Optimum Synthesis of Mechanisms For Path Generation Using a New Curvature Based Deflection Based Objective Function

Transcription

1 Proceedngs of the 6th WSEAS Internatonal Conference on Smulaton, Modellng and Optmzaton, Lsbon, Portugal, September -4, 6 67 Optmum Synthess of Mechansms For Path Generaton Usng a ew Curvature Based Deflecton Based Obectve Functon SOHEIL DAMAGI School of Mechancal Engneerng Sharf Unversty of Technology, Tehran IA daamangr@mech.sharf.edu GHAZALEH JAFAIJASHEMI TAYLO'S COLLEGE (subang aya) Taylor s College, Kuala Lumpur MALAYSIA ghazaleh.afar@gmal.com MOHAMMADHOSSEI MAMDUHI School of Mechancal Engneerng Sharf Unversty of technology, Tehran IA mamduh_m@mech.sharf.r HASSA ZOHOO Fellow, the Academy of Scences of I.. Iran. Member, Center of Excellence n Desgn, obotcs, and Automaton Sharf Unversty of Technology, Tehran IA zohoor@sharf.edu Abstract: - Ths paper proposed a new curvature based path-descrpton method for path-generaton of planar mechansm, n whch the poston vector of a pont on coupler, whch follows most closely a predefned path, s drectly calculated as a functon of coupler moton propertes. In addton ths curvature based method makes t possble to defne an obectve functon ndependent of rotaton and translaton transformatons. Therefore, n ths method, ntal poston, orentaton, and coupler pont vector parameters are not the optmzaton varable. These fve parameters could be calculated as a functon of other desgn parameters and they are found when these optmal values are obtaned. educng fve parameters of the optmal varables leads to a smaller desgn space and so less computaton tme to fnd the optmal result. Applcaton of the method s also shown and ts computaton tme and accuracy are compared wth the result reported n the lterature. It s found that ths method s superor as t converges faster wthout loosng any accuracy. Key-Words: - Mechansm synthess; Optmzaton; Path Generaton; Curvature based obectve functon; Deflecton based obectve functon; Introducton Path generaton s one of the mportant functonal requrements n mechancal engneerng desgn. The use of optmum synthess s nevtable whenever the mechansm needs to follows a number of postons exceedng a certan number, 6 ponts for path generaton of a planar four-bar lnkage. An obectve functon s generally defned as a functon of desgn parameters, whch t s then optmzed to fnd the optmal values of these parameters. For any optmzaton method, the calculaton tme depends on the number of desgn parameters. The lower the number of parameter, the faster s the calculaton.

2 Proceedngs of the 6th WSEAS Internatonal Conference on Smulaton, Modellng and Optmzaton, Lsbon, Portugal, September -4, 6 67 To reduce the calculaton tme, researchers try to reduce the number of the desgn parameters. To do so, they use dfferent mathematcal method to obtan the path characterstcs to measure the goodness of matchng between the desred and generated path. Ther nterest was manly n removng the rotaton, translaton and scale desgn parameters. Ullah et al[7]. used Fourer Descrptors and compare the ampltude of characterstc spectrum of generated and requred path as a measure of ther matchng goodness. In ths method the ponts on the curve need to be sampled n the constant tme ntervals. Yong Lu et al.[] resolved ths pont by employng efned umercal epresentaton. Other researchers have tred dfferent optmzaton methods n the synthess of rgd body mechansms. olle et al.[4] used gradent and least square method; Kunur et al[]. used genetc algorthm; Lu et al[] appled artfcal mmune system; and Smal et al.[] employed tabu-gradent-search to fnd optmum soluton to the mechansm synthess. Ths paper presents a curvature based path descrpton method to defne a new obectve functon whch s ndependent of the vector poston of coupler pont, translaton and rotaton translatons parameters. To the best of the authors knowledge ths s the frst attempt at usng such error functon to be ndependent of vector poston of the coupler pont. Here the basc dea of the method s ntroduced. Two llustratve examples are provded. In the frst example the result s compared wth what s reported by Smal et al.[]. In the second example the valdty of the assumpton s dscussed. Formulaton The man dea of ths artcle s to employ the curvature to defne the path, whch enable us to ntroduce a new error functon wth fewer number of desgn parameters. The curvature of a path s defned as: r r p& && p κ = p& () n p&,&& p are the velocty and acceleraton of the movng pont p tracng the path. When p s a pont on a rgd body ts poston vector, wth respect to the base, s obtaned as a functon of the poston of another pont f on the body, ts poston respect to f and the orentaton of the body (fg.): ( ) p= f t + e () eplacng f as a functon of ρ andα, the velocty and acceleraton vector of p are obtaned as follows: α p= ρe + e α p& = ( & ρ+ & αρ) e + & e ( ) α && p= && ρ+ && αρ+ && αρ & α ρ e + && e & e () ow we defne r = e θ, where θ = α (fg.), and ntroduce p& and p&& nto eq. we have: Fg. Parameters to defne a pont X κ = Y (4) X = r ρ( ρ+ ρ ) + r ( ρ ρ ρ ) snθ ( ρ + ρ+ ( ρ ρ )) + " cosθ Y = r r ( ρ snθ ρ cosθ) + ρ + ρ α&& was also assumed to be zero wthout loosng any generalty. Ths equaton could be wrtten as follows: 6 5 κ B κ B B κ κ 4 ( B B + B B) A ( B + 9BB B) A A ( ) ( ) κ B B B B A A A + + κ B B A A r+ κ B A = (5) A = ρ ρ+ ρ ( ) ( ρ ρ ρ ) snθ ( ρ ρ ( ρ ρ )) A = A = " cosθ B = ρ + ρ B = ( ρ snθ ρ cosθ) B = Supposng that p s the pont on coupler that follows the desred path wth curvature equal toκ at that pont, the above equaton could be employed to obtan r, when the knematcs characterstcs of the

