ts-6 study on mechanical behaviors of pneumatic artificial muscle

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Study on mechanical behaviors of pneumatic artificial muscle

Kanchana Crishan Wickramatunge, Thananchai Leephakpreeda *

School of Manufacturing Systems and Mechanical Engineering, Sirindhorn International Institute of Technology, Thammasat University, P.O. Box 22,

Thammasat-Rangsit Post Office, Pathum-Thanni 12121, Thailand

a r t i c l e i n f o

Article history:Received 23 July 2009

Accepted 19 August 2009

Available online 15 September 2009

Communicated by M. Kachanov

Keywords:

Empirical modeling

Mechanical spring system

Pneumatic Artificial Muscle

Stiffness parameter

a b s t r a c t

Nowadays, Pneumatic Artificial Muscle (PAM) has become one of the most widely-usedfluid-power actuators which yields remarkable muscle-like properties such as high force

to weight ratio, soft and flexible structure, minimal compressed-air consumption and low

cost. To obtain optimum design and usage, it is necessary to understand mechanical behav-

iors of the PAM. In this study, the proposed models are experimentally derived to describe

mechanical behaviors of the PAMs. The experimental results show a non-linear relationship

between contraction as well as airpressurewithin the PAMs anda pulling force of thePAMs.

Three different sizes of PAMs available in industry are studied for empirical modeling and

simulation. The case studies are presented to verify close agreement on the simulated

results to the experimental results when the PAMs perform under various loads.

Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction

A Pneumatic Artificial Muscle (PAM) is a pneumatic actuator for converting pneumatic power to pulling force. The PAM

has advantages over conventional pneumatic cylinders such as high force to weight ratio, flexible structure, variable instal-

lation possibilities, no mechanical wear, minimal compressed-air consumption, size availability, low cost and strong reliabil-

ity for human use. The minimal compressed-air consumption is inferred from the fact that the size of the PAM is relatively

small; therefore, according to the high force to weight ratio, the PAM can generate large pulling forces with the least amount

of compressed-air consumption. However, this PAM exhibits highly non-linear characteristics due to compressibility of air,

inherent properties of elastic-viscous material and geometric behaviors of the PAM shell. This makes the PAM difficult to

model and control [1,2]. Currently, existing models do not describe every stage of the mechanical behaviors well; therefore,

an improved model is still required.

The PAM was first invented in the 1950s by the physician, Joseph L. McKibben for artificial limb of the handicapped [3]. In

the 1980s, a redesigned and more powerful PAM was introduced by Bridgestone Company and was used for painting appli-

cations in industry and some applications to assist disabled individuals and service robotics. The PAM has also been applied

for different kinds of humanoid applications and advanced robots. The PAM is mainly used as a robotic actuator where com-

pliance and low power to weight ratio are important, e.g. walking/running machines or even humanoid robots [4,5]. For

instance, the Shadow Group [6] and FESTO [7] are producing different kinds of PAMs for robotic and industrial applications.

Typically, the PAM has been made of thin rubber tube (bladder) covered by a braided mesh shell and the two ends were

closed, one being air inlet and the other connected with the load. When the PAM is supplied with compressed air at the inlet

port, the internal bladder tends to increase its volume against the braided mesh shell but non-extensibility of the braided

mesh threads cause the actuator to shorten and produce pulling forces if it is connected with the load. A new type of

PAM has been introduced by FESTO recently. Its basic concept involves the wrapping of a watertight, flexible hose with

0020-7225/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijengsci.2009.08.001

* Corresponding author. Tel.: +66 2 9869009x2204; fax: +66 2 9869009x2201.

E-mail address: [email protected] (T. Leephakpreeda).

International Journal of Engineering Science 48 (2010) 188–198

Contents lists available at ScienceDirect

International Journal of Engineering Science

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http://dx.doi.org/10.1016/j.ijengsci.2009.08.001

mailto:[email protected]

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http://dx.doi.org/10.1016/j.ijengsci.2009.08.001



non-elastic fibers arranged in a rhomboidal fashion. This results in a three-dimensional grid pattern, and when compressed

air is fed into the PAM, the grid pattern is deformed. A pulling force is generated in the axial direction, resulting in a short-

ening of the muscle as air pressure is increased.

