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Review Article

# A Survey on Static Modeling of Miniaturized Pneumatic Artificial Muscles With New Model and Experimental ResultsPUBLIC ACCESS

[+] Author and Article Information
K. P. Ashwin

Robotics and Design Lab,
Department of Mechanical Engineering,
Indian Institute of Science,
Bangalore 560012, India
e-mail: ashwinkp@iisc.ac.in

A. Ghosal

Professor
Department of Mechanical Engineering,
Indian Institute of Science,
Bangalore 560012, India
e-mail: asitava@iisc.ac.in

1Corresponding author.

Manuscript received May 1, 2018; final manuscript received October 4, 2018; published online October 24, 2018. Assoc. Editor: Francois Barthelat.

Appl. Mech. Rev 70(4), 040802 (Oct 24, 2018) (20 pages) Paper No: AMR-18-1055; doi: 10.1115/1.4041660 History: Received May 01, 2018; Revised October 04, 2018

## Abstract

Pneumatic artificial muscles (PAMs) are linear pneumatic actuators consisting of a flexible bladder with a set of in-extensible fibers woven as a sheath on the outside. Upon application of pressure, the actuators contract or expand based on the angle of winding of the braid. Due to the similarity in properties of the actuators with biological muscles and the advantages thereof, these are increasingly being used in many robotic systems and mechanisms. This necessitates the development of mathematical models describing their mechanics for optimal design as well as for application in control systems. This paper presents a survey on different mathematical models described in the literature for representing the statics of PAM. Since it is observed that the validity of existing static models, based on energy balance methods, is not consistent with changes in parameters when applied to their miniaturized versions of pneumatic artificial muscles (MPAM), a new model has been proposed. The model takes into account material properties of the bladder as well as the end-effects which are prominent for MPAMs. Experiments conducted on fabricated MPAMs, of different diameters and lengths, show that the proposed model predicts the pressure-deformation characteristics of MPAMs with maximum error of less than 7%.

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## Introduction

In 1958, Richard H. Gaylord patented a “fluid actuated stroking device” which is “an expansible chamber device comprising a bladder confined within a braided sheath…adapted to be energized by a fluid” [1]. The bladder which is sealed on one end is made of flexible material and the braided sheath is usually woven using in-extensible fibers. The device is essentially a linear actuator with the interesting property that if the angle at which the outer sheath is braided differs from a particular locking angle, it contracts or expands upon pressurization of the fluid contained in the bladder. This invention gained popular attention when later used by McKibben in a design of orthotic wheelchair [2]. Due to the similarity of this flexible actuator with biological muscles, the device is often identified by the name “McKibben muscles” or “fluidic artificial muscles (FAMs).” For several years, these actuators did not achieve much commercial success probably due to the bulky accessories which are required to energize the system or due to the faster progress in the development of electric motors and other actuators. More recently since the 1980s, the actuator have regained its commercial and academic attention due to its unique advantages. The FAMs which make use of pressurized air are also called “pneumatic artificial muscles (PAMs)” and are now extensively studied by engineers especially in the field of bio-inspired [3] and medical robotics [4]. In Ref. [5], the authors have listed in detail the major developments toward the evolution of pneumatic artificial muscles.

Among different types of conventional actuating mechanisms such as electric motors, pneumatic pistons, and shape memory alloys as well as the flexible actuators used in robots [6], artificial muscles stand out due to their following advantages:

• High power to weight ratio: The earliest commercial PAM called “rubbertuator” by Bridgestone corporation and Hitachi weighed about 6 kg and could lift a mass of approximately 2 kg (refer Refs. [7] and [8]). At present, the PAMs manufactured by companies like Festo [4] have a lifting force of 6000 N while weighing only about 800 g.

• Flexibility and compliance: An unpressurized PAM exhibits the same flexibility as that of the bladder but it becomes stiff while remaining reasonably compliant upon pressurization. This compliance is a necessity for the development of medical devices such as minimal invasive surgical tools [911] and rehabilitation robots [7,12,13].

• Compatibility with human environment: The primary actuation mechanism of PAM is pressurized air or pressurized inert gas. Hence, it is safer compared to other devices which use electricity, heat, or chemically active substances. The only practical safety concern regarding PAM could be the rupture of inner bladder under high pressure. However, by controlling volume flow rate of air into PAMs, this issue can be addressed.

• Low cost in fabrication: A simple PAM could be fabricated from inexpensive off-the-shelf materials. Hence, the manufacturing cost of PAMs is very low compared to other actuators in the same functionality regime. However, it may be noted that PAMs require pneumatic circuitry which increases the initial cost.

Due to the abovementioned advantages, PAMs have found many applications in the robotic industry. A detailed survey on robots which make use of PAMs spanning the domains such as biologically inspired robots, rehabilitation devices, industrial robots, exoskeletons, and aerospace applications can be found in Ref. [14]. The application of PAM in haptic force sensing for laparoscopic surgery [15] and as a “dummy device” in pedestrian safety systems [16] suggest the scope of indirect application of PAMs in engineering. Particularly interesting are the miniaturized versions of pneumatic artificial muscles (MPAMs), where the diameter is less than 5 mm. Due to the small size, these actuators are used in different applications such as wearable hand exoskeletons [17], cardiac compression devices [18], and tool manipulation in surgical devices [10,19,20]. Miniaturized PAMs of diameter less than 2 mm can be bundled into an organized muscle structure for lifting heavy loads. An advantage of using multiple PAMs as bundle or in parallel is the ability to recruit selected muscles as per the load requirement. The variable recruitment technique of muscle bundles is more energy efficient compared to a single muscle of equivalent capacity and are studied in Refs. [2125]. It is shown in Ref. [26] that bundling MPAMs exhibit better contraction ratio compared to a single muscle of same diameter. By adjusting the braiding characteristics of PAM or by clubbing two PAMs with different characteristics, the PAM could generate a moment resulting in bending actuators [2729]. From these references, we can see that researchers across the world are putting considerable emphasis in this promising field.

With the widespread use of PAMs and MPAMs in robotics, accurate mathematical description of underlying mechanics has become a necessity. However, due to the complex interaction of forces in a PAM, this task is not trivial. Nonetheless, many attempts have been made in this regard due to two reasons. First, a mathematical model would help to improve the control system of robots, especially in implementing model-based control systems [15,30,31]. In such cases, however, it is desired to have an easily implementable and computationally efficient model to improve the response of the controller. Second, a model with sufficient parameters helps to efficiently choose or fabricate an actuator with optimized qualities intended for a particular task. In this case, an accurate model which describes the mechanics of PAM based on actuator dimensions, braiding characteristics, material properties, etc., is preferred. Like any other pneumatic systems, PAM exhibits hysteresis which is a major hindrance in modeling statics and dynamics of PAM. Due to static frictional forces and nonlinearity in the material of the bladder, a quasi-static contraction (or elongation) of PAM shows different curves for force versus length and pressure versus length plots for compression as well as decompression of air. In force modeling methods, this additional frictional force is generally added (or subtracted) from a mean curve for contraction (and elongation) of PAM. This frictional force, as may be seen in Sec. 2 of this paper, is mostly empirically calculated. For modeling the dynamics of PAM, the rate of change of state of PAM is related to the change in input parameters where kinetic friction is also included in the model.

