Dynamic Model of a Compliant Link with Large Deflection and Shear Deformation

Transcription

1 Proceedings of he 5 IEEE/ASME Inernaional Conference on Advanced Inelligen Mecharonics Monerey, California, USA, 4-8 July, 5 TB-3 Dynamic Model of a Complian in wih arge Deflecion and Shear Deformaion Chao-Chieh an and Ko-Meng ee Absrac: The dynamic model for lins in mos mechanisms has ofen based on small deflecion heory wihou considering shear deformaion. For applicaions lie ligh-weigh lins or high-precision elemens, i is necessary o capure he deflecion caused by shear forces. A complee dynamic model is presened here o characerize he moion of a complian lin capable of large deflecion wih shear deformaion. We derive he governing equaions from Hamilon s principle along wih he essenial geomeric consrains ha relae deformaion and coordinae variables, and solve hem using a semi-discree mehod based on he Newmar scheme and shooing mehod ha avoids he problem of shear locing ha occurs when using finie elemen mehod. The dynamic model has been validaed experimenally. We expec ha he dynamic model will serve as a basis for analyzing a wide specrum of complian muli-lin mechanisms. Keywords Complian in; Shear Deformaion; arge Deflecion; Dynamic Model; Shooing mehod. INTRODUCTION Dynamic analyses of complian lins have been a subjec of ineres for simulaion and conrol of flexible mechanical sysems. Examples include space robo arms and high-speed roboic manipulaors. These dynamic models are ofen based on he assumpion of small deflecion wihou considering shear deformaion. This assumpion is saisfacory provided ha he lin undergoes a small deflecion such ha he heory of linear elasiciy holds. However, for applicaions involving highly complian lins (such as rubber fingers in [], ligh-weigh arms, and high-precision elemens), he effecs of large deflecion wih shear deformaion on he lin moion canno be ignored. In order o predic more accuraely he defleced shape during ransien, here is a need o model he dynamics ha capure he shear deformaion of a defleced complian lin. In he las wo decades, several approaches have been developed o analyze complian lins undergoing large deflecion and overall roaion. Javier [] has divided his research field ino hree groups. The firs is he simplified elaso-dynamic mehod originally proposed by Winfrey [3]. This approach assumes ha small deformaion does no affec rigid body moion in order o decouple he rigid body moion Manuscrip received May 5, 5. This wor was suppored in par by he Georgia Agriculural Technology Research Program (ATRP) and he U.S. Poulry and Eggs Associaion. C.-C. an is a Ph.D. Candidae in he Woodruff School of Mechanical Engineering a he Georgia Insiue of Technology, Alana, Georgia , USA ( gg3j@mail.gaech.edu) *Corresponding auhor, K.-M. ee is a Professor of he Woodruff School of Mechanical Engineering a he Georgia Insiue of Technology, Alana, Georgia , USA ( omeng.lee@me.gaech.edu) from he lin deformaion. The second is he floaing frame mehod based on defining he deformaion relaive o a floaing frame which follows he rigid body moion of he lin. For example, see [4] and [5]. This mehod maes use of linear finie elemen (FE) heory since reliable FE pacages are widely available. Alhough his mehod can accoun for shear deformaion, he deflecion is assumed o be small in order for he linear heory of elasiciy o hold. The hird is he large roaion vecor mehod [6~7] based on defining he overall moion plus deformaion wih respec o he inerial frame. Unlie he floaing frame mehod, his mehod allows large deflecion of complian lin. As a resul, nonlinear FE mehod (FEM) has o be used. This mehod, when solved using FEM, can lead o excessive shear forces nown as shear locing [9] as poined ou by Shabana [8]. We presen here a dynamic model based on he generalizaion of classical beam heory so ha i can capure he shear deformaion of a large-defleced complian lin. The classical beam heory was originaed by Daniel Bernoulli, which assumes ha a sraigh line ransverse o he axis of he beam before deformaion remains sraigh, inexensible, and normal o he mid-plane afer deformaion. Anoher imporan bu implici assumpion for classical