Ercan U' Acar, Howie Choset

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Exploiting Critical Points to Reduce Positioning
Error for Sensor-based Navigation

• Ercan U. Acar, Howie Choset

Carnegie Mellon University
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Outline

• Morse Decompositions
• Exact cellular decompositions in terms of
  critical points for sensor-based coverage
• Topological Navigation Using Morse
  Decompositions
• Determining paths that do not heavily rely on
  dead-reckoning

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Morse Decompositions

• Slice function h(x,y) x, slice(x,y)
  h(x,y)?
• At a critical point (x,y) of
• where M (x,y) m(x,y)0

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1-connected
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2-connected
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1-connected
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2-connected
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• Connectivity of the slice in the free space
• changes at the critical points
• (Morse Theory, 1920)

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Each cell can be covered by back and forth
motions
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Incremental Construction in Unknown Spaces
(Acar et. al, ICRA00)

• Critical point sensing using range data

Surface normal
At a critical point (x,y),
At critical point (x,y)
slice
robot

• Complete sensor-based coverage algorithm
• Complete
• No limitations on obstacle configurations
• Any senor system that is sufficient to guide the
  robot
• along the obstacle boundaries
• Low computational power requirement

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Experiment

• Nomad Scout with 16 sonars
• Area 4m x 4.6m

Reeb Graph
(time compressed)
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Topological Navigation Using Morse Decompositions
Goal Navigation algorithm that is less sensitive
to dead-reckoning error
A
C
D
The robot needs to travel from one critical
point to another to reach an uncovered cell
B
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Navigation within a Cell
Each cell of a Morse Decomposition has none
intersecting upper and lower boundaries
Straight line following
A
Boundary following (robust to dead-reckoning
error)
B
Straight line following
Reverse boundary following (robust to
dead-reckoning error)
y
x
Assumption Dead-reckoning error is bounded
within each cell Critical
points are laterally far away from each other
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Types of Critical Points
IN
OUT
START
END
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Encoding Navigation Information Into Reeb Graph
Order of boundary following and line following
motions is determined by

• Types of critical points

• Relative position of cells
• with respect to critical points

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• Line following
• Boundary following

• Boundary following

• Line following
• Boundary following
• Line following
• Reverse Boundary following

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• Line following
• Boundary following

• Boundary following

• Line following
• Boundary following
• Line following
• Reverse Boundary following

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• Line following
• Boundary following

• Boundary following

• Line following
• Boundary following
• Line following
• Reverse Boundary following

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Navigation Between Cells without a Common Boundary
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Navigation Between Cells without a Common Boundary
Dead-reckoning error accumulates along the entire
path
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Experiment
Vinyl
Carpet
error

• Boundary following
• Line following
• Reverse Boundary following

The robot successfully reaches point B in the
presence of dead-reckoning error
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Conclusions

• Representations of the space with simple
  structures
• for navigation Morse Decompositions
• Decreased the dimension of the
• problem from three to two

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Future Work

• Limitations
• Orientation error
• Large cells
• Critical points that are laterally
• close to each other
• Critical point recognition
• Determining invariant features of critical points

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Complexity of a Morse Decomposition
Decomposition using critical points
Reeb Graph Representation (Nodes are critical
points, edges are cells, faces are objects)
Eulers formula for a planar graph of nodes -
of edges of faces 2
Modified Eulers formula for a Morse
Decomposition of critical points - of cells
of obstacles 1 of cells of
critical points of obstacles -1