Distance Computation between Non-convex Polyhedra at Short Range Based on Discrete Voronoi Regions

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1 Distne Computtion etween Non-onvex Polyhedr t Short Rnge Bsed on Disrete Voronoi Regions Ktsuki Kwhi nd Hiroms Suzuki Deprtment of Preision Mhinery Engineering, The University of Tokyo Hongo, Bunkyo-ku, Tokyo , Jpn {kwhi, suzuki}@im.pe.u-tokyo..jp strt n lgorithm for lulting the minimum distne etween non-onvex polyhedr is desried. polyhedron is represented y set of tringles. In lulting the distne etween two polyhedr, it is importnt to serh effiiently the pir of the tringles whih gives the pir of losest points. In our lgorithm disrete Voronoi regions re prepred s voxels round non-onvex polyhedron. Eh voxel is given the list of tringles whih hve possiility to e the losest to the points in the voxel. When tringle on the other ojet is interseting voxel, the losest tringles n e effiiently serhed from this list on the voxel. The lgorithm hs een implemented, nd the results of distne omputtions show tht it n lulte the minimum distne etween non-onvex polyhedr omposed of thousnd of tringles t intertive rtes. 1 Introdution Computing distne etween ojets is n importnt prolem in vrious field inluding omputer grphis nimtion nd simultion. In order to prevent the moving ojets from interferene, it is needed to detet if the ojets re olliding or not. In ddition to this interferene detetion, it is useful to lulte the minimum distne etween the ojets for prediting the instne of ollision in dynmis simultion. The lgorithms of distne omputtion n e lssified y their trgets s follows. Non-polyhedrl Ojets CSG models Curved surfes Clouds of points Polyhedrl Ojets without topologil informtion Sets of Polygons Polyhedrl Ojets with topologil informtion Convex polyhedr Non-onvex polyhedr In this pper we present n lgorithm for polyhedrl ojets. The ojet onsists of set of tringles nd is not neessrily needed to hve topologil informtion. 2 Relted Work For distne omputtion etween onvex polyhedrl models speified y oundry representtion, there re two mjor lgorithms: losest-feture lgorithms [7][8] nd simplex-sed lgorithms [2][1]. The Lin-Cnny losest feture lgorithm [7] defines Voronoi region on its fetures (fes, edges, nd verties). If two fetures re losest, eh feture is inside the Voronoi region of the other. In dynmis simultion the lgorithm n effiiently find the losest fetures y trking the pir of the losest fetures. V-Clip [8] employs the improved losest feture lgorithm whih n hndle the penetrtion etween ojets nd lulte the distne roustly. In these losest feture lgorithms the method of serhing the losest feture pir utilizes Voronoi regions defined round onvex polyhedr. Unfortuntely these lgorithms re sed on onvex polyhedr nd nnot e diretly pplied to non-onvex ones. In this pper, we del with the non-onvex prolem. We utilize Voronoi regions for piking up the losest tringles etween non-onvex polyhedr. Beuse it is diffiult to lulte urte struture of the Voronoi regions on nononvex polyhedron, they re defined y the disrete pproximtion method using voxels [6].

2 3 Disrete Voronoi Regions In this setion we desrie the definition of the disrete Voronoi regions nd the lgorithm of distne omputtion etween non-onvex polyhedr. R R R 3.1 Cost of Distne Computtion The totl ost of interferene detetion etween two polyhedr n e formulted s the following eqution [3]: Figure 1. Voronoi Regions Defined round Convex Polygon T = N v C v + N p C p, (1) where X T : N v : C v : N p : C p : totl ost funtion for interferene detetion numer of ounding volume pir overlp tests ost of testing pir of ounding volumes for overlp numer of feture pirs tested for interferene ost of testing pir of fetures for interferene s C p is expensive in generl, ounding volume with smller C v n sve T y deresing the numer N p. Typil lgorithms for defining ounding volumes inlude xisligned oxes, otrees [10], sphere trees [9], OBBTree [3], swept sphere volumes [5] et. voe ll, the PQP lirry utilizing swept sphere volumes n perform effiient distne lultion y trnsforming distne omputtion to intersetion detetion. In lulting the minimum distne etween non-onvex polyhedr the simplest lgorithm tests the ll pirs of fetures in them nd piks up the pir with the smllest distne. In the sme wy of ollision detetion, euse the ost of distne omputtion etween pir of fetures is expensive, the