beam heory is ha he deflecion mus be small. Rayleigh [] laer included he roaory ineria in he equaion of moion. Timosheno [] furher revealed ha he effec of roaory ineria is small for low frequency vibraion bu a high frequency he shear sress deformaion is comparable o roaory ineria. However, in order o derive a linear differenial equaion, he above classical heory and is subsequen modificaions approximae he curvaure of he lin: d d y () ds dx where is he angle of roaion of he lin; and s is he arc lengh from origin o poin (x,y) of he lin. The approximaion in () is only valid for dy/dx << and hence, he classical beam heory is limied o small lin deflecions. The difficuly o describe he moion of lins undergoing large deflecion lies on proper relaions beween angle of roaion and coordinae variables (x, y). This is because he curvaure d/ds is needed o describe he srain energy in addiion o he coordinae variables needed o express he ineic energy. In order o characerize he dynamics of a complian lin, a geomerically exac curvaure formula is necessary. The exac curvaure equaion ha can describe dynamics of a large-defleced lin can be found in mos calculus exboos: /5/$. 5 IEEE. 79

2 d ds d y / dx dy / dx 3 / When he deflecion is small, i.e., dy/dx <<, () reduces o (). Equaion () has been used in several papers o formulae he dynamic equaions of a lin, such as Reddy e al. [] and Monasa e al. [3]. However, as poined ou by Hodges [4], () defines he curvaure along coordinae x, which is on he original undefleced posiion of he beam. I does no ae ino accoun he well-nown shorening effec due o ransverse deflecions. This resuling error is ofen unaccepable in many applicaions where large deflecion is of paricular concern. In order o overcome his problem, we can parameerize x and y by he arc lengh s. This leads o anoher curvaure equaion; () d xy xy (3) ds x ; y y(s) ; and he prime denoes derivaive where x(s) wih respec o s. Equaion (3) has been used by Wagner [5] o derive he dynamic equaions of a large-defleced beam, where he square of (3) is subsiued ino he srain energy funcion of he beam in deriving he governing equaions based on Hamilon s principle. However, he resuling equaions are highly coupled and canno accoun for shear deformaion of he lin. Based on he above observaions, we develop a geomerically accurae relaion beween he angle of roaion and coordinae variables ha can be easily incorporaed in he dynamic model of large-defleced lins. While he previous angle of roaion is defined wihou considering shear effec (see (), (), and (3)), his paper provides wo consrain equaions in he derivaion of dynamical equaions so ha angle of roaion induced by bending and shear can boh be accommodaed. This paper exends he finger model in [6] o accoun for he effecs of lin compliance on he dynamic response. Specifically, his paper offers he following:. We presen a disribued-parameer dynamic model o predic he moion of a large-defleced lin, which accouns for bending, shear, and axial deformaion. The model incorporaes geomeric consrain equaions o characerize he nonlinear inemaics of a deformed lin.. We offer a mehod o numerically solve he governing equaion of a complian lin. In he ime domain we use Newmar algorihm while in spaial domain we use shooing mehod. As will be shown, he shooing mehod avoids he problem of shear locing. 3. We illusrae and validae experimenally he modeling of he lin dynamics wih a numerical example.. DYNAMIC MODE OF COMPIANT INK The dynamic model of complain lin is formulaed in wo seps. Firs, we develop wo geomeric consrain funcions o relae he deformaion and coordinae variables. Second, we incorporae he consrain equaions in he variaional form and apply Hamilon s principle o derive he governing equaions of he lin.. Geomeric Consrains Figure shows a defleced lin of lengh in he reference X-Y, where he local frame x-y is aached o he lin wih is x-axis poining o he iniially sraigh, undefleced axis of he lin. In order o fully describe he defleced shape, we define as he deflecion angle induced by bending, and as he shear angle. Hence he oal angle of roaion is +. Fig. (b) shows an infiniesimal segmen ds of he lin, he coordinae of which