Collision Detection: Algorithms and Applications
Ming C. Lin, U.S. Army Research Oce and University of North Carolina, Chapel Hil l, NC, USA
Dinesh Mano cha, University of North Carolina, Chapel Hil l, NC, USA
Jon Cohen, University of North Carolina, Chapel Hil l, NC, USA
Stefan Gottschalk, University of North Carolina, Chapel Hil l, NC, USA
http://www.cs.unc.edu/~geom/collide.html
Fast and accurate col lision detection between general The most challenging problem in a simulation,
geometric models is a fundamental problem in model- namely the collision phase, can b e separated into three
ing, robotics, manufacturing and computer-simulated parts: collision detection, contact area determination,
environments. Most of the earlier algorithm are either and collision resp onse. In this pap er, we address the
restricted to a class of geometric models, say convex rst two elements by presenting general a purp ose col-
polytopes, or are not fast enough for practical applica- lision detection and contact area determination algo-
tions. We present an ecient and accurate algorithm rithm for simulations. The collision resp onse is appli-
for col lision detection between general polygonal mod- cation dep endent. The algorithm rep orts the contact
els in dynamic environments. The algorithm makes area and thus enables the application to compute an
use of hierarchical representations along with frame to appropriate resp onse.
frame coherencetorapid ly detect col lisions. It is ro-
Our algorithm not only addresses interaction b e-
bust and has been implementedaspart of public domain
tween a pair of general p olygonal ob jects, but also large
packages. In practice, it can accurately detect al l the
environments consisting of hundreds of moving parts,
contacts between large complex geometries composedof
such as those encountered in the manufacturing plants.
hundreds of thousands of polygons at interactive rates.
Furthermore, we do not assume the motions of the ob-
jects to b e expressed as a closed form function of time.
1 Intro duction
Our collision detection scheme is ecient and accurate
to the resolution of the mo dels .
Collision detection is a fundamental problem in
Given the geometric mo dels, the algorithm pre-
rob otics, computer animation, physically-based mo d-
computes the convex hull and a hierarchical representa-
eling, molecular mo deling and computer simulated en-
tion of each mo del in terms of oriented b ounding b oxes.
vironments. In these applications, an ob ject's motion
At runtime, it uses tight tting axis-aligned b ounding
is constrained by collisions with other ob jects and by
boxes to pair down the numb er of ob ject pair interac-
other dynamic constraints. The problem has b een well
tions to only those pairs within close proximity [12].
studied in the literature.
For each pair of ob jects whose b ounding b oxes over-
A realistic simulation system, which couples geomet-
lap, the algorithm checks whether their convex hulls
ric mo deling and physical prototyping, can provide a
are intersecting based on the closest feature pairs [22].
useful to olset for applications in rob otics, CAD/CAM
Finally for each ob ject pair whose convex hulls over-
design, molecular mo deling, manufacturing design sim-
lap, it makes use of oriented b ounding b ox hierarchy
ulations, etc. Such systems create electronic represen-
OBBTree to check for actual contact [18].
tations of mechanical parts, to ols, and machines, which
need to b e tested for interconnectivity, functionality,
and reliability. A fundamental comp onent of sucha Organization: The rest of the pap er is organized as
system is to mo del ob ject interactions precisely. The follows: Section 2 reviews some of the previous work
interactions mayinvolve ob jects in the simulation envi- in collision detection. Section 3 outlines the algorithm
ronment pushing, striking, or smashing other ob jects. for pruning the numb er of ob ject pairs. We brie y
Detecting col lisions and determining contact points is a describ e the closest feature and contact determination
crucial step in p ortraying these interactions accurately. algorithms in Section 4. Finally,we describ e the imple-
leftmark
mentation and p erformance on di erent applications in at every time step can b ecome time consuming as N
Section 5 and M get large. Toachieveinteractive rates, the total
numb er of pairwise intersection tests must b e reduced
b efore p erforming exact collision tests on the ob ject
2 Previous Work
pairs, which are in the close vicinityofeach other. Sev-
eral metho ds dealing with this situation are found in
Collision detection has b een extensively studied in
[7,12, 16]. Most metho ds use some typ e of a hierarchi-
CAD, computer graphics, rob otics, and computational
cal b ounding b oxscheme. Ob jects are surrounded by
geometry. Since collision detection is needed in a wide
b ounding b oxes. If the b ounding b oxes overlap, indi-
variety of situations, many di erent metho ds have b een
cating the ob jects are near each other, a more precise
prop osed. Most of them make sp eci c assumptions
collision test is applied.
ab out the ob jects of interest and design a solution
based on ob ject geometry or application domain.
