International Journal of Engineering Science

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1 International Journal of Engineering Science 8 (1) Contents lists available at ScienceDirect International Journal of Engineering Science journal homepage: 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, Thammasat-Rangsit Post Office, Pathum-Thanni 111, Thailand article info abstract Article history: Received 3 July 9 Accepted 19 August 9 Available online 15 September 9 Communicated by M. Kachanov Keywords: Empirical modeling Mechanical spring system Pneumatic Artificial Muscle Stiffness parameter Nowadays, Pneumatic Artificial Muscle (PAM) has become one of the most widely-used fluid-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 behaviors 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 air pressure within the PAMs and a pulling force of the PAMs. 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 9 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 installation possibilities, no mechanical wear, minimal compressed-air consumption, size availability, low cost and strong reliability 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,]. 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 195s by the physician, Joseph L. McKibben for artificial limb of the handicapped [3].In the 198s, a redesigned and more powerful PAM was introduced by Bridgestone Company and was used for painting applications 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 compliance and low power to weight ratio are important, e.g. walking/running machines or even humanoid robots [,5]. For instance, the Shadow Group [] 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 * Corresponding author. Tel.: x; fax: x1. address: thanan@siit.tu.ac.th (T. Leephakpreeda). -75/$ - see front matter Ó 9 Elsevier Ltd. All rights reserved. doi:1.11/j.ijengsci.9.8.1

2 K.C. Wickramatunge, T. Leephakpreeda / International Journal of Engineering Science 8 (1) 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 shortening 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 virtual 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 [1] 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 simulated results when the PAMs perform the works under load conditions as case studies.. Experimental setup The experimental setup illustrated in Fig. 1 is capable of providing real-time data acquisition of the PAM length, the air pressure within the PAM and the pulling force during tension. The pulling forces were generated by a hydraulic cylinder. A 1-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 measurement 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 cm and the length of 3 cm was labeled as PAM-A, the nominal diameter of 1 cm and the length of 3 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 investigation of the following hypothesis with a single PAM was reported in [1]. 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 L. When the air pressure P within the PAM changes from low pressure ` Air Compressor Interface Pressure Regulator Pressure Transducer PAM Load Cell Linear Variable Differential Transformer Hydraulic Cylinder (a) (b) Fig. 1. Experimental setup: (a) diagram and (b) photograph.

3 19 K.C. Wickramatunge, T. Leephakpreeda / International Journal of Engineering Science 8 (1) P 1 to high pressure P, the PAM contracts until it reaches its new equilibrium length according to that new air pressure within the PAM as shown in Fig. a. In this study, this maximum contraction length dependent upon the air pressure is called the unstretched length and it is denoted as L u. It can be seen that the unstretched length change according to the air pressure as given in the graph of Fig. b. As shown in Fig. c, 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, L s 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.. 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 functionality of these two systems can be presented as shown in Fig. 3. Both the PAM and the mechanical spring system show L o P 1 P L u (a) Lu P (b) F L u L (c) L s Fig.. Illustration of PAM parameters:(a) initial and final length of the PAM at the operating air pressure without pulling force; (b) variation of unstretched length against the air pressure within PAM and (c) length definitions when PAM exerted by pulling force.

4 K.C. Wickramatunge, T. Leephakpreeda / International Journal of Engineering Science 8 (1) F F Fig. 3. Equivalent system diagram of PAM and spring system. 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 function of the operating air pressure, P (gage pressure) and the stretched length, L s. 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 L s F elastic ¼ KðP; L s ÞL s : ð1þ 5. Results and discussion The unstretched length of the PAM as mentioned in Section 3 was measured by increasing the pressure 5 bar (gage pressure) without applying the pulling force. The plots given in Fig. 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, Fig.. Plot of unstretched length against air pressure.

5 19 K.C. Wickramatunge, T. Leephakpreeda / International Journal of Engineering Science 8 (1) Table 1 Pressure dependent unstretched length; PAM-A, PAM-B and PAM-C. PAM L u = h P + h 1 P + h h h 1 h R A B C Force(N) 5 P(bar) 3 1 (a) Ls(cm) bar 7 bar 5 3 bar 3 1 bar bar Ls(cm) Fig. 5. Experimental results of contraction and elongation of the PAM (length = 3 cm, diameter = cm): (a) three-dimensional plots and (b) twodimensional plots. (b)

