Kinematic Synthesis of Planar, Shape-Changing Rigid-Body Mechanisms

Transcription

1 Adrew P. Murray Mechaical ad Aerospace Egieerig, Uiversity of Dayto, Dayto, OH James P. Schmiedeler Bria M. Korte Mechaical Egieerig, The Ohio State Uiversity, Columbus, OH Kiematic Sythesis of Plaar, Shape-Chagig Rigid-Body Mechaisms This paper presets a kiematic procedure to sythesize plaar mechaisms, composed of rigid liks ad revolute joits, capable of approximatig a shape chage defied by a set of curves. These morphig curves, referred to as desig profiles, differ from each other by a combiatio of rigid-body displacemet ad shape chage. Desig profiles are coverted to piecewise liear curves, referred to as target profiles, that ca be readily maipulated. I the segmetatio phase, the geometry of rigid liks that approximate the shapes of correspodig segmets from each target profile is determied. I the mechaizatio phase, these rigid liks are joied together at their ed poits with revolute joits to form a sigle chai. Dyads are the added to reduce the umber of degrees of freedom (DOF s) to ay desired value, typically 1. The approach ca be applied to ay umber of desig profiles that ca be approximated with ay umber of rigid liks, which ca the be used to costruct a mechaism with ay umber of DOF s. Naturally, greater difficulty is ecoutered for larger umbers of desig profiles ad/or liks ad for more dramatic chages i shape. The procedure is demostrated with examples of sigle-dof mechaisms approximatig shape chages betwee two ad three desig profiles. DOI: / Itroductio For a mechaical system whose fuctio depeds o its geometric shape, the ability to alter that shape i a cotrolled maer ca ehace performace ad expad applicatios. Example systems with such capabilities, ofte kow as adaptive or morphig structures, iclude morphig airfoils 1, active aperture ateas 2, ad deformable mirrors for adaptive optics systems 3. A sigificat amout of research has bee devoted to actuatig adaptive structures usig smart materials techology. Morphig airfoils have bee proposed with magetostrictive 4,5, piezoelectric 6, ad shape memory alloy 7 11 actuators. Piezoelectric actuators have bee proposed for atea reflectors ad bimorph mirrors 15. Compliat mechaisms provide a meas of structurally imposig a shape chage 16 regardless of the type of actuatio. Saggere ad Kota 17 developed a sythesis procedure for compliat four-bars that guide their flexible couplers through a sequece of discrete prescribed precisio shapes that ivolve both shape chage ad rigid-body displacemet. Lu ad Kota 18 itroduced ad the refied 19 a more geeral approach usig fiite elemet aalysis ad a geetic algorithm to determie a optimized compliat mechaism s topology ad dimesios give the target profiles, support locatios, exteral loads, actuator types ad locatios, desig domai, ad material properties. Stubbs et al. 20 desiged a sigle-degree-of-freedom DOF, rigid-body mechaism that could morph betwee two desired airfoil shapes. The sythesis techique applies specifically to a quaterary-biary cross-liked mechaism that was selected based o the uique costraits of the specific morphig wig problem. Erdma et al. 21 show diagrams of two shape-chagig rigidbody mechaisms: a tire cotractor mechaism p. 94 that collapses to approximate circles of two differet radii ad a Watt Cotributed by the Mechaisms ad Robotics Committee of ASME for publicatioithejournal OF MECHANICAL DESIGN. Mauscript received July 21, 2006; fial mauscript received July 13, 2007; published olie February 4, Review coducted by Thomas R. Chase. Paper preseted at the ASME 2006 Desig Egieerig Techical Cofereces ad Computers ad Iformatio i Egieerig Coferece DETC2006, Philadelphia, PA, September 10 13, six-bar air spoiler p. 75 i which just two liks provide a shape chage ad sythesis ivolves solvig two separate motio geeratio problems i sequece. Rigid-body mechaisms offer a umber of advatages for the practical implemetatio of adaptive structures. They provide a higher load-carryig capacity tha compliat mechaisms ad would likely require fewer actuators actig i parallel, such as alog the legth of a airfoil with chagig camber, to resist exteral loads. With rigid liks, sythesis ca be primarily kiematic, so the system ca be modeled without a priori kowledge of exact exteral loads. Also, rigid-body mechaisms ca typically achieve larger displacemets, eablig more dramatic shape chages. Furthermore, actuatio is ot a additioal developmet eed because existig techology rather tha, for example, smart materials techology is typically used to actuate rigid-body mechaisms. The primary disadvatage is that a large umber of rigid bodies is required to approximate a complex shape chage, which ca lead to sigificat frictio accumulated i the correspodigly large umber of joits, desig challeges to avoid mechaism sigularities, ad poor force trasmissio. This paper presets a methodology for the kiematic desig of plaar mechaisms, cosistig of rigid liks joied together by revolute joits, that ca approximate a desired chage betwee a arbitrary umber of shapes defied by curves termed desig profiles. The three curves i Fig. 1 a are example desig profiles. The basic desig strategy is to divide the desig profiles ito segmets such that oe edge of a sigle rigid body ca approximate the shapes of the correspodig segmets from all of the desig profiles. Sectio 3 explais how the segmetatio is accomplished, ad Sec. 2 provides the required mathematical defiitios ad operatios. The rigid bodies correspodig to adjacet segmets are joited together at their ed poits with revolutes to form a chai that ca be maipulated to approximate each desig profile. Figure 1 b shows a chai of four rigid bodies positioed to approximate each of the desig profiles i Fig. 1 a. Each rigid body is just a curve at this stage, which costitutes oe edge of a rigid body i the completed mechaism. Mechaizatio, the, as described i Sec. 4, ivolves defiig the full geometry of these rigid bodies ad addig more liks so that the motio of the coected set of edges ca be cotrolled to Joural of Mechaical Desig Copyright 2008 by ASME MARCH 2008, Vol. 130 /

