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1 This document contains the daft vesion of the following pape: R. Sinha, S.K. Gupta, C.J. Paedis, P.K. Khosla. Extacting aticulation models fom CAD models of pats with cuved sufaces. ASME Jounal of Mechanical Design, 4():06-4, 00. Reades ae encouaged to get the official vesion fom the jounal s web site o by contacting D. S.K. Gupta (skgupta@umd.edu).

2 Extacting Aticulation Models fom CAD Models of Pats with Cuved Sufaces Rajaishi Sinha *, Student Membe, ASME Institute fo Complex Engineeed Systems Canegie Mellon Univesity Pittsbugh, PA 53 Tel.: Fax: Satyanda K. Gupta, Membe, ASME Depatment of Mechanical Engineeing and Institute fo Systems Reseach Univesity of Mayland College Pak, MD Chistiaan J.J. Paedis, Membe, ASME Institute fo Complex Engineeed Systems and Depatment of Electical and Compute Engineeing Canegie Mellon Univesity Pittsbugh, PA 53 Padeep K. Khosla, Membe, ASME Depatment of Electical and Compute Engineeing and Institute fo Complex Engineeed Systems Canegie Mellon Univesity Pittsbugh, PA 53 * Coesponding autho

3 Abstact In an assembly, degees of feedom ae ealized by ceating mating featues that pemit elative motion between pats. In complex assemblies, inteactions between individual degees of feedom may esult in a behavio diffeent fom the intended behavio. In addition, cuent methods pefom assembly easoning by appoximating cuved sufaces as piecewise linea sufaces. Theefoe, it is impotant to be able to eason about assemblies using exact epesentations of cuved sufaces; veify global motion behavio of pats in the assembly; and ceate motion simulations of the assembly by examination of the geomety and mateial popeties. In this pape, we pesent a linea algebaic constaint method to automatically constuct the space of allowed instantaneous motions of an assembly fom the geomety of its constituent pats. Ou wok builds on pevious wok on linea contact mechanics and cuved suface contact mechanics. We enumeate the conditions unde which geneal cuved sufaces can be epesented using a finite numbe of constaints that ae linea in the instantaneous velocities. We compose such constaints to build a space of allowed instantaneous velocities fo the assembly. The space is then descibed as a set-theoetic sum of contact-peseving and contact-beaking subspaces. Analysis of each subspace povides feedback to the designe, which we demonstate though the use of an example assembly a 4-pat mechanism. Finally, the esults of the analysis of a 4-ba linkage ae compaed to those fom mechanism theoy. Intoduction and Motivation In cuent design pactice, when a device that contains pats with cuved sufaces is consideed, easoning about its behavio is usually done by analytical methods that ae difficult to implement [, ], o by appoximating all the cuves as piecewise plana sufaces [3]. The appoximation is usually based on past expeience o on heuistics. Howeve, such appoximations may invalidate the analysis by unintentionally constaining the degees of feedom in the device. These eoneous esults popagate thoughout the design pocess, influencing the decisions of pat and assembly designes, device analysts, pocess enginees, and opeatos. Anothe consideation is the euse of the deived device models. Even in a collaboative setting, when a pat o assembly fom a pevious design poject is eused, sometimes all that is known a pioi is the geomety of the pat o assembly. Only the CAD model of the pat o assembly pesists acoss designs. Behavioal models of the pat o assembly ae not eused. In addition, featue-based design of complex devices equies veification of the fact that actual kinematic behavio matches the equied behavio. If component models that captue the geomety as well as the physical behavio of the device can be ceated, it should be possible to constuct system-level simulation models meely by configuing the component models [4]. The above discussion indicates a need fo thee capabilities that will suppot all aspects of the design pocess - ceation of the ability to exactly eason with a boad class of cuved sufaces; the ability to obtain kinematic models fom assembly geomety; and the ability to encapsulate kinematic behavio models and CAD models in the epesentation of a pat o an assembly. Ou eseach suppots the fist two aspects, and we ae cuently woking on a famewok that suppots modula simulation-based design [4].

