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Configuration Space of 2T 1R Robot

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Title: Configuration Space of 2T 1R Robot


1
Configuration Space of 2T1R Robot
  • Gokul Varadhan

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

3
Polygonal robot translating in 2-D workspace
configuration space
workspace
4
Polygonal robot translating rotating in 2-D
workspace
configuration space
workspace
5
Polygonal robot translating rotating in 2-D
workspace
q
y
x
6
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
7
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)
8
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)
9
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
10
2D Translation and Rotation
Obstacles
Robot
11
Contact Surfaces
3,929 contact surfaces
12
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
13
Free Space and Contact Surfaces
  • F is bounded by the C-surfaces

C-surfaces
F
C-obstacle
14
Free Space Computation
  • To obtain the free space requires computing
    arrangement of the C-surfaces

15
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)
16
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
17
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

18
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

19
Free Space Approximation
3,929 contact surfaces
20
2T1R Gears
Goal
Start
21
2T1R Gears
22
2T1R GearsPath in Configuration Space
Path
Goal
Start
23
Next lecture
  • Probabilistic roadmap methods
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