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Design and Characterization of a Variable-Length Continuum Mechanism with Force Locking

Katelyn King, Veronica Fish, Allison M. Okamura

arXiv:2609.30759v1cs.RO

TL;DR

Few mechanical designs combine variable length and stiffness in a miniaturizable continuum mechanism, despite applications that call for both flexible and rigid behavior. This paper presents a rod-driven helical backbone that bends while extending or contracting, and reports that force locking increases axial stiffness and anisotropic bending stiffness. The authors identify a limited rigid workspace and single-plane bending as current boundaries.

  • Problem

    Few mechanical designs combine variable length and variable stiffness, and existing combined designs are not sufficiently miniaturized because of their complexity.

    Source: §I · p. 1

  • Method

    A helical backbone driven by two push-pull rods extends and contracts, while coil contact force-locks the backbone and allows bending at any length.

    Source: A. Working Principle · p. 2, B. Force Locking · p. 2

  • Results

    Force locking increases axial stiffness and contributes to anisotropic bending stiffness, which is highest along loading directions with an axial component.

    Source: A. Effects of Force Locking · p. 5

  • Takeaways & Limitations

    The continuous helical mechanism combines variable length with variable stiffness in a miniaturizable device.

    Source: §V · p. 6

    The rigid workspace is small relative to the flexible range, and the device is limited to single-plane bending.

    Source: §V · pp. 6–7

Abstract

from arXiv · show

The utility of flexible continuum mechanisms for dexterous navigation is often impaired by their low stiffness, making them ineffective at manipulation in high-force scenarios. To address this challenge, we propose a novel continuum mechanism that achieves both flexible and rigid behavior by antagonistic extension and contraction of a rod-driven continuum helical structure. The helical design combines variable-length capacity with force locking for workspace and stiffness enhancement. In this article, we present the detailed design of the proposed mechanism and characterize its performance through experiments that quantify bending and stiffness. The results demonstrate 180 degree bending range of motion with an average distal positioning error of <10%. Further tests demonstrate that force locking directly improves axial stiffness and thus indirectly increases bending stiffness anisotropically, with maximum bending stiffness along load paths with a large axial component. Tensioning the driving rods provides additional stiffness tunability in the force-locked state, where increasing rod tension proportionally increases bending stiffness with a dimensionless gain of 0.56.

I. INTRODUCTION

Continuum mechanisms trade flexibility for stiffness, yet many applications need both safe navigation and high-force manipulation. Few mechanical designs combine variable length and variable stiffness in a miniaturizable device, motivating a continuous, force-locking backbone.

I. INTRODUCTION

  • Flexible continuum mechanisms generally have low stiffness, while applications may require both delicate navigation and high-force manipulation.Source: §I · p. 1
  • Few mechanical designs combine variable length and variable stiffness, and existing combined designs are not sufficiently miniaturized because of their complexity.Source: §I · p. 1
  • Continuous backbones support length variation, whereas discrete backbones enable stiffness control through force locking, making their architectures difficult to unify.Source: §I · p. 1
  • A continuous helical cut can fully compress to enable force locking while retaining a continuous, extensible architecture.Source: §I · p. 1

II. MECHANISM DESIGN

The mechanism uses two push-pull rods to extend, contract, and bend a helical backbone. Coil contact creates a rigid mode without eliminating bending across the device’s length range.

A. Working Principle

  • Two rods routed 180° apart drive the helical backbone, with pushing or pulling them extending or contracting it until adjacent coils contact.Source: A. Working Principle · p. 2

B. Force Locking

  • Coil contact along the inner arc creates a rigid mode; with space between coils, inner-rod tension instead bends the flexible mechanism.Source: B. Force Locking · p. 2
  • Antagonistic push-pull actuation lets the device bend continuously at any length, including when fully contracted.Source: B. Force Locking · p. 2

C. Backbone Architecture

The backbone uses overlapping helical units to form a continuous, rod-routed structure. Its geometry sets the contracted length, while experiments characterize axial stiffness, bending range, and bending stiffness.

