## Abstract

Fiber reinforced elastomeric enclosures (FREEs) are soft pneumatic representative elements that can form the basis for building soft self-actuating structures/mechanisms. When placed in different configurations, they exhibit unique stroke amplification characteristics that can be leveraged to create interesting deformation patterns. Such deformations occur as a combination of axial and bending deflection due to internal pressurization and external forces. This paper presents a lumped reduced-order model that enables quick and accurate analysis of mechanisms made from FREEs grouped as a system. The model proposed is a modified four-spring pseudo-rigid-body (PRB) model that effectively captures the axial and bending stiffnesses of contracting FREEs. Parametric estimation of the model is performed using a multistart optimization routine to fit the PRB model with results from experiments and finite element analysis (FEA). The model is also generalized and statistically verified for FREEs with different fiber angles, length-to-diameter ratios, and different actuation pressures. Finally, efficacy of the approach is validated through three case studies that involve a planar arrangement of FREEs at different orientations.

## 1 Introduction

Soft robots [1,2] have generated interest in the recent past due to their energy density, light weight, safety, and adaptability. Their functionality results from soft constituents such as muscles, tendons, and tissue-like connectors. Designing a robot or a mechanism using these components have their challenges marked by high levels of structural and functional coupling. For example, conventional mechatronic systems have well-defined and distinct actuators, and transmission systems that tailor the force and impedance output. This decoupling leads to straightforward system design and controls. In contrast, soft mechanisms consist of several building blocks that simultaneously impart forces as actuators, deform or bend as structural members, and amplify displacements or forces as transmission members. This paper deals with the design of soft robotic systems composed specifically of fiber reinforced pneumatic actuators whose coupled behavior is captured by a unique reduced order model.

Fiber-reinforced elastomeric enclosures (FREEs) are hollow cylinders reinforced by a network of two families of fibers (α and β) as shown in Fig. 1(a) [3,4]. When pressurized with fluids, FREEs enable different motion patterns such as bending, extension, rotation, and spiral motion depending on the relative fiber orientations [4]. In this paper, we limit our attention to a specific class of FREEs that have symmetric fiber angles ($α=1−β$) and thus contract in length upon actuation as shown in Figs. 1(b) and 1(c). Known as McKibben pneumatic muscles, they have been used as biomimetic actuators in robotics [5,6] for over two decades. The motivation of this work stems from the increased performance benefits attained by arranging FREEs in a unique topological network. For example, Fig. 1(d) shows several FREEs arranged in a dyadic or pennate configuration similar to the fiber strands in a muscle [7]. The net actuation stroke is more than five times the stroke of a single FREE, implying large possible amplifications or gear ratios that can result from planar or spatial topological connections. The resulting performance benefits are due to the FREEs acting as both actuators that contract and structural members that bend or flex to ensure overall system compatibility.

Fig. 1
Fig. 1
Close modal

While several models exist to capture the axial contracting actuation of FREEs [812], few models take into account their bending behavior due to transverse loads or moments. Trivedi et al. [13] and Trivedi [14] treat the bending of extending FREEs as Cosserat rods [15] with linear elastic material properties and decouple actuation forces from the bending mechanics. Giri and Walker [16] and Walker et al. [17] introduced discrete or lumped approaches to model continuum arms. However, these models are accurate only when the systems of FREEs are co-axial as in continuum manipulators [13] and cannot be directly extended to a system of FREEs in a planar or a spatial network as shown in Fig. 1(d). Furthermore, these models do not take into account the changes in geometry (area of cross section and elastic properties) that result due to internal pressurization of contracting FREEs.

In this paper, we restrict our study to model planar topologies of contracting FREEs. We exploit a reduced-order model developed for simulating large deflections in planar compliant beams known as pseudo-rigid-body (PRB) model [1820]. The term reduced order is applicable in the context that in place of solving partial or ordinary differential equations to obtain deformation characteristics, the presented approach utilizes a nonlinear polynomial equations to capture the same behavior. Hence, there is a reduction in the order of complexity as compared to models derived using continuum mechanics. In this regard, the authors recently adapted a three-spring PRB model [21] for FREEs based on an existing model for soft elastomeric joints [22,23]. Two torsional springs at either end models the bending stiffness, while a central linear spring models the axial stiffness. In this paper, we present a kinematic inversion of this model by introducing two linear and two torsional springs.

Two reasons can be attributed to this choice of model: (1) the presence of linear springs at the tip helps to better capture the variations in the PRB parameters across different design parameters, and (2) such a model could potentially be used to capture interesting nonlinear behavioral characteristics that FREEs can exhibit. An example of this is FREEs having varying fiber angles along their length [11]. The scope and requirements of the PRB model presented in this work is as follows: (1) Each PRB model is specific to a FREE with regard to material properties and manufacturing method, (2) the PRB model should effectively capture the axial and bending characteristics of FREEs, and (3) the PRB model should be easily scalable to be used in topologies of interconnected FREEs to obtain quantitative results for global force and displacement characteristics. Toward achieving these requirements, the parameters of the model are optimized such that the tip deflections from the model match with values obtained from finite element (FE) analysis and experiments. Furthermore, to make the model more general, we also propose a second parametric model to capture the variations of the PRB parameters as a function of pressure, fiber angles, and geometry parameters.

