Adaptive Dynamic Collision Checking for Many Moving Bodies Mitul Saha Department of Computer Science, Stanford University.

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Adaptive Dynamic Collision Checking for Many
Moving BodiesMitul SahaDepartment of Computer
Science,Stanford University.

• NSF-ITR Workshop
• Collaborators Fabian Schwarzer,
• Jean-Claude Latombe

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Static vs. Dynamic Collision Checking
Static
Dynamic

• Static collision checking tests a single
  configuration q for collision
• Dynamic collision checking ensure that all
  configurations on a continuous path are
  collision-free.

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Motivation (1)
Bottleneck in PRM planning Efficient testing of
sampled configurations and connections for
collision.
qb
qa
(Usually people trade-off reliability for speed
in dynamic collision checking.)
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Motivation (2)
Static collision tests (for sampled
configurations) are done efficiently using
pre-computed bounding volume (BV) hierarchies
A
B
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Motivation (3)
But dynamic collision tests (for connections
betweensampled configurations) using BV
hierarchies are usually approximate.
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2
3
1
3
2
3

• ? too large ? collisions are missed
• ? too small ? slow test of local paths

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Motivation (3)
But dynamic collision tests (for connections
betweensampled configurations) using BV
hierarchies are usually approximate.
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Previous Approaches to Dynamic Collision Testing

• Bounding-volume (BV) hierarchies? Discretization
  issue
• Feature-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 segments
• Swept-volume intersection Cameron, 85 Foisy,
  Hayward, 93? Swept-volumes are expensive to
  compute. Too much data.? No pre-computed BV
  hierarchies
• Algebraic trajectory parameterization Canny,
  86 Schweikard, 91 Redon, Kheddar,
  Coquillard, 00? High-degree polynomials,
  expensive
• Combination Redon, Kheddar, Coquillard, 00 BVH
  algebraic parameterization Ehmann, Lin, 01
  BVH feature tracking ? Sequential strategy

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Desirables in Nutshell

• Adaptive resolution
• Exact collision checking
• Competitive with discrete checkers w.r.t. speed

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Definitions
Workspace
C-space
Ai (qb)
qb
?i (qa,qb)
qb
qa
qa
Ai (qa)

• ?i(qa,qb) ?curveLength(Ai) on qa qb?

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Definitions
Workspace
C-space
Ai (qb)
Aj (qb)
qb
?ij (qb)
qb
qa
qa

• ?ij(q) ?distance(Ai, Aj) ?
• ?ij(q) 0 ? Collision

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Key Lemma
Workspace
C-space
Ai (qb)
Aj (qb)
qb
?ij (qb)
?i (qa,qb)
?j (qa,qb)
qb
qa
?ij (qa)
qa
Aj (qa)
Ai (qa)
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Key Lemma
Aj
?i
Ai
?ij
?i lt ?ij Safe
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Collision Checking using the Lemma
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Collision Checking using the Lemma

• After the collision check a collision-free
  connection may look like
• Each sub-segment segment satisfies the lemma
• ?i (qa,qb) ?j (qa,qb) lt ?ij (qa) ?ij (qb)
• and hence the collision checking is exact
• The collision checking is adaptive

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GeneralizationLarge Number of Moving Bodies

• Robot(s) and static obstacles treated as
  collection of rigid bodies A1, , An.
• If ?i(q,q) ?j(q,q) lt ?ij(q) ?ij(q)then
  Ai and Aj do not collide between q and q

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Generalized Bisection Method for Large Number of
Moving Objects

• Each pair of bodies is checked independently of
  the others ? priority queue Q of elements
  qa,qbij
• Initially, Q consists of q,qij for all pairs
  of bodies Ai and Aj that need to be tested.

• Until Q is not empty do
• qa,qbij ? remove-first(Q)
• If ?i(qa,qb) ?j(qa,qb) ? ?ij(qa) ?ij(qb) then
• qmid ? (qaqb)/2
• If ?ij(qmid) 0 then return collision
• Else insert qa,qmidij and qmid,qbij into Q
• Return no collision

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Heuristic Ordering Q

• Goal Discover collision quicker if there is one.
  
