Rapid Deployment of Reconfigurable Assembly Fixtures using Workspace Synthesis and Visibility Analysis

Rapld Deployment of Reconflgurable Assembly Flxtures uslng Workspace Synthesls and Vlslblllty Analysls Zhenyu Kong, Darlusz Ceglarek (2) Department of Industrlal Englneerlng, Unkerslty of WlsconslwMadlson, Madlmn, Wlsconsln, USA

Abstract Due to rapld changes In recent market demands, shortened produdlon rampupAaunch and model changeover of new produds wlth slmuitaneous manufadurlng of famlly of produds on a slngle produdlon llne Is becomlng Inevitable. Thls requlres systematlc methods for rapld deslgn and anatysls of reconflgurable fixture workspace synthesls and vlslblllty. Thls paper presents an Integrated approach for rapld reconflgvable fixture deployment whlch Is based on (1) the procrustes analysls Integrated wlth a palmlse optlmbatlon for fixture mrrkspace conflguratlons sythesls; and (2) screen space transformatlonbased vlslblllty analysls for rapld fixture callkatlon. A case study and slmulatlons Illustrate the proposed approach. Keywords: Reconflgurable fixture, Workspace synthesls, Vlslblllty

1

INTRODUCTION

I.I Problem Descrlptlon In recent years due to slgnlflcant dkersmcatlon of market and overall Increim In produd quallty and durablllty, shortenlng lead tlme of new produdlon Is Inevitably becomlng the prevalllng trend. In order to satisfy the ever-lncreaslng demand In product varlety, produdlon systems need to be Improved, 1.e. to develop Reconflgurable Manufadurlng and Assembly Systems (RMSs and RASs) [I][2]. Therefore m r n mrrlrssynthesis for produdlon of a famlly of parts on a slngle assembly llne must be developed. Flxhrrn wlibrptlon Is of paramount Importance for produd quallty slnce a slgnkanl number of fixture related failures are related to fixture Installatlon and malntenance [3]. The appllcatlon of RASs brlngs new challenges to fixture callbratlon slnce reconflgurable toollng elements need to be callbrated In muitlple positlons. However, currently there are no methodologies to determlne the best positlons of the measurement system such as laser tracker to fully calibrate a given M u r e or mlnlmhe the number of setup positlons of a measurement system. The objedive of thls paper Is to develop an Integrated approach for rapld reconflgurable fixture deployment based on (1) Fixture Workspace Synthesls p S ) whlch Identmes the best superpositlon of locatlng layouts for a famlly of parts; and (2) Fixture Vlslblllty Anaiysls (FVA) for toollng elements M l c h mlnlmhes nunber of setups of measurement equlpment durlng fixture callbratlon. 1.2

Related Work

FixWm DeSm m d Analysis Many algorlthmk and heurlstlc methods w r e developed to syrthesbe and analyze setup plans and fixture conflguratlons for a slngle rlgld part [4] [5]. As for deslgn and anaiyds of fixture for slngle compllantmexlble part, Menassa and DeVrles [6] proposed optlmhatlon technlques to asslst In the d e s l g and evaluatlon d 3-2-1 M u m for prlmatlc parts. CaI &el. p'l proposed an W 2-1" locatlng layout for sheet metal parts fixturlng.

In the area of assembly @ems for a famlly of parts, Lee et el. [El presented a workspace synthesls anaiysls for fixturlng of famlly of stamped parts uslng the genetle algorlthm. FWrn UYsibiMy/~ibiMy Spyrldl, S p l h and Requlcha [9] [lo] developed both anaiytlcal and dlscrethed accesslblllty anaiysls methods. Chen and Woo [Ill Rrsl developed the concept of vlslbllEy map and provlded geometrlc algortthm. The accessbllltyMslblllty methods were applled to mlnlmhe the number of workplace setup In CNC machlnlng and CMM lnspedlon [Illand compute the dleopenlng dlredlons for removlng hbrlcated wrkpleces [12]. In the case of tmllng callbratlon addressed In thls paper, the positbn and orlentatlon of toollng elements are m e d , and thus, the problem Is related to Identifylng measurement system positlon and orlertatlon (setup) and thls Is dlllerent from the iaforementbned research. 2