totl ost n e sved y reduing the numer of pirs of fetures with some ounding volume. Figure 2. Closest Feture in Voronoi Regions R Figure 3. CFL (Closest Feture List) on Voxels 3.2 Voxels round Ojets Voronoi region n e defined for eh feture on polyhedron: Definition Voronoi region ssoited with feture is set of points loser to tht feture thn ny other. Figure 1 shows two dimensionl illustrtion of the Voronoi regions defined round onvex ojet. If feture X on nother polygon is inluded in region R (Figure 2), it is losest to the edge, nd it is needless to lulte its distne ginst the other fetures,. The regions re esily defined on onvex polyhedron, nd I-Collide [7] nd V-Clip [8] utilize the Voronoi regions for lulting the distne etween onvex polyhedr. If we n define Voronoi regions on the fetures of nononvex polyhedron, feture X of the other polyhedron n e losest to the feture Y whose Voronoi region inludes X. Beuse it is diffiult to lulte urte struture of the Voronoi regions of non-onvex polyhedron, we disretely define Voronoi regions y voxelizing the spe round the polyhedron nd do not expliitly lulte the struture of Voronoi regions. Eh voxel is given the list of the fetures whih hve possiility to e losest to the voxel (Figure 3). This list is lled Closest Feture List (CFL). If the feture X hs interferene with the voxel V, we need to lulte the distnes etween X nd the fetures in CFL on the voxel V. 3.3 Representtion of Non-onvex Polyhedr In this pper we present n lgorithm for lulting distne etween ojets with the following properties:

3 polyhedron is represented y set of tringles. In the losest feture lgorithms, polyhedron is onstituted y three fetures (fes, edges, nd verties). In our lgorithm edges nd verties re not regrded s independent fetures ut s prt of tringle. By unifying them to tringles, we n derese the numer of distne lultions. tht is, ny points in V re given smller minimum distne y the tringle thn B. Consequently the tringle B nnot e losest to ny points in V. Voxel V d (x min ) Tringle tringle does not neessrily hve topologil informtion out neighoring tringles ross the edges. If topology is known, it is utilized for ulling the voxels whih nnot inlude the tringle on their CFLs. Tringles in polyhedron my hve interferene eh other. x mx x min d (x mx ) These properties re useful for deling with polyhedr with oundries, messy polygon models et. s the lgorithm does not lulte the Voronoi regions expliitly, tringles in polyhedron n e freely rrnged. The lgorithm does not distinguish the front nd k of tringle. So the distne lulted y the lgorithm is lwys zero or more thn zero. 3.4 Closest Tringles to Voxel In reting CFL on voxels, we utilize the minimum nd mximum vlues of the distne etween point in voxel nd tringle. If point x nd tringles re given, one tringle whih is losest to the point x n e hosen y the reltive position of x to the tringles. If x exists inside voxel V, the losest tringle vries s x moves in V. CFL on voxel n e reted y piking up ll the losest tringles for the ll points in voxel V. If the tringle Y on the other polyhedron is interfering with V, the tringles in the list on V hve possiility to e losest to Y. Given point x nd tringle, we n lulte their minimum distne d (x). In voxel V there exists the points nd xmx whih give the minimum nd mximum vlue of d (x) (Figure 4). By this definition, for ll points x in the voxel V we n write x min d (x min ) d (x) d (x mx ), x V. (2) In the sme wy, the minimum distne d B (x) n e defined for nother tringle B, nd we get the points x min B nd x mx B stisfying the following ondition: d B (x min B ) d B(x) d B (x mx ), x V. (3) If the points x mx nd x min B stisfies d (x mx ) < d B (x min B ), the definitions (2) nd (3) leds: d (x) <d B (y), x V, y V, (4) B Figure 4. Minimum Distne etween Tringle nd Point in Voxel In piking up ll the losest tringles for the voxel V, the minimum vlue d min i nd mximum vlue d mx i of the distne d i etween V nd the tringle i re lulted for ll tringles t first. Then the tringles re sorted y their minimum vlues of distne. If some tringles hve the sme minimum vlue, they re sorted gin y the mximum vlues of distne. By this proedure we n get the tringle k with the smllest d min k nd d mx k. By the ondition (4), the tringle i stisfying d mx k <d min i nnot e losest to the voxel V nd is not inluded in CFL on V. In our lgorithm voxels re defined on the lol oordinte system of non-onvex polyhedron. The CFLs re lulted only one s preproess of distne omputtions. They re kept with the geometri informtion of the polyhedron nd re reused when the reltive orienttion of the ojets is hnged, euse the CFL lultion generlly tkes muh time. For instne, the CFLs on voxels round polyhedron with 1314 tringles requires out 40 minutes to ompute. The CFL lultion n e muh elerted if we utilize the omputtion method of Voronoi region using grphis hrdwre [4]. 