As for curved mo dels, algorithms based on inter-
val arithmetic for collision detection are describ ed in
Rob otics literature deals with collision detection in
[15, 17]. These algorithms exp ect the motion of the
the context of path planning. Using sophisticated
ob jects to b e expressed as a closed form function of
mathematical to ols, several algorithms have b een de-
time. Moreover, the p erformance of interval arithmetic
velop ed that plan collision-free paths for a rob ot in
based algorithms is to o slow for interactive applica-
restricted environments [8, 9]. However, in path plan-
tions. Coherence based algorithms for curved mo dels
ners based on p otential eld metho ds, collision detec-
are presented in [24].
tion and distance computation are still considered as
ma jor b ottlenecks [21,10].
In many CAD applications, the input mo dels are
given as collections of p olygons with no top ology in-
Most computational geometry literature deals with
formation. Such mo dels are also known as `p olygon
collision detection of ob jects in a static environment.
soups' and their b oundaries mayhave cracks, T-joints,
Ob jects are at a xed lo cation and orientation, and
or mayhave non-manifold geometry. In general, no
the algorithms determine whether they are intersecting
robust techniques are known for cleaning such mo dels.
[11, 13]. In most mo deling and graphics applications,
Many of the algorithms describ ed are not applicable
where many ob jects are in motion, such an approach
to such mo dels. Rather techniques based on hierarchi-
would b e inecient. Moreover, the ob jects move only
cal b ounding volumes and and spatial decomp osition
slightly from frame to frame and the collision detec-
are used on such mo dels. Typical examples of b ound-
tion scheme should take advantage of the information
ing volumes include axis-aligned b oxes of which cub es
from the previous frame to initialize the computation
are a sp ecial case and spheres, and they are chosen
for the current frame [3, 23]. Several solutions based
for to the simplicity of nding collision b etween two
on this idea of coherence have b een prop osed in [22].
suchvolumes. Hierarchical structures used for colli-
Approaches that combine collision resp onse with de-
sion detection include cone trees, k-d trees and o ctrees
tection can b e found in [3, 33, 25]. The metho ds in
[31], sphere trees [20,30], R-trees and their variants
[32,33] make use of the b oundary representation to
[5], trees based on S-b ounds [7] etc. Other spatial rep-
detect collisions.
resentations are based on BSP's [27] and its extensions
Collision detection for multiple moving ob jects has
to multi-space partitions [32], spatial representations
recently b ecome a p opular research topic with the in-
based on space-time b ounds or four-dimensional test-
creased interest in large-scaled virtual prototyping en-
ing [1,6,9, 20] and many more. All of these hierarchical
vironments. For example, a vibratory parts feeder can
metho ds do very well in p erforming \rejection tests",
contain up to hundreds of mechanical parts moving si-
whenever two ob jects are far apart. However, when the
multaneously under p erio dical force impulses in a vi-
two ob jects are in close proximity and can havemul-
bratory b owl or tray. In a general simulation environ-
tiple contacts, these algorithms either use sub division
ment, there maybe N moving ob jects and M station-
techniques or checkvery large numb er of b ounding vol-
ary ob jects. EachoftheN moving ob jects can collide
ume pairs for p otential contacts. In such cases, their
with the other moving ob jects, as well as the stationary
p erformance slows down considerably and they b ecome
N
ones. Keeping trackof +NM pairs of ob jects
a ma jor b ottleneck in the simulation, as stated in [19]. 2
Collision Detection: Algorithms & Applications
3 Collision Detection b etween Multiple b ounding b oxes can overlap if and only if their inter-
vals overlap in all three dimensions.
Moving Ob jects
We construct three lists, one for each dimension.
