Fiber Reinforced Elastomeric Enclosures

• FREEs are soft fluid-driven actuators featuring an elastomeric tube and helically wound fibers that program controlled deformation modes such as axial extension, twisting, and bending.
• Their design decouples force generation from kinematic constraints, making them ideal for soft robots, exoskeletons, continuum manipulators, and modular actuator systems.
• Modeling FREEs involves geometric analysis, fluid Jacobians, and constitutive comparisons, with experiments validating force predictions and control performance under various configurations.

Searching arXiv for recent and foundational FREE papers to ground the article. Fiber Reinforced Elastomeric Enclosures (FREEs) are soft, fluid-driven actuators consisting of an elastomeric tube reinforced by one or more families of helically wound fibers around a fluid-filled internal cavity. Under internal pressurization, the elastomer tends to expand, while the fibers impose geometric constraints that program coupled axial extension or contraction, twist, and radial expansion. Unlike rigid actuators, FREEs generate spatial forces and moments without imposing rigid kinematic constraints, which makes them particularly suitable for soft robots, continuum manipulators, exoskeletons, grippers, and modular parallel actuator arrangements (Bruder et al., 2018, Sedal et al., 2019).

1. Definition, structure, and distinguishing characteristics

In the formulation used across the cited literature, a FREE is a soft fluid-driven actuator built from an elastomeric cylindrical tube, a fiber reinforcement architecture, rigid end fittings, and an internal pressurized cavity. A common baseline design uses a single family of inextensible fibers wound helically at a prescribed fiber angle, denoted $\Gamma$ or $\alpha$, relative to the tube axis (Bruder et al., 2018, Sedal et al., 2019). Other variants employ two helical families or an additional longitudinal strain-limiting fiber to convert extension into bending (Danforth et al., 2020, Kim et al., 14 Jul 2025).

The defining mechanical feature of a FREE is that the fibers constrain the natural expansion of the elastomer in a geometrically programmed way. This coupling allows pressure-driven deformation modes including axial extension or contraction, twisting about the main axis, radial change, bending, coiling, or combinations thereof, depending on fiber layout and boundary conditions (Bruder et al., 2018, Danforth et al., 2020, Kim et al., 14 Jul 2025). In the single-family cylindrical idealization, the principal design parameter is the fiber angle; changing this angle alters the balance between axial force and torsional moment, and more generally the relationship between pressure, deformation, and generated wrench (Bruder et al., 2018).

FREEs differ from conventional rigid actuators in that they do not incorporate a rigid kinematic constraint such as a bearing-constrained shaft or lead screw. Instead, they can impart spatial forces and moments to attached bodies without restricting those bodies to the same directions of motion. This suggests a useful distinction between force generation and motion constraint: rigid actuators usually combine both in a single mechanism, whereas FREEs decouple them by remaining inherently compliant (Bruder et al., 2018). A plausible implication is that FREE arrays can be embedded into soft structures without introducing rigid joints while still producing multi-degree-of-freedom actuation.

The application space described in the literature is correspondingly broad. FREEs are reported as actuators in exoskeletons and wearable assistive devices, soft manipulators and tentacles, soft grippers and hands, locomoting soft robots and snake-like robots, vibration isolators, and parallel actuator arrangements (Sedal et al., 2019). In later assembly-level work, they are also treated as modular building blocks for soft continuum arms intended for assistive devices, agriculture, search applications, or surgery (Kim et al., 14 Jul 2025).

2. Geometry, kinematics, and deformation programming by fiber architecture

The simplest and most widely analyzed FREE geometry is a cylindrical tube with a single helical fiber family. Under the ideal assumptions of inextensible fibers, negligible wall thickness, cylindrical shape, and no bending or taper, the deformation state is commonly represented by generalized coordinates

$$\vec{q}= \begin{bmatrix} \Delta l\ \Delta \phi \end{bmatrix},$$

where $\Delta l$ is change in length and $\Delta \phi$ is twist about the main axis (Bruder et al., 2018). The associated generalized force vector is

$$\vec{\tau}= \begin{bmatrix} F\ M \end{bmatrix},$$

with $F$ the axial force and $M$ the torsional moment (Bruder et al., 2018).

