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Technical Brief

# A Continuum Model for Fiber-Reinforced Soft Robot ActuatorsOPEN ACCESS

[+] Author and Article Information
Audrey Sedal

Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: asedal@umich.edu

Daniel Bruder

Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: bruderd@umich.edu

Joshua Bishop-Moser

Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: joshbm@umich.edu

Ram Vasudevan

Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: ramv@umich.edu

Sridhar Kota

Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: kota@umich.edu

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received September 22, 2017; final manuscript received January 18, 2018; published online February 12, 2018. Assoc. Editor: Andreas Mueller.

J. Mechanisms Robotics 10(2), 024501 (Feb 12, 2018) (9 pages) Paper No: JMR-17-1314; doi: 10.1115/1.4039101 History: Received September 22, 2017; Revised January 18, 2018

## Abstract

Fiber-reinforced elastomeric enclosures (FREEs) generate sophisticated motions, when pressurized, including axial rotation, extension, and compression, and serve as fundamental building blocks for soft robots in a variety of applications. However, most modeling techniques employed by researchers do not capture the key characteristics of FREEs to enable development of robust design and control schemes. Accurate and computationally efficient models that capture the nonlinearity of fibers and elastomeric components are needed. This paper presents a continuum model that captures the nonlinearities of the fiber and elastomer components as well as nonlinear relationship between applied pressure, deformation, and output forces and torque. One of the key attributes of this model is that it captures the behavior of FREEs in a computationally tractable manner with a minimum burden on experimental parameter determination. Without losing generality of the model, we validate it for a FREE with one fiber family, which is the simplest system exhibiting a combination of elongation and twist when pressurized. Experimental data in multiple kinematic configurations show agreement between our model prediction and the moments that the actuators generate. The model can be used to not only determine operational parameters but also to solve inverse problems, i.e., in design synthesis.

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## Introduction

The fiber-reinforced elastomeric enclosure (FREE) is a pneumatic actuator, which consists of an elastomeric tube with fibers wound around it in specified helical configurations. Pneumatic actuators can be particularly useful in soft robotic applications due to their flexibility under loading, physical adaptability, and the ready availability of air, the working fluid [1]. Pneumatically actuated soft robots have demonstrated capabilities in biomimetic locomotion [24], assistive wearable devices [5,6], and manipulators [7,8]. Fiber reinforcement enables soft robots to create sophisticated motions and loading, including axial extension, rotation, and torsion [912], but poses additional complexity in modeling.

Soft roboticists have developed a variety of methods to determine the behavior of FREEs under internal pressure. Many of these models rely on kinematic or kinetic assumptions that constrain the design space, like fiber symmetry [13,14] and negligible external loading [15]. Others are built on assumptions that simplify the model formulation but demand lengthy experimental parameter determination like fitting shear moduli with an equibiaxial tension experiment [16], or separate design characterization for every prototype [17]. Finite element analysis can predict FREE movement [18]; however, the time-consuming nature of finite element analysis and its inability to predict trends make it intractable for design optimization or control. It is, therefore, important to develop a model that captures the nonlinear relationship between a FREE's material, loading, and deformation and the coupling behavior it exhibits between length and torsion.

In this paper, we relate a FREE's torsional stiffness to its deformation and internal fluid pressure by extending a known hyperelastic composite modeling framework [19]. A major advantage of this formulation is its ability to predict physical phenomena, like torsion and radial expansion, in a computationally tractable way with a minimum of experimental parameter determination. In Sec. 2, we define the continuum model for a FREE with one fiber family, which is the simplest system exhibiting a combination of elongation and twist when pressurized. In Sec. 3, we explain the model's computation and its experimental validation. Section 4 compares the model's predictions with the measured behavior, and quantifies error. Section 5 illustrates possible design cases where the model is useful. Section 6 discusses potential avenues for error reduction and Sec. 7 describes possible future work.

## Model

The properties of the FREE determine its behavior and impact decisions about how to model it. Our prototypes are made of latex rubber tubes with embedded woven threads. Therefore, we consider the FREE wall as a composite material with a hyperelastic matrix and embedded fibers.

We build the continuum model on four key assumptions:

1. (1)First, we employ a continuum approximation that ignores voids in the material and irregularity in the fiber deposition.
2. (2)Next, we neglect interaction effects like sliding friction between the fiber and elastomer, and local buckling. This enables us to superpose the stress contributions of the fiber and elastomer. Models considering these assumptions have captured the behavior of composites in a number of engineering contexts [19,20]; a detailed study of the effects of interaction effects in FREEs is left to future work.
3. (3)We assume that the composite material of the FREE wall is incompressible. This assumption is justified for FREEs since they are mainly composed of natural rubber, whose Poisson's ratio approaches 0.5 [21].
4. (4)Finally, we assume that the deformed FREE is roughly cylindrical. This assumption is justified by the photos taken of inflated FREEs, as shown in Fig. 1.

Together, these assumptions constitute a simple and efficient continuum framework that captures the key behaviors of FREEs, including nonlinear behavior of the fiber and elastomer, and length–torsion coupling.

###### Continuum Mechanical Framework.

Below, we explain the basics of the continuum framework used and present the key equations of the FREE actuator model. Detailed derivations may be found in the Appendix, and further foundational reading in Refs. [19] and [2123] Display Formula

(1)${end-to-end rotation,Φaxialstretch,λzexternalradius,ro}⇔{internalpressure,Ptwistingmoment,M(axialforce,F)}$

The FREE has three kinematic quantities and three kinetic quantities, which fully define its shape and loading. When any three of the quantities in Eq. (1) are fixed, the rest of the quantities are fully defined. So, we fix three quantities and use the continuum framework to create a system of equations that will allow us to solve for the others.

Since we are interested in the unique torsion-generating capabilities of FREEs, we solve for the twisting moment. To do this, we fix rotation $Φ$, axial extension λz, and internal pressure while measuring the twisting moment. Under the continuum and cylindrical assumptions noted above, we create a geometric definition of the FREE's change of shape, and relate it to the stresses stored in the FREE wall. Integrating these stresses across the surfaces on which they act gives the loading.