C. Backbone Architecture

  • Adjacent helical units alternate chirality, are offset by 180°, and overlap by half a coil to create a rigid segment between units.Source: C. Backbone Architecture · p. 2, Figure 2 · p. 2
  • The V-shaped inter-unit segment aligns active coils during compression, while the coil cross section forms a hinge when contacted coils bend.Source: Figure 2 · p. 2, C. Backbone Architecture · p. 3
  • Backbone geometry sets contracted length, but parameter combinations with equal relaxed and contracted lengths can still produce different mechanical properties.Source: §III · p. 3

III. EXPERIMENTAL SETUP & RESULTS

  • Experiments characterize axial stiffness, bending range of motion, and bending stiffness, examining effects of backbone geometry, length, and rod forces.Source: §III · p. 3

A. Hardware Setup

The testbed independently actuates two rods and measures tip forces during controlled displacement. Three prototype geometries were fabricated, and their measured contracted lengths differed from model predictions.

A. Hardware Setup

B. Prototype Fabrication

  • Three prototypes varied coils per unit across T=3, 4, and 5, while solid height and coil gap were adjusted to maintain common relaxed and contracted lengths.Source: B. Prototype Fabrication · p. 3
  • Axial stiffness was characterized by compressing each prototype from relaxed to contracted length and recording rod forces over five trials.Source: Figure 5 · p. 4
  • The contracted-length model assumes coil gaps and inter-unit flexures close fully; incomplete flexure closure likely explains its underestimation.Source: B. Prototype Fabrication · p. 4
  • Measured contracted lengths varied by up to 10% from prediction, while relaxed lengths were consistent across the samples.Source: B. Prototype Fabrication · p. 4

C. Axial Stiffness

Axial-stiffness characterization compresses three prototypes from relaxed to contracted length and plots the resulting force–displacement curves; Figure 6 also contrasts flexible and force-locked configurations.

C. Axial Stiffness

  • Three prototypes were compressed from relaxed to contracted length while the resulting rod forces were measured.The measurements characterize variation in axial stiffness across the samples.Source: C. Axial Stiffness · p. 4
  • Five repeated trials produced the force–displacement curves plotted in Figure 5.Source: C. Axial Stiffness · p. 4
  • In Figure 6, each rigid configuration has at least one fully contracted rod to achieve force locking.The figure contrasts relaxed, flexible and contracted, rigid backbones at various bending curvatures.Source: Figure 6 · p. 4

D. Bending Range of Motion

The prototypes were tested through 180° bending at four starting lengths, and the trajectories show increasing positioning error at higher curvature due to hysteresis.

D. Bending Range of Motion

  • The prototypes were driven through bending angles from 0° to 180° at four uncurved starting lengths, from fully contracted to relaxed.The outer rod was pushed and pulled while the inner rod remained in tension at a fixed length.Source: D. Bending Range of Motion · p. 4
  • At high curvatures, friction-related hysteresis increases positioning error relative to the theoretical constant-curvature trajectory.Figure 7 distinguishes rigid workspace at fully contracted length from flexible workspace at other lengths.Source: Figure 7 · p. 5
  • Table III reports positioning and hysteresis error, while the supplied passage gives no table values.Source: Table III · p. 5

E. Bending Stiffness

Bending stiffness was characterized across poses and loading directions in flexible and rigid states, including tests with increased inner-rod tension.

E. Bending Stiffness

  • Stiffness tests compared flexible, relaxed backbones with rigid, fully contracted backbones at 0°, 30°, 60°, and 90° along the −x-axis.Rigid-state trials used increasing inner-rod tension, beginning at the force of initial coil contact.Source: E. Bending Stiffness · p. 5
  • A cycle pushed the tip 3 mm along the +x-axis, pulled it 6 mm along the −x-axis, then pushed it 3 mm along the +x-axis; the same procedure estimated y-axis stiffness at 30°, 60°, and 90°.Source: E. Bending Stiffness · p. 5
  • Y-axis stiffness was tested at 30°, 60°, and 90°; the 0° test was omitted to avoid damaging the force sensor.Figure 8 compares flexible and rigid bending stiffness, while Figure 9 plots stiffness change with increased inner-rod tension.Source: E. Bending Stiffness · p. 5

IV. DISCUSSION

The discussion frames backbone design, applied rod force, and force locking as the main factors affecting mechanism performance.

