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%.

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

**K. P. Ashwin**

**A. Ghosal**

^{1}Corresponding 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

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 [9–11] 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. [21–25]. 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 [27–29]. 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.

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.

*l*_{0}, *r*_{0}, and *t*_{0} 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 *r _{i}* =

*r*

_{0}−

*t*

_{0}represents the initial inner radius of the bladder. Initial and final winding angles of braid are denoted as

*θ*

_{0}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

*P*represents the input pressure, $\u03f5=(l0\u2212l/l0)$ represents the strain in the bladder along the axial direction, and letters

_{i}*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]:

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.

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 (*P _{i} 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:

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 $\theta limit=54.7deg(54\xb044\u2032)$ 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

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.

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 “*k _{ϵ}*” which was intended to account for this noncylindrical tip effects. The force was given as

In their work, to match the experimental results, the factor *k _{ϵ}* is chosen as $k\u03f5=\kappa 1e\u2212Pi+\kappa 2$, where the constants

*κ*

_{1}and

*κ*

_{2}are experimentally calculated. In another work by Itto and Kogiso [38], the value of

*k*is chosen as $k\u03f5=\kappa 1e\kappa 2Pi+\kappa 3$ to add more flexibility. In Tondu's model, static frictional force is also included to improve the static characteristics

_{ϵ}where $Scontact(1)=2\pi r0l0(\u2009sin\u2009\theta 0/(1\u2212k\u03f5\u03f5)1\u2212cos2\theta 0(1\u2212k\u03f5\u03f5)2)$ is the contact surface between the strands of the braid, *μ _{s}* 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

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 (*θ*_{min})

where

*r*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

_{n}where *ν _{n}* and

*E*represent Poisson's ratio and Young's modulus of the braid strand, respectively. The modified expression for force takes the form

_{n}While Tondu added the correction factor term in the form of *k _{ϵ}* in Gaylord's equation, in Ref. [23], another correction term is added by Meller et al. The force takes the form

where the newly added correction terms $kf(1)$ as well as $k\u03f5(1)$ are determined as functions of input pressure as

The quantities *F*_{meas,max} = *κ*_{1}*P _{i}* +

*κ*

_{2}and

*ϵ*

_{meas,max}=

*κ*

_{3}ln(

*P*) 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].

_{i}While the factors *F*_{meas,max} and *ϵ*_{meas,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

where the factors $kf(2)$ and $k\u03f5(2)$ are not functions of pressure, but take the values

To get the values of constants, a test pressure *P*_{test} is applied with zero end load to get the displacement *l*_{0} − *l* = *x*_{max}. Then, *F*_{max} is the value of end force which will pull the actuator back to zero displacement position. Both quantities are experimentally determined.

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

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 *P _{u}* is the pressure required to inflate the unconstrained inner tube to a diameter equivalent to the device diameter at any value of

*θ*and

*μ*

_{st}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

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

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.

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

where

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

where *V* = *πr*^{2}*l* is the volume occupied by the device and *V _{b}* =

*πlt*(2

*r*−

*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:

where the quantity *E _{b}* 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

*F*= −

_{f}*μ*

_{f}F_{Kothera}sgn(

*v*) is added to Kothera's model, where

*v*is the tip velocity of PAM and the constant

*μ*(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.

_{f}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

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

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:

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 *F _{z}* and axial force

*F*acting on bladder during inflation are found analytically in terms of

_{θ}*λ*

_{1}and

*P*. These values in conjunction with the static force balance equations from braid give the final expression for blocked force (applied load) as

_{i}In the models described so far, the correction factor *k _{ϵ}* 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

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 (*σ*_{1} is Hoop's stress on the bladder). The symbol *l*_{cyl} refers to the length of cylindrical section of PAM, *l*_{cone} refers to the slant length of conical frustum at the ends, and *r _{c}* 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.

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 $\sigma \alpha =\sigma \alpha 1+\sigma \alpha 11$, where $\sigma \alpha 1$ is the stress component due to deformation of bladder, while $\sigma \alpha 11$ is the stress component due to the braid strands. The solution procedure consists of guessing an initial value of the transverse stretch ratio *λ*_{2} 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 *λ*_{2} will eventually predict the shape of outer surface of actuator, and hence, the final deformed length. The main equations used are

where the axial stress *σ*_{2}(0) and the radial stretch ratio *λ*_{2}(0) are at the initial configuration and *γ*(*λ*_{2}) 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

where

*P*as the pressure acting at the outer radius by the bladder on the sleeve, and

_{o}*σ*

_{1,2}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

*P*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].

_{o}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

For depressurization, the axial force is given as

where the detailed expression for moments, *M _{f}*,

*M*,

_{d}*M*, and

_{r}*M*, 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.

