Bending Fluidic Actuators

• Bending Fluidic Actuators are soft, pressurized devices that generate controlled, programmable curvature for applications such as shape morphing and grasping.
• They utilize design principles including pressurized networks, anisotropic strain engineering, and structural wrinkling to achieve precise, multi-modal bending profiles.
• Advances in materials, fabrication, and robust control algorithms are expanding BFA applications in soft robotics, adaptive devices, and biomedical systems.

Bending Fluidic Actuators (BFAs) are a class of soft actuators that exploit the coupling between mechanical compliance and pressurization to achieve controlled, programmable curvature in flexible structures. Their development is foundational to soft robotics, adaptive devices, and novel interface technologies, enabling shape morphing, grasping, and conformal locomotion that would be challenging or impossible with rigid-link mechanisms. The engineering, modeling, and application of BFAs span a range of materials and architectures, including pneumatic, hydraulic, ionic, and hybrid designs, and their operating principles are tightly connected to advanced concepts in continuum mechanics, nonlinear material science, and distributed control.

1. Structural Principles and Mechanisms of Bending

BFAs operate by leveraging internal pressure differentials to induce anisotropic deformation. The basic mechanism is the generation of curvature within a compliant beam or shell due to differential expansion, contraction, or wrinkling across the actuator cross-section. Three broad strategies are prominent:

1. Pressurized Networks and Chambers: In soft elastomeric or fiber-reinforced beams, embedded or surface-mounted fluidic networks generate local stresses that result in global bending profiles. The geometry and distribution of the channels (e.g., location relative to the neutral axis, network density, and cross-sectional area) determine the resultant curvature field. The relationship between network design and resulting shape deformation is formalized by equations such as

$$\frac{\partial^2 d_n}{\partial x^2} = -\varphi(x) \frac{p}{E} \left. \frac{\partial \psi}{\partial (p/E)}\right|_{p=0}$$

where $d_n(x)$ is the network-induced displacement, $\varphi(x)$ is the local channel density, $p$ is pressure, and $E$ is Young’s modulus (Matia et al., 2014).

1. Anisotropic Strain Engineering: By engineering strain anisotropy—such as tuning the in-plane ($s_{xx}$, $s_{yy}$) and through-thickness ($s_{zz}$) strains using doubly interdigitated electrodes in piezoelectric films—the ratio and distribution of deformation axes can be arbitrarily controlled. For example, the effective strains in a DIDE-structured PZT actuator are computed as

$$s_{xx} \simeq d_{33} \langle E_x^2/|E| \rangle - \frac{1}{2}d_{33} \langle (E_y^2 + E_z^2)/|E| \rangle$$

and

$$s_{yy} \simeq d_{33} \langle E_y^2/|E| \rangle - \frac{1}{2}d_{33} \langle (E_x^2 + E_z^2)/|E| \rangle$$

(Wapler et al., 2013). By adjusting the patterned electrode geometry, arbitrary bending profiles from saddle to spherical to unidirectional can be realized.

1. Structural Constraints and Wrinkling: In hybrid or shell-based actuators, forced partial wrinkling or geometric confinement selectively lowers stiffness in prescribed directions. For inflatable joints, the maximum restoring moment in the bending direction is programmably reduced via the angular width ($\Delta \theta$) of unwrinkled regions, with the restoring moment

$M_{max} = \pi P R^3 \sin(\Delta \theta/2)$

for internal pressure $P$ and beam radius $R$ (Wang et al., 16 Oct 2024).

These mechanisms allow precise programming of bending axis, amplitude, and shape by manipulating material, geometry, and boundary conditions.

