i}^{q} is damped toward zero. The backbone is reconstructed at the attitude update rate without integrating acceleration:
\mathbf{p}_{0}=\mathbf{0},\qquad\mathbf{p}_{i}=\mathbf{p}_{i-1}+L\hat{\mathbf{u}}_{i},\quad i=1,\ldots,N.III-C Closed-Loop Tip Height Control
The estimated tip height above the straight hanging configuration is
h=L\sum_{i=1}^{N}(1+\hat{u}_{i,z}).Height is independent of heading, so this task evaluates height feedback rather than heading-drift correction. A tendon is selected before each trial because bending in either direction raises the tip. The spool velocity is
v=k_{p}e+k_{i}I_{h}-k_{d}\dot{h},\qquad\dot{I}_{h}=e\,\mathbf{1}_{\{|e|<10\,\mathrm{mm}\}},\quad e=h^{*}-h.The measured height derivative is low-pass filtered. Conditional integration limits windup during slack take-up; velocity magnitude and its rate of increase are limited. Unwinding past neutral is prohibited because it reverses the height response.
After alignment, the robot settles for 2 s at zero command; median height and spool position establish trial references. Trials are rejected if initial height exceeds the straight configuration by 5 mm.
IV EXPERIMENTAL SETUP
IV-A Platform and Ground Truth
The robot comprised nine 29 mm segments (261 mm total), suspended from a rigid base and actuated by four antagonistic tendons using velocity-controlled servomotors (Dynamixel XM540-W270-R) in the 4 cardinal directions. Unless stated otherwise, one tendon produced unidirectional curling. The fused, quaternion, magnetic, and inertial estimators processed identical recordings at 16.7 Hz (Section III). The ground truth from the Optitrack Motive system recorded at 100 Hz was used. We report position RMSE across all segments and frames, and tip RMSE.
IV-B Pose Estimation Scenarios
We evaluated four conditions:
- Planar curl (2D curl): Slow, unobstructed single-tendon bending constrained to a plane on a flat surface.
- Planar curl with obstacle (2D obst.): The same motion against a box of rocks, whose location was unknown to the estimator, testing contact-induced deformation.
- Free curl in 3D (3D curl): Single-tendon actuation while freely suspended, testing spatial reconstruction and magnetic correction of quaternion heading error.
- Manual manipulation in 3D (3D hand): Arbitrary bending with slack tendons, including shapes beyond the actuated workspace, testing externally imposed deformation without actuator or tendon-length inputs.
IV-C Demonstrations
IV-C1 Cloth Grasping
Mounted on a UFactory Xarm7, the robot curled around and lifted a shirt without feedback control. We present the estimated backbone during the task, demonstrating onboard sensing during occlusion by the cloth and arm.
IV-C2 Closed-Loop Height Hold
The controller in Section III-C used fused feedback to maintain the tip 40 mm above its relaxed vertical position. Height, h=L\sum_{i}(1+u_{i,z}), is independent of heading error. We compare target, estimated, and independently measured heights, and assess disturbance recovery after attaching a 500 g tip load. Furthermore, to demonstrate the modular aspect of our paradigm, we used only 8 modules (9 including the base) for this experiment where as every other experiments we used 9 modules (10 , including the base)
IV-D Drift and Hysteresis
IV-D1 Cyclic Actuation
One tendon underwent 50 pull–release cycles of half a spool turn, each lasting 10 s (5 s per direction; 500 s total). Encoder feedback at 50 Hz tracked p^{*}(t)=p_{0}+\tfrac{A}{2}[1-\cos(2\pi t/T)], ensuring identical commanded strokes and return positions. Pulling and releasing estimates were compared at matched spool positions to assess hysteresis, repeatability and drift.
IV-D2 Static Hold
The robot remained in a fixed curled configuration for 30 min. Joint-angle and tip-position changes quantified estimator drift, particularly for the quaternion chain and fused estimate.
V RESULTS
- Fusion (this work)
- 7.6\pm 1.1
- 11.3\pm 1.1
- 13.3\pm 0.9
- 7.8\pm 4.0
- 10.0\pm 3.2
- 4.1
- 1.2
- Magnetics only
- 36.9\pm 2.2
- 18.0\pm 1.5
- 29.4\pm 0.9
- 14.1\pm 2.7
- 24.6\pm 9.5
- 51.1
- 3.3
- Constrained acceleration + quaternions
- 133.8\pm 3.4
- 163.0\pm 3.4
- 71.9\pm 2.1
- 90.5\pm 4.3
- 114.8\pm 36.8
- 54.6
- 17.1
- Acceleration only
- 189.0\pm 20.4
- 153.8\pm 15.3
- 177.7\pm 9.1
- 180.8\pm 21.1
- 175.3\pm 20.7
- 24.5
- 0.3
- —
- Tip RMSE (mm)
- Tip RMSE (mm)
- Tip RMSE (mm)
- Tip RMSE (mm)
- Tip RMSE (mm)
- Keine Daten
- Keine Daten
- Method
- 2D curl
- 2D obst.