For a relaxed configuration with length $L$, radius $R$, and fiber angle $\alpha$0, the fiber length and the number of fiber revolutions are given by

$\alpha$1

These quantities remain fixed under the inextensible-fiber assumption (Bruder et al., 2018). The current length is

$\alpha$2

and the fiber-length constraint yields the current radius

$\alpha$3

The internal volume then follows as

$\alpha$4

Thus, under the ideal cylindrical model, length, twist, radius, and volume are not independent but are all determined by $\alpha$5 and the design parameters $\alpha$6 (Bruder et al., 2018).

The fiber angle governs the qualitative actuation mode. Low $\alpha$7, with fibers more aligned to the axis, favors axial contraction under pressurization and relatively smaller twist; high $\alpha$8, with fibers more circumferential, allows more axial extension and often produces extension with twist (Bruder et al., 2018). In one formulation that includes pressure on the end caps, the ratio between axial force and twisting moment is

$\alpha$9

This establishes fiber angle as a direct programming variable for the force–moment balance (Bruder et al., 2018).

A related geometric result appears in work on biologically inspired bending FREEs. For an extending FREE with relaxed fiber angle $$\vec{q}= \begin{bmatrix} \Delta l\ \Delta \phi \end{bmatrix},$$0, the behavior transitions at the “neutral angle”

$$\vec{q}= \begin{bmatrix} \Delta l\ \Delta \phi \end{bmatrix},$$1

If $$\vec{q}= \begin{bmatrix} \Delta l\ \Delta \phi \end{bmatrix},$$2, the actuator is contracting under pressurization; if $$\vec{q}= \begin{bmatrix} \Delta l\ \Delta \phi \end{bmatrix},$$3, it is extending (Danforth et al., 2020, Danforth et al., 2020). The same geometric threshold is discussed in constitutive studies of fiber-reinforced elastomers as the “magic angle” $$\vec{q}= \begin{bmatrix} \Delta l\ \Delta \phi \end{bmatrix},$$4, the angle maximizing enclosed volume for fixed fiber length (Chatterjee et al., 2019). This suggests a deep connection between FREE actuation mode selection and angle-dependent anisotropic stress states in fiber-reinforced elastomers.

When a longitudinal strain-limiting fiber is added to an extending FREE, axial extension is converted into bending. Using the geometric relations

$$\vec{q}= \begin{bmatrix} \Delta l\ \Delta \phi \end{bmatrix},$$5

and the FREE radius–length relation

$$\vec{q}= \begin{bmatrix} \Delta l\ \Delta \phi \end{bmatrix},$$6

the curvature is

$$\vec{q}= \begin{bmatrix} \Delta l\ \Delta \phi \end{bmatrix},$$7

and the maximum curvature becomes

$$\vec{q}= \begin{bmatrix} \Delta l\ \Delta \phi \end{bmatrix},$$8

In that framework, $$\vec{q}= \begin{bmatrix} \Delta l\ \Delta \phi \end{bmatrix},$$9 is monotonically increasing in $\Delta l$0 for $\Delta l$1, so larger fiber angles yield larger attainable curvature (Danforth et al., 2020).

3. Force generation, fluid Jacobians, and pressure-to-wrench mappings

A central line of FREE research formulates actuation through a state-dependent fluid Jacobian. For a single cavity actuator with volume $\Delta l$2, volumetric flow satisfies

$\Delta l$3

For the cylindrical single-family FREE with $\Delta l$4,

$\Delta l$5

Using power balance between mechanical power and fluid power,

$\Delta l$6

one obtains the pressure-to-force relation

$\Delta l$7

For a given state, the generalized force direction is encoded in $\Delta l$8, while the magnitude scales linearly with the scalar pressure $\Delta l$9 (Bruder et al., 2018).

This model explicitly separates active fiber-generated forces from passive elastomeric forces: $\Delta \phi$0 The pressure model concerns $\Delta \phi$1, while passive elastomer response is treated separately or subtracted experimentally (Bruder et al., 2018).

The same work generalizes the approach to arrays of $\Delta \phi$2 FREEs mounted in parallel to a common end effector. For actuator $\Delta \phi$3, the local generalized force $\Delta \phi$4 is mapped into a 6D end-effector wrench by its attachment offset $\Delta \phi$5 and axis direction $\Delta \phi$6. The net wrench becomes

$\Delta \phi$7

where $\Delta \phi$8 is end-effector state and $\Delta \phi$9 is the stacked fluid Jacobian in end-effector coordinates (Bruder et al., 2018). This formulation treats a parallel FREE assembly as a state-dependent linear map from pressure inputs to spatial wrench.