The fit PRB model is beneficial for quick and accurate evaluation of design concepts composed of soft FREE-based systems, which would otherwise require computationally expensive high fidelity tools. The model can then be used to optimize several parameters such as FREE geometry, fiber angles, and topology and enable the generation of user insight in design. One such insight, presented in Sec. 4, is the ability of FREEs connected in a dyadic configuration (similar to the pennation of muscle fibers) to mechanically amplify the overall stroke. The PRB model is then used to optimize the dyadic configuration for actuating a compliant gripper. Similarly, an example is presented to optimize the topological arrangement of FREEs to obtain maximum angular rotation in a motor-like rotary configuration.

## 2 Method

In this section, we present the PRB model used to approximate the behavior of the FREEs and the method used to obtain its parameters.

### 2.1 Pseudo-Rigid-Body Model.

The PRB model of length L selected in this work is composed of two linear and torsional springs connected to a rigid segment (Fig. 2). The PRB model is described by four parameters: (1) torsional spring stiffness (Kth), (2) linear spring stiffness (Kex), (3) length of rigid segment as a fraction of undeformed FREE length (γ), and (4) actuation force acting on the linear springs (fa). The torsional springs behave like revolute joints and capture stiffness associated with the bending behavior of the FREEs. The linear springs are assumed to have an undeformed length of lo and model the stiffness associated with axial length changes. The actuation of FREEs results from the internal pressurization leading to contraction in length. This is modeled as an actuation force acting on the linear spring stiffness. A similar model was presented in Ref. [21] with differences primarily in the placement of the linear and torsional springs. We believe that this model will be more accurate in capturing the curvilinear deformation profiles of the FREE.

Fig. 2
Fig. 2
Close modal
We are interested in the tip position of the FREE as a function of internal actuation loads and transverse bending loads. The tip position is determined using forward kinematic equations (Eq. (1)) of the PRB model and is a function of the joint angles and linear spring displacements.
$xtip=(lo+δl)+γLcos(θ1)+(lo+δl)cos(θ1+θ2)ytip=γLsin(θ1)+(lo+δl)sin(θ1+θ2)θ=θ1+θ2$
(1)
θ1 and θ2 are the angular deflections at the joints and δl is the linear spring deflections. The joint deflections are calculated using energy equivalence (Eq. (2)).
$τ=JTFext$
(2)
where the joint torques $τ=(Kthθ1,Kthθ2,2Kexδl+2fa)$ are related to the tip loads Fext = (Fx, Fy, Mz) by the Jacobian of the PRB model, J (Appendix  B). The Jacobian can be obtained by differentiating the terms of Eq. (1) [23]. The PRB model is subjected to two forces: (1) an internal actuating force due to pressurization (fa) acting on the linear springs and (2) an external transverse load displacing the tip position (Fext) inducing joint deflections. To fit this PRB model for the FREE, we need to understand its axial and transverse bending behaviors. These can be obtained either experimentally [21] or using finite element analysis (FEA) [12]. While obtaining experimental data for different FREE sizes and shapes is tedious, we rely on fitting the PRB model on the FEA data. Toward this, we establish the accuracy of the FEA results by comparing it with experiments conducted on select FREEs. Details of the experimental testing method are explained in Appendix  A.

### 2.2 Finite Element Modeling.

As mentioned previously, the purpose of developing an FE model is to assist in pertinent data collection to fully define the PRB model for FREEs of varying fiber angles and slenderness ratios. All the parameters of the PRB model, except the actuation force fa, will be obtained by fitting the results with the FE model. The actuation force fa is equal to the blocked force of the FREE, which is the force required to restrain the pressurized FREE from axially contracting. The blocked force has been shown to be a function of FREE geometry and the fiber angles alone and is given by [6]
$fa=πPr2(3cos2α−1sin2α)$
(3)
where P is the actuation pressure, α is the fiber angle, and r is the radius of the FREE.

On the other hand, FE models present a realistic description for complex bending deformations, albeit at higher computation costs. The FE model (analyzed using ABAQUS/Standard) used in this work consists of the following parts (Fig. 3):

1. Elastomeric Silicone-Based Cylindrical Body: The nonlinear deformation behavior of the body is captured using a hyperelastic incompressible Neo–Hookean material model, with strain energy density W = C1(I1 − 3). The material model was chosen based on preliminary studies and comparison with experimental data similar to Ref. [24]. Meshing was performed using tetrahedral quadratic hybrid elements (ABAQUS element C3D10H), and actuation pressure was applied to the inner walls.

2. Inextensible Kevlar Fibers: The two families of fibers (eight starts per family) are modeled using quadratic beam elements (ABAQUS element B32).

3. Rigid End Cap: The end cap was modeled as a rigid steel object meshed using the same elements as the main body. The end cap serves as the region for application of external loads.

Fig. 3
Fig. 3
Close modal

Conventional pneumatic artificial muscles are made by constraining a hyperelastic bladder by means of a strain-limiting outer sheath fixed at the tips. On pressurization, the inner bladder expands radially coming in contact with the external sheath and in process generates an axial stroke. However, for FREEs, the fibers are embedded in the structure of the hyperelastic bladder. Therefore, there is no contact that needs to be accounted for when pressurizing FREEs, and all the components are attached together using tie constraints. A tie constraint ties two surfaces together so that no relative motion occurs between them. This aligns well with the fabrication of FREEs where the end cap is glued onto the body at the tip and the Kevlar fibers are attached to the body using a silicone. For each simulation, the FREE is pressurized internally to simulate axial deformation after which a transverse bending load is applied on the center of the end cap to simulate bending deformation.