Different models have been proposed to describe the behaviors of the PAMs [3,8–11]. Among these models, the Chou and

Hannaford model [8] and the Tondu and Lopez model [3] are widely used. These models are based on the hypothesis on vir-

tual works of an infinitely thin inner tube and continuously cylindrical shape. Although they are excellent initial descriptions

of the mechanical behaviors, these models still have limitations in predicting on behavior of the PAM, at least in no-load

conditions.

Furthermore, pulling force, length, air pressure, diameter and material properties are the major parameters of the PAM

and dynamic behaviors and the relationships between these parameters are greatly changed from one PAM to another. This

means that the parameters can be variable for purpose of usage. For example, the different sizes of the PAMs are designed for

different applications. In [12] and [13], physical configuration and the behavior of the PAM hinted at variable stiffness similar

to spring-like characteristics.

In this study, the PAM will be regarded as a mechanical spring system. With this assumption, an empirical modeling of

the PAM is proposed and discussed with both theoretical and experimental approaches. Three different sizes of the PAMs

will be used in this study to generate the empirical model and finally, experimental results will be compared with the sim-

ulated results when the PAMs perform the works under load conditions as case studies.

2. Experimental setup

The experimental setup illustrated in Fig. 1 is capable of providing real-time data acquisition of the PAM length, the airpressure within the PAM and the pulling force during tension. The pulling forces were generated by a hydraulic cylinder. A

1000-N load cell mounted at one fixed end of the PAM was implemented to measure the pulling forces. The other movable

end of the PAM was attached with a sliding part of a Linear Variable Differential Transformer (LVDT) for measurement of the

PAM length. A pressure transducer connected at the air inlet of the PAM was used to observe the air pressure. All measure-

ment devices were interfaced to the computer through data acquisition card. In this experiment, three FESTO branded PAMs,

available for industry in Thailand, were used: the nominal diameter of 2 cm and the length of 30 cm was labeled as PAM-A,

the nominal diameter of 1 cm and the length of 30 cm was labeled as PAM-B and the nominal diameter of 1 cm and the

length of 7.5 cm was labeled as PAM-C.

3. Description of the parameters

First, the working principles of the PAM described here give insight to derive the empirical model. A preliminary inves-

tigation of the following hypothesis with a single PAM was reported in [14]. For Practical usage and design, the PAMs with

different sizes in lengths and diameters are studied in details and discussion on effects from size is provided for choosing the

PAMs in this paper. Initially, the PAM keeps its original length at the zero gage pressure or the atmospheric pressure without

any pulling load and this nominal length is denoted as L0. When the air pressure P within the PAM changes from low pressure

(a) (b)

`

Interface

Air Compressor

Load Cell

PressureRegulator

PressureTransducer

Linear Variable Differential Transformer

HydraulicCylinder

PAM

Fig. 1. Experimental setup: (a) diagram and (b) photograph.

K.C. Wickramatunge, T. Leephakpreeda/ International Journal of Engineering Science 48 (2010) 188–198 189



P 1 to high pressure P 2, the PAM contracts until it reaches its new equilibrium length according to that new air pressure with-

in the PAM as shown in Fig. 2a. In this study, this maximum contraction length dependent upon the air pressure is called the

unstretched length and it is denoted as Lu. It can be seen that the unstretched length change according to the air pressure as

given in the graph of Fig. 2b. As shown in Fig. 2c, the contracted PAM tends to increase its length according to the pulling

force F , with which it is exerted. The instantaneous length of the muscle is considered L. Therefore, the stretched length,

Ls is defined as the length difference between instantaneous length and unstretched length. It should be noted that the

PAM has a certain working length within the stretched length for a given pressure. For example, the stretched length is zero

while the air pressure within the PAM is atmospheric pressure. The stretched length increases when the air pressure is

increased.

