Exact Cell Decomposition

1
Exact Cell Decomposition

• Professor Jean-Claude Latombe
• presentation by Ted Hwang

2
Exact Cell DecompositionPlanning Methods

• polygonal configuration space
• translation and rotation of a rod in the plane
• general path planning
• translation and rotation of concave polygons in
  the plane

3
Polygonal Configuration Space
4
Polygonal Configuration Space

• cells form channels instead of line paths
• channels give the robot more information to avoid
  obstacles encountered at run time

5
Polygonal Configuration Space
6
Polygonal Configuration Space
qfinal
qinit
7
Generalization to 3 Dimensions
8
Translation and Rotation in the Plane
obstacle
Work space, not configuration space
q
d
obstacle
(x,y)

• The robot is modeled as a line segment with
  length d, and called a ladder, rod, or bar.

9
Critical Curves
Type 0
E
10
Critical Curves
Type 1
d
E
11
Critical Curves
Type 2
X
d
12
Critical Curves
Type 3
d
E
X
13
Critical Curves
Type 4
X2
X1
d
14
Critical Curves
Type 5
d
X
E
15
Non-Critical Regions
Any open region enclosed by critical curves is a
non-critical region.
16
Adjacency of Cells

• two cells are adjacent if they share a critical
  curve section and as the rod crosses the critical
  curve it maintains a constant orientation in a
  small neighborhood of the crossing point

17
Connectivity Graph

• We can write a set of conditions called a
  crossing rule for each type of critical curve to
  test the adjacency of the two cells separated by
  the critical curve.
• The connectivity graph is the non-directed graph
  whose nodes are the all the cells in the space.
  A link connects two nodes if the corresponding
  cells are adjacent.

18
Connectivity Graph
19
Thats all!