# Reconsidering Fascicles in Soft Pneumatic Actuator Packs

Wyatt Felt Email<EMAIL_ADDRESS>
###### Abstract

This paper discusses and contests the claims of “Soft Pneumatic Actuator
Fascicles for High Force and Reliability” a research article which was
originally published in the March 2017 issue of the Journal Soft Robotics. The
original paper claims that the summed forces of multiple thin-walled extending
McKibben muscles are greater than a volumetrically equivalent actuator of the
same length at the same pressure. The original paper also claims that the
purported benefit becomes more pronounced as the number of smaller actuators
is increased. Using reasonable assumptions, the analysis of this paper shows
that the claims of the original paper violate the law of conservation of
energy. This paper also identifies errors in the original methodology that may
have led to the erroneous conclusions of the original paper. The goal of this
paper is to correct the record and to provide a more accurate framework for
considering fascicles used in soft pneumatic actuator packs.

##  I Introduction

The March 2017 issue of the journal Soft Robotics included a research article
[1] that considered the use of multiple Soft Pneumatic Actuators (SPAs) in
parallel combinations. While the original paper is interesting in many ways,
the central claims of the paper appear to be based on faulty analysis. The
present paper seeks to provide a more thorough analysis, to correct some of
the main conclusions of the original, and, thereby, to provide a sound
theoretical basis for the design of pneumatic actuator packs with multiple
individual actuators working in parallel.

The original paper [1] has several interesting contributions. It explores
combining extending McKibben muscle actuators together to sum their forces.
Each individual actuator is referred to as a “fascicle” and together they form
a “pack” (Fig. 1). The pack configuration is useful to achieve high forces
without increasing the size of the individual actuators. By using four
actuators, for example, the force output of the pack is quadruple that of the
individual fascicles. The authors also show how the relatively small
individual actuators can be arranged in patterns that make them less bulky.
For instance, arranging the actuators in a row creates a low-profile pack that
could be worn under clothing. This flat pack is certainly less cumbersome than
a single actuator with a larger diameter.

Though [1] considers extending McKibben muscles, others have looked at effect
of using multiple, thin contracting McKibben muscles. Suzumori’s group, for
example, has been experimenting with bundles of thin contracting McKibben
muscles since at least 2011 [2, 3]. Their work shows that, due to the large
diameter change in contracting McKibben muscles, a “mulifilament” muscle can
exhibit a greater contraction ratio than the individual actuators [3]. Others
have looked at how the principles of “variable recruitment” (which is common
in mammalian muscle) can be applied to groups of contracting McKibben muscles
to achieve gains in efficiency [4, 5, 6, 7, 8].

Figure 1: The original paper makes the erroneous claim that the force output
from a pack of $n$ individual actuators exceeds that of a single
volumetrically comparable actuator. The paper further claims that this benefit
becomes more pronounced as $n$ increases.

Though not without merit, the original paper,[1], presents specious claims
about the benefits of fascicles. The central claim could be summarized as
follows: By using multiple thin-walled soft pneumatic actuator “fascicles,” it
is possible to achieve higher forces in a pack than a single larger actuator
with an equivalent length and volume at the same pressure could produce. The
original paper further claims that the purported improvement in volumetric
force-density increases with increasing numbers of fascicles. These counter-
intuitive (and thus attractive) claims appear to be based on a faulty analysis
of the McKibben muscle force equations.

The analysis presented in this work demonstrates that the claims in the
original paper are not true. The primary contribution of this paper is to
provide a more thorough and correct analysis of the models incorrectly used in
[1]. It appears that errors in the analysis of these models led to the
erroneous claims of the original paper.

##  II Analysis

###  II-A Original claim

The original paper claims that the sum of forces from the individual actuator
“fascicles” in a pack is greater than the force from “an equivalent single SPA
of comparable volume.” The original paper further predicts that this benefit
would become more pronounced as the number of fascicles in the pack is
increased. Examples of the claims in the paper include:

From the abstract,

> “Experimental measurements show an SPA pack to generate over 112 N linear
> force, while the model indicates the benefit of parallel actuator grouping
> over a geometrically equivalent single SPA scale as an increasing function
> of the number of individual actuators in the group. For a module of four
> actuators, a 23 % increase in force production over a volumetrically
> equivalent single SPA is predicted and validated, while further gains appear
> possible up to 50 %.”

From the methods,

> “By grouping SPAs in parallel, actuator packs can be formed, which
> outperform individual SPAs of comparable size. Not only can higher force
> output be achieved with this configuration but also the benefit mutually
> increases with the number of constitutive actuator units utilized in the
> pack. This is to say that the more unit actuators are combined in parallel,
> the greater the gain in force output can be attained. This improvement in
> spatial-force efficiency allows for either stronger actuation in a given
> physical space or more compact actuation for a given required force.”

And from the discussion (emphasis original),

> “It is shown in the Results section that a fascicle arrangement of SPAs is
> capable of generating more linear force than an equivalent single SPA of
> comparable volume. Moreover, the gain in force production continues to
> increase within practical bounds as the number of units in a parallel
> configured SPA pack is increased. This model-based finding indicates that
> high-performance SPA design favors multiplicity and can be exploited as a
> new actuator design strategy, which takes into consideration the effect of
> multiple unit actuators in parallel coordination as well as the individual
> unit actuator design parameters to achieve desired performance.”