2. Materials and Fabrication Approaches

BFA realization is predicated on material systems that combine compliance, resilience, and—in some cases—integrated functionality such as sensing or anisotropy. Key material/fabrication paradigms include:

• Elastomers and Reinforcement: PDMS, TPE, NR, PUR, and other elastomers are common due to their high extensibility and processability. Fiber-reinforcement, often achieved by helically winding inextensible fibers, constrains radial or longitudinal expansion to favor bending (Kelageri et al., 2018, Kelageri et al., 2018).
• Porous Architectures via Additive Manufacturing: Graded porosity, tuned via control of nozzle height/velocity and layer coiling (as in InFoam printing), imparts spatially modulated stiffness and enables programmable bending when combined with vacuum actuation. Porosity ($\phi$) is governed by relationships such as

$$\phi = 100 \left[1 - \frac{G \alpha \pi d^2}{4Wh_c}\right]$$

with $G$ as extruder constant, $\alpha$ as extrusion multiplier, $d$ as nozzle diameter, $W$ as coil width, and $h_c$ as coil height (Willemstein et al., 2022, Willemstein et al., 2023).

• Low-cost Constructs: BPActuator and xBalloon architectures use ubiquitous, inexpensive materials (balloons, polyethylene film), with geometry imparted via sewing or thermal fusing of plastics. This promotes accessibility and democratizes experimentation (Qi et al., 2021, Xie et al., 2021).
• Hybrid and Constraint-based Designs: Actuators combining soft (latex balloons, custom elastomers) and rigid shells translate internal pressure into high-force output with simplified force-area relations, obviating the need for detailed modeling of soft material deformation in many cases (Wan et al., 2020).

A notable trend is the integration of actuation and sensing in a single structure, as realized by 3D-printed conductive porous elastomers capable of piezoresistive proprioception (Willemstein et al., 2023).

3. Modeling, Analysis, and Control

Accurate modeling of BFAs requires addressing geometric and material nonlinearities, distributed pressures, anisotropic elasticity, and—in dynamic cases—elastohydrodynamic coupling. Principal modeling approaches include:

• 3D Analytical Mechanics: For fiber-reinforced actuators, fully 3D Eulerian models under toroidal symmetry account for hyperelasticity, incompressibility, distributed lateral pressure, and variable tip torque. The kinematics (e.g., principal stretches $$\lambda_{(\rho)}, \lambda_{(\theta)}, \lambda_{(\phi)}$$) and static equilibrium (moment balance) are solved subject to boundary conditions (Cacucciolo et al., 2016).
• Multibody Dynamical Modeling: Discretizing the actuator into RSDA-linked segments allows for closed-form Lagrangian equations that capture actuation dynamics, including pressure-bending relationships and tip trajectory. This framework supports system identification and real-time control (Kelageri et al., 2018).
• Distributed Network Theory: For channel-embedded beams, continuum models relate spatial channel density and pressure patterns to bending or cancellation of external deflections, using derived integro-differential relationships (Matia et al., 2014).
• Feedforward Hysteresis Compensation and Reservoir Computing: Advanced control methods combine physical reservoirs (dual PAM actuators) with Takagi–Sugeno (T–S) fuzzy logic to implement low-latency, robust compensation of hysteresis and nonlinearities, outperforming Echo State Networks (ESNs) in both accuracy and computational efficiency (Shen et al., 11 Sep 2024).

These modeling strategies allow both precise open-loop design and robust closed-loop control in varied environmental and loading conditions.

4. Performance Characteristics and Application Domains

BFAs exhibit a wide spectrum of performance, highly contingent on their geometry, material systems, and drive mechanisms. Measured capabilities include:

Actuation Type	Max Bending Angle / Curvature	Force Output	Frequency	Power / Voltages	Notable Features
3D-printed Ionic Polymer	124°; 0.82 cm⁻¹	0.76 mN	≤ 2 Hz	2.5–4.5 V	Noiseless, low voltage, biocompatible
Piezo-DIDE/PZT	Tunable (saddle/sphere/uni-dir.)	Application dep.	>kHz (elec.)	>100 V (elec. fields)	Arbitrary strain ratio, rapid actuation
FFMS (Hydraulic)	>100% strain, programmable bending	>115× own wt.	>5 Hz	Up to ~12 kPa	Sheet layout, wearable safety
Balloon-Polyethylene	Up to 35° (BPActuator)	0.07 N @50 kPa	Low	Air pressure	Extreme low cost, very accessible
Inflatable wrinkled joint	Stiffness ratio ~sin(Δθ/2) tunable	Application dep.	Application dep.	Pressure-driven	Programmable anisotropy, energy saving

