Department of Mathematics Romanian Academy, Timisoara Branch
The 13th International Conference on Mathematics and its Applications
Timisoara, November 1-3, 2012
Proceedings
C o lectia "C O N F E R IN T E "
EDITURA POLITEHNICA
TIMISOARA - 2013
5
C o n ten ts
Conference details 9
I In v ited speakers 13
D . B u rd e - Derived length and nildecomposable Lie algebra ... 15 L. H ú son - Bivariate adaptive dose finding in a single cohort of patients with
repeated m ea su res... 25 J. Jaric, D . K u zm a n o v ic - On thermodynamics of anisotropic non-linear elastic
bodies ... 33 M . R a zza g h i - A hybrid approximation method for optimal control problems . 4 7
II M a th em a tica l A n a ly sis and A p p lica tio n s 57
A .M . A cu , D .F . S o fo n ea - Some optimal quadrature formulas in sense Nikolski and error b o u n d s ... 59 N . K . A g b e k o - Some mysterious facts by substituting the addition with the
maximum operation on Banach la ttic e s ... 65 P. B u rai - Generalized convexity and the simplest case of Friedrichs’-Poincaré
in e q u a lity ... 71 E. C o n sta n tin e sc u , A .N . B ra n g a - Solving a certain extremal problem with
p o ly n o m ia ls... 77 N . CriväJ - Almost periodic multifunctions with values in generalized uniform
spaces ... 83
A . H á z y - On (fc, h)-convex f u n c t io n s ... 89 D . L u p u lescu , N . C ofan , I. S ta n - Gagliardo’s completion for Fréchet couples 95 D . L u p u lescu , C . H ed rea, I. S ta n - Interpolation of Fréchet couples . . . . 101 T . M a tsu d a - Desingularization of mixture tetranomial distribution and its ap
plication for the detection of web application a t t a c k s ... 107 J .C . N d o g m o - A general result on the infinitesimal generators of the induced
equivalence g r o u p ... 113 C . Prat,a - The method of Lyapunov function for the study of the exponential
dichotomy of co-semigroups in Banach s p a c e s ... 119 M . Sadik u - Accelerating convergence of trigonometric series by means of Euler
type op erators... 125
C . S to ic a , L. B iri§ - Nonuniform behaviors for cocycles over multivalued non- autonomous dynamical systems in Banach s p a c e s ... 131 S .E . V la d - Binary signals: a note on the prime period of a p o i n t ... 137
III A lgebra and G eom etry, C om p u ter A lgeb ra S y stem s in
R esearch 143
L. K o v á cs, A . K o v á cs - Examples of symbol elimination in program verification 145 C . L äzurean u, T . B in za r - Some geometrical properties of the Maxwell-Bloch
equations with a c o n t r o l... 151 C . M ilici - On the Implication A lg e b r a s ... 159 C . P o p , C . P etri§ o r - A geometric approach of a smooth linear version of Chua’s
s y s t e m ... 165 M . S te fa n esc u , M . D u m itr u , P. A n g h e l What we are doing with algebraic
h y p erstru ctu res... 171 M . S to lo je s c u - Edge detection techniques for X-ray image segmentation . . . 179 G . T ig an - A case-study model for impact o sc illa to rs... 185
IV A p p lied M a th em a tics in E n gin eerin g and E con om ics 191
C. B o ta , B . C á ru n tu - Approximate solutions for the generalized pantograph delay differential equation by the square minimization m e t h o d ... 193 A .L . C iu rd ariu , M . N e a m p i - Solution moment stability in a stochastic model
with delay of financial a s s e t s ... 199 N . D a m lja n o v ic, A . P e to je v ic , M . Z izovic - Comparative application of
Lattice MCD-method with P R O M E T H E E -m eth od ... 205 F .I. D ra g o m ir esc u , I. M o isa - On convective/absolute instabilities quantifica
tion in swirling flo w s...211 M . D u m itra c h e , C . G h eld iu - Hyperbolic equation - the coefficient d = d{t) . 2 1 7 R . E n e, T . B in z a r - On the convergence of the solution of dynamic problem of
periodic elastic m e d i a ... 22 3 V . Farrokhi, L. P ok orád i — A comparative analysis of evaluation methods for
readiness of business intelligence project ...229 C. G h eld iu , R .M . G e o rg e scu - The convergence of the intern exact control for
reinforced reticulated s t r u c t u r e s ... 2 3 5