4. Method of modeling the PAM

From previous section, it can be concluded that functions of both the PAM and mechanical spring system show

significantly similar characteristics when operated with or without pulling forces in working conditions. The physical func-

tionality of these two systems can be presented as shown in Fig. 3. Both the PAM and the mechanical spring system show

P2

Lu

Lo

P1

(a)

Lu

P

(b)

F

Lu Ls

L

(c)

Fig. 2. Illustration of PAM parameters:(a) initial and final length of the PAM at the operating air pressure without pulling force; (b) variation of unstretchedlength against the air pressure within PAM and (c) length definitions when PAM exerted by pulling force.

190 K.C. Wickramatunge, T. Leephakpreeda / International Journal of Engineering Science 48 (2010) 188–198



similar behavior when the pulling force is applied. Conventionally, the stiffness of the spring system is constant and it is

dependent upon the material properties and the geometry of the spring. On the other hand, the stiffness of the PAM is a

variable parameter and it depends upon not only the above mentioned properties but also the operating air pressure within

the PAM.

Based on experimental observations, the pulling forces acting on the muscle can be modeled in the same way as the force

acting on the mechanical spring system. The stiffness parameter of the muscle is denoted as K and it is considered a functionof the operating air pressure, P (gage pressure) and the stretched length, Ls. The elastic force adversely generated by the PAM

is denoted by F elastic and the expression given in Eq. (1) shows the proposed model for the force acting on the PAM as a

function of K and Ls

F elastic ¼ K ðP ; LsÞLs: ð1Þ

5. Results and discussion

The unstretched length of the PAM as mentioned in Section 3 was measured by increasing the pressure 0–5 bar (gage

pressure) without applying the pulling force. The plots given in Fig. 4 show the behavior of the unstretched length of the

three PAMs according to the air pressure within the PAM. The lines indicate the simulation results of the second order

polynomial function. It can be seen that the unstretched length decreases as the air pressure within the PAM increases,

F

F

Fig. 3. Equivalent system diagram of PAM and spring system.

Fig. 4. Plot of unstretched length against air pressure.


http://-/?-

http://-/?-



0

2

4

6

8

1

23

4

56

0

200

400

600

800

1000

Ls(cm)P(bar)

F

orce(N)

(a)

0 1 2 3 4 5 6 7 80

100

200

300

400

500

600

700

800

900

1000

1 bar

2 bar

3 bar

4 bar

5 bar

Ls(cm)

Force (N)

(b)

Fig. 5. Experimental results of contraction and elongation of the PAM (length = 30 cm, diameter = 2 cm): (a) three-dimensional plots and (b) two-dimensional plots.

Table 1

Pressure dependent unstretched length; PAM-A, PAM-B and PAM-C.

PAM Lu = h2P 2 + h1P + h0

h2 h1 h0 R2

A 0.2620 À2.9900 30.1000 0.9825

B À0.0310 À0.9800 30.5000 0.9794

C À0.0030 À0.2700 7.5000 0.9779




as expected. The greater the air pressure, the greater the contraction. In Fig. 4, the dotted plots of experimental data can be

used to determine the functional relationship between Lu and P by fitting the data into the second order polynomial function.

Table 1 shows the coefficient values of functions after fitting the data for the three PAMs. These polynomial functions can be

used to determine the unstretched length for a given air pressure. The lines of the simulation results in Fig. 4 have slopes of

nearly zero at the low pressure. In this operating range, it may not be noticeable that the PAM contracts when the low pres-

sure of the compressed air is applied to the PAMs. This threshold-like behavior is strongly dominant in the PAM with short

length and small diameter. The hysteresis at various pressures was not observed in experiments. This fact can be explained

as follows. As defined in this study, the unstretched length is the length of the PAM under no load condition for a given air

pressure. Under an equilibrium condition, the PAM remains a given length, called the unstretched length.