2 Fig. 1 a Desig profiles that defie a shape chage. b Chai of four rigid segmets joited together show relative to each desig profile. c Solutio sigle-dof mechaism sythesized from the chai. pass through all of the desig profiles. I Fig. 1 b, the three positios of each segmet defie a three-precisio-positio motio geeratio problem that ca be solved with traditioal techiques. I Fig. 1 c, five liks were added betwee the existig chai ad the groud via such techiques to yield a sigle-dof mechaism. This same example is explaied i more detail i Sec. 5, ad Sec. 6 briefly describes a secod example of a mechaism that chages from a shape approximatig the letter U to oe approximatig the letter D. Coclusios are preseted i Sec Rigid Lik Geometry The procedure for geeratig rigid liks that compose a shapechagig likage ivolves covertig the desired curves, deoted as desig profiles, ito target profiles that are readily maipulated ad compared. This sectio presets methods for dealig with a etire desig profile. I practice, desig profiles are divided ito segmets as described i Sec. 3, so the methods preseted here are actually applied to idividual segmets of the desig profiles rather tha the etire profiles. 2.1 Desig Profiles. A set of p desig profiles specifies a shape-chage problem, where a desig profile is a curve defied such that a ordered set of poits o the curve ad the arclegth betwee ay two such poits ca be determied. The piecewise liear curves solid lies i Fig. 2 are simple examples of desig profiles. I practice, desig profiles are likely specified either i parametric form like the composite splies stadard i computatioal geometry 22 or as piecewise liear curves, the stadard i this work. If a desig profile is specified by the lie segmets coectig the ordered set of poits, a i,b i T,,...,m, the legth of the ith segmet of the desig profile is c i = a i+1 b i+1 a i b i = ai+1 a i 2 + b i+1 b i 2 ad the arclegth of the etire desig profile is m 1 C d = c i 1 2 q q+1 c i g 1 c s c i otig the case where q=0 ad the gth poit is o the first liear segmet of the desig profile. The, x g y g = a q b q+1 + g 1 c s c i aq+2 a q+2 b q+2 b q+2 a b q+1 a q+1 b q+1 4 ad all poits o the target profile may be determied. Thus, a target profile is the piecewise liear curve composed of the lie segmets coectig the ordered set of poits z i = x i,y i T, i =1,...,, ad ay desig profile ca be represeted by a target profile of two or more poits. Usig the defiitios of segmet ad curve legths i Eqs. 1 ad 2, the arclegth of the target profile is deoted as C. Sice poits o the target profile are separated by equal arclegths alog the desig profile, they are ot at equidistat itervals alog the target profile, as see i Fig. 2. The target profile arclegth is less Target Profiles. The desig profiles may be defied by ay umber of poits, but are coverted to target profiles that all have the same umber of poits to allow for idetificatio of correspodig poits o each. I Fig. 2, five poits geerate the target profiles, represeted by the dashed lies. The first ad last poits of a target profile ecessarily coicide with the first ad last poits of the desig profile: x 1,y 1 T = a 1,b 1 T ad x,y T = a m,b m T. The poits of the target profile are located at equal arclegths of c s =C d / 1 alog the desig profile. To locate the gth poit of the target profile, determie q such that Fig. 2 Two desig profiles solid lies ad their target profiles dashed lies. The top three-poit desig profile labeled a,b is represeted by a five-poit target profile labeled x,y. The five-poit, ulabeled bottom desig profile has the same arclegth as the top desig profile, but the two five-poit target profiles have differet arclegths / Vol. 130, MARCH 2008 Trasactios of the ASME

3 tha or equal to the correspodig desig profile arclegth, C C d. The most sigificat loss of shape iformatio occurs where the curvature is largest for a cotiuous desig profile or where the agle at a vertex is smallest for a piecewise liear desig profile. Large values of produce smaller variatios betwee the desig ad target profiles ad i the distaces betwee cosecutive poits o the target profile. The heuristic employed i this work is selectig such that the target profile arclegth is greater tha 99% of the desig profile arclegth, C 0.99C d. As a fuctio of, C is ot mootoic. I Fig. 2, the horizotal segmet of the top desig profile is twice the legth of the vertical segmet, so C=C d if is a multiple of 4. The choice of =5, as depicted, results i a target profile shorter tha that for =4. For this example, the at which C 0.99C d, with o icrease i resultig i C 0.99C d,is= Shifted Profiles. From a set of p desig profiles, let the jth target profile be defied by, z ji = x ji y ji T,,...,. A rigidbody trasformatio i the plae, Z ji =Az ji +d, where cos A = si ad d = d 1 si cos d 2 will relocate the profile preservig the respective