DETC2004/CIE ALGORITHMIC FOUNDATIONS FOR CONSISTENCY-CHECKING OF INTERACTION-STATES OF MECHATRONIC SYSTEMS

Transcription

1 Proceeding of DETC 04 ASME 2004 Deign Engineering Technical Conference and Compuer and Informaion in Engineering Conference Sal Lake Ciy, Uah, USA, Sepember 28-Ocober 2, 2004 DETC2004/CIE-79 ALGORITHMIC FOUNDATIONS FOR CONSISTENCY-CHECKING OF INTERACTION-STATES OF MECHATRONIC SYSTEMS Changxin Xu Mechanical Engineering Deparmen and Iniue for Syem Reearch Univeriy of Maryland College Park, MD Sayandra K. Gupa Mechanical Engineering Deparmen and Iniue for Syem Reearch Univeriy of Maryland College Park, MD ABSTRACT In order o reduce produc developmen ime, we need ofware ool ha can perform auomaed validaion of he propoed deign concep. Thee ool will enure ha only valid deign concep are ranferred o he deailed deign age for furher developmen. Thi paper provide a ep oward he auomaed validaion of propoed deign concep. We define he problem of coniency-checking of ineracionae a a key ep in he deign concep validaion. We preen a polynomial ime algorihm for olving he ineracion coniency-checking problem. We alo preen an algorihm for analyzing inconien ineracion-ae and idenifying he inconien ineracion. We believe ha he framework decribed in hi paper will provide he underlying foundaion for conrucing he nex generaion ofware ool for concepual deign of complex mecharonic yem. Keyword: deign validaion, coniency-checking, and minimum cu of nework. INTRODUCTION Mecharonic yem are referred o a yem ha inegrae elemen from mechanical, elecrical, elecronic, and informaion domain, which are deigned o provide beer oluion han would be poible if componen from only one domain are ued [Wal0]. A muliple ineracion-ae yem i a yem in which he ineracion beween elemen of ueenvironmen and elemen of he yem can have differen ineracion opologie depending upon he mode of yem operaion and he ae of he ue-environmen. For example, conider a hybrid vehicle. While he vehicle i going down a hill, he engine i oring energy ino he baerie. While he vehicle i going up a hill, boh he baerie and he engine are providing power o he wheel. In hi cae, he ineracion opology beween yem componen (baery, engine, and wheel) i changing depending upon he ae of he ueenvironmen (e.g., uphill or downhill). Increaing he auonomy and inelligence in mecharonic yem ofen require hem o be muliple ineracion-ae yem. The produc developmen proce i uually a hierarchical proce during which alernaive are generaed and evaluaed a muliple level of abracion. In order o peed up he produc developmen proce, only valid alernaive hould be paed from a higher level o a lower level. Mo of he commercial CAD yem for mechanical produc are aiding deigner only in he deailed deign age. Compuer aided deign ool for he concepual age of mechanical deign are eiher rericed o few pecific produc or only providing imple keching funcion. Hence, ool for he auomaed validaion of propoed deign concep currenly do no exi. To faciliae compuer-aided concepual deign, we have developed a framework for modeling deign concep behind mecharonic yem wih muliple ineracion-ae [Gup04]. We conider he deigned mecharonic yem a an arifac ha inerac wih he ue-environmen o produce behavior ha aify he cuomer requiremen. The diincion beween our approach and he radiional funcional repreenaion approache for concepual deign i a follow: Correponding Auhor Copyrigh 2004 by ASME

2 We ue ineracion inead inpu/oupu flow o decribe relaionhip beween objec. Thu non-flow baed relaionhip can alo be capured. We ue ineracion-ae o capure he operaing mode of he yem. Hence we can uppor yem wih muliple ineracion-ae. Deign concep modeled uing our framework can be evaluaed and imulaed qualiaively. For example, even can be ued o imulae he deign concep. Thi capabiliy enable deigner o idenify promiing deign concep. We believe ha hi new modeling framework could have he following benefi. Fir, i provide compuer inerpreable repreenaion cheme for muli-ae mecharonic yem deign concep. Hence i provide improved uppor for deign informaion archiving and reue. Second, formal repreenaion are expeced o provide he neceary foundaion for he developmen of compuer aided evaluaion uppor during concepual deign. In order o reduce he produc developmen ime, we need ool ha can perform auomaed validaion [Chan90] of he propoed deign concep. Thee ool will enure ha only valid deign concep are ranferred o he deailed deign age for furher developmen. The imporance of hi ep can be beer underood by examining he conequence of no performing he deign validaion a he deailed deign age. For example, he produc developmen ge ignificanly delayed if non-manufacurable hape are paed from he deailed deign ep o he manufacuring ep. Similarly, paing invalid deign concep o he deailed deign age lead o unneceary delay in he produc developmen. Many ool have been developed ha can perform validaion during deailed deign. Thee ool check variou feaure in he geomeric model of he propoed deign o ae heir validiy. Thee ool are ignificanly reducing he ime o carry ou he validaion ak. We are inereed in developing validaion ool for he concepual deign age. Developing uch ool require he following hree ep. Fir we need o develop a repreenaion o model deign concep. Thi i analogou o he developmen of feaure-baed repreenaion for modeling deailed deign. The