The Use of Harmoic Iformatio i the Optimal Sythesis of Mechaisms A.M.CONNOR, S.S.DOUGLAS & M.J.GILMARTIN SUMMARY This paper reviews several uses of harmoic iformatio i the sythesis of mechaisms ad shows that such iformatio ca be put to eve greater use i this field. Results are preseted for both sigle ad multi-degree of freedom systems which support this claim. I both cases, the iclusio of harmoic iformatio ito the objective fuctio aids the search to locate high quality solutios.. Itroductio I the sythesis of mechaisms the accepted methods is to apply kiematic optimisatio followed by possibly dyamic optimisatio. I this approach the kiematic desig requiremets to be optimised are usually structural error ad trasmissio agle. That is the emphasis is placed o satisfyig the desired motio. A major drawback to this approach is that the resultat geometry may lead to solutios that are dyamically uattractive This paper proposes a alterative to existig methods by also icludig harmoic iformatio ito the optimisatio criteria. The reaso for this is that by through the study of the harmoic cotet a isight ito the dyamic performace of mechaisms ca be obtaied. For example compare the case were two mechaisms have bee sythesised for some desig requiremet. Oe solutio possess a substatial harmoic cotet, over the other which mechaisms should be adopted? If the mechaism with the substatial harmoic cotet is chose it is reasoable to expect that these harmoics would excite uwated vibratios. If the other mechaism was chose the the potetial for adverse vibratios to be stimulated is much reduced ad a superior dyamic performace obtaied. This paper demostrates how sigle ad multi degree of freedom mechaisms ca be sythesised to exhibit good motios via the optimisatio of harmoic spectra. 2. Literature Review Much of the previously published work which utilises harmoic iformatio has bee cocered with the aalysis of the output motios i terms of harmoic series [3]. Aother example of early work [4] developed dyamic performace criteria for crak rocker likages. The use of harmoic aalysis revealed that the higher harmoics of the acceleratios were cosiderably smaller tha the fudametal harmoic. I all cases, the method foud results where the ratio of the iput crak to the groud lik was small. Some work has icluded such iformatio i the sythesis of mechaisms. Some examples of this [5,6,7] make use of the observatio that mechaisms with short iput craks are likely to produce output motios that are relatively small. These motios ca therefore be described as approximatios of simple harmoic fuctios of the iput crak agle. This work has also bee exteded to cover multi-degree of freedom mechaisms[8]. Other approaches have bee developed which iclude harmoic iformatio as a basis for mechaism sythesis. Oe example [9] of this is the use of harmoic coefficiets to store mechaism iformatio i a library. Comparig the harmoic coefficiets of the desired motio to those i the library eables a mechaism cofiguratio which approximates the required motio to be selected. This ca the be umerically optimised to reduce the error betwee the actual ad the desired output. A.M.Coor, Egieerig Desig Cetre, School of Mechaical Egieerig, Uiversity of Bath, Bath, BA2 7AY. +44 ()225 8263, esamc@bath.ac.uk, http://www2.bath.ac.uk/~esamc/
3. The Fourier Trasform Whilst the results preseted i later sectios are for differet applicatios, ad harmoic iformatio is used i differet ways, both methods utilise a discrete Fourier trasform (DFT). Essetially, the DFT represets time domai iformatio as a complex series. The DFT is derived from the aalytical Fourier trasform. The oly differece betwee the DFT ad aalytical techiques is that the coefficiets relatig to the Fourier series are calculated by umerical itegratio. The required series is deoted by; f x a a cosxb si x 2 The idividual coefficiets are calculated usig the followig expressios. a a b 2 2 2 f ( x) dx f ( x) cosxdx f ( x) si xdx Each harmoic, other tha the fudametal harmoic, cosists of a real ad imagiary part. The magitude of each harmoic is give as; 2 2 H a b Similarly, the phase of each harmoic is give as; ( H ) ta b a The importace of the magitude ad phase of each harmoic ca be realised by cosiderig a hypothetical series. Each term i the series represets a siusoid i the time domai. Addig the total umber of siusoids, takig ito accout phase differece, usig the priciple of superpositio results i the time domai fuctio. 