The pneumatic artificial muscle (PAM) being a new type of actuator is widespread used in engineering field because it possesses the attractive attributes of safe operation, low cost, high output force to volume. It is a prime requirement to establish an effective theoretical model to predict the performance of the PAM as well as other actuators. In this paper a new PAM model is established by employing conservation energy and the Mooney-Rivlin strain energy function for the rubber of PAM. This model is compared with the previous models and the experimental results and it shows that the model truly improves the assessment of forces and contraction that can be achieved by the PAM. However, the significant discrepancy still exists between the theoretical model and the experiment. A number of factors, like friction force, which result in the discrepancy are discusse

Jian, Jin

Qingwei, He

Songbo, Li

English

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L. Songbo, J. Jian, H. Qingwei: Modelling and assessment of pneumatic artificial muscleENGINEERING MODELLING 19 (2006) 1-4, 95-100 95SUMMARYThe pneumatic artificial muscle (PAM) being a new type of actuator is widespread used in engineering fieldbecause it possesses the attractive attributes of safe operation, low cost, high output force to volume. It is a primerequirement to establish an effective theoretical model to predict the performance of the PAM as well as otheractuators. In this paper a new PAM model is established by employing conservation energy and the Mooney-Rivlinstrain energy function for the rubber of PAM. This model is compared with the previous models and the experimentalresults and it shows that the model truly improves the assessment of forces and contraction that can be achieved bythe PAM. However, the significant discrepancy still exists between the theoretical model and the experiment. Anumber of factors, like friction force, which result in the discrepancy are discussed.Key words: pneumatic artificial muscle, strain energy, Mooney-Rivlin model, testing.UDC 519.85:531.2:539.3:616.77:611.73Preliminary communicationReceived: 28.05.2006.Modelling and assessment of pneumaticartificial muscleLi Songbo(1), Jin Jian(2) and He Qingwei(1)(1)School of Mechanical Engineering, Shanghai Jiaotong University, Shanghai, P.R. CHINAe-mail: Songbo.Listen@sjtu.edu.cn ; Heqinwei@sjtu.edu.cn(2)Department of Mechanical Automation, Shanghai University, Shanghai, P.R. CHINA1. INTRODUCTIONThe Pneumatic Artificial Muscle (PAM) (see Figure1) which was first developed in the 1950’s consists ofan internal rubber bladder surrounded by a braidedmesh shell.ratio the PAM has a wide range of applications, suchas potential projects, vibrators, conveyors, etc,especially in robots.In this paper we present a model employinghyperelastic theory to describe the inner elastomerbladder and do experiments to test its improvement inprediction of PAM’s output force. The previous modelsmostly ignored the elastic energy of bladder [1-4] andsome considered the elastic energy but they did notgive the details about the theory and experiment [5].This paper will give the detailed description oftheoretical model and experimental method; lastly, itwill provide the contrast between the model predictionsand the experimental results.2. THEORETICAL MODEL DERIVATION2.1 Basic definitionThe elastomer bladder can be described by (T0, R0,L0) in the undeformed state and (Ti, Ri, Li) in theFig. 1  The geometry of the PAMThe PAM generates longitudinal contractionaccording to the gas pressure loading on the internalrubber surface. Because of its high force to weightL. Songbo, J. Jian, H. Qingwei: Modelling and assessment of pneumatic artificial muscle96 ENGINEERING MODELLING 19 (2006) 1-4, 95-100deformed state, where the T0 and Ti stand for theundeformed and deformed thickness, the same asdiameter of R0 and Ri and the length of L0 and Li.The principal strains of the bladder are describedaccording to the cylinder coordinate, i.e., radial strain(λ1), circumferential strain (λ2) and axial strain (λ3).The corresponding principal strains are given by:; i20RRλ = ; i30LLλ = (1)Note that due to the material incompressibility,these