can herefore be described by is geomeric cener (x, y) and he orienaion +. We also inroduce he axial deformaion variable e so ha he disance beween wo adjacen infiniesimal segmens is ds+de. The variables x, y,,, and e are funcions of arc lengh s and ime. They can be expressed explicily as x( s, and y ( s,, ec. O Y dy p (x,y) y ds+de dx O f X + F s (x+dx,y+dy) Undefleced posiion (x, y) Figure Schemaic of a complian beam Y M h+dh v+dv (a) Trigonomeric relaion (b)infiniesimal segmen Figure Schemaic of an infiniesimal segmen h Since he plane moion has only hree degrees of freedom, hree of he five variables (x, y, e) are independen. The rigonomery relaing he coordinaes (x, y) o deformaion variables (e) in he x-y frame can be derived wih he aid of Fig. (a) and are saed as wo geomeric consrains: g x ( e ) cos( ) (4a) g y ( e ) sin( ) (4b) Compared wih () and (3), he shear angle can be embedded in he wo geomeric consrains in (4) easily. Noe ha Rao e al. [7] have similar consrain equaions bu again heir model canno capure shear deformaion. In order o express he relaions (4) in he X-Y frame, we need he following coordinae ransformaion equaion. p p f p (5) where X ( s, cos sin p ; Y ( s, sin cos X v (x,y) (x+dx,y+dy) ds+de x M+dM 73

3 X (, x( s, p ; and p f Y (, y( s, Combining (4) and (5), we have he following geomeric consrains in he X-Y frame: p ( e ) (6a,b) where X ( s) cos( ) p ; and Y( s) sin( ) A his poin, raher han deriving he explici expressions for and from (6), we show in Subsecion. ha hese wo equaions can be appended in he variaional form by muliplying wo agrange mulipliers, which resul from using more variables (five) han enough (hree).. Hamilon s Principle Wih (6), he equaion of moion of he lin can be sysemaically derived using Hamilon s principle, where he following variaional form holds: nc ( K V W ) d (7) where K and V are he ineic and poenial energy of he lin respecively; W nc is he virual wor done by nonconservaive forces; and and are wo arbirary insan of ime. Following he sandard procedure of Hamilon s principle, we firs form he oal ineic energy of he lin as K I ( ) A ( X Y ) ds (8) where I is momen of ineria of he lin; is he angle of roaion induced by bending momen; A is mass per uni lengh; and he do over he variable denoes he ime derivaive of he variable. Similarly, he poenial (srain) energy of he beam can also be expressed as V ) EI( ) GA EA( e ds (9) where A is cross-secion area and I is he momen of area; E and G are he modulus of elasiciy and he modulus of shear respecively; is he shear correcion facor; is he shear angle; and e is he axial elongaion. The s, nd, and 3 rd erm of (9) represen he srain energy due o bending, shear, and axial deformaion respecively. Equaions (8) and (9) express he ineic and poenial energy funcions in sandard quadraic forms. The nonconservaive forces applied a he lin include a prescribed roaion and an exernal force F a he origin O f. Dissipaive forces proporional o he angular velociy can also be accommodaed. As an illusraion, we use mass proporional damping model o formulae he virual wor as follows: nc T W EI (, ( F p I ds () where is he damping coefficien. Here, we muliply he geomerical consrains (6a,b) by he agrange mulipliers, h and v, respecively. Inegraing he sum of he agrange-muliplied consrains over he range [, ], he following inegral equaion idenical o zero is obained: hg vg d () Since () is equal o zero, we can subrac i from (7) so ha we have enough independen variables for he variaional procedure. By aing he dela operaor, we have nc K V W hg vg d () The resuling sysem of parial differenial equaions ha governs he dynamics of large-defleced lin can be obained using sandard manipulaions of variaional calculus [8]. We furher inroduce non-dimensional independen variable u s / [,] o replace s, for example, x(s, = x(u,. The equaions can be wrien as follows afer normalizaion. EI e I ( ) I ( ) T (3a) EA e T (3b) e ( ) T GA (3c) A p (3d,e) e p ( ) (3f,g) where h v T cos( ) ; and sin( ) Funcions h and v are agrange mulipliers corresponding o (3f) and (3g) respecively. They urn ou o be he reacion forces of an infiniesimal segmen in he X and Y direcion as shown