4. Sigle Degree of Freedom Systems The origis of this work was a experimetal study of idustrial sewig mechaisms. This revealed that the mechaisms sigificat harmoic cotet was causig uacceptable stitch patters at high speeds. The results preseted demostrate how harmoic sythesis ideas have bee applied to sigle degree of freedom idustrial sewig mechaisms. I both cases, the optimisatio method was a hill climbig techique. usig Powells method of cojugate directios [] ad a scaled lagragia pealty fuctio []. 4. Straight Lie Needle Mechaism This mechaism cosists of a four bar crak rocker coupled to a slider crak. The crak rotatio provides the iput motio ad the slider oscillatio provides the output motio. The kiematics objective of this desig was to determie lik geometry such that a desired traslatio of the eedle is produced for a fully rotatioal crak iput. To this objective the desig requiremet that the solutio possess smooth motio ad good trasmissio agle characteristics was added. This became the optimisatio criteria. The output traslatio 2
requiremets became the equality costraits ad the mechaism feasibility was moitored by the iequality costraits. Figure shows the trial mechaism geometry. Figure : Trial Mechaism Nie desig parameters were selected to be freed for the optimisatio. Figure 2 displays the two limit positios of the optimised solutio. It ca be see that the mechaism strokes betwee the specified equality costraits (show as diamods) ad also that good trasmissio agles have bee determied. Figure 2 : Optimised Solutio The table i Figure 3 shows the harmoic cotributios to the output motio. It ca be see that motio cosists priciplely of the fudametal harmoic. This clearly idicates that the optimisatio criteria which pealises the square of the secod to fifth harmoics has bee successful. 4.2 Right Had Looper Mechaism Order Magitude Phase Agle 2.2546 86.782 2.3852 2.837 3.38E- 295.258 4.236E-2 227.743 5.5629E-4 87.6564 Figure 3 : Harmoic Iformatio I this sythesis the objective was to desig a spatial - plaar mechaism. The spatial (RRRSR) mechaism coverted a cotiuously rotatig crak iput ito a orthogoal output oscillatio. The plaar mechaism desig requiremet was to geerate a coupler curve. The sythesis problem was partitioed ito two related optimisatio strategies. I the first the kiematics objective of the desig was to determie a four bar coupler curve mechaism which gave good approximatio to a specified segmet of a coupler curve. For this desig optimisatio criterio comprisig coupler curve structural error ad smooth output motio ad good trasmissio agles was selected. As the objective fuctio was multi fuctioal each criteria was weighted to eable a rage of searches to be udertake. I this sythesis formulatio the iput oscillatio is ukow ad iequality costraits were costructed to keep the mechaism feasible over its rage of operatio. Oce a optimal solutio has bee foud the derived iput oscillatio became the equality costraits for the spatial mechaism optimisatio. The objective criteria of the spatial drive was agai for smooth motio o the output oscillatig lik ad also 3
good spatial trasmissio agle o the output dyad. The iequality costraits have bee costructed to keep the solutio feasible. Figure 4 shows a trial mechaism ad the boudary defiig the feasible zoe for the coupler curve. The vertical dashed lie represets the output path of the first optimisatio ad the diamods represet the desired coupler curve. Figure 4 : Trial Mechaism I this sythesis because the iput oscillatio is ukow a iterative techique is applied to determie the iput agular throw which positios the trial mechaism as close as possible to the desired coupler curve. Figure 5 shows the optimal solutio obtaied whe six mechaism parameters were freed. I this optimisatio the relative weightig betwee the structural error ad the harmoic ad trasmissio fuctios was oe hudred to oe. Figure 5 : Optimised Solutio Figure 6 shows the optimised spatial drive. I this optimisatio the spatial trasmissio agle ad the smooth harmoic criteria were equally weighted. Figure 6 : Optimised Spatial Mechaism 5. Multi-Degree of Freedom Systems The results preseted i this sectio illustrate how harmoic iformatio ca be used i the sythesis of multi-degree of freedom mechaisms. The particular applicatio is that of a hybrid five bar path geeratig mechaism [2]. Hybrid mechaisms have bee defied [3,4,5] as a multi-degree of freedom mechaism, where differet motor types provide the iputs required. All research i this area to date cocers mechaisms with two degrees of freedom, where oe iput is provided by a costat velocity (CV) motor ad the other iput is provided by a programmable servo motor. The desired optimal coditio for such cofiguratios is cosidered to be whe the CV motor provides the bulk of the motio. Whe this occurs, the 4
