Extacting Aticulation Models fom CAD Models of Pats with Cuved Sufaces Rajaishi Sinha 1,*, Satyanda K. Gupta 2, Chistiaan J.J. Paedis 1, Padeep K. Khosla 1 1 Institute fo Complex Engineeed Systems, Canegie Mellon Univesity, Pittsbugh, PA 15213. * Coesponding autho; sinha@cs.cmu.edu 2 Depatment of Mechanical Engineeing, Univesity of Mayland, College Pak, MD 20742.
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TABLE OF CONTENTS TABLE OF CONTENTS... III LIST OF FIGURES...V ABSTRACT...VII 1. INTRODUCTION...1 2. REVIEW OF PREVIOUS WORK...2 2.1. EARLY WORK ON CONTACT MECHANICS... 2 2.2. CONTACT MECHANICS FOR PLANAR CONTACTS...2 2.3. EXTENDING CONTACT MECHANICS TO CURVED SURFACES... 3 3. GENERALIZED CONTACT MECHANICS...4 4. ARTICULATION IN ASSEMBLIES...9 4.1. SOLVING THE SET OF NON-PENETRATION CONDITIONS FOR INSTANTANEOUS ARTICULATION... 9 4.2. A CAD IMPLEMENTATION FOR INSTANTANEOUS ARTICULATION...10 4.3. GIVING FEEDBACK TO THE DESIGNER... 11 5. ILLUSTRATIVE EXAMPLES...13 5.1. EXAMPLE 1 (ARM): DEMONSTRATION OF THE FRAMEWORK... 13 5.2. EXAMPLE 2 (1-DOF 4-BAR LINKAGE): GLOBAL CONSTRAINT RESOLUTION... 17 6. DISCUSSION...19 7. CONCLUSIONS AND FUTURE WORK...21 ACKNOWLEDGEMENTS...22 REFERENCES...23 iii
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LIST OF FIGURES FIGURE 1. PLANE CONTAINING ALSO CONTAINS THE NORMAL VECTORS TO.... 6 FIGURE 2. 4-PART ASSEMBLY WITH 3 DEGREES OF FREEDOM... 13 FIGURE 3. FORMATION OF THE MATRIX REPRESENTATION... 14 FIGURE 4. MAPPING FEASIBLE SOLUTIONS TO ASSEMBLY JOINTS... 16 FIGURE 5. 4-BAR LINKAGE... 17 v
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ABSTRACT Degees of feedom in an assembly 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. 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 on ou pevious wok on cuved suface contact mechanics. We enumeate the conditions unde which geneal cuved sufaces can be epesented using a finite numbe of constaints 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 motion sub-spaces. Analysis of each subspace povides feedback to the designe, which we demonstate though the use of an example assembly a 4-pat am. Finally, the esults of the analysis of a 4-ba linkage ae compaed to those fom mechanism theoy. vii
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1. INTRODUCTION Assemblies ae composed fom pats. The geomety of the pats imposes cetain estictions on the way that they can be assembled, and also on the way that they move elative to one anothe. This in tun has a beaing on thei static stability, kinematic behavio, and dynamic pefomance chaacteistics. Howeve, none of this infomation exists initially. It must be deived fom the CAD models of the individual pats, and fom the elative positions of these pats in the final assembly. In ode that the collection of pats can be gouped togethe to fom the assembly epesentation of the atifact, joints between pats must be specified. The joints between pats ae defined by the designe at the conceptual design level to meet cetain functional equiements of the assembly. Thus, thee can be two types of constaints between pats, namely, constaints induced by the geomety of pats, and constaints intoduced by the designe to satisfy functional equiements of the assembly. Both these types of constaints inteact to poduce a esultant behavio of a joint. Aticulated devices ae ealized though contacts between pats. In ode to veify that a complex device is a coect spatial ealization of the intended functional design concept, we need to extact its behavio fom its geometic epesentation. Techniques have been developed to pedict the instantaneous degees of feedom fom the CAD models of pats composed of polygonal plana faces [Hiai and Asada, 1993; Mattikalli et al., 1994]. Howeve, these techniques handle only pats with plana faces; most engineeing devices have cuved pats. When cuved pats ae appoximated as piecewise plana pats, eoneous esults ae possible. In this eseach, we pesent a methodology that extends ou ealie wok on contact sufaces. That wok [Sinha et al., 1998] easons about the degees of feedom at each joint, based on suface mating constaints, that ae in tun obtained fom analyzing the natue of body to body contact. Non-penetation constaints ae imposed along the bounday of each contact suface in the fom of algebaic inequalities. It is shown that a finite numbe of non-penetation conditions ae epesentative of the entie suface in contact. Using linea pogamming methods, instantaneous velocities and acceleations fo each pai of bodies ae computed. In this wok, 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. Such a methodology is useful in that it 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 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). 1
