Kinetic Collision Detection: Algorithms and Experiments

Transcription

1 Kineti Collision Detetion: Algorithms n Experiments Leonis J. Guis Feng Xie Li Zhng Computer Siene Deprtment, Stnfor University Astrt Effiient ollision etetion is importnt in mny rooti tsks, from high-level motion plnning in stti environment to low-level retive ehvior in ynmi situtions. Espeilly hllenging re prolems in whih multiple roots re moving mong multiple moving ostles. There is extensive work on the ollision etetion prolem in rootis s well s in other fiels. Mny of the suessful pprohes exploit the ontinuity or oherene of the motion to reue the ollision heking overhe. In this pper, we present numer of ollision etetion lgorithms formulte uner the Kineti Dt Strutures (KDS) frmework, frmework for esigning n nlyzing lgorithms for ojets in motion. The KDS frmework les to event-se lgorithms tht smple the stte of ifferent prts of the system only s often s neessry for the tsk t hn. Erlier work hs emonstrte the theoretil effiieny of KDS lgorithms. In this pper we present new lgorithms n emonstrte their prtil effiieny s well, y n implementtion n iret omprison with lssil ro n nrrow phse ollision etetion tehniques. 1 Introution Collision etetion is n lgorithmi prolem rising in ll res of omputer siene eling with the simultion of physil ojets in motion. Exmples inlue motion plnning in rootis, virtul relity nimtions, omputer-ie esign n mnufturing, n omputer gmes. Though physil simultion involves severl other omputtionl tsks, suh s motion ynmis integrtion, grphis renering, n ollision response, ollision etetion remins still one of the most time onsuming in suh system. Often the ollision etetion prolem is roken up into two prts, the so-lle ro phse, in whih we ientify the pirs of ojets we nee to onsier for possile ollision, n the nrrow phse in whih we trk the ourrene of ollisions etween speifi pir of ojets [13, 16]. For the ro phse, lmost ll uthors use some kin of simple ouning volumes for the ojets themselves, or for portions of their E-mil: {guis,feng,lizhng}@s.stnfor.eu. The uthors wish to knowlege support from NSF grnts CCR , IRI , n US Army MURI grnt DAHH trjetories in spe or spe-time, so s to quikly eliminte from onsiertion pirs of ojets tht nnot possily ollie. Mny ifferent kins of ouning volumes hve een suggeste n trie, inluing xis-ligne or oriente ouning oxes, spheres, et. [19, 6, 11, 16]. The nrrow phse is more speilize, oring to the types of ojets eing onsiere. The simplest ojets to onsier re onvex polytopes, n this se hs een extensively stuie in the literture [15, 16, 18, 1]. More omplex ojets re then roken up into onvex piees, whih re teste pirwise. Algorithmilly, the onvex polytope intersetion prolem is speil se of liner progrmming; in two n three imensions even more effiient tehniques hve een evelope in omputtionl geometry, tht n e pplie fter suitle preproessing of the two polytopes [7]. The methos, however, tht hve proven to work est in prtie exploit the temporl oherene of the motion to voi oing n initio intersetion test t eh time step. Not surprisingly, the ollision etetion prolem is losely relte to the istne omputtion prolem. Sine the istne etween two ontinuously moving polytopes lso hnges ontinuously, mny well-known ollision etetion lgorithms, suh s those of Lin n Cnny [14, 15], Mirtih [16, 17], n Gilert et l. [1] (see lso [5]), re se upon trking the losest pir of fetures of the polytopes uring their motion. The effiieny of these lgorithms is se on the ft tht, in smll time step, the losest pir of fetures will not hnge, or will hnge to some nery fetures on the polytopes. Collision etetion nee not require ext istne mintenne, however. Furthermore ollisions, whether they e of ouning volumes or of the ojets themselves, ten to e very irregulrly spe over time n thus goo time step is hr to hoose. This is extly the setting in whih event-se methos ome in hny. We present elow new ollision etetion lgorithm for multiple moving onvex oies, tking vntge of these intuitions. The ro phse of our lgorithm uses spheres s the ouning volumes n mintins the ul power igrm [2] of the spheres s they move; this is gurntee to ontin n ege etween the losest pir of spheres, whih re the two tht n ollie next. For the nrrow phse, we mintin seprtion ertifites etween two moving onvex polytopes. These ertifites re upte s neessry uring the motion. These

