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Hierarchical Segmentation of Automotive
Surfaces and Fast Marching Methods
Prasad N. Atkar
David C. Conner
Aaron Greenfield
Howie Choset
Alfred A. Rizzi
BioRobotics
Lab
Microdynamic
Systems
Laboratory
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Automated Trajectory Generation
• Generate trajectories on
curved surfaces for material
removal/deposition
– Maximize uniformity
– Minimize cycle time and material
waste
Spray Painting
Complete Coverage
Uniform Coverage
Cycle time and Paint waste
Programming Time
Bone Shaving
CNC Milling
2
Challenges
• Complex deposition
patterns
Deposition
Pattern
Spray
Gun
Warping of
the
Deposition
Pattern
• Non-Euclidean surfaces
Target Surface
• High dimensioned searchspace for optimization
35.08
0 Micr
3
Related Research
• Index Optimization
– Simplified surface with simplified
deposition patterns (Suh et.al,
Sheng et.al, Sahir and Balkan, Asakawa
and Takeuchi)
• Speed Optimization
– Global optimization
(Antonio and
Ramabhadran, Kim and Sarma)
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Overview of Our Approach
• Divide the problem into smaller sub-problems
– Understand the relationships between the
parameters and output characteristics
– Develop rules to reduce problem dimensionality
– Solve each sub-problem independently
Dimensionality
Reduction
Constraints
Model Based Planning
Path Variables
Simulation
Output
System
Parameters
Rule Based
Planning
Output
Characteristics
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Our Approach: Decomposition
• Segment surface into
cells
– Topologically
simple/monotonic
– Low surface curvature
• Generate passes in each cell
y
Q Q
Q
a(t)
x
Q
Select start
curve
Optimize end
effector speed
Optimize index width
and generate offset
curve
Repeat offsetting
and speed
optimization
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Rules for Trajectory generation
Avoid painting holes
(cycle time, paint waste)
Select passes with minimal
geodesic curvature (uniformity)
Minimize number of turns
(cycle time, paint waste)
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Choice of Start Curve
• Select a geodesic
curve
– Select spatial
orientation
(minimizing number
of turns)
Average
Normal
– Select relative
position with respect
to boundary
(minimizing geodesic
curvature)
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Effect of Surface Curvature
Not a
geodesic
• Offsets of geodesics
are not geodesics in
general!!
geodesic
• Geodesic curvature of
passes depends on
surface curvature
– Gauss-Bonnet
Theorem
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Selecting position of Start Curve
• Select start curve as a
geodesic Gaussian
curvature divider
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Speed and Index Optimization
• Speed optimization
– Minimize variation in
paint profiles along the
direction of passes
• Index optimization
– Minimize variation in
paint deposition along
direction orthogonal to
the passes
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Offset Pass Generation (Implementation)
• Marker points
• Self-intersections
difficulty
Initial
front
• Topological changes
Images from http://www.imm.dtu.dk/~mbs/downloads/levelset040401.pdf
Front at a later instance
Marker pt. soln.
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Level Set Method [Sethian]
• Assume each front at is
a zero level set of an
evolving function of
z=Φ(x,t)
• Solve the PDE (H-J eqn)
given the initial front
Φ(x,t=0)
http://www.imm.dtu.dk/~mbs/downloads/levelset040401.pdf
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Fast Marching Method [Sethian]
• Φ(x,t)=0 is single valued in t if F
preserves sign
• T(x) is the time when front crosses x
• H-J Equation reduces to simpler
Eikonal equation
Г
T=0
given
T=3
• Using efficient sorting and
causality, compute T(x) at all x
quickly.
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FMM: Similarity with Dijkstra
• Similar to Dijkstra’s
algorithm
– Wavefront expansion
– O(N logN) for N grid
points
• Improves accuracy by
first order
approximation to
distance
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FMM Contd.
∞
∞
1
1
Dijkstra
FMM
First order approximation
For 2-D grid
In our example,
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FMM on triangulated manifolds
• Evaluate finite
difference on a
triangulated domain
– Basis: two linearly
independent vectors
C
5
2
Front
A
T(A)=10
5
4
B
T(B)= 8
grad.
Dijkstra: T(C)=min(T(A)+5, T(B)+5)=13
FMM: T(C)=8+4=12
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Hierarchical Surface Segmentation
• Segment surface into
cells
• Advantages
– Improves paint uniformity,
cycle time and paint waste
• Requirements
– Low Geodesic curvature of
passes
– Topological monotonicity
of the passes
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Geometrical Segmentation
• To improve uniformity
of paint deposition
– Minimize Geodesic
curvature of passes
– Restrict the regions of
high Gaussian
curvature to
boundaries
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Geometrical Segmentation
• Watershed Segmentation on
RMS curvature of the surface
– Maxima of RMS
sqrt((k12+k22)/2) ≈ Maxima of
Gaussian curvature k1k2
http://cmm.ensmp.fr/~beucher/wtshed.html
• Four Steps
– Minima detection
– Minima expansion
– Descent to minima
– Merging based on
Watershed Height
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Topological Segmentation
• Improves paint waste and
cycle time by avoiding holes
• Orientation of slices
– Planar Surfaces (cycle time
minimizing)
– Extruded Surfaces (based on
principal curvatures)
– Surfaces with non-zero
curvature (maximally
orthogonal section plane)
Symmetrized
Gauss Map
Medial Axis
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Pass Based Segmentation
• Improves cycle time
and paint waste
associated with
overspray
• Segment out narrow
regions
– Generate slices at
discrete intervals
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Region Merging
• Merge Criterion
– Minimize sum of lengths of
boundaries : reduce boundary illeffects on uniformity
• Merge as many cells as possible
such that each resultant cell is
– Geometrically simple
• Inspect boundaries
– Topologically monotonic (single
connected component of the
offset curve, and spray gun
enters and leaves a given cell
exactly once)
• Partition directed connectivity
graph such that each subgraph
is a trail
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Region Merging Results
Segmented
Merged
Segmented
Segmented
Merged
Merged
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Summary
• Rules to reduce dimensionality of the
optimal coverage problem
• Gauss-Bonnet theorem to select the start
curve
• Fast marching methods to offset passes
• Hierarchical Segmentation of Surfaces
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Future Work—Cell Stitching
• Optimize ordering in which cells are
painted
• Optimize overspray to minimize the
cross-boundary deposition
• Optimize end effector velocity
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Thank You!
Questions?
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