###### Defining Fiber-Reinforced Elastomeric Enclosure Movement.

Before finding the loading, we need to derive a mathematical description of the FREE's shape-changing activity. This section presents the deformation gradient for a FREE, defining its deformation in the continuum.

Consider a FREE with a single fiber family that is wound at an angle Γ from the central axis, and consider a point on the FREE skin with coordinates $R,Θ,Z$. When inflated, the FREE undergoes a length change $L→l$, a radius change $Ro,Ri→ro,ri$, and the rotation of one endcap relative to another in radians defined by $Φ$. Assuming that the FREE is not buckled, the transformation of any point in the FREE wall can be written as a change of cylindrical coordinates from $R,Θ,Z→r,θ,z$ as shown in Fig. 2. The sign conventions used here are shown in Fig. 3Display Formula

(2a)$r=R2−Ri2lL+ri2$
Display Formula
(2b)$θ=Θ+ZΦL$
Display Formula
(2c)$z=lLZ=λzZ$

Assuming incompressibility of the FREE skin, no end effects, and uniform stretching, the relationships of Eqs. (2a)(2c) define this shape change.

The deformation gradient F, shown below, is the gradient of the coordinate transformation in Eqs. (2a)(2c)Display Formula

(3)$F=[∂r∂R∂rR∂Θ∂r∂Zr∂θ∂Rr∂θR∂Θr∂θ∂Z∂z∂R∂zR∂Θ∂z∂Z]=[Rrλz000rRrΦL00λz]$

###### Fiber deformation.

As the FREE deforms, the helix created by the fiber changes. We define a unit vector M tangent to the fiber in the FREE's initial configuration as shown in Fig. 4, and we can see below that premultiplying it by F gives the deformed fiber direction Display Formula

(4)$M=[0 sin(Γ) cos(Γ)]m=FM=[0rRsin(Γ)+rΦLcos(Γ)λz cos(Γ)]$

This fiber transformation applies only to a taut fiber; buckled configurations are left to future work. These mathematical descriptions are used to find expressions for the stresses.

###### Relating Stress and Strain Through Strain Energy.

The stress stored inside the FREE wall is a function of its strain and its material properties. Here, we present the strain energy of the FREE, which enables us to relate the stress and strain.

###### Strain energy of a fiber-reinforced elastomeric enclosure.

Stress is connected to strain though a strain energy function $Ψ=f(F,Ci)$, where F is the deformation gradient and Ci are material properties. $Ψ$ models the Helmholtz free energy (in units of energy per reference volume) of the deformed body. We find the total strain energy $Ψ$ through superposition of elastomer and fiber energy Display Formula

(5)$Ψ=Ψelastomer+Ψfiber$

A modeling framework like this one enables a choice of any physically viable strain energy model. To demonstrate the viability of our continuum framework for modeling FREEs, we choose the simplest available strain energy models. The Neo–Hookean model [22] used for the elastomer and the standard fiber model for the fiber [23] enable the choice of any material moduli C1 for the elastomer and C2 for the fiber. These energy models are shown below: Display Formula

(6)$Ψelastomer=C12(I1−3)$
Display Formula
(7)$Ψfiber=C22(I4−1)2$
Display Formula
(8)$I1=tr(FTF)=λz2+r2R2+R2λz2r2+r2Φ2L2$
Display Formula
(9)$I4=(FM)·(FM)=M·(FTFM)=λz2 cos2(Γ)+r2Φ2 cos2(Γ)L2+2r2Φ cos(Γ)sin(Γ)LR+r2 sin2(Γ)R2$

In Eq. (9), “$·$” symbolizes a vector dot product. I1 and I4 are invariants related to the motion undergone by the FREE skin and the fibers (Eqs. (8) and (9)).

###### Cauchy stress.

The Cauchy or “true” stress tensor gives the magnitudes of differential stresses in the FREE in cylindrical coordinates. The relationship between Cauchy stress $σ$, strain energy, and deformation is shown below: Display Formula

(10)$σ=[σrrσrθσrzσθrσθθσθzσzrσzθσzz]=∂Ψ∂FFT−pI$

Here, the hydrostatic pressure variable p is a Lagrange multiplier arising from the incompressibility of the FREE wall.

When the strain energy is known, we can relate it to stress with the Cauchy stress expression shown in Eq. (10). A detailed explanation of the Helmholtz free energy, its relation to energy density in the material, and a derivation of the Cauchy Stress can be found in Ref. [24]. Further discussion of these models, the nature of the quantities I1 and I4, and an expanded calculation of $σ$ by replacing Eqs. (6)(9) into Eq. (10) can be found in Sec. A.2 of the Appendix.

###### Contribution of Internal Pressure.

The Cauchy stress depends on r, which is not a fixed quantity here. We find the new radii through the FREE's hydrostatic equilibrium equations, along with the incompressibility assumption of Eq. (2a) and fiber extensibility. The relationship between pressure P and the deformed FREE radii is shown below: Display Formula

(11)$−P=∫riro1r(σrr−σθθ)dr$

A detailed derivation of Eq. (11) is in Sec. A.3 of the Appendix.

Given the deformed radius (Eq. (11)) and the constitutive relationship between stress and deformation (Eq. (10)), we can now solve for each component of the FREE wall stress, visualized in Fig. 5.

Of specific interest here is the torsional load produced by the FREE, which we find by integrating the contribution to the moment caused by the shear stress $σθz$ direction across the deformed cross section of the FREE wall. Since $σθz$ is an off-diagonal entry of $σ$, p does not appear in the expression for the moment Display Formula

(12)$M=∬AσθzrdA$

where Display Formula

(13)$σθz=C1λzrΦL+2rC2λz cos(Γ)(I4−1)(Φ cos(Γ)L+ sin(Γ)R)$

The expression in Eq. (12) gives the torsional moment as a function of the FREE's initial state, deformation, and material parameters.