We review our previous algorithm for multiple mov-
Each list contains the values of the endp oints of the
ing convex p olytop es in complex environments. Co-
intervals in each corresp onding dimension. By sorting
herence combined with incremental computation is a
these lists, we can determine whichintervals overlap.
ma jor theme of our algorithms. By exploiting coher-
In the general case, such a sort would take O N log N
ence, we are able to incrementally trim down the num-
time, where N is the numb er of ob jects. We can re-
b er of pairwise ob ject pairs and feature tests involved
duce this time b ound bykeeping the sorted lists from
in each iteration.
the previous frame, up dating only the interval end-
p oints. In environments where the ob jects make rel-
De nition: Temporal and geometric coherence is
atively small movements b etween frames, the lists will
the prop erty that the state of the application do es not
b e nearly sorted, so we can re-sort using insertion sort
change signi cantly b etween successive time steps or
in exp ected O N time [26,3]. Graphs 1 { 6 are timings
simulation frames. The ob jects move only slightly from
taken from a multi-ob ject simulation where we com-
frame to frame. This slightmovement of the ob jects
pare the p erformance of using xed versus dynamic
translates into geometric coherence, since their spatial
sized b oxes [12]. Parameters such as the number of
relationship do es not change muchbetween frames.
ob jects, the p olygonal complexity of the ob jects, the
For a con guration of N ob jects, the worst case run-
velo city of the ob jects, etc. were varied as the graphs
2
ning time for any collision detection algorithm is O N
show.
N is the numb er of ob jects. However, evidence
where Graph 1 Graph 2
suggests that these cases rarely o ccur in simulations
0.18 0.23
[3, 12, 22, 28]. So our algorithm uses a Sweep and 0.16 Dynamic 0.20 Box
0.18
technique to eliminate testing ob ject pairs that
Prune 0.14 Fixed Cube
0.12 0.15
are far apart, and later we show that the technique can 0.10 0.13 b e extended to eliminate testing features that are far 0.08 0.10
0.06 0.08
apart b etween two colliding ob jects. Seconds per Frame
0.04 0.05
e use a b ounding b ox based scheme to reduce the W 0.02 0.03
2 0.00 0.00
O N b ottleneck of testing all p ossible pairs of ob jects 0 500 1000 1500 2000 0 100 200 300 400
for collisions. In most realistic situations, an ob ject has Number of Polytopes Number of Faces to b e tested against a small fraction of all ob jects in the
Graph 4 vironment for collision. For example, in a simulation en Graph 3
of a vibratory parts feeder most ob jects are in close 0.16 0.14
ximity to only a few other ob jects. It would b e
pro 0.14 0.12
tless and exp ensivetokeep track of all p ossible p oin 0.12
0.10
teractions b etween ob jects at each time step. in 0.10 0.08
0.08 xes surrounding the ob jects Sorting the b ounding b o 0.06 0.06
Seconds per Frame
ey to our Sweep and Prune approach [12]. It is the k 0.04
0.04
tuitively obvious how to sort b ounding b oxes
is not in 0.02 0.02
verlaps. We use a dimension in 3-space to determine o 0.00 0.00
0.0 2.0 4.0 6.0 8.0 10.0 0 102030405060708090
reduction approach. If two b ounding b oxes collide in 3-
% Density of Simulation Volume Rotational Velocity (degrees/frame)
D, then their orthogonal pro jections on the x, y , and z
axes must overlap. The sweep and prune algorithm b e-
gins by pro jecting each 3-D b ounding b ox surrounding
4 Exact Collision Detection
an ob ject onto the x, y , and z axes. Since the b ound-
Given two ob jects in close proximity, the algorithm ini- ing b oxes are axially-aligned, pro jecting them onto the
tially checks whether their convex hulls are overlap- co ordinate axes results in intervals. We are interested
ping. It incrementally computes their closest feature in overlaps among these intervals, b ecause a pair of leftmark
Graph 5
Collision Tests nely discretized paths. In most situations, the algo- 0.11
Sort Lists
rithm runs in expectedconstant time if the p olytop es
0.10 Update Boxes
are not moving at large discrete steps e.g. 180 degrees 0.09
0.08 of rotation p er step .
0.07
Voronoi Regions 0.06 4.1.1
0.05
Each convex p olytop e is pre-pro cessed into a mo d- 0.04
Seconds per Frame
i ed b oundary representation. The p olytop e data
0.03 structure has elds for its features faces, edges, and
0.02
ertices and corresp onding Voronoi regions. Each fea-
0.01 v y its geometric parameters and its 0.00 ture is describ ed b
0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45
neighb oring features, i.e. the top ological information
Rotational Velocity for Dynamic Box of incidences and adjacencies.