FIXTURE WORKSPACE SYNTHESIS (M) M€IH0D0L0DY The reconflgvable fixture uses reconflgvable toollng elements whlch can be freely reconflgured to any positlon M h l n its fundlonal work errvebpe Therefore, the positlon of a locator Is no longer at a slngle m e d polnt, but rather at any polnt Athln the FWE workspace. Fixture Workspace Synthesls p S ) for a famlly of parts baslcally entalls seeklng the best superposltlon of locators for all the parts. For a general case, assume a part famlly has N parts, each of whlch has k locatlng polnts, and deflne C, ( M , 2,...,4 as the mlrrlmum clrcle that contalns N conespondlng locatlng polnts Qlg. 1 deplcts a case wlth flM and b 6 ) . The problem of W S can be expressed as mlnlmhlng the mwlmum dlameter of C, ( M , 2,...,4 and thus the objedke can be formulated as: MUr(Mm(C,)) 1< l < k (11 Let us deflne (C&,wd as the largest drcle among the k clrcles obtalned based on Eq. (1). Then durlng fixture deslgn, the d l m e t e r of the FWE pFm) for a gken reconflgunble toollng element must be equal to or larger

m.

than so as to ensure that the conespondlng locatlng polnts can be contalned wlthln the M of the reconllgurable toolhg element. Thus, WB daRne the mwlmum Recdgurablltty Index (Rl) as: RI, = Dmd(cdMJw@ (2) Then for any M u r e wrkspace conflguratlon layout, Its RI must be Athln the followlng range: 15RI5RI-

obtalned, whlch lead to the besl match for the conflguratlons. 2.2 P a l W z e Conflguratlon Optlmlzatlon (PCO) Although procrustes analyds Is to mlnlmhe the dlstance between the conespondlng TEs of the conflguratlons, It Is based on the wm of the square of the dlstances of TEs, and Its ob]edke fundlon Is expressed as:

(31 I-1

The solutlon of Cibased on Eq. (8) Is represented as: (CI)= ( I = I , 2, ... k) (9) The dlameter of mwlmum clrcle obtalned based on Eq. (9) Is represented as ( C m ) L sBased . on the requlrement of reconllgurable M u r e wrkspace analysis (Eq(l)), the followlng holds:

Figure 1 Expected configuration fortwo part5

In order to overcome the challenges of slmuttaneousty optlmhe the mnflguratlons of muttlple parts, a M u r e conflguratlon synthesls method far a famlly of part Is developed by applylng procrustas-based p a M s e optlmhatlon. 2.1 Prellmlnary Conflguratlon Layout Uslng Procruztez A n m l z In reconflgurable M u r e deslgn, each tmllng element such as locator or clamp, Is not lust a polnt, rather tt Is a locatlng area. For slmpllctty. we use a clrcle to represent a potentlal toollng element posttbn. Consequently, WB can stlll utllhe a p o M that Is the center of the clrcle to represent a bcatlng area, along Ath dlameter of the clrcle.

For M u r e synthesls, each part Is represented by a set of locators or clamps, W c h are k n as Todlng ~ Elements (TEs). Two parts classltled by the same T E s are recognbad as Identlcal In the sense of M u r e conflgurdlon deslgn. The procrustes analysls for M u r e mrrkspace conflguratlon can be presented as follows: (1) mrdcrrpte Rn -. A centered coordlnate @em Is utllhed for coordlnate reglstratlon, firstly computlng "centered" polnts whose coordlnate Is the artthmetlc werage of the TE coordlnates of each conflguratbn. and then translatlng each conflguratlon to make the "centered" polnt colncldent Ath the coordlnate orlgln. (2) SLnlhrIty Tmnsfonnathns. Assume C, and C2 are two M u r e mnflguratlons M e r cmrdlnate reglstratlon and are represented by k x m (m=2 or 3 conespondlng to 2D and 3D parts respedkely) matrix. Let R m d T be the rotatlon and translatlon transfomatlons, respedkely. Then Iflttlng C, to C2 uslng R and T transfomatlons, WB can get: C2 = C,R+IkTr+ E (41 Were & I s a k x m matrix wlth a11 elements of one and E Is resldual error, whlch lndlcates the dWerence between two M u r e conflguratlons. 6y applylng the least square method, the E can be mlnlmbed. The sum of square of E Is: d(c,,c2)=lkrEll (51 The solutlon for mlnlmhlng the sum of square of the resldual enor cm be obhlned as: T=O (61 R= WT Were U and V can be obhlned by uslng the slngular

m

value decomposlUon Clr C1 = VAU ( U and V belong to speclal orthogonal group and AIs the vedor of elgen values of c:c,). By uslng thls approach, the necessary translatlon and rotatlon transfomatlons for each conflguratlon can be

/m

< ( G l M h ( k ) < (GILs

(10)

Accordlng to thls relatlonshlp. the scope of search domalns for the varlables can be narrowed to a small range. Then a slmple searchlng optlmhatlon method can be applled to Rnd the deslred solutlon rapldly. For thls optlmhatlon, the key polnt Is how to emclently accompllsh tt for muttlple mnflguratlons. We used a method called PalMse Conflguratlons Optlmhatlon (PCO) that slmuttaneously deds wlth only two conflguratlons at any tlme. For N parts, assume YJ, I=/ ,...,N are thelr conflguratlons that h w e been processed by uslng the procrustes analysls. For each conflguratlon Y, we compute the werage of other N-I

F(,). We then condud search optlmhatlon for the conRguratlon palr of Y , and F(,) based on objedke conflguratlons

fundlon represented In Eq. (1). Repeat thls process untll the obtalned (C&,wrer4 cannot be reduced any further. Then the best match for the N conflguratlons (parts) Is detemlned, whlch meets the requlrement of objedke fundlon presented In Eq. (1).