3.5 Clulting Distne with CFL With CFLs we ompute the distne etween non-onvex polyhedr nd B with the following four steps. t the first step, the voxels on whih hve interferene with the tringles in B re piked. The interferene test is done effiiently y defining BSP tree of voxels on nd n OBBTree [3] on the tringles of B. t the seond step, the losest feture tests re pplied for the interseting tringles nd voxels found in the first step. The Lin-Cnny losest feture lgorithm utilizes the neessry nd suffiient ondition tht eh feture is inside the

4 Voronoi region of the other. This ondition n e rewritten for the disrete Voronoi regions with CFLs s follows: Feture X intersets voxel V Y (the CFL of V Y inludes feture Y ) nd (5) Feture Y intersets voxel V X (the CFL of V X inludes feture X) Beuse the ondition(5) is not suffiient ut neessry for non-onvex polyhedr, if voxel V on interseting tringle T in B re found t the first step, they n not e losest nd re ignored t the seond step when the ondition(5) is not stisfied. In order to hek the ondition, the voxels on B whih hve interferene with the tringles in re piked t the seond step. If tringle in the CFL of V hs interferene with the voxel with the CFL inluding T re found, the ondition is stisfied. t the third step, the piked voxels re sorted y their minimum vlues of distne d min k, nd t the finl step, for eh voxel in the order of d min k, the minimum distne etween the tringles in the CFL nd the tringle interfering the voxel is lulted. By sorting the voxels t the third step, we n effiiently eliminte the exess voxels. If in the wy of finl step we hve voxel k with distne d min k whih is greter thn the minimum distne given y the previous voxels, we n ut off the distne omputtions for the following voxels, whih nnot e the losest to the polyhedron ny more. In the implementtion of our lgorithm, voxels re defined in the region whih is twie s lrge s n ojet s ounding ox. If the ojet is out of this region, the distne is pproximted y tht of the onvex hull of the originl polyhedr. 3.6 Voxel Culling If topologil informtions etween tringles re ville, the voxels where the tringle n e losest n e ulled y the pproximted Voronoi region on the tringle. The voxels whih hve no interferene with the region nnot e losest nd do not need to e lulted their distnes to the tringle. The Voronoi region of the tringle is pproximtely defined y the plnes ontining the edges on the tringle. Eh plne is prllel to the verge of the norml vetor of the tringle nd tht of the neighoring tringle. When the polyhedron is onvex, the pproximted region on tringle is identil to the urte Voronoi region. When non-onvex, it is not neessrily identil, ut it inludes the urte Voronoi region, so the pproximtion y neighoring tringles does not mke n inomplete CFL. s shown in Figure 5, the pproximted Voronoi region for tringle is prtitioned with plnes on edges (plotted with roken lines). The voxels interfering with the region n e the losest to the tringle. pproximted Voronoi Region on Tringle Polyhedron Tringle Figure 5. Voxel Culled y pproximted Voronoi Region 4 Results Bsed on the lgorithm of the disrete Voronoi regions we hve implemented progrm for lulting the minimum distne etween non-onvex polyhedr. 4.1 Length of CFL Tle 1 shows the numer of voxels nd the verge length of CFL defined on unny model of 1314 tringles (Figure 6). The verge length of CFL per voxel is ffeted y the size of the voxel. If the size of voxel is inresed, more tringles per voxel re listed. It n e sid tht smller voxels mke the length of CFL shorter nd tht the effiieny of distne lultion n e improved, ut we hve found tht the length of CFL do not dereses very rpidly when the size of voxels is smller thn the size of tringles. The hnge of tringles piked for distne lultion in some voxel sizes is shown in Figure 7 s drk tringles. The thik line etween two ojets indites the pir of losest points. CFL on voxels re lulted on eh polyhedron s preproess of distne lultion. For instne, drk tringle on the k of the unny is given in CFL on the lk voxels (Figure 8). Tle 1. Numer of Voxels nd Length of CFL Numer of Voxels verge Length of CFL