A closely related lumped model, developed for dynamic control of a single FREE, resolves pressure-induced axial force and torsional moment through thin-wall equilibrium. With linearized elastomer force and damping,

$$\vec{\tau}= \begin{bmatrix} F\ M \end{bmatrix},$$0

the governing equations become

$$\vec{\tau}= \begin{bmatrix} F\ M \end{bmatrix},$$1

$$\vec{\tau}= \begin{bmatrix} F\ M \end{bmatrix},$$2

Under low-pressure operating conditions for the studied actuator, simulations showed $$\vec{\tau}= \begin{bmatrix} F\ M \end{bmatrix},$$3 and $$\vec{\tau}= \begin{bmatrix} F\ M \end{bmatrix},$$4 change less than $$\vec{\tau}= \begin{bmatrix} F\ M \end{bmatrix},$$5, allowing a linearized control model with $$\vec{\tau}= \begin{bmatrix} F\ M \end{bmatrix},$$6 and $$\vec{\tau}= \begin{bmatrix} F\ M \end{bmatrix},$$7 (Habibian, 2019, Habibian et al., 2021).

In more recent continuum-arm simulation work, FREE physics is embedded into a Cosserat rod model. There, the fiber-angle kinematics are written in terms of axial and radial stretches

$$\vec{\tau}= \begin{bmatrix} F\ M \end{bmatrix},$$8

and current fiber angles satisfy

$$\vec{\tau}= \begin{bmatrix} F\ M \end{bmatrix},$$9

The corresponding axial force and torsional couple are then taken from analytical FREE theory as functions of $F$0, $F$1, $F$2, and $F$3, and applied as distributed loads to a Cosserat rod (Kim et al., 14 Jul 2025). This suggests a hierarchy of models: Jacobian-based quasi-static wrench models, lumped dynamic end-cap models, and rod-based assembly models all use pressure-induced force transmission constrained by fiber geometry, but at different levels of abstraction.

4. Constitutive modeling and model classes

The constitutive modeling of FREEs spans several levels of fidelity. A comparative study evaluates three model classes for single fiber-family FREEs: a linear lumped-parameter model, a nonlinear continuum-mechanics model, and a neural network model (Sedal et al., 2019). The benchmark comprises 8 FREE designs, 22,880 pressure–kinematics configurations, and approximately 12,575 unbuckled data points (Sedal et al., 2019).

The linear lumped-parameter model imposes an inextensible-fiber kinematic constraint and a thin-wall approximation, from which the volume

$F$4

and the fluid Jacobian

$F$5

are derived (Sedal et al., 2019). With pressure-generated generalized force

$F$6

and a linear wall stiffness

$F$7

the full model is

$F$8

This model uses only three experimentally identified scalars $F$9, but has a singularity at

$M$0

where the fibers become parallel to the axis (Sedal et al., 2019).

The nonlinear continuum model treats the wall as an incompressible hyperelastic composite with finite strain. Using the kinematic mapping

$M$1

and a deformation gradient

$M$2

it decomposes the Helmholtz free energy as

$M$3

with

$M$4

Axial force and moment then follow from stress integration: $M$5 This model uses only two physical parameters, $M$6 and $M$7, corresponding to elastomer and fiber stiffness (Sedal et al., 2019).

The neural network model maps $M$8 to $M$9 using a shallow fully connected network with one hidden layer of 6 neurons, hyperbolic tangent activation, and 62 total parameters (Sedal et al., 2019). It achieves the best actuator-specific accuracy when trained and tested on the same sample, but does not generalize well across design space (Sedal et al., 2019).

The comparison yields a characteristic trade-off. The neural net achieves the highest peak performance, while the first principles-based models generalize best across all actuator design parameters tested (Sedal et al., 2019). Quantitatively, the continuum model has the lowest average error across all training–test combinations, approximately $L$0, with moderate error range, whereas the lumped model has the lowest single-case error, $L$1, but an average error of approximately $L$2, and the neural network can reach diagonal errors around $L$3 but with errors up to approximately $L$4 under cross-design generalization (Sedal et al., 2019). A plausible implication is that constitutive structure is particularly valuable when FREE models are intended for new geometries rather than re-identification of one fabricated specimen.