size of the servo motor is small ad there is large potetial for eergy regeeratio o the CV motor axis by the iclusio of a flywheel. Previous work [5] has utilised objective fuctio criteria based o either torque requiremet, or power cosumptio, ad distributio. Such objective fuctios are computatioally expesive. This work attempts to replace such dyamic criteria with a approximatio which utilises frequecy domai iformatio. Miimisig the magitude of high order (higher frequecy) harmoics of the servo motor profile is likely to lead to high quality solutios for the followig reasos. As the magitude of low orders is reduced, the motio profile treds towards simple harmoic motio. Simple harmoic motio is ifiitely differetiable without itroducig discotiuities i higher derivatives. Therefore, a motor profile with low order harmoics is likely to exhibit desirable velocity, acceleratio ad torque profiles. The search method used i this istace is based aroud a Geetic Algorithm [7]. GAs are a ovel search ad optimisatio method based o atural selectio, populatio geetics ad the priciple of survival of the fittest. This method has bee show to be readily applicable to mechaism sythesis problems [2,2,8,9]. 5. Five Bar Mechaisms A fivebar mechaism is a two degree of freedom mechaism which requires two iputs to fully defie the mechaism motio. Figure 7 illustrates such a mechaism. r q s y p t x Figure 7 : Five Bar Mechaism I a hybrid mechaism of this type, oe iput is provided by a programmable servo motor whilst the other is provided by a CV motor. I Figure 7, is cosidered to be the CV motor iput whilst 5 is the servo motor iput. Lik t is the groud lik ad all motios are expressed relative to the x-axis.. However, if the sythesis is to search for the mechaism lik legths which result i servo motor profile beig the most desirable, the obviously the motor motio profile is ot kow. It is therefore ecessary to specify alterative iputs for use i the aalysis routie embedded i the sythesis method. This is achieved by cosiderig the ature of the mechaism ad relatig this to the desired output motio of the mechaism. For the sake of the aalysis ad sythesis of the five bar mechaism, it is assumed that the ed effector is the revolute joit betwee liks q ad r. The iput motio for lik p is kow, ad so by defiig the actual positio of the ed effector it is possible to fully describe the motios i the mechaism. 5.2 Output Motio Error Figure 8 illustrates how the positios of liks p ad q ca be used to defie the required agles of the mechaism so that it may be aalysed fully ad also calculate a error score. 5
Desired Actual Error q 3 p 2 Figure 8 : Calculatio of Error A vector is defied from the ed of lik p to the desired positio of the ed effector for the give value of 2. The actual positio of this poit is the calculated, give the legth of lik q, ad the error betwee the two poits foud. This error is summed aroud the cycle for the iput lik ad forms the basis of oe compoet of the objective fuctio. objerr 2 error 5.3 Mechaism Mobility The two liks, p ad q, form a dyad which ca be used to evaluate the error at a give poit for a give iput agle. Similarly, the remaiig two mobile liks, r ad s, form aother dyad which is used to calculate a pealty fuctio criteria for use i the overall objective fuctio. This pealty fuctio is based o the mobility of the mechaism. For a truly mobile mechaism, the dyad formed by the liks r ad s should be able to close for all give positios of the CV iput crak. This meas that the positio of the commo revolute joit, whe cosidered as part of the dyad formed by r ad s, should be able to reach the actual positio defied by the dyad formed by the liks p ad q. For each positio that this dyad caot close, the mobility couter is icreased. It is importat to realise that for each positio of the iput lik, there are two possible closures of the dyad. This is illustrated i Figure 9. q r p s t Figure 9 : Multiple Closures of the Mechaism The implicatio of this, at this stage, is that for each positio of the CV iput lik the mobility score may icrease by two. The mobility pealty score is summed throughout the cycle