2 re generliztion of the kineti ertifites use y [9] in two-imensions. In orer to tke the vntge of the temporl oherene of ontinuous motion, we stuy the lgorithms uner the Kineti Dt Strutures (or KDSs for short) frmework, introue in [4]. In the kineti setting we ssume tht the instntneous motion lws for our polytopes re known, though they n e hnge t will y ppropritely notifying the KDS. Our smpling of time is not fixe, ut is etermine y the filure of ertin onitions, lle ertifites. Using the known motion lws, these ertifite filure times re estimte n ple in n event queue. In the plinest form of KDS, the ertifites eing mintine prove (in the strit mthemtil sense) tht no ollision hs ourre. In our nrrow phse implementtion, for exmple, these re seprtion ertifites, proving tht the two polytopes re seprte y plne. The filure of seprtion ertifite nee not men tht ollision hs ourre; it n simply men tht the file ertifite hs to e reple y one or more others, still proving the non-intersetion of the polytopes. In generl, t kineti event orresponing to ertifite filure, the ertifite set eing mintine h to e repire, n in the proess the ttriute omputtion possily upte s well. A goo KDS will hoose ertifite set tht n e repire lolly n effiiently when one of its memers fils. This is possile euse KDS exploits ontinuity or oherene of the motion to otin ontinuity of the proof. A goo KDS is lle ompt if it requires little spe, responsive if it n e upte quikly fter ertifite filure, lol if it justs esily to hnges in the motion plns of the ojets, n effiient if the totl numer of events is smll. Our kineti ollision-etetion t strutures hve ll these esirle properties; they mintin only smll onstnt numer of ertifites per ojet, n the ost for proessing ertifite filure or motion pln upte is smll. Most importntly, KDS-se pproh llows ifferent prts of the system to e smple only t the rte neessry for the motions in eh prt. For exmple, fst moving ullet nee not fore fine time step in prts of the environment wy from the ullet s pth. In Setion 2, we first introue our seprtion mintenne lgorithm for two onvex polytopes. In Setion 3, we esrie our power igrm mintenne lgorithm for set of moving lls. Those two lgorithms together give us kineti ollision etetion lgorithm for set of moving onvex oies. The following Setion 4 isusses the mintenne of the kineti event queue n our methos for lulting event times. Setion 5 presents the results of our experiments on those kineti ollision etetion lgorithms n ompres kineti event-se sheuling with fixe time-step methos. () () Figure 1. The () vertex-fet n () ege-ege ses of the isoltion onition. 2 Convex olytopes In this setion we onsier the prolem of ertifying the seprtion (equivlently, non-intersetion) of two onvex polytopes n moving rigily in 3D. Two non-interseting onvex polytopes lwys hve seprting plne; suh plnes hve een use previously in omputer nimtion for ollision etetion [3]. It is lso known tht seprting plne n lwys e positione so tht it is either prllel to fet of one of the polytopes, or prllel to pir of eges, one from eh polytope. If we mke two opies of this seprting plne n trnslte them towrs n respetively, till they eh mke ontt, we will hve one of the two situtions epite in Figure 1. In oth situtions we en up with two prllel plnes, one supporting n one supporting. The two plnes prtition spe into three piees, hlfspe ontining, the empty sl etween the two plnes, n hlfspe ontining this is wht we ll the seprtion onition. In sitution (), the vertex-fet se, we hve ontt t vertex of n ontt fet of (this simpliility ssumption n e esily remove), while in sitution (), the ege-ege se, we hve ontt with ege of n ege of. The seprtion onition is esily expressile in terms of the oorintes of the verties of n in ontt with the two plnes n their immeite neighors. We first ssert tht the plne supporting is ove tht supporting this is just signe volume or orienttion test on the tetrheron, in oth situtions () n (). In terms of oorintes this is the eterminnt onition: x y z 1 x y z 1 x y z 1 x y z 1 >.

