Peano-HASEL and Curling-HASEL Architectures

• Peano-HASEL and Curling-HASEL architectures are soft actuators based on HASEL technology that utilize dielectric liquid redistribution for mechanical actuation.
• They consist of modular subsystems integrating electrical, hydraulic, and mechanical dynamics and are modeled using a rigorous port-Hamiltonian framework for precise energy management.
• Advanced control via IDA-PBC with integral action ensures rapid response, effective disturbance rejection, and robust performance as validated by simulations and experiments.

Peano-HASEL and Curling-HASEL architectures represent a class of soft actuators based on Hydraulically Amplified Self-healing Electrostatic (HASEL) technology. Curling-HASEL actuators, in particular, employ modular, electrically driven mechanisms where deformation is governed by the redistribution of dielectric liquid in response to electric fields and mechanically coupled elastic elements. Recent research provides a rigorous port-Hamiltonian modeling and control framework for these systems, enabling advanced performance in soft robotics and electrically controlled morphing structures (Cisneros et al., 2024).

1. Modular Subsystem Composition

Curling-HASEL actuators are constructed from the serial interconnection of $n$ identical elementary subsystems (typically $n=4$), each exhibiting a coupled electro-mechanical-hydraulic structure. Each subsystem consists of:

• Electrical branch: A variable-length capacitor $C_{s_i}$ (electrodes facing moving dielectric liquid) arranged in parallel with an inductor $L_i$ and a series resistance $r_{L_i}$, incorporating a leakage conductance $1/R_i$.
• Hydraulic constraint: A total constant liquid volume, distributed between the chamber (within electrodes) and the shell (outer envelope). Electrode zipping/unzipping is parameterized by length $l_{e,i}$, and this redistribution couples electrical charge to mechanical configuration through fluid displacement.
• Mechanical branch: Comprises a torsional spring (stiffness $K_{b,i}$, angle $\theta_i$) at the base film, a linear spring (stiffness $K_i$, length $l_{p,i}$) modeling the top film stretch, a lumped inertia $M_i$ about $\theta_i$, and viscous damping $b_i$.

Subsystems share a common high-voltage input $U_{in}$. Changes in capacitor area $l_{e,i}(\theta_i)$ induce Maxwell stress, pumping the dielectric liquid into the shell, increasing the triangular area $A_s(\theta_i, l_{p,i})$, and actuating the structure against the mechanical elements. All $l_{e,i}$ are subject to the fluid volume constraint: $A_s + X_h(L_e - l_{e}) = \mathrm{const}$ for prescribed $X_h$ (shell height) and $L_e$ (electrode length).

2. Port-Hamiltonian System Formulation

The port-Hamiltonian (PH) modeling framework encapsulates multi-domain dynamics within a structured energy-based formulation. For curling-HASEL, the state vector and input are:

$x = [\theta; l_p; p; \phi; Q] \in \mathbb{R}^{5n}$

• $\theta \in \mathbb{R}^n$: torsional angles,
• $l_p \in \mathbb{R}^n$: top-film lengths,
• $p \in \mathbb{R}^n$: angular momenta ($p = M \dot{\theta}$),
• $\phi \in \mathbb{R}^n$: inductor fluxes,
• $Q \in \mathbb{R}^n$: capacitor charges,
• $u = U_{in} \in \mathbb{R}$: the common actuation voltage.

The total energy (Hamiltonian) is:

$$H(x) = \frac12\,\theta^T K_b\,\theta + \frac14\,(l_p - L_p)^T K\,(l_p - L_p) + \frac12\,p^T M^{-1}p + \frac12\,\phi^T L^{-1}\phi + \frac12\,Q^T C^{-1}Q$$

State evolution follows:

$\dot x = [J(x)-R(x)]\nabla_x H(x) + G(x)u$

where $J(x)-R(x)$ is the interconnection-plus-dissipation matrix, $G(x)$ encodes the input structure, and $y = G(x)^T\nabla_x H(x) = i_e$ yields the total leakage current.

3. Subsystem Dynamics and Fluid-Volume Constraints

• Electrical Dynamics:

$\dot{\phi} = -r_L L^{-1} \phi + Q$

$\dot{Q} = -Q/R + g_a(\theta)\,U_{in}$

with $$g_{a,i}(\theta_i) = \gamma_1 \cos(\gamma_2 \theta_i)$$ governing electromechanical coupling.

• Hydraulic (Volume) Constraint:

For each $i$:

$$A_{s,i}(\theta_i, l_{p,i}) + X_h[L_e - l_{e,i}(\theta_i, l_{p,i})] = A_T \ (\mathrm{const.})$$

with

$$A_{s,i} = \frac14 l_{p,i} L_v \sin \delta_i, \quad \delta_i = \frac{\pi + \theta_i}{2} - \arcsin\left(\frac{L_v}{l_{p,i}} \sin\left(\frac{\pi - \theta_i}{2}\right)\right)$$

The parameter $d_i = 2A_{s,i}/l_{p,i}$ features in $J(x)-R(x)$.