2. REVIEW OF PREVIOUS WORK 2.1. Ealy Wok on Contact Mechanics Pevious wok on plana contact mechanics and scew theoy pepaed the foundation fo this wok. Ohwovoiole and Roth [1981] showed that unidiectional constaints can be modeled as scews. Hiai and Asada [1993] descibed the allowable motions of a pat using polyhedal convex cones to epesent the space of movement. Mattikalli and Khosla [1991] descibed a method to obtain degees of 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: 1. Only bodies descibed (o appoximated) by plana sufaces ae consideed. 2. 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. 2.2. 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 and limits 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 intoduce limits on the degees of feedom of the joint [Rajan et al., 1997; Rajan and Nof, 1997]. Reasoning about these constaints povides the designe with valuable insight into the instantaneous degees of feedom of the assembly. Othe eseaches ([Mattikalli et al., 1994], [Baaff and Mattikalli, 1993]) 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. This non-penetation condition at a point can be witten as [Baaff and Mattikalli, 1993]: ( v + ω ) n 0 Eq. 1 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 ]. Equation 1 implies that the genealized velocity vecto fo elative motion between two 2
points, one on each contact suface, should not have a component opposite to the nomal to the base suface. A component into the base suface will imply penetation. To pevent the penetation of one pat into the othe, Equation 1 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, see Baaff and Mattikalli [1993], whee the authos use non-penetation conditions to detemine the impending motion diection of polyhedal igid bodies in contact. A closed plana suface (o patch) is bounded by a finite set of cuves; these cuves may be staight line segments 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. 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 will be 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. 2.3. Extending Contact Mechanics to Cuved Sufaces In a pevious pape [Sinha et al., 1998], 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 which ae geat acs (fo spheical sufaces) o staight lines and cicula acs (fo cylindical sufaces). Spheical contact 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 cone fo the given spheical contact suface. Howeve, since the cost of computing the convex cone is high, we chose to geneate the non-penetation condition at the vetices of the spheical patch. This will not influence the final esult. Non-penetation at evey point in a contact patch on a cylindical suface with bounday segments that ae exclusively constant-z and constant-θ segments can be epesented entiely by nonpenetation at the vetices of these segments. 3
3. GENERALIZED CONTACT MECHANICS The discussion in Section 2 illustates that fo cetain cuved sufaces, if the bounday can be descibed by a finite numbe of segments all possessing cetain popeties, then the non-penetation condition at any point on the cuved suface is satisfied if the non-penetation condition is satisfied at the finite numbe of bounday vetices. The natue of the physical contact between a pai of pats in an assembly povides useful infomation about the types and limits 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 intoduce limits on the degees of feedom of the joint. When two pats ae in contact with each othe, it implies that thee is no inte-penetation between the pats at evey point on the contact sufaces. Non-penetation conditions can be witten as linea inequalities 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 1. Given: 3 1. A continuous cuve C( λ ) : λ [0,1] R. 2. C(0)=P 1 and C(1)=P 2 ; P 1,P 2 R 3. 3. C lies on a paametizable, diffeentiable contact suface S fomed between two bodies A and B. Then non-penetation (by Equation 1) at P 1 and P 2 implies non-penetation at any point on C, if and only if: 1. C is a cicula ac, possibly with infinite adius (limiting case of a staight line) 2. 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 1 by showing that given Equation 1 witten at P 1 and P 2, Equation 1 holds along a set of points between P 1 and P 2. Equation 1 witten at P 1 is: v + ω ) n 0 Eq. 2 ( 1 1 and at P 2 is: v + ω ) n 0 Eq. 3 ( 2 2 4