distaces betwee poits withi it. Ay profile relocated i this fashio is called a shifted profile. Target ad mea profiles described i Sec. 2.4 are both shifted to perform desig operatios without alterig the origial desig problem. The distace betwee target profiles j ad k is defied to be D = x ji x ki 2 + y ji y ki 2 = z ji z ki 2 Viewig the target profile s poits as a sigle poit i 2-dimesioal space, this distace is the square of the Euclidea orm i that space, so D is a appropriately defied metric. Determiig the rigid-body trasformatio that shifts target profile j to the locatio that miimizes D with respect to target profile k is referred to as the image registratio problem. This problem was origially partially solved i three dimesios 23,24 ad for a arbitrary umber of dimesios 25 utilizig sigular value decompositio. These solutios were later show to produce reflectios istead of rotatios for certai data sets, ad a corrective measure was proposed 26. The derivatio that follows restricts the problem to two dimesios, yieldig a closed-form solutio. To determie the rigid-body trasformatio that shifts target profile j to the locatio that miimizes D with respect to target profile k, oe must fid ad d such that D = 0 ad D d = 0 6 where D, that ow measures the distace betwee the shifted j ad the oshifted k, is D = Z ji z ki 2 = z T ji z ji + d T d + z T ki z ki +2d T Az ji 2z T ki Az ji 2d T z ki The result is a pair of simultaeous equatios i ad d: z ki d T si 5 7 cos ji =0 8 cos si z Fig. 3 a Three target profiles. b Two of the target profiles are shifted to miimize D relative to the third. c A mea profile solid lie. d The mea profile shifted to miimize D relative to each of the origial target profiles. Az ji + d z ki = 0 9 Itroducig the defiitio z ji =z jt = x jt,y jt T, Eq. 9 becomes d = 1 z k T Az jt 10 Substitutig Eq. 10 ito Eq. 8 elimiates d ad simplifies to a liear equatio i cos ad si. The solutio for is give by 1 x k T y jt x jt y kt x ki y ji x ji y ki ta = x ji x ki + y ji y ki 1 x j T x kt + y jt y kt 11 The value of is used to costruct the matrix A, ad the Eq. 10 yields d. 2.4 Mea Profiles. A mea profile is a sigle profile that approximates the shapes of all target profiles i a set. A mea profile z mi = x mi y mi T,,...,, is formed by shiftig target profiles 2 through p such that their respective distaces relative to referece target profile 1 are miimized. The, a ew piecewise liear curve defied by poits, each the geometric ceter of the set of p correspodig poits i the shifted target profiles, is geerated. That is, z 1i + p Z ji j=2 z mi = i =1,..., 12 p For example, two target profiles i Fig. 3 a are shifted i Fig. 3 b to their respective distace-miimizig positios relative to the first profile. Figure 3 c shows the mea profile that approxi- Joural of Mechaical Desig MARCH 2008, Vol. 130 /

4 mates the target profiles whe regarded as rigid bodies. I Fig. 3 d, this mea profile is shifted via a rigid-body trasformatio to approximate the shape ad locatio of the target profiles. Note that a ew trasformatio is calculated to shift the mea profile to each of the target profiles i their origial locatios. The described procedure of creatig target profiles, geeratig a mea profile, ad shiftig the mea profile to its D miimizig locatio relative to each of the target profiles potetially coverts a shape-chage problem to a rigid-body guidace problem. If the mea profile acts as a sufficiet approximatio to the desig profiles, as it may i the case of the three locatios of the mea profile i Fig. 3 d, the the problem is coverted to the desig of a mechaism for three fiitely separated positios of a movig lamia. Assumig that the mea profile is isufficiet i this capacity, a process for the segmetatio of the target profiles ito multiple approximatig rigid bodies is itroduced. 3 Segmetatio The procedure for geeratig target profiles from desig profiles is always performed for each desig profile s etire arclegth. The process for geeratig mea profiles, however, ca be accomplished across ay subset of the target profiles. For example, give five target profiles of 100 poits, poits may be used to geerate a mea profile, defiig a piecewise liear curve that approximates poits oly. I fact, the procedure for geeratig a mea profile may be applied to ay segmet of the target profiles. With the target profiles divided ito s segmets, correspodig segmets from all of the target profiles are used to geerate mea profiles that defie the geometry of the rigid liks. The key is to divide the target profiles so as to reduce the error i approximatig the desig profiles, producig the descriptio of a chai of two or more rigid liks coected by revolute joits to better approximate the shape chage. 3.1 Error Metric. The distace D is a poor error metric for segmetatio because it depeds o a segmet s umber of poits. A better metric, the error E, is defied as follows. For mea profile l, the error E lj associated with matchig the correspodig segmet i target profile j is the maximum distace betwee ay two correspodig poits o the two profiles whe the mea profile is shifted to the distace-miimizig locatio relative to the target profile segmet. The error E l associated with this mea profile is the maximum value of E lj for all target profiles j=1,...,p. The overall error E is the maximum value of E l for all mea profiles l=1,...,s. Geerally speakig, E is the distace that quatifies the sigle, worst-case poit-to-poit matchig betwee the mea profiles ad the target profiles. 