nex ep i o develop he definiion of validiy. Thi i analogou o defining wha feaure parameer will be conidered valid during he deailed deign age. For example, very hin wall or feaure wih zero-draf angle may no be conidered valid in he conex of injecion molding. Finally, we need algorihm ha can deermine if a propoed deign i no valid. Thi i analogou o he developmen of a geomeric algorihm ha can deec if he given deign conain a feaure wih zero draf angle. Thi paper provide a ep oward he auomaed validaion of a propoed deign concep. We formally define he problem of ineracion coniency-checking of ineracion ae a a key ep in he deign concep validaion. We alo preen polynomial ime algorihm for olving he ineracion coniency-checking problem and for idenifying inconien ineracion. We believe ha he framework decribed in hi paper will provide he underlying foundaion for conrucing he nex generaion ofware ool for concepual deign of complex mecharonic yem. The remainder of he paper i organized in he following manner. Secion 2 preen an overview of he modeling framework for repreening muli-ae mecharonic yem concep. Secion 2 preen alo he validaion problem formulaion and an overview of he relaed reearch. Secion 3 explain how he validaion problem can be convered ino he deerminaion of he ize of a minimum - cu problem of a nework. Secion 4 preen algorihm o olve he validaion problem along wih an analyi of he algorihm. Secion decribe he implemenaion and example. Finally, Secion preen concluding remark. 2 BACKGROUND AND PROBLEM FORMULATION 2. Overview of he modeling framework We call he reul of he concepual deign age a deign concep. A deign concep need o have he following hree main ingredien. Fir, he deign concep will need o idenify he major componen ha will be needed o mee he cuomer requiremen. Second, he deign concep will need o pecify he baic working principle behind every main componen o enure ha he componen can be fully realized. Third, he deign concep will need o pecify how he componen will inerac wih each oher. Figure how he overall concepual deign modeling framework and he main primiive ued in hi framework. The raionale behind he main primiive hown in hi figure i a follow. We need o be able o model even in he ueenvironmen o which he deign concep will repond. To enure he afey of he operaion, we need o be able o model unafe world-ae. Thee are he ae ino which he device hould never ener becaue i can caue ignifican operaional difficulie or creae hazardou condiion. For example, when he door of a machine ool i open, he pindle hould no roae. Engineering characeriic are needed o pecify he quaniaive conrain aociaed wih he operaion of he device being deigned. We need raniion diagram o model how he device (or he componen of he device) inerac wih he ue-environmen in repone o variou even in he ueenvironmen. Figure 2 how he primiive ued in defining he ineracion-ae raniion diagram. The dependency among variou primiive definiion i hown in Figure 3. The primiive a he ar of he arrow i needed for defining he primiive a he end of he arrow. In our framework, he fir ak i o define he iniial primiive of he deign world. The deign world uually include he device o be deigned and he objec in he ueenvironmen wih which he device inerac. For conrucing objec, we need o conruc aribue and aribueineracion for each objec. The device objec repond o even in he ue-environmen. Thu, given a deign problem, we hould alo define he even pace. Afer objec are creaed, we can ue hee objec o conruc a e of ineracion-ae by adding objec ineracion according o he cuomer requiremen. Afer ineracion-ae are conruced, we can define a e of ae raniion o conruc an iniial raniion diagram. Engineering characeriic hould alo be defined a par of he behavior pecificaion. Afer he iniial raniion diagram i conruced, he device objec may need o be furher decompoed uch ha hey can be realized via known 2 Copyrigh 2004 by ASME

3 Deigner Cuomer Requiremen Idenify Behavior Specificaion Behavior Specificaion Iniial Ineracion ae raniion diagram Even pace Unafe ae Engineering characeriic Concepual Deign (Performed by Elaboraion) Deign Concep Deailed Ineracion ae raniion diagram Even pace Unafe ae Figure : Uage of he modeling framework. Working principle Aribue Objec Aribue ineracion Objec ineracion Ineracion ae Sae raniion Traniion diagram Objec mapping Even pace Deign world ae Figure 2: Dependency relaionhip beween primiive. Ineracion Sae Traniion Diagram Ineracion Sae Objec Aribue Aribue-ineracion Aribue Objec Ineracion Sae Objec Aribue Objec-ineracion Objec-ineracion Traniion Ineracion Sae Traniion Aribue Aribue Aribue-ineracion Figure 3: Srucure of ineracion ae raniion diagram. 3 Copyrigh 2004 by ASME