3 e () () e Figure 2. The () vertex-fet n () ege-ege ses of push event. () () Figure 3. The () vertex-fet n () ege-ege ses of roll event. We ll this onition the isoltion ertifite. We lso nee to ssert tht the two plnes support n respetively, tht is ove the top plne n elow the ottom plne. Beuse n re onvex polytopes, it is suffiient to ssert this onition lolly for the verties of n tht re neighors of,,, n. We ll these itionl ertifites, whih n e expresse s orienttion tests s well, the support ertifites. There re four support ertifites in the ege-ege se, while in the vertex-fet se the numer of support ertifites equls the egree of the vertex. When polytopes n move ontinuously s rigi ojets, s long s the isoltion ertifite n the relevnt support ertifites remin vli, n nnot ollie. This is the motion oherene tht our kineti t struture will exploit. Note tht we o not moel seprting plne expliitly the existene of suh plne is implie y the seprtion onition. Our most importnt tsk is to see how we n repir the seprtion onition when one of its ertifites fils. A filure of the isoltion ertifite is lle push event orresponing to the eterminnt ove eoming zero. We ress the vertex-fet n ege-ege situtions seprtely. Figure 2 () shows the sitution when vertex of hs eome oplnr with fet of. When this event ours we nee to hek if is insie the tringle ; ifit is, n hve ollie n the ollision response oe hs to e invoke. If is outsie the tringle then the line supporting t lest one sie of, sy, seprtes from the tringle. Now imgine line through prllel to n suppose we strt rotting the plne roun tht line so tht it oes not enter. The rottion will e stoppe y enountering either vertex e of or vertex f of. In the former se, e must e neighor of n we n now use e n s two eges giving us new ege-ege isoltion ertifite. The ltter se is symmetri. Figure 2 () shows the sitution when the ege of hs eome oplnr with ege of. in this se s well, we n either report ollision or fin new isoltion ertifite y heking the fetures jent to n. We ll filure of support ertifite roll event. The sitution when vertex support ertifite fils is espeilly simple n is shown in Figure 3 (): neighor e of vertex of is now on the plne through n prllel to fet. We nee to reple the ol ginst vertex-fet isoltion ertifite y one of e ginst, elete ll the support ertifites for n the ones for e. An ege support ertifite for ege of fils when one of the fets jent to the ege, sy e, eomes prllel to the ege of see Figure 3 (). In this se the ol ege-ege isoltion ertifite etween n will e reple y either e ginst or e ginst. The hoie is itte y whether the iretion of is further from tht of e or e. This ompletes our isussion of how the seprtion onition is to e mintine s the polytopes move. We note tht, unlike lgorithms, suh s Lin-Cnny [15], in whih the losest pir of fetures of the polytopes is mintine, no preproessing (suh s the omputtion of Voronoi igrm) whtsoever is require y our lgorithm. Whenever one of our ertifites fils, either ollision hs tully ourre, or strightforwr lol test repirs the seprtion onition so tht the simultion n go on. 3 ower Digrm for Blls Blls re mong the most extensively use primitives in geometri moeling; for exmple, in severl ollision etetion or ry tring pkges, ojets re surroune y ouning spheres so tht preliminry intersetion tests n e quikly performe. In this setion, we esrie metho to mintin the power igrm of set of moving spheres

4 n utilize this struture in performing the ro phse of our ollision etetion. ower igrms [2] re generliztion of Voronoi igrms in whih the sites efining the igrm re not points ut lls. They erive their nme from the ft tht the istne use in their efinition is not the stnr Eulien istne, ut inste the lssil notion of the power of point with respet to ll. In the following, we enote B(o,r) the ll entere t the point o n with rius r. For point p n ll B(o,r), the power istne δ(p,b) is efine to e op 2 r 2. For set of lls B, the power igrm ell V(B) of B B then onsists of ll the points tht re loser to B thn to ny other lls in B, mesure using the power istne. It is esy to verify tht the isetor etween two lls is plne. Thus, eh power igrm ell is (possily unoune) onvex polytopes. This property mkes it esier to represent n mnipulte power igrms thn the stnr Voronoi igrms for lls uner the