Introduction to Probabilistic Roadmaps and Local Planners

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Introduction to Probabilistic Roadmaps and Local
Planners

• Ron Wein

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Probabilistic Roadmaps
Given two free configurations qinit and qgoal,
try to connect these two configuration to two
vertices u, v in G. Then try to find a path from
u to v.
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The Local Planner
Given two free configurations q and q, segment
the straight line between them to m segments
along the intermediate configurationsq0 q,
q1, , qm q such that qk1 - qk lt
?. Try to connect each pair of adjacent
configurations qk and qk1. This is done by
inflating the robot, and the query is reduced to
an interference query.
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Separation of Convex Polytopes
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Using Linear Programming
This problem can be solved using linear
programming If A p1, , pm and B q1, ,
qn, find a hyperplaneH ax by cz d 0,
such that
ap1x bp1y cp1z d gt 0 ap2x bp2y cp2z d
gt 0 apmx
bpmy cpmz d gt 0
aq1x bq1y cq1z d lt 0 aq2x bq2y cq2z d
lt 0 aqnx
bqny cqnz d lt 0
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The Dobkin-Kirkpatrick Hierarchy
Let P be a convex polytope with avertex set
V(P), where V(P) n.

• Define the hierarchy P1, P2, , Pkwhere
• Pi1 ? Pi and V(Pi1) ? V(Pi).
• The vertices of V(Pi) - V(Pi1) forman
  independent set in Pi.

Each face F of Pi1 that is not a face of Pi can
be associated with a unique vertex v of Pi, that
lies in the half-space opposite to Pi1 with
respect to the hyperplane supporting F.
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Polytope Hyperplane Separation
Let ?(Pi,H) be the separating distance of Pi and
a hyperplane H,obtained at some point ri ? V(Pi).
Let H be a hyperplane parallel to H that touches
ri. Then
Thus ?(P,H) can be computedin O(log n) time.
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Separating two Convex Polytopes
Using the Dobkin-Kirkpatrick hierarchies, the
interference query can by answered in O(log m ?
log n) time.
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Separation of Non-Convex Polytopes
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Bounding Volumes Hierarchy
We used simple geometric entities to bound our
polytopes. We keep different levels of bounding
volumes in a hierarchy tree.
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Types of Bounding Volumes

• Axes-Aligned Bounding Box (AABBs),
• Ellipsoids,
• Oriented Bounding Boxes (OBBs),
• Spherical shells,
• and more

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OBB Tree A Hierarchical Structure for Rapid
Interference Detection

• S. GottSchalk, M. C. Lin and D. Manocha

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Oriented Bounding Boxes
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Bounding a Set of Triangles
Compute the 3-dimensional mean vector, and the
3?3 covariance matrix
The matrix ? is symmetric, therefore its
eigenvectors are mutually orthogonal. Use the
normalized eigenvectors as the axes of the
bounding box.
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Separation of Bounding Boxes
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The Separating Axis Theorem
The separating axis of two oriented boxes is
either perpendicular to one of the faces, or can
be obtained as the vector multiplication of two
box axes.
The interference query between two oriented boxes
can be answered by checking 15 axes.
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Separating Axis (Vertex-Vertex)
We can show that there exists a vector L such
that all three terms have the same sign. Thus
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Interference of OBBs
Using careful analysis for the various 15
possible axes, it is possible to implement the
interference test for two OBBs using about 200
floating-point operations.
This method has been implemented and successfully
applied to large models.