whee v is the elative tanslational velocity between the bodies A and B, ω is the elative angula velocity between the two bodies, 1 and 2 ae the position vectos of P 1 and P 2, espectively. n 1 and n 2 ae the nomals to S at P 1 and P 2 espectively. Foming a linea combination of Equation 2 and Equation 3, we get: λ( v + ω 1) n1 + (1 λ)( v + ω 2 ) n2 0 with λ [ 0,1] Eq. 4 Reaanging tems in Equation 4, we get: ( n + (1 λ) n2) v + λn 1 ω 1 + (1 λ) n2 ω 2 λ 1 0 Eq. 5 Fo Equation 5 to be tue and of the fom of Equation 1, the following would have to be tue v, ω : n = λn 1 + ( 1 λ) n2 Eq. 6 and n ω + 1 λ n ω = n ω λ 1 1 ( ) 2 2 Eq. 7 Equation 6 is an expession which indicates that n spans all the nomals fom n 1 and n 2. Fo Equation 7 to be satisfied fo all ω, it is sufficient to show that: n + ( 1 λ) n = λ { λn + (1 n } λ 1 1 2 2 1 ) 2 Eq. 8 Equation 8 is an expession fo the geneatix o tace [Gay, 1998] of C which geneates a locus of points whee Equation 1 is satisfied, given that it is satisfied at P 1 and P 2. Taking the dot poduct of Equation 8 with n 1 and n 2 and eaanging tems, we get: { 1 } ( n1 n2 ) = 0 { } ( n n ) = 0 2 1 2 Eq. 9 Thus, lies in the plane containing both P 1 and P 2 and nomal to the vecto n 1 n2. Assuming that n 1 n 2, we can then paameteize as (see Figue 1): 5
n n = o + α and n = β n1 + (1 β ) n2 Eq. 10 Figue 1. Plane containing also contains the nomal vectos to. Whee o is an abitay oigin. Using Equation 10 to substitute fo, 1 and 2 in Equation 8 and expanding, we get: α n1 n2 n [ λ β ] = 0 Eq. 11 Fom Equation 11, it follows that since n 1 n2 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 βae equivalent. We now impose the following constaint on C: d n( s) = 0 ds Eq. 12 whee s is the ac length, and n (s) is the nomal vecto field. This condition equies the tangent vecto to the cuve at evey point on the cuve to be pependicula to the nomal to the suface, 6
effectively focing the cuve to lie on the specified suface S. It also equies the nomal vecto to the cuve to be paallel to the nomal vecto to the suface. Substituting fom Equation 10: d dα n = ds ds n + n α dn dβ 1 n n 2 n dn n dβ dβ ds Eq. 13 o, upon simplification and substitution of Equation 11 in Equation 12: d dα n( s) = n = 0 ds ds Eq. 14 which implies that: ( s) = constant α Eq. 15 With Equation 15, Equation 10 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 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 2 and 3 fo a cicula ac. Thus Equation 2 becomes: v + ω ( o + Rn )) n 0 Eq. 16 ( 1 1 and Equation 3 becomes: v + ω ( o + Rn )) n 0 Eq. 17 ( 2 2 whee R is the adius of the cicula ac. Expanding Equations 17 and 18 and foming a linea combination: v + ω o) ( λn + (1 λ) n ) 0 Eq. 18 ( 1 2 which is of the same fom as Equation 1. Theefoe, non-penetation at the points P 1 and P 2 implies non-penetation all along C. Note that this is simila to the poof of non-penetation along a cicula 7
ac on a ight cicula cylinde, pesented in Sinha et al. [1998]. This completes the poof of Poposition 1. Lemma 1.1. The cuve C exists on suface S when: 1. ( ) ( n n ) 0 1 2 1 2 = 2. The intesection of S with the above plane is a cicula ac. Poof. The above two conditions follow fom Poposition 1. Cuve C lies in a plane containing the points P 1 and P 2, as defined in Equation 9. C is also a cicula ac on S, as shown in Equation 15. Coollay 1.1. The staight line l( λ) : λ [0,1] λ 1 + (1 λ) 2 on a plane suface S satisfies Poposition 1. Coollay 1.2. The geat ac on a spheical suface S subtending an angle less than π satisfies Poposition 1. Coollay 1.3. 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 1. Poof. Coollaies 1.1 though 1.3 ae discussed and poved individually in Sinha et al. [1998]. Hee, we show that they emege as special cases of Poposition 1. As pe the Poposition, the only possible segment on a plana suface (Coollay 1.1), with n 1 equal to n 2, is the cicula ac with infinite adius, i.e. the staight line joining the points P 1 and P 2. The geat ac on a spheical suface also satisfies Poposition 1. 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 1 being paallel to n 2. Coollay 1.4. The staight vetical line stating at the apex of a ight conical suface S satisfies Poposition 1. 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 1 is satisfied. Note that cicula acs can be pesent on the cone, but do not satisfy Poposition 1 because the unit nomals along the ac ae not equal to the unit nomal vectos of the cone. 8