Beyond actuator-level models, continuum constitutive work on fiber-reinforced elastomers supplies angle-dependent material laws relevant to FREE walls. A hyperelastic model of incompressible fiber-reinforced elastomer sheets proposes

$L$5

with Cauchy stress

$L$6

Experiments on PDMS composites with fiber orientations from $L$7 to $L$8 show that fibers are extensible and that the inextensibility constraint is invalid in those specimens (Chatterjee et al., 2019). This does not invalidate inextensible-fiber FREE models as ideal design tools, but it indicates that high-fidelity FREE wall simulations may require extensible-fiber constitutive laws rather than hard geometric constraints.

A further extension introduces finite-strain viscoelastic models for fiber-reinforced composites using multiplicative decomposition of the deformation gradient and fiber stretch. That framework includes Mooney–Rivlin or Neo–Hookean matrix models, Holzapfel-type fiber potentials, modified potentials for compressive buckling and slackness, and efficient time stepping for matrix and fiber viscoelasticity (Tagiltsev et al., 2018). Although developed for pressurized multi-layer composite pipes, the formulation maps directly onto FREE-like structures under large deformation and offers a route to modeling creep, stress relaxation, and hysteresis (Tagiltsev et al., 2018).

5. Parallel combinations, modules, and force zonotopes

FREEs are especially important as components of parallel soft actuation systems because their wrenches superimpose without introducing rigid joints (Bruder et al., 2018). In a parallel array, each actuator contributes only along the half-ray generated by positive pressure. Because a single actuator cannot reverse its force direction without negative pressure, multi-dimensional controllability requires antagonistic arrangements (Bruder et al., 2018).

The wrench set achievable at a given state $L$9 under pressure bounds $R$0 is the force zonotope

$R$1

Equivalently, with actuator direction vectors

$R$2

$R$3

which is also the convex hull of the $R$4 extreme points generated by $R$5 (Bruder et al., 2018). The zonotope is used as a design and evaluation tool because it respects the directional nature of each actuator and the positivity of pressure bounds (Bruder et al., 2018).

Several design implications follow directly. Coverage of a desired wrench set $R$6 requires $R$7 at all relevant states. The zonotope also reveals whether an arrangement has sufficient antagonism to generate both signs of each wrench component, whether wrench capabilities are isotropic or anisotropic, and how actuation degenerates across the workspace as $R$8 changes (Bruder et al., 2018). The work argues that in an $R$9-dimensional wrench space, at least $\alpha$00 suitably arranged actuators are needed for full control under positive-pressure constraints (Bruder et al., 2018).

An experimental example uses three FREEs in parallel, all aligned with the end-effector $\alpha$01-axis and attached symmetrically around it. The three actuators are one extending FREE with $\alpha$02 and two contracting FREEs with $\alpha$03 and $\alpha$04, all with relaxed $\alpha$05 and $\alpha$06 (Bruder et al., 2018). This arrangement yields a well-balanced zonotope in the 2D wrench space $\alpha$07, enabling full 2D control with the minimal three actuators (Bruder et al., 2018).

Module-level studies in finite elements examine assemblies of four FREEs arranged at the corners of a square and attached to a common end-effector (Habibian, 2019, Habibian et al., 2021). In one LR configuration, diagonally opposite actuators have opposite handedness, allowing selective generation of extension, twist, or bending via different pressure patterns. In the reported cases, equal pressurization of all four FREEs produces pure axial motion, diagonal pairs produce net twist, and adjacent pairs produce approximate bending (Habibian, 2019). Finite element results further show that a 30-degree difference in winding angle dramatically alters the shape of the workspace generated by four FREEs assembled into a module (Habibian et al., 2021).

A more recent assembly framework uses multiple FREEs as Cosserat rods connected in parallel and serial combinations. It studies a BR2 arm, composed of one bending FREE and two twisting FREEs in parallel, and a two-segment BR2-B3 or B3-BR2 arm with six FREEs total (Kim et al., 14 Jul 2025). This work does not present a zonotope analysis, but it demonstrates the same general principle: distinct FREE types can be combined into modules that generate rich 3D bending, twisting, and stretching through coordinated pressure inputs (Kim et al., 14 Jul 2025).

6. Experimental validation, applications, and limitations

Experimental validation of FREE models spans force prediction, dynamic control, morphology emulation, and assembly-level reconstruction. In the foundational parallel-actuation study, a 2-DOF test platform evaluated 125 pressure triplets at each of four end-effector configurations for two independent sets of FREEs. The active wrench was isolated by subtracting the zero-pressure measurement,

$\alpha$08

and compared against $\alpha$09 (Bruder et al., 2018). The reported prediction errors were less than $\alpha$10 N force and less than $\alpha$11 Nm moment in RMSE (Bruder et al., 2018).