4 working principle. The elaboraion ep mu enure ha he device deired behavior i aified. In our framework, a deign concep i defined a an ordered e coniing of he iniial behavior pecificaion, he fully elaboraed raniion diagram, he even pace, and he e of unafe world-ae. In order for a deign concep o be valid, he underlying raniion diagram will have o be valid. In order for a raniion diagram o be valid, he ineracion-ae in he raniion diagram hould only conain conien ineracion. A even pace and unafe ae have been defined, we can acually imulae he deign concep in our framework and evaluae i performance. Thi paper focue on he problem of ineracion coniency checking. An ineracion-ae i defined by a e of objec and heir invarian ineracion. Conien ineracion will enure valid ineracion ae. The relaionhip among objec are called objec ineracion. The relaionhip among aribue are called aribue ineracion. The objec ineracion hould be reolved ino aribue ineracion, herefore we only dicu aribue ineracion. The aribue ineracion are decribed by he following properie: Ineracion Reaon indicae wha caue he ineracion. I can be energy flow, ignal flow, or ma flow ha pecify flow ype of relaionhip beween aribue. Or he reaon can be paial, which indicae paial conrain among a e of componen. Furhermore, he reaon can be law, which indicae he phyical law governing he relaionhip among phyical aribue of a componen. Ineracion Type indicae wheher he ineracion i caual. For non-caual ineracion, here i no need o pecify he dependence among aribue. For example, he ineracion of he ma and he volume of an objec wih uniform deniy i a non-caual ineracion. For caual ineracion, we need o pecify he dependen relaionhip beween aribue. Paricipaing Aribue decribe he name of he aribue ha inerac. Equaion decribe he equaion behind he ineracion. From he perpecive of he governing equaion behind he relaionhip, here are wo ype of ineracion: Simple Ineracion: Thee can be modeled uing algebraic or ordinary differenial equaion. For example, he ineracion of he ma aribue and he volume aribue of an objec wih uniform deniy i given by m = dv, where m i he ma value, d i he deniy, and v i he volume of he objec. Complex Ineracion: Thee canno be modeled explicily uing equaion during concepual deign. If he imulaion of he deign concep i neceary, hen a implified numerical imulaion can be ued a a model for hee ype of ineracion. For example, he ineracion among a ligh ource, a camera, and a peron (in which he ligh from he ource reflec from he peron face and form he image a he camera len) canno be modeled by algebraic equaion or ordinary differenial equaion. However, qualiaive relaionhip can be defined beween he aribue. In he concepual deign age, deigner uually are no much concerned abou he deail of he equaion behind an aribue ineracion. Inead, hey are inereed in he qualiaive relaionhip beween aribue. 2.2 Problem formulaion Le X be he e of aribue belonging o all he objec in an ineracion-ae. Le F be he e of ineracion in ae defined over X. Each f in F i a ube of X and decribe an ineracion. During he concepual deign age we are only concerned wih he qualiaive naure of ineracion. For example, conider he hybrid car example. Le u aume we only conider major objec: engine, baery, moor, ranmiion and he wheel. The hybrid car i required o ener differen ineracion-ae when he road condiion change. When he vehicle ravel uphill or accelerae, he engine and he baery boh provide power o he wheel hrough he ranmiion and moor repecively. The road can be modeled a a ue-environmen objec. In hi cae, he ineracion-ae coni of hee objec and heir ineracion. Figure 4 graphically how he ineracion opology. Baery Moor Wheel Road Engine Tranmiion Figure 4: Example of an ineracion-ae for hybrid car. We ue he following noaion o repreen he main aribue paricipaing in he ineracion: x = Baery Power Oupu, = Moor Power Oupu, x 3 = Engine Power Oupu, x 4 = Tranmiion Power Oupu, x = Wheel Power Inpu, x = Road Slope. Then we can li he paricipaing aribue in ineracion when he vehicle i going uphill a follow: f = {x, }, = {x 3, x 4 }, f 3 = {, x 4, x }, f 4 = {x, x }, Each of he above-decribed e of aribue implie ha here exi a pecific relaionhip among he aribue in he e and hence all he aribue in he e canno be aigned value independenly. Pleae noe ha we are no concerned abou he pecific equaion ha i aociaed wih he ineracion. In mo cae, uch equaion are no available a he concepual deign age. So we are only concerned abou he e of aribue ha paricipae in an ineracion. We alo model he conrain on he value of individual aribue a e of paricipaing aribue coniing of only one member. Since here i a maximum power conrain on he engine power oupu, we have f = {x 3 }. The lope of he road i deermined by he ue-environmen; herefore we model 4 Copyrigh 2004 by ASME

5 i a f = {x }. Therefore, in hi cae we have ix variable and ix ineracion in hi ineracion-ae. We formulae he ineracion coniency problem in he following manner. Given, Se X = {x,, x n } Se F = {f,, f 3, f m }, where each f i X and F = X n m The problem of ineracion coniency i o deermine if here exi F F uch ha cardinaliy(f ) > cardinaliy( F ). If uch F exi, hen he given e of ineracion i conidered inconien. Le u conider he following example, X = {x,, x 3, x 4, x } F = {f = {x 3, x 4, x }, = {x, x 3 }, f 3 = {x, }, f 4 = {x,, x 3 }, f = {, x 3 }} Alhough here i a oal of five aribue and only five ineracion, he la four ineracion (i.e., = {x, x 3 }, f 3 = {x, }, f 4 = {x,, x 3 }, f = {, x 3 }) only involve hree variable (i.e., x,, x 3 ). Therefore, hee ineracion are overconrained. Thu, he ineracion in hi ae are inconien and hi ae i invalid. If n < m, he e of ineracion i obviouly inconien. Thu we only deal wih cae in which n m. Here we aume ha no redundan equaion will be ubequenly ued in he deailed deign age o realize he e of ineracion. A redundan equaion can be deduced from a e of oher equaion. For example, aume ha we have he following wo equaion: x + =3, + x 3 =. Then he equaion x 3 - x =2 can be derived from he fir wo equaion and hence i i a redundan equaion. If he e of ineracion i inconien, a naural problem ha arie i idenifying he ineracion ha lead o he inconiency. Deigner need o locae he ube of inconien ineracion and modify hem o enure ha he modified ineracion are conien. 