strains are related by:λ1λ2λ3=1 (2)In the context of the bladder deformation, thisrelation is used to calculate the radial strain with theuse of the strains λ2 and λ3:0 012 3 i iR L1R Lλλ λ= = (3)2.2 Constitutive equationNon-linear elastomer materials are classicallymodeled by hyperelastic constitutive equation [6].Thus, the material behaviour is represented by a strainenergy function, denoted W. To isotropic hyperelasticmaterials, the strain energy function is given by:( ) ( ) ( )N Ni j 2iij 1 2ii j 1 i 11W C I 3 I 3 J 1D+ = == − − + −∑ ∑ (4)In the case of material incompressibility the secondterm is zero. Then, the Eq. (4) can be simplified as:( ) ( )Ni jij 1 2i j 1W C I 3 I 3+ == − −∑ (5)Let N=1, we get the linear strain energy function,i.e., Mooney-Rivlin model [6]. This model is one ofthe simplest rubber models and it will simplify thefollowing discussion. It is defined by two materialconstants, C10 and C01, and the corresponding strainenergy function is given by:( ) ( )10 1 01 2W C I 3 C I 3= − + − (6)where I1 and I2 which are the reduced strain invariantscan be described by the principal strains; then the strainenergy function is given as:( )( )2 2 210 1 2 32 2 2 2 2 201 1 2 2 3 3 1W C 3C 3λ λ λλ λ λ λ λ λ= + + − ++ + + − (7)So, incorporating strain definitions given by Eqs.(1) and (3) into Eq. (7), the strain energy function canbe written as:( )2 2 2 2 2 2 2 2 20 0 0 i10 01 102 2 2 2 22 2 2i 0i i4n R L 4n R B LW C C CB L 4n RL B Lπ ππ−= + + +−−( )2 2 2 2 2i i i 001 10 012 2 2 2 2 20 0 0 iL B L L LC C 3 C 34n R L L Lπ− ⎛ ⎞ ⎛ ⎞+ + − + −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠(8)2.3 Model derivationUsing conservation of energy equation, i.e., the inputwork should equal output work, when PAM’s lengthhas a minute change:bP' V F L V W∆ ∆ ∆= + (9)where P' is the relative gas pressure of the bladder, ∆Vis the volume change, F is the axial tension, ∆L is theaxial displacement, Vb is the elastomer volume, takingdirectly into account the incompressibility assumptiongiven by Eq. (2), which is a constant, and ∆W is thechange of the strain energy function. The Eq. (9) canbe given by differential form:bdV dWF p' VdL dL= − (10)where dVdL can be written as:2 2 20 i 02 20 0R 3L cos adV 1dL sin a Lπ ⎛ ⎞−= × −⎜ ⎟⎜ ⎟⎝ ⎠(11)The relation of R, L, B and a0 is [4]:2 2i00i0 0L1 cos a ( )LRR sin a−= (12-a)002n RsinaBπ= ; 00Lcos aB= (12-b)where a0 is the angle between the braided thread andthe bladder axis, B is the total length of the thread. So,using the Eq. (12), Eq. (11) can be expressed simply as:( )2 2i2dV 1 3L BdL 4 nπ= × − (13)At the same time, dWdL can be written as:( )( ) ( )( )2 2 2 2 2 2 2 2 2 210 0 0 i 01 0 i2 22 2 3 2 2i i i2 2 201 i i10 i 10 i 01 02 2 2 2 2 2 2 2 30 0 0 0 i8C n R L 2L B 8C n R LdWdL B L L B LC L B 2LC L 2C L 2C L2n R 2n R L L Lπ ππ π−= + −− −−− + + −(14)L. Songbo, J. Jian, H. Qingwei: Modelling and assessment of pneumatic artificial muscleENGINEERING MODELLING 19 (2006) 1-4, 95-100 97Thus, using the Eqs. (13) and (14), the final modelcan be expressed as:( ) ( )( )( )( )2 210 0 i2 2i b2 22 2 3i i2 22 2 01 i i01 0 i 10 0 i22 2 2 0 00 i210 i 01 02 30 i8C C 2L BP'F 3L B V4 n B L LC L B 2L8C C L C L LC 2CL B L2C L 2C LL Lπ⎡ −⎢= × − − +⎢⎢ −⎣−+ − + +−⎤+ − ⎥⎥⎦(15)where 2 2 2 20 0 0C n R Lπ=  is a constant for specific PAM.3. RUBBER TENSION EXPERIMENTTo determine the material constants of the Mooney-Rivlin model, C10 and C01, the uniaxial tension test iscarried out at room temperature using a classicaldumbbell-shaped specimen (20x4x1.65 mm). After thecurve fitting, a comparison of fitted and theexperimental (nominal stress-nominal strain) data isplotted in Figure 2.Fig. 3  The PAM testing setupThe hardware includes a personal computer,Advantech integrated board card (PCI 1711), directcurrent source (24V), two FESTO proportionalpressure regulators (MPPE-3, 8 bar max.), tension andcompression force sensor (±5V), two FESTO pressuresensors (SDE-10-5V/20mA), Celesco compact stringpot displacement sensor (50 inches max.), FESTO aircylinder (DNA-2 1/2", 200 kg max.), air compressor.Testing system frame was built with aluminium alloysections. We made the finite element analysis (FEA) ofthe frame in order to check whether it has enoughstiffness to bear the maximum load of air cylinder. Theanalysis result shows that the maximal displacement ofthe frame is less than 0.2 mm, Figure 4. 