Eulien istne. The power igrm is useful s proximity struture. In [12], it is shown tht mong set of isjoint lls, the losest pir re lwys neighors in the power igrm. Thus, if we n mintin the power igrm for isjoint lls, we re then le to mintin their losest pir s well. By trking tht pir, we n onveniently etet ollisions. When lls my penetrte eh other, we n then mintin the onnete omponents of the moving lls n use them to filter the pirs neee to e psse to the nrrow phse. For the onveniene of mintenne, we onsier the ul simpliil omplex, lle the power omplex, of the power progrm. For lls in generl position, the power omplex is tringultion of the onvex hull of the ll enters. For equl size lls in two imensions, the power omplex is just the sme s the well-known Deluny tringultion of their enters. The ruil property is tht the power igrm ells of two lls re jent if n only if there exists n ege etween them in the power omplex. In the following, we will first review the metho for mintining the Deluny tringultion (in two imensions) n then exten it to power igrms in ny imension [1]. The InCirle test is the most importnt primitive use in the omputtion of Deluny tringultions. For four points,,, in the plne, InCirle(,,,)is true if is insie the oriente irumirle of. Suppose tht n re two tringles in tringultion T. Then, the ege is si to e lolly Deluny if the test InCirle(,,,)is flse. It is known tht the Deluny tringultion mits lol ertifition, i.e. T is the Deluny tringultion if eh ege in T is lolly Deluny. Suh lol ertifition mkes it very esy to mintin the Deluny tringultion s we n simply ertify the lol Deluny-hoo of eh ege n fix the tringultion lolly whenever suh ertifite fils. The InCirle ertifite fils when the points,,, eome o-irulr. A flip op () () Figure 4. Flip events () in two imensions n () in three imensions. ertion is then invoke to fix the tringultion, where the flip is to elete the ege n insert the ege, or equivlently, to elete n n n (Figure 4()). The ove proeure extens to higher imensions where the InCirle test is reple y the InSphere test. But the flipping opertion eomes more sutle. In three imensions, when five points eome o-spheril, there re extly two wys to tringulte those five points. The lol flip opertion is then to reple tringultion of five o-spheril points with its ul tringultion (Figure 4()). This property tully hols in ny imension, i.e. in imensions, there re extly two wys to tringulte + 2 points in onvex position. The power omplex n e mintine in similr wy to the Deluny tringultion. First, we nee to reple the InCirle tests y the Inower tests. In three imension, we sy tht ll B(,r ) is insie the power sphere of lls B(,r ), B(, r ), B(, r ) n B(e,r e ), if the following eterminnt onition hols. x y z x 2 + y2 + z2 r2 1 x y z x 2 + y2 + z2 r2 1 x y z x 2 + y2 + z2 r2 1 x y z x 2 + y2 + z2 r2 1 x e y e z e x 2 e + y2 e + z2 e r2 e 1 <. For set of isjoint lls, the sme lol property hols for power omplexes. Thus, the power omplex n e ertifie y sserting tht eh fet in the omplex stisfies the lol Inower test with respet to its two jent verties. An importnt ifferene etween power omplexes n Deluny tringultions is tht ll might not present in the power omplex. For this to hppen, it must e the se tht the ll is ompletely ontine in the union of four other lls. In prtie, suh situtions only hppen rrely. Thus, we n tret them seprtely. In our metho, we trk the lls tht over eh sent ll n prevent it from presenting in the power omplex. Suh trking n e etter

5 unerstoo y lifting mp L, whih we o not isuss here. By mintining the power omplex, we n mintin the losest pir of lls for set of isjoint lls, or the onnete omponents of the lls when they n penetrte eh other. This gives us wy to perform ro phse ollision etetion s we only nee to invoke our nrrow phse test for those ojets with overlpping ouning spheres. A reson tht we hoose the power igrm s our ro phse struture is tht it typilly hs linerly mny eges n unergoes suqurtilly mny hnges for lgeri motions (lthough, in the worst se, the numer of eges n e qurti, n the numer of hnges might e ui or higher). Atully, the theoretil oun on the numer of hnges of the power omplex is still fmous open question. Inste of performing theoretil nlysis, we stuy those omplexity mesures y experiments n ompre them