2.3 Overview of our approach Given he e of ineracion, we ue he following approach o olve he problem: ) Conruc an ineracion nework from he e of ineracion. Secion 3. how how he nework i conruced. Then we how ha he coniency problem i equivalen o checking he ize of he minimum - cu problem in he ineracion nework. Secion 3.2 preen he proof for hi equivalence. 2) We ue he algorihm FINDMINIMUMSTCUTSIZE o compue he ize of he minimum - cu of he nework and find ou wheher he e of ineracion i conien. Secion 4. preen hi algorihm. If he ineracion are found o be inconien, hen we deermine he e of ineracion ha lead o inconiency. Secion 4.2 decribe he algorihm FINDINCONSISTENTINTERACTIONS defined for hi ak. 2.4 Relaed work The uual approach o olve he minimum cu problem i o ue i cloe relaionhip o he maximum flow problem. Ford and Fulkeron howed he dualiy of he maximum flow and he minimum --cu in heir famou Max-Flow-Min-Cu-Theorem [Ford]. They alo gave a imple algorihm for olving he problem. Finding a minimum cu wihou pecifying he verice o eparae can be done by finding minimum --cu for a fixed verex and all V - poible choice of V - {} and hen elecing he malle one. Goldberg and Tarjan ued puh-relabel algorihm o achieve a faer compuaion. They do no mainain a valid flow during he operaion; each node may have a poiive flow exce, and he algorihm rie o puh i o neighboring node. Many modificaion baed on hee wo ype of approache have been made o achieve faer algorihm. Algorihm ha are no baed on flow have alo been developed. Nagamochi and Ibaraki gave a procedure ha repeaedly idenifie and conrac edge ha are no in he minimum cu unil he minimum cu become apparen. I applie only o undireced graph wih non-uniform edge weigh [Naga92]. The approach by Gabow i baed on a maroid characerizaion of he minimum cu problem. According o hi characerizaion, he minimum cu in a graph i equal o he maximum number of dijoin direced panning ree ha can be found in i. Gabow algorihm find he minimum cu by finding uch ree [Gabo9]. Karger and Sein give a randomized algorihm ha find he minimum cu in an arbirarily weighed undireced graph [Karg9]. 3 MAPPING CONSISTENCY CHECKING PROBLEM TO MINIMUM S-T CUT PROBLEM IN INTERACTION NETWORK 3. Conrucion of ineracion nework We build an ineracion nework G ha decribe how ineracion F and aribue X are relaed o each oher. There are four kind of node in G: -node: Source node. -node: Sink node. x-node: Node correponding o an aribue in X. f-node: Node correponding o an ineracion in F. There are hree ype of edge in G: f-edge: Edge connecing he -node o an f-node. The capaciy of hi edge i uni. fx-edge: Edge connecing an f-node o an x-node. The capaciy of hi edge i n+ uni. x-edge: Edge connecing an x-node o he -node. The capaciy of hi edge i uni. Now we preen he algorihm for conrucing he ineracion nework G. Algorihm CONSTRUCTINTERACTIONNETWORK Inpu: Syem of ineracion F wih repec o X. There are n variable in X and m ineracion in F. Oupu: Ineracion nework G Sep: ) Creae an empy nework G. 2) Iner node ino nework G. Label hi node a - node. 3) Iner node ino nework G. Label hi node a - node. 4) Iner a node for every f F ino G. Label hee node a f-node. Creae an edge from he -node o every f- Copyrigh 2004 by ASME

6 node. Label hee edge a f-edge. Se he capaciy of every f-edge o. ) Iner a node for every x X ino G. Label hee node a x-node. Creae an edge from every x-node o he - node. Label hee edge a x-edge. Se he capaciy of every x-edge o. ) For every f, iner an edge from he f-node o an x- node if x belong o f. Label hee edge a fx-edge. Se he capaciy of every fx-edge o n+. Figure how nework G for he following aribue and ineracion: X = {x,, x 3, x 4, x } n = F = {f = {x 3, x 4, x }, = {x, x 3 }, f 3 = {x, }, f 4 = {x,, x 3 }, f = {, x 3 }} m = f f 3 f 4 f f = { x 3, x 4, x }, = { x, x 3 }, f 3 = { x, } f 4 = { x,, x 3 }, f = {, x 3 } Figure : Ineracion nework conruced from he above relaionhip. 3.2 Mapping Coniency-Checking Problem o Minimum Cu Problem In hi ecion we will how ha he ineracion coniency-checking problem can be mapped o he problem of checking he ize of he minimum - cu in nework G. Le G = (V, E) be an edge-weighed direced graph (digraph) wih a finie e of verice V and a e of ordered pair of verice, E V V, called edge. We ypically ue e or (u, v) o denoe an edge e = (u, v). c(e) i called he capaciy of e. A nework i a digraph in which wo verice are diinguihed a he ource and he arge where, and in which each edge ha a non-negaive capaciy. A flow in a nework i defined o be a funcion f ha aign a real number o each edge, ubjec o wo conrain: Flow of an edge i non-negaive and le han or equal o he capaciy; For each verex oher han he ource and he arge, he flow ino he verex equal he flow ou of i. The value of a flow i he ne flow ino he ink. Given a nework, a flow i a maximum flow provided i ha he large x x 3 x 4 x value among all flow. A direced - pah in G i a equence of verice and edge of he form, (, v ), v, (v, v 2 ), v 2,..., v k-, (v k-, ),. An - cu i a pariion of he node e V ino wo ube S and T = V-S. Alernaively, we can define a cu a he e of edge whoe endpoin belong o he differen ube S and T. A cu i referred o a an - cu if S and T. The ize of an - cu i he um of he capaciie of all he forward edge (edge from S o T) in he cu. An - cu i a minimum - cu provided i ha he malle ize among all - cu. A pah of a nework i a equence, e 0, v, e,, e k, wih, v,, V, and e 0, e,, e k E, uch ha i ar in, end in and doe no conain any verex wice. The reidual capaciy of an edge e i =(v i, v i+ ) i given by re(e i ) = c(v i, v i+ ) - f(v i, v i+ ) Given a flow nework G = (V, E) and a flow f, he reidual nework of G induced by f i G r = (V, E r ), where E r = {(u, v) V V: re(u, v)>0}. Each edge of he reidual nework, or reidual edge, can admi a ricly poiive ne flow. A reidual edge may no be an edge in E. An augmening pah wih repec o a nework G and a flow f i a imple pah from o in he reidual nework G r [Corm90]. Figure (a) and (b) illurae a nework G and he nework wih a flow value of. An augmening pah P can be formed by, (, f ), f, (f, ),, (, ),. The reidual capaciy of hi pah i he minimum re(p) = min{re((, f )), re((f, )), re((, ))}. Thu re(p) = min{2,, 4} = 2. The reidual nework for he nework wih a flow value of i hown in Figure (c). 3 /3 0/ (a): Original nework f f f 0/ 0/ / x Figure : Reidual nework x 0/ / (c): Reidual nework of he flow hown in (b) x (b): Nework wih flow = Now we preen mahemaical preliminarie ha prove ha he coniency-checking problem can be mapped o he problem of finding he ize of he minimum - cu in a nework. Lemma. The ize l* of he minimum - cu in nework G i le han or equal o he number of ineracion m. Proof: A cu of G can be creaed by elecing all edge wih an f-edge label from he nework (for example, ee edge 4 Copyrigh 2004 by ASME

7 in doed line in Figure 7). The ize of hi cu i equal o he um of he capaciie of all edge wih an f-edge label. There are m uch edge in G and he capaciy for each uch edge i uni. Therefore, he ize of hi cu i m. Therefore, we can conclude ha he ize l* of a minimum cu in G i le han or equal o m. f f 3 f 4 f f = { x 3, x 4, x }, = { x, x 3 }, f 3 = { x, } f 4 = { x,, x 3 }, f = {, x 3 } m=, n = Figure 7: A cu of he nework. Lemma 2. A minimum - cu of nework G canno conain an edge wih an fx-edge label. Proof: According o Lemma, he ize of he minimum - cu of G i le han or equal o m. Since he capaciy of fx-edge i n+, any cu ha conain an edge wih an fx-edge label mu have a ize of a lea n+. Since n m, any cu ha conain an fx-edge canno be a minimum - cu due o Lemma. Lemma 3. If he ize l* of he minimum - cu of nework G i le han m, hen he minimum cu mu conain a lea one f-edge and one x-edge. Proof: According o lemma 2, minimum - cu C* doe no conain any edge wih an fx-edge label. Le C* be a minimum - cu of G uch ha l* < m. Cu C* can be of he following hree ype: ) all edge in he cu are f-edge; 2) all edge in he cu are x-edge; 3) edge in he cu conain boh ype of edge. In cae and 2, we can find a pah from o. Therefore, C* canno be a cu. Thu only cae 3 produce a valid cu. Theorem. If here exi a ube of ineracion F F uch ha cardinaliy(f ) > cardinaliy( F ) (i.e. he number of ineracion i greaer hen he number of variable in he ineracion), hen here would exi a minimum - cu in nework G of a ize le han m. Proof: Fir le u conruc he ineracion nework a hown in Figure 8 according o algorihm CONSTRUCTINTERACTIONNETWORK. We define a he e of f-edge ha connec he -node wih f-node ha correpond o F-F, a he e of x-edge ha connec x- x x 3 x 4 x node ha correpond o F wih he -node, a he e of f-edge ha connec he -node wih f-node ha correpond o F, and a he e of x-edge ha connec x-node ha correpond o X- F wih he -node. And we define he cardinaliie of hee e of edge a he following: F- F F X- (F ) (F ) Figure 8: A cu illuraing erminology ued in Theorem. l f = cardinaliy( ), l x = cardinaliy( ), l f = cardinaliy( ), l x = cardinaliy( ) Since for every f-node F, here i only one f-edge ha connec i wih he -node, cardinaliy(f ) = cardinaliy( ), hu l f = cardinaliy(f ). Similarly, ince for every x-node F, here i only one xedge ha connec i wih he -node, cardinaliy( F ) = cardinaliy( ), hu l x = cardinaliy( F ). Cu C = i an - cu (hown in doed line in Figure 8) baed on i conrucion. We define l = cardinaliy (C). According o he conrucion of he nework, we have l f + l f = m () According o he definiion of C we have l = l f + l x We are given cardinaliy(f ) > cardinaliy( F ), ha i l f > l x (2) Hence l = l f + l x l < l f + l f (by 2) l < m (by ) Since, cardinaliy(c) < m and cardinaliy(c*) cardinaliy(c), cardinaliy(c*) < m. Figure 9 how an example furher illuraing erminology ued in hi Theorem. Theorem 2. Le C* be a minimum - cu of he ineracion nework G, and he ize of he cu l* be le han m. In hi cae here would exi F F uch ha cardinaliy(f ) > cardinaliy( F ). Proof: According o Lemma 2 and Lemma 3, he cu mu be formed in he manner hown in Figure 0. We define a he e of f-edge ha connec he -node wih f-node ha correpond o F, a he e of f-edge ha connec he - node wih f-node ha correpond o F-F, a he e of xedge ha connec x-node correponding o (F-F ) wih he -node, and a he e of x-edge ha connec x-node ha correpond o X- (F-F ) wih he -node. And we define he cardinaliie of hee e of edge a he following: l f = cardinaliy( ), l x = cardinaliy( ), l f = cardinaliy( ), l x = cardinaliy( ) 7 Copyrigh 2004 by ASME