With regard to the word “validated” in the abstract, [1] reports no
experimental comparison between a “volumetrically equivalent single SPA” and a
“module of [n] actuators.” The force output of a four-actuator pack is tested
experimentally, but the force data from this pack are not compared against
forces from a volumetrically equivalent actuator. The predicted “23 % increase
in force [from a four actuator pack compared to a single equivalent actuator]”
is “validated” only against an erroneous prediction from the model.

The erroneous claims of [1] have been repeated in subsequent publications. For
example, [9] states

> “[The authors of [1]] have proved that the arrangement of multiple smaller
> soft pneumatic actuators produces larger output forces compared to one soft
> actuator with the same total volume.”

Because the original paper uses only a simple model to establish its claims,
this paper uses only the same simple model (correctly analyzed) to refute it.
This paper presents no challenge to the experimental data of the original
paper, only to the paper’s untested predictions and conclusions.

###  II-B Actuator model

This work seeks to mirror the assumptions of the original paper insofar as
possible without duplicating errors. The model used in [1] was presented by
Chou and Hannaford [10] and is common in the literature. The model assumes
that the McKibben muscle is cylindrical and that its kinematics are governed
by an inextensible fiber braid. The outer dimensions of the actuator are given
by the axial length $L$ and diameter $D$ of the braid

$L=b\cos{\theta}$ (1) $D=\frac{b\sin{\theta}}{N\pi}$ (2)

where $b$ is the unwound length of the fibers in the braid, $N$ is the number
of times they circle the axis and $\theta$ is a measure of their angle with
respect to the long axis.

The diameter $D$ can also be expressed as a fraction of $D_{0}$, where
$D_{0}=\frac{b}{N\pi}$ is the theoretical diameter of the fibers if the braid
angle were 90 ∘ (i.e. $\theta=\pi/2$)

$D=D_{0}\sin{\theta}.$ (3)

The values of $D_{0}$, $N$ and $b$ are determined by the fabrication of the
actuator. Unlike the length $L$, diameter $D$ and braid angle $\theta$, they
do not change as the actuator creates motion.

Neglecting the wall-thickness of the actuator (thin-walled approximation), the
force output $F_{\text{thin}}$ of the actuator is approximated in [10] as

$F_{\text{thin}}=\frac{\pi
P^{\prime}}{4}{D_{0}}^{2}\left(3\cos^{2}\\!\left(\theta\right)-1\right)$ (4)

where P’ is the internal gauge pressure of the actuator.

This equation is presented in [10] for contractile McKibben muscles (i.e.
$\theta<54.7^{\circ}$). Accordingly, a positive value of $F_{\text{thin}}$
corresponds to a tensile force. The extending McKibben muscles considered in
this work and in [1] thus have braid angles larger than 54.7 ∘ and
correspondingly negative values of $F$. Nevertheless, when a force is said to
be “higher” or have an “increase” compared to another, it is understood that
it is larger in magnitude.

The analysis in [1] is based on a form of the equation in [10] that takes into
account the thickness $t_{k}$ of the elastomeric actuator walls. For clarity,
the force output predicted from this approximation is designated
$F_{\text{thick}}$ here. In [10] it has the following form

$\displaystyle F_{\text{thick}}=$ $\displaystyle\frac{\pi
P^{\prime}}{4}{D_{0}}^{2}\left(3\cos^{2}\\!\left(\theta\right)-1\right)$ (5)
$\displaystyle+\pi
P^{\prime}\left(D_{0}t_{k}\left(2\sin{\theta}-\frac{1}{\sin{\theta}}\right)-{t_{k}}^{2}\right).$

This approximation takes into account the volumetric effect of the wall-
thickness but not its elastic forces that resist the actuator motion.

When presented in [1], the authors use the identity of $D_{0}$ to rewrite the
equation as follows

$\displaystyle F_{\text{thick}}=$ $\displaystyle\frac{b^{2}P^{\prime}}{4\pi
N^{2}}\left(3\cos^{2}\\!\left(\theta\right)-1\right)$ (6) $\displaystyle+\pi
P^{\prime}\left(\frac{b}{N\pi}t_{k}\left(2\sin{\theta}-\frac{1}{\sin{\theta}}\right)-{t_{k}}^{2}\right).$

They also provide a valid expression for the fiber length $b$ based on the
Pythagorean theorem and the circumference of the cylinder

$b=\sqrt{L^{2}+\left(D\pi N\right)^{2}}.$ (7)

###  II-C Individual actuator parameters

The individual actuator parameters presented in the original paper are not
self-consistent. That is, they do not satisfy the model equations. This
inconsistency is corrected in this work by identifying the value of $\theta$
that preserves the consistency of the rest of the parameters.

TABLE I: Individual actuator parameter values as defined in the original paper (not self-consistent) Parameter | Value
---|---
$L$ | 145 mm
$D$ | 17 mm
$\theta$ | 75.2 ∘
$N$ | 16

The parameters of the individual actuators given in the original paper are
listed in Table I. These values are not self-consistent. That is, using the
listed parameters to calculate $b$ results in different values from each of
the three equations, Eq. (1), Eq. (2) and Eq. (7).

Assuming that the easy-to-measure values of $L$, $D$ and $N$ are correct, this
work uses Eq. (7) to calculate the value of $b$ as 866.7 mm. This value of $b$
results in the same value of $\theta$ from both Eq. (1) and Eq. (2), $\theta=$
80.4 ∘. The resulting set of self-consistent parameters are listed in Table
II.