3.2 Uiform Segmetatio. To geerate a likage composed of s rigid liks, a iitial solutio might divide the target profiles ito s segmets of roughly equal umbers of poits, the last poit of a segmet beig the first poit of the ext segmet. A mea profile is geerated for each set of segmets. For example, give target profiles of =102 poits, if s=4, the segmets are composed of poits 1 26, 26 51, 51 76, ad The first three segmets ad their correspodig mea profiles each have 26 poits, ad the last has 27. Oce geerated, each mea profile ca be shifted idividually to the locatio relative to its correspodig segmet i each target profile that miimizes D. The positios of the s mea profiles relative to each other will differ as they are superimposed o, or shifted to, each target profile. The ed poits of the mea profile segmets i geeral will ot coicide i ay of the positios at this stage. They will later be joied together with revolute joits i the mechaizatio stage. 3.3 Segmetatio to Meet Specified Error. A alterative to uiform segmetatio for iitiatig a desig is to specify a acceptable error E a istead of the umber of segmets. I this case, segmets are grow poit by poit alog the target profiles, startig with segmets of oly two poits. A mea profile is geerated from the set of target profile segmets, the error E l for the mea profile is computed, ad provided that E l is less tha E a, the process is repeated by addig aother poit to the segmets. Ultimately, the poit that causes E l to exceed E a for the set is excised from the segmets. The process the begis agai with a ew segmet formed from the last poit of the just completed segmet ad the excised poit. This geerates a ukow umber of segmets, the last of which geerally has the smallest error. This discussio of segmetatio holds fixed ad, therefore, places a lower boud o E. If o limits are placed o or s, a arbitrarily small error ca be achieved by the segmetatio process. Note that this is true eve i the presece of desig profiles of differet arclegths. At this stage i the desig process, the segmets are separate rigid bodies ad ot yet coected via revolute joits. Oce coected, the error is see to grow. Cosider the logest target profile as havig arclegth C L, ad the shortest C S. Give that the rigid bodies i the chai ca be made arbitrarily short i theory, a chai of arclegth C L +C S /2 may be geerated that exactly matches the target profiles over the portios for which the chai ad profiles alig. Aligig the chai at oe ed of either target profile, ad the aligig the chai itself with the target profile, produces a error at the other ed of C L C S /2. However, the error miimizig locatio of the chai would distribute the error equally at both eds, ot etirely at oe ed. Thus, the theoretical miimum error for the coected chai is observed to be oe-fourth the differece i arclegth betwee the logest ad shortest target profiles. 3.4 Error-Reducig Segmetatio. To reduce E, the segmetatio locatios o the target profiles are moved regardless of how the iitial segmetatio was accomplished. Segmets o the profiles where the shape chage is most dramatic are shorteed to distribute the error more equally across all of the segmets the goal i this work, but ot a requiremet. Startig with segmet 1, the umber of poits i each segmet l, except for the last segmet s, is icreased by 1 if E l Ē ad decreased by 1 if E l Ē, where Ē is the average of all the E l s. The umbers of poits i segmets 1 ad s chage by 1 ad i the other segmets by zero or 2. Although E s does ot explicitly determie whether segmet s icreases or decreases i legth, its effect o Ē does so idirectly. As a example, let four mea profiles be used to approximate a set of target profiles each defied by 102 poits. With uiform segmetatio, the mea segmets approximate poits 1 26, 26 51, 51 76, ad If E 1,E 3 Ē E 2,E 4, the ew segmets cotai poits 1 25, 25 52, 52 75, ad Segmet 2 icreases by two poits, ad segmet 3 retais the same umber of poits, although they are differet poits from the previous iteratio. With the target profile segmets redefied through this method, a ew mea profile is geerated for each set, the error E is recomputed, ad the process is repeated util the value of E ceases to decrease. Because the error may actually icrease i the short term, possibly idicatig oly a local miimum, the process cotiues for several iteratios after E icreases, ad each E is compared to several previous iteratios istead of just the immediate predecessor. The segmetatio that provides the smallest value of Ē is the error-reducig segmetatio, ad the correspodig mea profiles defie the geometry of the rigid liks that compose the likage. Because the target profiles typically cotai hudreds or thousads of poits, alterig segmets by two poits is a modest chage, ad exhaustive approaches that examie sigle-poit alteratios are ulikely to offer sigificat beefit. The procedures preseted i Secs. 2 ad 3 work without regard for the differeces i arclegth i the origial desig profiles. There is o explicit eed for the p desig profiles specifyig a shape-chage problem to have the same arclegth. The error i approximatio is geerally smaller if all p profiles have roughly equal arclegth. From the desiger s stadpoit, the errors itro / Vol. 130, MARCH 2008 Trasactios of the ASME