Graph 6
De nition: A Voronoi region asso ciated with a fea-
0.05
ture is a set of p oints closer to that feature than any
0.04
other [29].
0.03 oronoi regions form a partition of the space 0.02 The V
0.01 outside the p olytop e, and they form the generalized Seconds per Frame
0.00 Voronoi diagram of the p olytop e. Note that the gener-
0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45
oronoi diagram of a convex p olytop e has linear
Rotational Velocity for Fixed Cube alized V
numb er of features and consists of p olyhedral regions.
A cel l is the data structure for a Voronoi region of a
pairs and check for overlap. If the convex hulls are in-
single feature. It has a set of constraint planes which
tersecting, it checks whether the overlapping features
b ound the Voronoi region with p ointers to the neigh-
b elong to the original mo del or are intro duced by the
b oring cells which share a constraint plane with it in
convex hull computation. Eventually it makes use of
its data structure. If a p oint lies on a constraint plane,
hierarchy of oriented b ounding b oxes to check for exact
then it is equi-distant from the two features which share
contact.
this constraint plane in their Voronoi regions. For more
details on this construction and its prop erties, please
4.1 Collision Detection b etween Convex Poly-
refer to [22].
top es
4.1.2 Closest Feature Tests
The algorithm computes the convex hull of all ob jects
as part of pre-pro cessing. Furthermore, it classi es the
Our metho d for nding closest feature pairs is based
feature of convex hulls into red and green features. The
on Voronoi regions. We start with a candidate pair of
red features corresp ond to the features of the original
features, one from each p olytop e, and check whether
mo del and the green features are intro duced by the
the closest p oints lie on these features. Since the p oly-
convex hull computation.
top es and their faces are convex, this is a lo cal test
involving only the neighb oring features of the current
We use the algorithm describ ed in [22]tokeep track
candidate features. If either feature fails the test, we
of closest features for a pair of convex p olytop es. The
step to a neighb oring feature of one or b oth candidates,
algorithm maintains a pair of closest features for each
and try again. As the Euclidean distance b etween fea-
convex p olytop e pair and calculates the Euclidean dis-
ture pairs must always decrease when a switch is made,
tance b etween the features to detect collisions. This
cycling is imp ossible for non-p enetrating ob jects. An
approach can b e used in a static environment, but is es-
example of the algorithm is given in Fig. 1.
p ecially well-suited for dynamic environments in which
ob jects move in a sequence of small, discrete steps. The Given a pair of features Face1and vertex V ,on
b
metho d takes advantage of coherence: the closest fea- ob jects A and B , as the closest features we test to see
tures change infrequently as the p olytop es move along if vertex V lies within Cel l 1 of of Face1. V violates
b b
Collision Detection: Algorithms & Applications
troid of each convex p olytop e { the
Object B puting the cen
weighted average of all vertices. Then a hyp erplane
from each edge is extended towards the centroid. The
yp erplane tap ers to a p oint, forming a
Cell 1 extended h
yramid-typ e cone over each face. Each of the faces of
CP p
en p olytop e is now used as a constraint plane. V the giv
b t imp osed by
Cell 2 If a candidate feature fails the constrain
the face indicating the closest feature pair lies p ossi-
bly b ehind this face , the algorithm stepping \enters"
wn in Fig. 2 . If at any
Face 1 Pa inside of the p olytop e as sho
time we nd one p oint on a feature of one p olytop e
tained within the pseudo internal Voronoi region
Cell 3 is con
this p oint satis es the constraints p osed byan
Object A i.e.
pseudo internal Voronoi region of the other p olytop e ,
it corresp onds to a p enetration. A detailed discussion
is presented in [28]
4.1.4 Feature Classi cation
Figure 1: A walk across Voronoi cel ls.