ad, for twety four precisio poits, may rage betwee zero ad forty eight. As with the error score value, it has bee foud that raisig the mobility pealty to a power acts so that the search is directed more towards feasible solutios. The followig expressio is used to calculate the cotributio of the mobility pealty fuctio to the overall objective fuctio. obj mob mobility 3 Oce two iputs to the mechaism are kow, i this case 2 ad 3, it is possible to calculate ay other agle i the mechaism. I this case, 5 is calculated usig the equatios derived i sectio 4.3. The values for 5 aroud the cycle of the mechaism ca be used to calculate a umber of figures of merit which describe the quality of the motio profile. However, the situatio is complicated by the presece of 6
multiple closures i the mechaism. For each iput positio, two values for 5 ca be calculated ad the questio which arises is which value should be chose?. A algorithm has bee developed to choose the set of servo motor displacemets for a give mechaism. The algorithm is based o the fact that it is desirable to have smooth velocity profiles for the servo motor iput. 5.4 Closure Trackig Algorithm The algorithm miimises the fluctuatio i servo motor velocity i the followig way. Give that the mechaism is assumed to start i the ope closure, the algorithm chooses the ext displacemet agle (from a choice of two) based o the magitude of the differece i displacemet betwee each possible agle ad the startig agle. For each subsequet poit, the algorithm chooses a displacemet by cosiderig the differece i magitude ad directio betwee each possible velocity for the curret stage ad kow velocity of the previous stage. The effect of this is that the at each step of the cycle, the closure is chose so that the tred i displacemets cotiues with the smallest chage i displacemet. The tred is oly broke if the agle i the ext step produces a step chage may times larger tha the alterative which results i a chage of directio. 5.5 Assessmet of Quality Give that the servo motor displacemet is ow kow, it is possible to calculate a figure of merit which describes the quality of the motio profile. Previous work [2] has show that the use of oly harmoic cotet does ot force the search towards dyamically optimal mechaisms. This is because the method locates a mechaism where the closig dyad (lik r ad s) has log liks whe compared to the CV iput crak. This occurs because such log liks produce servo motor motio profiles where the a large cartesia displacemet is produced by a small agular displacemet. Because of this the fudametal harmoic, which describes the magitude of the displacemets i the time domai, becomes sigificat whe compared to the values of low order harmoics. To overcome this, a secod figure of merit has bee developed which works i cojuctio with the harmoic cotet. These two desig objectives ca ow be described i more detail. 5.5. Harmoic Cotet Oe of the desig objectives used is based o the magitude of the harmoics of the motor displacemet profile. The cotributio to the overall objective fuctio is calculated by summig the value of each harmoic raised the power of its order plus oe. This effectively pealises low order harmoics with high magitudes. obj a b harm 2 2 i 5.5.2 Motio Swept Area Usig oly the harmoic cotet of the motio profile as a desig objective does ot lead to approximate dyamically optimum mechaisms. This is because the method is proe to fidig solutios that have log liks i the closig dyad which require greater torque to move them. Whilst short iput liks are desirable, it is also importat to take ito accout the motio profile i time domai form. This is achieved by calculatig the swept area of the servo motor iput. The swept area is defied as beig the RMS of the motor displacemet multiplied by the legth of lik s. obj swept s 2 5 i i 7
