CS691G Computational Geometry Motion Planning

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CS691G Computational GeometryMotion Planning

• Ileana Streinu Oliver Brock
• Fall 2005

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Why Motion Planning?
Robert Bohlin
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Why Motion Planning?
Dynamic Simulation
Virtual Prototyping
Molecular Biology
Haptics/Teleoperation
Character Animationfor Games and Movies
James Kuffner
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Structural Molecular Biology
Lydia Kavrakis research group
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Visibility Graph
Goal
Start
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Trapezoidal Decomposition
R
S is a set of n non-crossing line segments
slightly customized to look like a robotics
problem
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Some Counting
n line segments
at most 6n4 vertices
42n2(2n)
at most 3n1 trapezoids
Consider leftmost vertex in trapezoid that is
endpoint of line segment
Rectangle ! 1
Left endpoints of segments ! 2n
Right endpoints ! 1n
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Connection to Path Planning

• Description
• O(n log n)
• Point location
• O(n)
• Search
• O(n)

We should be able to do better with point
location
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Randomized Construction
A
A
B
A
D
s1
q1
A
B
C
p1
C
E
D
s2
p2
q2
C
F
G
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Randomized Construction
B
A
A
C
D
B
s1
q1
A
p1
G
E
F
E
C
F
G
D
D
s2
p2
q2
C
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Complexity

• Expected running time O(n log n)
• Expected point query time O(log n)
• Expected search structure O(n)

Our first motion planning algorithm in 2D!
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Visibility Map
Goal
Shakey, SRI, 1968
Start
Workspace ? Configuration Space
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Workspace / C-Space
position
Workspace
Local frame Robot frame
orientationor heading
Configuration Space
Global frame World frame Reference frame
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Computing C-SpaceGrowing Obstacles
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Sliding Along the Boundary
Workspace
Configuration Space
How about changing ? ?
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Translational Case (Fixed ?)
Obstacle
Robot
C-Space Obstacle
Reference Point
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C-Obstacle Construction
? positive linear combination
? positive linear combination
? positive linear combination
-r3
o2
o2
o3
o3
o1
-r1
-r2
o4
o1
o5
o4
o5
r2
r1
r3
From previous slide
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C-Obstacle Construction
Jean-Claude Latombe
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C-Obstacle Construction in 3D
Jean-Claude Latombe
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C-Obstacle in 3D
Jean-Claude Latombe
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Configuration Space for Arms
Jean-Claude Latombe
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Four Categories

• Exact Cell Decomposition
• Approximate Cell Decomposition
• Roadmap Methods examples
• Visibility
• Voronoi
• PRM
• Potential Field Methods

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Exact Cell Decomposition
Jean-Claude Latombe
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Exact Cell Decomposition cont.
Jean-Claude Latombe
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Exact Cell Decomposition cont.
Jean-Claude Latombe
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Exact Cell Decomposition cont.
Jean-Claude Latombe
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2n-Tree
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Approximate Cell Decomposition
Again we build a graph and search it to find a
path!
Jean-Claude Latombe
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Approximate Cell Decomposition
Fortunately there are PRM Methods!
Jean-Claude Latombe
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Retraction
A retraction is a continuous mapping of
Cfreeonto a one-dimensional network of curves R
½ Cfree.
Voronoi Retraction
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Roadmap Methods
Roadmap
?
?
How to obtain the roadmap?
Jean-Claude Latombe
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Generalized Voronoi Diagram
Jean-Claude Latombe
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Artificial Potential Field MethodsAttractive
Potential
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Repulsive Potential
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Total Potential Function