CS B659: Principles of Intelligent Robot Motion
Collision Detection
Probabilistic Roadmaps
How to test forcollision?
Collision Detection Methods• Many different methods• In particular:
• Grid method: good for many simple moving objects of about the same size (e.g., many moving discs with similar radii)
• Closest-feature tracking: good for moving polyhedral objects• Bounding Volume Hierarchy (BVH) method: good for few moving
objects with complex and diverse geometry
Grid Method
Subdivide space into a regular grid cubic of square bins
Index each object in a bin
d
Grid Method
d
Running time is proportional tonumber of moving objects
Useful also to compute pairs of objects within some distance (vision,sound, …)
Closest-Feature Tracking(M. Lin and J. Canny. A Fast Algorithm for Incremental Distance Calculation. Proc. IEEE Int. Conf. on Robotics and Automation, 1991)
The closest pair of features (vertex, edge, face) between two polyhedral objects are computed at the start configurations of the objects
During motion, at each small increment of the motion, they are updated
Efficiency derives from two observations: The pair of closest features
changes relatively infrequently When it changes the new closest features will
usually be on a boundary of the previous closest features
Closest-Feature Test for Vertex-Vertex
VertexVertex
Application: Detecting Self-Collision in Humanoid Robots(J. Kuffner et al. Self-Collision and Prevention for Humanoid Robots. Proc. IEEE Int. Conf. on Robotics and Automation, 2002)
BVH with spheres:S. Quinlan. Efficient Distance Computation Between Non-Convex Objects. Proc. IEEE Int. Conf. on Robotics and Automation, 1994.
BVH with Oriented Bounding Boxes:S. Gottschalk, M. Lin, and D. Manocha. OBB-Tree: A Hierarchical Structure for Rapid Interference Detection. Proc. ACM SIGGRAPH '96, 1996.
Combination of BVH and feature-tracking:S.A. Ehmann and M.C. Lin. Accurate and Fast Proximity Queries Between Polyhedra Using Convex Surface Decomposition. Proc. 2001 Eurographics, Vol. 20, No. 3, pp. 500-510, 2001.
Adaptive bisection in dynamic collision checking:F. Schwarzer, M. Saha, J.C. Latombe. Adaptive Dynamic Collision Checking for Single and Multiple Articulated Robots in Complex Environments, manuscript, 2003.
Bounding Volume Hierarchy Method
Enclose objects into bounding volumes (spheres or boxes) Check the bounding volumes first Decompose an object into two
Bounding Volume Hierarchy Method
Enclose objects into bounding volumes (spheres or boxes) Check the bounding volumes first Decompose an object into two Proceed hierarchically
Bounding Volume Hierarchy Method
Enclose objects into bounding volumes (spheres or boxes) Check the bounding volumes first Decompose an object into two Proceed hierarchically
Bounding Volume Hierarchy Method
• BVH is pre-computed for each object
Bounding Volume Hierarchy Method
BVH in 3D
Collision Detection
Two objects described by their precomputed BVHs
A
B C
D E F G
A
B C
D E F G
Collision Detection
AASearch tree
AA
pruning
Collision Detection
AA
CCCBBCBB
Search tree
AA
A
B C
D E F G
Collision Detection
CCCBBCBB
AASearch tree
pruning
A
B C
D E F G
If two leaves of the BVH’s overlap(here, G and D) check their contentfor collision
Collision Detection
CCCBBCBB
AASearch tree
GEGDFEFD
A
B C
D E F G
G D
Variant
AA
CCCBBCBB
Search tree
AA
A
B C
D E F GAA
CABA
Collision Detection• Pruning discards subsets of the two objects that are separated
by the BVs
• Each path is followed until pruning or until two leaves overlap
• When two leaves overlap, their contents are tested for overlap
Search Strategy and Heuristics
If there is no collision, all paths must eventually be followed down to pruning or a leaf node
