In this work, we present a closed-form model, which describes the kinematics of fiber-reinforced elastomeric enclosures (FREEs). A FREE actuator consists of a thin elastomeric tube surrounded by reinforcing helical fibers. Previous models for the motion of FREEs have relied on the successive compositions of “instantaneous” kinematics or complex elastomer models. The model presented in this work classifies each FREE by the ratio of the length of its fibers. This ratio defines the behavior of the FREE regardless of the other parameters. With this ratio defined, the kinematic state of the FREE can then be completely described by one of the fiber angles. The simple, analytic nature of the model presented in this work facilitates the understanding and design of FREE actuators. We demonstrate the application of this model in an actuator design case study.

## Introduction

Soft, fluid-driven actuators use structured compliance to create motion from the expansion of flexible volumes. Actuators based on these principles may, for example, contract along a central axis like biological muscles [1]. The McKibben muscle is a popular variety of such “pneumatic artificial muscles.” Each McKibben muscle consists of an elastomeric tube surrounded by a sleeve of braided helical fibers. The braid is made up of equal numbers of right-handed and left-handed helices of the same pitch. They were developed in the 1950 s by their namesake, Joseph Laws McKibben [2]. Since that time, they have been used in numerous applications such as legged robots and human assistive devices [3].

Bishop-Moser [4] extended the functional principle of McKibben muscles to a generalized class of cylindrical soft actuators known as fiber-reinforced elastomeric enclosures (FREEs). Like McKibben muscles, FREE actuators are formed from two sets of identical helical fibers. A “set” or “family” of fibers is a group of fibers characterized by the same angle with respect to the cylinder axis (e.g., α or β). In the McKibben muscle, the two sets maintain equal and opposite angles (α = −β). In a FREE, the fibers are wound with fiber angles selected by a designer to result in a desired behavior (Fig. 1). This choice of configuration permits complex actuated behaviors such as twisting while extending or contracting.

The kinematics of FREEs was initially developed only for small deformations. Krishnan et al. [5] described the motion of FREEs with instantaneous strain equations. These equations described the transformation of the FREE as a small change relative to its current configuration. To calculate the evolution of a FREE actuator over large deformations, the instantaneous strains were successively composed. This fiber-only kinematic model allowed the designer to consider the kinematics of the FREE fibers under tension without considering the elastomer.

Combined fiber-elastomer methods model the motion of specific FREE actuators with specified geometries and defined elastomeric properties. For instance, finite element solvers can be used to model FREEs [6]. Recent work has begun to explore the use of constitutive models to predict the motion of unloaded contracting, twisting, or bending fiber-reinforced actuators [7] and torsionally loaded fiber-reinforced actuators [8]. These constitutive methods rely on models of strain energy in the elastomers and fibers to relate internal pressures to predicted deformations for specific actuator geometries. The governing equations of Ref. [5] can also be adapted to consider the elastomer [9].

The model presented in this work allows the designer to consider the kinematics of FREE fibers independent of specific geometry and material choices. For instance, in an finite element analysis model, one would need to define the inner and outer diameter of the elastomer, the elastic material, a model for the elastic behavior, the length of the actuator, the two pitches of the fiber families, the fiber material, the number of reinforcing fibers in each family, the pressure ranges, etc. Once these choices have been made, a computationally intensive process must be conducted to evaluate the behavior of the specified FREE. After conducting a number of these numerical experiments, it is perhaps possible to extract some design heuristics from the observed behavior, but it is difficult to generalize the results.

In the model presented in this work, the behavior of the FREE actuators is described in terms of a single variable. This variable, η, is the ratio the lengths of the fibers in the two sets and is independent of the specific actuator geometry and materials. η is a design choice that is conserved over the course of actuation. The introduction of η allows this work to extend models developed for McKibben muscles [10] into the broader class of FREE actuators. The mathematical model presented in this work describes large-deformation kinematics of FREEs in closed-form and without the need to compose a succession of instantaneous strains.

The kinematic model presented here describes the length, diameter, rotation, and volume of FREE actuators as functions of the fiber angle β. The structure of this model provides a common language to describe every cylindrical, two-helical-fiber-set FREE by parameters that define the behavior (η) and state (β) of the actuator. The size and geometry of the actuator is given by the length bβ of the β fibers and the diameter D0 (when β = −π/2). The simple, analytic nature of the model presented in this work facilitates the understanding and design of FREE actuators.