- 3D curl
- 3D hand
- Avg.
- Keine Daten
- Keine Daten
- Fusion (this work)
- 8.8\pm 1.1
- 16.5\pm 1.9
- 21.1\pm 1.8
- 13.0\pm 9.5
- 14.9\pm 6.5
- Keine Daten
- Keine Daten
- Magnetics only
- 59.9\pm 2.6
- 36.2\pm 3.0
- 48.0\pm 1.4
- 24.9\pm 5.9
- 42.3\pm 13.8
- Keine Daten
- Keine Daten
- Constrained acceleration + quaternions
- 193.7\pm 9.2
- 246.0\pm 5.4
- 113.1\pm 6.8
- 151.8\pm 7.8
- 176.2\pm 51.1
- Keine Daten
- Keine Daten
- Acceleration only
- 297.5\pm 35.1
- 232.6\pm 32.1
- 279.4\pm 12.9
- 289.1\pm 32.8
- 274.7\pm 37.4
- Keine Daten
- Keine Daten
V-A Pose Estimation Against Motion Capture
Across four scenarios and 20 trials, fusion achieved scenario-averaged backbone and tip RMSEs of 10.0 and 14.9 mm, respectively, with tip error equal to 5.5% of the 270 mm robot length. Link angular RMSE was 9.3^{\circ} (Table I, Fig. 4).
Magnetic-only backbone and tip RMSEs averaged 24.6 and 42.3 mm, partly reflecting sparse updates during motion. At coil activation, magnetic angular errors were 5.4^{\circ} median, 9.7^{\circ} RMS, and 17.8^{\circ} at the 95th percentile. The constrained and unconstrained inertial variants saturated in every trial, with backbone RMSEs of 114.8 and 175.3 mm, consistent with joint accelerations below residual accelerometer bias. Position errors generally accumulated toward the tip.
V-A1 Scenario Comparison
Planar curling produced fused backbone and tip RMSEs of 7.6 and 8.8 mm, with 6.3^{\circ} angular RMSE. Obstacle contact increased these to 11.3 and 16.5 mm and 9.7^{\circ}. Compression violated the fixed-length approximation, while marker reconstruction in 10–14% of frames contributed to an 8.8 mm metric floor, compared with 2.0 mm without contact.
Free 3D curling produced the largest errors: 13.3 mm backbone and 21.1 mm tip RMSE. Joint bends reached 20^{\circ}–40^{\circ}, exceeding the 25^{\circ}–30^{\circ} training cap. Magnetic angular errors reached a 7.4^{\circ} median and 31.7^{\circ} 95th percentile; errors decreased as the robot straightened.
Manual manipulation achieved 7.8 mm backbone and 13.0 mm tip RMSE despite occlusion, twisting, and compression, without actuator inputs. The best trial achieved 3.5 mm backbone RMSE against a 1.4 mm metric floor.
V-A2 Model Variants and Alignment
Five training and architecture variants differed by less than 0.4 mm in fused backbone RMSE and 0.2^{\circ} in magnetic angular error. Receiver alignment had a larger effect, with its omission increasing error for both fused and magnetic-only estimation.
V-B Demonstrations
V-B1 Cloth Grasping
Estimation continued during occlusion by the cloth and arm (Fig. 5(a)). Across stationary intervals lasting 7, 20, and 14 s, tip estimates varied by at most 6 mm for fusion, 40 mm for magnetic-only estimation, and 50–280 mm for the inertial variants. These measurements characterize stability rather than absolute accuracy.
V-B2 Closed-Loop Height Hold
Fused feedback at 18.6 Hz, using 8 modules, regulated tip height to 40 mm (Fig. 5(b)). The reconstructed response reached the target in 3.4 s, peaked at 46 mm (15% overshoot), and settled within \pm 2 mm after 5.5 s, subsequently holding 40.6\pm 0.04 mm for 10 s.
A 500 g load applied at 24 s lowered the tip and redistributed bending from distal to proximal joints. Height recovered to within 2 mm of the target 5.1 s after its minimum and subsequently held 39.0\pm 0.3 mm. Furthermore, in this scenario we showed that our paradigm work just as well with one less module that the other scenarios, demonstrating our modular ability.
V-C Long-Term Estimation
V-C1 Cyclic Actuation
Across 49 analyzed cycles over 500 s, the final phase-matched tip difference reached 29.1 mm for fusion; magnetic-only differences ranged from 29 to 44 mm without a clear trend (Fig. 5(c)(ii)). 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. Magnetic measurements thus showed less baseline change but greater variability. Without ground truth, these changes cannot separate sensor drift from creep or imperfect motion repetition.
V-C2 Static Hold
During a 31.9 min interval with all gyroscopes below 0.02 rad/s, 5258 magnetic fixes were recorded. Fused and magnetic-only tip displacements reached 11.1 and 12.7 mm, respectively, with linear-trend displacements of 8.7 and 8.4 mm and RMS scatter of 0.6 and 1.3 mm (Fig. 5(c)(i)). The shared downward trend was consistent with curl relaxation, but does not establish an absolute drift bound without independent ground truth.