FIXTURE VlSlBlLrrY ANALYSIS (Fvw USING SCREEN SPACE TRANSFORMATION FOR MINIMIZATION OF NUMBER OF S m P S Flgure 2 Illustrimtes the vlslblltty problem as applled to Inllne M u r e callbratlon along wlth the conespondlng termlnology. Observation Space (0s)Is a predeRned space In Wlch the Measurement Equlpment (ME)Is allocated. Any posttlon lnslde 0s Is called an Obsenwtlon Posttlon (00. Measurement Targets (MTs) are those feature polnts of the toollng elements (Iocatodclamps) that need to be measured and callbratad by the ME. Any objed that may block llnes of slgM between ME and MTs Is called Measurement Obstacle (4 Examples . of MOs are: materlal handllng devlce(s), robots and other toollng elements. A slngle ME setup Is descrlbed as a slngle posttlon of ME Athln 0s. For lwllne M u r e callbratlon, each MT represents a slngle geometrlcal feature of a g k e n toollng element (locator or clamps). In order to soke the problem of mlnlmum number of setups, the vlslblllty should be checked for every MT and a11 the posslble OPs Athln the 0s to verify If the llne of slgM between ME and MT Is blocked by s u m M O s or not. H M v e r , thls method has extremely Intenske computatlons of (1) h e m y computatlon of Intersedlons between strdgM llnes and planes In 3D space, and (2) the number of surface polygons representhg obstacles Is qutte large. In order to overcome these shortcomlngs, the 3

Screen Space Transformation (SST) and a sirrplified modeling of measurement o b d e s are presented in the ensuing sections.

Flgure 2: Setup of local vlew coordlnate system (LVCS) for dilTerent M s

Figure 2 also illu&iates M exanple fw which the proposed visibillty algwithm is applied. All the objects are located in a W r l d Cowdinate System ( K s ) , represented by Xw, Yw, and Zw. It is assumed that all measurement obstades are cmvex pdyhe&ons. First we s d p the l m i l v i m cmrdnate systems that take each measurement bug& as wigin. Each observation space is represented as a sphere which can lead to a convenient setup fw a local v i m cmrdnate system. In Fig. 2 two local view cowdinate systems LVCS, and LVCS2 take MT, and MT2 as orign respectively. Due to the spherical shape of the OS, the mientation of the axis of each l m i l view cmrdnate system can be convenientiy debermined.

(Eq. (3)). It CM be observed that fw all RI 2 1, the selected fixture is capable of assembling a given part family. However, the o w n e d minimized wwkspice m y potentially increase folture calibration time and decrease accuracy due to tooling element visibilty lirrritations. Thus the propased integration of fixture wwkspice synthesis and visibillty analysis is cmducted in two steps: (1) rrrinimize forture wwkspice synthesis, as measured by RI; (2) maximize tooling element visibillty by reducing RI, brt still meet Eq. (3) (use full workspce capability of the Mure). This can be integiated as identifyng tooling elements visibilty within the range of 1 < RI < R ,,I as shrmn in Fig. 4.

I

1

I

l

RI m

b a

Figure 4: Relationship bedween the visibiky and RI

As shown in Fig. 5, visibillty is defined by whether the lines of s i a t bedween measurement equipment and measurement bug& ( t d i n g elements) are blocked by measurement obshcles w not. M e n one fixture element is being calihted, all other tooling elements are considered as o b d e s . Since all ob&icles such as robots and material handng, are m n g e d in relation to M u r e configuration, is the folture configuration that determines the results ofvisibillty.