## 3 Fitting the PRB Model

In this section, we detail the steps used to validate the PRB models by comparing them with FE simulations. The PRB model parameters will be fit such that the difference between the tip deflection of the FREE evaluated from modeling and FE simulation is minimized. In our implementation, we use a multistart gradient-based interior point search optimization routine (implemented in Matlab® R2016a) to optimally fit the PRB model parameters. An L2-norm-based error function is minimized subject to constraints that ensure the physical realizability of the PRB model. The fitting algorithm is run for a total of 27 times for every applied pressure, fiber angle, and slenderness ratio.
$minf(γ,Kth,Kex)=Errst.0.1≤γ≤0.850≤Kth≤100≤Kex≤10$
(4)
where
$Err=∑i=0N(xtipi−xtipisim)2+(ytipi−ytipisim)2N$
(5)
where $xtipi$ and $ytipi$ are the x and y coordinates, respectively, of the tip position for ith tip load and N is the number of loads applied. The goodness of fit, between FEA and the PRB model, for varying length-to-diameter ratios (λ), actuation pressures (Pa), and fiber angles (α) is tabulated in Table 1. When loaded, the FREEs experience a maximum deflection in the range of 0–45 mm along the x-axis and 0–95 mm along the y-axis. The residuals indicate that the PRB models perform well in capturing the axial and bending behaviors of FREEs over varying FREE parameters.
Table 1

PRB model fit residuals (mm)

λPa(psi)α = 25 degα = 30 degα = 40 deg
6200.330.880.39
17.50.300.870.73
15.00.270.230.38
8.5201.050.581.38
17.50.250.741.46
15.00.431.121.57
10200.950.271.56
17.50.660.490.97
15.00.210.571.87
λPa(psi)α = 25 degα = 30 degα = 40 deg
6200.330.880.39
17.50.300.870.73
15.00.270.230.38
8.5201.050.581.38
17.50.250.741.46
15.00.431.121.57
10200.950.271.56
17.50.660.490.97
15.00.210.571.87

### 3.1 Effect of Elastic Properties on Model Parameters.

The purpose of using the PRB model approach is to capture the deformation behavior (both axial and transverse) of FREEs using a lower order simplified model within acceptable error bounds. Since a FREE is an inflatable, its elastic properties are a function of the (i) actuation pressure, (ii) fiber angle α, and (iii) length-to-diameter ratio λ. The effects of varying elastic properties on the PRB model parameters is explained below:

1. Dependence on Actuation Pressure: The spring constants Kex and Kth are in general expected to increase with actuation pressure Pa. Figure 4 captures the increase in the stiffness values as a function of pressure in the range of testing pressures (0.14 Mpa). Furthermore, the kinematic parameter γ decides the fraction of the FREE length that deforms as a rigid body. With actuation, the center of compliance of FREEs is pushed toward the ends due to the necking of the cross sections. The necking effect results due to constraining the ends from radial expansion, resulting in a curvilinear deformation profile [8,24]. The dependence of γ on actuation pressure is plotted in Fig. 4 and is shown to be relatively invariant with applied pressure.

2. Dependence on Fiber Angle: The variation of the axial stiffness Kex as a function of the fiber angle is well documented in the previous research [25]. In general, smaller fiber angles have larger axial and bending stiffness because of the resistance it poses against increase or decrease in FREE length due to contraction or extension. This increase in stiffness is observed in Fig. 4. Furthermore, the kinematic parameter γ depends on the increase in radius of the inflated FREE. It is well known that FREEs with smaller fiber angles lead to a larger radial bulge upon pressurization as shown in Fig. 5, thus leading to an increase in γ. Figure 4 clearly demonstrates the increase of γ as fiber angle decrease. A noticeable anomaly is observed for slenderness ratio λ = 6, where α = 30 deg has a lower value of γ than α = 40 deg. This could be attributed to dominant shearing effects present during bending that is common for short thick beams.

3. Dependence on Length-to-Diameter Ratio: FREEs considered in this paper are slender cylinders that behave like elastic beams upon pressurization. Like beams, FREEs undergo bending under the action of transverse loads or bending moments. The bending stiffness of beams under the Euler–Bernoulli assumption decreases nonlinearly with an increase in λ. We expect similar dependence of FREE stiffness on slenderness ratio. In Fig. 4, the values of the stiffness are smaller for larger slenderness ratio, keeping fiber angle and pressure the same. The kinematic parameter γ remains relatively constant with changing λ with the exception being very short beams with dominant shearing effects.

Fig. 4
Fig. 4
Close modal
Fig. 5
Fig. 5
Close modal

It is important to note the significance of using a four-spring PRB model here. While the three-spring PRB model proposed by the authors [21] was able to approximate the behavior of FREE deformation well, it was unable to capture variations in the location of torsional springs effectively. The locations of the torsional springs became largely dependent on the design variable γ, while the other design variables ($KthandKex$) appeared to have little impact. This resulted in γ varying nonlinearly across different FREE designs. However, by incorporating linear springs at the extremal ends, the position of torsional springs becomes dependent on both γ and Kex, resulting in a more linear trend in the variation of γ.