A . K o v á cs, L. K o v á cs, L. K ovács - The boundary element method in the study of the fluid’s non-stationary motion through network profiles . . . 241 Y u. M en sh ik o v - Improvement of the forecast of static economic processes . . 249 L. M ih o n - Advanced modeling and diagnosis of a powertrain through specific
p a r a m e t e r s ... 255 L. M o leriu (M o a tä r) - Inferring evolution models from experimental data on
populations of th y m o c y te s ... 263 L. P ok orád i - Modular fault tree sensitivity a n a ly s is ... 271 T . P o rtik , L. P ok orád i - Fuzzy rule based risk assessment with summarized
defuzzyfication ...27 7 I. Sprint,u - Mathematical model used in the analysis of orthotropic plates . . 283 I. S p r in ju , D . B arti§ - Analysis of composite rectangular plates based on the
classical laminated plate theory ... 28 9 J. S ta n o je v ic , T . L evajk ovic - On the Cramer-Lundberg model with stochastic
premia and the Panjers recu rsion ...295 D . T en en g, K . P á r n a - Modeling and forecasting index prices with normal
inverse g a u s s ia n ... ...3 0 3 T . V arga, L. P ok orád i, T . P o rtik - Computational method of process audit
evaluation based on VDA 6.3 standard by fuzzy a p p lic a tio n ...3 0 9 M . Z izovic, N . D a m lja n o v ic - Main Advantage of Lattice MCD-Method . . 315
V P ro b a b ility and S ta tistic s, A p p lica tio n s to H ea lth and
C linical R esearch 321
I. Golet,, I. G o le t - On contraction types in probabilistic metric spaces . . . . 3 2 3 L. K ovács, T . F eren ci, B . B e n y ó , A . K o v á cs, G .J . C h a se - Short- and
long-term evolution of insulin sensitivity variability in critically ill patients 3 2 9 E. K o v á c s, T . S zá n ta i - Hypergraphs in the characterization of regular-vine
copula s t r u c t u r e s ...3 3 5 J. S árosi, Z. F ab u lya - Mathematical analysis of the newest model for the force
generated by fluidic m u sc le... 3 4 5 I. T o th , S. P a ra lesc u , §. M ihäicutjä - Statistical confirmation of intuitive
assumptions in sleep apnea syn d rom e... 351
7
Sponsors 357
University ” Politehnica” of Tim isoara November, 1-3, 2012
MATHEMATICAL ANALYSIS OF THE NEWEST MODEL FOR THE FORCE GENERATED BY FLUIDIC MUSCLE
József SÁROSI and Zoltán FABULYA University of Szeged
A b s t r a c t
The less well-known type of pneumatic actuators is the pneumatic artificial muscles (PAMs). Different designs have been developed, but the McKibben muscle is the most popular and is made commercially available by different companies (e. g. Fluidic Muscle manufactured by Festo Company). Many researchers have investigated the relationship of the force, length and pressure to find a good theoretical approach for the equation of force produced by PAMs.
In most cases, significant differences have been noticed between the theoretical and experimental results. In this paper our goal is to present our newest precise approximation algorithm for the force of Fluidic Muscle.1
1 In tr o d u ctio n
The working principle of different pneum atic muscles is well described in literature (Caldwell et al. (1993), Chou and H annaford (1996), Daerden (1999), Tondu and Lopez (2000), Daerden and Lefeber (2002), B alara and Petk (2004)). PAMs have various names: Pneum atic Muscle A ctuator, Fluid A ctuator, Fluid-Driven Tension A ctuator, Axially C ontractible A ctuator, Tension A ctuator, etc. (D aerden (1999), D aerden and Lefeber (2002), Ram asam y et al. (2005)). M ost types of PAMs consist of a rubber bladder enclosed w ithin a helical braid th a t is clam ped on bo th ends. A PAMs energy source is gas, usually air. The muscle will expand radially and contract axially when inflated, while generating high pulling forces along the longitudinal axis.
T he tensile force depends on th e contraction and the applied pressure. This feature is totally different from pneum atic cylinders, because a cylinder develops a force th a t depends on the applied pressure and piston surface area and independent from
1 Keywords and phrases: Fluidic Muscle, Static Model, Force Function, MS Excel Solver
346 J. Sárosi and Z. Fabulya
displacem ent (Daerden and Lefeber (2002)). Typically, the air muscle can contract by about 25 % of its initial length.