VI DISCUSSION
z displacement relative to the first cycle.- Baaij et al. [20]
- 110
- 3 magnetometers + magnet, NN
- —
- 4.5% (config.)
- Adamu et al. [21]
- 240
- Hall + magnet/segment, MLP
- <5%L
- max 11 mm per marker
- Guo et al. [19]
- —
- embedded magnets + sensors
- 0.23–9.34 mm
- —
- Li et al. [24]
- —
- Hall + base magnet + IMUs
- —
- <1 mm
- Pittiglio et al. [25]
- 64
- ball chain + Hall array
- 2.9%L (max 7.1%)
- —
- Thuruthel et al. [26]
- 120
- cPDMS strain, LSTM
- 2.4\pm 2.2 mm
- —
- Vicari et al. [27]
- 220
- base pressure, BiLSTM
- 6.0\pm 4.7 mm (2.7%L)
- —
- Martin et al. [12]
- 660
- 8 IMUs, PCC
- median <10%L
- —
- Stella et al. [28]
- 750
- 2 IMUs, PCC filter
- \approx7%L
- —
- Han et al. [29]
- 583
- bend + IMUs, Kalman
- RMSE 17.0 mm (2.9%L)
- —
- Sue et al. [13]
- 650
- coils + magnetometers, NN
- 47.4 mm (7.3%L)
- RMSE 44.2 mm
- This work
- \approx270
- PCB coils + IMUs, NN + ESKF
- 14.9 mm (5.5%L)
- RMSE 10.0 mm
VI-A Effect of Magnetic Fusion
Magnetic corrections offered little benefit during 13–30 s trials. Magnetic estimates had 5.4^{\circ} median angular error and updated every 3 s per joint, whereas attitudes updated at 16.7 Hz with heading drift of approximately 0.65% of angular travel. Greater magnetic weighting may therefore degrade short-term accuracy while potentially helping after larger accumulated motion.
Across 49 cycles, fused and magnetic-only baseline changes were approximately 16 and 4 mm, with cycle-to-cycle variations of 2.3 and 7.9 mm. Filter parameters came from holdout residuals and measured heading drift without further tuning. 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.
Fusion reconstructed the backbone without acceleration integration, avoiding the 100–300 mm errors developed by inertial variants within 1 s. Fused feedback also supported height regulation and disturbance recovery, with a reconstructed settling time of 5.5 s to \pm 2 mm. However, height is heading-insensitive, and the response was reconstructed from logged attitudes because the controller’s estimate was not recorded.
Overall tip RMSE was 14.5 mm (5.4% of length), with 3.3% during planar curling and 5–8% in 3D. These values are comparable to reported small magnetic systems and below the approximately 7–10% reported for longer IMU-only arms, but exceed the 2.9% reported for bend-sensor–IMU fusion (Table II). Our system uses no strain or bend sensors and shares one trained joint model, although differing platforms and metrics limit direct comparison.
VI-B Measurement and Evaluation Limitations
RMSE includes reference, reconstruction, and alignment uncertainty. Planar references used single markers with possible center offsets; 3D references used three markers, inferring missing rotation from chain pitch when only two were visible.
Obstacle contact compressed joints and required reconstruction of interior segments in 10–14% of frames. Restricting evaluation to measured positions reduced fused RMSE from 11.3 to 10.5 mm, indicating that reference limitations contributed to contact error. Friction, actuation speed, marker occlusion, and manual placement introduced further variation. Tight curls exceeded the training range, while manual forces could violate the no-twist and fixed-length assumptions. Hardware and calibration also varied between sessions.
A shared time offset and world heading were fitted from segment attitudes. 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 also used attitudes from other trials in the same scenario, excluding the scored trial and motion capture. The sensing channels are therefore not fully independent, and a single world heading cannot correct relative segment drift.
Without independent ground truth, long recordings cannot separate creep from estimator drift, and height-control accuracy remains unverified. These limitations are insufficient to explain the much larger inertial errors but constrain conclusions about differences of a few millimetres between methods.
VII CONCLUSIONS
We presented a modular continuum robot combining PCB coils, magnetometers, and onboard attitude sensing for shape estimation without actuator models, strain sensors, or external cameras during deployment. A shared learned joint model, per-segment calibration, and quaternion–magnetic fusion achieved pooled backbone and tip RMSEs of 10.0\pm 3.2 and 14.9\pm 6.5 mm, respectively, across 20 trials in four scenarios. Mean tip RMSE corresponded to 5.5% of robot length, with estimation demonstrated under both tendon actuation and manual manipulation. The estimate also supported tip-height control and disturbance recovery, although control accuracy requires independent ground-truth validation. Future work will improve drift correction, expand the training workspace, extend feedback control from tip height to backbone shape, and incorporate force sensing for more comprehensive proprioception.
ACKNOWLEDGMENT
This work used large language models (LLMs) such as ChatGPT and Claude to assist in proofreading and revising the manuscript text. In addition, LLMs were also used to assist in the creation of code segments/scripts used for experimental data analysis, figure generation and implementation.