### 3.2 Parametric Model Fitting.

We propose linear and quadratic regression models to capture the dependencies of the PRB parameters on actuation pressure, fiber angle, and slenderness ratio as shown below (Eq. (6)). Here, we assume that the number of fibers, external diameter, and wall thickness of the FREEs are assumed to be constant.
$γ=a0+a1λ+a2α+a3Pa+a4λ2+a5α2+a6Pa2+a7Paλ+a8λα+a9αPaKth=b0+b1λ+b2α+b3Pa+b4λ2+b5α2+b6Pa2+b7Paλ+b8λα+b9αPaKex=c0+c1λ+c2α+c3Pa+c4λ2+c5α2+c6Pa2+c7Paλ+c8λα+c9αPa$
(6)

The linear model will use the first three coefficients, while the nonlinear model will use all nine coefficients. The best fit for the PRB model parameters is chosen based on the Akaike's information criterion (AIC). AIC provides a measure of model quality with the most accurate model having the smallest AIC value. The AIC values for first- and second-order models for the three parameters are tabulated in Table 2.

Table 2

AIC-based model selection

AICModel order = 1Model order = 2
γ−88.24−114.68
Kth−4.160.87
Kex17.81−26.44
AICModel order = 1Model order = 2
γ−88.24−114.68
Kth−4.160.87
Kex17.81−26.44

The AIC value for Kth is smaller for a linear model, indicating a better fit that accurately capturing the parametric behavior. However, the AIC values for γ and Kex was observed to be smaller for a quadratic model, indicating a more nonlinear trend that requires second-order terms to capture the parametric behavior.

The parameters obtained from Eq. (4) are used to fit the nonlinear models defined in Eq. (6). The coefficient obtained from this fit are tabulated in Table 3. The goodness of fit can be determined by plotting the standardized residual plot of the data. As a rule of thumb, a good model presents a symmetric distribution of standardized residuals about 0 and is bounded within ±2 as shown in Fig. 6.

Fig. 6
Fig. 6
Close modal
Table 3

Parametric model coefficients

i = 0i = 1i = 2i = 3i = 4i = 5i = 6i = 7i = 8i = 9
ai−0.7180.1110.0150.07900−0.001−0.002−0.0020
bi5.402−0.073−0.1230.079000000
ci48.317−2.592−1.7260.1240.0620.0170.006−0.0230.042−0.002
i = 0i = 1i = 2i = 3i = 4i = 5i = 6i = 7i = 8i = 9
ai−0.7180.1110.0150.07900−0.001−0.002−0.0020
bi5.402−0.073−0.1230.079000000
ci48.317−2.592−1.7260.1240.0620.0170.006−0.0230.042−0.002

To demonstrate the efficacy of the model, we need to assess its capability to generalize FREEs defined within or outside the manifold from which the data were collected. We consider two configurations and mean tip position errors between the PRB model and FEM simulations for this study. The first configuration represents an interpolated FREE that lies within the boundaries of the domain from which training data were collected, while the second configuration lies outside this region, i.e., an extrapolated FREE. The results are tabulated in Table 4. It can be observed that the model error for an interpolated FREE is less than that of the extrapolated FREE. This indicates that by collecting data points uniformly over the workspace, one can approximate the deformation characteristics of FREEs within that manifold with great degree of accuracy, thereby circumventing the need for expensive simulations. This can become useful while creating conceptual mechanism designs having soft elements like FREEs.

Table 4

Generalization error

InputsPRB model parametersMean tip position error (mm)
λ = 8.5, α = 35 deg, Pa = 15 psiγ = 0.62, Kth = 1.68, Kex = 2.721.46
λ = 8.5, α = 45 deg, Pa = 22.5 psiγ = 0.70, Kth = 1.05, Kex = 2.723.40
InputsPRB model parametersMean tip position error (mm)
λ = 8.5, α = 35 deg, Pa = 15 psiγ = 0.62, Kth = 1.68, Kex = 2.721.46
λ = 8.5, α = 45 deg, Pa = 22.5 psiγ = 0.70, Kth = 1.05, Kex = 2.723.40

## 4 Case Study

The PRB model is deemed to be useful in the design and analysis of mechanisms that comprise a combination or a network of FREEs. A simple combination of two FREEs arranged in a dyadic configuration was presented and analyzed in our earlier work [21]. In this section, we summarize the salient features of this simple system and present two other examples: a compliant gripper and a rotor.

### 4.1 Pennate Arrangement of FREEs.

Pennation (Fig. 7) is commonly seen in the physical arrangement of the muscle fibers, which are inclined to the line of action of the muscle by an angle (also called pennate angle) [7]. As the pennate angle ξ of the muscle fibers increases, the muscles begin to favor resisting external forces over producing larger strokes. In this case study, we study a two-stage pennate mechanism comprising four FREEs inclined at a specified angle. Stage 1 is composed of two FREEs, whose one end is connected at the bottom with a pin connector O, and the other end is fixed by a floating hinge support as shown in Fig. 7. Stage 2 FREEs emanate from the hinge support and culminate in connection B. The deflection of the stage 1 adds to the deflection of stage 2, resulting in an amplified net deflection at B. Due to the symmetric nature of the problem, only one half of the mechanism is analyzed. The PRB analysis consists of three steps.