4. ARTICULATION IN ASSEMBLIES The existence of an unconstained degee of feedom will cause the genealized velocity space epesentation geneated fom the non-penetation conditions to be non-empty. Theefoe, the velocity space can be analyzed to detect and classify unconstained degees of feedom. 4.1. 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 nonpenetation 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: n ( v + ) 0 i = 1KTotal nume of vetices inall patches i ω i Eq. 19 whee v and ω ae the elative velocities between the two pats, and i is the position of each vetex. n i is the nomal at 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 which 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: v ω vb ω 0 J A patch Eq. 20 A B whee J patch is the Jacobian 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: AB v ω A A J v = J B ω B J v M ω p patch1 patch2 patch A A v B 0 ω B J Eq. 21 whee p is the numbe of patches in which this body-body pai paticipates. Using expessions such as Equation 22 witten fo all the body-body contact pais in an assembly, we get: 9
v A L ω A J AB J AB 0 v = B L 0 J BC J BC ω B M M M O M v A 0 ω A 0 0 vb M ω B M J assembly Eq. 22 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 body-body pais simultaneously. The simplex is 6N-6-dimensional (3 vaiables fo tanslational velocity and 3 vaiables fo angula velocity fo each of N-1 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 [1956] and othes. Hiai and Asada [1993] used cones to descibe the possible contact-peseving and contact-beaking motions between two polyhedal bodies. 4.2. 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 which 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. 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 nodes. Each pat is scaled by a measue popotional to its bounding box dimensions, so that the esult of the intesection is a egula solid (o a set of egula solids). {( A, B, I ): A, B Set of pats; A B; = Result of intesection between A and B} G = I Eq. 23 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 1 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 1. The set I can be patitioned as: I = Eq. 24 S1U S2 U S3UKU S 144424443 n finite n whee S 1 though S n ae a finite numbe of pimitive sufaces (o potions theeof). On each S, thee exists a finite set of bounday segments Ω: 10
Ω = { σ : σ Set of bounday segments of S} s.t. σ σˆ = α1uα 2 UKUα m 1442443 finite m Eq. 25 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 now 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 which satisfy Poposition 1. On othe sufaces such as splines, nonpenetation conditions would have to be witten fo evey point on the suface. 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. [1994] 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 0 ω = 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 which peseve the contact ( J V = 0 assembly ), and the space of motions which beak the contact ( V > 0 ), whee V is the vecto of genealized velocities: J assembly S allowed = S S Eq. 26 contact peseving + contact beaking 11
S contact-peseving is the space of possible genealized velocity vectos which cause all contacts to be maintained, o: S = Nullspace( J assembly ) contact peseving Eq. 27 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 [1994], Avis and Fukuda [1991, 1996], Motzkin et al. [1953], Bemne et al. [1996], Fukuda and Podon [1996] 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 [1988] show that the exteme ay membeship poblem fo a cone is in NP. Simila conclusions ae dawn by Avis [1998]. Fo a discussion of the complexity class of enumeation poblems see Fukuda [1998]. Given a cone, it is possible to veify the feasibility of a given solution in polynomial time [Nemhause and Wolsey, 1988]. Theefoe, we popose the following heuistic to constuct a finite set of feasible solutions fom the geomety of the assembly: 1. 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. 2. 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. 12
5. ILLUSTRATIVE EXAMPLES 5.1. Example 1 (Am): Demonstation of the Famewok In ode to illustate the famewok defined in the pevious sections, we pesent an example assembly. The assembly in Figue 2 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. Figue 2. 4-pat assembly with 3 degees of feedom. 13