Equation (13) affords us a heuristic way to understand several facets of FREE behavior in torsion without fabricating a FREE or doing extensive experimental parameter determination. We can understand:

1. (1)Sensitivity of a given design to the deformation, including the FREE's length change.
2. (2)Coupling effects on torsion.
3. (3)Relative tension storage in the fiber and elastomer, respectively.
4. (4)The effect of proposed material properties on the design's torque generation ability.

###### Model Discussion.

One key advantage of this model is its ability to be used as a heuristic framework to help designers understand FREE behavior in situations where various behaviors are fixed and others are unknown. Here, we outline some of the insights that this model captures.

###### Deformation-Force Coupling.

The problem setup described above has fixed P, $Φ$, and λz. We then need Eqs. (11)(13) to find the torsional moment. However, as noted in Sec. 2.1, the FREE is fully determined when only three of the kinematic and force quantities are fixed. Below, we examine potential variations of the problem where input P remains fixed but the axial stretch λz and/or the twist angle $Φ$ may be unconstrained, or the radius r is constrained. Validating each variation of the problem experimentally is left to future work, as is a detailed study of the force generation capabilities of FREEs.

###### Unconstrained length change.

First, we can contrast the problem setup described in Sec. 2.1 with one in which the length change of the FREE is not fixed (e.g., if the end of the FREE were on a roller). Then, the net axial force would be zero by default. Since the axial stretch of the FREE λz is still a part of the shear stress expression in Eq. (13), we need to solve an additional equation to find the moment.

Axial force is found by integrating the axial stress and sum it with the force due to pressure on the ends of the FREE to find the force Display Formula

(14)$F=∬AσzzdA+2πri2P$

where Display Formula

(15)$σzz=−p+C1λz2+2C2λz2 cos2(Γ)(λz2 cos2(Γ)+r2Φ2 cos2(Γ)L2+2r2Φ cos(Γ)sin(Γ)LR+r2 sin2(Γ)R2)$

Since the FREE's length is unconstrained, we can expect that it produces no axial force. Solving Eq. (14) with F = 0 in conjunction with the hydrostatic pressure as described in Sec. A.3 of the Appendix gives the FREE's axial stretch λz. Plugging λz back in to Eqs. (11)(13) gives the moment. A similar procedure can be used for other fixed, nonzero values of axial force by setting F to a different value in Eq. (14).

###### Unconstrained twisting.

When the twist angle $Φ$ is unconstrained, there will be no resultant torsion on the FREE. So, we can then find $Φ$ by setting M = 0 in Eq. (12) with the pressure–radius relationship defined in Eq. (11). Similarly, setting $M=Mknown$ in Eq. (12) gives the twist at that fixed moment.

If length and twist are unconstrained, then the FREE is moving without any constraints and both F and M are fixed to zero.

If the radius of the FREE is fixed (e.g., if the FREE were inside a pipe), it becomes straightforward to solve for P, F, and M under fixed $Φ$ and λz since the kinematic quantities already appear in Eqs. (11), (13), and (15). If either or both of $Φ$ or λz are unconstrained instead, we can again use the procedures outlined above to find them.

###### Significance of Fiber Extensibility.

Fiber extensibility expands the design space for FREEs past what is afforded by the inextensibility assumption: we can now design FREEs with new kinds of helical constraining elements such as polymers. The expanded design space includes fibers, which may be more readily available, stronger, or have desired properties for the required working environment of the FREE without aligning closely with an inextensibility assumption.

We can further contrast this model with a version in which the fiber is assumed to be inextensible. An inextensible fiber will have $I4→1$ and will store stress without lengthening or shortening, i.e., $C2→∞$. However, a fully inextensible fiber will primarily store tension when the interior volume of the FREE has a noncylindrical shape, which changes the kinematics and the pressure–radius relationships from what is presented here. Then, we may either (1) solve for the deformed shape with a series of partial differential equations with known boundary conditions at the FREE endcaps, which is not computationally expedient or (2) assume a different deformed shape of the FREE, though our photos of inflated FREEs indicate that a cylinder most closely approximates the deformed FREE. A detailed mathematical description of the hydrostatic pressure balance for inextensible fibers can be found in Ref. [25].

## Model Implementation and Testing

The model results were compared with experimental measurements of torsion of FREEs at various deformations and pressures.

###### Model Computation.

The model was implemented on Wolfram Mathematica for a sample set of single fiber-family FREEs with reference lengths of 9.7–10.1 cm, initial external radii varying from 6.3 to 7.4 mm, at fiber angles of $Γ=60 deg, Γ=40 deg$, and $Γ=30 deg$. Fabrication error causes variation in the radius of each sample: the FREEs are made by layering elastomer and fiber over an extruded elastomeric tube by hand, so there is variation from sample to sample and some variation along the wall of each FREE sample.

Each of the elastomer and fiber properties, C1 and C2, can be observed in a simple tension test. The elastomeric matrix was tested between 0% and 300% strain. Fitting to the Neo-Hookean model gave $C1=0.5$ MPa, which agrees with values from the literature [21]. The fiber was found to have $C2=1$ MPa. The total computation time for this data set was about 14 s.

###### Experimental Validation.

The FREE's axial stretch, rotation, and interior pressure were fixed while the torsional loading was measured. The position-controlled test bed that performed the experiment is shown in Fig. 6. The FREE was clamped at each end while a linear actuator at the left end extended the FREE axially, a servo rotated one end, and the air inlet enabled pressurization. The load cells shown measured axial force and moment at a rate of 1 Hz. (The force measurements were used only for error analysis, which is explained further below.) The camera photographed the FREE in its deformed and pressurized state, allowing buckled configurations to be observed. The scheme of imposing deformation and measuring loading is frequently used in mechanics experimentation.