8 F- F (F ) f f 3 f 4 f x x 3 x 4 x F X- (F ) Figure 9: An example of a cu for illuraing Theorem. F X- (F- F ) F- F (F- F ) Proof: I direcly follow from Theorem 2. Theorem 3. Le C* be a minimum cu of ize le han m. Le F be he e of f-node ha are conneced o -node by edge ha are no in C*. Le F be he e of all f-node. Then F (F-F ), cardinaliy(f ) cardinaliy ( F ). Proof: We will prove hi heorem by conradicion. Aume here exi F (F-F ) uch ha cardinaliy (F ) > cardinaliy ( F ). We define a he e of f-edge ha connec he -node wih f-node ha correpond o F-F, a he e of x-edge ha connec x-node correponding o F wih he -node, a he e of f-edge ha connec he -node wih f-node ha correpond o F, a he e of x-edge ha connec x-node ha correpond o X- F wih he -node, a he e of fedge ha connec he -node wih f-node ha correpond o F, and a he e of x-edge ha connec x-node correponding o F wih he -node. Since for every f-node F, here i only one f-edge ha connec i wih he -node, cardinaliy(f ) = cardinaliy( ). Similarly, ince for every x-node F, here i only one xedge ha connec i wih he -node, cardinaliy( F ) = cardinaliy( ). Then he aumpion can alo be repreened a cardinaliy( ) cardinaliy( ) < 0 () We eparae F from F-F a hown in Figure. Obviouly ( - ) i alo a cu C of he nework. - F-F -F X- (F ) - (F ) - Figure 0: A cu illuraing erminology ued in Theorem 2. According o he conrucion of he nework, we have l f + l f = m (3) Since cu C* =, cardinaliy (C*) = l f + l x (4) According o Lemma 3, we alo have: l f > 0 and l x >0 We are given cardinaliy (C*) < m, hu l f + l x < m (by 4) l f + l x < l f + l f (by 3) Then we have l x < l f Tha ae ha cardinaliy( ) < cardinaliy ( ) Since for every f-node F, here i only one f-edge ha connec i wih he -node, cardinaliy(f-f ) = cardinaliy( ). Similarly, ince for every x-node F, here i only one xedge ha connec i wih he -node, cardinaliy( (F-F )) = cardinaliy( ). Therefore, cardinaliy( (F-F )) < cardinaliy (F-F ). We rename (F-F ) a F, hen we have cardinaliy(f ) > cardinaliy( F ) Corollary. Le C* be a minimum - cu of ize le han m and be he e of f-edge ha are no in C*. The e of inconien ineracion i repreened by he f-node ha are conneced o he -node by edge in. F F F (F ) Figure : A cu illuraing erminology ued in Theorem 3. According o he definiion of cu, cardinaliy(c*) = cardinaliy( ) + cardinaliy( ) () cardinaliy(c ) = cardinaliy( - ) + cardinaliy( ) + cardinaliy( ) (7) cardinaliy( - ) = cardinaliy( ) cardinaliy( ) (8) Thu, cardinaliy(c ) = cardinaliy( ) cardinaliy( )+ cardinaliy( ) + cardinaliy( ) (9) (by 7 and 8) Then cardinaliy(c ) - cardinaliy(c*) = cardinaliy( ) cardinaliy( ) < 0 (by and 9) Therefore, cardinaliy (C ) < cardinaliy (C*) Thu C* i no a minimum - cu. Thi conradic wih he Theorem aemen. 8 Copyrigh 2004 by ASME

9 From he above heorem and corollary, we can conclude ha he coniency-checking problem can be olved by finding he ize of he minimum - cu of G. Theorem 3 help in enuring ha here are no oher inconien ineracion ha are no covered by Corollary. 4 ALGORITHMS FOR FINDING MINIMUM S-T CUT AND IDENTIFYING INCONSISTENT INTERACTIONS 4. Algorihm for finding minimum - cu in nework G According o he dualiy beween maximum flow problem and minimum cu problem, he ize of he minimum - cu can be found by compuing he maximum flow beween and. Our algorihm i baed on Ford and Fulkeron baic maximum flow algorihm of finding he augmening pah. Algorihm FINDMINIMUMSTCUTSIZE Inpu: A direced nework G Oupu: The ize of he minimum cu of G and he reidual nework G r of G Sep: ) Se ize of minimum cu o 0. 2) Iniialize flow of he nework, e f(e) = 0, e E. 3) Se G r = G. 4) Find an augmening pah from he -node o he -node in G r a. If a pah i found, hen i. Augmen flow along hi pah. ii. Increae he ize of he minimum cu by. iii. Generae new reidual nework G r. iv. Go o Sep 4. b. Ele, reurn he ize of he minimum cu and reidual nework. The working of hi algorihm i illuraed in Figure 2. Figure 2a how he original nework. Iniially, he reidual nework i he ame a hi nework (ee Sep 3 of he above algorihm). Figure 2b how an - pah a, (, f ), f, x, (x, ),. Sending a uni flow along hi pah will aurae he flow capaciie in edge (, f ) and (x, ) a hown in Figure 2c. The reidual nework wih repec o hi flow i hown in Figure 2d. A new pah hown in Figure 2e i found a, (, ),, (, x ), x, (x, f ), f, (f, ),, (, ),. Now we analyze he complexiy of hi algorihm. Sep can be execued in ime O(). Sep 2 can be done in ime O(E) where, E = Number of f-edge + Number of fx-edge + Number of x-edge. E ha an upper bound of n + nm + m. Thu, Sep 2 ake ime O(nm). Sep 3 ake O(V + E). Since O(V) = O(n+m), ep 3 ake O(nm). Sep 4 will be execued a mo m ime. For a deph-fir earch, Sep 4a ake ime O(E) + O() + O(V + 2E) = O(nm). Sep 4b ake O() ime. Thu in he wor cae, Sep 4 ake O(nm 2 ). Thu he wor cae ime complexiy for hi algorihm i O(nm 2 ). For he nework hown in Figure, we find he ize of he minimum - cu of he nework. In hi cae C* = 4 a hown in Figure 3. The maximum flow of he nework i alo hown in Figure 3. Since m =, he e of he ineracion i no conien. The reidual nework wih repec o he maximum flow i hown in Figure 4. Ford and Fulkeron algorihm find maximum flow by finding all he augmening pah in he nework from o and auraing he flow along he pah. However, here are everal characeriic of our problem ha can be ued o reduce he complexiy of he algorihm direcly. ) The nework in our problem i acually a pecial nework. Nework G = (V, E) ha a node e V pariioned ino wo ube V and V 2 o ha for every edge e i =(v i, v i+ ) E, eiher v i V and v i+ V 2 or v i V 2 and v i+ V. Thu any - pah follow he paern, f, x, f, x, f, x,, in which f-node and x-node appear in a pair wie manner. 