_{tr}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

where *α _{l}*, Δ

*T*, 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

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

where *l*_{corr} and *P*_{i,corr} are factors used to correct length and dead-band pressure, respectively. The term *F*_{min} is used as a threshold so that the PAM does not generate pushing forces. Finally, the term *k _{b}*(

*l*−

*l*

_{b}_{0}) 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 (*σ*_{1}) and axial stress (*σ*_{2}) terms which considers the elasticity of material [42]. The force is obtained as

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

To the above model, friction is added as

One modification in the friction term compared to the other models is that *μ _{f}* 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 $Picorr=Pi\u2212Pc$, where

*P*is calculated from experiments. Similarly, the tip effect is considered by using a corrected length, $Lcorr=l\u22122[(\pi 2\u22121)(r\u2212r0)]$, 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

_{c}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 ($P\xafi$) required to initialize the contraction. The final force expression takes the form

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

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

*n*and also the parameter

*q*

_{1}are later empirically determined to be

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

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

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, *κ*_{1} and *κ*_{2}, are experimentally determined.

Purely empirical formulations are also presented in the works of [15,63–67] 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

In almost all the models described in Secs. 2.1–2.9, 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 *F*_{fstat} is calculated from the static force term *F*_{stat} obtained from phenomenological models as

While the term *F*_{stat} 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 *k*_{1} and *k*_{2} 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

where $ft0fit=\kappa 0\u03f5\u22121+\kappa 1+\kappa 2\u03f5+\kappa 3\u03f52+\kappa 4\u03f53$ is the approximated mean curve of force–displacement hysteresis. The function $W[\u03f5s]$ is the output of Preisach model which is a weighted summation of small discrete hysteresis relays and the function $W(\u03f5s)fit$ represents a curve fitted between the two curves generated by $W[\u03f5s]$ and *κ*_{scale} 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 (*F _{m}* and

*ϵ*), 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

_{m}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, *F*_{hys}, is the sum $\u22111nFi$.

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

where *h*(*t*) is a solution of the equation

*n*,

*k*,

*k*

_{1},

*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

In the above, **H*** _{r}* are the backlash (play) operators of PI model and

*k*is the sampling number of the operator. The weights

*w*and threshold

_{i}*r*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]).

_{i}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

where $Hria,d$ are different for ascending and descending; *κ*_{1} is a constant. The two operators are subject to constraints

In the above, there are (2*n* + 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

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.

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.

In our study, we use two MPAMs consisting of an inner silicone tube with *r _{o}* = {0.55, 0.75} mm and

*r*= {0.25, 0.25} mm braided on the outer surface using nylon cords of radius ∼ 50

_{i}*μ*m 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

*r*and the inner surface of braid with the radius

_{o}*r*. 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.

_{b}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).

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 $P\xafi$. 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].

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]

where *u _{r}* and

*u*are the displacements of silicone tube in the radial and axial directions, respectively.

_{z}^{3}Solving the equations, we get the displacements

where *c _{i}*,

*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

*P*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 $Ps=Pi(ri2/ro2\u2212ri2)+Psil$, where

_{i}*P*

_{sil}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

where Λ_{1} and Λ_{2} are Lame's parameters. Substituting the constants, we get the displacements as

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

where *r _{b}* =

*r*

_{0}+

*δ*is the initial radius of the braided sleeve and the quantities $u\u0302r$ and $u\u0302z$ 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

Since the braid and sleeve are sealed at the tips, the axial displacement of the sleeve and the tube is the same. Hence, $u\u0302z=uz(l=l0)$ 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

and we have

Simplifying and substituting for $u\u0302z$, we get

In the above expression, the constants *c*_{1}, *c*_{2}, and *c*_{3} depend only on applied *P _{i}* which is the inflection pressure $P\xafi$. Substituting the values of constants, we get the following equation:

where $Ps=P\xafi(ri2/ro2\u2212ri2)+(F/\pi ri2)$. This equation can be numerically solved to find the inflection pressure. For values of applied pressure below $P\xafi$, Eq. (62) can be used to find the end-point elongation of the MPAM.

For values of pressure above $P\xafi$, 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., $ur|ro=u\u0302r$ and $uz|lo=u\u0302z$. Then from the kinematics of braid, Eq. (64), and from Eq. (59), we can write

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, *F _{e}*, 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 $FPi=Pi(\pi ri2)$, 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

*F*. This total axial force

_{u}*F*is borne unequally by the axial end of silicone tube as well as the nylon braid (ref Fig. 9)

_{e}where *F*_{sil} represents the axial force acting on silicone tube and *F*_{nyl} 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 *P _{o}* 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 Δ_{nyl} as well as the silicone tube Δ_{sil}, it is possible to write the individual components of forces in terms of the end force *F _{e}*. We use the material properties of the tube and sheath to calculate the axial displacement of nylon sheath

where $F\u0302=(Fnyl/m)cos\u2009\theta 0$ is the force acting on a single strand of braid and $A\u0302nyl$ is the area of cross section of single nylon strand, and *E*_{nyl} is the modulus of elasticity of nylon (refer Fig. 9). Similarly, the axial displacement of silicone tube can be written as

where *A*_{sil} and *E*_{sil} 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