###### Measurement Techniques.

After deformation and internal pressure were fixed, the FREE was held in place for 20 s to allow it to approach static equilibrium. Pressure was fixed using a pressure regulator (Wilkerson ER1). Force and torque measurements were sampled at 1 Hz over the 20 s period in several such conditions with strain-gauge sensors (LoadStar RAS1-25 lb and LoadStar RST1-6 N·m, respectively), and pressure feedback was taken at 1 Hz. Each data point in Figs. 79 represents a sample over 1 s. Measurements were taken at various rotations for each FREE sample at extensions between 1 mm and 2 mm. These small pre-extensions were applied to limit the possibility of reaching critical buckling conditions. Since the cylindrical assumption is violated if the FREE wall is buckled, analysis of buckled configurations is left to future work. The resulting axial stretches $1.009<λz<1.021$ were used in the inputs to Eq. (13) when the model predictions were calculated, and each sample had the same λz throughout the tests. Test inclusion criteria were measurements in which the pressure was increased relative to the previous measurement and the FREE did not buckle. A detailed analysis of the associated axial force is left to future work. Moment produced by the mounting of the FREE to the grips was subtracted from the measured torque.

###### Measurement Error.

Sensor noise, sensor cross-talk, parasitic friction, FREE wall irregularity, and stress relaxation are the known potential sources of error.

###### Sensor error.

The torque and force sensors employed here have a quoted accuracy of 0.02% for the force sensor and 0.2% for the torque sensor. However, since they are mounted in series, we need to account for cross-talk—that is, the possibility that an axial force imposed on a torsional load cell may change its resistance reading. To characterize cross-talk error, we imposed axial forces between 0 and 12 N without torsion on the RST1. We measured 0.6 Nmm/N of cross-talk error due to axial force on the torque sensor. While testing the FREEs, we took force measurements (none of which exceeded 11 N) throughout, enabling us to calculate the cross-talk error of each FREE in each specific configuration. Parasitic friction was an additional concern. To avoid bending moments on the force and torque sensors, a Delrin bushing was placed at the interface between the torque sensor and air inlet as shown in Fig. 6. Friction from the torque sensor sliding against the bushing may introduce error to the force measurements. To estimate this error, we used a spring scale to cycle between tensile loads of 6 N and 0 N at a rate of roughly 0.05 Hz (comparable to the 20 s time that the FREEs were held fixed). Comparing the force sensor readings at 0 N after six cycles gave a range of 0.08 N. Carrying this force sensor error forward into the cross-talk error, we found 0.6 Nmm/N × 0.08 N $=0.0048$ Nmm of additional error. The sensor error, cross-talk error, and parasitic friction error are included in the error bars of the measurements plotted in Figs. 79.

###### Fiber-reinforced elastomeric enclosure surface irregularity.

During fabrication, irregularities occur in the wall thickness between FREEs and along wall of each FREE. The radius inputs to the model are the mean of three measurements on the same sample. The shaded lines in Figs. 79 represent the model predictions at ±1 standard deviation of radius. The standard deviations were: 0.14 mm for the $Γ=60 deg$ FREE, 0.1 mm for the $Γ=40 deg$ FREE, and 0.13 mm for the $Γ=30 deg$ FREE.

###### Stress relaxation.

Since FREEs are made partly from rubber, we expected that experimental error would arise from relaxation of the FREEs over time. However, upon observation of our torsion measurement over 20 s, we did not see changes in torsion measurements that exceeded sensor noise and cross-talk as characterized above. Therefore, time-dependent behavior of FREEs is left for future study.

## Results

The model quantifies the relationship between torsional loading on the FREE and FREE deformation in the R, Θ, and Z directions. Figures 79 show the experimental data plotted against the moment predictions for each FREE sample at various internal pressure and deformation states.

The root-mean-square error between the measured data and the model predictions is shown in Tables 13. Some predictions do not fall within the experimental and fabrication error as described above. We expect that higher accuracy could be achieved by using more refined strain energy models which take shears, interaction between the fiber and elastomer, and the woven nature of the fibers into account. Without these refinements, the model captures the general concave-up relationship between pressure, torque, and deformation.

## Design Case Studies

Our continuum model is useful for finding not only the particular operating conditions leading to a desired FREE torque, but also for exploring the space of available designs and operating conditions to satisfy a variety of constraints. Two case studies are presented. The first shows the model as an analysis tool for determining operational parameters, and second uses the model for design synthesis. In the first case study, the pressure is found for a given FREE to produce the desired torque at a given torsion. The second case study shows the fiber design angle Γ of a FREE to produce the desired torque for a given operating pressure and torsion.

###### Selection of Operating Parameters.

Once a FREE is manufactured at a given fiber angle Γ, it is important to control the behavior. In many instances, this means controlling the pressure to derive a desired moment and twist. The problem is formulated in the below equation: Display Formula

(16)$choosePs.t.M=MspecΦ=Φspecλz=λz,spec$

Equation (12) is used to relate the pressure to torque for a given torsion. Geometry was assumed to be Γ = 40 deg, and relaxed length, interior radius, and wall thickness 79 mm, 4 mm, and 1 mm, respectively. Axial stretch λz was assumed to be 1.013 (that is, a length change of +1 mm). Material properties were $C1=0.64$ MPa and $C2=4$ MPa. To obtain Mspec = 30 N-mm and $Φspec$ = 60, we find the required pressure to be 34.1 kPa. This analysis is shown in Fig. 10, along with the operational space of pressure, torque, and twist.

###### Design Synthesis of Fiber Angle.

If operating parameters are known, we can determine which design angle Γ of FREE will satisfy the task requirements. In this case study, we apply the model to find the design angle Γ that produces a desired range of twist angles $Φ$ and a desired torque at a given pressure. This problem is stated in the below equation: Display Formula

(17)$findallΓs.t.M=MspecP=Pspecλz=λz,specΦ∈{Φmin,Φmax}$

We use identical initial dimensions and material properties as the previous case study. Equation (12) is again used to define the relationship between torque, rotations ($Φ$), and design angle (Γ) for a fixed pressure and 10 deg steps of design angle. $Mspec=30 Nmm$. Pspec = 40 kPa, $Φmin=40 deg$, and $Φmax=95 deg$ are used for this case study.

Figure 11 shows the torque–rotation angle $Φ$ relationships for each design angle Γ. The areas where the dashed and solid lines intersect demonstrate the feasible design angles Γ that meet the required rotation angle $Φ$ and torque constraints. The feasible designs are then $Γ=20 deg, Γ=40 deg$, and $Γ=50 deg$ with deformations of 51.08 deg, 51.06 deg, and 90.1 deg, respectively.