The comparative actuator-modeling study generated benchmark data from eight FREE samples spanning a broad design space, with fiber angles

$\alpha$12

and approximately $\alpha$13 mm, $\alpha$14 mm, $\alpha$15 mm (Sedal et al., 2019). Each sample was tested over a grid of axial displacement, twist, and pressure, then all buckled configurations were excluded, leaving approximately $\alpha$16 of the configurations as usable unbuckled data (Sedal et al., 2019). Buckling occurred in approximately $\alpha$17 of tested configurations, with no simple pattern (Sedal et al., 2019).

A dynamical control study demonstrates that a simple lumped-parameter model can support pressure-based rotational control of a single FREE. With a closed-loop controller embedded in the system, the lumped model predicts the actual rotational motion of a FREE with at most $\alpha$18 error (Habibian et al., 2021). In thesis-level detail, step-input rotation experiments yield approximately $\alpha$19 RMS deviation between simulation and experiment, while polynomial trajectory following yields approximately $\alpha$20 RMS deviation (Habibian, 2019). This suggests that low-order models remain effective for control design even when they abstract away much of the wall mechanics.

Application studies illustrate how FREE geometry can be tuned to match biological motion. In a snake-mimic platform, extending-type FREEs with strain-limiting fibers were configured as head, midsection, and tail segments to emulate anti-predator thrashing in three snake genera. The curvature values of the fabricated soft-robotic head, midsection, and tail segments overlap with those exhibited by live snakes, and robot motion durations were less than or equal to those of snakes for all three genera (Danforth et al., 2020). That work uses fiber angles $\alpha$21 and $\alpha$22 in different modules, together with longitudinal strain limiters, to produce straight, kinked, curved, or coiled segments (Danforth et al., 2020).

Assembly-level experiments validate rod-based simulation against optical motion capture and physics-informed reconstruction. In serial twist–twist and bend–bend assemblies, as well as a full BR2-B3 arm, errors in tip position, total twist, total bending, and total elongation are typically below $\alpha$23 (Kim et al., 14 Jul 2025). The same digital twin framework was used for interactive forward control of a six-FREE arm manipulating through a constrained opening, with successful retrieval in 20 out of 20 real-world trials using replayed pressure sequences (Kim et al., 14 Jul 2025).

Several limitations recur across the literature. Ideal cylindrical models fail when FREEs bend, buckle, kink, or taper (Bruder et al., 2018). Buckling was observed experimentally, especially for a $\alpha$24 FREE at higher pressures, reducing effective axial force transmission (Bruder et al., 2018). More generally, buckled states are excluded from many benchmark analyses because the standard models assume cylindrical, unbuckled geometries (Sedal et al., 2019). Passive elastomeric forces, hysteresis, fluid compressibility, rate dependence, and friction or fiber slip are also typically neglected or approximated (Bruder et al., 2018, Sedal et al., 2019, Habibian, 2019).

A common misconception is that fiber geometry alone determines FREE behavior. The evidence is more specific. Fiber angle is the dominant kinematic programming parameter, but finite element results indicate that variations in the material properties of the elastic enclosure of a FREE are more significant than variations in fiber properties (Habibian et al., 2021). In module studies and single-actuator FEA, elastomer stiffness changes deformation much more strongly than fiber stiffness once the fibers are already much stiffer than the matrix (Habibian, 2019, Habibian et al., 2021). Another misconception is that inextensible-fiber models are universally exact; planar constitutive experiments on fiber-reinforced elastomers show measurable fiber extensibility and angle-dependent fiber strain (Chatterjee et al., 2019). The idealized inextensible-fiber assumption is therefore best understood as a modeling abstraction that can be highly useful but is not universally literal.

The broader trajectory of FREE research points toward increasingly integrated models. Quasi-static fluid Jacobians support wrench design and controllability analysis (Bruder et al., 2018). Continuum and constitutive models support generalization across geometries (Sedal et al., 2019). Finite-strain viscoelastic and wave models indicate how damping, compressive buckling, and transient wall mechanics might be incorporated (Tagiltsev et al., 2018, Cheviakov et al., 2019). Rod-based assembly simulation enables digital twins for modular continuum arms (Kim et al., 14 Jul 2025). This suggests a layered methodology for future work: use low-order geometric models for design intuition and control allocation, continuum constitutive models for design transfer and material selection, and assembly-level simulators for task-level behavior in complex soft robotic systems.