In the review paper by Tondu [32], the author meticulously reviewed the major static and dynamic modeling improvements carried out by well-known researchers in the field. Starting from the simple and arguably the first static model proposed by Schulte [33], the author carefully addressed different physical considerations which could improve the basic model such as the inclusion of material properties, noncylindrical end-effects, representation of PAM as a fiber-reinforced membrane model as well as muscle hysteresis. The author lists the necessary considerations, reasonable assumptions as well as precautions to be taken in deriving the static and dynamic formulae representing the actuation of PAM. However, since the objective of the paper is to identify accurate means to describe the physics of PAM, a few models in the literature which rely on empirical formulations as well as the models which provide only minor improvements from the standard equations are not detailed in Tondu's article. In this paper, we update the review paper by Tondu in two ways: first, in our review, we include some modeling considerations that are not mentioned in Tondu's paper in Sec. 2; second, it has been noted in the literature as well as from the experiments conducted by the authors that static characteristics of MPAMs are not always consistent with the models used for normal sized PAMs due to the larger ratio between the volume occupied by bladder and the internal volume of the bladder and the end effects. Hence, most models used in describing MPAM statics necessitates correction factors to be included in models for larger PAMs. In order to address this gap in the literature, this paper also presents a new approach in modeling statics of MPAM. A detailed model and its comparison to the existing models is presented in Sec. 3. In Sec. 4, we present the details of the experiments done to validate the model. We present experimental results for MPAMs with two different diameters and of three different lengths each. It is shown that the computed results obtained from the proposed model lie within the range of experimental results and the maximum error is less than 7%. Finally, conclusions of this work are presented in Sec. 5.

## Review of Static Modeling of Pneumatic Artificial Muscles

In this section, we introduce models used by different research teams to describe the statics of PAM. Many models which assume quasi-static motion do not consider hysteresis into account since accurate phenomenological description of hysteresis is not yet available and many control system strategies use the mean value between contraction and extension profiles [16,30,34]. As described before, in the case of force balance formulations, these hysteresis forces can be added (or subtracted) in case the hysteresis effect is non-negligible. Also, extensile PAMs are not commonly used compared to the contractile PAMs since additional arrangements are required to avoid the buckling effect. In Ref. [35], the authors compare the performance differences between contractile and extensile muscles. It is shown that the derived mathematical models are valid for extensile muscles as well. Hence, most models stated here will assume the primary actuation mode of PAMs as contractile. Finally, a few fundamental models mentioned in Tondu's paper are also discussed here for completeness.

We will use the following nomenclature in this paper, unless stated otherwise (refer Fig. 1):

l0, r0, and t0 represent the length, outer radius, and thickness of the bladder before deformation, respectively. After deformation, these quantities change to l, r, and t. The quantity ri = r0 − t0 represents the initial inner radius of the bladder. Initial and final winding angles of braid are denoted as θ0 and θ, respectively. The symbols N, m, and b represent the number of turns of the braid along the length, number of strands of braid as well as the length of a single braid strand, respectively. The symbol Pi represents the input pressure, $ϵ=(l0−l/l0)$ represents the strain in the bladder along the axial direction, and letters F and κ will be used to represent force and constants in general.

If we assume that the PAM remains cylindrical after deformation, i.e., if the tapering effects at the ends are not considered, we can write the following equations [36]: Display Formula

(1)$l0=b cos θ0,2πr0N=b sin θ0$
Display Formula
(2)$l=b cos θ,2πrN=b sin θ$

In the above equations, it is assumed that the braid is in contact with the outer surface of the tube at all times and the thickness of the braid is neglected. It is also assumed that the braid material is inextensible.

###### Basic Modeling Strategies.

The earliest mathematical model which takes into account the mechanics of a PAM can be found in Gaylord's patent [1]. For static equilibrium of the compressed muscle, the energy provided by the applied pressure (Pi dV) must be balanced by the work done by the PAM which is carrying the load applied at the tip to a particular distance (F dz). By expressing the change in volume enclosed by the braided sheath dV and the displacement of the PAM tip dz in terms of the angle of winding, the energy balance formula yield the following expression for force: Display Formula

(3)$FGaylord(1)=b24πN2Pi(3 cos2θ−1)$

In the above equation, the initial cylindrical shape of the bladder is assumed to stay cylindrical even after deformation and the simple kinematic equations of the braid given by Eqs. (1) and (2) are used. The above relation gives the value of final braid angle θ for the applied pressure and axial loading from which we can find the final length of PAM using Eq. (2). The limiting value of braid angle (and hence, the length) for which the force exerted is maximum can be found by differentiating the above equation with respect to θ and setting to zero. The value $θlimit=54.7deg(54°44′)$ hence becomes a locking angle for deformation of PAM, and a PAM wound with a braid at any initial winding angle will theoretically approach this locking angle with an increase in pressure. This model is also found in the literature in its alternate form Display Formula

(4)$FGaylord(2)=πr02Pi[q1(1−ϵ)2−q2]q1=3 tan2θ0,q2=1 sin2θ0$
which shows the primary behavior of PAM as a nonlinear spring. In this simplistic and first approximation of PAM statics, the volume occupied by air inside the bladder is assumed to be the same as the volume enclosed by the braided sheath. However, this assumption is an overestimation of pressure energy since the volume of air inside the bladder is only the volume enclosed by the cylinder formed by its inner radius. In Ref. [36], Chou gives an expression for the force taking into account the thickness of bladder Display Formula
(5)$FChou(1)=b24πN2Pi(3 cos2θ−1)+πPi[bt0Nπ(2 sin θ−1 sin θ)−t02]$

The experimental comparisons shown in Chou's paper suggest that even though the model is derived based on simplified assumptions, this is a good first approximation. In the coming years, researchers improvised on this basic model by adding correction factors, relaxing the modeling assumptions or adding force terms arising from other physical phenomena contributing to the statics of a PAM.

###### Correction Factors for Gaylord's Model.

Gaylord's model assumes the initial cylindrical shape of the PAM to remain cylindrical even after deformation. However, since one end of the PAM is connected to the pressure inlet system and the free end is always sealed, the radial expansion of bladder will be nonuniform. In the clamped ends, the radius of bladder will be the initial radius after deformation. Hence, on either ends of a PAM, the cylinder takes approximately the shape of a conical frustum. In Ref. [37], Tondu modified the basic equation by Gaylord to include a factor “kϵ” which was intended to account for this noncylindrical tip effects. The force was given as Display Formula

(6)$F=πr02Pi[q1(1−kϵϵ)2−q2]$

In their work, to match the experimental results, the factor kϵ is chosen as $kϵ=κ1e−Pi+κ2$, where the constants κ1 and κ2 are experimentally calculated. In another work by Itto and Kogiso [38], the value of kϵ is chosen as $kϵ=κ1eκ2Pi+κ3$ to add more flexibility. In Tondu's model, static frictional force is also included to improve the static characteristics Display Formula

(7)$Ffstat=μsPi(Scontact(1)Sscale(1))$

where $Scontact(1)=2πr0l0( sin θ0/(1−kϵϵ)1−cos2θ0(1−kϵϵ)2)$ is the contact surface between the strands of the braid, μs is the coefficient of friction between the braid strands, and $Sscale(1)$ is a correction factor for the surface area of contact $Scontact(1)$, since the formulation of the contact surface area assumes flat strands of braid. Taking into account these considerations, the force is given as Display Formula

(8)$FTondu=πr02Pi[q1(1−kϵϵ)2−q2]±μsPi[Scontact(1)Sscale(1)]$

where the quantity $Sscale(1)$ is experimentally determined. This model is applied in many works such as in the control system design of a multi joint arm in Ref. [34] as well as in variable recruitment of PAM bundles in Refs. [22] and [25].