These actuators enable applications including, but not limited to, wearable assistive robots, adaptive microfluidic systems, biomedical devices, haptic interfaces, soft grippers, continuum robots, and aquatic soft robots (Zhu et al., 2019, Zhang et al., 2023). For instance, the rolling dielectric elastomer actuator-based mantas robot achieves 1.25 body lengths per second in water (Zhang et al., 2023), while the hybrid balloon-shell actuator delivers >100 N block force at only 50 kPa (Wan et al., 2020).

5. Comparative Analysis and Limitations

BFAs are distinct, and often superior to, conventional rigid or unimodal actuators in terms of shape programmability, compliance, and safe human interaction. However, energetic and structural trade-offs are pronounced:

• Displacement vs. Versatility Trade-off: Architectures enabling arbitrary curvature or highly anisotropic bending (for instance, DIDE-structured PZTs) sometimes experience a reduction in maximum attainable displacement compared to isotropic or unimodal equivalents (Wapler et al., 2013).
• Efficiency and Force Generation: Hybrid rigid-soft shell approaches vastly improve efficiency and output force relative to fully soft actuators, though allowable stroke and cycle life may be limited by soft material fatigue (Wan et al., 2020).
• Programmability: Selective structural wrinkling and porosity grading introduce programmable anisotropy and compliance, allowing for energy-efficient and "sequential" or multi-axis actuation, but require precise fabrication control (Wang et al., 16 Oct 2024, Willemstein et al., 2022).
• Control Complexity: Nonlinearities, hysteretic response, and environmental interactions necessitate advanced control schemes, including feedforward compensators and physical or hybrid reservoir computing (Shen et al., 11 Sep 2024).

A plausible implication is that material selection and system design must be closely aligned to the operational target—e.g., human safety and rapid response in assistive wearables favor low-voltage ionic or hydraulic actuators, whereas high-precision, high-speed shaping (e.g., adaptive optics) favors piezoelectric or dielectric systems.

6. Prospects and Future Directions

Progress in BFA research is increasingly multidimensional:

• Integrated Sensing and Actuation: Embedded piezoresistive elements, enabled by 3D-printed graded porosity cTPE, facilitate proprioceptive sensing with hysteresis-compensated models (e.g., Wiener–Hammerstein cascades), enabling real-time closed-loop operation (Willemstein et al., 2023).
• Programmable, Distributed Architectures: Modular, multi-degree-of-freedom BFAs ("MuA-Ori" and cascaded bistable chains) enable switching between deformation modes and binary sequencing of actuation events, all with minimal control input (Ben-Haim et al., 2019, Forte et al., 2021).
• Rapid, Accessible Prototyping: Low-cost, minimal-tooling actuators (BPActuator, xBalloon) democratize soft robotics research, expanding it into education, interaction design, and DIY communities (Qi et al., 2021, Xie et al., 2021).
• Advanced Manufacturing: Techniques such as InFoam printing and direct ink writing allow both mechanical and functional property tuning (porosity, conductivity, biocompatibility) in a single manufacturing step, accelerating application-tailored design (Willemstein et al., 2022, Trümpler et al., 21 Jan 2025).
• Robust Control Paradigms: Integrating physical reservoir computing with soft actuation dynamics enables real-world robust feedforward hysteresis compensation for tracking and manipulation tasks, with execution speed and generalization advantages over classical network approaches (Shen et al., 11 Sep 2024).

This suggests ongoing evolution toward BFAs that are multifunctional, programmable at multiple scales, and seamlessly integrated with sensing and intelligent control—forming the backbone of next-generation soft machines.