Fig. 7
Fig. 7
Close modal
Kinematics: Kinematics requires expressing the coordinates of the two points A and B in terms of PRB joint angles and linear spring deformation.
$xA=(l0+δl1)cos(ξ)+γL1cos(ξ+θ11)+(lo+δl1)×cos(ξ+θ11+θ12)yA=(l0+δl1)sin(ξ)+γL1sin(ξ+θ1)+(lo+δl1)×sin(ξ+θ11+θ12)xB=xA+(l0+δl2)cos(α)+γL2cos(α+θ22)+(lo+δl2)×cos(α+θ21+θ22)yB=yA(l0+δl2)sin(α)+γL2sin(α+θ22)+(lo+δl2)×sin(α+θ21+θ22)$
(7)
The above equation is linearized for small changes in the coordinates and PRB parameters is given as
$δx+Jδq=0$
(8)
where the coordinate x = (xA, yA, xB, yB) and $q=(θi1,θi2,li)$, for i = 1…2. J is the Jacobian matrix.
Principle of Virtual Work: The principle of virtual work equates the virtual work performed by external forces and the internal virtual strain energy due to deformations in the torsional and linear springs. Two external forces are acting on the mechanism, force FxA acting at A in the x-direction constrains A against moving in this direction. This constraint is physically enforced due to the adjustable hinge support. Similarly, a force FxB acting at B prevents x-displacement enforced by the symmetry condition. Furthermore, force Fa acts at the linear springs to simulate FREE pressurization.
$FTδx+FaTδl=∑i=1n∂U∂qiδqi$
(9)
where $δl=[δli]$ for i = 1, 2 is the virtual change in the extensional springs of the FREEs, and $q=[θi1,θi2,δli]$ for i = 1, 2 are the virtual changes in the rotations and extensional spring deflections of the FREEs. F = [FxAFxB] represents external constraint loads acting on the system. The strain energy stored in the system is due to the torsional and extensional springs and is given by
$U=∑i=1212Kthiθi12+12Kthiθi22+Kexiδli2$
(10)
Numerical Solution: The virtual work formulation yields six equations, equal to the number of PRB parameters. However, there are eight unknowns that include q and constraint forces FxA and FxB. The two additional equations are obtained from constraining x-displacements of points A and B
$xA−xA0=0xB−xB0=0$
(11)

Figure 7 also details the results from analysis and corresponding experimental validation for the two-stage dyad configuration composed of FREEs having $(α=30,λ=6,Pa=20psi)$. We observe that the stroke generated increases with pennate angle up to a threshold value of $ξ≈22deg$ and then decreases. The maximum deflection at the threshold pennate angle is three times the axial contraction of the FREE (taken at $θ=90deg$). However, the actuation force generated at B increases nonlinearly with pennate angle and reaches a maximum value at $ξ=90deg$. These studies indicate that the dyad topology of FREEs can be an effective building block for soft robots, as it can modulate between high stroke-low force and low stroke-high force scenarios by varying the pennate angle.

### 4.2 Compliant Gripper.

Soft grippers are increasingly popular because of their safety and adaptability [26,27] by distributing forces evenly along the object being gripped. In this section, we present the design of a soft gripper that consists of a network of FREEs as shown in Fig. 8. The gripper is composed of two parts: the soft region comprising four FREEs that actuate the gripper and form a shape morphing gripping surface and a rigid portion made from stainless steel wires that support the soft components and provide more gripping force. The gripper leverages the combined action of the FREE deformation and the kinematics of the geometric arrangement toward the gripping action. Analysis of the gripper involves modeling both the FREEs and wires using the PRB parameters. The PRB model for the wire is similar to Fig. 2 with two torsional springs, but with a rigid member instead of a linear spring simulating large axial stiffness, and no actuation force implying passive deformation. The torsional spring constants are evaluated similar to the procedure in Eq. (4). Just as in the previous example, the analysis is carried out in three steps.

Fig. 8
Fig. 8
Close modal
Kinematic Analysis: For analysis, a symmetric half of the gripper is considered, which is further bifurcated into two loops 1 and 2 as shown in Fig. 9, leading to end points A and B. The bifurcation is required because the gripper represents a parallel mechanism. We formulate the kinematic loop closure equations [28] to obtain the coordinates of points A and B as a function of internal deflection parameters. The coordinates can then be expressed as a nonlinear function of the PRB angles using forward kinematics.
$x=f(q)δx=Jδq$
(12)
where x = (xA, yA, xB, yB), $q=(θ1,θ2,δl,θ3,θ4,θ5,θ6)$, and J is the jacobian matrix. The first three parameters characterized the deformation of the FREE, while the last four characterize the deformation of the beam.
Fig. 9
Fig. 9
Close modal

Virtual Work Formulation: Virtual work is applied for both loops separately where the external forces acting in loop 1 are FxA, that constrains point A along x–direction, and reaction FyA exerted by the rigid members to resist downward deformation. Similarly the external forces acting in loop 2 are FxB exerted to constrain point B along y-direction, and a force FxB is felt by the rigid members that seek to pull point B along the x-axis.

Numerical Solution: The virtual work equations are numerically solved along with the following kinematic constraints that equate the displacements of points A and B.
$xA−xB=0xB−xinitial=0yA−yB=0$
(13)

To maximize the stroke, a pennation angle of $ξ=30∘$ is chosen. The inclinations of the gripping FREEs are chosen to be 120 deg. To validate the analysis, a prototype was constructed with $ξ=30deg$ and FREE parameters ($α=30deg$, $λ=6,Pa=0.121Mpa$). The rigid stainless steel wires are assumed to have a Young's modulus of E = 2.3 GPa. The accuracy of the model is validated by comparing the stroke generated at A and C between the simulation and prototype. Experimental results as reported in Table 5, and errors can be partly attributed to inconsistencies in the prototype.