• Mechanical Dynamics:

$\dot{\theta} = p/M$

$\dot{l}_p = d\, (p/M)$

$$\dot{p} = -K_b\,\theta - K\,(l_p-L_p) - b\, (p/M) + \frac{\partial}{\partial \theta}\left(\frac12 Q^2/C_{s}(\theta, l_p)\right)$$

The last term effects electro-mechanical feedback due to charge-storage dependence on geometry.

4. Parameter Identification and Model Validation

For a typical subsystem, the following parameters were empirically identified:

Parameter	Symbol	Value
Top-film length	$L_p$	$15\,\mathrm{mm}$
Vertical gap	$L_v$	$15\,\mathrm{mm}$
Electrode length	$L_e$	$15\,\mathrm{mm}$
Shell height	$X_h$	$2\,\mathrm{mm}$
Mass	$m$	$0.047\,\mathrm{kg}$
Width	$w$	$50\,\mathrm{mm}$
Film thickness	$t$	$18\,\mu\mathrm{m}$
Relative permittivity	$\epsilon_r$	$2.2$
Vacuum permittivity	$\epsilon_0$	$8.85 \times 10^{-12}\,\mathrm{F/m}$
Resistance	$R_i$	$10\,\Omega$
Series resistance	$r_{L_i}$	$20\,\Omega$
Inductance	$L_i$	$150\,\mathrm{H}$
Linear spring	$K_i$	$400\,\mathrm{N/m}$
Torsional spring	$K_{b,i}$	$0.202\,\mathrm{N{\cdot}m/rad}$
Damping	$b_i$	$0.0199\,\mathrm{N{\cdot}s/m}$
Coupling coeff. 1	$\gamma_1$	$104.33$
Coupling coeff. 2	$\gamma_2$	$7.67$

The model fit was approximately 90% on identification data and 85–89% on validation (Cisneros et al., 2024).

5. Control Strategy: IDA-PBC with Integral Action

Position regulation of curling-HASEL is achieved via Interconnection and Damping Assignment-Passivity Based Control (IDA-PBC) supplemented with integral action for disturbance rejection. The closed-loop PH form is:

$\dot{x} = [J_d - R_d]\nabla H_d$

with desired energy function

$$H_d = \frac12(\theta - \theta^*)^T \tilde{K}_b(\theta - \theta^*) + \frac14(l_p - l_p^*)^T \tilde{K}(l_p - l_p^*) + \frac12 p^T M^{-1} p + \frac12(\phi - \phi^*)^T \tilde{K}_{\phi}(\phi - \phi^*) + \frac12(Q - Q^*)^T \tilde{K}_Q(Q - Q^*)$$

The state-feedback law is:

$$u(x) = (\bar{R} g_a)^{-1} \bigg\{-\tilde{K}_b(\theta-\theta^*) -M^{-1}p - r_{55}\,\tilde{K}_Q(Q-Q^*) + L^{-1}\phi + (\bar{R} C^{-1} Q) \bigg\}$$

Integral action introduces an additional controller state $x_c \in \mathbb{R}^n$ and augments the closed-loop energy:

$H_{cl} = H_d + \frac12 K_{int}\|Q - x_c\|^2$

The integral control law is:

$$u_{int} = -r_{55} K_{int} (Q - x_c), \quad \dot{x}_c = -\alpha_1 \tilde{K}_b (\theta - \theta^*) - \alpha_3 M^{-1} p$$

with design constants $\alpha_1=1$, $\alpha_3=1$, $K_{int}>0$. This structure-preserving scheme effects rejection of unknown load torques $d_u$ and input disturbances $d_a$.

6. Simulation and Experimental Outcomes

Model-in-the-loop simulations confirm endpoint regulation to $h^* = 20\,\mathrm{mm}$ for the actuated beam under IDA-PBC. Increasing $\tilde{K}_b$ raises closed-loop bandwidth, resulting in a rise time of $0.1 - 0.3\,\mathrm{s}$. With integral action, the system exhibits robust rejection of disturbances, restoring the endpoint with less than $\pm 1\,\mathrm{mm}$ error following (i) an external torque $d_u = -0.04\,\mathrm{N{\cdot}m}$ at $t = 3\,\mathrm{s}$ and (ii) an input-voltage drop $d_a = -30\,\mathrm{V}$ at $t = 7\,\mathrm{s}$. The control voltage remains below $10\,\mathrm{kV}$.

Experimental validation on a four-link curling-HASEL prototype, employing laser-profile feedback and a Trek 610E amplifier, yields less than $5\%$ steady-state tracking error and robust disturbance rejection, closely matching simulation results (Cisneros et al., 2024).

7. Implications and Application Domains

The modular, port-Hamiltonian description of curling-HASEL actuators enables scalable modeling, precise energy accounting, and structure-preserving feedback. This facilitates the integration of soft actuators in robotics, compliant mechanisms, and adaptive structures where disturbance rejection, dynamic regulation, and safety by design are paramount.

A plausible implication is that these modeling and control techniques could inform future soft actuator platforms, especially where robust interaction with unstructured environments and distributed control are required.