• Sort Q by decreasing values of
  li(qa,qb) lj(qa,qb) hij(qa) hij(qb)

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Generalized Bisection Method for Large Number of
Moving Bodies

• Things to note
• Different pair of bodies tested with different
  resolutions, i.e. usually close pair of objects
  are checked with finer resolution than the
  distant ones.
• Usually close pair of objects are considered
  before the distant ones.

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Calculating Terms in the Key Lemma
?i(qa,qb)
Ai(q)
Ai(qa)
Aj(qb)
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Lower bound on Distance between Bodies

• Need to find ?ijs
• Exact and approximate distance computations are
  very expensive compared to pure collision
  checking
• So is there any alternative?

Aj
Aj
?ij
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Lower bound on Distance between Bodies

• Use a classical BVH collision checker with a
  slight modification
• Compute distance between BVs instead of just
  testing BV overlap ? BVs are RSSs

?ij(q)

• returns values often much larger than ½
  distances
• much faster than BV-based exact or ½-approximate
  distance computation
• almost as fast as pure collision checking

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Calculating Terms in the Key Lemma
?i(qa,qb)
Ai(q)
Ai(qa)
Aj(qb)
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Bounding Motions in Workspace ?i(qa,qb)
Upper Bound Path Length
Rigid Body
A1
A2
A3
A4
??
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Comparative Experiment

• SBL PRM planner (single-query,
  bi-directional, lazy in cc) with
  fixed-discretization collision checker
• A-SBL Same planner, with new collision
  checkerExperiment
• Run SBL 10 times on same planning problem with
  some resolution e
• If a collision has been missed, reduce e and
  repeat
• If no collision has been missed, return average
  planning time
• Run A-SBL 10 times and return average planning
  time

Robot 2,502 triangles Obstacles 432
Triangles SBL ? 17 sec A-SBL ? 4.8 sec
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Some Result
Robot 2,502 triangles Obstacles 432
Triangles SBL ? 17 sec A-SBL ? 4.8 sec
Robot 2,991 triangles Obstacles 432
Triangles SBL ? 83 sec A-SBL ? 44 sec
Robot 2,991 trianglesObstacles 74,681
triangles SBL ? 1.20 sec A-SBL ? 0.81 sec
Robot 2,502 trianglesObstacles 34,171
triangles SBL ? 3.2 sec A-SBL ? 2.1 sec
Robots 6 x 2,991 trianglesObstacles 19,668
triangles SBL ? 85 sec A-SBL ? 52 sec
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In Action
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In Action
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Summing up..

• New dynamic collision checker which
• Never misses a collision
• Eliminates need for determining resolution
  factor e
• Is even faster than the discrete checker in PRM
  planning
• Techniques that considerably improve efficiency
• Greedy Distance Computation algorithm, nearly as
  efficient as pure collision checker
• Fast technique for bounding lengths of paths
  traced out in workspace
• Heuristics for ordering collisions tests
• Represent deformable objects with a large number
  of small rigid objects

vs
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Summing up..

• A free downloadable c implementation of the
  new dynamic collision checker is on the web
  http//robotics.stanford.edu/mitul/mpk

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Adaptive Dynamic Collision Checking for Many
Moving ObjectsMitul SahaDepartment of Computer
ScienceStanford University
Any Question?

• NSF-ITR Workshop
• Collaborators Fabian Schwarzer,
• Jean-Claude Latombe

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SBL
X
Sanchez-Ante, Latombe, 01
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SBL
Sanchez-Ante, Latombe, 01
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Our New Collision Checker

• Based on Adaptive Bisection
• Ideas
• Relate configuration changes to path lengths in
  workspace
• Use distance computation rather than pure
  collision checking

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Current Research

• Extension to deformable objects
• Large number of small bodies can be used to
  approximate
• deformable objects