An improvement in Tondu's model was proposed by Davis and Caldwell [39] by including a more detailed derivation for the surface area of contact between the braid strands. For a PAM of given dimensions and braiding pattern, the surface area is defined in terms of the minimum angle of winding possible in the stretched state (θmin) Display Formula

(9)$Scontact(2)=b2 sin θmin cos θminN sin θ cos θ$

where Display Formula

(10)$θmin=12 sin−1(rnmπr0)$
with rn denoting half the diameter of a single strand. Since the contact area calculated in the above equations is also that of flat strands, a scaling factor is used just like in Ref. [37]. However, while the scaling factor is empirically determined by Tondu, an attempt to quantify this factor can be found in their work, and the scaling factor is taken as the ratio of surface areas formed by two flat strands in contact to two spheres in contact. The contact between spheres is calculated using Hertz's contact theory and the final scaling factor is given as Display Formula
(11)$Sscale(2)=rn1.442[Pirn3(1−νn2)En]13$

where νn and En represent Poisson's ratio and Young's modulus of the braid strand, respectively. The modified expression for force takes the form Display Formula

(12)$FDavis=πr02Pi[q1(1−kϵϵ)2−q2]±μsPi[Scontact(2)Sscale(2)]$

While Tondu added the correction factor term in the form of kϵ in Gaylord's equation, in Ref. [23], another correction term is added by Meller et al. The force takes the form Display Formula

(13)$FMeller=πr02Pikf(1)[q1(1−kϵ(1)ϵ)2−q2]$

where the newly added correction terms $kf(1)$ as well as $kϵ(1)$ are determined as functions of input pressure as

$kf(1)=kf(Pi)(1)=Fmeas,max(Pi)πr02Pi(q1−q2),kϵ(1)=kϵ(Pi)(1)=1ϵmeas,max(Pi)(1−13 cos θ0)$

The quantities Fmeas,max = κ1Pi + κ2 and ϵmeas,max = κ3ln(Pi) are calculated using curve fit on experimental data. This model is applied in the analysis of a climbing robot actuated using FAM in the work of Chapman et al. [40].

While the factors Fmeas,max and ϵmeas,max contributed by Meller are empirical functions, in Ref. [41], Andrikopoulos et al. used constant values to these functions for simplicity. The expression for the force then takes the form Display Formula

(14)$FAndrikopoulos(1)=πr02Pikf(2)[q1(1−kϵ(2)ϵ)2−q2]$

where the factors $kf(2)$ and $kϵ(2)$ are not functions of pressure, but take the values

$kf(2)=Fmaxπr02Ptest(q1−q2),kϵ(2)=l0xmax(1−q2/q1)$

To get the values of constants, a test pressure Ptest is applied with zero end load to get the displacement l0 − l = xmax. Then, Fmax is the value of end force which will pull the actuator back to zero displacement position. Both quantities are experimentally determined.

###### Inclusion of Material Properties for Bladder: linear Elastic Model.

The earliest model which takes into account the material property of the bladder and thickness of the tube is probably that of Schulte [33] mentioned in the Appendix of National Research Council's report on the application of external power in prosthetics and orthotics. The force according to this work is given by Display Formula

(15)$FSchulte=b24πN2Pi(3 cos2θ−1)+bEN[l0 sin θ− cos2θ sin θ(bNsin θ−2πr0)]−l0bN(Pi−Pu)(μs+μst)sin θ$

The first term on the right-hand side of Eq. (15) represents the original pull equation by Gaylord. The second term is the resultant of considering the material properties of the bladder—the constant E being Young's modulus of bladder material. The third term represents the friction force where Pu is the pressure required to inflate the unconstrained inner tube to a diameter equivalent to the device diameter at any value of θ and μst is the coefficient of friction between the braid and the tube. Many researchers have used this model in their study (see, for example, Ref. [42]) in a different form Display Formula

(16)$FFerraresi=Pi4πN2(3l2−b2)−Et0l(1Nl2−b2−12πN2ri)+EA(ll0)$

where A is the cross-sectional area of the cylinder.

This model by Ferraresi can also be seen in another format in the works of Kothera et al. [43]. The model derived in their paper, using force balance techniques, is essentially Chou's model which accounts for the thickness of bladder and Schulte's model which considers its linear elasticity. The force according to Kothera et al. is taken as Display Formula

(17)$FKothera(1)=Pi4πN2(3l2−b2)+Pi(Vbl−tl22πrN2)+EVb(1l0−1l)+El2πrN2(tl−t0l0)$

From the experimental data shown in the works of Kothera et al., it is observed that modeling bladder as linear elastic material, even though this appears to be a simple approach, is fairly accurate. This is an interesting observation since this shows that the PAM operation is limited to within the linear regime of deformation of bladders which otherwise usually belong to hyper-elastic material category. Since determining accurate values for constants in nonlinear models often requires precise experimentation, linear material model may suffice for PAMs undergoing small deformation. However, if the material properties can be accurately described and if extensive computation can be afforded, then nonlinear material modeling may provide better accuracy.

###### Inclusion of Material Properties for Bladder: Nonlinear Elastic Model.

If the bladder material is considered nonlinear elastic, then obtaining analytical expressions of force using force-balancing techniques is difficult (if not impossible). The stress components are obtained from strain energy density functions and are directly used in the balance equations. In the paper by Delson et al. [44], the authors used a nonlinear Mooney–Rivlin material [45,46] to account for the elastic properties of the bladder. The strain energy density of a Mooney–Rivlin material takes the form Display Formula

(18)$W=C10(λ12+λ22+λ32−3)+C01(1λ12+1λ22+1λ32−3)$

where Display Formula

(19)$λ1=ll0,λ2=2r−t2r0−t0,λ3=tt0$
are the three stretch ratios. It may be noted that the median diameter is used in this formulation and the thickness is accounted for, unlike in the model used in Ref. [47], where the stretch ratios were defined assuming an incompressible material model and is given by Display Formula
(20)$λ1=ll0,λ2=rr0,λ3=1λ1λ2$

The final expression for force that is derived from the energy balance equation can be written as Display Formula

(21)$FDelson=PidVdl+VbdWdlandπlt(2r−t)=πl0t0(2r0−t0)$

where V = πr2l is the volume occupied by the device and Vb = πlt(2r − t) represents the volume of the bladder. Since the above equation cannot be directly integrated, a numerical integration scheme may be required for the solution.