4 Famewoks fo the detection and epesentation of aticulation behavio in assemblies can be estictive. In ode to geneate assembly o disassembly plans fo such assemblies, the designe needs to take aticulation infomation into consideation. Howeve, cuent methods of epesenting aticulation ae esticted to systems that equie complete specification by the use [5] o ae based on featue ecognition [6]. The fome ae open to incoect input by the use esulting in illegal aticulation behavio. The latte do not account fo incomplete geomety and incidental contacts. Othe methods exist to obtain aticulation models by easoning on the geometic epesentation of the atifact. Techniques have been developed to pedict the instantaneous degees of feedom fom the CAD models of pats with only polygonal plana faces [3, 7]. Howeve, when the cuved pats that exist in most engineeing devices ae appoximated as piecewise plana pats, eoneous esults ae possible. In this eseach, we pesent a methodology that genealizes ou ealie wok on contact sufaces [8]. Peviously, the instantaneous degees of feedom at each joint wee based on suface mating constaints that wee in tun obtained fom analyzing body to body contacts. We imposed nonpenetation constaints along the bounday of each contact suface in the fom of algebaic inequalities. One can show that a finite numbe of non-penetation conditions ae epesentative of the entie suface in contact. Using linea pogamming methods, we computed instantaneous velocities and acceleations fo each pai of bodies. In this aticle, we obtain a set of popeties that must be satisfied by a geneal contact suface in ode to peseve the lineaity of the model. We descibe a method by which the space of allowable motions in the assembly can be descibed concisely. We then descibe a heuistic that can find feasible joints fom the space of allowable motions. Such a methodology can povide useful feedback to the designe. He o she can detemine which components ae fee to move in the assembly. The pocedue can be completely automated, so that thee ae no eos induced by use inteaction. This eliminates the possibility of input eos. In addition, since the method is algebaic and uses linea pogamming, it is elatively fast and is valid fo all possible suface contacts, unlike ule-based systems that opeate on a featue level. This method will also account fo contact sufaces with incomplete geomety (such as potions of planes, cylindes, o sphees). Ou method is subject to the following estictions. The CAD models used in the analysis ae assumed to model suface contacts at joints. The joint contacts induce the instantaneous degees of feedom. Since the algoithms detect the instantaneous degees of feedom, they ae applicable fo a single configuation of a mechanism, and cannot be used to pedict all possible configuations of the mechanism. Review of Pevious Wok. Ealy Wok on Contact Mechanics Pevious wok on plana contact mechanics and scew theoy pepaed the foundation fo this wok. Ohwovoiole and Roth [9] showed that unidiectional constaints can be modeled as scews. Hiai and Asada [7] descibed the allowable motions of a pat using polyhedal convex cones to epesent the space of movement. Mattikalli and Khosla [0] descibed a method to obtain degees of 3

5 feedom fom component mating constaints, wheein they use a unit sphee to epesent the space of all available degees of feedom. Some issues have not yet been addessed satisfactoily in these famewoks. Cuent shotcomings in aticulation eseach include:. Only bodies descibed (o appoximated) by plana sufaces ae consideed.. Cuent techniques ae local; global inteaction (popagation of constaints beyond the point whee they ae induced) is not satisfactoy. 3. Cuent simulation techniques do not detect incoect/incomplete inputs; thee is no veification fo coectness of the aticulation epesentation and fo the compound effect of geometic inteactions and physics-based inteactions.. Contact Mechanics fo Plana Contacts A pat in an assembly is in physical contact with one o moe othe pats. The natue of these contacts can povide useful infomation about the types of the degees of feedom at these contact points. Some of these contacts induce suface mating constaints, leading to the fomation of a joint. Othe contacts ae incidental, in that they may bound the values of the degees of feedom of the joint [6, ]. Reasoning about these constaints povides the designe with valuable insight into the instantaneous degees of feedom of the assembly. Othe eseaches [3, 0] have woked with polygonal bodies and polygonal sufaces of contact. They appoximate cuved plana boundaies using staight lines, and use linea pogamming techniques to solve the contact poblem. When a pai of pats ae in contact with each othe, it implies that thee is no inte-penetation between the pats at the contact sufaces. Penetation of one pat into the othe equies that the elative velocity at all the points of contact between a pats This non-penetation condition at a point can be witten as [3]: ( v + ω ) n 0 () whee v is the elative tanslational velocity between two pats, ω is the elative angula velocity between two pats, is the position of the point and n is the nomal at a point of contact on the suface of contact. This equation is linea in v and ω. We define the genealized velocity vecto as [ v ω ]. In Figue, we show the opeation of the non-penetation condition at a point of contact between two plana sufaces and, that ae pat of body and body, espectively. Point P is on suface, and is in contact with point P on suface. n is the suface nomal at P, v and ω ae the elative tanslational and otational velocities of the body pai. is the position vecto of the contact point P (o P ). 4

6 n v + ω P, P O P, Figue. Non-penetation condition at a point. Planes and ae coplana but ae shown as sepaated fo claity. Figue shows the non-penetation condition along a line segment fom P to Q. Fo nonpenetation at evey point on this line segment, it is sufficient that Equation () be satisfied at P and Q. O P P n v + ω P, Q P Q Figue. Non-penetation condition along a line. is the position vecto of an abitay point along the contact line between P and Q. Planes and ae coplana but ae shown as sepaated fo claity. To pevent the penetation of one pat into the othe, Equation () must be satisfied at evey point on the suface of contact and on the bounday of the suface of contact. Such a method of expessing plana contact between two bodies has been used befoe. Fo example, Mattikalli et al. [3] use non-penetation conditions to detemine the impending motion diection of polyhedal igid bodies in contact. 5