In this section, the mechanical behaviors of the PAM are also drawn by using the same experimental setup, which has

been mentioned in Section 2. Initially, the PAM is maintained to be steady at the unstretched position by supplying com-

pressed air at a constant pressure. After that, the pulling force is applied increasingly from zero pulling force by the hydraulic

cylinder. The PAM is stretched in such a way that the pulling force balances against the elastic force within the working

range. The length of the PAM L is recorded until the L reaches the Lo. In the same way for the reverse direction, the L is re-

corded by reducing the pulling force until the L reaches to the Lu again. Fig. 5 shows the experimental results of the stretched

length in the working range of air pressure in both forward and backward direction of the PAM-A.

The stretched lengths increase as shown in Fig. 5 with solid lines with upward heading arrows. On the other hand, the

stretched lengths decrease as shown in Fig. 5 with solid lines with downward heading arrows. Fig. 5a and b present

three-dimensional plot and two-dimensional plot, respectively. It can be observed that the hysteresis non-linearity takes

place from increasing or decreasing the pulling force at various air pressures. The stretched length at the low pressure is

shorter than that at the high pressure because the unstretched length is longer at low pressures as seen in Fig. 4. These re-

sults indicate that the stiffness parameters K in Eq. (1) are different for cases of increasing and decreasing the pulling forces.

In Eq. (1), the stiffness parameter K is explicitly taken as a second order polynomial function of Ls and P and it is expressed

in Eq. (2). However, a universal approximation method with fuzzy logic or neural network can be used as an alternative

choice for improved accuracy [15].

K ¼ c 3P 2 þ c 2PLs þ c 1L

2

sþ c 0; ð2Þ

where c 0, c 1, c 2 and c 3 are constant coefficients, which can be obtained from experimental data.

The second order polynomial for the function of the stiffness parameter in Eq. (1) is selected initially for purpose of study

since it can be used as the lowest order non-linear model of the stiffness parameter. The higher order polynomial functions

can be applied further with the same approach if some requirements, e.g. accuracy are to be fulfilled.

To yield the best fit, the range of the air pressure is divided into the operating conditions where the air pressure of

0–2.5 bar is considered the low-pressure range and the air pressure of 2.5–5.0 bar is considered the high-pressure range.

In general, the selection of pressure range might be varied according to experimental data. This technique is used to

strengthen the accuracy of the non-linear model due to extensive ranges. The unknown coefficients of Eq. (2) are determined

by applying the least squared method to the experimental data of the three PAMs. The values of coefficients are obtained and

given in Table 2.

According to Table 2, the plots of the stiffness parameter, K in Eq. (2) for the PAM-A and PAM-B and the PAM-C are

illustrated in Fig. 6. The cases of elongation and contraction are presented in Fig. 6a,c,e and b,d,f of PAM-A, PAM-B and

PAM-C, respectively. It should be noted that plots in the section A show the results from Eq. (2) within the working length

while the section B is out of the stretched length. From observation, the value of the stiffness parameter increases while the

Table 2

Values of coefficient in K . Values of coefficients in K ; PAM-A, AM-B and PAM-C.

PAM Constant _L > 0 elongation _L < 0 contraction

0–2.5 bar 2.5–5.0 bar 0–2.5 bar 2.5–5.0 barA c 3 À2.2990 4.1820 À2.9770 3.8370

c 2 À5.7410 À3.1270 À5.8900 À2.3890

c 1 2.7410 1.8180 3.2030 1.8500

c 0 119.3040 70.3770 104.8240 53.0210

R2 0.9690 0.9968 0.9512 0.9967

B c 3 À21.1140 0.4950 À19.7340 0.2390

c 2 À4.1280 À3.8940 À3.3550 À3.7790

c 1 3.2680 3.2150 3.2900 3.4710

c 0 210.5100 73.9000 191.1340 67.1860

R2 0.9989 0.9865 0.9951 0.9809

C c 3 À15.6580 0.2660 À9.7570 0.1220

c 2 À7.6990 À12.4690 À13.3090 À19.3640

c 1 73.6480 80.0420 109.5880 118.5010

c 0 271.0390 175.8510 214.1010 156.4060

R2 0.9982 0.9911 0.9958 0.9881




stretched length increases for given air pressure. However, for a given stretched length, simulation results show that the

stiffness parameter decreases when the air pressure within the PAM increases at a range of low pressure. On the other hand,

the stiffness parameter increases when the air pressure within the PAM gets higher after passing a minimum value. This

unexpected non-linear behavior might be caused by the complex structures within the PAM, geometry and the rubbery