The system is implemented in C++, using ACIS as a solid modelle, and with Open Invento fo 3- D visualization. Contact analysis indicates that this is an open-chain assembly, with 9 lineabounday plana contacts (base-to-am, am-to-slide and slide-to-oto), 1 cuved-bounday plana contact (base-to-am), 8 patial cylindical contacts (base-to-am, am-to-slide and slide-to-oto). This geneates a total of 81 inequalities, linea in 18 vaiables foming the elative tanslational and otational velocities (assuming the base is gounded). Figue 3. Fomation of the Matix Repesentation. The 81 constaints define a polyhedal convex cone in 18-dimensional space (see Figue 3). The matix epesentation can be patitioned into two: the nullspace and the inteio bounday enumeation epesentation. Singula value decomposition etuns 3 singula values, indicating that thee ae thee completely unconstained degee of feedom (and as a esult, the nullspace is 3 - dimensional). The nullspace basis [N 1, N 2, N 3 ] is shown in Equation 29. 14
N N N N 1 2 3 i = = = whee [ 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 17 0 1 0 1] [ 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1] = v x v y vz ω x ω y ω z 1444442 444443 Am vx v y vz ω x ω y ω z 1444442444443 Slide v x v y vz ω x ω y ω z 1444442444443 Roto Eq. 28 N 3 Nullspace basis vecto N 1 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 velocities in the x and z diections. Basis vecto N 2 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 17 0] tanslational component stems fom the fact that the instantaneous otation is not about the oigin, but instead about the axis [1 0 1] though the point [18 0 1]. Fo the instantaneous velocity of the body to be zeo at the point [18 0 1], o: v = v + p = 0 total ω Eq. 29 v = p ω = Eq. 30 [ 0 17 0] whee v is the tanslational component, namely [0 17 0]; ω is the instantaneous angula velocity component, namely [1 0 1]; and p is the locus of positions which satisfies Equation 30. Thus, we get an axis [1 0 1] passing though [18 0 1]. Thus, nullspace analysis descibes the bounday of the cone. Feasible solutions ae constucted fom the contact patches. The 18 pimitive patches esult in 216 candidate feasible solutions. Of these, fou solutions ae found to be valid. Of the 4 valid solutions, 3 ae identical to the nullspace basis vectos. The fouth is [0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0], coesponding to the contact-beaking motion (vetical tanslation along the z-axis) between the base and the othe 3 pats. 15
Figue 4. Mapping feasible solutions to assembly joints. Once valid feasible solutions ae available, it is possible to goup them togethe to define joints between pats in the assembly. The two feasible solutions in this example esult in a evolute joint being fomed between the two subassemblies (see Figue 4). Joint infomation is added to the epesentation of the assembly. Such automatic detection of joints is useful when pefoming motion simulation. 16
5.2. Example 2 (1-DOF 4-Ba Linkage): Global Constaint Resolution To veify the coectness of ou implementation, we compaed the esults of ou system against the analytical fomulation fo a 4-ba linkage. The example in this subsection is a 4-ba assembly with 1 degee of feedom (see Figue 5) with each ba having a joint axis-to-joint axis distance of 39 units. Contact analysis indicates that thee ae 8 cuved-bounday plana contacts (2 between each pai of bas) and 4 complete cylindical contact (1 between each pai of bas). Ba 3 Ba 2 Ba 4 Ba 1 Figue 5. 4-Ba Linkage This geneates 460 inequalities in 18 elative tanslational and otational velocity vaiables (assuming that Ba 1 is gounded). The inequality coefficient matix A (Figue 3) has a ank of 17. Theefoe, the nullspace N is 1-dimensional, indicating the pesence of 1 completely unconstained degee of feedom. A basis vecto fo the nullspace of A is: N1 = whee N 1 [ 0 39 0 0 0 1 39 0 0 0 0 0 0 0 0 0 0 1] = v x v y vz ω x ω y ω z 1444442 444443 Ba 2 vx v y vz ω x ω y ω z 1444442444443 Ba3 v x v y vz ω x ω y ω z 1444442444443 Ba 4 Eq. 31 Removing one of the bas 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 1- dimensional nullspace. Examining the nullspace, we see that fo an unit instantaneous otation of Ba 4 with espect to Ba 1 about the z-axis, Ba 2 undegoes a unit instantaneous otation about the z-axis, but at a position of [39 0 0] (using Equation 29). 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 [Paul, 1979] (see pp. 233-234), we see that the velocity-loop equation witten fo ou 4-ba linkage is: 17