7 n v + ω P, P R Q O P Q Figue 3. Non-penetation condition in a polygonal suface. is the position vecto of an abitay point within the polygon bounded by P, Q and R. Planes and ae coplana but ae shown as sepaated fo claity. We define a pimitive contact patch to be a contact suface that is pat of a single type of suface. A compound contact patch is an aggegation of two o moe pimitive patches. A closed plana patch is bounded by a finite set of cuves; these cuves may be staight line segments (Figue 3) o cuved line segments. Any point in the inteio of the patch can be expessed as a linea combination of points at the vetices of the bounday of the convex hull of the patch. Theefoe, the nonpenetation condition at any point in the inteio can be expessed as a linea combination of the non-penetation conditions at the vetex points. As long as the convex hull has a finite numbe of vetices, thee will be a finite numbe of non-penetation conditions, all of which ae linea in v and ω. When extending to cuved suface contacts, it is desiable to peseve the lineaity of the fomulation fo easons of computational efficiency; lineaity allows fo easie seach and bounday enumeation..3 Extending Contact Mechanics to Cuved Sufaces In a pevious pape [8], we extended the esults obtained fo plana suface contacts by showing that simila esults can also be obtained fo spheical and cylindical sufaces defined by edges that ae geat acs (fo spheical sufaces) o staight lines and cicula acs (fo cylindical sufaces). Spheical sufaces in contact always esult in unconstained otations, because they shae a common cente. As befoe, the non-penetation conditions must be witten at the vetices of the convex hull fo the given spheical contact suface. Howeve, since the cost of computing the convex hull is high, we chose to geneate the non-penetation condition at the vetices of the spheical patch. This will not influence the final esult. Fo cylindical sufaces, non-penetation at evey point in a contact patch with bounday segments that ae exclusively staight line (constant-z) and cicula (constant-θ) segments can be epesented entiely by non-penetation at the vetices of these segments. R 6

8 3 Genealized Contact Mechanics In the pevious section, we indicated that the linea popeties of plana contact mechanics models can be peseved when cuved suface contacts ae modeled. This is possible if the contact suface bounday can be descibed by a finite numbe of segments such that satisfying the non-penetation condition the end points of the segments is necessay and sufficient to satisfy the condition at any point on the cuved suface. This section defines and poves the necessay conditions. The natue of the physical contact between a pai of pats in an assembly povides useful infomation about the types of the degees of feedom. Some of these contacts induce suface mating constaints, leading to the fomation of a joint. Othe contacts ae incidental, in that they bound the values of the degees of feedom of the joint. A contact implies that non-penetation between the pats at evey point on the contact suface. Non-penetation conditions can be witten as inequalities that ae linea in the instantaneous velocity, which when taken togethe, descibe a linea subspace. The following Poposition will be used to establish a theoetical basis fo the linea teatment of cuved sufaces in assembly modeling. Poposition. Given: 3. A continuous cuve C( λ): λ [0,] R. 3. C(0) = P and C() = P; P, P R 3. C lies on a paametizable, diffeentiable contact suface S fomed between two bodies A and B. Then non-penetation (by Equation ()) at P and P implies non-penetation at any point on C, if and only if:. C is a cicula ac, possibly with infinite adius (limiting case of a staight line). The unit nomal to C at any point along C is equal to the unit nomal to S at that point. Poof. We pove Poposition by showing that given Equation () witten at P and P, Equation () holds along a set of points between P and P. Equation () witten at P is: and at P is: v + ω ) n 0 () ( v + ω ) n 0 (3) ( whee v is the elative tanslational velocity between the bodies A and B, ω is the elative angula velocity between the two bodies, and ae the position vectos of P and P, espectively. n 7

9 and n ae the nomals to S at P and P espectively. Foming a linea combination of Equation () and Equation (3), we get: λ( v + ω ) n + ( λ)( v + ω ) n 0 with λ Reaanging tems in Equation (4), we get: λ n + ( λ) n ) v + λn ω + ( λ) n ω 0 (5) ( Fo Equation (5) to be tue and of the fom of Equation (), the following would have to be tue v,ω : + ) [ 0,] n = λn ( λ n (6) (4) and n ω + ( λ n ω = n ω (7) λ ) Equation (6) is an expession which indicates that n only spans the nomals fom n to n. By substituting Equation (6) in Equation (7) and using the vecto identity A B C = B C A to eaange tems, we get: ω λ + λ = ω λ + λ { } ( n ( ) n ) ( n ( ) n ) Equation (8) is tue if ω = 0 (the tivial case fo no otational motion). Fo Equation (8) to be also tue fo ω 0, it is sufficient to show that: λ n + ( λ) n = λ (9) { λn + ( n } ) Equation (9) is an expession fo the geneatix o tace [] of C that geneates a locus of points whee Equation () is satisfied, given that it is satisfied at P and P. Taking the dot poduct of Equation (9) with n and n and eaanging tems, we get: { } ( n n ) = 0 { } ( n n ) = 0 Thus, fo non-plana contacts, lies in the plane containing both P and P and nomal to the vecto n n. Fo plana contacts (the tivial case), n = n = n and Equation (5) educes to: v + + ( ) n 0 ( ω ( λ λ ) ) Which is of the same fom as Equation (). Popeties and ae then satisfied, and Poposition is poved. Fo the nontivial case, assuming that n n is a non-zeo vecto, we can then paameteize as (see Figue 4): (8) (0) () 8