The Mooney–Rivlin material model is also found in the work of Kothera et al. [43] where an energy balance method is used to derive the following expression for force: Display Formula

(22)$FKothera(2)=Pi4N2π(3l2−b2)−Vb(2C10[λ1dλ1dl+λ2dλ2dl+λ3dλ3dl]+2C01[λ1(λ22+λ32)dλ1dl+λ2(λ32+λ12)dλ2dl]+2C01[λ3(λ12+λ22)dλ3dl])−Pi2b3l4π2mrnEbN4$

where the quantity Eb is Young's modulus of braiding material. An application of this model can be seen in the works of Wereley's team [48] where PAM is used to produce large trailing edge flap in a helicopter. To this model, a friction force of the form Ff = −μfFKothera sgn(v) is added to Kothera's model, where v is the tip velocity of PAM and the constant μf (which is not the coefficient of friction) is found out from experiments. It is worth noting that the authors suggest the use of derivations using force balance method [43] compared to energy balance method since the former was shown to have a better performance compared to the latter.

In Ref. [49], a neo-Hookean material [50] is assumed for the bladder which gives the strain energy density in terms of the stretch ratios as Display Formula

(23)$W=E6(λ12+λ22+λ32−3)$

Due to the relative simplicity in the material definition as opposed to the Mooney–Rivlin model, the authors could derive analytical expressions for pressure in terms of deformation as Display Formula

(24)$Pi=E3(r02ri2−1)×(λ18 cos4θ0 cos 2θ0−2λ16 cos2θ0 cos 2θ0+λ14 cos 2θ0+2λ12 cos2θ0 cos 2θ0−cos 2θ0−2λ12 cos6θ0+cos4θ0)λ13(1−5λ12 cos2θ0+7λ14 cos4θ0−3λ16 cos6θ0)$

From the above equation, for a given value of input pressure, the axial stretch ratio is calculated numerically. This is then used in the calculation of the axial force using the following formulation: Display Formula

(25)$FTrivedi=πE(r02−ri2)(λ1−1)$

The above expression, however, makes the assumption that the material is linear elastic in the axial direction, which is inconsistent with the initial assumption.

Another description of static model considering Mooney–Rivlin material model can be found in Ref. [51]. In their model, Hoop's force Fz and axial force Fθ acting on bladder during inflation are found analytically in terms of λ1 and Pi. These values in conjunction with the static force balance equations from braid give the final expression for blocked force (applied load) as Display Formula

(26)$FWang=Fz−Piπri2−Piril2−Fθl22πN2ro2$

###### End-Effects Consideration.

In the models described so far, the correction factor kϵ was used to take care of the effects of noncylindrical ends. A few researchers have attempted to quantify this effect hoping to obtain better static characteristics of PAM. For example, in Ref. [52], the model developed considers the end-effect of PAM with ends modeled as conical frustums. The mathematical model derived takes the form Display Formula

(27)$FDoumit=m[(Pi(r−t0−2rn)−σ1t0mN sin θ)lcyl+(Pi(r+rc−2t0−2rn)cos β−2σ1t0mN sin(θ+θc2))lcone−Pi(r−2rn−t0)22mr cos θlNtan θ]×cos β cos θc−PScontact(2)μs$

The first term in the expression refers to the model taking into account the characteristics of conical ends, while the second term is the frictional force component (σ1 is Hoop's stress on the bladder). The symbol lcyl refers to the length of cylindrical section of PAM, lcone refers to the slant length of conical frustum at the ends, and rc refers to the radius at the clamped end of the PAM. The quantity $Scontact(2)$ is the effective area of contact between braids which is calculated by assuming the contact to be same as the contact between two cylinders and applying Hertz's contact theory. The validity and propriety of this assumption is, however, criticized in Tondu's review paper [32].

A more involved formulation for end tapering can be found in Ref. [30] where the force model used is same as the one suggested by Ferraresi and found in Refs. [43] and [53]. At the ends, the bladder is assumed to take the shape of a section of elliptic toroid instead of conical frustum. The section of ellipse from π/3 radians to π/2 radians measured from the major axis is assumed to be the shape of PAM at the clamped ends. An expression relating the deformed radius of PAM with the eccentricity of ellipsoid is derived. Making use of this expression, a theoretical estimate for the length of a single strand of braid is formulated. By reducing the error between the actual length of braid strand and the numerically calculated value of the same for a given contracted length, the radius profile of deformed PAM is estimated. The obtained radius profile f(z) at the outer surface is assumed to differ from the profile at the inner surface g(z) by a constant thickness t. Then using incompressibility condition (the volume of bladder at rest and the volume of bladder after deformation are same), the error between the initial and final volume is set to zero. This gave the final value of thickness of bladder, and hence, the deformed shape of PAM. The estimate of thickness predicted from this approximation is compared with the models assuming fully cylindrical tips and is shown to have better conformation with results at higher contraction, where the end-effect is prominent.

###### Bladder as a Thin-Walled Tube.

A few modeling attempts considering the bladder as a thin walled tube can also be found in the literature. In Ref. [54], the authors use large deformation theory on the deflation of a fiber-reinforced thin cylinder to determine the statics of PAM [55,56]. To reflect the embedding of the nylon braiding cords, the stress resultant of thin cylinder is resolved as $σα=σα1+σα11$, where $σα1$ is the stress component due to deformation of bladder, while $σα11$ is the stress component due to the braid strands. The solution procedure consists of guessing an initial value of the transverse stretch ratio λ2 and iteratively adjusting the guess by comparing the value of initial length of tube obtained from formulation with the actual initial length. The closest choice of λ2 will eventually predict the shape of outer surface of actuator, and hence, the final deformed length. The main equations used are Display Formula

(28)$FLiu=2πEr0(2σ2(0)λ2(0)−PiEr0), l0=−∫λ2(0)1dλ2λ1 sin γ(λ2)$

where the axial stress σ2(0) and the radial stretch ratio λ2(0) are at the initial configuration and γ(λ2) is the angle made by the meridian of PAM (on the surface) with the z-axis given as a function of the stretch ratio. However, it may be noted that in most PAMs, the fiber is not embedded inside the bladder, but forms a sheath on the outer surface. Hence, the application of this model on a general PAM structure is debatable.

Another model by Ball and Garcia [57] also considers thin wall approach in modeling PAM. In this case, the expression for force is given as Display Formula

(29)$FBall=Fstrands+Fpressure+Felastic±μfPeff$

where Display Formula

(30)$Fstrands+Fpressure=Pol22πN2−Pi(b2−l24πN2−Vbl),Po=Pi−σ2(r0−ri)λ1λ2b2−l2n2π2−4πVbl,Felastic=σ1Vbl$
with Po as the pressure acting at the outer radius by the bladder on the sleeve, and σ1,2 is given in terms of material constants and stretch ratios. In the case of thick-walled bladder as well as prestrained bladders, the thin wall tube model is applied sequentially as if the thick cylinder is an array of concentric nested thin tubes. The computational method calculates the pressure Po of the innermost layer and works sequentially outward. The derivations based on thin film approach on the same lines of Ref. [54] may also be found in the fiber-reinforced electro-pneumatic PAM shown in Ref. [58].