Table 5

Stroke comparison for compliant gripper

Stroke (mm) atSim xExp xSim yExp y
A40.8742.025.322
C0025.024.0
Stroke (mm) atSim xExp xSim yExp y
A40.8742.025.322
C0025.024.0

### 4.3 Soft Rotary Motor.

The presence of rotary servomotors is almost ubiquitous in modern day industries. Lightweight motors with large load torques are particularly attractive for aerospace applications like deployment mechanisms used in solar arrays and antenna drives. In this section, we design a rotary motor to maximize its angular displacement by optimal placement of the FREEs, similar to the pennate angle.

The proposed concept is shown in Fig. 10. The design is composed of a central rotor assembly that is connected to four symmetrically distributed FREEs. The FREEs act as actuating members, and when pressurized, they bend and provide torque and angular displacement ($θm$) at the rotor. Furthermore, the angular displacement is a function of the pennation angle (α) at which the FREEs are connected to the rotor. Each FREE connected to the rotor has four joint variables $q=(θ1,θ2,δl,θ3)$ and two forces F = (Fx, Fy) working to keep the tip position of the floating link fixed. Four constraints are obtained from the vitual work formulation and the remaining two from Eq. (14). The rotor angle is calculated as follows: $θm=π/2−(θ1+θ2+θ3)$.
$xinit−xfinal=0yinit−yfinal=0$
(14)
Fig. 10
Fig. 10
Close modal

Figure 11 illustrates the effect of pennation angle on the angular displacement. It is observed that the angular displacement increases with an increase in pennation angle and then decreases beyond a threshold. The maximum value of $θm$ is 48 deg for $ξ=27$ deg at Pa = 20 psi. Figure 11 also shows a comparison between results obtained from analysis and experiments for two pennation angles, $ξ=0$ deg, 30 deg. It can be observed that the error is greater at lower pressures (<15 psi). In addition to inaccuracies present in any prototype due to geometrical faults, this error can be attributed to the configuration being present outside the testing manifold from which data were collected. As reported earlier, the error for such points is usually larger compared to interpolated points within the test manifold. However, at higher pressures, these errors reduce as the operating configuration transitions within the test manifold.

Fig. 11
Fig. 11
Close modal

## 5 Conclusion

Analysis of soft self-actuating mechanisms is challenging because of the nonlinearity due to large deformations and hyperelastic material properties. These result in a set of coupled partial differential equations that are typically solved using computationally expensive finite element package. This paper aims to formulate a lumped PRB model to analyze a specific soft building block known as FREEs. We lay out an elaborate framework to fit the model parameters such that the transverse deflection can be captured with less than 10%–15% accuracy. Using the PRB model, a mechanism composed of a planar combination of FREEs can be analyzed quickly, as they reduce to find roots for a set of nonlinear equations. Perhaps the most important advantage of the PRB model is the ability to quickly evaluate design concepts and incorporate it within an optimization framework. The examples illustrated in the paper generate insight on how inclined FREE topologies (or pennate configurations) can be combined to increase available stroke, which can be considered the soft robotic equivalent of gears. While the PRB model in this paper captures quasi-static deformation, it can be easily extended to include dynamic characteristics [29] in future improvements. A dynamic model will aid in better design and enable controls. Furthermore, the model can be extended to capture a spatial combination of FREEs.

### Appendix A

This section details the experiments conducted to determine the bending behavior of FREEs, which in turn is used to validate the FE model used for the data collection. Three categories FREEs with different fiber angles α = (25 deg, 30 deg, 40 deg) were fabricated. Under each category, three FREEs with length l = (78, 110.5, 130)mm, diameter d = 13 mm, and slenderness ratio (ratio of FREE length to diameter) $λ=(6,8.5,10)$ were fabricated and subjected to three actuation pressures, $Pa=(15,17.5,20)psi$, along with transverse bending loads. The experimental setup is shown in Fig. 12. FREEs are constrained at one end, and the other end is subjected to a planar transverse end point load, and the FREE's tip deflection is recorded. Excessive loading is avoided to ensure that no part of the FREE undergoes buckling or kinking. To account for hysteresis, each data point is recorded as the mean of multiple trials. This study is used to validate the FEM model, which can then be used as a suitable surrogate for experimental data collection with reasonable confidence. The axial parameters of the FREE such as the actuation force fa on the linear springs have already been verified by the authors’ previous publications [24] and other researchers [6,25]. For the finite element simulation, a maximum axial load of 50 N and bending load of 1.5 N are used to obtain the deformation profiles. The parameters used for FEA are listed in Table 6.

Fig. 12
Fig. 12
Close modal

Results of the comparison between experiments and FEA for $λ=6,8.5,10$ are tabulated in Table 7. Each observation corresponds to the mean error between experimental and simulated tip position for three transverse loads divided by the tip displacement obtained from experiments. The relatively large errors can be explained as follows. When FREEs are subjected to small loads, the tip deflections are also relatively small. In such cases, even a small deviation (∼2 mm) compounded by the resolution of readings from experiments (±2 mm) can reflect as large errors. An example in this regard would be the FREE, $λ=6,$$α=25deg$, Pa = 15 psi subjected to a transverse load of 0.098 N. We observe (xexp, yexp) = (16, 1) and (xfem, yfem) = (13.1, 2.16), which translates to an error of 19.48%. Another important observation is the reduction in error with an increase in λ. This occurs because for the same transverse load, the tip deflection increases with λ, thereby reducing the percent error.