to lssil ouning-ox methos in Setion Sheuling Kineti Events The min loop of kineti t struture onsists of eteting n proessing kineti ertifite filures. These pening kineti events n e orgnize in numer of wys, of whih the simplest is to just mintin them in glol priority queue. Eh tive kineti ertifite sheules n event in the queue, whih its erliest preite filure time lrger thn the urrent time. When ertifite fils, the proof, n with it the ssoite ertifite set s well s the event queue, nees to e upte. The lultion of n event time is frequently non-trivil omputtionl tsk. In the exmple ertifites of the previous setion, the ertifite itself is n lgeri inequlity on the oorintes of few feture points (verties) of the ojets. Yet even in the simplest se of llisti rigi motions for the ojets with no outsie fores (suh s grvity), vertex trjetories re still polynomil mixtures of lgeri n trigonometri funtions of time. The evlution of the roots of lgeri forms on these trnsenentl funtions is hllenging numeril prolem. Furthermore, the ost of suh omputtions nnot lwys e justifie in terms of the finl result tht nees to e ompute. For exmple, lmost ll kineti simultions involve the e-sheuling of events these re events tht will not hppen euse the ssoite ertifites were remove from the proof, n the omputtionl resoures tht went into their event time lultion will e wste. We ress this prolem y the use of onservtive pproximtions to the ertifite filure times. We exploit the geometri struture of our ertifites to get esier to ompute pproximte filure times tht re gurntee to e less thn or equl to the tul filure time (this is wht we men y onservtive pproximtion). We ll these pproximtions reonfirmtion times. Eh ertifite then sheules reonfirmtion time in the event queue. When tht reonfirmtion time is rehe, the ertifite is evlute gin, new reonfirmtion time is ompute, n the ertifite is resheule. This ontinues until we re lose enough to the tul filure time y some riterion. In the simplest se we just vne to the next frme time for the simultion when the reonfirmtion intervl is less thn the interfrme intervl. In more elorte ses, sine we re y now very lose to the tul filure time, we swith to n nlyti pproximtion n estimte tht time more preisely. We hve experimente with polynomil pproximtions to the trigonometri funtions rising in our motion plns n otine goo results. In oth ses n event my e proesse t time slightly ifferent from the tul time when it ours; this n use ertin iffiulties tht we hve to ress. Overll, however, this metho hs mny vntges euse few esy onservtive steps get us quite lose to the tul event time. 4.1 Computing Certifite Filure Times To illustrte the use of reonfirmtion events we fous on the onvex polytope seprtion ertifites presente in Setion 2. We nee to tret isoltion n support ertifites ifferently, s the ltter epen only on the rottionl n not the trnsltionl motion of the onvex oies. Let us look gin in Figure 1, where we hve two prllel plnes, one supporting from elow n the other from ove. Suppose this is snpshot tken t the time when the isoltion ertifite ws first rete. Let us imgine in ition n intermeite plne prllel to these two n hlf-wy etween them. For now imgine this intermeite plne s fixe in spe, while n ontinue their motion from the snpshot shown. It is ler tht s long s none of the verties,,, n involve in the isoltion ertifite ross this plne, the isoltion ertifite nnot fil. If we hve upper ouns on the trnsltionl n rottionl veloities of n, we n use them to esily erive upper ouns on the spees of the four verties in the ertifite. If D enotes the seprtion of the two prllel plnes t the snpshot in Figure 1 n v, v, v, n v the mximum spees of,,, n respetively, then onservtive oun on the isoltion ertifite filure time is { min{d/2v,d/2mx{v t =,v,v }}, min{d/2 mx{v,v },D/2mx{v,v }} for ses () n () respetively. Better ouns n e otine y giving the intermeite plne some motion s well, sy y verging the liner spees of the two oies t the time of the snpshot, or y etter lning the position of the intermeite plne with respet to the reltive spees of the two oies. We note tht when the oies n hve no rottionl motion, roll events o not our n the support ertifites nnot fil. Thus eh support ertifite is est onsiere in the reltive frme of the oy where the roll event hppens.