6 K.C. Wickramatunge, T. Leephakpreeda / International Journal of Engineering Science 8 (1) as expected. The greater the air pressure, the greater the contraction. In Fig., the dotted plots of experimental data can be used to determine the functional relationship between L u 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. 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 pressure 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. Initially, the PAM is maintained to be steady at the unstretched position by supplying compressed 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 L o. In the same way for the reverse direction, the L is recorded by reducing the pulling force until the L reaches to the L u 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.. These results 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 L s and P and it is expressed in Eq. (). 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 3 P þ c PL s þ c 1 L s þ c ; ðþ where c, c 1, c 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.5 bar is considered the low-pressure range and the air pressure of.5 5. 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. () 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. According to Table, the plots of the stiffness parameter, K in Eq. () for the PAM-A and PAM-B and the PAM-C are illustrated in Fig.. The cases of elongation and contraction are presented in Fig. a,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. () 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 Values of coefficient in K. Values of coefficients in K; PAM-A, AM-B and PAM-C. PAM Constant _ L > elongation _ L < contraction.5 bar.5 5. bar.5 bar.5 5. bar A c c c c R B c c c c R C c c c c R

7 19 K.C. Wickramatunge, T. Leephakpreeda / International Journal of Engineering Science 8 (1) Fig.. Plots of stiffness parameter, K of the PAM-A, B and C: (a), (c), (e) elongation and (b), (d), (f) contraction. 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

8 K.C. Wickramatunge, T. Leephakpreeda / International Journal of Engineering Science 8 (1) 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 1 9 Elongation PAM-A 1 9 Contraction PAM-A bar bar bar bar (a) (b) Elongation PAM-B Contraction PAM-B bar bar bar bar (c) (d) 35 Elongation PAM-C 35 Contraction PAM-C bar bar bar bar (e) (f) 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.

9 19 K.C. Wickramatunge, T. Leephakpreeda / International Journal of Engineering Science 8 (1) 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 (a). PAM-A (increasing force) (b). PAM-A (decreasing force) (c). PAM-B (increasing force) (d). PAM-B (decreasing force) (e). PAM-C (increasing force) (f). PAM-C (decreasing force) Fig. 8. Case studies on elongation and contraction of PAM under loading conditions.

10 K.C. Wickramatunge, T. Leephakpreeda / International Journal of Engineering Science 8 (1) 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 N. Fig. 8b,d, and f shows the experimental results of the three PAMs when the force is decreased to 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 experimental results very well.. 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 behavior of the PAM for design and usage. The empirical model is based on the characteristics of a mechanical spring, which is a fundamental 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 interpretation 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 selections 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 given length 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 simulation 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-rsa References [1] T. Kerscher, J. Albiez, J.M. Zöllner, R. Dillmann, Evaluation of the dynamic model of fluidic muscles using quick-release, in: Proceedings of the First IEEE/ RAS-EMBS International Conference on Biomedical Robotics and Biomechatronics, BioRob,, pp. 37. [] K. Balasubramanian, K.S. Rattan, Trajectory tracking control of a pneumatic muscle system using fuzzy logic, in: Annual Conference of the North American Fuzzy Information Processing Society NAFIPS, 5, pp [3] B. Tondu, P. Lopez, Modeling and control of McKibben artificial muscle robot actuators, IEEE Control Syst. Mag. () [] F. Daerden, D. Lefeber, Pneumatic artificial muscles: actuators for robotics and automation, Eur. J. Mech. Environ. Eng. 7 () [5] B. Tondu, V. Boitier, P. Lopez, Naturally compliant robot-arms actuated by McKibben artificial muscles, in: Proceedings of the IEEE International Conference on Systems, Man and Cybernetics, Part 1 (of 3) 3, San Antonio, TX, USA, 199, pp. 35. [] Shadow Robot Group (London), The SHADOW Air Muscle, < [7] FESTO, Fluidic Muscle, < [8] C.P. Chou, B. Hannaford, Measurement and modeling of McKibben pneumatic artificial muscles, IEEE Trans. Rob. Autom. 1 (199) 9 1. [9] N. Tsagarakis, D.G. Caldwell, Improved modelling and assessment of pneumatic muscle actuators, in: Proceedings IEEE International Conference on Robotics and Automation,, pp [1] G.K. Klute, B. Hannaford, Accounting for elastic energy storage in McKibben artificial muscle actuators, ASME J. Dyn. Syst. Meas. Contr. 1 () [11] S. Davis, N. Tsagarakis, J. Canderle, D.G. Caldwell, Enhanced modelling and performance in braided pneumatic muscle actuators, Int. J. Rob. Res. (3) [1] C.-P. Chou, B. Hannaford, Static and dynamic characteristics of McKibben pneumatic artificial muscles, in: Proceedings IEEE International Conference on Robotics and Automation, 199, pp

11 198 K.C. Wickramatunge, T. Leephakpreeda / International Journal of Engineering Science 8 (1) [13] D.B. Reynolds, D.W. Repperger, C.A. Phillips, G. Bandry, Modeling the dynamic characteristics of pneumatic muscle, Ann. Biomed. Eng. 31 (3) [1] K.C. Wickramatunge, T. Leephakpreeda, Empirical modeling of pneumatic artificial muscle, in: Proceedings - International Multi Conference of Engineers and Computer Scientists, vol., 9, pp [15] T. Leephakpreeda, Novel determination of differential-equation solutions: universal approximation method, J. Comput. Appl. Math. 1 () 3 57.