SaddleSense: Saddle-Shaped Modules for 3D Shape Sensing with Single-Axis Off-the-Shelf Bend Sensors

Dong Heon Han; Runze Zuo; Chankyo Kim; Saima Jamal; Daniel Bruder

cs.ROOct 31, 2026

SaddleSense: Saddle-Shaped Modules for 3D Shape Sensing with Single-Axis Off-the-Shelf Bend Sensors

Authors: Dong Heon Han, Runze Zuo, Chankyo Kim, Saima Jamal, Daniel Bruder

Organizations: Robotics Department, University of Michigan, Ann Arbor, MI, USA. · Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, USA.

Abstract

Accurate posture feedback is essential for precise control of continuum robots. Off-the-shelf bend sensors measure curvature about only one axis; without a suitable mounting arrangement, they can restrict robot motion and provide incomplete posture information. SaddleSense introduces a 3D-printed thermoplastic polyurethane (TPU) saddle-shaped lattice that holds one commercial bend sensor along each curved edge. Because each sensor flexes within its own channel rather than spanning the segment axially, the lattice converts multi-axis body deformation into sensor curvature without tying segment length to sensor length. Each four-sensor module accommodates two-axis bending, contraction, and extension while quantifying all three deformation modes across its four readings. A hybrid calibration maps the sensor readings to piecewise constant curvature (PCC) parameters using a learned model paired with a physics-based fallback for out-of-distribution inputs. We demonstrate a six-module prototype with 24 off-the-shelf bend sensors on a 430~~mm continuum body. Using only each module's four local readings, the physics-based model alone achieves a tip position root mean square error (RMSE) of 29.9~~mm (7.0% of body length). The data-driven model reduces the tip RMSE to 13.2~~mm (3.1%) on held-out samples of motions similar to the training set and 15.8~~mm (3.7%) on separately recorded trials. Reducing each module from four sensors to one nearly triples the tip error.

Figures & tables

Fig. 1: Overview of SaddleSense. (A) Modular saddle-shaped rings are mounted along a deformable continuum body, with each curved channel holding one commercial single-axis bend sensor (four per module around the circumference). (B) A hybrid calibration—combining a learned model with a physics-based fallback for inputs far from the training distribution—maps the sensor signals $\mathbf{v}$ to piecewise constant curvature (PCC) parameters $\mathbf{q}$ , from which PCC kinematics reconstructs the 3D shape $\mathbf{x}$ .

Fig. 2: SaddleSense module geometry and deformation. (A) Under segment bending, each sensor (purple) flexes within its circumferential channel while the segment length $d_{segment}$ remains independent of the sensor length. Bending widens the outer channel radius ( $r_{1}$ ) and tightens the inner channel radius ( $r_{2}$ ). (B) Close-up of the boxed module in (A), illustrating how bending induces distinct curvature changes ( $\kappa_{1}$ to $\kappa_{3,4}$ ) across the four sensors.

Fig. 3: Piecewise constant curvature (PCC) model of one segment. (A) Planar bending with angle $\theta$ and centerline arc length $L_{i}$ . The inner and outer arcs at radial offset $d_{i}$ have curvatures $\kappa_{\max}$ and $\kappa_{\min}$ . (B) Spatial bending in a plane oriented at angle $\phi$ . (C) Segment cross-section showing the four circumferential arcs $L_{i,1}$ to $L_{i,4}$ at projected offsets $d^{\prime}_{i,k}$ from the neutral bending axis. (D) Sequential segment transformations are chained in $\mathrm{SE}(3)$ to reconstruct the 3D backbone.

Fig. 4: Experimental setup. Six SaddleSense modules cover a 430 mm flexible duct hose. An Arduino Due reads the 24 bend sensors through two analog multiplexers and streams the data to a PC over USB, while OptiTrack motion-capture markers (M0–M6) on the rigid connectors measure ground-truth segment poses. Insets: (1) sliding a bend sensor into the internal pouch of a 3D-printed TPU channel, (2) the assembled channel with only the sensor leads exposed, and (3) a channel mounted circumferentially between two connectors, with arrows indicating its in-plane flexing direction.

Fig. 5: Validation of the module sensing principle. (A) Normalized reading change of the four sensors in segment 2 (divided by bending magnitude

\Delta

) versus bending direction

\phi

for samples with

\Delta

above the median (circles: binned means; curves: sinusoidal fits). (B) Peak sensitivity directions of all 24 sensors across the six segments (radial coordinate indicates segment index). (C) Sensor-count ablation showing tip position RMSE as the number of sensors per module is reduced from four to one (evaluated with an identical single-frame

128\times 128

MLP; error bars show standard deviation across 3 repeats).