6 Consier the se of vertex-fet roll event, s in Figure 3 (), in the lol frmework of oy. Let θ e the ngle etween the ege e n its projetion on the plne supporting t when the support ertifite ws rete. Though the oies n re rotting out ifferent xes, we fous on wht hppens lolly t vertex n speifilly roun n xis through on the support plne n perpeniulr to e. If ω n ω re upper ouns on the ngulr veloities of oies n respetively, then it is not hr to show tht θ/( ω + ω ) is onservtive oun on the time when e n roll onto the support plne. By tking the minimum of these times over ll neighor verties of, onservtive oun for the roll event n e otine. In the ege-ege se of Figure 3 (), onservtive oun n e similrly erive. 4.2 Initiliztion, Composition, n Reovery In our onvex polytope ollision etetion system, the power igrm of ouning lls for the polytopes is use for the ro phse, n the seprtion ertifites etween pirs of onvex polytopes for the nrrow phse. The ro phse psses pir of polytopes to the nrrow phse when the istne of the ouning lls rops elow some speifie frtion of the sum of their rii. For two isjoint lls the plne of points hving equl power to the lls seprtes the lls, n fortiori their enlosing polytopes. This plne n e use to initilize the seprtion ertifite etween the polytopes we omit the strightforwr etils. A polytope pir is roppe out of the nrrow phse when the istne of their ouning lls exees some other lrger speifie frtion of the sum of their rii (so s to uil some hysteresis into the system). Initiliztion for the nrrow phse my lso e reinvoke in mi-simultion, ut ue to lk of spe we omit this isussion. 5 The Experiments We hve implemente our lgorithms on entium II C using Mirosoft Visul C++. We hve lso ompre our 3D onvex polytope methos to some of the stte-of-thert ollision etetion pkges ville toy. Beuse of the ler istintion etween the ro n nrrow phses, we hve onut our omprisons for the two phses oth jointly n seprtely. Animtions se on our lgorithms n e seen t feng/kds_lsn/ 5.1 Bro hse We n lwys perform pirwise heking to etet ll ojet pirs with overlpping ouning volumes. Suh rute-fore metho works well for smll numer of ojets n requires no t strutures, ut its qurti growth rte limits its slility. To reue the numer of pirs eing heke, mny lgorithms use xis-ligne ouning oxes n test the intersetion of the projetion of the ojets in lower (one or two) imensions [6, 16]. Only those pirs tht hve interseting projetions nee to e heke further. We hve implemente metho to mintin the sorte orer of the projetion of the oxes on eh of the x-, y- n z-xes thus eteting potentil ollisions mong the oxes when the orering in one projetion hnges. As for the test t, we first generte set of isjoint lls with vrile rii rnomly, with uniform istriution. These lls were use s the input t to test the power igrm methos. Then, for eh ll, we rete n xis-ligne ox with the sme position n veloity, n omprle size. These oxes form the input t to the ouning ox methos. During the motion, ojets oune ginst eh other when they ollie. We then rn the lgorithms on t sets rnging from 1 to 1 ojets for suffiiently long simultion times to otin relile verge numer of events n ollisions per seon. The t re summrize in Figure 5. In the three figures, the x-xis mesures the numer of ojets in the t file, n the y- xis mesures () the numer of events per seon, () the numer of ertifites in the struture, n () the numer of ollisions per seon, respeitively. From Figure 5(), we n see tht the numer of events when using the ouning ox metho exhiits qurti growth in the numer of ojets, while in the power igrm metho it is signifintly smller n pproximtely liner. However, the time svings re not s signifint s the reution in events woul inite. In the power igrm metho, we hve to ompute eterminnt, whih is egree five polynomil for liner motions, n then solve the resulting eqution y using numeril methos. Also, it tkes more time to upte the power igrm when n event hppens. In our experiments, eh event uptes out 1 ertifites on the verge n for eh ertifite, it tkes out 1 3 floting point opertions to ompute the filure time. Still the power igrm metho eomes