The model can be used to compute a feasible design set or optimize parameters in both the continuous and discrete domains for a variety of operating conditions.

## Discussion

The general concave-upward trends of the measurements are captured by the model, demonstrating the viability of the model as a heuristic tool for FREE design synthesis and further justifying the model assumptions. Furthermore, the material properties characterized for the elastomer and fiber indicate that fiber extensibility does contribute to the FREE behavior.

The root-mean-square error exceeds the estimated measurement error. More features may be added to the model to improve its accuracy such as higher fidelity elastomer models [21], fiber–matrix interaction models [20], a fiber model specifically for woven materials [26], and corrections for irregularities in the geometry. The fabrication process may produce minor perturbations in the fiber angle, which may cause the model to disagree with the experiments. Or, local bulging may occur between the fibers, breaking the framework's assumption of evenly distributed fibers throughout the FREE wall.

## Conclusion

The model presented in this paper has the potential to be the basis for a heuristic, top-down design methodology for pneumatic soft robots, in which material properties, fiber angles, and tube dimensions can be chosen for any given application, deformation, or loads. We use a continuum mechanical framework to relate the displacement, torsional loading, and internal pressure of fiber-reinforced soft pneumatic mechanisms. The model captures the fundamental trends of experimental data at each of the configurations tested as well as the nonlinearity of a FREE's behavior due to coupling between loading and deformation. The assumptions of this continuum framework can be used to calculate FREE torsion and design FREEs with minimal experimental parameter determination, in an expanded design space that includes extensible fibers. The moment–pressure relationship of a proposed FREE and its moment–rotation relationship are presented as specific examples of the use of the model in a design context.

We illustrated the possibility of such a model both to find operating conditions for an existing FREE design which allow it to meet a given torque requirement, and to find the feasible design space for a FREE under known conditions.

Future directions of this work include experimental verification on the complete single-fiber FREE design space, an exploration of axial loading on FREEs, and a study of how material defects and manufacturing variation affect FREE torsion.

## Acknowledgements

The authors would like to acknowledge Professor Alan Wineman for useful conversations about mechanical modeling and sources of error, and Professor C. David Remy for helpful comments. We would also like to thank Marie Rice for pointing us to helpful reading and assisting with debugging code.

## Funding Data

• Toyota Research Institute.

## Appendices

###### Invariant-Based Models

Previous successful models for hyperelastic fiber-reinforced tubes use invariant-based strain energy functions to approximate this Helmholtz free energy [19]. In contrast with statistical or stretch-based models, invariant-based models are built from quantities which are invariant to changes in the FREE's configuration that reflect the directional properties of the material. For example, a tensile force in any direction in space on an isotropic material will create the same stresses. Or, if our anisotropic FREE has rotated in space but not undergone any stretching, it will not store any strain energy. A detailed discussion of invariants can be found in Ref. [22], and a derivation of all of the invariants of a pressurized fiber-reinforced tube can be found in Ref. [19].

In order to demonstrate the usefulness of a continuum framework in this soft actuator design context, we use the simplest available strain energy models—that is, the ones that depend on the fewest possible invariants. Out of the several possible invariants for a single fiber-family tube, the Neo–Hookean model (Eq. (6)) and standard fiber model (Eq. (7)) depend on one invariant each. For designers seeking greater accuracy, or using elastomer and fibers with different (e.g., shear) behavior, models including additional invariants may be used in the same framework presented here. Many of these models are discussed in Refs. [19], [21], and [22].

###### Relating Deformation to Stress With Strain Energy

Using deformation gradient (Eq. (3)) and the strain energy models (Eqs. (6) and (7)), we can find the Cauchy stress tensor with the constitutive relation in Eq. (10)Display Formula

(A1)$C=FTF,b=FFT$

Equation (A1) introduces the right (C) and left (b) Cauchy–Green deformation tensors, which are useful quantities for computing the stress. Below, we use the superposition of the fiber and matrix with the chain rule and Eqs. (6)(9) to evaluate Eq. (10)Display Formula

(A2)$σ=∂Ψelast+Ψfiber∂FFT−pI=∂Ψelast∂I1∂I1∂FFT+∂Ψfiber∂I4∂I4∂FFT−pI=C12∂I1∂FFT+C2(I4−1)∂I4∂FFT−pI=C12∂tr(FTF)∂FFT+C2(I4−1)∂(FM)·(FM)∂FFT−pI=C122FFT+C2(I4−1)2FM(FM)T−pI$

An alternate expression of Eq. (A2) using Eq. (A1) and the definition of the tensor product ⊗ [22] is shown below: Display Formula

(A3)$σ=−pI+C1b+2C2(I4−1)FM⊗FM$

###### Finding Radius With Hydrostatic Equilibrium

In FREE applications, length and rotation are often controlled, but radius change is left free. Under the fixed pressure, stretch and twist, the new radii ri and ro need to be determined.

Radius can be determined by assuming the deformed FREE stresses satisfy hydrostatic equilibrium; that is, that on an arbitrarily chosen subsection of the FREE wall, there is no net stress. The FREE wall needs to be incompressible for this equilibrium to hold [22,23]. This assumption is justified for FREEs since their elastomer is natural rubber, which has a Poisson's ratio close to (but less than) 0.5 [21], and the fibers have less volume overall than the elastomeric matrix Display Formula

(A4)$∇·σ=0$

In cylindrical coordinates, Eq. (A4) becomes the system of partial differential equations shown below: Display Formula

(A5)$∂σrr∂r+1r∂σrθ∂θ+∂σrz∂z+1r(σrr−σθθ)=0∂σrθ∂r+1r∂σθθ∂θ+∂σθz∂z+2rσrθ=0∂σrz∂r+1r∂σθz∂θ+∂σzz∂z+1rσrz=0$