###### Advanced Modeling of Pneumatic Artificial Muscles.

A few recently developed models try to capture the forces in PAM in greater detail. For example, in Chen and Ushijima [59], two expressions for pressurization as well as the depressurization of a MPAM are derived. For pressurization, the axial force is given as Display Formula

(31)$FChen(1)=πPi4(bNπ)2(3 cos2θ−1)−4m2Nb sin θ(Mf+Md+Mr+Mtr)$

For depressurization, the axial force is given as Display Formula

(32)$FChen(2)=πPi4(bNπ)2(3 cos2θ−1)+4m2Nb sin θ(Mf−Md−Mr+Mtr)$

where the detailed expression for moments, Mf, Md, Mr, and Mtr, represents the effects of friction between strands of threads, bending deformation of thread strand, bulging of bladder between the threads in braided sleeve, and the friction between threads and bladder, respectively. The highly detailed model requires numerical integration tools and the accuracy of the model may depend heavily on the coefficients of friction between the braid strands, between the braid and the tube as well as the guess on the contact surface area between the strands.

Another example is the description of statics in Ref. [41], where the model considers the effect of thermal expansion in PAM during actuation. The improved model from Ref. [41] takes the form Display Formula

(33)$FAndrikopoulos(2)=πr02Pikf(2)[q1(1−kϵ(2)(ϵ+αlΔT))2−q2]−(2πr0l0μsSscale(1))× sin θ0(1−kϵ(ϵ+αlΔT))1−cos2θ0(1−kϵ(ϵ+αlΔT))2×Pisgn(v)$

where αl, ΔT, and v represent the coefficient of thermal expansion of bladder, the change in temperature, and the velocity of MPAM tip, respectively.

Apart from the usual methods which focus on finding an exact analytical expression to relate pressure, force, and displacement of a PAM, a few models use numerical methods such as finite element methods to solve the statics of PAM. For example, in Ref. [60], the authors use finite element analysis to analyze the dynamics of PAM used in parachute systems. The preliminary model used is Display Formula

(34)$ϵ=1−F tan2θ02πr02Pi$

where F is the applied force. The application of finite element analysis can also be found in Ref. [61] as well as the analysis of a pneumatic bending fiber reinforced actuator in Ref. [27]. Such analyses could be proven useful especially for actuators with nonuniform physical structure.

###### Modeling for Miniaturized Versions of Pneumatic Artificial Muscles.

In the case of modeling miniaturized PAMs, thin-walled tube approximation is not appropriate since the ratio between the bladder material volume and the inner volume of bladder is usually high. Also, it has been found that many models for normal sized PAM need to be adjusted by adding correction factors so as to include the effects of forces which are difficult to measure. De Volder et al. [19] show the analysis of a miniaturized FAM with outer diameter 1.5 mm and length between 22 mm and 62 mm intended to use in a fluidic actuated surgical tool. The equation for the force used is Display Formula

(35)$Fde Volder=max[(Fmin,(Pi−Pi,corr)b24πN2)(3(l−lcorr)2b−1)]+max[0,kb(l−lb0)]$

where lcorr and Pi,corr are factors used to correct length and dead-band pressure, respectively. The term Fmin is used as a threshold so that the PAM does not generate pushing forces. Finally, the term kb(l − lb0) is added to generate a linear spring force equivalent in the model.

Another analysis and validation of statics of a MPAM with outer diameter between 3.02 mm and 4.19 mm is shown by Hocking et al. [53]. The basic force equation derived from Ferraresi has Hoop's stress (σ1) and axial stress (σ2) terms which considers the elasticity of material [42]. The force is obtained as Display Formula

(36)$F=Pi4πN(3l2−b2)+σ1Vbl−σ2tl22πN2r$

In Hocking's paper, these stresses are considered as nonlinear (polynomial) functions of strain and the equation is modified as Display Formula

(37)$FHocking(1)=Pi4πN(3l2−b2)+Vbl∑i=1nEi(ll0−1)i−tl22πN2r∑i=1nEi(rr0−1)i$

where the material constants Ei are empirically identified from experimental results.

To the above model, friction is added as Display Formula

(38)$FHocking(2)=FHocking(1)±μfFHocking(1)sgn(v)$

One modification in the friction term compared to the other models is that μf is assumed to vary with pressure. A dead-band pressure which is the threshold value of pressure up to which contraction does not start is usually observed in the case of MPAMs. In this paper, correction to account for dead-band pressure is made as $Picorr=Pi−Pc$, where Pc is calculated from experiments. Similarly, the tip effect is considered by using a corrected length, $Lcorr=l−2[(π2−1)(r−r0)]$, in the above equation. A similar strategy is used in Ref. [3] where the model used is essentially that of Hocking et al. [53] with the thickness term included from Chou's model [36]. In this model, the stress is empirically related to strain as a function of pressure as Display Formula

(39)$σ=∑j=1n(EjI+EjSPi)ϵj$

where the constants are experimentally determined.

In another paper by Sangian et al. [62], the authors characterize miniaturized FAM of outer diameter 5.6 mm taking into account the pressure dead-band. Gaylord's model is modified to include the threshold pressure ($P¯i$) required to initialize the contraction. The final force expression takes the form Display Formula

(40)$FSangian=πr02[q1(1−ϵ)2−q2]×[Pi−P¯i+Et0b2πNr02{(1−l2l02 cos2θ0)12+sin θ0}]$

The use of empirical model formulation for MPAM (outer diameter 1.8 mm) can also be found in Ref. [26] where the static model used is Display Formula

(41)$ϵl=1−1κ1(F−κ2Pi−κ3)$
Display Formula
(42)$ϵr=κ4ϵl2+κ5ϵl+κ6κ6(1−ϵl)−1$

###### Empirical Considerations.

As mentioned in Sec. 1, advanced and more involved models are often quite difficult to implement in real-time control systems. Additionally, the measurement of exact values for parameters necessary for these advanced models will not be possible in all cases—it is hard to measure the deformed outer diameter of miniaturized muscles and axial strain of PAMs which are already employed in a robot. Hence, many models use empirical formulation derived from the basic models for practical purposes. In Ref. [5], Eq. (6) is modified to obtain Display Formula

(43)$FTakosoglu=4πr02Pi[q1(1−ϵ)n−q2]$
the factor n and also the parameter q1 are later empirically determined to be
$n(Pi)=κ1e−Piκ2+κ3,q1(Pi)=κ4e−Piκ5+κ6$

In a model in Ref. [38], the expression is further empirically adjusted to Display Formula

(44)$FItto=πr02Pi[q1{1−κ1(1+eκ2Pi)ϵ}2−q2]$

and the above model so formed is seen to agree well with experimental values.

In Ref. [16], the authors analyze the static model of a PAM used as “pedestrian dummy device” in the test setup of pedestrian safety system. The model derived takes the form Display Formula

(45)$FDoric=FChou−FPAM, e−FPAM, swhereFPAM, e=(1−ww0)κ1,FPAM, s=Pϵl0κ2$

The second and third terms take into account the effects of thickness, elasticity of bladder as well as the form of PAM. The correction factors for elasticity as well as the shape of PAM, κ1 and κ2, are experimentally determined.