TABLE II: Individual actuator parameter values considered here (self-consistent) Parameter | Value
---|---
$L$ | 145 mm
$D$ | 17 mm
$\theta$ | 80.369 ∘
$N$ | 16
$b$ | 866.728 mm

The original paper’s parameters do not result in a physically meaningful braid
description. This work uses the angle in Table II to avoid duplicating this
small error in the original work.

###  II-D Generating “equivalent” actuator parameters

A more serious problem in the original paper is found in the prescribed
methods for generating the parameters of the “equivalent” actuator. Following
the methodology described in the original paper results in another
inconsistent parameter set for the equivalent actuator. By failing to adjust
the the number of fiber turns with the change in diameter, the original paper
appears to consider a geometrically impossible equivalent actuator.

The original paper claims that a pack with many fascicles can produce more
force at the same pressure than an “equivalent” actuator of “comparable
volume.” The volume is preserved by using an “equivalent” actuator with the
same length as the pack but with a cross-sectional area that has been scaled
to match the total cross-sectional area of the actuators in the pack.

The original paper states,

> “For a cross-sectional area, $A$, of an individual unit actuator, the
> “equivalent SPA” is defined to have equal volume to a pack of $n$ unit
> actuators, by defining its cross-sectional area as $A_{\text{eq}}=nA$. This
> simple relationship for the equivalent area defines the necessary diameter
> for an equivalent single SPA. All other actuator parameters, as shown in
> Table 1, are held equal.”

Note that, in this work, the parameters of the “individual unit actuator” and
“equivalent” actuators are designated with the subscripts “ind” and “eq”
respectively.

The problem with the methodology described in the original paper is the
holding of “all other actuator parameters” equal while $D$ is scaled. That is,
problems arise when $L_{\text{ind}}=L_{\text{eq}}$,
$\theta_{\text{ind}}=\theta_{\text{eq}}$, $N_{\text{ind}}=N_{\text{eq}}$ and
$D_{\text{ind}}\neq D_{\text{eq}}$. Attempting to generate a set of equivalent
braid parameters in this way results in a set that is not self-consistent.

The number of turns $N$ must be defined differently if the diameter of the
actuator is different and the other parameters are kept constant. It is
possible that this error in calculation led the authors of the original paper
to their erroneous conclusions.

It is reasonable to hold the length and braid angles constant (i.e.
$L_{\text{ind}}=L_{\text{eq}}$ and $\theta_{\text{ind}}=\theta_{\text{eq}}$).
Indeed, for the volume to be “comparable” with equivalent cross-sectional
areas, the lengths must be equal. Similarly, the braid angle, which changes
over the actuation stroke, largely dictates the force output of a McKibben
muscle actuator [10] (see also Eq. (4) and Eq. (5) in this work). Two McKibben
muscle actuators with the same outer dimensions but different braid angles
would produce different forces for the same pressure. To compare apples to
apples, as they say, the braid angle must be the same in the equivalent
actuator.

With these two parameters equal, Eq. (1) makes it clear that the length $b$ of
the fibers in the individual and equivalent actuators must also be equal

$\displaystyle\text{Eq.~{}\eqref{eq:length}: }l$ $\displaystyle=b\cos{\theta}$
(8) $\displaystyle\text{Assuming }l_{\text{ind}}$
$\displaystyle=l_{\text{eq}}$ $\displaystyle
b_{\text{ind}}\cos{\theta_{\text{ind}}}$
$\displaystyle=b_{\text{eq}}\cos{\theta_{\text{eq}}}$ $\displaystyle
b_{\text{ind}}\cos{\theta_{\text{ind}}}$
$\displaystyle=b_{\text{eq}}\cos{\theta_{\text{ind}}}$ $\displaystyle
b_{\text{ind}}$ $\displaystyle=b_{\text{eq}}.$

If $b_{\text{ind}}=b_{\text{eq}}$ and
$\theta_{\text{ind}}=\theta_{\text{eq}}$, then the number of fiber wraps in
the two actuators are equal (i.e. $N_{\text{ind}}=N_{\text{eq}}$) if and only
if the diameters are equal (i.e. $D_{\text{ind}}=D_{\text{eq}}$)

$\displaystyle\text{Eq.~{}\eqref{eq:diameter}: }D$
$\displaystyle=\frac{b\sin{\theta}}{N\pi}$ (9) $\displaystyle\text{Assuming
}D_{\text{ind}}$ $\displaystyle\neq D_{\text{eq}}$
$\displaystyle\frac{b_{\text{ind}}\sin{\theta_{\text{ind}}}}{N_{\text{ind}}\pi}$
$\displaystyle\neq\frac{b_{\text{eq}}\sin{\theta_{\text{eq}}}}{N_{\text{eq}}\pi}$
$\displaystyle\frac{b_{\text{ind}}\sin{\theta_{\text{ind}}}}{N_{\text{ind}}\pi}$
$\displaystyle\neq\frac{b_{\text{ind}}\sin{\theta_{\text{ind}}}}{N_{\text{eq}}\pi}$
$\displaystyle N_{\text{ind}}$ $\displaystyle\neq N_{\text{eq}}$ .

Thus, if $L$ and $\theta$ are kept the same, it is unreasonable to hold $N$
constant while scaling the diameter $D$ (as the original paper does).