3 Proceedngs of the 6th WSEAS Internatonal Conference on Smulaton, Modellng and Optmzaton, Lsbon, Portugal, September -4, Set n=n+ o? Select the dmensons of mechansm Calculate knematcs propertes of coupler Calculate (umercally) F Problem defnton and ts constrans Calculate the curvature of the path Set n= Evaluate n n the neghborhood of n- Select a value for Is n=? Yes? ( ) Imag + = = Aabs( ) resultng r s found as a functon of and α, and therefore s calculated from = re θ. Thus for any value of, a set of values are obtaned. A semanalytcal procedure s used to fnd from the above relatons. At the frst step s found numercally from eq.5 and analytcal method s used to fnd the subsequent values of. For all mechansms that generate path around to the desred one, s weakly vared wthα. If n s the answer α for n, n + s obtaned analytcally as a functon of α n + n the neghborhood of n. For ponts on the desred path, values for are calculated by the above procedure, but must be real and remans constant as the coupler s a rgd body wthout any deformaton. Therefore we defne the error functon as: F = = ( ) ( ) = Imag + Aabs and = = I and A are weghtng factors here. Ths error functon s an ndcaton of the angular and longtudnal deformatons of the coupler and ts mnmum correspond to the best mechansm that generates a path most close to the desred one. Any optmzaton technque could be use to fnd the optmum answer. The calculaton procedure to obtan the desgn varables ncludng s shown n fg.. Meanwhle, numercal experment of ths procedure shows that the mnmum value of error corresponds approxmately to the pont wth maxmum value Fnal soluton Calculate obectve functon o? Stoppng crtera met? Yes? Fg. Flowchart for Curvature base obectve functon solve algorthm = dα+ coupler are known. Lettng, the Fg. Desgn parameter needed to defne a four bar mechansm

