Helically Wound Structured Electrostatic Layer Jamming

• The paper demonstrates that HWS-ELJ employs a helical electrode design to exponentially amplify stiffness via voltage-induced frictional forces.
• The mechanism uses flexible copper foil electrodes laminated with polyimide on a cylindrical core to achieve rapid, reversible control of stiffness.
• Experimental validation shows that HWS-ELJ can achieve over 12× mechanical amplification and a fourfold increase in stiffness in robotic finger prototypes.

Helically Wound Structured Electrostatic Layer Jamming (HWS-ELJ) is a variable stiffness mechanism that employs electrostatic attraction in a helical geometry to modulate interlayer friction, facilitating tunable stiffness in robotic and articulated devices. By leveraging a helically wound configuration of flexible electrodes on a cylindrical substrate, HWS-ELJ achieves an exponential amplification of effective stiffness with respect to the wrap angle, outperforming conventional planar electrostatic layer jamming approaches in both range and compactness. The mechanism is voltage-controlled, enabling rapid, reversible, and programmable transitions between compliant and stiffened states, and has been validated both theoretically and experimentally, including integration in robotic finger prototypes (Bai et al., 25 Dec 2025).

1. Structural and Material Design

HWS-ELJ's architecture consists of two flexible copper foil electrodes, each laminated with polyimide (PI) dielectric films and arranged in direct contact along a cylindrical core of radius $R$. Electrode 1 is affixed within a singular helical groove on the cylinder, providing a stationary path, while Electrode 2 is helically wound atop the core with constant pitch $H$, ensuring $H >$ electrode width $\omega$ to preclude overlaps. The total helical wrap is characterized by angle $\Phi$ (for example, $\Phi \approx 450^\circ$ in the reference prototype). The spatial trajectory of the winding can be described as

$$\vec{r}(\phi) = (R \cos \phi,\, R \sin \phi,\, (H/2\pi)\phi),\ \phi \in [0,\, \Phi].$$

When a high voltage $V$ is applied between the electrodes, they act as a distributed parallel-plate capacitor separated by a combined dielectric thickness $d_e = d_1 + d_2$ (PI films). The resulting electrostatic normal pressure substantially enhances frictional resistance along the helical interface, with the frictional force directly tunable via $V$. For the construction described in (Bai et al., 25 Dec 2025), typical specifications include 8 mm (fixed) and 7 mm (wound) electrode widths, $9$–$10$ µm PI lamination, and a helical core fabricated using additive manufacturing techniques.

2. Physical Principles and Theoretical Modelling

The principle of operation relies on voltage-induced normal pressure amplifying interlayer static friction, thereby increasing resistance to sliding under external loads—a mechanism termed "layer jamming." In the HWS-ELJ configuration, the geometrically extended helical interface provides an exponential scaling effect on the attainable stiffness.

For an infinitesimal helical segment parameterized by arc length $s$, local curvature $\kappa = R/(R^2 + (H/2\pi)^2)$, and $a = \sqrt{R^2 + (H/2\pi)^2}$, the force balance includes:

• Tension $T(s)$ tangential to the helix,
• Normal support force $N(s)$,
• Electrostatic per-unit-length load $q_e = \epsilon_0 \epsilon_e \omega V^2/(2 d_e^2)$,
• Static friction $F_f = \mu dN$ with $\mu$ the measured static friction coefficient (e.g., $\mu \approx 0.22$ for PI-PI contact).

The coupled equilibrium equations are: \begin{align*} & \frac{dT}{ds} = \mu \frac{dN}{ds}, \ & \frac{dN}{ds} = \kappa T + q_e. \end{align*}

Elimination and integration yield:

$$T(s) = T_0 e^{\mu \kappa s} + \left[\frac{\epsilon_0 \epsilon_e \omega V^2}{2 d_e^2 \kappa}\right] (e^{\mu \kappa s} - 1),$$

where $T_0$ is preload. At the terminal end $s=L=a\Phi$, the resulting tension—and thus effective sliding resistance—increases exponentially with wrap angle and geometric parameters, in contrast to the linear scaling observed in planar ELJ configurations.

Effective stiffness, interpreted as resistance to sliding under displacement, scales proportionally to the exponential term $e^{\mu \kappa L}$, conferring substantial amplification for modest geometric increases.

3. Experimental Characterization and Quantitative Performance

Prototypes were constructed using 3D-printed cylindrical cores with 450° helical grooves, copper foil electrodes as described above, and dielectric-laminated PI films. Experimental evaluation examined three specimens subject to varying preloads ($T_0 \approx 0.25\,\mathrm{N}$, $0.50\,\mathrm{N}$, $1.00\,\mathrm{N}$), realized by hanging masses of 25 g, 50 g, and 100 g, respectively.