Equation (A4) shows hydrostatic equilibrium, which is expanded in cylindrical coordinates in Eq. (A5). With our previous assumptions of uniformity in z and around the FREE wall, we can conclude that none of the stresses vary in the θ- or z directions along the FREE wall. Additionally, we can see by the form of F that $σrθ=σrz=0$ and Eq. (A5) reduces to Display Formula

(A6)$∂σrr∂r+1r(σrr−σθθ)=01r∂σθθ∂θ=0∂σzz∂z=0$

Equations (A5) then simplify to Eq. (A6). Because of axial symmetry and uniform stretch, we are primarily interested in the first hydrostatic equilibrium equation. Re-arranging it, we have Display Formula

(A7)$−∂σrr∂r=1r(σrr−σθθ)$

We use the boundary conditions $σrr(ri)=P$ and $σrr(ro)=Patm=0$. Integrating Eq. (A7) in r, and then applying a change of variables from r to R (using Eqs. (2a) and (2b)) gives Display Formula

(A8)$−σrr(ri)=∫riro1r(σrr−σθθ)dr=∫RiRoRr1R(σrr−σθθ)drdRdR=∫RiRoRr1R(σrr−σθθ)RλzrdR=∫RiRoRλzR2−Ri2(σrr−σθθ)dR$

Here, using Eqs. (A3) and (2a), and substituting 3 with the FREE dimensions Display Formula

(A9)$σrr−σθθ=C1R2r2λz2−2C2(I4−1)(rΦ cos(Γ)L+r sin(Γ)R)2$

Applying the boundary condition gives Display Formula

(A10)$−P=−p(ri)=−σrr(ri)=∫RiRoRλzR2−Ri2(σrr−σθθ)dR$

Under incompressibility of the FREE wall Display Formula

(A11)$ro=ri2+Ro2−Ri2λz and r=ri2+R2−Ri2λz$
due to volume conservation.

Equations (A10) and (A11) enable us to find the new interior and exterior radii ri and ro. Once Eq. (A10) is solved, we know all of the kinematic quantities in the FREE's deformed configuration and can solve for the loads. Note that this procedure yields different results when the fibers are inextensible; Liu and Rahn [27] have an example of such a model for McKibben actuators with inextensible fibers.

We can find the expression for p by adding $dp/dr$ to both sides of Eq. (A7)Display Formula

(A12)$dpdr=ddr(σrr+p)+σrr−σθθr$

We can note that using Eq. (10) that $σrr+p$ in Eq. (A12) is only a function of the deformation. Then, we can replace $σrr+p$ with Qrr and perform a change of coordinates from r to RDisplay Formula

(A13)$dpdR=dQrr(R)dR+σrr−σθθRrRdrdR$

Integrating Eq. (A13) under the same boundary condition (Eq. (A10)) gives the hydrostatic pressure expression necessary to find the radial, axial, and circumferential forces Display Formula

(A14)$p(R)=Qrr(R)−P+∫RiRRλzR2−Ri2(σrr−σθθ)dR$

Thus, using Eqs. (A10), (A11), and (A14), we can find the FREE's deformed radius and its hydrostatic pressure.