0 50 100 150 20002468Norminal StrainNorminal Stress  Uniaxial TestUniaxial PredictFig. 2  Fitting curve of Mooney-Rivlin modelFrom Figure 2, we noticed that the fitted databasically describe the experimental data under the lowand median strain. The fitting mechanical constants ofrubber is C10=0.0142 MP, C01=0.1008 MP.4. PAM EXPERIMENT4.1 Testing setupThe following testing system which can produceand record the tension, length, and pressure of the PAMis shown in Figure 3.Fig. 4  FEA deformation contour of the testing frameThe software system can control the pressure ofthe PAM and the air cylinder and save the acquisitiondata of sensors about the force and the displacementgenerated by the PAM. The main testing control userinterface is shown in Figure 5.Fig. 5  The user interface of the testing systemL. Songbo, J. Jian, H. Qingwei: Modelling and assessment of pneumatic artificial muscle98 ENGINEERING MODELLING 19 (2006) 1-4, 95-1004.2 Tested PAMsIn order to verify the predictive force of the modelunder different conditions, four types of PAM whichall belong to FESTO Company were tested in theexperiments, the type numbers are MAS-20-500,MAS-10-500, MAS-20-200, MAS-10-200, and allPAM’s max. permitted contractions are 25%. Thedetailed dimensions are listed in Table 1.Table 1 The dimensions of the PAMsFrom Figure 6 it is seen that with the increment ofthe pressure the linear force-length phenomenon isenhanced, but at 0-10 mm contraction region thenonlinear force-length relationship is very obvious.In order to assess the theoretical models, the modelof this paper and the model developed by Chou [1] werecompared with the experimental results, Figures 7-12.Type num MAS -20-500 MAS -10-500 MAS -20-200 MAS -10-200 L0 (mm) 500 500 200 200 R0  (mm) 20 10 20 10 T0 (mm) 2.8 1.9 2.5 1.7 N (turns) 3.3 2.4 2.1 1.8 4.3 Experimental resultsOur focus is to test the output force-contraction ofthe PAM, therefore, only the constant pressure testingis applied. In the testing the PAM was fixed at one endand free to move subject to the constraint of the aircylinder at the other end. The pressures of air input toPAM and cylinder were controlled by two proportionalpressure regulators, a pressure sensor was introducedat the inlet of the tested PAM to make pressureacquisition data recorded on the PC in each instancethrough the Advantech board card, the same as aircylinder, the only difference was that the software mustconvert the pressure data to force data using theequation Fa=P⋅Sa (Sa is the cross-sectional area of theair cylinder), the tensile force collected by the tensionand compression force sensor subtracted Fa is theoutput force of the PAM, the contraction of the PAMwas recorded by the displacement sensor.The experimental results were obtained at activationpressures of 1 to 6 bars, the testing results of force-contraction profile at varying pressure of 1 bar to 6bars are shown in Figure 6. 0 0.05 0.1 0.15 0.2050010001500Output Force (N)Contraction Length (M)123456Fig. 6  Force-contraction profile at varying pressure 0 0.05 0.1 0.15 0.20100200300400Output Force (N)Contraction Length (M)  FpapermodelFchoumodelFexperimentFig. 7  Force-contraction curve of PAM at 1 bar 0 0.05 0.1 0.15 0.20200400600800Output Force (N)Contraction Length (M)  FpapermodelFchoumodelFexperimentFig. 8  Force-contraction curve of PAM at 2 bars 0 0.05 0.1 0.15 0.2050010001500Output Force (N)Contraction Length (M)  FpapermodelFchoumodelFexperimentFig. 9  Force-contraction curve of PAM at 3 bars 0 0.05 0.1 0.15 0.2050010001500Output Force (N)Contraction Length (M)  FpapermodelFchoumodelFexperimentFig. 