Lω Lω 4 3 Lω = 0 2 = 0 Eq. 32 whee L is the length of the ba, ω 2 is the angula velocity of Ba 2 with espect to Ba 1, ω 3 is the angula velocity of Ba 3 with espect to Ba 1, and ω 4 is the angula velocity of Ba 4 with espect to Ba 1. Thus, ω 2 is equal to ω 4 and ω 3 is zeo. Since the links ae igid, fo evey δθ otation of Ba 2 and Ba 4, thee must be a δθ L x-axis displacement of Ba 3. This is identical to the pevious esult. This validates ou implementation of degee of feedom extaction. 18
6. DISCUSSION In cuent design pactice, when a device which contained 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 [Mattikalli et al., 1994]. The appoximation is made on the basis of past expeience (i.e. selecting the numbe of planes to epesent a cuved suface is a heuistic). Such an appoximation popagates thoughout the design pocess - pat and assembly designes must ensue that the appoximation does not constain the degees of feedom in the device; analysts must ensue that assemblies which ae stable ae not modeled as unstable; toleancing specialists must account fo the fact that the appoximated sufaces may have modeled toleances that ae diffeent fom the desied specifications; pocess enginees may discove poblems of fit and/o intefeences in the model when actually no poblem occus (and vice vesa). Assembly pocess enginees, tool enginees and opeatos may notice that some assembly tasks ae impossible to pefom due to this appoximation. Even with an acceptable appoximation, it is difficult to popagate the physical behavio though the diffeent aspects of the design pocess. Even in a collaboative setting, when a pat o assembly used in a pevious design poject was eused in a new design poject, all that was known a pioi was the geomety of the pat o assembly. Only the CAD model of the pat o assembly pesisted acoss designs. No physical model-elated infomation about the pat o assembly was eused. In addition, featue-based design of complex devices equies veification of the fact that actual kinematic behavio matches the equied behavio. 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 cuved sufaces; the ability to obtain kinematics fom assembly geomety; and the ability to package kinematic behavio into the epesentation of a pat o an assembly. A change in the geomety of a pat o a change in the elative positions of pats necessitates a complete egeneation of the models of the mechanical behavio of the assembly. If an acceptable model that mimics the mechanics behavio of an atifact can be extacted fom the geomety of the atifact, and if mechanics models ae meged when the coesponding geomety is meged, then the designe will be able to: ceate new designs out of existing atifacts o pats and euse potions o all of olde designs in new design pojects. In addition, the mechanical models will ensue physically coect behavio fo the integated atifact; this implies that any automatically geneated simulation will be faithful to the models. Theefoe, the designe will not have to eceate a simulation each time a new poduct is designed fom peviously ceated atifacts o pats. At the conceptual level, the designe knows the type and behavio of the joints in the concept design. Howeve, the geomety still needs to be defined. At the peliminay design stage, the geometic infomation is defined, and the elative positions and oientations of components in the assembly ae specified. Following this stage, component and joint epesentations ae eniched to achieve a final efinement of the geomety. 19
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 [ADAMS, 1998] o ae featue ecognition based [Rajan et al., 1997]. 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. Ou method 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. 20
7. CONCLUSIONS AND FUTURE WORK This eseach fowads the state-of-the-at in the following ways: 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 (see [Mattikalli et al, 1994] and [Baaff and Mattikalli, 1993]). 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 motion limits to 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 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 discetization of cuved plana contact boundaies, on the final esult of the aticulation analysis. 21
ACKNOWLEDGEMENTS This eseach was funded in pat by DARPA unde contact ONR #N00014-96-1-0854, 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. We would like to thank D. Raju Mattikalli of The Boeing Company and Pof. Benoit Moel of Canegie Mellon Univesity fo useful feedback. 22
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