Purely empirical formulations are also presented in the works of [15,6367] for its relative ease in control system design. In these papers, the empirical expressions for blocked force as a function of applied pressure and axial strain take different forms such as Display Formula

(46)$F(Pi,ϵ)=(κ1+κ2ϵ+κ3ϵ2)Pi+(κ4+κ5ϵ+κ6ϵ2+κ7ϵ3+κ8ϵ4)$
Display Formula
(47)$F(Pi,ϵ)=κ1+κ2ϵ+κ3ϵ2+κ4Pi+κ5ϵPi$
Display Formula
(48)$F(Pi,ϵ,ϵ˙)=(κ1Pi2+κ2Pi+κ3)ϵ+κ4Pi+κ5+κ6ϵ˙$
Display Formula
(49)$F(ϵ)=κ1Fmax(1−ϵϵmax)$

where constants κ are determined from prior experimentation.

###### Modeling Hysteresis.

In almost all the models described in Secs. 2.12.9, hysteresis is accounted by adding or subtracting a frictional force term to the static equation for axial force. A convincing representation of the added frictional force term is not yet developed to the best of our knowledge. In most cases, an approximating function is chosen to represent this frictional force term which is empirically determined. For example, in Ref. [30], this additional frictional force term Ffstat is calculated from the static force term Fstat obtained from phenomenological models as Display Formula

(50)$Ffstat=−μfFstat=(κ1+κ2Pi)Fstat$

While the term Fstat gives the mean curve of force-deformation plot, adding or subtracting this frictional force term will give the pressurizing and the depressurizing curve. In the above equation, the coefficient of friction is assumed to be linearly dependent on applied pressure and the constants k1 and k2 are determined from experiments.

In a few research works, empirical formulations are derived for force–displacement curves for expansion and compression of a PAM in a manner different from the method mentioned earlier. In cases where accurate hysteresis modeling is required—especially for practical applications, force-length and pressure-length hysteresis profiles of PAM are found out for compression and expansion curves separately. For example, in Ref. [68], van Damme et al. derived a hysteresis profile for pleated PAM using Preisach hysteresis model [69]. The math model takes the form Display Formula

(51)$Fhyst=Pil02ft0fit(1+κscale(W[ϵs]−W(ϵs)fit))$

where $ft0fit=κ0ϵ−1+κ1+κ2ϵ+κ3ϵ2+κ4ϵ3$ is the approximated mean curve of force–displacement hysteresis. The function $W[ϵs]$ is the output of Preisach model which is a weighted summation of small discrete hysteresis relays and the function $W(ϵs)fit$ represents a curve fitted between the two curves generated by $W[ϵs]$ and κscale is a scaling factor. The proposed model is shown to estimate hysteresis phenomenon in PAM for contractile range below 20%.

A Maxwell slip model [70] for hysteresis is described in Refs. [71] and [72]. In this method, the force-length hysteresis of the PAM—the hysteresis component in PAM force curve due to the motion of PAM as well as the stretching of bladder—is experimentally determined and modeled. In order to achieve this, at first, the force is measured from a constrained model where the motion of PAM is arrested. Then, isobaric experiments are carried out where the pressure is kept constant and force value corresponding to change in length is obtained. The difference between the two values gives the force-length hysteresis in PAM. This component of hysteresis appears to be qualified as “nonlocal memory hysteresis” which can be modeled using Maxwell slip model. In nonlocal memory hysteresis modeling, when the PAM is actuated towards a particular contracted length (following a particular force-length curve) and is allowed to dilate (following a different curve), by “remembering” the parameters of return points (Fm and ϵm), the subsequent contraction and dilation can be modeled by knowing the characteristic curve called the “virgin curve.” In mathematical form, this procedure can be written as Display Formula

(52)$Fhys=Fm+2f((ϵ−ϵm)/2);f=y(ϵ),v≥0,f=−y(−ϵ),v≤0$

where y(⋅) represents the virgin curve. In their papers, this virgin curve is identified as piecewise linear curve. For each piece of the curve, a slip element with stiffness “k” and maximum saturation force “w” can be attributed. The piecewise continuous stiff elements can also be visualized as a parallel arrangement of spring systems with each element having different values of stiffness and a saturation force limit (representing the pressure in pressure-length hysteresis plot) beyond which displacement does not take place for that particular element. From the knowledge of each slip elements, the hysteretic force can be calculated for any choice of length by intuitively choosing the right number of elements that would contribute to the section of curve. The total hysteresis force, Fhys, is the sum $∑1nFi$.

In another paper by Lin et al. [73], the authors show a Bouc–Wen model [74] to represent the pressure-length hysteresis of PAM for use in control system design. In this work, the hysteresis loop for pressure-length curve is represented using the expression Display Formula

(53)$l(t)=k(k1Pi(t)−h(t))+ρ$

where h(t) is a solution of the equation Display Formula

(54)$h˙=αPi˙(t)−γPi˙(t)|h|n−βz|Pi˙(t)||h|n−1$
with the parameters n, k, k1, ρ, α, β, and γ identified by minimizing the least square error between the model and the experimental data. The paper also presents a Prandtl–Ishlinskii (PI) [75] model for pressure-length hysteresis representation where the loop is given by the equation Display Formula
(55)$l(k)=wTHr[Pi,l0](k)=∑i=0n−1wi·max{Pi(k)−ri,min{Pi(k)+ri,l(k−1)}}$

In the above, Hr are the backlash (play) operators of PI model and k is the sampling number of the operator. The weights wi and threshold ri are found out using least square error minimization as mentioned in the case of Boruc–Wen model. The application of Prandtl–Ishlinskii model on trajectory control of PAM can be found in Ref. [76] (see also Ref. [77]).

The models suggested in Ref. [73], however, are suitable mostly for symmetric hysteresis loops. In Ref. [78], a modification to this model which can be used in asymmetric hysteresis loop is proposed. Here, in the basic PI model, the backlash operator is divided into two, one for ascending and one for descending curves and is written as Display Formula

(56)$l(t)=κ1Pi(t)+∑i=1nwia(Hria[Pi,l0](t)−Pi(t))+∑i=1nwid(Hrid[Pi,l0](t)−Pi(t))$

where $Hria,d$ are different for ascending and descending; κ1 is a constant. The two operators are subject to constraints

$Hria[1,l0](t)=1,Hria[1,l0](k)=1,κ1+∑i=1nwiaHria[1,l0](t)+∑i=1nwidHrid[1,l0](t)=1$

In the above, there are (2n + 3) parameters that need to be identified—the additional three parameters compared to the classic PI model are from determining coefficients of a quadratic function used in the descending play operator. Another variant in PI hysteresis model is shown in Ref. [79], where an “extended unparallel PI” model is proposed. Here, the PI model is modified so that the ascending and descending edges are multiplied with factors α and β, which change the respective slopes. The final backlash operator becomes Display Formula

(57)$Hri,αi,βi[Pi](k)=max{αi(Pi(k)−ri),min{βi(Pi(k)+ri),Hri,αi,βi[Pi](k−1)}}$

###### Summary of Models in the Literature.

The improvement put forward by different phenomenological models in the literature from the basic model by Gaylord and the key equations used in the same is shown in Table 1. Table 2 compares the major modeling considerations in the models. The models shown in rows marked with {}a are experimentally validated in the literature on miniaturized versions of PAMs or FAMs. From the earlier part of this section, we see that most of the models for MPAMs require prior experimentation to accurately determine the correction factors, friction coefficients as well as the empirical constants used in the stress equations. The simpler model proposed by Sangian et al. (Eq. (40)) is quite inaccurate in predicting the pressure-deformation characteristics of MPAM as shown in Sec. 4 of this paper, while the numerical iterative method used by Ball et al. is computationally expensive and nontrivial to implement. Moreover, it is also observed by the authors that many models proposed in the literature are inconsistent to the changes in initial parameters when applied on MPAMs. For measurements taken from specimens belonging to same fabricated lot, the accuracy of theoretical models varies considerably when only the initial length or braid angle is different, while keeping all the other material and fabrication parameters constant. This anomaly and the gap in the literature necessitates the development of an improved statics model for MPAM. Additionally, as shown in Sec. 4, numerical solutions of the various models do not match very well with the experimental results of MPAMs (see Figs. 13 and 14 in Sec. 4). In Sec. 3, we propose a new approach to model the statics of MPAM, which is shown to be consistent with the variation in MPAM parameters and in reasonable agreement with experiments done on MPAMs.