10 n P n n O P Figue 4. Plane containing also contains the nomal vectos to. o n = + α n and n = βn + ( β ) n () Whee o is an abitay oigin. Using Equation () to substitute fo, and in Equation (9) and expanding, we get: n n (3) α [ λ β ] = 0 n Fom Equation (3), it follows that since n n cannot be zeo (as pe ou assumption), λ is equal to β. In all of the subsequent analysis, we will use β, with the undestanding that the paameteizations by λ and β ae equivalent. The cuve descibed in Equation () must lie on the specified suface S. This is equivalent to saying that the tangent vecto to the cuve at evey point on the cuve must be pependicula to the nomal to the suface and the nomal vecto to the cuve must be paallel to the nomal vecto to the suface. We enfoce this by the following constaint on C: d n( s) = 0 ds whee s is the ac length, and n (s) is the nomal vecto field. Substituting fom Equation (): (4) d dα = ds ds n n + α n dn n dβ n n dn n dβ dβ ds (5) o, upon simplification and substitution of Equation (5) in Equation (4): 9

11 d dα (6) n( s) = n = 0 ds ds which implies that: ( s) = constant α (7) With Equation (7), Equation () educes to that of a cicula ac in the plane. The unit nomal vecto to any point on this ac is equal to the unit nomal vecto to the suface S at that point. This established popeties and, and poves Poposition. To pove the convese, i.e. given a cicula ac lying on the suface S with the unit nomal vecto field to the ac equal to the unit nomal vecto field of the suface, we wite Equations () and (3) fo a cicula ac. Thus Equation () becomes: v + ω ( o + Rn )) n 0 (8) ( and Equation (3) becomes: v + ω ( o + Rn )) n 0 (9) ( whee R is the adius of the cicula ac. Expanding Equations (8) and (9) and foming a linea combination: v + ω o) ( λn + ( λ) n ) 0 (0) ( which is of the same fom as Equation (). Theefoe, non-penetation at the points P and P implies non-penetation all along C. Note that this is simila to the poof of non-penetation along a cicula ac on a ight cicula cylinde, pesented in Sinha et al. [8]. This completes the poof of Poposition. Lemma.. The cuve C exists on suface S when:. ( ) ( n n ) 0 =. The intesection of S with the above plane is a cicula ac. Poof. The above two conditions follow fom Poposition. Cuve C lies in a plane containing the points P and P, as defined in Equation (0). C is also a cicula ac on S, as shown in Equation (7). Coollay.. The staight line l( λ) : λ [0,] λ + ( λ) on a plane suface S satisfies Poposition. Coollay.. The geat ac on a spheical suface S subtending an angle less than π satisfies Poposition. 0

12 Coollay.3. The staight vetical line paallel to the axis and the cicula ac subtending an angle less than π on a ight cicula cylindical suface S satisfies Poposition. Poof. Coollaies. though.3 ae discussed and poved individually in Sinha et al. [8]. Hee, we show that they emege as special cases of Poposition. As pe the Poposition, the only possible segment on a plana suface (Coollay.), with n equal to n, is the cicula ac with infinite adius, i.e. the staight line joining the points P and P. The geat ac on a spheical suface also satisfies Poposition. The possible segments on a ight cicula cylindical suface ae the vetical staight line and the cicula ac. The subtended angle is equied to be less than π to pevent n being paallel to n. Coollay.4. The staight vetical line stating at the apex of a ight conical suface S satisfies Poposition. Poof. Upon examination of a ight cicula conical suface, we see the staight line (o meidian lines) of the cone has its unit nomal vecto equal to the unit nomal vecto of the suface of the cone. Theefoe, Poposition is satisfied. Note that cicula acs can be pesent on the cone, but do not satisfy Poposition because the unit nomals along the ac do not point in the same diection as the nomal vectos of the cone. 4 Aticulation in Assemblies This section descibes how the space of allowed motion is computed fom the non-penetation conditions, and how the model of the space can be queied to povide designe feedback. When an unconstained degee of feedom exists, the space of allowed instantaneous motion geneated fom the non-penetation conditions will be non-empty. The space can be analyzed to povide feedback to the designe. 4. Solving the Set of Non-Penetation Conditions fo Instantaneous Aticulation Each pimitive patch induces a non-penetation condition at each of its (finite) vetices. Since penetation must not occu at any point at any time, the inequalities fo all the non-penetation conditions fo the all the patches of a pai of bodies consideed simultaneously fom the linea pogam: whee v ni ( v + ω i ) 0 i = KTotal nume of vetices in all patches () and ω ae the elative velocities between the two pats, n i is the nomal at each vetex and i is the position of each vetex. Since at any time, all the non-penetation conditions fo all the pats must be satisfied, it is possible to solve all the inequalities fo all the vetices of all the pats in the same linea pogam. This will esult in a solution that is globally valid. Using a single linea pogam, it is possible to obtain all the instantaneous degees of feedom fo the assembly. If we wite the non-penetation condition fo a patch concisely as:

13 0 B A B A patch v v ω ω J () whee J patch is the non-penetation inequality coefficient matix fo that paticula patch. Then the non-penetation conditions fo all the patches in a body-body contact pai fomed between bodies A and B can be witten as: 0 M = B A B A patch patch patch B A B A AB v v v v p ω ω ω ω J J J J (3) whee p is the numbe of patches in which this body-body pai paticipates. Witing Equation (3) fo all the body-body contact pais in an assembly, we get: 0 M M O M M M L L M = B B A A BC BC AB AB B B A A assembly v v v v ω ω ω ω 0 J J J J J (4) whee J assembly is a complete epesentation of the assembly with instantaneous aticulation. Solving this global simplex povides all the tanslational and angula velocities fo all the bodybody pais simultaneously. The simplex is 6N-6-dimensional (3 vaiables fo tanslational velocity and 3 vaiables fo angula velocity fo each of N- ungounded bodies in the assembly). Since the oigin is a vetex of this high-dimensional space, this stuctue is also called a Polyhedal Convex Cone. Such stuctues have been studied extensively by Goldman and Tucke [3] and othes. Hiai and Asada [7] used cones to descibe the possible contact-peseving and contact-beaking motions between two polyhedal bodies. 4. A CAD Implementation fo Instantaneous Aticulation Having established a theoetical famewok fo teating cuved suface contacts in the pevious sections, we now descibe the system that extacts non-penetation conditions fom the CAD models of pats in an assembly. A contact gaph stuctue G can be used to epesent the assembly (Figue 5). In the contact gaph, pats ae epesented as nodes, and contacts between pats ae epesented as edges between the coesponding nodes. Edges between nodes ae automatically deived by pefoming intesections between the coesponding pats.

14 {( A, B, I ): A, B Set of pats; A B; = Result of intesection between A and B} G = I (5) whee G is the contact gaph that contains a finite numbe of 3-tuples. Each 3-tuple contains two pats and an intesection set. Base Am Roto Slide Figue 5. Contact gaph fo a 4-pat assembly. Nodes epesent the fou pats. Edges epesent the contacts. The contact suface that foms the intesection set is shown in each edge. The intesection infomation in each edge is examined fo featues that could indicate the pesence of suface mating constaints. Each element of I can be thought of as a constaint patch on the mating suface between A and B. Fo a paticula element, all bounday segments that do not satisfy Poposition ae discetized into pimitive segments (staight lines on planes, cicula acs and staight lines on a cylinde fo constant z and constant θ espectively, and geat acs on a sphee), each of which satisfy Poposition. The set I can be patitioned as: I = S U S U S3 UKU S n (6) finite n whee S though S n ae a finite numbe of pimitive sufaces (o patches). On each S, thee exists a finite set of bounday segments Ω: Ω = { σ : σ Set of bounday segments of S} s.t. σ ˆ σ = α Uα UKUα m finite m whee σ is eithe a pimitive segment, o can be appoximated as σˆ which is a union of a finite numbe of pimitive segments α. Thus, the intesection set I is composed of a finite numbe of pimitive bounday segments. The non-penetation condition fo each end-point o vetex of each pimitive bounday segment α is witten as a constaint in the linea pogam. The linea space can now be descibed (o enumeated) using standad bounday enumeation techniques. Note that this technique will wok only fo those sufaces that satisfy Poposition. On othe sufaces such as splines, non-penetation conditions would have to be witten fo evey point on the suface. (7) 3