Figure 2: The original paper evaluates a larger diameter actuator while
holding the length, braid angle and the number of turns of the fiber equal.
This is not possible in a physical actuator. The number of fiber turns must
change with the diameter. This can be demonstrated with a simple household
experiment. Shown are two equal-length strings wrapped around the same axial-
length of cylinders with different diameters. The angles of the fibers with
respect to the cylinder axis are approximately equal. It is clear, however,
that the fiber wraps around the thin cylinder more times than the larger-
diameter cylinder.

This relationship between $D$ and $N$ can be easily verified by wrapping and
taping two equal-length strings around two respective household cylinders of
different diameters (Fig. 2). If the strings are wrapped around the same axial
length of the cylinders, their angle with respect to the axis will be
approximately equal. If the diameters of the cylinders are different, the
number of turns around the cylinder will be different.

###  II-E Considering the claims without wall-thickness

In this work, the original paper’s problematic claims are first discussed
without taking the elastomeric wall-thickness into account. The neglect of the
wall-thickness is a reasonable approximation for actuators whose walls are
thin relative to their diameters. To indicate that the wall-thickness has been
neglected, the variables are marked with the subscript “thin.”

Neglecting the wall-thickness, the external cross-sectional area
$A_{\text{eq}}$ of the equivalent actuator is $n$-times greater than that of
the individual actuator $A_{\text{ind}}$. The outer diameter $D_{\text{eq}}$
of the equivalent actuator is the product of the square-root of the number of
actuator fascicles $n$ and the outer diameter of the individual actuator
$D_{\text{ind}}$

$\displaystyle A_{\text{eq}}$ $\displaystyle=nA_{\text{ind}}$ (10)
$\displaystyle\left(\pi/4\right){D_{\text{eq}}}^{2}$
$\displaystyle=n\left(\pi/4\right){D_{\text{ind}}}^{2}$ $\displaystyle
D_{\text{eq}}$ $\displaystyle=\sqrt{n}D_{\text{ind}}.$

The maximum diameter $D_{0}$ of the equivalent actuator scales in the same way
as $D$. This can be seen by examining Eq. (3) while considering that the braid
angles between the two actuators are the same

$D_{0\text{,eq}}=\frac{D_{\text{eq}}}{D_{\text{ind}}}D_{0\text{,ind}}.$ (11)

By Eq. (10) and Eq. (11), the maximum diameter of the equivalent actuator
$D_{0\text{,eq}}$ is similarly scaled from that of the individual actuator
$D_{0\text{,ind}}$

$D_{0\text{,eq}}=\sqrt{n}D_{0\text{,ind}}.$ (12)

####  II-E1 Method 1, Force Equation

The total force output $F_{\text{thin,pack}}$ from a “pack” of identical
individual fascicles in parallel is the sum of their individual forces given
by Eq. (4) (recall that the lowercase “$n$” is the count of the fascicles in
the pack)

$\displaystyle F_{\text{thin,pack}}$ $\displaystyle=nF_{\text{thin,ind}}$ (13)
$\displaystyle F_{\text{thin,pack}}$
$\displaystyle=n\frac{\pi{\left(D_{0\text{,ind}}\right)}^{2}P^{\prime}}{4}\left(3\cos^{2}\\!\left(\theta_{\text{ind}}\right)-1\right).$

The force output from the single equivalent actuator is also given by Eq. (4)

$F_{\text{thin,eq}}=\frac{\pi{\left(D_{0\text{,eq}}\right)}^{2}P^{\prime}}{4}\left(3\cos^{2}\\!\left(\theta_{\text{eq}}\right)-1\right).$
(14)

Substituting for $D_{0\text{,eq}}$ from Eq. (12) and considering that
$\theta_{\text{eq}}=\theta_{\text{ind}}$, the force from the equivalent
actuator is

$\displaystyle F_{\text{thin,eq}}$
$\displaystyle=\frac{\pi{\left(\sqrt{n}D_{0\text{,ind}}\right)}^{2}P^{\prime}}{4}\left(3\cos^{2}\\!\left(\theta_{\text{ind}}\right)-1\right)$
(15) $\displaystyle F_{\text{thin,eq}}$
$\displaystyle=n\frac{\pi{\left(D_{0\text{,ind}}\right)}^{2}P^{\prime}}{4}\left(3\cos^{2}\\!\left(\theta_{\text{ind}}\right)-1\right).$

Eq. (15) is equivalent to Eq. (13).

Thus, when neglecting the wall-thickness, the force from an equivalent
actuator at the same pressure is predicted to be identical to the force from
the actuator pack, regardless of the number of fascicles used

$F_{\text{thin,eq}}=F_{\text{thin,pack}}.$ (16)

This contradicts the claims of the original paper.

####  II-E2 Method 2, Conservation of Energy

The law of conservation of energy can also be used to evaluate the claims.

Consider an ideal actuator without losses or energy storage. The input energy
$E_{\text{in}}$ to the actuator is equal to the output energy $E_{\text{out}}$

$\displaystyle E_{\text{in}}$ $\displaystyle=E_{\text{out}}$ (17)
$\displaystyle P^{\prime}{\Delta}V$ $\displaystyle=F_{\text{avg}}{\Delta}L.$

The input is flow energy, that is, the product of the gauge pressure in the
actuator $P^{\prime}$ (considered to be constant) and the change in volume
${\Delta}V$. The output energy is mechanical work, that is, the product of the
average extension force $F_{\text{avg}}$ and the distance over which that
force is applied (written as a change of length ${\Delta}L$) [11].

