Exact Collision Detection of Two Moving Ellipsoids under Rational Motions

Transcription

1 Proceedings of he 2003 IEEE Inernaional Conference on Roboics & Auomaion Taipei, Taiwan, Sepember 1-19, 2003 Exac Collision Deecion of Two Moving Ellipsoids under Raional Moions Yi-King Choi, Wenping Wang Deparmen of Compuer Science, The Universiy of Hong Kong, Hong Kong, P.R. China Myung-Soo Kim School of Compuer Science Eng., Seoul Naional Universiy, Seoul, Souh Korea Absrac In his paper, we describe an exac mehod for deecing collision beween wo moving ellipsoids under pre-specified raional moions. Our mehod is based on an algebraic condiion ha deermines he separaion saus of wo saic ellipsoids he condiion iself is described by he signs of roos of he characerisic equaion of he wo ellipsoids. To deal wih moving ellipsoids, we derive a bivariae funcion whose zero-se possesses a special opological srucure. By analyzing he zero-se of his funcion, we are able o ell wheher or no wo moving ellipsoids under prespecified raional moions are collision-free; if no, we can deermine he inervals in which hey overlap. I. INTRODUCTION Designing analyzing moions of objecs in 3D space are of imporance o roboics, compuer graphics, 3D compuer games [] CAD applicaions. To simulae a dynamic environmen, i is criical o ensure ha rigid objecs do no penerae each oher impulsive response is properly hled when objecs collide. Collision deecion is o deermine wheher or no wo moving objecs come ino conac wih each oher. To perform direc collision deecion for general objecs is compuaionally complicaed. Using bounding volumes o enclose complex shaped objecs reduces he compuaional cos by firs deecing collision on he bounding volumes. Ellipsoids are ofen chosen as bounding volumes for roboic arms in collision deecion because of heir flexibiliies in shape [8], [1], [20]. Enclosing enclosed ellipsoids are sudied for collision avoidance beween convex polyhedra in [8], [16] using heurisic soluions. Rimon Boyd [1] presen an efficien numerical echnique for compuing he quasi-disance, called margin, beween wo separae ellipsoids using an incremenal approach. In [17], Sohn e al. compue he disance beween wo ellipsoids using line geomery. Using he Lagrange condiions, Lennerz Schömer [11] presen an efficien algorihm for compuing he disance for quadraic curves surfaces. Ellipsoids are also used o represen soil paricles in geo-mechanics he isopoenial surface of a molecule in compuaional physics. The problem of deermining wheher or no wo ellipsoids overlap is also sudied in hese fields [12], [13]. However, he proposed soluions are also numerical echniques for saic ellipsoids, canno be applied o exac (i.e., nonincremenal) collision deecion beween wo moving ellipsoids following pre-specified closed-form moion pahs. To deal wih moving ellipsoids, a brue-force approach is o perform inerference esing beween wo saic ellipsoids along he moion pah a discree ime inervals. However, errors may be incurred due o inadequae emporal sampling resoluion. In his paper, we presen an exac collision deecion mehod for wo ellipsoids moving under pre-specified raional moions. We make use of an algebraic condiion given in [19] ha decides wheher or no wo saic ellipsoids are separae by roo characerizaion of he characerisic equaion of he wo ellipsoids. We exend he characerisic equaion o ake ino accoun he ime parameer hen perform collision deecion using he zero-se of he bivariae characerisic equaion of he wo moving ellipsoids. We firs presen in Secion II he preliminaries for formulaing he collision deecion problem of wo moving ellipsoids