15 Finding one solution to the simplex is easy; finding all solutions is a moe difficult poposition. Howeve, useful infomation can still be obtained by pojecting the simplex on eithe the v, o the ω space. Such linea pogamming methods have peviously been used by Mattikalli et al. [3] to obtain solutions to the stability poblem fo assemblies. Solutions ae etuned in the fom of allowable instantaneous tanslational and angula velocities. Tanslational velocities of zeo indicate that tanslation is constained fo that body-body pai. Angula velocities of zeo indicate that otation is constained fo that body-body pai. 4.3 Giving Feedback to the Designe In ode to completely descibe all the possible elative motions of the assembly, it is necessay to completely descibe the bounday of the polyhedal convex cone in 6N-6-dimensional space. Useful feedback can be povided to the designe in the fom of questions such as: What degees of feedom exist when the otations of a paticula pat ae constained? This question can be answeed by adding ω = 0 fo the body in question, to the set of constaints and evaluating the linea pogam. Othe possible what-if analyses include gounding a pat (i.e. setting 0 v = and ω = 0 fo that pat) and obtaining the instantaneous degees of feedom fo all the othe pats. The space of allowed motions can be epesented by the set-theoetic sum of the space of motions that peseve the contact ( J V = 0 assembly ), and the space of motions that beak the contact ( V > 0 ), whee V is the vecto of genealized velocities: J assembly S allowed = S S (8) contact peseving + contact beaking S contact-peseving is the space of possible genealized velocity vectos that cause all contacts to be maintained, o: S contact peseving = Nullspace( J assembly ) (9) The basis vectos of the nullspace completely descibe the possible contact-peseving motions. A singula value decomposition of J assembly is used to compute the nullspace. Computing the bounday of the space of contact-beaking motions is a much hade poblem. Avis [4], Avis and Fukuda [5, 6], Motzkin et al. [7], Bemne et al. [8], and Fukuda and Podon [9] have all poposed methods to enumeate the bounday of a polyhedal convex cone. Howeve, time equiements fo these methods quickly explode when confonted with cones of inceasing dimensionality. Nemhause and Wolsey [0] show that the exteme ay membeship poblem fo a cone is in NP. Simila conclusions ae dawn by Avis []. Fo a discussion of the complexity class of enumeation poblems see Fukuda []. Given a cone, it is possible to veify the feasibility of a given solution in polynomial time [0]. Theefoe, we popose the following heuistic to constuct a finite set of feasible solutions fom the geomety of the assembly: 4

16 Figue 6. 4-pat assembly with 3 degees of feedom.. Fo evey pimitive contact patch, pick the axis and a position on the axis - fo planes, the nomal to the plane; fo cylindes and cones, the pincipal axis; fo sphees, any axis; in addition, pick a second axis tangent to the diection of cuve paameteization fo planes, an axis lying in the plane, fo cylindes and cones, the axis of the base; fo sphees, any axis pependicula to the fist axis.. See if a tanslational velocity along and/o a otational velocity about these axes ae feasible solutions to the set of non-penetation conditions. 3. Since thee ae a finite numbe of patches, theefoe a finite numbe of axes, the numbe of possible feasible solutions will be finite. Note that this heuistic will fail to detect degees of feedom that ae not along axes of symmety. Intoduction of domain-specific knowledge (geometic infomation, in this case), enables us to daw useful infeences about the space of allowed motions. The feasible solutions that emege fom this test, along with the nullspace of the cone, fom a epesentation of the cone. Feasible solutions fo a paticula pai of pats can be combined to fom joints between pats. 5

17 5 Illustative Examples 5. Example (Am): Demonstation of the Famewok In ode to illustate the famewok defined in the pevious sections, we pesent an example assembly. The assembly in Eo! Refeence souce not found. is a 4-pat assembly, with thee functional degees of feedom. We choose to ende the base immobile (i.e. we gound it), by constaining all the six degees of feedom. The system is implemented in C++, using ACIS as a solid modelle, and with Open Invento fo 3- D visualization. This example poblem was solved on a SUN Ulta- compute unning Solais 7, using 6 seconds of CPU time, fom input of the CAD models to output of the joint infomation fo the assembly. Contact analysis indicates that this is an open-chain assembly, with 9 linea-bounday plana contacts (base-to-am, am-to-slide and slide-to-oto), cuved-bounday plana contact (baseto-am), 8 patial cylindical contacts (base-to-am, am-to-slide and slide-to-oto). This geneates a total of 8 inequalities, linea in 8 vaiables foming the elative tanslational and otational velocities (assuming the base is gounded). The 8 constaints define a polyhedal convex cone in 8-dimensional space (see Figue 7). The matix epesentation can be patitioned into two: the nullspace and the inteio bounday enumeation epesentation. Singula value decomposition etuns a ank of 5, indicating that thee ae thee completely unconstained degees of feedom (and as a esult, the nullspace is 3- dimensional). The nullspace basis [N, N, N 3 ] is shown in Equation (30). N N N 3 = [ ] [ ] [ ] = = whee Ni = vx vy vz ωx ωy ωz vx vy vz ωx ωy ωz vx vy vz ωx ωy ω z Am Slide Roto (30) Nullspace basis vecto N indicates the pesence of a tanslational degee of feedom fo the slide and oto with espect to the base. This feedom is at a 45 angle to the hoizontal plane, indicated by the equal infinitesimal elative tanslational motions in the x and z diections. Basis vecto N indicates that thee is a elative instantaneous otation of the oto with espect to the base at a 45 angle to the hoizontal plane. The [0 7 0] tanslational component stems fom the fact that the instantaneous otation is not about the oigin, but instead about the axis [ 0 ] though the point [8 0 ]. Fo the instantaneous velocity of the body to be zeo at the point [8 0 ], 6