5 in 3D sene of VRML1.0. Figure 6. Non-onvex Polyhedron with 1314 Tringles voxels voxels Figure 8. Voxels ssoited with Tringle 5 Conlusion nd Future Work voxels voxels Figure 7. Piked Tringles nd Size of Voxels 4.2 Distne Clultion with CFL In lulting the distne with voxels, the totl numer of distne lultions etween pir of tringles is diminished to 172 on verge, whih is out 1/10000 ompred with the numer of ll omintions of tringles. The verge of CPU time onsumed in distne omputtion is 0.06 seond with MIPS R MHz. The tringles whih re piked from CFL on voxels nd lulted their distne re shown in drk olor (Figure 9). The thik line etween two ojets indites the pir of losest points. The lgorithm enles us fst omputtion of the distne etween non-onvex polyhedr whih onsist of thousnd of tringles, nd we utilize this lgorithm for our intertive rigid ody dynmis simultor nmed pvrml, whih n simulte the dynmis of physilly-sed models defined We hve desried n the disrete Voronoi regions for lulting the distne etween non-onvex polyhedr. By defining the minimum nd mximum vlues of distnes etween voxel nd tringles, we n effiiently eliminte the voxels whih nnot e losest. The lgorithm enles us to lulte the distne etween non-onvex polyhedr mde of thousnd of tringles t intertive rtes, nd we utilize it for n intertive dynmis simultor. It is found tht the verge length of CFL is ffeted y the size of the voxel. In order to hieve more effiient distne omputtion lgorithm, method for deiding the optiml size of voxels for n polyhedron is needed. If the polyhedr re interfering, our lgorithm nswers zero s the distne etween them. In terms of dynmis simultion, it n e useful to extend the lgorithm to tret penetrtion s minus distne in estimting the instne of ollision. Tle 2 shows the omprison of the distne omputtion ost with the PQP lirry [5]. Our implementtion is out 3.4 times s slow s PQP. The polyhedron in Figure 6 is used for this test, nd CFLs re defined on voxels. This result is minly used y the lrge N v, i.e. the numer of intersetion tests nd losest feture tests etween tringles nd voxels performed in the first, seond nd third step desried in the setion 3.5, nd these tests ost more thn 70% of ll omputtion time. By onstruting hierrhil struture of tringles in CFLs of voxels, they

6 n e more effiient nd it is likely to e possile to redue N v s smll s tht of PQP. (1) () Tle 2. Performne of Distne Computtion lgorithm v. Computtion Time v. N p v. N v PQP se CFL se (2) () Referenes [1] S. Cmeron. Enhneing GJK: Computing minimum penetrtion distnes etween onvex polyhedr. In IEEE Interntionl Conferene on Rootis nd utomtion, pril [2] E. G. Gilert, D. W. Johnson, nd S. S. Keerthi. fst proedure for omputing the distne etween omplex ojets in three-dimensionl spe. IEEE Journl of Rootis nd utomtion, 4(2): , pril [3] S. Gottshlk, M. C. Lin, nd D. Mnoh. OBBTree: hierrhil struture for rpid interferene detetion. In Computer Grphis Proeedings, nnul Conferene Series, pges CM SIGGRPH, [4] K. E. H. III, T. Culver, J. Keyser, M. Lin, nd D. Mnoh. Fst omputtion of generlized voronoi digrms using grphis hrdwre. In Computer Grphis Proeedings, nnul Conferene Series, pges CM SIGGRPH, [5] E. Lrsen, S. Gottshlk, M. Lin, nd D. Mnoh. Fst proximity queries with swept sphere volumes. Tehnil Report TR99-018, Deprtment of Computer Siene, University of North Crolin, Chpel Hill, [6] D. Lvender,. Bowyer, J. Dvenport,. Wllis, nd J. Woodwrk. Voronoi digrms of set-theoreti solid models. IEEE Computer Grphis nd pplitions, 12(5):69 77, Septemer [7] M. C. Lin nd J. F. Cnny. Effiient lgorithm for inrementl distne omputtion. In IEEE Conferene on Rootis nd utomtion, [8] B. Mirtih. V-Clip: Fst nd roust polyhedrl ollision detetion. Tehnil Report TR-97-05, MERL, July [9] J. Pitt-Frnis nd R. Fetherstone. utomti genertion of sphere hierrhies from d dt. In Proeedings of IEEE Interntionl Conferene on Rootis nd utomtion, [10] H. Smet. Sptil Dt Strutures: Qudtree, Otree nd Other Hierrhil Methods. ddison Wesley, (3) () (4) (d) (5) (e) (1 5), ( e): Sequenes of Distne Clultion Drk Tringles: Piked Tringles in Clultion Thik Line: Pir of Closest Points Figure 9. Closest Points etween Two Moving Non-onvex Polyhedr