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Collision prediction for polyhedra under piecewise screw motions Byung-Moon Kim and Jarek Rossignac GVU Center and College of Computing Georgia Tech, Atlanta, USA The problem Compute the time and place of collision between moving bodies MOTIVATION

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Collision prediction for polyhedra under piecewise screw motions

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Collision prediction for polyhedra under piecewise screw motions

Byung-Moon Kim and Jarek Rossignac

GVU Center and College of Computing

Georgia Tech, Atlanta, USA

Compute the time and place of collision between moving bodies

MOTIVATION

- Increase speed & accuracy of 3D animation and simulation
SCOPE

- Limited to polyhedral (triangulated) shapes
- Limited to rigid body motions (no deformation)
- (see [Von Herzen & Zatz’ 90] for collision of deforming shapes)

- S. Udupa, Collision detection and avoidance in computer controlled manipulators. Proc. 5th Int. Conf. Artif. lntel.,1977.
- J. W. Boyse. Interference detection among solids and surfaces. Communications of ACM, 22(1):3–9, January 1979.
- N. Ahuja, R. T. Chien, R. Yen, and N. Bridwell. Interference detection and collision avoidance among three dimensional objects. Conference on AI, Stanford University, August 1980.
- D.P. Dobkin, D.G. Kirkpatrick, Fast detection of polyhedral intersection, Theoret. Comput. Sci. 27, 1983.
- J. U. Korein. A Geometric Investigation of Reach. The MIT Press, 1984.
- S. A. Cameron and R. K. Culley. Determining the minimum translational distance between two convex polyhedra. In Proceedings of IEEE International Conference on Robotics and Automation, April 1986.
- J. F. Canny. Collision detection for moving polyhedra. IEEE Transactions on Pattern Analysis and Machine Intelligence, 8(2), March 1986.
- …
- P. Jimenez, F. Thomas, and C. Torras. 3D collision detection: a survey. Computers and Graphics, 25(2), 2001.

Most approaches simulate the motions of all the objects and after each time step, check if any pair of objects interfere

- O(n2) static interference detections between pairs of objects
- Each checks whether an edge of one stabs the face of another

- Quick rejections of distant pairs of objects
- Use bounds (boxes, spheres) around each object [Rimon&Boyd’97]
- Use velocity and distance [Culley&Kempf’86]
- Track minimum distances over time [Lin&Canny’91]

- Quick rejection of disjoint portions of the objects
- Decompose shapes into convex parts [Bajaj&Dey’92]
- Use hierarchy of bounds around object or its surface [Hubbard’96]
- Partitionspace [Bandi&Thalmann’95][Gottschalk&Lin&Manocha’96]
- Track mobile data [Basch&Guibas&Hershberger’97]

t1

t2

t3

- Detection: Simulate motion step-by-step and test for static interference between parts at each key-frame
- Stop when interference is detected
- Search for correct collision time
- Binary split of last time-step

- Expensive (O(n2) per time step)
- Can easily miss collisions

- Prediction: Compute time when the objects will first collide
- Test all pairs of surface elements that could collide
- Vertex-triangle, triangle-vertex, edge-edge

- Report the first collision to occur
- Fast
- Exact (can’t miss)

- Test all pairs of surface elements that could collide

- Assume solid A (bus) moves by a(t)
- Assume solid B (taxi) moves by b(t)
- a(t) and b(t) are parameterized rigid body transformations
- Can be represented by 4x4 matrix or pose (origin + orthonormal basis)

- Express everything in the moving CS of A (the bus)
- See the accident from the perspective of a passenger of the bus
- A (the bus) is now static
- The pose of B (the taxi) is defined by M(t)=b(t)*a–1(t)

- Two body collision problem may be reduced to the detection of the collision of a single moving body with a static obstacle

From Boyse’79

- Assume solids A and B are initially disjoint
- Assume A is static and B moves by rigid-body motion M(t)
- First collision occurs at time t
- The boundary of A and of [email protected](t) intersect
- The intersection must contain either:
- a vertex of A in a face of [email protected](t) or
- a vertex of [email protected](t) in a face of A or
- the intersection of an edge of A in an edge of [email protected](t)

