Configuration Space of 2T 1R Robot

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Configuration Space of 2T1R Robot

• Gokul Varadhan

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Last Lecture

• Minkowski sum
• Configuration space of 2T, 3T robots

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Polygonal robot translating in 2-D workspace
configuration space
workspace
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Polygonal robot translating rotating in 2-D
workspace
configuration space
workspace
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Polygonal robot translating rotating in 2-D
workspace
q
y
x
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Contact Surfaces (C-surfaces)

• A C-surface arises from a contact between
  features of the robot and the obstacle

R
O
R
O
Type A contact
Type B contact
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Type A Contact Surface
APPLAi,j
vi R(q) . (bj-1 bj ) ? 0 ?
Contact is feasible when
vi R(q) . (bj1 bj ) ? 0
bj-1
ai1(q)
O
R
bj
bj1
vi R(q)
ai(q)
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Type A Contact Surface
C-surface is a ruled surface
Contact surface
fi,jA(q) 0
fi,jA(q) viR(q) . (bj ai(q) )
where
bj-1
ai1(q)
O
R
bj
bj1
vi R(q)
ai(q)
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Type A C-constraint
CONSTAi,j (q)
APPLAi,j(q) ? fi,jA(q) ? 0
Similarly, you can define a C-constraint for Type
B contact Type B C-surface is also a ruled surface
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2D Translation and Rotation
Obstacles
Robot
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Contact Surfaces
3,929 contact surfaces
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Representation of C-obstacle

• How can we represent C-obstacle in terms of
  C-surfaces?
• Non-convex case
• Resort to convex decomposition

For the case of a convex robot and a convex
obstacle,
CONSTAi,j (q) ? CONSTBi,j(q) is true for all
contacts (edge-vertex pairs)
q ? CO
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Free Space and Contact Surfaces

• F is bounded by the C-surfaces

C-surfaces
F
C-obstacle
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Free Space Computation

• To obtain the free space requires computing
  arrangement of the C-surfaces

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Arrangement

• Arrangement A(S) of a set S of geometric objects
  Halperin 1997 Agarwal Sharir 2000

Decomposition of space into relatively open
connected cells of dimensions 0,...,d
Arrangement of lines (clipped within a window)
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Free Space Computation

• Compute an arrangement of the C-surfaces
• Compute intersections between the C-surfaces
• Retain the appropriate portions of the arrangement

F
C-obstacle
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Free Space Computation

• Arrangement computation is difficult
• Computing surface-surface intersection is prone
  to robustness problems
• Typically O(n2) number of contact surfaces
• Contact surfaces are non-linear

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Free Space Approximation

• We have developed an accurate and efficient
  approximate algorithm Varadhan and Manocha 2004
  
• Provides certain geometric and topological
  guarantees on the approximation
• Approximation is close to the boundary of the
  free space
• It has the same number of connected components
  and genus as the exact Minkowski sum

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Free Space Approximation
3,929 contact surfaces
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2T1R Gears
Goal
Start
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2T1R Gears
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2T1R GearsPath in Configuration Space
Path
Goal
Start
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Next lecture

• Probabilistic roadmap methods