5.6 Problem Represetatio The required motio is defied by twety four discrete poits. The curve defied by this poits exhibits two cusps, or poits of zero velocity, ad so may be for a applicatio such as a pick ad place mechaism. The parameters used i the search for the optimal mechaism are the legths of liks p,q,r ad s. The legth of lik t is defied by icludig i the search the co-ordiates of the groud iputs. These are defied as local (x,y) co-ordiates with i a global costrait evelope defied by the user. The search was termiated whe the structural error dropped below a give threshold, but oly if the lik legths of the located mechaism were such that o lik was more tha five times loger tha the legth of the CV motor iput crak. 5.7 Results Five trial rus of the problem were carried out. I each case a very rapid iitial reductio i objective fuctio value was followed by a slow, but defiite, improvemet. Each of the solutios foud was differet due to the probabilistic ature of the GA used as the search algorithm. However, each solutio had similar objective fuctio values. The best solutio was obtaied i the fifth ru ad had the followig parameter set. (x,y) cv =, (x,y) servo = 27,-2 p = 6 q = 24 r = 3 s = 6 The graph i Figure shows the required servo motor iput motio. 2 Servo Iput 8 6 4 2 3 6 9 2 5 8 2 24 27 3 33 36 CV Iput Figure : Servo Motor Displacemet Figure shows the actual output motio of the ed effector i relatio to the desired motio. 3 25 2 5 5 Actual Desired - 2 3 Figure : Ed Effector Motio 8
The graph i Figure 2 shows the torque requiremet for the mechaism. The mechaism ca be see to be operatig as a valid hybrid device as the torque requiremet of the servo motor is lower tha that for the CV motor. Torque (Nm).8.6.4.2 -.2 -.4 -.6 -.8 CV Iput Torque (CV) Torque (Servo) 3 6 9 2 5 8 2 24 27 3 33 36 Figure 2 : Mechaism Torque Requiremets 6. Discussio The results preseted i the previous sectios show that the use of harmoic iformatio i the sythesis of mechaisms ca force the search method to fid high quality solutios. I the sigle degree of freedom examples, reducig the harmoic cotet of the output motio allowed higher operatig speeds to be achieved. I the multi-degree of freedom example, itroducig harmoic iformatio has elimiated the ecessity for computatioally expesive dyamic optimisatio criteria ad has produced a mechaism which exhibits a very desirable torque distributio betwee the two axes of cotrol. 7. Coclusios The results preseted show the usefuless of harmoic iformatio i the sythesis of mechaisms. This iformatio has bee applied i two examples to show how improved dyamic performace ca be achieved. 8. Refereces [] Douglas, S.S. (98) Desig of a High Speed Sewig Machie, PhD Thesis, Liverpool Polytechic [2] Coor, A.M. (996) The Sythesis of Hybrid Mechaisms Usig Geetic Algorithms, PhD Thesis, Liverpool Joh Moores Uiversity [3] Freudestei, F. (959) Harmoic Aalysis of Crak ad Rocker Mechaisms with Applicatios, ASME Joural of Applied Mechaics, Vol 26, [4] Fuabashi & Freudestei, F. (979)Source to be cofirmed [5] Farhag, K., Midha, A. & Cipra, R.J. (985) Some Sythesis Problems i Harmoic Motio Geeratig Likages, Proceedigs of the 9th OSU Applied Mechaisms Coferece, Vol [6] Farhag, K., Midha, A. & Bajaj, A.K. (988) Sythesis of Harmoic Motio Geeratig Likages - Part I : Fuctio Geeratio, ASME Joural of Mechaisms, Trasmissios ad Automatio i Desig, Vol., No. [7] Farhag, K., Midha, A. & Hall, A.S. (988)Sythesis of Harmoic Motio Geeratig Likages - Part II : Path ad Motio Geeratio, ASME Joural of Mechaisms, Trasmissios ad Automatio i Desig, Vol., No. [8] Basu, P.S. & Farhag, K. (992)Kiematic Aalysis ad Desig of Two Iput Five Bar Mechaisms Drive by Relatively Small Craks, ASME 22d Bieial Mechaisms Coferece, Vol. DE v47 [9] McGarva, J.R. (994) Rapid Search ad Selectio of Path Geeratig Mechaisms from a Library, Mechaisms ad Machies Theory, Vol. 29, No. 2 [] Powell, M.J. (964) A Efficiet Method for Fidig the Miimum of a Fuctio of several variables without calculatig Derivatives, Computer Joural, Vol 7 9
[] Schuldt, P. & Gabrielle, G. (977) Applicatio of a New Pealty Fuctio Method, Joural of Egieerig for Idustry [2] Coor, A.M., Douglas, S.S. & Gilmarti, M.J. (995) The Sythesis of Hybrid Five Bar Path Geeratig Mechaisms Usig Geetic Algorithms, Proceedigs of the st IEE/IEEE Coferece o Geetic Algorithms i Egieerig Systems : Iovatios ad Applicatios [3] Tokuz, L.C. (992) Hybrid Machie Modellig ad Cotrol, PhD Thesis, Liverpool Joh Moores Uiversity [4] Bradshaw, W.K. (993) Cotrol of Hybrid Machies, Iteral Research Report, Liverpool Joh Moores Uiversity [5] Greeough, J.D. (993) Desig of Hybrid Machies, Iteral Research Report, Liverpool Joh Moores Uiversity [6] Goldberg, D.E. (987)Geetic Algorithms i Search, Optimisatio ad Machie Learig, Addiso-Wesley [7] Coor, A.M. (994) The Use of Geetic Algorithms i Optimisatio, MSc Dissertatio, Liverpool Joh Moores Uiversity [8] Coor, A.M., Douglas, S.S. & Gilmarti, M.J. (995) The Sythesis of Path Geeratig Mechaisms Usig Geetic Algorithms, Proceedigs of the th Iteratioal Coferece of Applicatios of Artificial Itelligece i Egieerig