under raional moions. Our collision deecion soluion is hen described in deails in Secion III. A numerical example is provided o illusrae how our mehod works. We hen give some experimenal resuls in Secion IV finally he paper is concluded in Secion V where we ouline some open problems fuure research direcions. II. PRELIMINARIES Throughou his paper, we assume ha an ellipsoid A is given by a quadraic equaion X T AX = 0 in E 3, where X = (x,y,z,w) T are he homogeneous coordinaes of a poin in 3D space. The symmeric coefficien marix A is normalized so ha he inerior of A is given by X T AX < 0. We also use A o denoe he closed poin se comprising of he boundary poins inerior poins of he ellipsoid, use In(A ) o denoe he inerior poins of A. Given wo ellipsoids A : X T AX = 0 B : X T BX = 0, hey are said o be separae or disjoin if A B = /0, overlapping if A B /0. Then A B are said o be ouching exernally if A B /0 In(A ) In(B) = / /03/$ IEEE 39

2 In his secion, we firs give he separaion condiion for wo ellipsoids in E 3 which is rigorously proved in [19]. Then, we inroducd raional moions ha are he assumed moions aken by wo moving ellipsoids in his sudy. A. Algebraic condiion for he separaion of wo ellipsoids For wo ellipsoids A : X T AX = 0 B : X T BX = 0 in E 3, he quaric polynomial f() = de(a B) is called he characerisic polynomial f() = 0 is called he characerisic equaion of A B. Fig. 1 shows wo ellipsoids heir corresponding characerisic polynomial f(). The relaionship beween he geomeric configuraion of wo ellipsoids he roos of heir characerisic equaion is sudied in [19] is saed as follows: Theorem 1 (Separaion Condiion of Two Saic Ellipsoids in E 3 ) Le A B be wo ellipsoids wih characerisic equaion f() = 0. Then, 1) A B are separae if only if f() = 0 has wo disinc negaive roos; 2) A B ouch each oher exernally if only if f() = 0 has a negaive double roo. Remark: Noe ha he heorem given in [19] assumes ha he characerisic equaion be given in he form of f() = de(a + B) = 0 herefore he resul here is saed in erms of posiive roos. We make he changes here in consisency wih he classic lieraure in linear algebra. The above heorem enables us o deermine wheher or no wo ellipsoids are separae by considering only he exisence of disinc negaive roos, wihou he need of solving he quaric equaion. This can be done by using he mehod of Surm s sequence [3] ha compues he number of real zeros of a polynomial of any degree in a given inerval. B. 3D raional Euclidean moions A raional Euclidean moion in E 3 is given by M() = R() V (), (1) where V () E 3, R() is a 3 3 orhogonal marix, can be considered as a parameer of ime. The moion is a composiion of a roaion R() acing upon a poin in E 3, followed by a ranslaion V (). All raional moions can be represened in (1) wih ( v0 V () =, v 1, v ) T 2 v 3 v 3 v 3 e e2 1 2e 1 e 2 2e 0 e 2 e 2 2 e2 2e e 1 R() = 1 2e 0 e 2 0 E e2 1 2e 2 (2) + 2e 1 e 2 + e 2 2 e2 2e 3 0 e 1 2e 1 2e 0 e 1 e 2 0 e2 1 2e 0 e 2 + 2e 2 e e2 3 where E = e e2 1 + e2 2 + e2 3 v 0,...,v 3,e 0,..., are polynomials in [10]. Noe ha e 0,e 1,e 2, are he Euler parameers ha describe a roaion abou a vecor in E 3. We call hem normalized Euler parameers when E = 1. Readers are referred o [15] for a survey on raional moion design [6], [9], [10] for inerpolaing a se of posiions in E 3 using piecewise B-spline moions. When he enries of V () R() are polynomials of maximal degree k, we called M() a raional moion of degree k. A moving ellipsoid A () underaking a raional moion M() is represened as X T A()X = 0, where A() = (M 1 ()) T AM 1 (). Assume ha he maximal degree of he enries in R() V () are k R k V, respecively. Then, M 1 () A() are given by M 1 () = R()T <k R > R()T V () <kr +k V >, A() = P() <2k R > U() <2kR +k V > (3) U() T <2k R +k V > s() <2(k R +k V )> Fig. 1. Two ellipsoids heir characerisic polynomial f(). for som 3 marix P(), 3-vecor U(), scalar funcion s(). Here, he brackeed subscrip represens he maximal degree of he enries of he associaed eniy. 350