5 Fig. 4 Portios of two rigid segmets solid light gray ad black lies show at top i their idividual distacemiimizig positios relative to a portio of a target profile dashed medium gray lies ad at bottom assembled together as a chai with a revolute at their ed poits. This is a portio of target profile 1 ad the solutio segmets from the example i Sec. 5 show i a differet orietatio. The error for the two segmets i combiatio icreases slightly after they are assembled, primarily because of the displacemet of the left segmet light gray. duced by these methods, ad certaily durig the mechaizatio process i the ext sectio, may ot yield satisfactory mechaical desigs. Table 1 Rigid segmet ed poits prior to ad followig assembly ito a chai for positio 1 of the example mechaism described i Sec. 5 i Iitial segmet i ed poit Iitial segmet i+1 start poit Fial revolute coectio , , , , , , , , , Mechaizatio Oce a set of rigid-body segmets has bee geerated through the segmetatio process, these segmets are joied together at their ed poits with revolute joits to form a likage. The, liks are added to costrai the likage to have a reduced umber of DOF s. The goal is to achieve a mechaism of a chose umber of DOF s without icreasig the error i the shape-chage approximatio beyod a acceptable value. 4.1 Morphig Chai of Rigid Segmets. Sice the segmeted mea profiles are geerated idividually, their ed poits will geerally ot coicide i their distace-miimizig positios relative to the target profiles, as show i the top portio of Fig. 4 ad the middle colums of Table 1. Coectig the rigid segmets at their ed poits with revolute joits requires movemet away from these positios. Therefore, the error i approximatig the shape chage with the rigid-body segmets icreases due to their beig joied i a chai, as show i the bottom portio of Fig. 4. If the error is excessive, oe optio is to icrease the umber of rigid bodies geerated via the segmetatio process. Typically, the drop i maximum error dimiishes with each additioal segmet, as show for the example i Sec. 5. Other optios iclude weightig the poit-to-poit error metric, weightig the distace betwee mea ad target profile segmets, ad coectig the rigid-body segmets with revolute joits located at poits other tha the ed poits. While umerical optimizatio techiques could be applied, the authors curretly utilize a ad hoc approach to alig the coected rigid-body chai with the target profiles. The rigid segmets are costructed i the sketchig mode of a parametric desig software package, ad geometric costrait programmig GCP 27 techiques are employed. Adjacet rigid segmets are costraied to have coicidet ed poits, which is equivalet to their coectio with revolute joits. The etire chai is maipulated to approximate the desig or target profile so as to visually miimize the error, ad the process is repeated for each profile. The result is a set of p arragemets of the coected rigid-body segmets to which additioal liks of a mechaism ca be added to reduce the umber of DOF s. The coordiates of the revolute coectio poits for positio 1 of the mechaism described i Sec. 5 are give i the last colum of Table Addig Biary Liks. Biary liks may be added betwee the existig chai ad the groud util a sigle-dof mechaism is achieved. Multi-DOF mechaisms may provide a better match to the target profiles, recogizig that reduced actuatio requiremets may come at a cost to accuracy of the shapechage approximatio. To costrai a s-lik ope chai to be a M-DOF mechaism, s+2 M biary liks must be added betwee segmets of the chai ad the groud, per Gruebler s equatio 21. To covert a ope chai ito a 1-DOF mechaism, s+1 such biary liks are required, so at least oe rigid segmet must have two biary liks to groud idetifyig a four-bar sublikage. The other segmets i the chai may have zero except for the ed segmets, oe, or two biary liks to groud. I fact, ay cosecutive i liks of the origial s-lik ope chai may have at most i+1 biary liks to groud. A strategy that has prove effective is to idetify oe segmet for a four-bar sublikage ad add a sigle biary lik to the other segmets, formig a series of coected five-bar sublikages. The selectio of the segmet for the four-bar sublikage is arbitrary; however, experiece shows that selectig a iterior segmet ofte makes sigularity avoidace, as described i Sec. 4.3, easier. 4.3 Locatig Biary Liks. If the umber of target profiles is less tha or equal to 5, it is theoretically possible to add biary liks betwee the existig chai ad the groud without icreasig the error i approximatig the shape chage. The p locatios of each segmet defie a p fiitely separated positio problem. For p 5, p-positio circle ad ceter poits ca be foud for each segmet 21, while for p 6, least-squares approximatios such as those developed by Sarkisya et al. 28 ca be used to locate circle ad ceter poits. Certaily, additioal freedoms exist whe p is small. I the three-desig-profile example i Sec. 5, ay poit i the plae ca be selected as a circle/ceter poit ad the correspodig ceter/circle poit will be uiquely defied. With oly two desig profiles, as i the example i Sec. 6, a ifiity of ceter/circle poits exists for each circle/ceter poit chose i the plae. Regardless of the umber of target profiles, the desiger has the flexibility to choose circle ad ceter poits that do ot represet exact solutios to the p fiitely separated positio problem. I such a case, the error i the shape-chage approximatio icreases due to the additioal costraits of the added liks. A key cosideratio i locatig the additioal biary liks is to avoid sigular cofiguratios betwee the desig positios that would prevet the mechaism from matchig all of the target profiles, preferably with a sigle actuator. Desig for multiple actuators is ot explicitly cosidered