- Vertex/face collision
- V(t)[email protected](t) is a parametric curve.
- Find its intersection with plane PV(t)•N=0: solve for t
- Complexity of finding the roots ti depends on nature of M(t)
- Then check which V(ti) lie inside the face

- Face/vertex collision
- Swap the role of A and B

- Edge/edge collision
- When does edge ([email protected](t),[email protected](t)) collide with edge (c,d)?
- They are coplanar when cd•(([email protected](t))([email protected](t)))=0
- Solve for roots ti (more complex than vertex/face)
- Then check that ([email protected](ti),[email protected](ti)) intersects with (c,d)

- Complexity of root finding depends on nature of M
- Translation [Boyse’79, Cameron’85]
- Rotation (both objects around same axis) [Schomer&Thiel’95]
- Linear translation+variable speed rotation
- [Canny’86, Jimanez&Torras’85, Schomer & Thiel95]

- Assume A moves with constant velocity v and B is fixed
- Collision may occur between
- A vertex p of A and a triangle T of B
- Intersect Ray(p,v) with T

- A triangle T of A with a vertex p of B
- Intersect Ray(p,-v) with T

- An edge (a,b) of A with an edge (c,d) of B
- Check when the volume of tetrahedron (a+tv,b+tv,c,d) becomes zero
- solve (cd(ca+tv))(cb+tv)=0 for t
- (cd(ca+tv))cb + (cd(ca+tv))tv)=0
- (cdca+t(cdv))cb + (cdca+t(cdv))tv)=0
- (cdca)cb +t(cdv)cb + (cdca)tv +t2(cdv)v = 0
- (cdca)cb +t(cdv)cb + (cdca)tv = 0, because (cdv)v = 0
- t = (cacd)cb / ((cdv)cb - (cdv)ca)
- t = (cacd)cb / (abcd)v

- Make sure that, at that time, the two edges intersect
- Not just the lines

- Check when the volume of tetrahedron (a+tv,b+tv,c,d) becomes zero

- A vertex p of A and a triangle T of B

d

b

v

a

c

Q

E

S

K

- Screw motions are great!
- Uniquely defined by start pose S and end pose E
- Independent of coordinate system
- Subsumes pure rotations and translations
- Minimizes rotation angle & translation distance
- Natural motions for many application

- Simple to apply for any value of t in [0,1]
- Rotation by angle tb around axis Axis(Q,K)
- Translation by distance td along Axis(Q,K)
- Each point moves along a helix

- Simple to compute from poses S and E
- Axis: point Q and direction K
- Angle b
- Distance d

Screw Motion

(Ceccarelli [2000] Detailed study of screw motion history)

- Archimede (287–212 BC) designed helicoidal screw for water pumps
- Leonardo da Vinci (1452–1519) description of helicoidal motion
- Dal Monte (1545–1607) and Galileo (1564–1642) mechanical studies on helicoidal geometry
- Giulio Mozzi (1763) screw axis as the “spontaneous axis of rotation”
- L.B. Francoeur (1807) theorem of helicoidal motion
- Gaetano Giorgini (1830) analytical demonstration of the existence of the “axis of motion” (locus of minimum displacement points)
- Ball (1900) “Theory of screws”
- Rodrigues (1940) helicoidal motion as general motion
- ….
- Zefrant and Kumar (CAD 1998) Interpolating motions

P’

EL

U’

O’

V’

d

(O+O’)/2

P

axis

K

b

SL

V

Q

U

O

I

From initial and final poses M(0) and M(1)

K:=(U’–U)(V’–V);

K:=K / ||K||;

b := 2 sin–1(||U’–U|| / (2 ||KU||) );

d:=K•OO’;

Q:=(O+O’)/2 + (KOO’) / (2tan(b/2));

To apply screw motion:

Translate by –Q;

Rotate K to Z;

Rotate around Z by tb;

Translate by (0,0,td);

Rotate Z to K;

Translate by Q;

- Split: Insert a new vertex in the middle of each edge
- Cubic B-spline tweak: Tuck old vertices in
- 4-point tweak: Bulge new vertices out
- Jarek tweak: Do half of each

- Polyscrew motion: interpolates consecutive poses by screws
- Subdivide using Split&Tweak on screws

Computing and visualizing pose-interpolating 3D motions

Jarek R. Rossignac and Jay J. Kim (Hanyang University, Seoul, Korea), CAD, 33(4)279:291, April 2001.