18 Figue 7. Fomation of the Matix Repesentation. v = v + p = 0 total ω (3) o: v = p ω = (3) [ 0 7 0] whee v is the tanslational component, namely [0 7 0]; ω is the instantaneous angula velocity component, namely [ 0 ]; and p is the locus of positions that satisfies Equation (3). Thus, we get an axis [ 0 ] passing though [8 0 ]. Thus, nullspace analysis descibes the bounday of the cone. Feasible solutions ae constucted fom the contact patches. The 8 pimitive patches esult in 6 candidate feasible solutions. Each candidate is tested to detemine if it is within the space of allowed instantaneous motion. Fou candidates ae found to be valid solutions. Of the fou valid solutions, thee ae identical to the nullspace basis vectos. The fouth is [ ], coesponding to the contact-beaking motion (vetical tanslation along the z-axis) between the base and the othe 3 pats. Once valid feasible solutions ae available, it is possible to goup them togethe to define joints between pats in the assembly. One of the feasible solutions in this example esults in a evolute joint being fomed between the base and the am (see Figue 8). This joint infomation is added to 7

19 Figue 8. Mapping feasible solutions to assembly joints. the epesentation of the assembly. Once joints ae detected automatically, infomation about the mateial popeties can be combined with kinematic infomation to pefom motion simulations. 5. Example (-DOF 4-Ba Linkage): Global Constaint Resolution To veify the coectness of ou implementation, we compaed the esults of ou system to the analytical fomulation fo a 4-ba linkage. The example in this subsection is a 4-ba assembly with degee of feedom (see Figue 9) with each ba having a joint axis-to-joint axis distance of 39 units. Computing the contact sufaces fom the CAD model indicates that thee ae 4 contacts between the bas, each shaped like a top hat. Each top hat contact has plana contact sufaces bounded by a cuved bounday and complete cylindical contact suface. The two plana contact sufaces constitute the top and the im of the hat. The cylindical contact suface constitutes the side of the hat. The non-penetation condition is satisfied eveywhee on the cylindical contact suface if it is satisfied at 6 points at the top and bottom bounday of the suface. On the two plana contact sufaces, we select 09 points that appoximate the boundaies of the sufaces, and impose the nonpenetation condition at these points. This esults in a total of 4 inequalities pe pai of bas, fo a gand total of 460 inequalities in 8 elative tanslational and otational velocity vaiables (assuming that Ba is gounded). 8

20 Ba 3 Ba Ba 4 Ba Figue 9. 4-Ba Linkage The inequalities ae used to constuct the inequality coefficient matix A (simila to the one in Figue 7). Each inequality becomes a ow in the matix, esulting in a matix of size The matix has a ank of 7. Theefoe, the nullspace N is -dimensional, indicating the pesence of completely unconstained degee of feedom. A basis vecto fo the nullspace of A is: N = whee N [ ] = vx v y vz ω x ω y ω z Ba v x v y v z ω x ω y ω z Ba3 v x v y vz ω x ω y ω z Ba 4 Beaking one of the ba-ba contacts would have esulted in an open-chain assembly, with 3 moe degees of feedom. The closed-loop eliminates these exta degees of feedom, and this is eflected in the -dimensional nullspace. Examining the nullspace, we see that fo an unit instantaneous otation of Ba 4 with espect to Ba about the z-axis, Ba undegoes a unit instantaneous otation about the z-axis, but at a position of [39 0 0] (using Equation (3)). Ba 3 undegoes an instantaneous pue tanslation along the x-axis of a magnitude 39 times that of the otation. The 4-ba linkage has been extensively studied by mechanism theoists. Its behavio is well known. Fom [3] (see pp ), we see that the velocity-loop equation witten fo ou 4-ba linkage is: L ω 4 Lω = 0 (34) Lω = 0 3 whee L is the length of the ba, ω is the angula velocity of Ba with espect to Ba, ω3 is is the angula velocity of Ba 4 with the angula velocity of Ba 3 with espect to Ba, and ω 4 espect to Ba. Thus, ω is equal to ω 4 and ω3 (33) is zeo. Since the links ae igid, fo evey δθ otation of Ba and Ba 4, thee must be a L δθ x-axis displacement of Ba 3. This is identical to the pevious esult. This validates ou implementation of degee of feedom extaction. 9