Considering two closed-volume fluid-powered actuators actuated by the same
constant pressure $P^{\prime}$ and acting over the same distance ${\Delta}L$,
the average force of one actuator can only exceed the other if it undergoes a
larger change in volume ${\Delta}V$. For the claims of the original paper to
be valid, the sum of the volume changes of the individual actuators would need
to be larger than the volume change of the equivalent actuator. This, however,
is not the case.

To illustrate this, consider an extending McKibben muscle at an initial state
before extension (subscript “1”) and final state (“2”) after extension. The
change of volume ${\Delta}V$, with the same assumptions as the force model, is

$\displaystyle{\Delta}V$ $\displaystyle=V_{2}-V_{1}$ (18)
$\displaystyle=\frac{\pi}{4}\left({D_{2}}^{2}{L_{2}}-{D_{1}}^{2}{L_{1}}\right).$

Though the actuator extends axially, the value of $D_{0}$ remains the same.
Accordingly, Eq. (3) can be re-arranged to find the relationship between the
relaxed diameter $D_{1}$ and the post-extension diameter $D_{2}$. $\theta_{1}$
and $\theta_{2}$ are the respective fiber angles in the relaxed and extended
states.

$\displaystyle D_{0}=\frac{D_{1}}{\sin{\theta_{1}}}$
$\displaystyle=\frac{D_{2}}{\sin{\theta_{2}}}$ (19) $\displaystyle{D_{2}}$
$\displaystyle=\frac{\sin{\theta_{2}}}{\sin{\theta_{1}}}{D_{1}}$
$\displaystyle{D_{2}}$ $\displaystyle=\gamma{D_{1}}$ $\displaystyle\gamma$
$\displaystyle=\frac{\sin{\theta_{2}}}{\sin{\theta_{1}}}.$

When the actuator extends from $L_{1}$ to $L_{2}$, $D_{2}$ is scaled down from
the initial diameter $D_{1}$ by a factor of $\gamma$.

The value of $\gamma$ depends on $\theta_{1}$ and $\theta_{2}$ which, in turn,
can be found with length of the actuator $L$ at the respective angles and the
constant fiber-length $b$ (Eq. (1)). Because the individual and equivalent
actuators in this analysis have the same values for $L_{1}$, $L_{2}$ and $b$,
they share the same values of $\theta_{1}$, $\theta_{2}$ and $\gamma$.

With Eq. (19), the change in volume from Eq. (18) is

$\displaystyle{\Delta}V$
$\displaystyle=\frac{\pi}{4}\left({D_{2}}^{2}{L_{2}}-{D_{1}}^{2}{L_{1}}\right)$
(20) $\displaystyle=\frac{\pi}{4}\left({\left(\gamma
D_{1}\right)}^{2}{L_{2}}-{D_{1}}^{2}{L_{1}}\right)$
$\displaystyle=\frac{\pi}{4}{D_{1}}^{2}\left({\gamma}^{2}{L_{2}}-{L_{1}}\right).$

The total change in volume for the pack of $n$ actuators
${\Delta}V_{\text{pack}}$ is $n$-times the change of the individual actuators
${\Delta}V_{\text{ind}}$

$\displaystyle{\Delta}V_{\text{pack}}$ $\displaystyle=n{\Delta}V_{\text{ind}}$
(21)
$\displaystyle=n\frac{\pi}{4}{D_{1\text{,ind}}}^{2}\left({\gamma}^{2}{L_{2}}-{L_{1}}\right).$

By Eq. (10), the change in volume for the equivalent actuator
${\Delta}V_{\text{eq}}$ is

$\displaystyle{\Delta}V_{\text{eq}}$
$\displaystyle=\frac{\pi}{4}{D_{1\text{,eq}}}^{2}\left({\gamma}^{2}{L_{2}}-{L_{1}}\right)$
(22)
$\displaystyle=\frac{\pi}{4}{\left(\sqrt{n}D_{1\text{,ind}}\right)}^{2}\left({\gamma}^{2}{L_{2}}-{L_{1}}\right)$
$\displaystyle=n\frac{\pi}{4}{D_{1\text{,ind}}}^{2}\left({\gamma}^{2}{L_{2}}-{L_{1}}\right).$

The change volume is the same

${\Delta}V_{\text{pack}}={\Delta}V_{\text{eq}}.$ (23)

By Eq. (17), which assumes ideal actuators, the average force from a pack of
individual actuators $F_{\text{avg,pack}}$ is predicted to be equal to the
average force from an equivalent actuator $F_{\text{avg,eq}}$ at the same
pressure, undergoing the same length change

$F_{\text{avg,pack}}=F_{\text{avg,eq}}.$ (24)

This is true regardless of how many actuators are used to form the pack.
Accordingly, it contradicts the claims of the original paper.

###  II-F Assumptions about wall-thickness

It has been shown, without considering the wall-thickness, that there is no
force benefit from increasing the number of fascicles (compared to an
equivalent actuator). Both the force equation used by the original work and
the conservation of energy contradict the claims of the original work. This
work now considers the wall-thickness and shows, under reasonable assumptions,
that the addition of wall-thickness does add support for the claims of the
original paper.

One thing that is not explored in the original work is the effect of the
elastomeric bladder inside the McKibben muscle. The walls of the bladder
create non-trivial elastic forces that resist actuator motion and reduce the
output forces. These elastic forces were not considered in the original paper
and they are not considered here. What is considered in the original paper is
the effect of the wall-thickness $t_{k}$ on the internal volume of the
actuator. Because the force output of the actuator is related to the rate of
volume change, this thickness term $t_{k}$ also appears in the force equation
Eq. (5). Its effect is to reduce the force-output of the actuator.