The algorithm highlighted ab ove returns all pairs of
overlapping features b etween the convex hulls. The
the constraint plane imp osed by CP of Cel l 1. The
resulting feature pairs can b e classi ed into:
constraint plane CP has a p ointer to its adjacent cell
Cel l 2, so the walk pro ceeds to test the containmentship
Red-red feature overlap as shown in Fig. 3 a :
of V within Cel l 2. In similar fashion, vertex V has
This corresp onds to an actual collision b etween
b b
a cell of its own, and we see if the nearest p oint P
the original mo dels.
a
on the edge to the vertex V lies within V 's Voronoi
b b
Red-green feature overlap as shown in Fig. 3 b :
cell. Basically,we are p erforming the containmentship
This mayormay not corresp ond to a collision.
tests of a p oint within a Voronoi region de ned by the
The algorithm presented in the next section is used
constraint planes of the region. The constraint plane
to check for exact contact.
causing the containmentship test to fail p oints the next
Green-green feature overlap as shown in Fig.
direction for the algorithm to advance in the searchof
3 c : Same as ab ove. The algorithm presented
a new and closer feature. Eventually,wemust reach
in the next section is used to check for exact con-
the closest pair of features.
tact.
4.1.3 Penetration Detection for Convex Polytop es
4.2 Exact Contact Determination
The key to detecting p enetrations lies in partition-
In this section, we present a robust, ecient and gen-
ing the interior as well as the exterior of the convex
eral purp ose algorithm to compute all contacts b etween
p olytop e. For internal partitioning, internal Voronoi
geometric mo dels comp osed of p olygons. The algo-
regions can b e used. The internal Voronoi regions can
rithms computes a hierarchical representation using
b e constructed for any convex p olytop e by computing
orientedbounding boxes OBBs . An OBB is a rect-
all the equi-distanthyp erplanes b etween two or more
angular b ounding b ox at an arbitrary orientation in 3-
facets on the p olytop e. However the general construc-
space. The resulting hierarchical structure is referred
tion of the internal Voronoi regions is a non-trivial com-
to as an OBBTree. The idea of using OBBs is not new
putation [29]. To detect a p enetration { as opp osed to
and many researchers have used them extensively to
knowing al l the closest features { it is unnecessary to
sp eed up ray tracing and interference detection compu-
construct the exact internal Voronoi regions.
tations [2]. In this pap er, we brie y describ e the algo-
Rather we use an approximation, lab eled as pseudo- rithms for computing tight- tting OBBs and checking
internal Voronoi region. It is calculated by rst com- them for overlap. More details are given in [18]. leftmark
V Inside External Cell
V
V Inside Internal Cell
V
Figure 2: Walk from external to internal Voronoi regions.
4.2.1 Building an OBBTree The eigenvectors of a symmetric matrix, suchasC,
are mutually orthogonal. After normalizing them, they
In this section we describ e algorithms for building an
are used as a basis. We nd the extremal vertices along
OBBTree. The tree construction has two comp onents:
each axis of this basis, and size the b ounding b ox, ori-
rst is the placement of a tight tting OBB around a
ented with the basis vectors, to b ound those extremal
collection of p olygons, and second is the grouping of
vertices. Two of the three eigenvectors of the covari-
nested OBB's into a tree hierarchy.
ance matrix are the axes of maximum and of minimum
Wewant to approximate the collection of p olygons variance, so they will tend to align the b ox with the
with an OBB of similar dimensions and orientation. geometry of a tub e or a at surface patch.
We triangulate all p olygons comp osed of more than
three edges. The OBB computation algorithm makes
use of rst and second order statistics summarizing the
vertex co ordinates. They are the mean, , and the
covariance matrix, C, resp ectively [14]. If the vertices
i i i
of the i'th triangle are the p oints p , q , and r , then
the mean and covariance matrix can b e expressed in
vector notation as:
n
X
1
i i i
= p + q + r ;
3n
i=0
Figure 4: Building the OBBTree: recursively partition the
n
b ounded p olygons and b ound the resulting groups.