in Fig. (b). The physical inerpreaions of each equaion in (3) are saed as follows.. Equaion (3a) is he momen balance equaion. The roaional ineria in he erm is ofen small, and can be negleced in srucural applicaions. Wihou deformaion, his equaion can be reduced o he one ha governs rigid-body roaion.. Equaion (3b) is he force balance equaion in he deformed axial direcion. Wihou he angle of roaion, i reduces o he familiar nd order differenial equaion ha governs he axial deformaion of a lin. 3. Equaion (3c) saes he shear sress-srain relaion where he shear sress comes from he reacion forces h and v. 4. Equaions (3d,e) are he resuls of applying Newon s nd law o each infiniesimal segmen direcly. 5. Equaions (3f,g) are he normalizaion of (6). They have o be solved simulaneously wih (3a)~(3e). When he complian lin governed by (3) is subjec o a prescribed roaion a O f and free a he oher end, he geomeric boundary condiions can be given as follows. (, ), e(, (4a) wih X (, and Y (, prescribed From calculus of variaion, we can deduce he naural boundary condiions of he lin from (4a). 73

4 v (,, h (,, (, T (4b) [ EAe ] u Hence we now have enough (eigh boundary condiions in order o solve (3). 3 NUMERICA APPROXIMATIONS Equaion (3) wih he boundary condiions (4) is a sysem of nonlinear hyperbolic equaions wih differenial consrain equaions. We presen here a semi-discree mehod o solve (3) and (4) numerically. Specifically, he spaial domain u is solved using shooing mehod while he emporal domain is solved wih Newmar family of inegraion schemes. For clariy of illusraion wih limied space, we focus on he model of a free-vibraing lin wih he following assumpions: (a) The origin O f is clamped a O wih x-y parallel o X-Y, i.e., =I. Hence, only frame x-y is needed o describe he lin. (b) The lin is inexensible and has no damping, e= and =. The governing equaions of a free-vibraing lin can hen be reduced from (3) o (5): EI I cos( ) sin( ) v h (5a) A x h (5b) A y v (5c) x cos( ) ; y sin( ) (5d,e) v cos( ) hsin( ) GA (5f) While developed for a vibraing lin, he exension of he numerical scheme o he complee governing equaions (3) is raher sraighforward. 3. Temporal Approximaion Moivaed by sabiliy consideraions, we use he Newmar family of ime inegraion schemes [9] for emporal discreizaion. e he posiion Z, is velociy Z, and acceleraion Z denoe he approximae soluion o z(,u), z (, u),and z (, u) a ime level and u [,] respecively. Assume he soluions of Z, Z, and Z have been obained, he Newmar mehod is an implici scheme ha finds he approximae soluion a nex ime level + according o he following formulae Z Z Z Z Z ( ) ( ) a( a a (6a) Z Z ( a) Z az (6b) where = + - denoes he ime sep size and (a, a ) are Newmar parameers ha deermine he sabiliy and accuracy of he scheme. By applying (6a), he erms involving ime derivaives in (5) can be discreized in he ime domain. Following he same convenion as above, we use capial leers o represen he approximae soluions, for example, (, u). The discreized differenial equaions a + can be wrien as follows. EI V cos A a I ( a a a H sin ) a a X X X ( ) X H A a a a X cos, sin cos H sin( ) GA Y Y Y ( ) Y V (7a) (7b) (7c) Y (7d,e) V (7f) Equaion (7) is a sysem of ime-independen differenial and algebraic equaions involving unnown funcions +,, H +, V +, X +, Y +, and +. The mehod o solve (7) will be presened in he nex secion. A he end of ime sep +, he approximae funcions (, ), ( X, X ), and ( Y, Y ) will be compued by using (6). Noe ha he calculaion of (7) requires nowledge of he iniial condiions,, ), X, X, ) and Y, Y, ). The ( ( X ( Y iniial posiions and velociies will be given and he iniial acceleraions can be obained by assuming zero applied force a = for free vibraion of a complian lin. EI, X, Y (8) I 3. Spaial Approximaion Afer emporal discreizaion, he governing equaion reduces o he nonlinear boundary value problem represened by (7). Nonlinear finie elemen mehod (NFEM) has ofen been adoped o solve he BVP numerically. However, he formulaion of NFEM is ofen complicaed. In addiion, when using FEM o solve problems wih shear deformaion, here is a numerical problem nown as shear locing [9] caused by inadmissible inerpolaion