## References

Rus, D. , and Tolley, M. T. , 2015, “ Design, Fabrication and Control of Soft Robots,” Nature, 521(7553), pp. 467–475. [PubMed]
Marchese, A. D. , Onal, C. D. , and Rus, D. , 2014, “ Autonomous Soft Robotic Fish Capable of Escape Maneuvers Using Fluidic Elastomer Actuators,” Soft Rob., 1(1), pp. 75–87.
Suzumori, K. , Endo, S. , Kanda, T. , Kato, N. , and Suzuki, H. , 2007, “ A Bending Pneumatic Rubber Actuator Realizing Soft-Bodied Manta Swimming Robot,” IEEE International Conference on Robotics and Automation (ICRA), Rome, Italy, Apr. 10–14, pp. 4975–4980.
Shepherd, R. F. , Ilievski, F. , Choi, W. , Morin, S. A. , Stokes, A. A. , Mazzeo, A. D. , Chen, X. , Wang, M. , and Whitesides, G. M. , 2011, “ Multigait Soft Robot,” Proc. Natl. Acad. Sci., 108(51), pp. 20400–20403.
Park, Y.-L. , Chen, B.-R. , Pérez-Arancibia, N. O. , Young, D. , Stirling, L. , Wood, R. J. , Goldfield, E. C. , and Nagpal, R. , 2014, “ Design and Control of a Bio-Inspired Soft Wearable Robotic Device for Ankle–Foot Rehabilitation,” Bioinspiration Biomimetics, 9(1), p. 016007. [PubMed]
Sasaki, D. , Noritsugu, T. , and Takaiwa, M. , 2005, “ Development of Active Support Splint Driven by Pneumatic Soft Actuator (Assist),” IEEE International Conference on Robotics and Automation (ICRA 2005), Barcelona, Spain, Apr. 18–22, pp. 520–525.
Martinez, R. V. , Branch, J. L. , Fish, C. R. , Jin, L. , Shepherd, R. F. , Nunes, R. , Suo, Z. , and Whitesides, G. M. , 2013, “ Robotic Tentacles With Three-Dimensional Mobility Based on Flexible Elastomers,” Adv. Mater., 25(2), pp. 205–212. [PubMed]
McMahan, W. , Chitrakaran, V. , Csencsits, M. , Dawson, D. , Walker, I. D. , Jones, B. A. , Pritts, M. , Dienno, D. , Grissom, M. , and Rahn, C. D. , 2006, “ Field Trials and Testing of the Octarm Continuum Manipulator,” IEEE International Conference on Robotics and Automation (ICRA 2006), Orlando, FL, May 15–19, pp. 2336–2341.
Kota, S. , 2014, “ Shape-Shifting Things to Come,” Sci. Am., 310(5), pp. 58–65. [PubMed]
Bishop-Moser, J. , Krishnan, G. , Kim, C. , and Kota, S. , 2012, “ Design of Soft Robotic Actuators Using Fluid-Filled Fiber-Reinforced Elastomeric Enclosures in Parallel Combinations,” IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Vilamoura, Portugal, Oct. 7–12, pp. 4264–4269.
Bishop-Moser, J. , Krishnan, G. , and Kota, S. , 2013, “ Force and Moment Generation of Fiber-Reinforced Pneumatic Soft Actuators,” IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Tokyo, Japan, Nov. 3–7, pp. 4460–4465.
Bishop-Moser, J. , and Kota, S. , 2015, “ Design and Modeling of Generalized Fiber-Reinforced Pneumatic Soft Actuators,” IEEE Trans. Rob., 31(3), pp. 536–545.
Tondu, B. , and Lopez, P. , 2000, “ Modeling and Control of McKibben Artificial Muscle Robot Actuators,” IEEE Control Syst., 20(2), pp. 15–38.
Shan, Y. , Philen, M. P. , Bakis, C. E. , Wang, K.-W. , and Rahn, C. D. , 2006, “ Nonlinear-Elastic Finite Axisymmetric Deformation of Flexible Matrix Composite Membranes Under Internal Pressure and Axial Force,” Compos. Sci. Technol., 66(15), pp. 3053–3063.
Connolly, F. , Walsh, C. J. , and Bertoldi, K. , 2017, “ Automatic Design of Fiber-Reinforced Soft Actuators for Trajectory Matching,” Proc. Natl. Acad. Sci., 114(1), pp. 51–56.
Singh, G. , and Krishnan, G. , 2017, “ A Constrained Maximization Formulation to Analyze Deformation of Fiber Reinforced Elastomeric Actuators,” Smart Mater. Struct., 26(6), p. 065024.
Bruder, D. , Sedal, A. , Bishop-Moser, J. , Kota, S. , and Vasudevan, R. , 2017, “ Model Based Control of Fiber Reinforced Elastofluidic Enclosures,” IEEE International Conference on Robotics and Automation (ICRA), Singapore, May 29–June 3, pp. 5539–5544.
Krishnan, G. , Bishop-Moser, J. , Kim, C. , and Kota, S. , 2015, “ Kinematics of a Generalized Class of Pneumatic Artificial Muscles,” ASME J. Mech. Rob., 7(4), p. 041014.
Holzapfel, G. A. , Gasser, T. C. , and Ogden, R. W. , 2000, “ A New Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models,” J. Elasticity Phys. Sci. Solids, 61(1–3), pp. 1–48.
Heinrich, C. , Aldridge, M. , Wineman, A. , Kieffer, J. , Waas, A. M. , and Shahwan, K. , 2012, “ The Influence of the Representative Volume Element (RVE) Size on the Homogenized Response of Cured Fiber Composites,” Modell. Simul. Mater. Sci. Eng., 20(7), p. 075007.
Gent, A. N. , 2012, Engineering With Rubber: How to Design Rubber Components, Carl Hanser Verlag GmbH Co KG, Cincinnati, OH.
Ogden, R. W. , 1997, Non-Linear Elastic Deformations, Courier Corporation, North Chelmsford, MA.
Demirkoparan, H. , and Pence, T. J. , 2007, “ Swelling of an Internally Pressurized Nonlinearly Elastic Tube With Fiber Reinforcing,” Int. J. Solids Struct., 44(11–12), pp. 4009–4029.
Fung, Y.-C. , and Tong, P. , 2001, Classical and Computational Solid Mechanics, World Scientific, Singapore.
Pipkin, A. , and Rivlin, R. , 1963, “ Minimum-Weight Design for Pressure Vessels Reinforced With Inextensible Fibers,” ASME J. Appl. Mech., 30(1), pp. 103–108.
Peng, X. , and Cao, J. , 2005, “ A Continuum Mechanics-Based Non-Orthogonal Constitutive Model for Woven Composite Fabrics,” Composites, Part A, 36(6), pp. 859–874.
Liu, W. , and Rahn, C. , 2003, “ Fiber-Reinforced Membrane Models of McKibben Actuators,” ASME J. Appl. Mech., 70(6), pp. 853–859.
View article in PDF format.