4 Proceedngs of the 6th WSEAS Internatonal Conference on Smulaton, Modellng and Optmzaton, Lsbon, Portugal, September -4, xd yd.5.5 α Table Desred traectory ponts of. Thus the value of s calculated such that the mnmum value of error occurs when the value of s maxmum. To smplfy the calculaton, s obtaned by mnmzng the magnary part of at the pont wth maxmum value of, as ths part s domnant at ths condton. Ths value s desgnated. as Examples Method s coded n MATLAB and appled to the optmum synthess of path generatng planar four bar mechansms. Ten desgn parameters generally nvolve descrbng these mechansms as shown n fg.. In the abovementoned procedure only fve parameters, namely L, L, L, L4 and are needed to fnd the soluton. Two examples are ntroduced. In the frst example the result of ths method s compared wth the result of Smal et al[]. In the second one an approxmaton procedure to estmate s studed. In these two examples the weghtng factors I and A put equal to and the Genetc Algorthm appled to fnd the optmum soluton. Example. The planar four bar mechansm that follows prescrbe path n coordnate wth crank rotaton s desgned. The desred ponts are gven n table. The result of ths method s shown n table and fg. 4and compared wth the result of Smal et al[]. 499 teratons are needed to fnd the optmum result that took s run tme on a Pentum IV PC. Smal et al. ntroduce ths example to compare tabu-search and the tabu-gradent algorthms. It was reported that 8 teratons wth 7 s run tme on Pentum IV are needed. Ths ndcates that the method proposed here s faster, ever thought, as reported n [], the tabu-gradent-search clamed to be the fastest optmzaton method n mechansm synthess, and converges much faster than genetc algorthm. Example. In ths example we compare the result of the Table Classc obectve functon TS soluton and curvatre based obectve functon GA soluton Classc O.F. TS C.B. GA L L L L Fg. 4 Desred coupler ponts and coupler curve generated wth and wthout assumng = Fg. 5 Coupler curve generated by Classc obectve functon, TS and Curvature based obectve functon, GA solutons optmal mechansm optmzaton, wth and wthout lettng = dscussed n the prevous secton. Heren the 4-pont path s provded. Consderng as one of the optmzaton parameters the calculaton tme was 5s wth 55 teratons. Meanwhle, the optmzaton needs about 89 teratons and takes s of run tme, by lettng =. Fg.5 compares the results of these optmzatons. Ths example ndcates that assumng = the method could be made much faster but accuracy s slghtly lower. 4 Concluson Ths paper presents an error functon whch could measure the goodness of matchng between the predefned curve and generated curve, as a functon of coupler moton propertes but ndependent of vector poston of coupler pont. Ths error functon defned on the bass of curvature of the desred and generated curve. As a result, the mechansm, and so the desgn parameters, wll be ndependent of rotaton and translaton. The method s appled to four bar planar mechansm and showed that t needs lesser calculaton tme. Ths s acheved due to lower number of desgn

5 Proceedngs of the 6th WSEAS Internatonal Conference on Smulaton, Modellng and Optmzaton, Lsbon, Portugal, September -4, parameters needed to defne the obectve functon. Although the llustratve examples were four bar planar mechansm, the obectve functon s constructed so that the algorthm can be easly appled to other path generaton mechansms. 5 eference Ahmad A. Smal, adm A. Dab, and a A. Atallah, 5, Optmum Synthess of Mechansms Usng Tabu-Gradent Search Algorthm, ASME J. Mech. Des. 7, pp 97-9 Irfan, Ullah, and Srdhar, Kota, 997, Optmal Synthess of Mechansms for Path Generaton Usng Fourer Descrptors and Global Search mthods, ASME J. Mech. Des., 9, pp Kunur, A., and Krshnamurty, S., 997, Genetc Algorthms n Mechancal Synthess, J. Appl. Mech., 4, pp olle, H., and Hunt, K., 97, Optmum Synthess of Planar Lnkages to Generate Coupler curves, J. Mech., 6, pp Yao, J., and Angeles, J.,, Computaton of All Optmum Dyads n the Approxmate Synthess of Planar Lnkages for gd-body Gudance, Mech. Mach. Theory, 5_8_, pp Sancbran,., Vadero, F., Garcoa, P., and Fernandez, A., 4, Gradentbased optmzaton of path synthess problems n planar mechansms, Mech. Mach. Theory, 9 _to be publshed_. Ulah, I., and Kota, S., 997, Optmal Synthess of Mechansms for Path Generaton Usng Fourer Descrptors and Global Search Methods, ASME J. Mech. Des., 9, pp Martnez-Alfaro, H., Valdez, H., and Ortega, J., 998, Lnkage Synthess of a Four-Bar Mechansm for n-precson Ponts Usng Smulated Annealng, DETC98/ MECH 594, Proc. of the 998 ASME Desgn Engneerng Techncal Conferences, Atlanta, Georga, USA. Kunur, S., and Krshnamurty, S., 997, Genetc Algorthms n Mechansm Synthess, Journal of Appled Mechansms and obotcs, 4, pp Camuto, M., and Knzel, G., 998, Path Generaton Usng a andom Walk Algorthm, DETC98/MECH 599, Proc. Of the 998 ASME Desgn Engneerng Techncal conferences, Atlanta, Georga, USA. Lu, Y., and Xao,., 5, Optmal Synthess of Mechansms for Path Generaton Usng efned umercal epresentaton Based Model and AIS Based Searchng Method, ASME J. Mech. Des.