The original paper does not discuss the effect of the wall-thickness on the
definition of the “equivalent” actuator. An increased wall-thickness reduces
the internal cross-sectional area of the actuator. The original paper does not
specify whether the cross-sectional area used to define equivalence is the
internal or external cross-sectional area. However, the original paper does
state that the the equivalent actuator has a “comparable volume” to total
volume of individual actuators in the pack. Accordingly, this work assumes
that the external cross-sectional area of the equivalent actuator is equal to
the total external cross-sectional area of the individual actuators in the
pack. This assumption makes the diameter relationship between the equivalent
and individual actuators identical to the relationship found in Eq. (10) (i.e.
$D_{\text{eq}}=\sqrt{n}D_{\text{ind}}$).

The fraction of the external diameter $D$ made up by the thickness of each
wall $t_{k}$ is designated $\hat{t}_{k}$

$\displaystyle 0\leq\hat{t}_{k}\leq 0.5$ (25)
$\displaystyle\hat{t}_{k}=\frac{t_{k}}{D}.$

This quantity is bounded between zero (no wall thickness) and one half (no
hollow interior).

For the sake of simplicity, this paper assumes that the relative wall-
thickness of the equivalent actuator $\hat{t}_{k\text{,eq}}$ is equal to the
relative wall-thickness of the individual actuator $\hat{t}_{k\text{,ind}}$.
That is, if the thickness of the original actuator walls made up 10 % of the
overall diameter, the walls of the equivalent actuator would make up 10 % of
the equivalent actuator’s overall diameter

$\displaystyle\hat{t}_{k\text{,ind}}$ $\displaystyle=\hat{t}_{k\text{,eq}}$
(26) $\displaystyle\frac{t_{k,\text{ind}}}{D_{\text{ind}}}$
$\displaystyle=\frac{t_{k,\text{eq}}}{D_{\text{eq}}}.$

When $\hat{t}_{k\text{,ind}}=\hat{t}_{k\text{,eq}}$, the equivalent actuator
has the same total volume of elastomeric material as the pack of individual
actuators. This assumed relationship is necessary because the original paper
does not discuss the wall-thickness of the equivalent actuator.

###  II-G The effect of wall-thickness

Eq. (5) can be manipulated algebraically to provide insight into the
volumetric effect of the thickness on the actuator force output. The equation
is first normalized by the gauge pressure $P^{\prime}$ and the expression in
(3) is substituted for $D_{0}$

$\displaystyle\frac{F_{\text{thick}}}{P^{\prime}}=$
$\displaystyle\frac{\pi}{4}D^{2}\left(2\csc^{2}\\!{\theta}-3\right)$ (27)
$\displaystyle+\pi\left(Dt_{k}\left(2-\csc^{2}\\!{\theta}\right)-{t_{k}}^{2}\right).$

See the appendix for the derivation of Eq. (27). Note that this form of the
equation needs to be used with caution. The form in Eq. (5) is parameterized
by $D_{0}$, which does not change during actuation. The form in Eq. (27) is
parameterized by the instantaneous external diameter $D$ which is itself a
function of the braid angle $\theta$ and changes over the course of the
actuation.

The expression in Eq. (27) can be further normalized by the external cross-
sectional area $A$ of the actuator defined by

$A=\frac{\pi}{4}D^{2}$ (28)
$\displaystyle\frac{F_{\text{thick}}}{P^{\prime}A}=$
$\displaystyle\left(2\csc^{2}\\!{\theta}-3\right)+4\left(2-\csc^{2}\\!{\theta}\right)\frac{t_{k}}{D}-4{\left(\frac{t_{k}}{D}\right)}^{2}.$
(29)

The normalized force in Eq. (29) is designated $\hat{F}_{\text{thick}}$, a
function of $\theta$ and the relative wall-thickness $\hat{t}_{k}$

$\displaystyle\hat{F}_{\text{thick}}\\!\left(\theta,\hat{t}_{k}\right)=\left(-4\right){\hat{t}_{k}}^{2}+\left(8-4\csc^{2}\\!{\theta}\right)\hat{t}_{k}+\left(2\csc^{2}\\!{\theta}-3\right).$
(30)

This function is plotted in Fig. 3.

Figure 3: Shown is the relationship between the normalized extension force
$\hat{F}_{\text{thick}}$, the braid angle $\theta$ and the relative
elastomeric wall-thickness $\hat{t}_{k}$. The force shown is normalized by
pressure and the current external cross-sectional area of the actuator. This
idealized extension force is highest at the theoretical braid angle of 90 ∘
and decreases with the braid angle. An extending McKibben muscle, such as
those considered in the original paper, is fabricated with some initial braid
angle greater than 54.7 ∘. This angle decreases as the actuator extends. When
there is no thickness, the maximum normalized force is one (equivalent to a
piston with the same area). Increasing the relative thickness $\hat{t}_{k}$
decreases the maximum extension force and increases the angle for which the
force is predicted to reach zero.