Fig. 6: Continuous 3D shape reconstruction on two held-out trials (left of vertical divider: in-distribution trial; right: separately recorded trial). (A) Ground-truth (black) and reconstructed (purple) 3D backbones at eight representative large-bending instants (four per trial), along with the preceding 1.2 s tip trajectory. Labels 1–8 correspond to the vertical markers in (C)–(D) and the photographs in (E). (B) Mean position RMSE and relative rotation error across segments 1 to 6 for each trial category. (C) Ground-truth (black) and estimated PCC parameters for segment 1 (purple) and segment 6 (teal). (D) Standardized sensor readings for segment 1 (purple) and segment 6 (teal). (E) Reference camera frames at instants 1–8.

Fig. 7: Shape estimation under out-of-distribution external loads. (A) Schematic of applied hand forces $F$ and curvature signs $(+,-)$ alongside photographs of the J, C, and S load cases. (B) Ground-truth (black) and reconstructed (purple) 3D shapes at the peak-bending frame of each case (light gray: straight rest pose; tip errors listed below each plot). (C) Distribution of tip position errors across each load shape for the hybrid combined estimator (purple) versus the learned model alone (teal), showing quartiles (boxes), 5th–95th percentiles (whiskers), and RMSE (diamonds and numerical labels).

Model	Params	In-dist.	Sep. rec.	Ext. load
Linear	450	33.0	32.9	–
Physics-based	330	29.9	30.0	43.6
MLP ens.	103k	14.1	18.2	83.8
GRU ens.	55k	13.2	16.1	66.3
Combined (ours)	56k	13.2	15.8	42.3

TABLE I: Tip position RMSE (mm) across calibration models. In-dist. and Sep. rec. report LOTO cross-validation over 13 trials (mean of 3 repeats). Ext. load models are trained on all 13 free-motion trials (mean of 3 repeats; –: not evaluated). Linear: global linear map of the 24 readings. MLP:

128\times 128

units over 5 frames. GRU: 64 units over 15 frames. Ens.: ensemble mean of 3 networks. Combined: GRU ensemble blended with the physics-based fallback.

Work	Sensors	3D	$\delta L$	Tip error
Martin et al. [ 27 ]	IMUs	$\checkmark$	$\times$	6% of length
Peng et al. [ 31 ]	IMUs	$\checkmark$	$\times$	20 mm
Stella et al. [ 32 ]	IMUs	$\checkmark$	$\times$	7% of length
Han et al. [ 26 ]	IMUs, bend	$\times$	$\times$	2.91% of length
SaddleSense	Bend	$\checkmark$	$\checkmark$	3.1% / 3.7% of length ∗

TABLE II: Comparison of continuum shape-estimation methods using embedded sensors. Tip errors are reported by each study; robot bodies, motion regimes, and evaluation metrics differ across works.

Continuum Robot Shape Estimation

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School for Engineering of Matter, Transport and Energy, Arizona State University, Tempe, AZ 85281, USA
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Oct 28, 2026J/K move · Enter open · S save

Controlling Continuum Robot Shape in Enclosed Space with Embedded Sensing using IMU and Actuator Feedback

Jiahe Wang, Sultan Haidar Ali, Eric Weissman, Jiefeng Sun

School for Engineering of Matter, Transport and Energy, Arizona State University, Tempe, AZ 85281, USA

Continuum Robot Control Continuum Robot Shape Estimation Real-Time Robot Control

Open paper AskPDF arXiv

Controlling continuum robots' shape to operate in confined, enclosed environments is challenging, since it cannot rely on external observation. Existing embedded shape sensing methods, although they can provide shape reconstruction, are insufficient to enable real-time shape control due to limitations in reconstruction accuracy, computational speed, and compatibility with shape-state representation. In this work, we present an embedded shape-sensing framework that uses readily available IMU and actuator feedback to reconstruct the continuum-robot shape with a neural network, providing accurate shape feedback at

400~\mathrm{Hz}

for physical closed-loop control. We systematically investigate four key reconstruction design choices: shape-state representation, network architecture, training objective, and sensing inputs, through more than

5{,}000

trained models on one- and three-segment continuum robots. The selected framework achieves global shape errors of

0.9\%L

and

1.7\%L

(of the robot length) for the one- and three-segment systems, respectively. In closed-loop control, one-segment tip tracking achieves a

6.1\%L

error and three-segment shape control achieves a

2.3\%L

global shape error, whereas IMU--PCC reconstruction using the same uncompensated measurements fails. Enclosed-space experiments further demonstrate operation without external observation. These results provide a practical framework for translating readily available embedded sensing into high-rate feedback for continuum robot closed-loop control.

Abstract

Continuum Robot Shape Estimation

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