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

See all 20 papers

Oct 28, 2026

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

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.

Jiahe Wang, Sultan Haidar Ali, Eric Weissman +1

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

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Sep 14, 2026

Tendon-Driven Continuum Robot with Modular Stiffness and In-Situ Self Pose Estimation

Continuum robots enable smooth shape morphing and safe interaction in confined environments. However, most existing systems are task-specific and depend on external sensing infrastructure, limiting their adaptability and real-world deployment. This paper presents a self-contained modular continuum robotic platform that combines mechanical reconfigurability with onboard pose estimation. The robot is constructed from interchangeable continuum joints with analytically precomputed stiffness, allowing rapid assembly and direct programming of the robot shape. Proprioceptive sensing is achieved using magnetic sensors and a modular learning-based framework, where a single model is trained per joint and reused across configurations. The system is experimentally validated in real world, demonstrating self-sensing capabilities and adaptation without external tracking.

Guo Ning Sue, Zheng Cao, Junzhe Hu +5

Carnegie Mellon University

Modular Robotics Robotics Robot State Estimation Robotic Perception Continuum Robot Design Continuum Robot Shape Estimation

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Nov 4, 2026

Curvature by Design: A Support Structure for Continuum Robots

Constant- and piecewise-constant-curvature models are widely used to simplify the kinematics of continuum robots, although the assumed curvature profiles are not necessarily inherent mechanical properties of the robot and may therefore be violated during operation. Rather than assuming a prescribed curvature profile in the kinematic model and estimating deviations from it, this paper presents a multi-segment scissor-based support structure that mechanically constrains a continuum robot to exhibit prescribed constant or spatially varying curvature profiles. The design uses Taylor approximations to enable an overconstrained multi-joint linkage to predictably distribute global bending among the small-angle movements of multiple flexures. The geometric coupling is realized using compliant flexure joints, enabling the support structure to be manufactured as a single piece. The structure is integrated with a tendon-actuated continuum robot with a flexible backbone. Experiments show that the structure makes the robot resistant to deformations other than the prescribed curvature, significantly enhancing shape precision compared to classical flexible-backbone designs. This constrains the robot from making S-shapes when external loads are applied in the opposite direction of the bending. By mechanically constraining the admissible deformation, the proposed structure reduces the number of independent shape variables that must be estimated.

Justus Sagemüller, Harald Minde Hansen, Kristin Y. Pettersen

Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden · Mechatronics, Robotics and Operation Section, European Organization for Nuclear Research (CERN), Meyrin, Switzerland · Department of Engineering Cybernetics, Norwegian University of Science and Technology (NTNU), Trondheim, Norway

Continuum Robot Design Mechanism Design Robot Kinematics

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Abstract

Continuum Robot Shape Estimation

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