here sice it ca be dealt with by applyig the sigle-dof methods ad sice reductio to a sigle DOF is the more difficult problem. This is particularly challegig, potetially prohibitively so, for dramatic shape chages ad those requirig large umbers of liks. Note, however, that a subtle shape chage defied by as may as eight or more very similar desig profiles is much easier to achieve with a sigle-dof mechaism tha is a dramatic shape chage defied by just two very disparate desig profiles. Sigularities are more easily avoided through a small rage of motio. For four or more desig profiles, selectig oexact or o-least-squares solutio circle/ceter-poit pairs may be required to avoid sigularities. Therefore, accurately approximatig dramatic chages betwee may disparate desig profiles would likely be extremely difficult, if ot impossible. A graphical method, based o stadard motio limit aalysis Joural of Mechaical Desig MARCH 2008, Vol. 130 /

6 Fig. 5 a Proximity to sigularity of the four-bar sublikage of the example i Sec. 5. The maximum trasmissio agle is show with solid lies, ad the miimum with dashed. b Sigular cofiguratios betwee desig positios with a cadidate circle/ceter-poit pair for additioal rigid segmet. 21, ca be used to check for the sigularities as each additioal biary lik is added. Begiig with the lik i the ope chai to be coected to groud by the two biary liks the four-bar sublikage, selectio of the first biary lik provides a kow path betwee the desig positios for its circle poit. A locatio is the eeded for a secod circle poit for which o sigular cofiguratio is reached as the kow path is traced. Two vectors are costructed i each of the target positios, oe betwee the existig circle poit ad the potetial ew circle poit ad oe from the potetial ew circle poit to its correspodig ceter poit. The mechaism is i a sigular cofiguratio whe these two vectors are colliear, so a sig chage i the agle betwee the two vectors whe cosiderig all of the target profiles idicates that a sigular cofiguratio occurs betwee them. Thus, o sig chage betwee ay two cofiguratios is a ecessary coditio but is ot sufficiet because two sig chages could occur. A advatage of usig GCP techiques for the mechaizatio is that the mechaism ca be readily maipulated through its full rage of motio after each biary lik is added to verify that the agle does ot exhibit a sig chage betwee ay of the target cofiguratios. A useful heuristic is to costruct two circles cetered at the potetial ew ceter poit. Oe circle has a radius equal to the sum of the distace from the existig circle poit to the ew circle poit ad the distace from the ew circle poit to the ew ceter poit. The other circle has a radius equal to the differece of these two distaces. If the path of the existig circle poit approaches either circle as the mechaism is maipulated, it is approachig a sigularity, so this provides a straightforward graphical check of the proximity to sigularity, as show i Fig. 5 a. With the four-bar sublikage i place, the coupler poit that attaches to the ext lik of the ope chai ow follows a kow path. A locatio is the eeded for the circle poit o this ext lik for which o sigular cofiguratio is reached as the kow path is traced. Two vectors are costructed i each of the target positios, oe betwee the existig coupler coectio poit ad the potetial ew circle poit ad oe from the potetial ew circle poit to its correspodig ceter poit. Agai, o sig chage i the agle betwee the two vectors whe cosiderig all of the target profiles idicates a potetially sigularity-free choice of biary lik, ad maipulatig the mechaism through its rage of motio cofirms the choice. The same circular costructios provide the graphical check of proximity to sigularity, ad i all cases, operatio farther from sigularity is preferred. Figure 5 b shows the sigular cofiguratios associated with a cadidate circle/ceter-poit pair that would prevet the mechaism from movig betwee the desig positios with the iput i the four-bar sublikage. Takig trasmissio agle i its broadest sese to mea the smallest agle betwee ay uactuated lik joited to groud ad the floatig lik to which it is joited, a sigularity occurs i a shape-chagig mechaism whe ay trasmissio agle goes to 0 deg or 180 deg. The procedure detailed above assumes that the iitially selected biary lik of the four-bar sublikage is the iput ad eables sequetial additio of biary liks such that each associated trasmissio agle does ot become 0 deg or 180 deg throughout the motio, otig that the iitially selected biary lik may ot be the best driver of the mechaism upo completio of the desig. The threshold of fidig a mechaism that avoids trasmissio agles of 0 deg ad 180 deg throughout the motio is deemed sufficiet to justify the methodologies preseted i this paper. Typically, performace is better if trasmissio agles are betwee 30 deg ad 150 deg. I the examples i Secs. 5 ad 6, trasmissio agles are well outside of this rage, ad drivability problems would likely result i mechaism prototypes. As a further sythesis cosideratio, because the trasmissio agle varies betwee the kow profiles, ad ot always favorably, selectig biary liks with a stricter guidelie tha deg at the precisio positios icreases the likelihood of locatig a practical solutio. A graphical procedure that could be implemeted to cotrol the trasmissio agle of the four-bar sublikage for sythesis problems ivolvig two ad three desig profiles is give by Chase 29 based o Su ad Waldro s 30 approach for up to four profiles. Note that the iput lik eed ot be part of the four-bar sublikage, but could be ay biary lik joited to groud that moves mootoically betwee all of the desig positios. I the example i Sec. 6, ay oe of five biary liks, oe of which are part of the four-bar sublikage, could be the iput. Ufortuately, the authors are ot aware of ay rigorous techiques for sigularity / Vol. 130, MARCH 2008 Trasactios of the ASME