3 III. COLLISION DETECTION OF TWO MOVING ELLIPSOIDS In his secion, we presen a collision deecion scheme for deermining wheher or no wo moving ellipsoids under raional moions are collision-free. PSfrag replacemens Given wo moving ellipsoids A () : X T A()X = 0 B() : X T B()X = 0 under raional moions M A () M B (), [0,1], respecively, A () B() are said o be collision-free if A () B() are separae for all [0,1]; oherwise, A () B() collide. The characerisic equaion of A () B(), [0, 1], is hen given by f(;) = de(a() B()) = 0. A any ime 0 [0,1], if A ( 0 ) B( 0 ) are separae, hen f(; 0 ) = 0 has wo disinc negaive roos; oherwise, A ( 0 ) B( 0 ) are eiher ouching exernally or overlapping, f(; 0 ) = 0 has a double negaive roo or no negaive roo, respecively. Hence, we need o generae he zero-se of f(;) = 0, i.e., all he poins (,) ha saisfy he equaion f(;) = 0. Wrie f(;) = g () + g 3 () 3 + g 2 () 2 + g 1 () + g 0 (). Suppose he maximum degrees of enries in R A () V A () of he moion M A () are k R k V, respecively; he maximum degrees of enries in R B () V B () of he moion M B () are also k R k V, respecively. Then each enry in A() B() will have he same degree as he corresponding enry in A() given by (3), he degree of he coefficiens g i (),i = 0,...,, is herefore = 3(2k R ) + 2(k R + k V ) = 8k R + 2k V. Noice ha he g i (),i = 0,...,, are coninuous funcions in as described in [1], he four roos of f(;) = 0 are also coninuous funcions in. Therefore, he zero-se of f(;) = 0 is a collecion of curve segmens or loops in he - plane. Le Z denoe such a collecion conaining only segmens or loops wih negaive. Since f(;) = 0 has special roo paerns, Z has special opological srucure, i.e., here can only be eiher 2 disinc, 1 double or no negaive for any ; or, geomerically, a line = 0 inersecs he zero-se of f(;) = 0 in a mos wo poins wih < 0. Fig. 2 shows four ypical layou of he curve Z, assuming ha he wo ellipsoids are separae iniially. Now he curve Z always sars wih wo separae branches a = 0. These wo branches remain separae if he wo moving ellipsoids are collision-free, as shown in Fig. 2(a). These wo branches merge ino one poin a some = 0, i.e., here is a double negaive roo for f(; 0 ) = 0 (see Fig. 2(b)), if he wo moving ellipsoids make conac only a ime = 0. Finally, if here is no curve segmen in he srip (,0) ( 0, 1 ), hen here is no negaive roo wihin he inerval ( 0, 1 ), so A () (a) (c) Fig. 2. Typical opologies of he zero ses for wo moving ellipsoids ha are (a) separae because here are wo disinc negaive roos for all ; (b) separae unil = 0 when hey ouch each oher hen separae again; (c) overlapping when [ 0, 1 ] indicaed by he absence of negaive roos; (d) overlapping when [ 0, 1 ] [ 2, 3 ]. B() are overlapping in he ime inerval ( 0, 1 ) (see Fig. 2(c) & (d)). Hence, we can deermine wheher wo given moving ellipsoids are collision-free or no; if hey collide, we may also repor all collision inervals hrough an analysis of he opology of he zero-se Z. In order o exrac he zero-se of f(;) = 0 in a robus manner, we firs conver f(;) = 0 ino a Bernsein represenaion by he variable subsiuion (b) (d) = u 1 u, = v 1 v, which maps he domain (,) (,0] [0,1] o (u,v) [0,1] [0, 1 2 ], accordingly, f(;) = where = = 1 d = 1 d 1 d S(u,v), a i j i j a i j ( u v 1 u )i ( 1 v ) j a i j ( 1) i u i (1 u) i v j (1 v) j b i j B,i (u)b, j (v) a i j R, b i j = ( ( 1)i )( k )a i j, d = (1 u) (1 v), f i j 351