Table 6

Finite element model parameters

ParameterValue
C100.0988 Mpa
Ekevlar31 Gpa
Esteel210 Gpa
νkevlar0.36
νsteel0.30
Seed Densityfibers1.5
Seed Densitycap2
Seed Densitybody2
ParameterValue
C100.0988 Mpa
Ekevlar31 Gpa
Esteel210 Gpa
νkevlar0.36
νsteel0.30
Seed Densityfibers1.5
Seed Densitycap2
Seed Densitybody2
Table 7

FEM model error (%)

λPa(psi)α = 25 degα = 30 degα = 40 deg
62015.1217.8817.23
17.513.1417.2617.96
15.017.7017.3812.84
8.5208.527.959.91
17.510.7711.038.33
15.013.459.036.77
10202.458.017.84
17.55.886.214.75
15.06.8610.206.92
λPa(psi)α = 25 degα = 30 degα = 40 deg
62015.1217.8817.23
17.513.1417.2617.96
15.017.7017.3812.84
8.5208.527.959.91
17.510.7711.038.33
15.013.459.036.77
10202.458.017.84
17.55.886.214.75
15.06.8610.206.92

### Appendix B

The total strain energy (U) stored in a PRBM due to internal pressurization and external tip loading is given by
$U=12Kthθ12+12Kthθ22+Kexδl2+2faδl$
(B1)
According to the principle of virtual work for deformable bodies, the virtual work done by externally applied forces and by internal forces is equal to zero.
$δWext=FextδXδWint=−δU$
(B2)
where $δX$ corresponds to the virtual displacements at the PRBM tip. Therefore, we obtain
$δWext+δWint=0FextδX=δUFextδX=Kthθ1θ̇1+Kthθ2θ̇2+2Kexδ̇l+2faδ̇l$
(B3)
From the forward kinematics, we know that the Jacobian of the PRBM is given as
$J=[−γLsin(θ1)−(lo+δl)sin(θ1+θ2)−(lo+δl)sin(θ1+θ2)1+cos(θ1+θ2)γLcos(θ1)+(lo+δl)cos(θ1+θ2)(lo+δl)cos(θ1+θ2)sin(θ1+θ2)110]$
and relates the tip velocity of the PRBM to its joint velocities as
$δX=JδΘ$
(B4)
where Θ is the joint positions of PRBM. Substituting Eq. (B4) in Eq. (B3), we obtain Eq. (2).