funcions. While several procedures have been made o overcome his problem, we propose in his secion an alernaive numerical formulaion called shooing mehod ha does no suffer from he problem of shear locing. The basic idea of shooing mehod is o rea boundary value problems as iniial value problems. Consider he following sysem of n differenial equaions q f ( u, q) (9) where u [,] ; q () is a (n-r)x vecor of nown iniial values; q u () is a rx vecor of unnown iniial values; and q () is a rx vecor of nown erminal values. In order o inegrae (9) as an iniial value problem, we have o mae r guesses for unnown iniial values q u (). The IVP can be inegraed numerically using MATAB ODE solver. Afer obaining he rajecories of q, her given erminal values q () have o be mached in order for he soluion q o be rue. Hence he procedure is similar o solving r nonlinear algebraic equaions excep ha he explici forms of he algebraic equaions are no nown. Ieraive procedures used for he shooing mehod can be found in [~]. 73

5 By seing q [,, H, V, X, Y ], we can recas (7a)~(7e) in he form of a nonlinear ODE coupled wih an algebraic equaion (7f). The nown iniial and erminal values of a vibraing lin are q T T () X () Y () () H () V () T T ( ) q ( ) T () Hence we have o mae iniial guesses for q u () o mach he given erminal values q (). The ODE s from (7a)~(7e) can be inegraed by Runge-Kua mehods. Noe ha (7f) is no a differenial equaion. Hence we canno incorporae i ino he ODE s. However, we sill need o now + a each R-K sep. Since (7f) is rue for he enire spaial domain, we can solve i separaely a every R-K sep j. As shown in (), he values V +, j, H +, j, +, j are nown a he j h sep. Hence () is a nonlinear equaion wih one variable +, j. I can be easily solved by bisecion mehod. Afer obaining he value of +, j, we can hen proceed he nex R-K sep j+ from (7) and ge he valuesv +, j+, H +, j+, +, j+. V, j cos, j, j () H, j sin(, j, j ) GA, j In summary, he seps for solving he vibraing lin problem are oulined as follows. Compuaional Seps:. Given (a,a ) wih iniial condiions (,, ), ( X, X, X ) and ( Y, Y, Y ),. For =~# of ime seps (I) Given iniial guesses q u, solve for q by an ieraive mehod. A each Runge-Kua sep j, solve () o obain +,j by bisecion mehod. The value +,j will be used for he nex sep j+. (II)Afer obaining q, calculae (, ), ( X, X ), and ( Y, Y ) from (6). End 4 SIMUATIONS AND EXPERIMENT VAIDATION By using he numerical schemes described in he previous secion, we simulae he free vibraion of a flexible seel rod whose governing equaions are expressed in (5). The simulaion parameers are lised in Table. The Newmar parameers (a, a ) are (.5,.5), which is nown as he consan-average acceleraion mehod and can be proved o be uncondiionally sable for any ime sep. Fig. 3 shows he ip displacemen and Fig. 4 shows he snapshos of he vibraing beam, which has a period approximaely equal o.49 seconds. Table Simulaion parameers and values for a seel rod Simulaion Parameers Values Densiy 785 g/m 3 Dimension (xwxt).x.7x.3 m Young s Modulus E GPa Shear Modulus G 8GPa (a, a ) (/, /) Time sep size. sec Iniial ip locaion (x,y) = (.77m,.46m) Iniial velociy (, X, Y ) (,,) y ip displacemen (m) y (m) ime (sec) Figure 3 Beam ip displacemen in one cycle = =.5 =. =.5 =. -.3 = x (m) Figure 4 Snapsho of a free-vibraing beam Figure 5 shows he ineic energy disribuion in one cycle. Clearly, he ineic energy is dominaed by he ranslaional energy in he y direcion, which is much larger han he roaional energy. For his reason he effec of roaional ineria has always been negleced in srucural mechanic problems. Figure 6 shows he energy disribuion beween ineic and poenial. I is worh noing ha here is no energy loss during he emporal inegraion. Energy (J).5.5 x ranslaional energy y ranslaional energy roaional energy (sec) Figure 5 Kineic energy of he vibraing beam 733