The predicted force output of an actuator at a given diameter is the product
of the function in Eq. (30) with the internal gauge pressure $P^{\prime}$ and
the external cross-sectional area $A$

$F_{\text{thick}}=\hat{F}_{\text{thick}}\\!\left(\theta,\hat{t}_{k}\right)P^{\prime}A.$
(31)

###  II-H Considering the claims with wall-thickness

Eq. (31) allows one to consider the claims of the original paper while
considering the thickness of the elastomeric walls. The force
$F_{\text{thick,pack}}$ from a pack of $n$ individual actuators is given by

$\displaystyle F_{\text{thick,pack}}$ $\displaystyle=nF_{\text{thick,ind}}$
(32)
$\displaystyle=n\hat{F}_{\text{thick}}\\!\left(\theta,\hat{t}_{k\text{,ind}}\right)P^{\prime}A_{\text{ind}}.$

The force $F_{\text{eq,thick}}$ from an actuator with an equivalent external
cross-sectional area ($A_{\text{eq}}=nA_{\text{ind}}$) and the same relative
wall-thickness ($\hat{t}_{k\text{,ind}}=\hat{t}_{k\text{,eq}}$) is given by

$\displaystyle F_{\text{thick,eq}}$
$\displaystyle=\hat{F}_{\text{thick}}\\!\left(\theta,\hat{t}_{k\text{,eq}}\right)P^{\prime}A_{\text{eq}}$
(33)
$\displaystyle=\hat{F}_{\text{thick}}\\!\left(\theta,\hat{t}_{k\text{,ind}}\right)P^{\prime}\left(nA_{\text{ind}}\right)$
$\displaystyle=n\hat{F}_{\text{thick}}\\!\left(\theta,\hat{t}_{k\text{,ind}}\right)P^{\prime}A_{\text{ind}}.$

Eq. (32) is equivalent to Eq. (33). Thus, even when considering the wall-
thickness, the force from an equivalent actuator at the same pressure is
predicted to be identical to the force from the actuator pack, regardless of
the number of fascicles used

$F_{\text{thick,eq}}=F_{\text{thick,pack}}.$ (34)

This contradicts the claims of the original paper.

###  II-I Numerical verification

As a final verification, the claims are considered through numerical
experimentation. The force from an SPA pack made up from individual actuators
with the parameters in Table II is calculated for a wall-thickness of 1 mm.
This is compared to the force from two equivalent actuators, one with the same
relative elastomeric wall-thickness and one with the same absolute wall-
thickness. This is repeated for various values of $n$ (i.e. various counts of
individual actuators in the pack). All the calculated forces come from Eq. (4)
and MATLAB code for the verification is included in the appendix. The results
are listed in Table III.

| SPA-Pack | Equivalent Actuator | Equivalent Actuator
---|---|---|---
$n$ | $nF_{\text{ind}}/P^{\prime}$ | Same relative thickness | Same absolute thickness
| ($t_{k,\text{ind}}=$ 1 mm) | ($\hat{t}_{k,\text{eq}}=\hat{t}_{k,\text{ind}}$) | ($t_{k,\text{eq}}=t_{k,\text{ind}}=$ 1 mm)
1 | -0.165 | -0.165 | -0.165
2 | -0.33 | -0.33 | -0.358
4 | -0.661 | -0.661 | -0.755
8 | -1.32 | -1.32 | -1.57
16 | -2.64 | -2.64 | -3.22
32 | -5.29 | -5.29 | -6.55
64 | -10.6 | -10.6 | -13.3

TABLE III: Numerical Verification. The predicted force output per unit
pressure (N/kPa). From Eq. (5) and Table II.

As expected, the force from the pack is identical to that from the equivalent
actuator with the same relative wall-thickness. This does not change with
increasing numbers of individual actuators. The equivalent actuator with the
same absolute wall-thickness of 1 mm shows the opposite trend of what is
claimed by the original paper. That is, it creates a higher magnitude force
than the pack and this disparity grows with an increasing $n$.

##  III Conclusion

This paper demonstrates that the central claims of the popular original paper,
“Soft Pneumatic Actuator Fascicles for High Force and Reliability,” [1] are
not true. The analysis of this paper refutes the original paper’s claim that
parallel combinations of extending McKibben muscles can produce more force-
per-unit-volume than a single actuator of the same kind. It also refutes the
original paper’s claim that this benefit becomes more pronounced as the number
of actuators in the pack increases. These claims have been refuted
analytically, with the original paper’s force model and with the law of
conservation of energy. These claims have also been refuted by direct
numerical calculation.