4 B m,n (u) = ( m n) u n (1 u) m n is he Bernsein basis funcion, is he degree of he coefficiens of f(;) in. Le S denoe he zero-se of S(u, v). Clearly, he zerose of f(;) he zero-se of S(u,v) have he same opology. For he exracion of he zero-se of S(u, v), we used he IRIT sysem, a geomeric modeling package developed a Technion, Israel. (See Elber Kim [5] for deails of he implemened algorihm.) The zero-se S was compued by IRIT organized ino conneced componens. By projecing hese componens o he -axis, we can deermine all he inervals of, if any, over which S does no have a curve segmen. This enables us o deermine wheher or no he wo ellipsoids are collisionfree, repor all he collision inervals when hey are no. (a) A Numerical Example Consider wo ellipsoids A : x2 5 + y2 8 + z2 10 = 1 B : x y2 5 + z2 = 1 moving under raional moions (Fig. 3(a)) given by wih V A () = M A () = M B () = R A () V A () R B () V B (), ( ) ( ) ,V B () = The roaional marices R A () R B () ake he form as in (2) wih Euler parameers {e (A) 0,e(A) 1,e(A) 2,e(A) 3 } {e (B) 0,e(B) 1,e(B) 2,e(B) }, respecively, where 3 e (A) 0 e (A) 2 e (B) 0 e (B) 2 = 0 e (A) 1 = e (A) 3 = 0 e (B) 1 = e (B) 3 = = = = Since he zero-se of heir characerisic equaion f(;) = 0 (as shown in Fig. 3(b)) conains curve segmens in he inervals [0,0.1699], [0.365,0.7055] [0.85,1] (corr. o sig. digis), we conclude ha he wo moving ellipsoids collide, in paricular, hey are overlapping or ouching wihin he ime inervals [0.1699, 0.365] [0.7055, 0.85]. PSfrag replacemens (b) Fig. 3. Two ellipsoids moving under quaric raional moions. (a) The moion pahs; (b) he corresponding zero-se of heir characerisic equaion. IV. EXPERIMENTAL RESULTS We have esed our collision deecion mehod on ellipsoids moving under raional moions of differen degrees. The compuaions are carried ou in double precision on a Penium 2GHz CPU. The resuls are shown in Table I. Higher order moion leads o an increase in he degree in of he coefficiens of he characerisic polynomial f(;), which in urn resuls in a longer execuion ime. For quaric moions, a single collision deecion akes abou seconds. I is worh noing ha a considerable amoun of ime has been spen on he zero-se exracion module which compues he inersecion of a Bézier surface a plane, as compared o he ime used in obaining he characerisic polynomial. Therefore he use of an efficien geomeric inersecion scheme is criical o he overall performance of our collision deecion mehod. I is also found ha he ime aken for compuing he zero-se for collision-free cases is in general longer han ha for colliding cases, because of he exisence of longer inersecion curves in he zero-se when wo moving ellipsoids do no collide. As have been menioned before, he degree of he coefficiens of f(;) can be up o = 8k R + 2k V in. In fac, when we apply our collision deecion mehod o ellipsoids moving under degree 6 raional moions, numerical insabiliy becomes a major problem for ob- 352

