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

Katelyn King; Veronica Fish; Allison M. Okamura

cs.ROSep 25, 2026

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

Authors: Katelyn King, Veronica Fish, Allison M. Okamura

Organizations: Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA

Abstract

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.

Figures & tables

Fig. 1: Working principle of the variable-length, force-locking continuum mechanism. (a) Equally pulling both rods contracts the device. The minimum length of the device when it is fully contracted is $L_{c}$ . (b) Equally pushing both rods extends the device. (c) Shortening the inner rod relative to the outer rod bends the device.

Fig. 2: Backbone design showing cross section geometry. As shown, one unit has four coils and adjacent units overlap by half a coil; a complete unit and two partial units are shown. (a) Backbone cross section geometry in the $x$ - $z$ plane and in the bending ( $x$ - $y$ ) plane. Holes through the center of each cross section facilitate rod-routing. (b) Contact between coils during bending indicating coil cross section geometry.

Symbol	Description	Dimension
$x_{w}$	Coil cross section width	$2$ mm
$x_{h}$	Coil cross section solid height	$1.29$ mm
$x_{g}$	Coil gap	$0.5$ mm
$r_{t}$	Coil cross section top radius	$1.25$ mm
$r_{b}$	Coil cross section bottom radius	$2.5$ mm
$d_{h}$	Channel diameter	$0.6$ mm

TABLE I: Geometric Parameters and Nominal Dimensions

Fig. 3: Experimental testbed showing curved poses ( $\theta\in[30^{\circ},60^{\circ},90^{\circ}]$ ) and test samples ( $T\in[3,4,5]$ ). The gantry-mounted force sensor pushes/pulls the tip of each sample $3$ mm along the $x$ - and $y$ -axes. Stepper-driven lead screws (not pictured) proximal to the fixed base drive the rods.

Fig. 4: Manufactured samples showing differences in coil geometry. Small variations in contracted length were observed, but are not visible at this scale.

	$T=3$	$T=4$	$T=5$
$x_{h}$	$1.98$ mm	$1.29$ mm	$0.90$ mm
$x_{g}$	$0.81$ mm	$0.5$ mm	$0.33$ mm

TABLE II: Geometric Dimensions of Test Samples

Fig. 5: Force-displacement curves quantifying the axial stiffness of three backbone prototypes with varied coils per unit. Gray lines represent the sum of the tension applied by the inner and outer rod while pulling the backbone from its relaxed length to its fully contracted length in each of the five trials. The black points are the average of all five trials. Compressing the backbone first deforms the helical segments which behave like a Hookean spring with stiffness $k_{f}$ indicated by the slope of the solid red line. Once the coils are fully compressed, the flexures begin to close, introducing non-linearity in the curve. The horizontal line estimates the minimum force ( $F_{c}$ ) required to fully contract each backbone. After this point, all the notches are closed, requiring more force to further compress the material, quantified by the slope of the dashed red line ( $k_{r}$ ).

Fig. 6: Relaxed (flexible) v. contracted (rigid) backbone ( $T=4$ ) at various bending curvatures. In the rigid state, at least one rod is fully contracted in each case to achieve the force-locked state.

Fig. 7: Bending curvature in the $x$ - $y$ plane. Pushing and pulling the outer rod bends the mechanism in the range of [ $0^{\circ}$ , $180^{\circ}$ ]. The fixed length ( $L$ ) of the inner rod (equal to the uncurved starting length of the backbone) is indicated for each trajectory. The fully contracted length ( $L_{c}$ ) maps the rigid workspace, while all other lengths are within the flexible workspace. The dashed lines indicate the theoretical constant curvature bending trajectory. High friction results in large hysteresis error at high curvatures.

$T$		Average position error (mm)	Maximum position error (mm)	Average hysteresis error (mm)	Maximum hysteresis error (mm)
$3$	$L$	$3.12$	$9.62$	$1.44$	$3.66$
	$L_{c}$	$5.10$	$9.13$	$0.99$	$3.15$
$4$	$L$	$3.22$	$7.91$	$2.72$	$6.48$
	$L_{c}$	$5.60$	$7.86$	$1.12$	$2.73$
$5$	$L$	$3.99$	$7.16$	$3.57$	$7.01$
	$L_{c}$	$3.05$	$6.56$	$1.38$	$3.68$

TABLE III: Positioning and Hysteresis Error

Fig. 8: Bending stiffness in each curved pose ( $30^{\circ},60^{\circ},90^{\circ}$ ) in each loading direction ( $\pm x,\pm y$ ) for backbone geometry with $T=4$ . Stiffness is higher in the rigid state compared to the flexible state in all conditions. Stiffness is anisotropic, with higher forces measured along the $y$ -direction than the $x$ -direction. In the $x$ -direction, pulling stiffness is generally higher than pushing. In the $y$ -direction, stiffness decreases with increasing bending angle, consistent with decreasing axial load.

Fig. 9: Effect of increasing rod tension on the stiffness of the uncurved ( $0^{\circ}$ ) mechanism loaded along the $x$ -axis. Tension is normalized relative to the minimum inner rod tension required to fully contract the backbone ( $F_{i,c}$ ); in the uncurved state, equal tension is applied to the inner and outer rods. $k_{x,c}$ is the stiffness measured for $F_{i,c}$ . Data are aggregated across backbone geometry and push/pull loading directions, as all conditions exhibit the same positive trend.

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Continuum robots are well suited for constrained environments but suffer from low distal stiffness, resulting in large posture errors under external loads. In this paper, we propose a novel structural primitive, the Distal-Stable Beam, which achieves a strong stiffness gradient through purely geometric design, maintaining compliance in the intermediate section while ensuring high distal rigidity. The structure consists of two parallel rods and one convergent rod constrained by guide disks, introducing geometric coupling that suppresses deformation modes and preserves distal posture. Experiments show that the distal stiffness is 12 times higher than at the center, corresponding to an approximately 100-fold improvement over a conventional cantilever beam. The proposed mechanism enables simultaneous compliance and distal stability without active stiffness modulation, providing a new design approach for continuum robots requiring both safety and precision.

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