Fig. 6. Plots of stiffness parameter, K of the PAM-A, B and C: (a), (c), (e) elongation and (b), (d), (f) contraction.




materials itself. With the proposed empirical modeling, such non-linearity can be captured so as to predict the complicated

mechanics of the PAM. It can be observed that the values of the stiffness parameter of the PAM-A are higher than the PAM-B

at the same stretched length and air pressure where the diameter of the PAM-A is larger than the PAM-B. Additionally, the

0 1 2 3 4 5 6 7 80

100

200

300

400

500

600

700

800

900

1000

1bar

2bar

3bar

4bar

5bar

0 1 2 3 4 5 6 7 80

100

200

300

400

500

600

700

800

900

1000

1bar

2bar

3bar

4bar

5bar

(a) (b)

0 1 2 3 4 5 6 70

100

200

300

400

500

600

1bar

2bar

3bar

4bar

5bar

0 1 2 3 4 5 60

100

200

300

400

500

600

1bar

2bar

3bar

4bar

5bar

(c) (d)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

50

100

150

200

250

300

350

400

1bar

2bar

3bar

4bar

5bar

0 0.2 0.4 0.6 0.8 1 1.2 1.40

50

100

150

200

250

300

350

400

1bar

2bar

3bar

4bar

5bar

(e) (f)

Elastic force , F elastic

(N)


(N)


(N)


(N)


(N)


(N)

Ls (cm) Ls (cm)

Ls (cm) Ls (cm)

Ls (cm)Ls (cm)

Elongation PAM-A

Elongation PAM-B

Elongation PAM-C Contraction PAM-C

Contraction PAM-B

Contraction PAM-A

Fig. 7. Comparison between experimental results and simulation results (solid lines) of the PAM-A, PAM-B and PAM-C: (a), (c), (e) elongation and (b), (d), (f)contraction.




values of the stiffness parameter of the PAM-C are also higher than the PAM-B at the same stretched length and air pressure

where the length of the PAM-C is shorter than the PAM-B with the same diameter. However, it should be noted that the

Pressure,P(bar)

Force(N)

Length,L(cm)

0 0.5 1 1.5 2 2.5 3 3.5 40

2

4

6

0 0.5 1 1.5 2 2.5 3 3.5 40

500

1000

0 0.5 1 1.5 2 2.5 3 3.5 40

10

20

30

P

ressure,P(bar)

Force(N)

Length,L(cm)

0 0.5 1 1.5 2 2.5 3 3.5 4

0

2

4

6

0 0.5 1 1.5 2 2.5 3 3.5 40

500

1000

0 0.5 1 1.5 2 2.5 3 3.5 40

10

20

30

(a). PAM-A (increasing force) (b). PAM-A (decreasing force)

Time (s)Time (s)

Time (s) Time (s)

Time (s) Time (s)

Pressure,P(bar)

Force(N)

Length,L(cm)

0 0.5 1 1.5 2 2.5 3 3.5 40

2

4

6

0 0.5 1 1.5 2 2.5 3 3.5 40

200

400

600

0 0.5 1 1.5 2 2.5 3 3.5 40

10

20

30

40

Pressure,P(bar)

Force(N)

Length,L(cm)

0 0.5 1 1.5 2 2.5 3 3.5 40

2

4

6

0 0.5 1 1.5 2 2.5 3 3.5 40

200

400

600

0 0.5 1 1.5 2 2.5 3 3.5 40

10

20

30

40

(c). PAM-B (increasing force) (d). PAM-B (decreasing force)

Pressure,P(bar)

Force(N)

Length,L(cm)