10  Force-contraction curve of PAM at 4 barsL. Songbo, J. Jian, H. Qingwei: Modelling and assessment of pneumatic artificial muscleENGINEERING MODELLING 19 (2006) 1-4, 95-100 99Fig. 12  Force-contraction curve of PAM at 6 barsA number of conclusions can be drawn from theabove figures. Firstly, either our or the model fromChou does not coincide well with the experimentalresults, the discrepancy produced by neglecting thefriction, pressure loss, the tip-effects of the PAM’s ends(a violation of the theoretical right circular cylinderassumption) and other factors is significant. Secondly,the new model is more accurate than the model fromChou, especially at mid to high contraction. Lastly, atthe low pressure (Figures 7 and 8) the theoreticalresults including the magnitude and shape of the curvecannot correspond with the testing results; but at thehigh pressure (Figures 11 and 12) the theoretical modelcurve shapes basically correspond well with the testingcurves. In other words, due to ignoring the frictionforce and other factors, the approximation oftheoretical model to experiment is only theapproximation of curve shape unless the output forceis large enough to ignore the existence of the friction.5. CONCLUSIONS ANDRECOMMENDATIONSIn this paper the theoretical PAM model whichconsiders the elastic energy described byincompressible Mooney-Rivlin equation was derivedand assessed. The more reasonable model would befound very useful to describe the static displacementand force performance of PAM based systems.In the present paper only the elastic energy factorwas considered and the significant effects of thefriction and other energy loss can be clearly found inFigures 7 upto 12. In order to perfect the PAM model,using the sound approach to account for effect of thefriction is the main aim of the further work.ACKNOWLEDGEMENTSThe authors would like to express their gratitude toDan Wenjiao and to Zhang Guoxing, EngineeringDepartment Manager of the FESTO China inShanghai, for the constant support of this research.6. REFERENCES[1] C.P. Chou and B. Hannaford, Measurement andmodeling of McKibben pneumatic artificialmuscles, IEEE Transactions on Robotics andAutomation, Vol. 12, No. 1, pp. 90-102, 1996.[2] C.P. Chou and B. Hannaford, Static and dynamiccharacteristics of McKibben pneumatic artificialmuscles, Proceedings of the IEEE InternationalConference on Robotics and Automation, SanDiego, IEEE, Vol. 1, pp. 281-286, 1994.[3] N. Tsagarakis and D.G. Caldwell, Improvedmodeling and assessment of pneumatic muscleactuators, Proceedings of the ICRA 2000 IEEEInternational Conference on Robotics andAutomation, San Francisco, IEEE, Vol. 4, pp.3641-3646, 2000.[4] B. Tondu and P. Lopez, Modeling and control ofMcKibben artificial muscle robot actuators, IEEEControl Systems Magazine, Vol. 20, No. 2, pp.15-38, 2000.[5] G.K. Klute and B. Hannaford, Accounting forelastic energy storage in McKibben artificialmuscle actuators, ASME Journal of DynamicSystems, Measurements and Control, Vol. 122,pp. 386-388, 2000.[6] L.R.G. Treloar, The Physics of Rubber Elasticity,Clarendon Press, Oxford, 1975.[7] E. Verron and G. Marckmann, Numerical analysisof rubber balloons, Thin-Walled Structures, Vol.41, pp. 731-746, 2003. 0 0.05 0.1 0.15 0.20500100015002000Output Force (N)Contraction Length (M)  FpapermodelFchoumodelFexperiment 0 0.05 0.1 0.15 0.205001000150020002500Output Force (N)Contraction Length (M)  FpapermodelFchoumodelFexperimentFig. 11  Force-contraction curve of PAM at 5 barsL. Songbo, J. Jian, H. Qingwei: Modelling and assessment of pneumatic artificial muscle100 ENGINEERING MODELLING 19 (2006) 1-4, 95-100MODELIRANJE I PROCJENA PNEUMATSKOG UMJETNOG MIŠI]ASA@ETAKPneumatski umjetni miši} (PAM), kao novi tip, uvelike se koristi u tehnici budu}i da ima atraktivna svojstvasigurnog rada, malog troška i veliki izvor snage u odnosu na volumen. Prvi zahtjev je, kao i kod drugih pobu|iva~a,napraviti efektivni teorijski model kojim bi se predvidio rad PAM-a. Ovaj rad prikazuje novi PAM model koriste}i seodr`anom energijom i funkcijom Mooney-Rivling-ove energije deformiranja za gumeni materijal PAM-a. Ovaj PAMmodel se uspore|uje s prethodnim modelima i rezultatima eksperimenata te se pokazuje da ovaj PAM model uistinupoboljšava procjenu sila i kontrakcija. Ipak, postoji zna~ajan nesklad izme|u teorijskog modela i eksperimenata.Raspravlja se o brojnim ~imbenicima, poput sile trenja, koji ukazuju na taj nesklad.Klju~ne rije~i: pneumatski umjetni miši}, energija deformiranja, Mooney-Rivlin-ov model, ispitivanje.

Modelling and assessment of pneumatic artificial muscle

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