## Kinematic Modeling

### Assumptions and Definitions.

Our model assumes that the fiber-reinforced actuators are made from two sets of helical, inextensible fibers with respective angles α and β (Fig. 1). The fibers surround an elastomeric bladder containing the pressurized fluid. We assume that sufficient fibers are used to prevent the bladder from bulging between the gaps in the fibers. The elastomer is assumed to be infinitely extensible with negligible stiffness. The fibers are assumed to be always under tension from the internal pressure in the bladder. Because the individual fibers within the families are identical, the kinematics of only one fiber in each family need to be considered.

We also assume that the profile of the FREE actuator remains cylindrical. This approximates an unbent actuator away from its ends. At the actuator ends, the diameter tapers to match the fixed-diameter of the end [11]. The cylindrical assumption allows us to adapt simple helical formulas that have long been used to describe McKibben muscles [10]. The length of this cylinder is l and its diameter is D. All fibers wrap around it in a helical fashion. The individual fibers in their respective families behave identically. They all have the same axial length l and diameter D as the cylinder. Under these assumptions, we can relate the length of the cylinder l to the “unwound” length b of the fibers in a family via the cosine of their angle
$l=bα cos(α)=bβ cos(β)$
(1)
Similarly, we can express the diameter of the cylinder D as a function of the fiber angles and fiber lengths
$D=bα sin(α)nαπ=bβ sin(β)nβπ$
(2)
The diameter is additionally a function of the number n of times that the fiber circles the axis. For example, nα = 0.75 signifies that, at the current diameter, the fibers of the α family circle the axis three-quarters of one time. The sign of n indicates the handedness of the helix (positive for right handed) and matches the sign of the corresponding fiber angle
$sign(nα)=sign(α)sign(nβ)=sign(β)$
(3)

The cylinder diameter and length will change as the actuator volume expands. These changes will correspond to changes in the fiber angles α and β. The number n of fiber turns may also change as the actuator ends rotate relative to one another about the cylinder axis. The unwound length b of the fibers, however, remains constant.

Krishnan et al. [5] analyzed the instantaneous kinematics of FREEs with values of α and β between −90 deg and 90 deg. As they noted, however, the symmetry of FREEs makes this formulation redundant. We eliminate this redundancy, without loss of generality, by deliberately specifying which fiber family is labeled by α and which is labeled by β. We use α to describe the family with the greater or equal unwound fiber length b (i.e., bαbβ, and thus $|α|≥|β|$). Furthermore, we can restrict our analysis to helices formed by the α fibers that are right handed and thus maintain a positive value of α. Because α is always positive, and α is strictly greater than β (β is negative when $|α|=|β|$), both angles cannot be equal to zero simultaneously
$0<α≤π2−α≤β<α$
(4)

The behavior of FREE actuators with left-handed (negative α) α fibers will simply be mirror-symmetric to the behavior modeled here (e.g., rotation that is modeled as counterclockwise will be clockwise in the actuator with left-handed α fibers).

### Behavior Described by η, State by β.

The behavior of a FREE actuator is defined by the ratio η of the unwound lengths b of the fibers from the two sets
$η=bβbα0<η≤1$
(5)
Under the assumptions of this model, the angle α follows uniquely from the angle β. With η, a simple analytic relationship between the fiber angles can be developed from Eqs. (5) and (1)
$α=cos−1(η cos(β))$
(6)

Equation (6) provides a clear functional relationship between the angles of each FREE (Fig. 2) parameterized by η.

The relationship between the fiber angles described by Eq. (6) has been observed previously but never defined in such an explicit form. Krishnan et al. [5] noted that each two-fiber-family FREE belongs to “a one-dimensional family of fiber angle configurations.” With the model presented here, it is now clear that the ratio η and Eq. (6) can be used to define this one-dimensional family of fiber angle configurations.

### Size Described by bβ and D0.

The length, diameter, and volume of a FREE actuator can be described as functions of η and β and variables that describe the dimensions of the particular actuator, bβ and D0. The first of these, bβ, is the unwound length of the individual fibers in the β family (Figs. 1 and 3). The diameter D0 is a standardizing measurement used for McKibben muscle actuators [10]. This quantity can also be defined for FREE actuators (Fig. 3). D0 is calculated from Eq. (2). It is the diameter of the helices if β = −π/2 (and thus, α = π/2)
$D0=bαn0,απ=−bβn0,βπ$
(7)

where n0 is the number of fiber turns at that diameter.

Thus, each FREE actuator can be defined by the actuator-specific quantities η, bβ, and D0. The state of the actuator is given by β. Examples of FREEs with the same values of bβ and D0 (but different values of η and β) are shown in Fig. 4.