X
1
i i i i i i
p p + q q + r r ; 1 j; k 3 C =
jk
j k j k j k
3n
i=0
The basic failing of the ab ove approach is that ver-
i i
i
where n is the numb er of triangles, p = p , q = tices on the interior of the mo del, which ought not in-
i
i i
q , and uence the selection of a b ounding b ox placement, can r = r . Each of them is a 3 1vector,
i i i i
T
have an arbitrary impact on the eigenvectors. For ex- p = p ;p ;p and C are the elements of the e.g.
jk
1 2 3
ample, a small but very dense planar patchofvertices 3by3 covariance matrix. Collision Detection: Algorithms & Applications
(a)
(b)
(c)
Figure 3: Feature classi cation based on the overlap b etween convex hulls
in the interior of the mo del can cause the b ounding b ox abox with a plane orthogonal to one of its axes, par-
to align with it. titioning the p olygons according to which side of the
plane their center p oint lies on a 2-D analog is shown
We improve the algorithm by using the convex hull
in Figure 4 . The sub division co ordinate along that
of the vertices of the triangles. The convex hull is the
axis was chosen to b e that of the mean p oint, ; of
smallest convex set containing all the p oints and e-
the vertices. If the longest axis cannot not b e sub di-
cient algorithms of O n lg n complexity and their ro-
vided, the second longest axis is chosen. Otherwise, the
bust implementations are available as public domain
shortest one is used. If the group of p olygons cannot
packages [4]. This is an improvement, but still suf-
b e partitioned along any axis by this criterion, then the
fers from a similar sampling problem: a small but very
group is considered indivisible.
dense collection of nearly collinear vertices on the con-
vex hull can cause the b ounding b ox to align with that
4.2.2 Fast Overlap Test for OBBs
collection.
Given OBBTrees of two ob jects, the interference al-
Given an algorithm to compute tight- tting OBBs
gorithm typically sp ends most of its time testing pairs
around a group of p olygons, we need to represent
of OBBs for overlap. A simple algorithm for testing
them hierarchically. Most metho ds for building hierar-
the overlap status for two OBB's p erforms 144 edge-
chies fall into two categories: b ottom-up and top-down.
face tests. In practice, it is an exp ensive test. OBBs
Bottom-up metho ds b egin with a b ounding volume for
are convex p olytop es and therefore, algorithms based
each p olygon and merge volumes into larger volumes
on linear programming and closest features computa-
until the tree is complete. Top-down metho ds b egin
tion can b e applied to check for overlap. However, they
with a group of all p olygons, and recursively sub divide
are relatively exp ensive.
until all leaf no des are indivisible. In our current imple-
mentation, wehave used a simple top-down approach.
One trivial test for disjointness is to pro ject the
Our sub division rule is to split the longest axis of boxes onto some axis not necessarily a co ordinate axis
leftmark
in space. This is an `axial pro jection.' Under this pro- when the b oxes are not overlapping , the algorithm
jection, eachbox forms an interval on the axis. If the takes ab out half the op erations in practice. This algo-
intervals don't overlap, then the axis is called a `sep- rithm for overlap detection b etween OBBs is ab out one
arating axis' for the b oxes, and the b oxes must then order of magnitude faster than previous algorithms and
b e disjoint. If the intervals do overlap, then the b oxes implementations to check for overlap b etween OBBs.
mayormay not b e disjoint { further tests may b e re-
quired. We make use of the separating axis theorem
5 Implementation and Performance
presented in [18]tocheck for overlaps. According to
it, two convex p olytop es in 3-D are disjoint i there
All these algorithms have b een implemented and avail-
exists a separating axis orthogonal to a face of either
able as part of general purp ose public domain packages.
p olytop e or orthogonal to an edge from each p olytop e.
These are:
Eachbox has 3 unique face orientations, and 3 unique
I COLLIDE collision detection package available at
edge directions. This leads to 15 p otential separating
http://www.cs.unc.edu/~geom/I COLLIDE.html.
axes to test 3 faces from one b ox, 3 faces from the
It contains routines for the sweep and prune as well
other b ox, and 9 pairwise combinations of edges . If
as the closest distance pair algorithm for convex p oly-
the p olytop es are disjoint, then a separating axis ex-
top es. This package is applicable to environments,
ists, and one of the 15 axes mentioned ab ove will b e a
which can b e describ ed as union of convex p olytop es.
separating axis. If the p olytop es are overlapping, then
It has b een widely used for dynamic simulation, archi-
clearly no separating axis exists. So, testing the 15
tecture walkthrough and other applications.
given axes is a sucient test for determining overlap
status of two OBBs.
RAPID interference detection package available at
http://www.cs.unc.edu/~geom/OBB/OBBT.html.