But if there is collision, it is desirable to detect it as quickly as possible
Greedy best-first search strategy with f(N) = d/(rX+rY)
[Expand the node XY with largest relative overlap (most likely to contain a collision)]
rX
rYd
X
Y
Recursive (Depth-First) Collision Detection Algorithm
Test(A,B)1. If A and B do not overlap, then return 12. If A and B are both leaves, then return 0 if their contents overlap
and 1 otherwise3. Switch A and B if A is a leaf, or if B is bigger and not a leaf4. Set A1 and A2 to be A’s children5. If Test(A1,B) = 1 then return Test(A2,B) else return 0
Performance• Several thousand collision checks per second for 2 three-
dimensional objects each described by 500,000 triangles, on a 1-GHz PC
Distance Computation
M
> M, prune
Greedy Distance Computation
Greedy-Distance(A,B,M)1. If min-dist(A,B) > M, then return M2. If A and B are both leaves, then return distance between their
contents 3. Switch A and B if A is a leaf, or if B is bigger and not a leaf4. Set A1 and A2 to be A’s children5. M min(max-dist(A1,B), max-dist(A2,B), M)6. d1 Greedy-Distance(A1,B,M) 7. d2 Greedy-Distance(A2,B,M) 8. Return Min(d1,d2)
M (upper bound on distance) is initialized to infinity
Approximate Distance
Approx-Greedy-Distance(A,B,M,a)1. If (1+a)min-dist(A,B) > M, then return M2. If A and B are both leaves, then return distance between their
contents 3. Switch A and B if A is a leaf, or if B is bigger and not a leaf4. Set A1 and A2 to be A’s children5. M min(max-dist(A1,B), max-dist(A2,B), M)6. d1 Approx-Greedy-Distance(A1,B,M,a) 7. d2 Approx-Greedy-Distance(A2,B,M,a) 8. Return Min(d1,d2)
M (upper bound on distance) is initialized to infinity
Desirable Properties of BVs and BVHsBVs:• Tightness• Efficient testing• Invariance
BVH: Separation Balanced tree
?
Spheres• Invariant• Efficient to test• But tight?
Axis-Aligned Bounding Box (AABB)
Axis-Aligned Bounding Box (AABB) Not invariant Efficient to test Not tight
Oriented Bounding Box (OBB)
Invariant Less efficient to test Tight
Oriented Bounding Box (OBB)
Comparison of BVs
Sphere AABB OBB
Tightness - -- +
Testing + + o
Invariance yes no yes
No type of BV is optimal for all situations
Desirable Properties of BVs and BVHsBVs:• Tightness• Efficient testing• Invariance
BVH: Separation Balanced tree ?
Desirable Properties of BVs and BVHsBVs:• Tightness• Efficient testing• Invariance
BVH: Separation Balanced tree
Construction of a BVH • Top-down construction • At each step, create the two children of a BV• Example:
For OBB, split longest side at midpoint
Computation of an OBB[Gottschalk, Lin, and Manocha, 96]
N points ai = (xi, yi, zi)T, i = 1,…, N
SVD of A = (a1 a2 ... aN) A = UDVT where
D = diag(s1,s2,s3) such that s1 s2 s3 0
U is a 3x3 rotation matrix that defines the principal axes of variance of the ai’s OBB’s directions
The OBB is defined by max and min coordinates of the ai’s along these directions
Possible improvements: use vertices of convex hull of the ai’s or dense uniform sampling of convex hull
x
y
X
Yrotation described bymatrix U
Static vs. Dynamic Collision Detection
Static checks Dynamic checks
Usual Approach to Dynamic Checking (in PRM Planning)1) Discretize path at some fine resolution e2) Test statically each intermediate configuration
< e e too large collisions are missed e too small slow test of local paths
1
2
32
3
3
3
Testing Path Segmentvs. Finding First Collision
PRM planning Detect collision as quickly as possible Bisection strategy
Physical simulation, haptic interactionFind first collision Sequential strategy
e too large collisions are missed e too small slow test of local paths