## Proposed Model for Miniaturized Versions of Pneumatic Artificial Muscles

In this section, we introduce a novel phenomenological approach in modeling statics of MPAM. Before the modeling is discussed in detail, characteristics of the MPAMs fabricated as well as the setup used for experimentation are discussed.

###### Fabricated Miniaturized Versions of Pneumatic Artificial Muscles Characteristics.

In our study, we use two MPAMs consisting of an inner silicone tube with ro = {0.55, 0.75} mm and ri = {0.25, 0.25} mm braided on the outer surface using nylon cords of radius ∼ 50μm at an angle of α = {36 deg, 38 deg, 40 deg} (refer Fig. 2). Since the angle of winding is less than 54.7 deg, the actuator contracts upon application of pressure [36]. The overall outer diameter of MPAM is 1.2 mm and 1.6 mm. For braiding, we used a standard Horn gear braiding machine used in the fabrication of coaxial communication cables. Most commercially available braiding machines are designed for braiding the tubes up to a minimum of 5 mm. However, by manually adjusting the configuration of machine, it was possible to braid the silicone tube so that the gap between silicone tube and braid is minimized. In spite of the care taken during fabrication, in the MPAMs used for experiments, there exists a small gap δ between the outer radius of the silicone tube ro and the inner surface of braid with the radius rb. The fabrication process also limits the range of helix angles with which the nylon fibers could be braided—in the fabricated MPAMs, we could get helix angles between 36 deg and 40 deg. The actual fabricated MPAMs are shown in Fig. 2. It maybe mentioned that the ends are larger as an epoxy adhesive is applied to seal the ends for experimentation.

The layout of pneumatic circuit used to actuate MPAM is shown in Fig. 3. A pneumatic compressor of maximum output pressure 1034 kPa (150 psi) is connected to a 1 l air (at NTP) reservoir which is used to deliver high pressure air to the MPAM. A pressure regulating circuit operates the compressor when the value of pressure in reservoir falls below certain threshold thereby maintaining availability of 827 kPa (120 psi) pressure at all times. Two proportional valves are used to control pressure inside air muscle—one for pressurizing the MPAM and the other for bleeding. A Honeywell pressure transducer (with range of 0–1034 kPa) is connected in series with MPAM to measure the inner pressure. An ATmel ATMega2560 microcontroller board interfaced with matlab controls the proportional valves through a current driver circuit to maintain user-defined value of pressure inside the MPAM. To keep a straight alignment of the MPAM, a 5 g weight is applied on the free end. For a 40 mm air muscle, the maximum deformation of 15 g end-loading varies from 5 g by less than 0.3 mm (less than 3% of total deformation). Since this variation in deformation is comparable to the error bounds of the measurements in the experiments, the effect of this small end-loading is ignored in the formulations. The experimental setup used is shown in Fig. 4.2

The deformation of MPAM is captured using a high-resolution camera and changes in length are computed using image processing. The measurement method consists of taking images of MPAMs in its operational state using high-resolution camera and identifying the length of MPAM by measuring the displacement between the image pixels corresponding to the tips of MPAM. At first, the size of each pixel in the HD camera image is calculated based on a benchmarking with a standard object with known dimensions. Then, the distance between two markers set in the either ends of MPAM is calculated in terms of pixels and using the scale mentioned earlier, it is converted in terms of millimeters. The possible error in this method is in identifying the marker pixels which is not more than 2 pixels size in each ends. For the scale and measurement setup used, this value is about 0.2 mm. We have also carefully avoided any perspective issues in measurement by conducting the scale determination as well as the MPAM operation in the same focal plane of camera. All measurements are repeated more than five times and the results are reported as mean of the obtained values and measurement errors shown in error bars.

Figure 5 shows the end-point displacement of MPAM during inflation as well as deflation with dead load (F = 0.05 N) attached at the end. The MPAM clearly shows hysteresis. The maximum error due to measurement is about 0.1 mm and the error bars in the plot are obtained from at least five sets of experiments. In the comparisons with the existing approaches, the hysteresis is not shown and the mean value, between the inflation and deflation, is used (see Fig. 5). We performed experiments with a 40 mm MPAM and a 60 mm MPAM using the experimental setup described earlier and compared the experimental results with theoretical values obtained from various models available in the literature—see Figs. 6 and 7. Since our focus is on identifying the mechanics of MPAM, the comparison plots are limited to only phenomenological models as opposed to the models which rely on empirical data as well as parameters which require sophisticated measurement setup for identification. Also, models which can be easily implemented and do not use correction factors are only considered. The MPAMs used for validation have same physical characteristics and differ only by their lengths. We can see that except Hocking's model, other models are not accurate when predicting the deformation of fabricated MPAM keeping all the parameters constant except the initial length. Even though Hocking's model is able to predict the unloaded displacement accurately, it gives large error when predicting the stiffness of a pressurized MPAM (actual stiffness is about ten times larger than the predicted stiffness value).

###### Characterization of Pressure Dead-Band.

In Fig. 8, we can see the pressure deadband which is the range of pressure below which contraction of MPAM is not apparent. This pressure deadband is mentioned in Ref. [53] as due to Mullin's effect, which is unlikely in the case of MPAM used in this work. In our case, the MPAM was prestretched and inflated multiple times, so as to form a permanent set before it is braided on the outer surface. This ensured the repeatability of bladder inflation characteristics while employed in the MPAM. It is also observed that the unbraided bladder inflates considerably at values of pressure within this dead-band range. On closer observation, it is found that the MPAM expands instead of contracting in this range (see the inset of Fig. 8) and this is due to the small gap δ (of the order of 0.04 mm) between the tube and the nylon sleeve during fabrication, as mentioned in earlier section. It may be noted that this expansion is not usually seen in commercial PAMs as well as fiber embedded PAMs where this gap is unlikely to occur, while it was prominent in the fabricated braided sleeve PAMs as in the case of the MPAM used in this work and in the work presented in Ref. [53]. Due to this gap, the initial stage of pressurization results in the expansion of silicone tube till the outer surface of silicone tube makes contact with the nylon sleeve. The pressure at which contact occurs is termed the critical dead-band pressure $P¯i$. Since the forces acting on MPAM before and after the critical deadband pressure are different, we consider this as two phases of contraction which has to be treated separately. We assume the bladder material as linear elastic for simplicity in derivations and also since the linear elastic model is shown to be sufficient to capture model characteristics as observed from Kothera's model [43].