### Calculating Rotation Δn, Diameter D, Length l, Volume V, and Surface Area Asurf.

The model presented in this work allows the length, rotation, diameter and volume of a FREE actuator to be described as functions of the actuator state β. The expression for the axial length l(β) (Fig. 5) is straightforward and is given by Eq. (1)
$l(β)=bβ cos(β)$
(8)

Note that the length of the actuator scales linearly with the length bβ and that the maximum length achievable by the actuator is bβ.

The axial rotation of the actuator is designated by Δn and is conserved in the fiber turns nα and nβ at D
$nα=n0,α+Δnnβ=n0,β+Δn$
(9)
where Δn is zero when D = D0. Equation (2) leads to an expression for Δn
$Δn=−n0,β sin(β)+sin(α) sin(α)−η sin(β)$
(10)
Note that there is no rotation when η = 1 (McKibben muscle)
$η=1→β=−α→ sin(β)+sin(α)=sin(−α)+sin(α)=0→Δn=0$
(11)
Substituting Eq. (6) into Eq. (10) gives the axial rotation Δn(β) in radians
$Δn(β)=−n0,β1−η2 cos2(β)+sin(β)1−η2 cos2(β)−η sin(β)$
(12)

The rotation Δn scales linearly with $n0,β=(−bβ/πD0)$ (Fig. 6).

The diameter D(β) is found by substituting Eqs. (6), (7), (9), and (12) into Eq. (2) (Fig. 5)
$D(β)=D01−η2 cos2(β)−η sin(β)1+η$
(13)

The diameter scales linearly with D0.

Assuming that the thickness of the elastomer inside the fibers is negligible, the internal volume of the actuator is given by the volume V(β) of the cylinder contained within the fiber helices (Fig. 7)
$V(β)=π4D(β)2l(β)$
(14)
Substituting Eqs. (8) and (13) into Eq. (14) yields
$V(β)=π4D02bβ cos(β)(1−η2 cos2(β)−η sin(β))2(1+η)2$
(15)
From Fig. 7, it is clear that the volume of each FREE type has a unique maximal point. For example, for the McKibben muscle case when η = 1 and α = −β, the volume is maximized when β ≈ −54.7 deg. This has long been known [10]. Here, the value of β that maximizes the volume is designated βLM. Smaller values of η correspond to less negative values of βLM. Because internal pressures drive the volume to expand, an actuator fabricated with a β value greater than βLM will decrease in β under actuation. Similarly, actuators with β less than βLM will increase in β under actuation. The derivative of the volume with respect to β is given by
$dVdβ=−π4D02bβ(γ−η sin(β))2(sin(β)γ+2η cos2(β))(1+η)2γγ=1−η2 cos2(β)$
(16)
The maximum volume occurs when $dV/dβ$ is zero. Equation (16) yields the following invertible expression relating η to the angle βLM that maximizes the volume
$dVdβ=0→η=1 cos(βLM)4 cot2(βLM)+1−2 tan−1(2−3)≤βLM<0$
(17)
The α and β combinations that maximize FREE volume were described by Krishnan et al. [5] as a “locked manifold” (Fig. 2)
$1+2 cot(αLM)cot(βLM)=0$
(18)

where the corresponding angles are designated with the subscript LM. “Locked” refers to the fact that internal pressure can no longer drive the actuator to deform because the volume is already maximized. Substituting Eq. (6) into Eq. (18) yields Eq. (17).

## Example of Model Application: Actuator Design

Our model enables simple, closed-form design analyses for FREE actuators. As an example, this section applies the proposed model to design an actuator to meet specified kinematics.

Consider a FREE that is specified to contract from an unpressurized length l1 = 5 cm to l2 = 4 cm, while rotating a quarter-of-a-turn about its axis. The diameter of the actuator at the contracted state is to be D2 = 2.5 cm. To ensure that the contracted configuration will be achievable through pressurization, the angle β2 of the fibers in the contracted state is specified to be 10 deg greater than the angle βLM that maximizes the volume.

By inspection, for an axially contracting actuator, the following must be true:
$−54.7 deg<βLM<β2<β1<0$
(19)
The initial unpressurized angle β1 can be selected numerically such that the quarter-turn rotation constraint is satisfied
$n1,β−n2,β=bβ sin(β1)D1π−bβ sin(β2)D2π=0.25$
(20)
where the rotation comes from the difference in the number of fiber turns given by inverting Eq. (2). The unknown values in Eq. (20) are functions of the specified kinematics and β1. bβ comes from inverting the length expression in Eq. (1)
$bβ=l1 cos(β1)$
(21)
The angle β2 of the fibers in the contracted state is given by inverting Eq. (1) and considering that β2 must be negative due to Eq. (19)
$β2=−cos−1(l2bβ)$
(22)
The angle βLM is given by the constraint that β2 be 10 deg greater than βLM
$βLM=β2−10π180$
(23)
which, by Eq. (17), can be used to calculate η. The value of D0 is found by inverting Eq. (13) with the values of D2, β2 and η
$D0=D21+η1−η2 cos2(β2)−η sin(β2)$
(24)
which allows D1 to be calculated with Eq. (13), η and β1.