7 Fig. 6 Sigle-DOF mechaism chagig betwee three desig profiles free sythesis whe the iput is ot withi the four-bar sublikage, so ad hoc methods are employed i such cases. 5 Three Desig Profile Example To demostrate its effectiveess, the desig methodology is applied to a example of a shape chage defied by the three desig profiles show i Fig. 3 a. The poits defiig these targets are listed i Table 2. No uits are listed because ay uit of legth would be appropriate. Profiles 2 ad 3 each cosist of eight liear segmets, while profile 1 cosists of six liear segmets, so it is clear that o rigid body of fiite legth could exactly match all three profiles. The arclegths differ by less tha 1%. Target profiles of 1800 poits are geerated from these three desig profiles to satisfy the 99% arclegth criterio i Sec Usig the segmetatio process of Secs. 3.3 ad 3.4, four segmets resulted i a acceptably small error E of 0.047, which is less tha 0.7% of the arclegth of the shortest desig profile. With five segmets, E=0.043, so the small error reductio does ot warrat the additioal complexity of havig the extra segmet. Employig oly three segmets, however, resulted i a sigificat Table 2 6 Defiig poits of the desig profiles i Figs. 3 a ad Profile 1 Profile 2 Profile 3 2.3, , , , , , , , , , , , , , , , , , , , , , , , ,2.4 Joural of Mechaical Desig MARCH 2008, Vol. 130 /

8 Fig. 7 Sigle-DOF mechaism chagig from U to D icrease i error to These errors oly apply to the segmetatio process, so the achievable error followig mechaizatio is larger. To achieve a sigle-dof mechaism, the methods of Sec. 4.1 were used to assemble the four segmets ito a chai with revolute joits at their ed poits, ad the methods of Secs. 4.2 ad / Vol. 130, MARCH 2008 Trasactios of the ASME

9 were used to add five biary liks betwee the chai ad the groud, as show i Fig. 6. Care was take to locate all of the ceter poits withi the regio roughly bouded by the desig profiles to avoid havig liks of extreme legth. Two biary liks were added to oe of the iterior shape-chage liks to establish a four-bar sublikage for actuatig the mechaism. As idicated i Fig. 6, this four-bar is a crak rocker, so it offers actuatio betwee the three desig profiles with a cotiuously rotatig iput. The coordiates of the revolute joits of this mechaism whe it is i positio 1 are give i Table 3. The mechaism is show i the desig positios i Figs. 6 a, 6 d, ad 6 g. Itermediate positios are show betwee each of the desig positios. As show most clearly i Fig. 6 f, the liks of the mechaism do overlap betwee the secod ad the third desig positio such that several of the liks would eed to be offset i parallel plaes i a prototype of this mechaism. Icreasig the allowable desig space for the ceter poits might allow the desiger to avoid this circumstace. Followig mechaizatio, the value of E for the mechaism was As aticipated, this value was larger tha that followig just the segmetatio stage, but it is still small less tha 0.9% of the arclegth of the shortest desig profile. 6 Two Desig Profile Example Desig profiles represetig the letters U ad D would seem to require a more dramatic shape chage. Figure 7 shows a sigle- DOF solutio mechaism for this problem defied by the poits listed i Table 4 agai without uits. Note that the U is taller tha the D because the two have the same arclegth. For each desig profile, 2600 poits were used to geerate the target profile, agai to satisfy the 99% arclegth criterio described i Sec Although there are oly two desig profiles, eight segmets were required to accomplish the shape chage with a error E of about 0.15% of the total arclegth usig the segmetatio process i Secs. 3.3 ad 3.4. As such, ie additioal biary liks were added to achieve a sigle-dof mechaism. I this case, care was take to locate all of the ceter poits withi the D profile. The coordiates of the revolute joits of this mechaism whe it is i positio 1 are give i Table 5. Table 3 Coordiates of the revolute joits of the mechaism show i Fig. 6 a Ceter poits Circle poits Morphig chai revolutes 3.132, , , , , , , , , , , , ,1.874 Table 4 Defiig poits of the desig profiles i Fig. 7 U desig profile 1.000, , , , , , , , , , ,7.500 D desig profile 1.500, , , , , , , , , , , , ,7.000 Table 5 Coordiates of the revolute joits of the mechaism show i Fig. 7 a Ceter poits A sigle four-bar sublikage was costructed with the segmet that forms the upper left portio of each letter, but this is ot where the mechaism is actuated because the shorter crak of the four-bar is ear a sigularity i the D cofiguratio. Ay of the biary liks attached to the segmets that form the right side of the U could serve as the iput sice all of them rotate mootoically betwee the desig cofiguratios. The mechaism does ot, however, have a crak iput that ca make a full rotatio, so the shape chage caot be executed with a cotiuously rotatig actuator uless a additioal likage is added. Followig mechaizatio, the error E grows to The fivefold icrease from the segmetatio error is ot surprisig give the larger umber of liks to be joied together i this example compared to the previous example. The error is still small eough that the two letters are clearly deoted with the mechaism, which is the metric for this example. Acceptable errors i other mechaisms will be applicatio depedet. 