The test apparatus included:

• Mounting on a multi-axis force/torque sensor,
• Vertical suspension of the winding electrode’s free end under load,
• Stepper motor actuation for controlled winding,
• Application of $0–3800\,\mathrm{V}$ across electrodes (in $400\,\mathrm{V}$ increments).

The force sensor measured vertical and torque responses, with derived total tension $F_f=\sqrt{F_{z1}^2 + (T_z/r)^2}$, sampled at $10\,\mathrm{Hz}$. Data processing involved outlier removal and extraction of steady-state means.

Results demonstrated a quadratic dependence of tension on voltage, confirming theoretical predictions:

$$T(V) = \alpha V^2 + T_0,\quad \alpha \approx 5.14 \times 10^{-8}\,\mathrm{N}/\mathrm{V}^2,$$

with fitting examples:

• $$T(V) = 5.138 \times 10^{-8} V^2 + 0.902\,\mathrm{N}$$ (preload $0.25\,\mathrm{N}$),
• $$T(V) = 5.138 \times 10^{-8} V^2 + 1.804\,\mathrm{N}$$ (preload $0.50\,\mathrm{N}$),
• $$T(V) = 5.138 \times 10^{-8} V^2 + 3.608\,\mathrm{N}$$ (preload $1.00\,\mathrm{N}$).

At $V \approx 3800\,\mathrm{V}$ and $T_0 = 1\,\mathrm{N}$, achieved tension exceeded $12\,\mathrm{N}$, corresponding to over $12 \times$ mechanical amplification. Empirical outcomes consistently surpassed theoretical baselines, an effect attributed to edge-field enhancements and atmospheric pressure acting at asperity contacts.

4. Comparative Analysis: HWS-ELJ Versus Planar ELJ

The helical geometry central to HWS-ELJ induces an exponential scaling in normal force and thus frictional resistance, differentiating it fundamentally from planar ELJ systems, which offer only linear scaling as a function of applied voltage and electrode area. For a given electrode contact area, the HWS-ELJ provides significantly higher stiffness enhancement, reducing the mechanical and electrical footprint required for equivalent modulation. This geometric amplification, manifest as the $e^{\mu \kappa L}$ term, establishes HWS-ELJ as more suitable for compact, integrated designs in variable-stiffness joints.

A plausible implication is that highly miniaturized actuators and compliant mechanisms benefit disproportionately from the HWS-ELJ architecture compared to planar approaches, as they can exploit the exponential stiffness gain for constrained volumes (Bai et al., 25 Dec 2025).

5. Integration with Robotic Systems and Functional Demonstration

A two-link robotic finger prototype was fabricated, employing a 3D-printed design with Segment 1 (fixed) and Segment 2 (articulated). The HWS-ELJ core, incorporating Electrode 1 and the fully-wound Electrode 2 (360° wrap), was embedded in the finger, with mechanical preloading via a coil spring. High-voltage routing was integrated into the base structure, with housing grounding via the negative terminal.

Performance was evaluated by suspending tip loads ($0$–$180\,\mathrm{g}$) and tracking the resulting bending angle using electromagnetic sensors. Bending stiffness was quantified as $k=F_\text{pull}/\Delta\theta$. Results indicated a stiffness increase from $k \approx 0.06\,\mathrm{N}/\mathrm{deg}$ (0 V) to $k \approx 0.25\,\mathrm{N}/\mathrm{deg}$ (3000 V), a greater than fourfold amplification. System response time was limited only by the rise time of the high-voltage supply ($<100\,\mathrm{ms}$), indicating suitability for real-time robotic applications. The finger demonstrated the ability to alternately lift fragile objects under compliant and stiffened conditions by switching the applied voltage (Bai et al., 25 Dec 2025).

6. Practical Significance and Application Scope

HWS-ELJ enables rapid, reversible, voltage-programmable modulation of joint or structure stiffness, presenting a compact solution for next-generation variable-stiffness mechanisms. Empirical validation confirms that devices with an active area on the order of $7 \times 16\pi\,\mathrm{mm}^2$ can realize more than tenfold stiffness modulation with response times under a few hundred milliseconds, even at moderate preloads (0.25–1 N) and voltages (1–3.8 kV) (Bai et al., 25 Dec 2025). This performance supports diverse applications in robotic manipulation, safe human-robot interaction, and adaptive actuation, where both compliance and rigidity are required within the same device architecture.

In summary, HWS-ELJ advances electrostatic jamming by exponentiating the mechanical amplification achievable with helical geometries, offering an experimentally substantiated, voltage-controlled, compact, and integrable variable stiffness paradigm.