IV. DISCUSSION

  • The discussion identifies backbone design, applied rod force, and force locking as the main performance factors.Source: §IV · p. 5
  • The discussion considers how these factors affect axial stiffness and bending stiffness.Source: §IV · p. 5
  • Bending curvature is also among the performance outcomes considered.Source: §IV · p. 5

A. Effects of Force Locking

Force locking raises bending stiffness anisotropically, with the greatest stiffness along loading directions that have an axial component. Increasing rod tension further increases stiffness in the uncurved state.

A. Effects of Force Locking

  • Rigid-state bending stiffness is higher than flexible-state stiffness in every tested condition, with higher stiffness along y than x and the greatest stiffness along directions with an axial component.The y-direction stiffness decreases as bending angle increases, consistent with decreasing axial load.Source: A. Effects of Force Locking · p. 5, Figure 8 · p. 6
  • Backbone contraction increases stiffness through both force locking and the shortened backbone's reduced moment arm.Pulling along the x-axis is generally stiffer than pushing in curved poses because pulling with the curvature further compresses the backbone; uncurved push and pull stiffness is similar.Source: A. Effects of Force Locking · p. 5

B. Effect of Rod Forces

  • In curved poses, inner-rod force primarily resists x-axis pushing, while force locking between inner coils helps resist x-axis pulling.The outer-rod tension must remain below inner-rod tension to curve the backbone.Source: B. Effect of Rod Forces · pp. 5–6
  • Increasing rod tension improves uncurved-state stiffness across the tested backbone geometries and push/pull directions.Fig. 9 aggregates these conditions and reports the same positive trend for each.Source: B. Effect of Rod Forces · p. 5, Figure 9 · p. 6

C. Effect of Backbone Geometry

Backbone geometry sets the contracted rigid length, but its parameters also shape stiffness and positioning accuracy. More coils improve the approximation of a continuous bending arc, while fewer coils are associated with greater maximum positioning error.

C. Effect of Backbone Geometry

  • Backbone geometry sets the rigid contracted length, while coil parameters must be tuned for acceptable axial and bending stiffness.Minimizing helical-unit spring rate increases the range of bending stiffness, whereas increasing coils per unit greatly reduces axial stiffness.Source: C. Effect of Backbone Geometry · p. 6
  • Maximum positioning error increases as coils per unit decrease, with the T = 3 prototype deforming mainly at its base as friction limited tension transfer to the tip.More coils provide additional notches that better approximate a continuous arc and help reduce positioning error.Source: C. Effect of Backbone Geometry · p. 6

V. CONCLUSIONS AND FUTURE WORK

The paper presents a rod-driven helical continuum mechanism that combines variable length with force locking for flexible and rigid behavior. Its stiffness is anisotropic, and its current rigid workspace and bending freedom remain limited.

V. CONCLUSIONS AND FUTURE WORK

  • The rod-driven helical mechanism combines variable length and force locking to provide flexible and rigid behavior in a continuous backbone.Coil compression increases axial stiffness, which anisotropically improves bending stiffness along loading directions with an axial component.Source: §V · p. 6
  • The rigid workspace is smaller than the flexible range because curvature and stiffness control are coupled, and the device currently bends in a single plane.The authors identify 3D navigation and multiple curvatures as future work.Source: §V · pp. 6–7
  • Future work includes assessing multi-rod locking and coil geometry, and reducing friction to mitigate bending hysteresis.The authors also propose studying alternating segment chirality for torsional rigidity and exploring low-friction sheaths.Source: §V · p. 7
  • The authors identify minimally invasive surgery as a potential application, where flexible navigation and force application may be needed in the same procedure.They describe endoscopic overtube use and note that gastrointestinal end-effectors may need to bend up to 180°.Source: §V · p. 7
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