7 Coclusios This work itroduces a systematic procedure to sythesize mechaisms composed of rigid liks joied together with revolute joits to approximate a desired shape chage defied by a arbitrary umber of morphig curves. The procedure ivolves comparig piecewise liear curves to defie the geometry of the rigid liks that approximate the shapes with a miimum amout of error. Additioal biary liks ca be added to the chai of shapechage liks to reduce the umber of degrees of freedom of the mechaism. The procedure is applied to two examples of sigle- DOF mechaisms desiged for shape chages defied by ope curves. It is equally applicable to multi-dof mechaisms. The authors believe that the procedure will eable desigers to leverage the advatages rigid-body mechaisms offer i developig ad implemetig morphig structures. Ackowledgmet This material is based o the work supported i part by the Natioal Sciece Foudatio uder Grat No to A. Murray. Refereces Circle poits Morphig chai revolutes 1.827, , , , , , , , , , , , , , , , , , Moer, H. P., 2001, Realizatio of a Optimized Wig Camber by Usig Formvariable Flap Structures, Aerosp. Sci. Techol., 5, pp Washigto, G. N., 1996, Smart Aperture Ateas, Smart Mater. Struct., 5 6, pp Marti, J. W., Mai, J. A., ad Nelso, G. C., 1998, Shape Cotrol of Deployable Membrae Mirrors, ASME Adaptive Structures ad Materials Systems Coferece, Aaheim, CA, pp Austi, F., Siclari, M. J., Va Nostrad, W., Weisesel, G. N., Kottamasu, V., ad Volpe, G., 1997, Compariso of Smart Wig Cocepts for Trasoic Cruise Drag Reductio, Proc. SPIE, 3044, pp Austi, F., ad Va Nostrad, W., 1995, Shape Cotrol of a Adaptive Wig for Trasoic Drag Reductio, Proc. SPIE, 2447, pp Ameduri, S., Esposito, C., ad Cocilio, A., 2001, Active Shape Airfoil Cotrol Through Composite Piezoceramic Actuators, Proc. SPIE, 4327, pp Joural of Mechaical Desig MARCH 2008, Vol. 130 /

10 7 Bart-Smith, H., ad Risseeuw, P. E., 2003, High Authority Morphig Structures, Proceedigs of the ASME Iteratioal Mechaical Egieerig Cogress, Washigto, DC. 8 Jardie, P., Flaaga, J., Marti, C., ad Carpeter, B., 1997, Smart Wig Shape Memory Alloy Actuator Desig ad Performace, Proc. SPIE, 3044, pp Marti, C. A., Jasmi, L., Flaaga, J., Appa, K., ad Kudva, J. N., 1997, Smart Wig Wid Tuel Model Desig, Proc. SPIE, 3044, pp Quackebush, A., Bilai, A., Batcho, P., Mckillip, R., ad Carpeter, B., 1997, Implemetatio of Vortex Wake Cotrol Usig SMA-Actuated Devices, Proc. SPIE, 3044, pp Webb, G. V., Lagoudas, D. C., ad Kulkarii, M., 1999, Adaptive Shape Cotrol for a SMA-Actuated Aerofoil Rib Structure, Proceedigs of the ASME Iteratioal Mechaical Egieerig Cogress, Nashville, TN, pp Agelio, M., ad Washigto, G., 2001, Poit Actuated Aperture Atea Developmet, Proc. SPIE, 4334, pp Yoo, H. S., ad Washigto, G., 1998, Piezoceramic Actuated Aperture Ateae, Smart Mater. Struct., 7 4, pp Yoo, H. S., Washigto, G., ad Theuisse, W. H., 2000, Aalysis ad Desig of Doubly Curved Piezoelectric Strip-Actuated Aperture Ateas, IEEE Tras. Ateas Propag., 48 5, pp Marti, J. W., Redmod, J. M., Barey, P. S., Heso, T. D., Wehlburg, J. C., ad Mai, J. A., 2000, Distributed Sesig ad Shape Cotrol of Piezoelectric Bimorph Mirrors, J. Itell. Mater. Syst. Struct., 11, pp Saggere, L., ad Kota, S., 1999, Static Shape Cotrol of Smart Structures Usig Compliat Mechaisms, AIAA J., 37, pp Saggere, L., ad Kota, S., 2001, Sythesis of Plaar, Compliat Four-Bar Mechaisms for Compliat-Segmet Motio Geeratio, ASME J. Mech. Des., 123 4, pp Lu, K. J., ad Kota, S., 2003, Desig of Compliat Mechaisms for Morphig Structural Shapes, J. Itell. Mater. Syst. Struct., 14, pp Lu, K. J., ad Kota, S., 2005, A Effective Method of Sythesizig Compliat Adaptive Structures Usig Load Path Represetatio, J. Itell. Mater. Syst. Struct., 16 4, pp Stubbs, M. D., Whittier, W. B., ad Reiholtz, C. F., 2004, Sigle Degreeof-Freedom Morphig Wig Desig ad Sythesis, Proceedigs of the ASME 2004 Desig Egieerig Techical Cofereces, Salt Lake City, UT. 21 Erdma, A., Sador, G., ad Kota, S., 2001, Mechaism Desig: Aalysis ad Sythesis, 4th ed., Pretice-Hall, New York. 22 Faux, I. D., ad Pratt, M. J., 1985, Computatioal Geometry for Desig ad Maufacture, Wiley, New York. 23 Hor, B. K. P., 1987, Closed-Form Solutio of Absolute Orietatio Usig Orthoormal Matrices, J. Opt. Soc. Am. A, 5 7, pp Hor, B. K. P., 1987, Closed-Form Solutio of Absolute Orietatio Usig Uit Quaterios, J. Opt. Soc. Am. A, 4 4, pp Aru, K. S., Huag, T. S., ad Blostei, S. D., 1987, Least-Squares Fittig of Two 3-D Poit Sets, IEEE Tras. Patter Aal. Mach. Itell., PAMI-9 5, pp Umeyama, S., 1991, Least-Squares Estimatio of Trasformatio Parameters Betwee Two Poit Patters, IEEE Tras. Patter Aal. Mach. Itell., 13 4, pp Kizel, E. C., Schmiedeler, J. P., ad Peock, G. R., 2006, Kiematic Sythesis for Fiitely Separated Positios Usig Geometric Costrait Programmig, ASME J. Mech. Des., 128 5, pp Sarkisya, Y. L., Gupta, K. C., ad Roth, B., 1973, Kiematic Geometry Associated with the Least-Square Approximatio of a Give Motio, ASME J. Eg. Id., 95 2, pp Chase, T. R., 2006, A Note o the Waldro Costructio for Trasmissio Agle Rectificatio, ASME J. Mech. Des., 128 3, pp Su, J. W. H., ad Waldro, K. J., 1981, Graphical Trasmissio Agle Cotrol i Plaar Likage Sythesis, Mech. Mach. Theory, 37 4, pp / Vol. 130, MARCH 2008 Trasactios of the ASME