T he layout of this paper is as follows. Section 2 (M aterials and M ethods) presents th e static modelling of PAMs and our newest force equations. Section 3 (Results and Discussion) compares th e m easured d a ta and calculated data. Finally, Section 4 (Conclusion and Future Work) gives the investigations we plan. Fluidic Muscle type DMSP-20-200N-RM-RM (with inner diam eter of 20 mm and initial length of 200 mm) produced by Festo Com pany is selected for our newest study.
2 M a teria ls and M e th o d s
T he general behaviour of PAMs w ith regard to shape, contraction and tensile force when inflated depends on th e geometry of the inner elastic p a rt and of the braid at rest (Figure 1), and on th e m aterials used (Daerden (1999)). Typical m aterials used for the m em brane construction are latex and silicone rubber, while nylon is norm ally used in the fibres.
Figure 1: Geom etry param eters of PAMs
W here F the pulling force, ro, lo, £*o the initial inner radius, length of th e PAM and th e initial angle between the thread and th e muscle long axis, r, Z, a th e inner radius and length of th e PAM and angle between th e th read and the muscle long axis when the muscle is contracted, h the constant th read length, n th e num ber of tu rn s of thread.
G ood description of the general static m odel of PAMs can be found in Chou and H annaford (1996), Daerden (1999), Tondu and Lopez (2000) and Kerscher et al. (2005). On the basis of them the force equation has been found:
F(p, k) = r l ■7T • p ■ (a ■ (1 - k)2 - b) (1) W here: a = 7tg z a-j —, 07 b = -r-i— , sin^oco7 k = to and
p - applied pressure,
k - contraction.
Equation (1) was modified w ith correction factors e (Tondu and Lopez (2000)) and p (Kerscher et al. (2005)):
F(p, k) = p • rg • 7T • p • (a • (1 - £ • k)2 - b) (2) Significant differences between the theoretical and experim ental results using (1) and (2) have been proved in Sárosi and Fabulya (2012). To elim inate the differences new approxim ation algorithm s w ith six and five unknown param eters have been introduced for the force generated by Fluidic Muscles:
F( p, k) = (a- p + b) ■ expCK + d- p- n + e - p + f (3)
F( p, k) = (p+ a) ■ expb'K + c- p- K + d- p + e (4) Equation (3) can be generally used w ith high accuracy for different Fluidic Mus
cle independently from length and diam eter under different values of pressure and (4) can be used w ith high accuracy for Fluidic Muscle w ith inner diam eter of 20 mm, only.
The unknown param eters of (3) (a, b, c, d, e and / ) and (4) (a, b, c, d and e) can be found by Solver in MS Excel 2010.
3 R e su lts and D iscu ssio n
O ur newest investigations were carried out in MS Excel. Firstly, the m easured d a ta and calculated d a ta using (1) were compared. As it is shown in Figure 2, the m easured force always drops from its highest value at full muscle length to zero at full inflation and position and there is only one intersection point between the m easured and calculated results and no fitting.
-5 0 5 10 15 20 25 30
Contraction p i]
--- 0 kPa (mea<urad) --- 0 kPa (calculated) ---100 kPa (meaaured)--- 100 kPa (calculated) --- 200 kPa (meam o d ) --- 200 kPa (calculated)--- 300 kPa (meaaured)--- 300 kPa (calculated) --- 400 kPa (m eanaed)--- 400kPa (calculated)---300kPa (meaaured) . . . . SOOkPa(calculaied)
Figure 2: Com parison of m easured d a ta and calculated d a ta using (1)
348 J. Sárosi and Z. Fabulya
R2 = 0.5397 => R = 0.7346 correlation index proves the inaccurate fitting for th e m easured data.
In the interest of fitting the sim ulation was repeated w ith (2) (Figure 3). The coefficients of (2) were found using Solver in MS Excel.
-5 0 5 10 15 20 25 30
Contraction [%]
--- 0 kPa (m easured) --- 0 kPa (calculated) ---100 kPa (measured) - - - - 100 kPa (calculated) --- 200 kPa (m e a s u r e d ) --- 200 kPa (calculated) --- 300 kPa (measured) --- 300 kPa (calculated) --- 400 kPa (m e a s u r e d ) --- 400 kPa (calculated) --- 500 kPa (m e a s u r e d ) --- 500 kPa (calculated)
Figure 3: Com parison of m easured d a ta and calculated d a ta using (2)
Figure 3 shows the m easured and calculated results still do not fit. B etter fit
ting was attained, b u t a t a pressure of 0 k P a we still have a rath er substantial inconsistency. R2 = 0.8888 =► R = 0.9427 correlation index proves this difference.