21 6 Discussion The geomety of the pats in an assembly esticts how they can be assembled, and how they move elative to one anothe. This in tun influences thei static stability, kinematic behavio, and dynamic pefomance chaacteistics. Howeve, models fo these phenomena do not exist initially. The designe povides joint infomation at the conceptual design stage to meet cetain functional equiements of the assembly. This infomation is povided eithe by specifying kinematic constaints between pats as annotations to the assembly CAD model, o by specifying mating conditions between specific featues on the pats. It can also be deived fom the CAD models of the individual pats, and fom the elative positions of these pats in the final assembly. O the infomation could be obtained by some combination of designe input and automatic deivation. Theefoe, thee can be two types of constaints between pats, namely, constaints intoduced by the designe to satisfy functional equiements of the assembly, and constaints induced by the geomety of pats in the assembly. Both these types of constaints inteact to poduce a esultant behavio of a joint. It is impotant to be able to deive the joint behavio fom the CAD model, as well as to veify that designe input is consistent with the CAD model. Ou method easons on the CAD model of the assembly to obtain the instantaneous degees of feedom. It is able to handle incomplete cuved geomety, while at the same time esolving global (i.e. multi-pat) constaint inteactions. Linea algeba-based constaint models ae deived diectly fom CAD models, and then conveted into aticulation epesentations suitable fo assembly planning and motion simulation. A poposition and othe suppoting theoems enumeate the popeties of the contact sufaces between the pats that must be satisfied to constuct a linea algebaic constaint model. The linea algeba-based constaint model can be used to povide designe feedback. The model descibes the space of allowable instantaneous motion in the assembly. Although solving fo the complete bounday of this space is thought to be in NP, whethe a paticula instantaneous degee of feedom is within the space can be veified in polynomial time. Theefoe, ou famewok also suppots the ability to quey the model with the designe specified joint infomation to veify that the infomation is coect. We validated ou method by applying it to the 4-ba linkage, and compaing the esults with those obtained fom mechanism theoy. Unde cetain cicumstances, ou method beaks down: contact sufaces that ae epesented by polynomial o tanscendental functions have to be appoximated as piecewise plana, cylindical o spheical contacts this appoximation may emove a degee of feedom. A valid candidate degee of feedom may not be detected if the feedom does not lie along an axis of symmety. Ou method is applicable only to assembly geomety that is modeled by exact CAD models without fit toleances. The method is also only applicable to the paticula mechanism configuation that is captued in the CAD model. 7 Conclusions and Futue Wok This eseach fowads the state-of-the-at in the following ways: 0

22 Exact Teatment of Cuved Geomety. Pevious wok on the application of linea pogamming techniques to the contact mechanics between two sufaces was esticted to handling plana sufaces only [3, 0]. Cuved sufaces wee handled by appoximating them as plana facets, o by using gaphical methods. This esult was extended to cuved sufaces whose boundaies possess cetain popeties. The vast majoity of manufactued pats have contacts that possess such boundaies. Theefoe, the wok pesented hee allows a designe to analyze and design devices that contain pats (o components) that in tun contain a boad class of cuved sufaces. Modeling Suppot Fo Kinematics Behavio. Solving the contact conditions at the bounday will enable the designe to obtain the instantaneous degees of feedom of the device. Using geneateand-test methods and easoning about the geomety of the atifact will eveal the bounds on the values of the degees of feedom. Such infomation, when combined with infomation about the mass distibution and with fiction models, will allow the designe to detemine the stability of the device, handle aticulation duing assembly planning, duing synthesis of pat geomety, as well as duing the opeation of the device. Accuate and Automatic Simulation Synthesis. Simulation of the opeation of the electomechanical device equies kinematic and dynamic infomation. Since the CAD model of each component incopoates such infomation, high-fidelity simulation can be ceated and executed with minimal use inteaction. Inteesting eseach issues emain with egad to the handling of model uncetainty and in the quantification of the effect of model appoximations, such as the discetiation of cuved plana contact boundaies, on the final esult of the aticulation analysis. Acknowledgements This eseach was funded in pat by DARPA unde contact ONR #N , by the Raytheon Company, the National Institute of Standads and Technology, the Robotics Institute, and the Institute fo Complex Engineeed Systems at Canegie Mellon Univesity. In addition to the eviewes, we would like to thank D. Raju Mattikalli and Pof. Benoit Moel fo thei insightful feedback. Refeences [] Ge, Q. and McCathy, J. M., "Functional Constaints as Algebaic Manifolds in a Cliffod Algeba," IEEE Jounal Robotics and Automation, vol. 7, pp , 99. [] Liu, Y. and Popplestone, R., "A Goup Theoetic Fomalization of Suface Contact," Intenational Jounal of Robotics Reseach, vol. 3, pp. 48-6, 994. [3] Mattikalli, R., Baaff, D., Khosla, P., and Repetto, B., "Gavitational Stability of Fictionless Assemblies," IEEE Tansactions on Robotics and Automation, vol., pp , 995. [4] Paedis, C. J. J., Diaz-Caldeon, A., Sinha, R., and Khosla, P. K., "Composable Models fo Simulation-Based Design," Engineeing with Computes, vol. in pess, 00. [5] ADAMS, " 000.

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