## References

Rus, D. , and Tolley, M. T. , 2015, “ Design, Fabrication and Control of Soft Robots,” Nature, 521(7553), pp. 467–475. [PubMed]
Marchese, A. D. , Onal, C. D. , and Rus, D. , 2014, “ Autonomous Soft Robotic Fish Capable of Escape Maneuvers Using Fluidic Elastomer Actuators,” Soft Rob., 1(1), pp. 75–87.
Suzumori, K. , Endo, S. , Kanda, T. , Kato, N. , and Suzuki, H. , 2007, “ A Bending Pneumatic Rubber Actuator Realizing Soft-Bodied Manta Swimming Robot,” IEEE International Conference on Robotics and Automation (ICRA), Rome, Italy, Apr. 10–14, pp. 4975–4980.
Shepherd, R. F. , Ilievski, F. , Choi, W. , Morin, S. A. , Stokes, A. A. , Mazzeo, A. D. , Chen, X. , Wang, M. , and Whitesides, G. M. , 2011, “ Multigait Soft Robot,” Proc. Natl. Acad. Sci., 108(51), pp. 20400–20403.
Park, Y.-L. , Chen, B.-R. , Pérez-Arancibia, N. O. , Young, D. , Stirling, L. , Wood, R. J. , Goldfield, E. C. , and Nagpal, R. , 2014, “ Design and Control of a Bio-Inspired Soft Wearable Robotic Device for Ankle–Foot Rehabilitation,” Bioinspiration Biomimetics, 9(1), p. 016007. [PubMed]
Sasaki, D. , Noritsugu, T. , and Takaiwa, M. , 2005, “ Development of Active Support Splint Driven by Pneumatic Soft Actuator (Assist),” IEEE International Conference on Robotics and Automation (ICRA 2005), Barcelona, Spain, Apr. 18–22, pp. 520–525.
Martinez, R. V. , Branch, J. L. , Fish, C. R. , Jin, L. , Shepherd, R. F. , Nunes, R. , Suo, Z. , and Whitesides, G. M. , 2013, “ Robotic Tentacles With Three-Dimensional Mobility Based on Flexible Elastomers,” Adv. Mater., 25(2), pp. 205–212. [PubMed]
McMahan, W. , Chitrakaran, V. , Csencsits, M. , Dawson, D. , Walker, I. D. , Jones, B. A. , Pritts, M. , Dienno, D. , Grissom, M. , and Rahn, C. D. , 2006, “ Field Trials and Testing of the Octarm Continuum Manipulator,” IEEE International Conference on Robotics and Automation (ICRA 2006), Orlando, FL, May 15–19, pp. 2336–2341.
Kota, S. , 2014, “ Shape-Shifting Things to Come,” Sci. Am., 310(5), pp. 58–65. [PubMed]
Bishop-Moser, J. , Krishnan, G. , Kim, C. , and Kota, S. , 2012, “ Design of Soft Robotic Actuators Using Fluid-Filled Fiber-Reinforced Elastomeric Enclosures in Parallel Combinations,” IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Vilamoura, Portugal, Oct. 7–12, pp. 4264–4269.
Bishop-Moser, J. , Krishnan, G. , and Kota, S. , 2013, “ Force and Moment Generation of Fiber-Reinforced Pneumatic Soft Actuators,” IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Tokyo, Japan, Nov. 3–7, pp. 4460–4465.
Bishop-Moser, J. , and Kota, S. , 2015, “ Design and Modeling of Generalized Fiber-Reinforced Pneumatic Soft Actuators,” IEEE Trans. Rob., 31(3), pp. 536–545.
Tondu, B. , and Lopez, P. , 2000, “ Modeling and Control of McKibben Artificial Muscle Robot Actuators,” IEEE Control Syst., 20(2), pp. 15–38.
Shan, Y. , Philen, M. P. , Bakis, C. E. , Wang, K.-W. , and Rahn, C. D. , 2006, “ Nonlinear-Elastic Finite Axisymmetric Deformation of Flexible Matrix Composite Membranes Under Internal Pressure and Axial Force,” Compos. Sci. Technol., 66(15), pp. 3053–3063.
Connolly, F. , Walsh, C. J. , and Bertoldi, K. , 2017, “ Automatic Design of Fiber-Reinforced Soft Actuators for Trajectory Matching,” Proc. Natl. Acad. Sci., 114(1), pp. 51–56.
Singh, G. , and Krishnan, G. , 2017, “ A Constrained Maximization Formulation to Analyze Deformation of Fiber Reinforced Elastomeric Actuators,” Smart Mater. Struct., 26(6), p. 065024.
Bruder, D. , Sedal, A. , Bishop-Moser, J. , Kota, S. , and Vasudevan, R. , 2017, “ Model Based Control of Fiber Reinforced Elastofluidic Enclosures,” IEEE International Conference on Robotics and Automation (ICRA), Singapore, May 29–June 3, pp. 5539–5544.
Krishnan, G. , Bishop-Moser, J. , Kim, C. , and Kota, S. , 2015, “ Kinematics of a Generalized Class of Pneumatic Artificial Muscles,” ASME J. Mech. Rob., 7(4), p. 041014.
Holzapfel, G. A. , Gasser, T. C. , and Ogden, R. W. , 2000, “ A New Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models,” J. Elasticity Phys. Sci. Solids, 61(1–3), pp. 1–48.
Heinrich, C. , Aldridge, M. , Wineman, A. , Kieffer, J. , Waas, A. M. , and Shahwan, K. , 2012, “ The Influence of the Representative Volume Element (RVE) Size on the Homogenized Response of Cured Fiber Composites,” Modell. Simul. Mater. Sci. Eng., 20(7), p. 075007.
Gent, A. N. , 2012, Engineering With Rubber: How to Design Rubber Components, Carl Hanser Verlag GmbH Co KG, Cincinnati, OH.
Ogden, R. W. , 1997, Non-Linear Elastic Deformations, Courier Corporation, North Chelmsford, MA.
Demirkoparan, H. , and Pence, T. J. , 2007, “ Swelling of an Internally Pressurized Nonlinearly Elastic Tube With Fiber Reinforcing,” Int. J. Solids Struct., 44(11–12), pp. 4009–4029.
Fung, Y.-C. , and Tong, P. , 2001, Classical and Computational Solid Mechanics, World Scientific, Singapore.
Pipkin, A. , and Rivlin, R. , 1963, “ Minimum-Weight Design for Pressure Vessels Reinforced With Inextensible Fibers,” ASME J. Appl. Mech., 30(1), pp. 103–108.
Peng, X. , and Cao, J. , 2005, “ A Continuum Mechanics-Based Non-Orthogonal Constitutive Model for Woven Composite Fabrics,” Composites, Part A, 36(6), pp. 859–874.
Liu, W. , and Rahn, C. , 2003, “ Fiber-Reinforced Membrane Models of McKibben Actuators,” ASME J. Appl. Mech., 70(6), pp. 853–859.

## Figures

Fig. 1

FREE behavior when relaxed (left) and pressurized (right) [17]

Fig. 2

Motion of a point on the FREE wall. As the FREE inflates, the reference point changes coordinates (in the cylindrical coordinate system shown) from (R, Θ, Z) to (r, θ, z).

Fig. 3

Torsional moment, rotation, and fiber angle sign conventions

Fig. 4

Fiber direction on a continuum element of the FREE wall

Fig. 5

Differential element of the FREE wall and associated stresses

Fig. 6

The position-fixed test bed, which measures torsion at various deformation and pressure states

Fig. 7

Predicted (-) and measured moment generation by pressure 60 deg design angle FREE

Fig. 8

Predicted (-) and measured moment generation by pressure 40 deg design angle FREE

Fig. 9

Predicted (-) and measured moment generation by pressure 30 deg design angle FREE

Fig. 10

Torque by pressure curve for a FREE with design angle Γ=40 deg at various deformations

Fig. 11

Torque-displacement profiles of a design set of FREEs at 40 kPa with feasible designs emphasized

## Tables

Table 1 Torque error for $60 deg$ design angle
Table 2 Torque error for $40 deg$ design angle
Table 3 Torque error for $30 deg$ design angle

## Discussions

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