To improve fitting quality under different values of pressure including 0 k P a new approxim ation algorithm s have been introduced w ith 6 and 5 param eters. In this paper result of (4) is presented, only. The unknown param eters of (4) can be found using Solver in MS Excel, too. Values of these unknown param eters: a = 286,17, b = - 0 ,3 3 , c = - 9 ,1 4 , d = 288,47, e = -2 7 1 ,3 5 .
The accurate fitting of (4) can be seen in Figure 4 and Figure 5. R 2 — 0.9994 R — 0.9997 correlation index proves the tight relationship between the m easured d a ta and calculated data.
-9 0 3 10 13 20 25 30 Contraction (%)
--- 0 kPi (meawrsd) --- 0 kPa icatcuUwd) ---lOOkPi ( m a n u n d ) ---100 kPa (cakóinál) --- 200 U% ( m o u n d ) --- 200 kPa (calculated) --- 300 kPi (m M tured)---300kP» (calcutMoD --- 400kPa (measured)--- 400kPa (calculoed) --- 500kPi (m eaw rcd)--- S00kPa (calculated)
Figure 4: Com parison of m easured d a ta and calculated d a ta using (4)
♦ F calculated [N] as a function o f F m easured [N]
---Linear (F calculated [N] as a function o f F m easured [N])
Figure 5: Relationship between th e m easured force and calculated force using (4)
4 C o n clu sio n and F utu re w ork
In this work new functions for the force produced by Festo Fluidic Muscle have been introduced. The acchracy of the approxim ation algorithm w ith 5 unknown param eters (R = 0.9997 correlation index) was proved here. The analysis was done in MS Excel. O ur m ain aim is to develop a new general m athem atical model for pneum atic artificial muscles applying our new models and results.
R eferen ces
[1] M. Balara, A. Petk, The Proporties of the A ctuators w ith Pneum atic Artificial Muscles, Journal o f Cybernetics and Inform atics, Vol. 4, 1-15, (2004).
350 J. Sárosi and Z. Fabulya
[2] D. G. Caldwell, A. Razak, M. J. Goodwin, Braided P neum atic Muscle Ac
tuators, IFAC Conference on Intelligent A utonom ous Vehicles, Southam pton, U nited Kingdom, 18-21 April, 1993, 507-512
[3] C. P. Chou, B. H annaford, M easurement and M odelling of M cKibben Pneum atic, Artificial Muscles, IE E E Transactions on Robotics and A utom ation, Vol. 12, No.
1, 90-102, (1996).
[4] F. Daerden, Conception and Realization of P leated Artihcial Muscles and Their Use as Com pliant A ctuation Elements, PhD D issertation, Vrije Universiteit Brussel, Faculteit Toegepaste W etenschappen Vakgroep W erktuigkunde, 5-19, (1999)
[5] F. Daerden, D. Lefeber, Pneum atic Artificial Muscles: A ctuator for Robotics and A utom ation, European Journal of Mechanical and Envirom ental Engineering, Vol. 47, No. 1, 11-21, (2002).
[6] T. Kerscher, J. Albiez, J. M. Zllner, R. Dillm ann, FLU M UT - Dynamic M od
elling of Fluidic Muscles using Quick-Release, 3rd International Symposium on Adaptive M otion in Animals and Machines, Ilm enau, Germany, 25-30 Septem ber, 2005, 1-6
[7] R. Ramasary, M. R. Juhari, M. R. M am at, S. Yaacobs, N. F. Mohd Nasir, M.
Sugisaka, An A pplication of Finite M odelling to Pneum atic Artificial Muscle, Am erican Journal o f Applied Sciences, Vol. 2, No. 11, 1504-1508, (2005).
[8] J. Sárosi, Z. Fabulya, New Function A pproxim ation for the Force G enerated by Fluidic Muscle, International Journal o f Engineering, Annals o f Faculty of Engineering Hunedoara, Vol. 10, No. 2, 105-110, (2012).
[9] B. Tondu, P. Lopez, Modelling and Control of M cKibben Artificial Muscle Robot A ctuator, IE E E Control System Magazine, Vol. 20, No. 2, 15-38, (2000).
József Sárosi - Technical Department Faculty of Engineering, University of Szeged,, Mars tér 7, H-6724 Szeged, Hungary
E-mail: sarosi@mk.u-szeged.hu