superior when there re more thn few hunre ojets, s is the se in most uses of suh simultions. Furthermore, in pplitions suh s omputtionl moleulr iology, spheres re nturlly the prefere ouning volumes. Figure 5() shows tht lthough the numer of ertifites use in the power igrm struture is lrger thn tht in the ouning ox metho, oth of them re liner s funtion of the numer of ojets. Figure 5() shows the numer of ollisions (per seon) re out the sme in the two experimentl moels, whih justifies our experimentl setup. It is lso ler from the t tht in oth ses, the numer of ollisions per seon hs liner reltionship to the numer of ojets for the rnom istriutions teste. The ove experiments provie some insight on how our metho ompres to fixe time step methos s well. One iffiulty in ompring event-se methos with fixe time-

7 x 14 4 #power igrm #ouning ox #power igrm #ouning ox 4 35 #power igrm #ouning ox numer of events numer of ertifites numer of ollisions numer of ojets numer of ojets numer of ojets () # of events/seon () # of ertifites () # of ollisions/seon Figure 5. Comprisons etween the power igrm n ouning ox methos. step methos is the hoie of the time step size. For fixe time-step metho to e ompletely preise, we woul hve to hoose the time step oring to the minimum interollision time (the time etween two ollisions) in orer to llow ll the ollisions to e pture. While most pplitions o not require tht mount of preision, resonle time step hoie woul still nee to e t lest proportionl to the verge inter-ollision time. We then expet the ost of the fixe time-step metho to e proportionl to the numer of ollisions times the upte ost of the struture t eh step. In the methos given in [6, 16], the upte ost is typilly liner in the numer of ojets, if the motion of the ojets is slow reltive to the time-step size, n worse otherwise. Furthermore, s our experiments show (Figure 5()), the numer of ollisions per seon is liner in the numer of ojets, n therefore the overll running time of the ouning ox metho using fixe time steps will e t lest qurti in the numer of ojets. 5.2 Nrrow hse We hve ompre our seprtion se metho to some of the well-known proximity se lgorithms. The losest pir trking methos presente in [15, 17] hve een quite suessful in exploiting the temporl oherene of motion. Our seprtion se metho is losely relte to these proximity methos, sine oth they n we mintin pir of fetures, one from eh polytope, tht n e use to otin seprting plne. In ll methos, the most importnt qulity mesure is the numer of times the pir of fetures use to otin the seprting plne hnges uring the simultion. Although the methos in [15, 17] o not trk every hnge in the losest pir of fetures, the numer of hnges in the losest pir is resonle mesure of the ost of these lgorithms, euse t eh time step they perform lol wlk long the polytopes to the next losest pir, whose length n e expete to e omprle to the numer of hnges to tht pir (were it to e trke). We hve onute experiments to ompre the numer of hnges of the losest pir n the numer of hnges in numer of hnges numer of hnges verties of ojets #losest pir #seprtion pir () ojets with vrying sizes ojets spee #losest pir #seprtion pir () ojets with vrying spee Figure 6. Comprisons etween the proximity-se n seprtion-se methos. the seprtion ertifites mintine y our metho. We rn t in two settings. In one set of t, the ojets hve the sme initil veloity ut ifferent omintoril sizes, rnging from 2 to 1 verties. In the other set of t, the ojets lwys hve 4 verties, ut ifferent veloities were use. We ounte the numer of events (i.e. numer of feture pir hnges for the losest pir metho n numer of seprtion ertifite filures for the seprtion plne metho) per ollision for eh metho. The experimentl results re summrize in Figure 6. As n e seen from the t, the numer of events in our metho is onsistently smller thn the numer of events