## References

1.
Laschi
,
C.
, and
Cianchetti
,
M.
,
2014
, “
Soft Robotics: New Perspectives for Robot Bodyware and Control
,”
Front. Bioeng. Biotechnol.
,
2
, p.
3
. 10.3389/fbioe.2014.00003
2.
Rus
,
D.
, and
Tolley
,
M. T.
,
2015
, “
Design, Fabrication and Control of Soft Robots
,”
Nature
,
521
(
7553
), pp.
467
475
. 10.1038/nature14543
3.
Krishnan
,
G.
,
Bishop-Moser
,
J.
,
Kim
,
C.
, and
Kota
,
S.
,
2012
, “
Evaluating Mobility Behavior of Fluid Filled Fiber-Reinforced Elastomeric Enclosures
,”
ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
,
Chicago, IL
,
Aug. 12–15
, American Society of Mechanical Engineers, pp.
1089
1099
.
4.
Krishnan
,
G.
,
Bishop-Moser
,
J.
,
Kim
,
C.
, and
Kota
,
S.
,
2015
, “
Kinematics of a Generalized Class of Pneumatic Artificial Muscles
,”
J. Mech. Robot.
,
7
(
4
),
041014
. 10.1115/1.4029705
5.
Chou
,
C.-P.
, and
Hannaford
,
B.
,
1996
, “
Measurement and Modeling of McKibben Pneumatic Artificial Muscles
,”
IEEE. Trans. Rob. Autom.
,
12
(
1
), pp.
90
102
. 10.1109/70.481753
6.
Kothera
,
C. S.
,
Jangid
,
M.
,
Sirohi
,
J.
, and
Wereley
,
N. M.
,
2009
, “
Experimental Characterization and Static Modeling of McKibben Actuators
,”
J. Mech. Des.
,
131
,
91010
. 10.1115/1.3158982
7.
Azizi
,
E.
,
Brainerd
,
E. L.
, and
Roberts
,
T. J.
,
2008
, “
Variable Gearing in Pennate Muscles
,”
,
105
(
5
), pp.
1745
1750
. 10.1073/pnas.0709212105
8.
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
. 10.1016/j.compscitech.2006.01.002
9.
Peel
,
L.
,
1998
, “Fabrication and Mechanics of Fiber-Reinforced Elastomers,” Ph.D. thesis,
Brigham Young University
,
Provo, UT
.
10.
Klute
,
G. K.
,
Czerniecki
,
J. M.
, and
Hannaford
,
B.
,
1999
, “
McKibben Artificial Muscles: Pneumatic Actuators With Biomechanical Intelligence
,”
1999 IEEE/ASME International Conference on Advanced Intelligent Mechatronics
,
Atlanta, GA
,
Sep. 19–23
, pp.
221
226
.
11.
Singh
,
G.
, and
Krishnan
,
G.
,
2015
, “
An Isoperimetric Formulation to Predict Deformation Behavior of Pneumatic Fiber Reinforced Elastomeric Actuators
,”
IEEE International Conference on Intelligent Robots and Systems
,
Hamburg, Germany
,
Sept. 28–Oct. 2
, pp.
1738
1743
.
12.
Connolly
,
F.
,
Polygerinos
,
P.
,
Walsh
,
C. J.
, and
Bertoldi
,
K.
,
2015
, “
Mechanical Programming of Soft Actuators by Varying Fiber Angle
,”
Soft Robotics
,
2
(
1
), pp.
26
32
. 10.1089/soro.2015.0001
13.
Trivedi
,
D.
,
Lotfi
,
A.
, and
Rahn
,
C. D.
,
2008
, “
Geometrically Exact Models for Soft Robotic Manipulators
,”
IEEE. Trans. Robot.
,
24
(
4
), pp.
773
780
. 10.1109/TRO.2008.924923
14.
Trivedi
,
D.
,
2010
, “
Modeling, Optimal Design, Shape Estimation and Workspace
,” Ph.D. thesis,
The Pennsylvania State University
,
Pennsylvania, PA
.
15.
Antman
,
S. S.
,
1995
,
Nonlinear Problems of Elasticity
,
Springer-Verlag
,
New York
.
16.
Giri
,
N.
, and
Walker
,
I. D.
,
2011
, “
Three Module Lumped Element Model of a Continuum Arm Section
,”
2011 IEEE/RSJ International Conference on Intelligent Robots and Systems
,
San Francisco, CA
,
Sept. 25–30
, pp.
4060
4065
.
17.
Walker
,
I. D.
,
Mattfeld
,
R.
,
Mutlu
,
A.
,
Bartow
,
A.
, and
Giri
,
N.
,
2012
, “
A Novel Approach to Robotic Climbing Using Continuum Appendages in In-Situ Exploration
,”
2012 IEEE Aerospace Conference
,
Big Sky, MT
,
Mar. 3–10
, IEEE, pp.
1
9
.
18.
Midha
,
A.
, and
Howell
,
L. L.
,
1994
, “
A Method for the Design of Compliant Mechanisms With Small-Length Flexural Pivots
,”
ASME J. Mech. Des.
,
116
, pp.
280
290
. 10.1115/1.2919358
19.
Midha
,
A.
, and
Howell
,
L. L.
,
1996
, “
Evaluation of Equivalent Spring Stiffness for Use in a Pseudo-Rigid-Body Model of Large-Deflection Compliant Mechanisms
,”
J. Mech. Des.
,
118
, p.
127
.
20.
Midha
,
A.
,
1995
, “
Parametric Deflection Approximations for End-Loaded, Large-Deflection Beams in Compliant Mechanisms
,”
J. Mech. Des.
,
117
, p.
157
.
21.
Satheeshbabu
,
S.
, and
Krishnan
,
G.
,
2017
, “
Designing Systems of Fiber Reinforced Pneumatic Actuators Using a Pseudo-Rigid Body Model
,”
2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)
,
,
Sept. 24–28
, IEEE, pp.
1201
1206
.
22.
Venkiteswaran
,
V. K.
, and
Su
,
H.-J.
,
2016
, “
Extension Effects in Compliant Joints and Pseudo-Rigid-Body Models
,”
J. Mech. Des.
,
138
(
9
),
092302
. 10.1115/1.4034111
23.
Venkiteswaran
,
V. K.
, and
Su
,
H.-J.
,
2016
, “
A Three-Spring Pseudorigid-Body Model for Soft Joints With Significant Elongation Effects
,”
J. Mech. Robot.
,
8
(
6
),
61001
. 10.1115/1.4032862
24.
Singh
,
G.
, and
Krishnan
,
G.
,
2017
, “
A Constrained Maximization Formulation to Analyze Deformation of Fiber Reinforced Elastomeric Actuators
,”
Smart Mater. Struct.
,
26
(
6
),
065024
. 10.1088/1361-665X/aa6dc6
25.
Hocking
,
E. G.
, and
Wereley
,
N. M.
,
2013
, “
Analysis of Nonlinear Elastic Behavior in Miniature Pneumatic Artificial Muscles
,”
Smart Mater. Struct.
,
22
(
1
),
14016
. 10.1088/0964-1726/22/1/014016
26.
Manti
,
M.
,
Hassan
,
T.
,
Passetti
,
G.
,
D’Elia
,
N.
,
Laschi
,
C.
, and
Cianchetti
,
M.
,
2015
, “
A Bioinspired Soft Robotic Gripper for Adaptable and Effective Grasping
,”
Soft Rob.
,
2
(
3
), pp.
107
116
. 10.1089/soro.2015.0009
27.
Deimel
,
R.
, and
Brock
,
O.
,
2015
, “
A Novel Type of Compliant and Underactuated Robotic Hand for Dexterous Grasping
”.
Int. J. Robotics Res.
, 0278364915592961.
28.
Erdman
,
A. G.
,
Erdman
,
A. G.
,
Sandor
,
G. N.
, and
Kota
,
S.
,
2001
,
Mechanism Design 1
,
Prentice-Hall
,
.
29.
Yu
,
Y.-Q.
,
Howell
,
L. L.
,
Lusk
,
C.
,
Yue
,
Y.
, and
He
,
M.-G.
,
2005
, “
Dynamic Modeling of Compliant Mechanisms Based on the Pseudo-Rigid-Body Model
,”
J. Mech. Des.
,
127
(
4
), pp.
760
. 10.1115/1.1900750