@) *een spice ( 3 Vim spice Figure 3: View spice and s c r m space. Next, SST is applied in each l o 4 view cmrdnate system. Figure 3(a) illustrates the view space which is the miginal 3D spice befwe SST transfwmation. The Screen Space Transfwmation [13] can be utilized to transform the 3D view s p c e to a 2D screen space where the wignal lines of s i a t bewme p d l e l to each other, and the perspective projection is converted to an orthogonal projection, as shrmn in Fig. 3@).Thus, the conputation can be sinplified to a 2D domin that is perpendcular to the projection mientiation. Therefwe, a line of s i a t in 3D spice is allwiated to a point in 2D space. Consequentiy, the conphition of intersection between a line of sight and an ob&ide polygon in 3D space can be cmverted to verifying if a point is within a polygon in 2D spice. Thus, the conphition CM be simplified significantiy. Furthermwe, aiter the SST, the measurement obshcle of 3D cmvex polyhehon bewmes a 2D pdygon by the orthogonal projection. Then the number of geunebrical elements used to represent the MO CM be sigrificimtly reduced by identifying the boundary edge of the 2D polygon. Therefwe the conphitional mplemty can be alleviated.

4

INTECRATlON OF FIXTURE W O R K P A C E SYNTHEslSAND FIXTURE VlslBlLlTY ANALYslS As dscussed earlier, the applications of rewnfigurable foltures instead of dedcated foltures, bing new challenges fw folture calibration and related visibilty analysis. The rewnfigurable tooling elements need to be calihted in multiple pasitions (within the FWE of the reconfigumble tool) instead of a sindehed pasition(s). The presented FWS mebhd in section 2 r r r i n i m k necessary volume space (in 2D case - circles CI as shrmn in Fig. 1) for each t d i n g element to assemble a family of piarts in a wen folture. The resulting wwkspace synthesis of the reconfigumble forture is measured as RI

Flgure 5: Relatlonshlp between FWS and WA.

Figure 6 illu&ates the relationship between MIA and FVS. Firsf as dscussed in Section 2, the Mls determines the l q w t of all the tooling elements, and then it wes the forture laywt as M input to the W A analysis. Based on the folture laywt, the FVA conputes the mrespondng visibilty property which is presented in ,,I Section 3, subject to the conshint of 1 < RI < .R

Fkhre Visibility Analysis subject to MIS con&iain Figure 6: Integration between FWA and W S

5 CASE STUDY The two parts in Fig. 7 are represented by their TEs ia and bl respectively, and they are arbitrarily placed. The centers of the TEs are the feiahres that need to be calihted. The rewnfigurable folture to be used for the reconfigumble folture has a circular FWE with dimeber of 3.5. First we perform the folture wwkspice synthesis. The result that meets the requirement of Eq. (1) is o k i n e d and shown in Fig. 8. The maximum circle of all the cirdes that wrap the conespondng l o d i n g point, i.e. (Cdrr,ncrrM, is okined, which is equal to 3.0. Based on Eq. (2), then Rlm=3.Y3=1.167. Assume that under some constraints, there are only three obsewiation positions available (Fig. E). It CM be observed that d t h w a , Fig. E wes the best forture wwkspace configuration, the visibilty for the measurement equipment is not g w d since none of the three

observatbn posttlons makes the measurement equlpment vlslble to a11 the measurement targets (locaton). In thls case, two setups are needed.

7 ACKNOWLEDGMENTS The authors grateblly acknowledge the Rnanclal support of the API Inc., State of Wlsconsln's Industrlal & Economlc Development Research Program ( I E D R P ] and the NSF Englneerlng Research Center for Reconflgurable Manuladurlng Systems (NSF Grant EEC95-92125). 8

Figure 7 Two parts w t h arbitrary positions

For any other alternatthrely feaslble M u r e mrrkspace conflgurmtlon. thelr RI must meet: 1 5 RI 5 1.167 @q.(3)), based on M l c h some adjustment can be made to the FWS module. Folowlng the algorlthm descrbed In Flg. 4 and 6 WB obtaln the M u r e workspace conflguratlon shorm In Flg. 9. The value of Is 3.18, and conespndlngly the RI Is 3.5fl.18 = 1.1 M l c h Is smaller than the prevlous conflguratlon. However, the resuttlng vlslblsty Is Improved slnce a11 the measurement targets are vlslble to the measurement equlpment from OP2 (only the T E s of the same part wlll block each other). That means only one setup Is needed, so the best vlslblllty Is achleved.

6 SUMMARY Thls paper presents an Integrated q p r o a c h for rapld reconllgurable M u r e deployment M l c h Is based on (1) M u r e workspace synthesis whlch allocates the best superposttlon of locatlng layouts for a famlly of parts to be produced on a slngle reconflgurable M u r e , and (2) M u r e vlslblllty analysis for toollng elements whlch allows to mlnlmhe number of setups of measurement equlpment durlng M u r e callbratlon. The lntegratlon of FWS m d FVA provldes an anaiytlcal tool for rapld M u r e deploynent In a new assembly system and allows for the optlmhdlon of reconflgurable M u r e capabilty to produce a famliy of parts and M u r e vlslblsty to mlnlmhe number of setups. Provlded case study also Illustrates the proposed method.

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