5 TABLE I AVERAGE CPU TIME TAKEN FOR RATIONAL MOTIONS OF VARIOUS DEGREES. THE RATIONAL MOTION M() IS COMPOSED OF R() AND V () GIVEN BY (1). THE COLUMNS (I) AND (II) SHOW THE TIME TAKEN IN POLYNOMIAL SET-UP AND ZERO-SET EXTRACTION, RESPECTIVELY. Degree in Average CPU ime (sec) R() V () M() f(;) (I) (II) overall aining an accurae answer. Errors come from boh he evaluaion of he high degree characerisic polynomial he exracion of he zero-se. The possibiliy of ackling his limiaion for high degree moions by using muli-precision compuaions remains o be explored. We have also applied a sampling echnique o approximae he zero-se of he bivariae characerisic equaion f(;) = 0. Using a marching square echnique applied o a regular grid of cells, we have experienced a considerable speed-up of a leas 250 imes in compuing he zero-se of f(;) = 0. This sampling-based approach is no exac; here is some danger of missing small loops in he zero-se. However, in many pracical siuaions, his approximae echnique would be quie useful. Almos all conemporary algorihms for collision-deecion are based on cerain sampling echniques. In he fuure work, we shall invesigae various ways of speeding-up he zero-se compuaion, which include mehods for approximaion as well as for exac compuaion of he zero-se. V. CONCLUSIONS We have presened an exac collision deecion mehod for wo ellipsoids moving under raional moions. The mehod is based on an algebraic separaion condiion for wo saic ellipsoids. Alhough he separaion condiion involves he characerizaion of he roos of he characerisic equaion of wo ellipsoids, here is in fac no need o solve for he roos of he quaric equaion. Our collision deecion mehod is based on his algebraic condiion uses he characerisic equaion f(; ) = 0 of wo moving ellipsoids. By exracing he zero-se of his bivariae funcion, we can easily deermine wheher or no wo moving ellipsoids are collision-free during heir moions. Alhough we consider only Euclidean raional moions in his sudy, our collision deecion mehod can in fac be applied wihou any modificaions o ellipsoids moving wih affine spline moions [7]. These ellipsoids may also change heir shapes during moion herefore are very suiable for applicaions such as compuer animaions morphing. VI. ACKNOWLEDGMENTS The work of Wenping Wang is suppored by he grans (HKU 7033/98E HKU 7032/00E) from he Research Gran Council of Hong Kong. VII. REFERENCES [1] R. Bhaia, Marix Analysis, Graduae Texbook of Mahemaics, vol. 169, Springer, New York; [2] T.J. IÁ. Bromwich, Quadraic Forms Their Classificaion by Means of Invarian-Facors, Hafner Publishing Co., New York; [3] L. Dickson, Elemenary Theory of Equaions, Wiley, New York; 191. [] D. H. Eberly, 3D Game Engine Design, Academic Press; [5] G. Elber M.-S. Kim, Geomeric consrain solver using mulivariae raional spline funcions, in Proceedings of he 6h ACM Symposium on Solid Modeling Applicaions, Ann Arbor, Michigan, USA, June 8, 2001, pp [6] T. Horsch B. Jüler, Caresian spline inerpolaion for indusrial robos, Compuer Aided-Design, vol. 30, no. 3, 1998, pp [7] D.E. Hyun, B. Jüler M.-S. Kim, Minimizing he disorion of affine spline moions, Graphical Models, vol. 6, no. 2, 2002, pp [8] M. Ju, J. Liu, S. Shiang, Y. Chien, K. Hwang, W. Lee. A novel collision deecion mehod based on enclosed ellipsoid, in Proceedings of 2001 IEEE Conference on Roboics Auomaion, 2001, pp [9] B. Jüler M.G. Wagner, Compuer-aided design wih spaial raional B-spline moions, ASME Journal of Mech. Design, vol. 116, 1996, pp [10] B. Jüler M. Wagner, Kinemaics Animaion, in G. Farin, J. Hoschek, M.-S. Kim, ediors, Hbook of Compuer Aided Geomeric Design, Norh-Holl; 2002, pp [11] C. Lennerz E. Schömer, Efficien disance compuaion for quadraic curves surfaces, in Proceedings of 2nd Conference on Geomeric Modeling Processing, Saiama, Japan, July 10-12, 2002, pp [12] X. Lin T. Ng, Conac deecion algorihms for hree-dimensional ellipsoids in discree elemen modeling, Inernaional Journal for Numerical Analyical Mehods in Geomechanics, vol. 19, 1995, pp

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