6 Energy (J) Toal energy Poenial energy Kineic energy ime (sec) Figure 6 Energy balance of he beam An experimen has also been conduced o measure he naural frequency of he seel rod whose maerial properies are lised in Table. The x direcion of he rod is parallel o he direcion of graviy so ha he effec of weigh is minimized. A proximiy sensor (Keyence EZ8T) is placed a he undefleced ip posiion such ha i is ON if he ip of he rod approaches and OFF if no. The period of vibraion can be recorded by adding wo adjacen OFF ime inervals. As shown in Fig., he period is approximaed.485 second which is very close o ha prediced by he previous simulaion resul (.49 second). Period (sec) Graviy Figure 7 Experimen seup Proximiy sensor 5 5 Couns Figure 8 Period of a free vibraing seel rod 5 CONCUSIONS A complee dynamic model, which accouns for bending, shear and axial deformaions wih no geomeric approximaion, has been presened for analyzing complian lins capable of large deflecion wih large overall moion. Specifically, we showed how he effec of shear angles on large deformaion can be incorporaed as geomeric consrains in he governing equaions derived using Hamilon s principle for predicing he ransien response of a complian lin. In addiion, we demonsraed a numerical mehod ha combines a Newmar scheme wih shooing mehod o solve he equaions, and ha he shooing mehod can avoid he shear locing- a common problem when using finie elemen procedures. Finally, an illusraive example of a free-vibraing seel rod has been given o show he applicaion of he model. The simulaion resul of he vibraing rod has mached well wih he experimen daa. In his paper we presen he dynamic model for predicing he moion of one complian lin, is exension o analyze a sysem wih muliple complian lins is been sudied. REFERENCES [] ee, K.-M., 999, On he Developmen of a Complian Grasping Mechanism for On-line Handling of ive Objecs, Par I: Analyical Model, In. Conf. on Advanced Inelligen Mecharonics Proc. (AIM 99), Alana, Sepember 9-3, pp [] Javier, G. D. J., Eduardo, B., 994, Kinemaic and Dynamic Simulaion of Mulibody sysem, Spring-Verlag. [3] Winfrey, R. C., 97, Elasic in Mechanism Dynamics, ASME Journal of Engineering for Indusry, 93, pp [4] asin, R. A., iins, P. W., ongman, R. W., 983, Dynamical Equaions of a Free-Free Beam Subjec o arge Overall Moions, The Journal of he Asronauical Sciences, 3(4), pp [5] Boo, W. J., 984, Recursive agrangian Dynamics of a Flexible Manipulaor Arm, The Inernaional Journal of Roboics Research, 3, pp [6] Simo, J.C., Vu-Quoc,., 986, On he dynamics of flexible beams under large overall moions--the plane case: Par I, ASME Journal of Applied Mechanics, Vol.53, pp [7] Simo, J.C., Vu-Quoc,., 986, On he dynamics of flexible beams under large overall moions--the plane case: Par II, ASME Journal of Applied Mechanics, Vol.53, pp [8] Shabana, A. A., 998, Dynamics of mulibody sysems, Wiley, New Yor. [9] Bahe, K.-J., 996, Finie Elemen Procedures, Prenice Hall, N.J. []Rayleigh, J. W. S., 945, Theory of Sound, Vol., Dover, New Yor. []Timosheno, S. P., 9, On he Transverse Vibraions of Bars of Uniform Cross-Secion, Philosophical Magazine, 43, pp.5-3. []Reddy, J. N., Singh, I. R., 98, arge Deflecions and arge-ampliude Free Vibraions of Sraigh and Curved Beams, Inernaional Journal for Numerical Mehods in Engineering, 7, pp [3]Monasa, F., ewis, G., 983, arge Deflecions of Poin oaded Canilevers wih Nonlinear Behavior, Journal of Applied Mahemaics and Physics, 34, pp [4]Hodges, D. H., 984, Proper Definiion of Curvaure in Nonlinear Beam Kinemaics, AIAA Journal,, pp [5]Wagner, H., 965, arge-ampliude Free Vibraions of a Beam, Journal of Applied Mechanics, 3, pp [6]Yin, Xuecheng, ee, K.-M., and an, Chao-Chieh, 4, Compuaional Models for Predicing he Defleced Shape of a Non-Uniform, Flexible Finger, IEEE Inernaional Conference on Roboics and Auomaion, 3, pp [7]Rao, B. N., Rao, G. V., 989, arge Ampliude Vibraions of a Tapered Canilever Beam, Journal of Sound and Vibraion, 7(), pp [8]Weinsoc, Rober, 974, Calculus of Variaions, Dover Publicaions Inc., N.Y. [9]Newmar, N. M., 959, A Mehod of Compuaion for Srucural Dynamics, ASCE J. of he Eng. Mechanics division, pp []Rao, B. N., Shasry, B. P., Rao, G. V., 986, arge Deflecions of a Canilever Beam Subjeced o a Tip Concenraed Raional oad, The Aeronauical Journal, 9, pp []an, C.-C., ee, K.-M., 5, Generalized Shooing Mehod for Analyzing Complian Mechanisms, IEEE Inernaional Conference on Roboics and Auomaion, Barcelona, Spain 734