The design specifications are achieved by fabricating an actuator with an unpressurized initial angle β1 of −8.2 deg (Fig. 8, η = 0.714, bβ = 5.05 cm, D0 = 3.4 cm).

## Discussion

The model presented in this work provides a closed-form framework for kinematic analysis and design of FREE actuators. The introduction of η and the analytical relationship between the fiber angles given in Eq. (6) is one of the major contributions of this work. Previously published fiber-only models have relied on “instantaneous” kinematics [5] to incrementally update fiber angles. To solve for large deformations with instantaneous kinematics, the nonlinear equations had to be iteratively solved and composed. The model presented in this work provides analytic functions describing the actuator rotation, length, diameter, and volume. These functions are parameterized by the kinematic state of the actuator given by the angle β. The simplicity of the presented model simplifies the design and understanding of FREEs. In this work, for example, we have shown how the model can be used to design a FREE that achieves desired kinematic behavior.

In addition to the closed-form kinematics, the model presented in this work has several improvements to previous FREE fiber-only models. The deliberate designation of the longer set of fibers with α allows the model presented in this work to describe FREEs with just a diagonal quadrant of the α-β coordinate frame. The ratio η leads to a simple parametric functional relationship between α and β (the first of its kind) allows the state of a FREE with a particular η value to be parameterized by a single angle β (rather than describing the state with potentially infeasible combinations of α and β). Previous work discovered that “every FREE belongs to a one-dimensional family of fiber angle configurations” [5]. This work provides the first analytic description of these configurations.

The model presented in this work shares the assumptions of previously published models fiber models [4,5]. The reformulation presented here is a simplification of the kinematic description in the previous models. Accordingly, the experimental verification of the previous models can be considered verification of the present work.

Like the models that have preceded it, the present model has limitations. External loading, for example, could buckle one or both of the fiber families. This would violate the assumption that the fibers are under tension. The model presented in this work is limited to rotation and/or length changes. Additional fiber families on FREEs can create planar [7] or helical [4,12] bends. The angles of the fibers in this model are constrained to be nonzero. So-called “straight-fiber” actuators are not governed by the equations presented here [13]. The governing equations of Ref. [5] have been adapted to include elastomer effects and noncylindrical deformation [9].

The model presented in this work does not take into account noncylindrical deformations or strain in the elastomers. The unmodeled elastomer strain will limit the motion of a FREE actuator to a small section of the possible fiber-angle-combinations defined by η. To account for the effects of the elastomer, a designer could take the insights from this fiber-only model and further explore them with fiber-elastomer models (e.g., see Refs. [69]).

The model presented in this work will facilitate the growing understanding of FREEs. The identification of the descriptor η allows the behavior of FREEs to be described independent of actuator-specific geometry. The linearly scaling functions make this behavior simple to predict and understand. As the understanding of FREEs grows, engineers will find new opportunities for these unique actuators to expand the functional ability of soft, fluid-driven systems.

## Acknowledgment

The authors would like to acknowledge Joshua Bishop-Moser and Daniel Bruder for their thoughtful reviews of this work prior to submission and Audrey Sedal for helping the authors understand the constitutive models that have been used to describe FREEs in other works. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE 1256260. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

## Funding Data

• National Science Foundation Graduate Research Fellowship (Grant No. DGE 1256260).

• Toyota Research Institute.

## Nomenclature

• bα, bβ =

unwound length of the individual helical fibers in the family

•
• D =

diameter of the cylindrical actuator

•
• D0 =

the diameter of the zero-height helical fibers when β = −π/2

•
• l =

axial length of the cylindrical actuator

•
• n0,α, n0,β =

the number of turns around the axis made by each fiber at the diameter D0

•
• nα, nβ =

the number of turns around the axis made by each fiber at the current diameter

•
• V =

volume of the cylindrical actuator

•
• α =

angle of the family of helical fibers with equal or longer unwound length

•
• αLM, βLM =

values of α and β on the locked manifold for a given value of η

•
• β =

angle of the other family of helical fibers

•
• γ =

the quotient of the sine of α and η

•
• Δn =

the axial rotation between the actuator ends (β = −π/2 → Δn = 0)

•
• η =

ratio of the lengths of the β fibers to that of the α fibers

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