Configuration Space of an Articulated Robot

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Configuration Space of an Articulated Robot
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Idea Reduce the Robot to a Point? Configuration
Space
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Two-Revolute-Joint Robot

• A configuration of a robot is a list of
  non-redundant parameters that fully specify the
  position and orientation of each of its bodies
• In this robot, one possible choice is (q1, q2)
• The configuration space (C-space) has 2
  dimensions

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?
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?
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Every robot maps to a point in its configuration
space ...
40 D
15 D
6 D
12 D
65-120 D
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Every robot maps to a point in its configuration
space ...
40 D
15 D
6 D
12 D
65-120 D
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... and every robot path is a curve in
configuration space
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Issues!!

• Dimensionality of configuration space
• Geometric complexity of free region
• ? Plan in configuration space, but compute in
  workspace

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Probabilistic Roadmaps (Sampling-Based Planning)
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• The cost of computing an exact representation of
  the configuration space of a multi-joint
  articulated object is often prohibitive.
• But very fast algorithms exist that can check
  if an articulated object at a given
  configuration collides with obstacles.
• ? Basic idea of Probabilistic Roadmaps (PRMs)
  Compute a very simplified representation of the
  free space by sampling configurations at
  random.

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Probabilistic Roadmap (PRM)
Space ?n
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Probabilistic Roadmap (PRM)
Configurations are sampled by picking coordinates
at random
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Probabilistic Roadmap (PRM)
Configurations are sampled by picking coordinates
at random
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Probabilistic Roadmap (PRM)
Sampled configurations are tested for collision
(in workspace!)
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Probabilistic Roadmap (PRM)
The collision-free configurations are retained as
milestones
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Probabilistic Roadmap (PRM)
Each milestone is linked by straight paths to its
k-nearest neighbors
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Probabilistic Roadmap (PRM)
Each milestone is linked by straight paths to its
k-nearest neighbors
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Probabilistic Roadmap (PRM)
The collision-free links are retained to form the
PRM
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Probabilistic Roadmap (PRM)
The start and goal configurations are included as
milestones
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Probabilistic Roadmap (PRM)
The PRM is searched for a path from s to g
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Basic PRM Algorithm
- FreeConf(q) tests if the configuration q is
collision-free - FreePath(q1,q2) tests if the
straight-line segment between q1 and q2 is
collision-free
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Collision Checking

• Check if objects overlap

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Hierarchical Collision Checking

• Enclose objects into bounding volumes (spheres
  or boxes)
• Check the bounding volumes

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Hierarchical Collision Checking

• Enclose objects into bounding volumes (spheres
  or boxes)
• Check the bounding volumes first
• Decompose an object into two

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BVH of a 3D Triangulated Cat
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Monte Carlo Integration
x1
x2
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Why Does PRM Work?

• In most feasible spaces, every configuration
  illuminates a significant fraction of the
  feasible space

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Narrow-Passage Issue
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Experimental Data
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Experimental Convergence Rate of Basic PRM
Algorithm
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Experimental Convergence Rate
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Key Topics for Future Lectures

• Collision checking
• Sampling strategy