0 0.5 1 1.5 2 2.5 3 3.5 40

2

4

6

0 0.5 1 1.5 2 2.5 3 3.5 40

200

400

0 0.5 1 1.5 2 2.5 3 3.5 40

2

4

6

8

Pressure,P(bar)

Force(N)

Length,L(cm)

0 0.5 1 1.5 2 2.5 3 3.5 40

2

4

6

0 0.5 1 1.5 2 2.5 3 3.5 40

200

400

0 0.5 1 1.5 2 2.5 3 3.5 40

2

4

6

8

(e). PAM-C (increasing force) (f). PAM-C (decreasing force)

Fig. 8. Case studies on elongation and contraction of PAM under loading conditions.




working range of the PAM-C is shorter than the PAM-B. This means the PAM-C with higher stiffness parameter can be

worked in the shorter stretched length compared with the stretched length of the PAM-B.

Fig. 7 shows the comparison between the validating data from experiment and results of the empirical model determined

from Eq. (1) in both cases of elongation and contraction from changing the pulling forces. It can be seen that the results of the

empirical model can be fitted to validating data well.

To see the performance of the proposed model, the dynamic behaviors of the PAM are studied with the three PAMs. The

empirical model is implemented to determine the position/length of the PAM when the pulling force exerts on the PAM in

time. In the case study, the PAM is initially kept at its unstretched position by the air pressure of 5 bar as depicted in Fig. 8.

The PAM is extracted by the hydraulic cylinder. Fig. 8a,c,e shows the experimental results of the PAM-A, PAM-B, and PAM-C

in sequence when the pulling force is increased from 0 N. Fig. 8b,d, and f shows the experimental results of the three PAMs

when the force is decreased to 0 N. It can be observed that the air pressure increases since the volumes of the PAMs decrease

and in turn the air pressures rise when the pulling force is applied and vice versa. The air pressure and the pulling force are

inputted to the empirical model in Eq. (1). The simulation results of change in the length of the PAMs can be obtained and

compared to the experimental results in the solid-line plots as shown in Fig. 8. The simulation results agree with the exper-

imental results very well.

6. Conclusion

In certain situations, the importance of the PAM is higher than the conventional pneumatic cylinders such as in humanoid

applications. The developed empirical model gives a concrete and effective description to understand the mechanical behav-

ior of the PAM for design and usage. The empirical model is based on the characteristics of a mechanical spring, which is afundamental element in engineering mechanics. In general, a mathematical model obtained from experimental data is called

an empirical model. The approach of empirical modeling focuses on the concepts of observation and data fitting from real

experiments. Nowadays, this method has been more useful and practical in system identification for control design than

the classical methods, e.g., lookup tables when an analytical model obtained by applying principles of mechanics can not

be applied to describe the complex mechanics, e.g., the PAM behavior. The proposed model of the PAM is based on a

well-known model of a mechanical spring but it is much more complicated. The stiffness parameter is dependent upon

not only the stretched length but also the air pressure within the PAM. Besides empirical modeling, the concrete interpre-

tation obtained in this study is an important contribution of this work as well. The agreement of simulation results on the

experimental results confirms the viability of the proposed techniques. Generally, it can be concluded that the suitable selec-

tions on PAM require information on the pulling force within the working range. The longer PAM yields the larger working

range for movement. The PAM with the larger diameter is suitable for the higher force application rather than small diameter

with the same length. However, the PAM with smaller diameter can be used for the compact applications. Control of the

compressed air can be used to adjust the stiffness parameter of the PAM in order to handle the pulling force for a givenlength of the PAM. This knowledge can be applied to regulate the dynamic behavior of the PAM. The proposed empirical

model of the PAM also provides a better description of the mechanical behaviors. This developed model can be used in sim-

ulation of the dynamic behavior of the PAM easily when it is compared with the currently available models. Furthermore,

this proposed method can be generalized to implementation for the PAMs which are available in the market.

Acknowledgment

Financial support for this research was provided by the Thailand Research Fund: TRF Research scholars-RSA5180011.

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http://www.festo.com/

http://www.shadowrobot.com/



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ts-6 study on mechanical behaviors of pneumatic artificial muscle

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