8 numer of verge time per frme verties (in milliseons) Tle 1. Seprtion running time for smple t sets. in the losest pir metho. Wht is more interesting is tht the rte of hnges to the seprting pir is very stle n mostly inepenent of the size n spee of the ojets, while the rte of hnges to the losest pir is signifintly more vrile n unstle. A possile explntion of this phenomenon is tht the losest pir sserts stronger geometri onition on the pir of fetures involve thn our seprtion ertifites o. This reltive stility mkes our kineti metho ttrtive for rel-time environments. On entium II C, with our unoptimize implementtion we otine the following running time sttistis for trking the seprtion plne etween two onvex oies. For eh t set, the initil veloities, spet rtio n ouning sphere re the sme ut the omplexity (numer of verties) iffers. The results re summrize in Tle??. The simultion is for 4 ojets with verge size 3, verge initil spee.9 n verge omplexity of 3 (resp. 6, 11) verties, ouning roun in ue of size 6. The totl verge time per frme is.316 seons when the power igrm is use for the ro phse n seprtion plnes re trke t the nrrow phse for ollision etetion. The totl verge per time per frme is.386 seons when simple pirwise ouning sphere test (n n) is use in the ro phse while sepertion plnes re trke in the nrrow phse. 6 Conlusions We hve presente new results on the esign n implementtion of ollision etetion lgorithms, using kineti t strutures. These strutures le to novel methos for exploiting temporl oherene in ollision etetion n relte proximity prolems. For the first time, kineti t strutures mke it possile to rigorously ompre lternte strtegies of mintining ttriutes of n evolving system in the mnner tht hs een so suessfully use in the nlysis of lgorithms. We expet to see mny more KDS pplitions in rootis in the future. Referenes [1] G.Alers, L. J. Guis, J. S. B. Mithell, n T. Roos. Voronoi igrms of moving points. Internt. J. Comput. Geom. Appl., 8:365 38, [2] F. Aurenhmmer. ower igrms: properties, lgorithms n pplitions. SIAM J. Comput., 16:78 96, [3] D. Brff. Anlytil methos for ynmi simultion of nonpenetrting rigi oies. Computer Grphis (roeeings of SIGGRAH 89), 23(3): , July [4] J. Bsh, L. J. Guis, n J. Hersherger. Dt strutures for moile t. In ro. 8th ACM-SIAM Sympos. Disrete Algorithms, pges , [5] S. Cmeron. Enhning GJK: omputing minimum n penetrtion istnes etween onvex polyher. In ICRA, [6] J. D. Cohen, M. C. Lin, D. Mnoh, n M. K. onmgi. I-ollie: An intertive n ext ollision etetion system for lrge-sle environments. In ro. ACM Intertive 3D Grphis Conf., pges , [7] D.. Dokin n D. G. Kirkptrik. Fst etetion of polyherl intersetion. Theoret. Comput. Si., 27(3): , De [8] H. Eelsrunner. Smooth surfes for multi-sle shpe representtion. In ro. 15th Conf. Foun. Softw. Teh. Theoret. Comput. Si., volume 126 of Leture Notes Comput. Si., pges Springer-Verlg, [9] J. Erikson, L. J. Guis, J. Stofi, n L. Zhng. Seprtionsensitive kineti ollision etetion for onvex ojets. In ro. 9th ACM-SIAM Sympos. Disrete Algorithms, [1] E. G. Gilert, D. W. Johnson, n S. S. Keerthi. A fst proeure for omputing the istne etween omplex ojets. Internt. J. Root. Autom., 4(2):193 23, [11] S. Gottshlk, M. C. Lin, n D. Mnoh. OBB-tree: A hierrhil struture for rpi interferene etetion. Comput. Grph., , ro. SIGGRAH 96. [12] L. Guis n L. Zhng. Eulien proximity n power igrm. In ro. 1th Cnin Conferene on Computtoinl Geometry, [13]. M. Hur. Collision etetion for intertive grphis pplitions. IEEE Trns. Visuliztion n Computer Grphis, 1(3):218 23, Sept [14] M. C. Lin. Effiient Collision Detetion for Animtion n Rootis. h.d. thesis, Dept. Ele. Engin. Comput. Si., Univ. Cliforni, Berkeley, CA, [15] M. C. Lin n J. F. Cnny. Effiient lgorithms for inrementl istne omputtion. In ro. IEEE Internt. Conf. Root. Autom., volume 2, pges , [16] B. Mirtih. Impulse-se Dynmi Simultion of Rigi Boy Systems. h.d. thesis, Dept. Ele. Engin. Comput. Si., Univ. Cliforni, Berkeley, CA, [17] B. Mirtih. V-Clip: Fst n roust polyherl ollision etetion. Tehnil Report TR97-5, MERL, 21 Browy, Cmrige, MA 2139, USA, July [18] B. Mirtih n J. Cnny. Impulse-se ynmi simultion. In K. Golerg, D. Hlperin, J. C. Ltome, n R. Wilson, eitors, The Algorithmi Fountions of Rootis. A. K. eters, Wellesley, MA, [19] S. uinln. The Rel-Time Moifition of Collision-Free ths. h.d. thesis, Dept. Comput. Si., Stnfor Univ., lo Alto, CA, 1994.