###### Model for First Phase–Expansion.

In the first phase of deformation, the bladder expands without the constraint of the outer braid sheath. In this phase, we use the linear thick cylinder approach to find the displacements in the axial and radial directions. The equilibrium equations for inflation of thick cylinder are given by [80] Display Formula

(58)$∂∂r(1r∂(rur)∂r)=0,∂2uz∂z2=0$

where ur and uz are the displacements of silicone tube in the radial and axial directions, respectively.3 Solving the equations, we get the displacements Display Formula

(59)$ur=c1r+c2r,uz=c3z+c4$

where ci, i = 1, 2, 3, 4, are constants. In the initial phase of deformation, since the braid has not come in contact with the tube, the outer surface will be pressure free. The applied pressure Pi will act in the inner cylindrical surface, while in the axial ends, inflation pressure as well as the pressure due to applied axial load will act. This pressure component will be $Ps=Pi(ri2/ro2−ri2)+Psil$, where Psil is the pressure acting on the silicone tube due to the applied axial load F. Using these boundary conditions as well as the zero displacement condition on the fixed end of the MPAM, we get the values of constants as Display Formula

(60)$c1=(Λ1+2Λ2)2Λ2(3Λ1+2Λ2)[Piri2ro2−ri2+Poro2ro2−ri2−PsΛ1Λ1+2Λ2]c2=12Λ2[ri2ro2ro2−ri2](Pi−Po)c3=Λ1Λ2(3Λ1+2Λ2)[−Piri2ro2−ri2+Poro2ro2−ri2+PsΛ1+Λ2Λ1]c4=0$

where Λ1 and Λ2 are Lame's parameters. Substituting the constants, we get the displacements as Display Formula

(61)$ur|r=ro=roΛ2(3Λ1+2Λ2)[2(Λ1+Λ2)ri2(ro2−ri2)Pi−λ2(Piri2ro2−ri2+Psil)]$
Display Formula
(62)$uz|l=l0=Λ1l0Λ2(3Λ1+2Λ2)[−Piri2ro2−ri2+(Piri2ro2−ri2+Psil)Λ1+Λ2Λ1]$

The MPAM expands according to the above equations till the tube makes contact with the braid. The pressure components at this point remain the same as that of the initial expansion phase, since there is no radial pressure on the outside surface of the silicone tube at the onset of contact. As the tube expands, the braid deforms as per the kinematics rule given in Eqs. (1) and (2). Taking into account the gap between the braid and the tube, the modified kinematics model of the braided sleeve can be written as Display Formula

(63)$l0=b cos θ0,2πrbN=b sin θ0$
Display Formula
(64)$l0+ûz=b cos θ,2π(rb+ûr)N=b sin θ$

where rb = r0 + δ is the initial radius of the braided sleeve and the quantities $ûr$ and $ûz$ represent the radial as well as the axial displacements of the braided sleeve. The above equations can also be written as a single expression which relates the radial and axial displacements of the sleeve Display Formula

(65)$ûr=rb{1 sin θ01−cos2θ0(1+ûzl0)2−1}$

Since the braid and sleeve are sealed at the tips, the axial displacement of the sleeve and the tube is the same. Hence, $ûz=uz(l=l0)$ at all times. At the critical inflection pressure, the tube makes contact with the braid surface. This is the point where the radius of deformed bladder becomes equal to the radius of the displaced sleeve. Hence Display Formula

(66)$ro+ur|ro=rb+ûr=ro+δ+ûr$

and we have Display Formula

(67)$ro+c1ro+c2ro=ro+δ+rb{1 sin θ01−cos2θ0(1+ûzl0)2−1}$

Simplifying and substituting for $ûz$, we get Display Formula

(68)$c1ro+c2ro=δ+rb{1 sin θ01−cos2θ0(1+c3)2−1}$

In the above expression, the constants c1, c2, and c3 depend only on applied Pi which is the inflection pressure $P¯i$. Substituting the values of constants, we get the following equation: Display Formula

(69)$rb{1 sin θ0×1−cos2θ0(1+Λ1Λ2(3Λ1+2Λ2)[−P¯i2ro2−ri2+PsΛ1+Λ2Λ1])2−1}−roΛ2(3Λ1+2Λ2)[2(Λ1+Λ2)ri2(ro2−ri2)P¯i−Λ12Ps]+δ=0$

where $Ps=P¯i(ri2/ro2−ri2)+(F/πri2)$. This equation can be numerically solved to find the inflection pressure. For values of applied pressure below $P¯i$, Eq. (62) can be used to find the end-point elongation of the MPAM.

###### Model for Second Phase-Contraction.

For values of pressure above $P¯i$, the contact is established, and in this phase, the radial as well as axial displacement of braided sheath will be same as that of the outer surface of silicone tube, i.e., $ur|ro=ûr$ and $uz|lo=ûz$. Then from the kinematics of braid, Eq. (64), and from Eq. (59), we can write Display Formula

(70)$c1+c2ro2=( sin θ sin θ0−1),c3=( cos θ cos θ0−1)$

The above equations represent the constraint on the motion of silicone tube imposed by the braided sleeve.

In this phase, an axial pull on sleeve generates a radial pressure on the outer surface of the silicone tube and vice versa. The total axial end force on the MPAM, Fe, has contributions from three components which are (1) the manually applied axial load F, (2) force acting on the walls due to the applied inner pressure $FPi=Pi(πri2)$, and (3) any other unaccounted forces such as the static frictional force between the threads and the axial component of force due to the conical shape at the ends which are essential to maintain the static equilibrium of the MPAM. These unaccounted force components are collectively termed Fu. This total axial force Fe is borne unequally by the axial end of silicone tube as well as the nylon braid (ref Fig. 9) Display Formula

(71)$Fe=F+FPi+Fu=Fsil+Fnyl$

where Fsil represents the axial force acting on silicone tube and Fnyl represents the axial force acting on the nylon braid. The force component acting on the braided sleeve is then converted into a radial force based on the kinematics of the braid. The pressure generated by this radial force will constitute the component Po in Eq. (60). Derivation of this radial pressure is detailed in what follows.

Since the displacement of the free end of MPAM Δ is same as the deformation of the nylon sleeve Δnyl as well as the silicone tube Δsil, it is possible to write the individual components of forces in terms of the end force Fe. We use the material properties of the tube and sheath to calculate the axial displacement of nylon sheath Display Formula

(72)$Δnyl=F̂bÂnylEnylcos θ0=Fnyll0 cos θ0mÂnylEnyl$

where $F̂=(Fnyl/m)cos θ0$ is the force acting on a single strand of braid and $Ânyl$ is the area of cross section of single nylon strand, and Enyl is the modulus of elasticity of nylon (refer Fig. 9). Similarly, the axial displacement of silicone tube can be written as Display Formula

(73)$Δsil=Fsill0AsilEsil$

where Asil and Esil are the cross section area and Young's modulus of silicone tube, respectively. From Eqs. (71) to (73), we get the individual components of forces in terms of end force acting on MPAM as Display Formula

(74)$Fnyl=FemÂnylEnylmÂnylEnyl+cos θ0AsilEsil$