## References

  * [1] M. A. Robertson, H. Sadeghi, J. M. Florez, and J. Paik, “Soft pneumatic actuator fascicles for high force and reliability,” _Soft robotics_ , vol. 4, no. 1, pp. 23–32, 2017.
  * [2] S. Wakimoto, K. Suzumori, and J. Takeda, “Flexible artificial muscle by bundle of mckibben fiber actuators,” in _2011 IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM)_ , July 2011, pp. 457–462.
  * [3] S. Kurumaya, H. Nabae, G. Endo, and K. Suzumori, “Design of thin mckibben muscle and multifilament structure,” _Sensors and Actuators A: Physical_ , vol. 261, pp. 66 – 74, 2017. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0924424716308792
  * [4] S. A. DeLaHunt, T. E. Pillsbury, and N. M. Wereley, “Variable recruitment in bundles of miniature pneumatic artificial muscles,” _Bioinspiration & biomimetics_, vol. 11, no. 5, p. 056014, 2016.
  * [5] M. Meller, J. Chipka, A. Volkov, M. Bryant, and E. Garcia, “Improving actuation efficiency through variable recruitment hydraulic mckibben muscles: modeling, orderly recruitment control, and experiments,” _Bioinspiration & biomimetics_, vol. 11, no. 6, p. 065004, 2016.
  * [6] M. Bryant, M. A. Meller, and E. Garcia, “Variable recruitment fluidic artificial muscles: modeling and experiments,” _Smart Materials and Structures_ , vol. 23, no. 7, p. 074009, 2014.
  * [7] R. M. Robinson, C. S. Kothera, and N. M. Wereley, “Variable recruitment testing of pneumatic artificial muscles for robotic manipulators,” _IEEE/ASME Transactions on mechatronics_ , vol. 20, no. 4, pp. 1642–1652, 2015.
  * [8] E. M. Chapman and M. Bryant, “Bioinspired passive variable recruitment of fluidic artificial muscles,” _Journal of Intelligent Material Systems and Structures_ , vol. 29, no. 15, pp. 3067–3081, 2018.
  * [9] C. Tawk, M. in het Panhuis, G. M. Spinks, and G. Alici, “Bioinspired 3d printable soft vacuum actuators for locomotion robots, grippers and artificial muscles,” _Soft Robotics_ , vol. 5, no. 6, pp. 685–694, 2018, pMID: 30040042. [Online]. Available: https://doi.org/10.1089/soro.2018.0021
  * [10] C.-P. Chou and B. Hannaford, “Measurement and modeling of mckibben pneumatic artificial muscles,” _IEEE Transactions on robotics and automation_ , vol. 12, no. 1, pp. 90–102, 1996.
  * [11] W. Felt, M. A. Robertson, and J. Paik, “Modeling vacuum bellows soft pneumatic actuators with optimal mechanical performance,” in _2018 IEEE International Conference on Soft Robotics (RoboSoft)_ , April 2018, pp. 534–540.

## Appendix

### Derivation of Eq. (27) from Eq. (5)

$\displaystyle F_{\text{thick}}=$ $\displaystyle\frac{\pi
P^{\prime}}{4}{D_{0}}^{2}\left(3\cos^{2}\\!\left(\theta\right)-1\right)$ (35)
$\displaystyle+\pi
P^{\prime}\left(D_{0}t_{k}\left(2\sin{\theta}-\frac{1}{\sin{\theta}}\right)-{t_{k}}^{2}\right)$
$\displaystyle\frac{F_{\text{thick}}}{P^{\prime}}=$
$\displaystyle\frac{\pi}{4}{D_{0}}^{2}\left(3\cos^{2}\\!\left(\theta\right)-1\right)$
$\displaystyle+\pi\left(D_{0}t_{k}\left(2\sin{\theta}-\frac{1}{\sin{\theta}}\right)-{t_{k}}^{2}\right)$
$\displaystyle\frac{F_{\text{thick}}}{P^{\prime}}=$
$\displaystyle\frac{\pi}{4}{\left(\frac{D}{\sin{\theta}}\right)}^{2}\left(3\cos^{2}\\!\left(\theta\right)-1\right)$
$\displaystyle+\pi\left(\frac{D}{\sin{\theta}}t_{k}\left(2\sin{\theta}-\frac{1}{\sin{\theta}}\right)-{t_{k}}^{2}\right)$
$\displaystyle\frac{F_{\text{thick}}}{P^{\prime}}=$
$\displaystyle\frac{\pi}{4}D^{2}\left(3\frac{\cos^{2}\\!\left(\theta\right)}{\sin^{2}\\!\left(\theta\right)}-\frac{1}{\sin^{2}\\!\left(\theta\right)}\right)$
$\displaystyle+\pi\left(Dt_{k}\left(2-\frac{1}{\sin^{2}{\theta}}\right)-{t_{k}}^{2}\right)$
$\displaystyle\frac{F_{\text{thick}}}{P^{\prime}}=$
$\displaystyle\frac{\pi}{4}D^{2}\left(2\csc^{2}\\!{\theta}-3\right)$
$\displaystyle+\pi\left(Dt_{k}\left(2-\csc^{2}\\!{\theta}\right)-{t_{k}}^{2}\right).$

### MATLAB code for numerical verification

    close all
    clear all

    n_vec = [1,2,4,8,16,32,64]

    L = .145;
    D_ind = 17e-3; s
    theta = 80.3693712115*pi/180;
    N = 16;
    b = 0.866728223;
    t_k_ind = 1E-3;
    D_0_ind = b/(N*pi);

    F_thick_over_P = @(D_0,t_k) ...
    (pi/4)*D_0^2*(3*cos(theta)^2 - 1) ...
     + pi*(D_0*t_k*(2*sin(theta) - ...
     1/sin(theta)) - t_k^2)

    F_over_P_ind = ...
    F_thick_over_P(D_0_ind,t_k_ind);

    for i = 1:length(n_vec)

      n = n_vec(i);
      D_0_eq = sqrt(n)*D_0_ind;

      t_k_eq1 = sqrt(n)*t_k_ind;
      t_k_eq2 = t_k_ind;

      F_over_P_pack(i) = F_over_P_ind*n;

      F_over_P_eq1(i) = ...
      F_thick_over_P(D_0_eq,t_k_eq1);

      F_over_P_eq2(i) = ...
      F_thick_over_P(D_0_eq,t_k_eq2);

      disp([num2str(n), ...
      ’ & ’,num2str(F_over_P_pack(i)*1000,3),...
      ’ & ’,num2str(F_over_P_eq1(i)*1000,3),...
      ’ & ’,num2str(F_over_P_eq2(i)*1000,3),...
      ’ \\’])
    end