2) Every f-edge and x-edge ha capaciy of. Tha mean ha once uch an edge i ued in a pah, i won be ued in anoher pah. Meanwhile, an f-node or an x-node alo can only be ued in one pah. 4.2 Algorihm for finding inconien ineracion Algorihm FINDINCONSISTENTINTERACTIONS Inpu: Ineracion reidual nework G r correponding o he maximum flow Oupu: e of f-node correponding o inconien ineracion Sep: ) Ue deph-fir earch o find all node in reidual nework G r ha are reachable from -node and pu hee node in e R. 2) Remove x-node from R and reurn R. Now we will how ha R correpond o he f-node ha are conneced o he -node by edge in a aed in Corollary. We denoe he node e ha i reachable from in G r a V, and he e of he remaining node a V 2 =V V. There i no pah in he reidual nework uch ha he -node reache he -node. Oherwie, an augmening flow could have been generaed and hence flow would have no been maximum. Thu, V and V 2. Therefore, cu C = {V, V 2 } i an - cu. Since he flow i maximum, according o he dualiy beween maximum flow and minimum cu, C i a minimum - cu [Ford]. Therefore, we conclude ha inconien ineracion can be found by finding reachable node in he reidual nework correponding o he maximum flow. Since we are only concerned abou he inconien ineracion, we remove x-node in he reachable node e. Now we analyze he complexiy of hi algorihm. For a deph-fir earch, Sep can be execued in ime O(E + V) = O(nm). Sep 2 ake ime O(n+m). Therefore, hi algorihm run in O(nm). For he nework hown in Figure, he reidual nework wih repec o he maximum flow i hown in Figure 4. Now we can find he reachable node from -node a {, f 3, f 4, f, x,, x 3 } a hown in Figure. Thu he e of ineracion node {, f 3, f 4, f } i inconien. One can eaily verify ha here are only hree variable {x,, x 3 } involved in four ineracion {, f 3, f 4, f }. IMPLEMENTATION AND EXAMPLES We have implemened he algorihm decribed in hi paper uing C++ and ha been eed on he Window 2000 plaform. We ran he program on a PC wih he following configuraion: () AMD Ahlon XP700+ CPU and (2) GB Memory. 9 Copyrigh 2004 by ASME

10 f x f x (a): Original nework (b): Pah: (, f, x, ) / 0/ f 0/ / 0/ x 0/ / f x (c): Add a flow along imple pah: (, f, x, ) (d): Reidual nework of he above nework f x (e): Anoher pah : (,, x, f,, ) Figure 2: Illuraion of algorihm FINDMINIMUMSTCUTSIZE. f 0/ x 0/ / / / / / f 3 0/ / 0/ / / / 0/ 0/ x 3 / / / 0/ f 4 0/ 0/ x 4 0/ f x Figure 3: Maximum flow of he ineracion nework. 0 Copyrigh 2004 by ASME

11 f x f 3 x 3 f 4 x 4 f x Figure 4: Reidual nework correponding o he maximum flow. f x f 3 f 4 x 3 x 4 f x Figure : Finding inconien relaionhip. Copyrigh 2004 by ASME

12 Figure how deign aleraive A behind a device baed on a planar mechanim. There are acive objec ha repreen variou link in he device (he ground objec i no couned). Every objec can be decribed by hree aribue (x, y, θ). Thee aribue preen he x and y coordinae of he cener of he objec, and i orienaion. In hi device, objec inerac wih each oher via join. We aume ha all join in hi cae are pivo join. The preence of a pivo join reduce wo degree of freedom beween wo link. Thi mean ha while (x, y, θ) aribue for one of he link can be aigned independenly, only one variable for he econd link can be aigned independenly. Therefore, a per our erminology, here are wo ineracion among objec due o he preence of he pivo join. Boh of hee ineracion involve he ame e of variable. However, he equaion behind each ineracion will be differen and can only be found afer aigning dimenional parameer o he link. A menioned before, we do no care abou he acual equaion involved bu raher he e of aribue ha paricipae in an ineracion. A Join Join B Join Join C f 7 = {θ A }, f 8 = {θ C } Then he ineracion coniency problem for hi device i formulaed a he following: X = {x A, y A, θ A, x B, y B, θ B, x C, y C, θ C, x D, y D, θ D, x E, y E, θ E, x F, y F, θ F } F = {f,, f 3,, f 8 } n = 8 and m = 8 By running our ofware, we ge he following reul: The ize of he minimum - cu i 7 < m, hu he ineracion are inconien. The e of inconien ineracion are idenified a {f,, f 3, f 4, f 3, f 4, f, f, f 7, f 8 }. Thee en ineracion only involve nine variable. Hence hi deign concep i no valid. Now le u conider anoher deign alernaive. Thi deign alernaive called alernaive B i hown in Figure 7. Thi alernaive ha he ame number of objec and join. However, he ineracion opology i differen. Ineracion in hi deign can be modeled a he following: Join B Join C Join Join Ground Join A Join Join D F Join E Join D Join F Ground Join E Join Figure : Deign alernaive A. Therefore, ineracion among objec due o he preence of join can be decribed by he following e of paricipaing aribue: f = {x A, y A, θ A, x B, y B, θ B }, = {x A, y A, θ A, x B, y B, θ B } Similarly, for oher join we ge f 3 = {x B, y B, θ B, x C, y C, θ C }, f 4 = {x B, y B, θ B, x C, y C, θ C } f = {x C, y C, θ C, x D, y D, θ D }, f = {x C, y C, θ C, x D, y D, θ D } f 7 = {x D, y D, θ D, x E, y E, θ E }, f 8 = {x D, y D, θ D, x E, y E, θ E } f 9 = {x E, y E, θ E, x F, y F, θ F }, f 0 = {x E, y E, θ E, x F, y F, θ F } f = {x F, y F, θ F, x A, y A, θ A }, = {x F, y F, θ F, x A, y A, θ A } Objec A and C are conneced o he ground via pivo join, o we need o model he following ineracion: f 3 = {x A, y A, θ A }, f 4 = {x A, y A, θ A } f = {x C, y C, θ C }, f = {x C, y C, θ C } We wan o have wo degree of freedom in hi device. Thee conrain are modeled a ineracion a well. However, only one aribue paricipae in hee wo ineracion. Therefore, we ge Figure 7: Deign alernaive B f = {x A, y A, θ A, x B, y B, θ B }, = {x A, y A, θ A, x B, y B, θ B } f 3 = {x B, y B, θ B, x C, y C, θ C }, f 4 = {x B, y B, θ B, x C, y C, θ C } f = {x C, y C, θ C, x D, y D, θ D }, f = {x C, y C, θ C, x D, y D, θ D } f 7 = {x D, y D, θ D, x E, y E, θ E }, f 8 = {x D, y D, θ D, x E, y E, θ E } f 9 = {x E, y E, θ E, x F, y F, θ F }, f 0 = {x E, y E, θ E, x F, y F, θ F } f = {x F, y F, θ F, x A, y A, θ A }, = {x F, y F, θ F, x A, y A, θ A } Objec A and objec D are conneced o he ground, o we have he following ineracion: f 3 = {x A, y A, θ A }, f 4 = {x A, y A, θ A } f = {x D, y D, θ D }, f = {x D, y D, θ D } We again wan o have wo degree of freedom in he yem. So we ge, f 7 = {θ A }, f 8 = {θ C } By running our ofware, we ge he following reul: The ize of he minimum - cu i 8 = m, hu he ineracion are conien. Thi example illurae ha he ineracion opology can have ignifican influence on he validiy of a deign concep. 2 Copyrigh 2004 by ASME

13 CONCLUSIONS The concepual deign age for mecharonic produc currenly lack compuer-uppored deign ool compared o he deailed deign age. The problem lie in he lack of formal repreenaion and evaluaion cheme. Developmen of a heoreical foundaion in he area of repreenaion of mecharonic yem concep wih muliple ineracion-ae will grealy faciliae he developmen of compuer-aided deign ool for he concepual deign age, hu reamlining he enire deign proce. Our paper preen a yemaic approach o checking he coniency of a e of ineracion in an ineracion-ae of a mecharonic yem. We alo provide an algorihm o find he e of ineracion ha caue he inconiency. During he concepual deign age, he acual equaion decribing he ineracion are uually no known. Therefore our algorihm only uilize he informaion on paricipaing aribue o carry ou i analyi. We have hown boh he oundne and compleene of our algorihm. Thi implie ha when our algorihm find a e of ineracion o be inconien, hey are acually inconien. Furhermore, when our algorihm find a e of ineracion o be conien, hey are acually conien. Even hough he coniency-checking problem ha an appearance of a combinaorial problem, we have found an algorihm ha work in polynomial ime and doe no require exhauive enumeraion. The algorihm decribed in hi paper preen a ep oward auomaed validaion of a propoed deign concep. We believe ha he framework decribed in hi paper will provide he underlying foundaion for conrucing he nex generaion ofware ool for concepual deign of complex mecharonic yem. Poible direcion for fuure exenion include: ) Minimum alernaive inconien ineracion e: Thi paper decribe an algorihm o find an inconien ineracion e. However, here could be many differen alernaive inconien ineracion e. We will need o have an algorihm ha can find he minimum e. 2) Faer algorihm: The algorihm preened in he paper are baed on claic Ford and Fulkeron algorihm. Oher reearcher have propoed many faer algorihm o find he minimum cu. We believe ha by combining ome of he approache and he pecial characeriic of he underlying ineracion nework, we hould be able o develop a much faer algorihm. 3) General Validaion Problem: Thi paper focue only on one apec of he deign validaion problem. We will need o develop algorihm for oher apec of he validaion problem uch a realizabiliy of ae raniion, and validiy of ae raniion diagram. REFERENCES [Chan90] B. Chandraekaran. Deign Problem Solving: A Tak Analyi. AI Magazine, Vol., 990. [Corm90] T. H. Cormen, C. E. Leieron and R. L. Rive. Inroducion o Algorihm. The MIT Pre, 990. [Ford] L. R. Ford Jr., and D. R. Fulkeron. Maximal Flow hrough a Nework. Can. J. Mah. 8, , 9. [Gabo9] H. N. Gabow. A Maroid Approach To Finding Edge Conneciviy And Packing Arborecence. J. Compu. Sy. Sci. 0(2):29 273, 99. [Gold88] A. V. Goldberg and R. E. Tarjan. A New Approach To The Maximum-Flow Problem. J. ACM 3:92 940, 988. [Gup04] S. K. Gupa, C. Xu and Z. Yao. A Framework For Concepual Deign of Muliple Ineracion-Sae Mecharonic Syem. In Proceeding of he Fifh Inernaional Sympoium on Tool and Mehod of Compeiive Engineering, Lauanne, Swizerland, April 3-7, [Karg9] D. R. Karger and C. Sein. A New Approach o Minimum Cu Problem. Journal of he ACM, 43(4): 0 40, July 99. [Naga92] H. Nagamochi, and T. Ibaraki. Compuing Edge Conneciviy In Muligraph And Capaciaed Graph. SIAM J. Dic. Mah. ():4, February 992. [Wal0] R. M. Waler. Overview Of The Deign And Developmen Of Mecharonic Syem. In Proceeding of he h World Muliconference on Syemic, Cyberneic and Informaic / The 7h Inernaional Conference on Informaion Syem Analyi and Synhei, Orlando, U.S.A., July 200. ACKNOWLEDGMENT The auhor graefully acknowledge he uppor provided by he Manufacuring Syem Inegraion Diviion of he Manufacuring Engineering Laboraory a NIST. Opinion expreed in hi paper are hoe of auhor and do no necearily reflec opinion of he ponor. 3 Copyrigh 2004 by ASME