SweepTrimmer: Boundaries of regions swept by sculptured solids during a pose-interpolating screw motion

Jarek R. Rossignac and Jay J. Kim

EL

f(d)

P

SL

1

OL

d

0

1

“Twister: A space-warp operator for the two-handed editing of 3D shapes”, Llamas, Kim, Gargus, Rossignac, and Shaw. Proc. ACM SIGGRAPH,July 2003.

Decay function

- For each pair of objects A and B do
- Approximate relative motion by a piecewise screw motion
- Insert intermediate poses as needed adaptively

- For each screw motion segment do
- Use quick rejection test to quickly identify collision-free situations
- If collision may not be discarded, then do
- For each vertex of A and each triangle of B do
- If collision cannot be discarded using bounds
- Then find time of first collision (if one occurs)

- For each vertex of B and each triangle of A do
- If collision cannot be discarded using bounds
- Then find time of first collision (if one occurs)

- For each edge of B and each edge of A do
- If collision cannot be discarded using bounds
- Then find time of first collision (if one occurs)

- Stop if collision was found and report time of first collision

- For each vertex of A and each triangle of B do

- Approximate relative motion by a piecewise screw motion

- Helix is V(t) = rcos(tb)i+rsin(tb)j+tdk in screw coordinates
- where V(0) lies on the i axis and the k-axis is parallel to s

- The screw intersects plane d+V(t)•n= 0 for values of t satisfying
- d+(rcos(tb), rsin(tb),td)•n= 0

- We compute all roots and check if they correspond to points in triangle
- Reduces to finding roots of f(t)=A+Bt+Ccos(bt+c)
- Separate roots using f’(t)=0, which requires solving B/bC=sin(bt+c)
- We use Newton iterations

- Requires roots of f(t)=A+(B+Ct)cos(bt+c)+(D+Et)sin(bt+c)
- We use Newton iterations from carefully computed seeds
- Angle or rotation < 180 degrees

- We use Newton iterations from carefully computed seeds
- Check which roots corresponds to true E/E intersections

- Decide early that some pairs of objects cannot intersect
- Use simple bounds on objects and their swept regions
- Balls, cylindrical annuli

- Avoid most root-findings by rejecting pairs of elements
- Use bounds on elements and their swept regions
- Vertex (helix), edge (annulus)

–

- Build (minimum) bounding spheres around objects
- Region swept by B lies in half of a cylindrical annulus

A

B

If B lies outside of this CSG solid: no collision

- Triangle separated from helix by plane or cylinder

Too high along axis: above plane

Not in screw angle: outside wedge

Too far from axis: outside cylinder

Too close to axis: inside cylinder

Too low along axis: below plane

- No collision if green edge lies outside of (wedge-portion of) the annulus containing region swept by red edge

above

Inside inner cylinder

Outside outer cylinder

Outside wedge

below

- Test setup
- A move along a fixed screw motion
- B is randomly placed and oriented in a in a box
- 50,000 different poses were tested

- Actual collision happened in about 10% of cases
- 50% cases rejected using bounding spheres around objects

- 50% of V/T cases and 66% of E/E cases rejected early
- A and B have about 160 triangles vertices
- 26,540 triangle/vertex and 58,266 edge/edge pairs
- Takes average of 4x10–7 sec per V/T or E/E rejection test
- Exact collision computation takes about 10–5 sec

Actual collisions only

50% cases are rejected by cylinder/sphere test

- Perform exact prediction, rather than interference detection
- Approximate relative motion by screws (better than other types of simple motions)
- Uses simple geometric rejection tests to identify cases where objects do not collide, they reduce overall cost by half
- Uses simple geometric rejection test to discard more than half of the V/T and E/E collision candidates
- Uses Newton to solve for exact collision time when needed: 10–5 sec per V/T, T/V, or E/E collision
- Could be combined with hierarchical culling and other speed-ups

Thank you

Questions?

Tring

http://tring.powelltown.com/