It contains routines for building the OBBTree data
verlap tests b etween two OBB- A structure and fast o
b22B
Trees. It has b een used for virtual prototyping and ulation-based design applications. More details on
T b11B B sim
their p erformance and robustness issues are given in
a11A , 18].
a22A [12
In practice, the algorithms based on OBBs asymp-
totically p erform much b etter than hierarchies based
rA
on sphere trees or axis-aligned b ounding b oxes like Oc- COLLIDE routines are able to compute
rB trees . The I
all contacts b etween environments comp osed of hun-
vex p olytop es at interactive rates ab out
TL dred of con
=20 of a second . The OBBTree based interference
L 1
detection algorithm available as part of RAPID has
b een applied to two complex synthetic environments to
~
Figure 5: L is a separating axis for OBBs A and B b ecause
demonstrate its eciency as highlighted in Table 1 .
~
A and B b ecome disjoint intervals under projection onto L.
These gures are for an SGI Reality Engine 90 MHz
R8000 CPU, 512 MB .
To p erform the test, our strategy is to pro ject the
A simple dynamics engine exercised the collision de-
centers of the b oxes onto the axis, and also to compute
tection system. At each time step, the contact p oly-
the radii of the intervals. If the distance b etween the
gons were found by the collision detection algorithm,
box centers as pro jected onto the axis is greater than
an impulse was applied to the ob ject at each contact
the sum of the radii, then the intervals and the b oxes
b efore advancing the clo ck.
as well are disjoint. This is shown in 2D in Fig. 5.
In the rst scenario, the pip es mo del was used as In practice, this corresp onds to at most 200 arithmetic
b oth the environment and the dynamic ob ject, as op erations in the worst case [18]. Due to early exit
Collision Detection: Algorithms & Applications
Wehave also applied the RAPID library to detect
Scenario Pip es Torus
collision b etween a moving torp edo on a pivot mo del
Environ Size 143690 pgns 98000 pgns
as shown in Fig. 6 . These are parts of a torp edo stor-
Ob ject Size 143690 pgns 20000 pgns
age and handling ro om of a submarine. The torp edo
Num of Steps 4008 1298
mo del is 4780 triangles. The pivot structure has 44921
Num of Contacts 23905 2266
triangles. There are multiple contacts along the length
Num of Box-BoxTests 1704187 1055559
of the torp edo as it rests among the rollers. A typical
Num of Tri-Tri Tests 71589 7069
collision query time for the scenario shown in Fig. 6 is
Time 16.9 secs 8.9 secs
100 ms on a 200MHz R4400 CPU, 2GB SGI Reality
Ave. Int. Detec. Time 4.2 msecs 6.9 msecs
Engine.
Ave. Time p er BoxTest 7.9 usecs 7.3 usecs
Ave. Contacts p er Step 6.0 1.7
6 Acknowledgements
Table 1: Timings for simulations
Thanks to Greg Angelini, Jim Boudreaux, and Ken
Fast at Electric Boat for the mo del of torp edo stor-
age and handling ro om. Wewould like to thank
shown in Fig. 7. Both ob ject and environment con-
John Canny, Brian Mirtich and Krish Ponamgi for in-
tain 140,000 p olygons. The ob ject is 15 times smaller
teresting discussions. Brian Mirtich and Krish Pon-
in size than the environment. We simulated a grav-
COLLIDE amgi also help ed in the implementationofI
itational eld directed toward the center of the large
library. This research has b een supp orted in part
cub e of pip es, and p ermitted the smaller cub e to fall
by a Sloan foundation fellowship, ARO Contract
inward, tumbling and b ouncing. Its path contained
P-34982-MA, DARPA contract DABT63-93-C-0048,
4008 discrete p ositions, and required 16:9 seconds to
NSF grant CCR-9319957, NSF Grant CCR-9625217,
determine all 23905 contacts along the path. This is a
ONR contract N00014-94-1-0738, NSF/ARPA Science
challenging scenario b ecause the smaller ob ject is en-
and Technology Center for Computer Graphics &
tirely emb edded within the larger mo del. The mo d-
Scienti c Visualization and NSF Prime contract No.
els contain long thin triangles in the straight segments
8920219.
of the pip es, which cannot b e eciently approximated
by sphere trees, o ctrees, and trees comp osed of axis-
aligned b ounding b oxes , in general. It has no obvious
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msec on SGI Reality Engine