e too large collisions are missed e too small slow test of local paths
Other Approaches to Dynamic Collision Detection
Bounding-volume (BV) hierarchies Discretization issueFeature-tracking methods
[Lin, Canny, 91][Mirtich, 98] V-Clip[Cohen, Lin, Manocha, Ponamgi, 95] I-Collide[Basch, Guibas, Hershberger, 97] KDS
Geometric complexity issue with highly non-convex objects Sequential strategy (first collision) that is not efficient for PRM path segmentsSwept-volume intersection
[Cameron, 85][Foisy, Hayward, 93]
Swept-volumes are expensive to compute. Too much data. No pre-computed BV hierarchiesAlgebraic trajectory parameterization
[Canny, 86][Schweikard, 91] [Redon, Kheddar, Coquillard, 00]
High-degree polynomials, expensive Floating-point arithmetics difficulties Sequential strategyCombination
[Redon, Kheddar, Coquillard, 00] BVH + algebraic parameterization [Ehmann, Lin, 01] BVH + feature tracking Sequential strategy
Exact Collision Detection with Adaptive Bisection
Idea: Cover line segment with collision free C-space neighborhoods
Use distance computation instead of collision checking
How do you show a C-space neighborhood is collision free?Relate changes in C-space to changes in workspace
Distance R
When moving from (x,y,q) to (x’,y’,q’), no point traces out more than distance|x-x’| + |y-y’| + R|q-q’|
For any q and q’ no robot point traces a path longer than:
l(q,q’) = 3|dq1|+2|dq2|+|dq3|
q = (q1,q2,q3)q’ = (q’1,q’2,q’3)dqi = q’i-qi
q1
q2
q3
How do you show a C-space neighborhood is collision free?Relate changes in C-space to changes in workspace
If l(q,q’) < d(q) + d(q’)then the straight path betweenq and q’ is collision-free
d(q)
d(q) = Euclidean distance between robot and obstacles (or lower bound)
q1
q2
q3
How do you show a C-space neighborhood is collision free?Relate changes in C-space to changes in workspace
q
q’
{q” | l(q,q”) < d(q)}
{q” | l(q’,q”) < d(q’)}
l(q,q’) < d(q) + d(q’)
q
q’
{q” | l(q,q”) < d(q)}
{q” | l(q’,q”) < d(q’)}
l(q,q’) < d(q) + d(q’)l(q,q’) = l(q,qint) + l(qint,q’) < d(q) + d(q’)
qint
q
q’
{q” | l(q,q”) < d(q)}
{q” | l(q’,q”) < d(q’)}
Bisectionl(q,q’) > d(q) + d(q’)
Generalization• A bound based on point that moves the most may be too
restrictive• Some links move much less than others• Some links may be closer to obstacles than others• There might be several interacting robots
• Instead look at each link individually
Generalization• Robot(s) and static obstacles treated as collection of rigid
bodies A1, …, An.
• li(q,q’): upper bound on length of curve segment traced by any point on Ai when robot system is linearly interpolated between q and q’
l1(q,q’) = |dq1|l2(q,q’) = 2|dq1|+|dq2| l3(q,q’) = 3|dq1|+2|dq2|+|dq3|
q1
q2
q3
Generalization• Robot(s) and static obstacles treated as collection of rigid
bodies A1, …, An.
• li(q,q’): upper bound on length of curve segment traced by any point on Ai when robot system is linearly interpolated between q and q’
• If li(q,q’) + lj(q,q’) < dij(q) + dij(q’)then Ai and Aj do not collide between q and q’
Generalized Bisection Method
I. Until Q is not empty do:1. [qa,qb]ij remove-first(Q)2. If li(qa,qb) + lj(qa,qb) dij(qa) + dij(qb) then
a. qmid (qa+qb)/2b. If dij(qmid) = 0 then return collisionc. Else insert [qa,qmid]ij and [qmid,qb]ij into Q
II. Return no collision
Each pair of bodies is checked independently of the others priority queue Q of elements [qa,qb]ij
Initially, Q consists of [q,q’]ij for all pairs of bodies Ai and Aj that need to be tested.
Heuristic Ordering Q
Goal: Discover collision quicker if there is one.
Sort Q by decreasing values of: [li(qa,qb) + lj(qa,qb)] – [dij(qa) + dij(qb)]
Possible extension to multi-segment paths(very useful with lazy collision-checking PRM)