Hierarchical Segmentation of Automotive Surfaces and Fast Marching Methods

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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

Microdynamic Systems Laboratory
BioRobotics Lab
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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
CNC Milling
Bone Shaving
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Challenges
Deposition Pattern

• Complex deposition patterns
• Non-Euclidean surfaces
• High dimensioned search-space for optimization

Warping of the Deposition Pattern
Target Surface
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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
Model Based Planning
Simulation
Constraints
Path Variables
Output Characteristics
Rule Based Planning
System Parameters
Output
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Our Approach Decomposition

• Segment surface into cells
• Topologically simple/monotonic
• Low surface curvature

• Generate passes in each cell

Repeat offsetting and speed optimization
Select start curve
Optimize index width and generate offset curve
Optimize end effector speed
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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)
• Select relative position with respect to boundary
  (minimizing geodesic curvature)

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Effect of Surface Curvature

• Offsets of geodesics are not geodesics in
  general!!
• 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
• Topological changes

Initial front
Front at a later instance
Marker pt. soln.
Images from http//www.imm.dtu.dk/mbs/downloads/l
evelset040401.pdf
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Level Set Method Sethian

• Assume each front at is a zero level set of an
  evolving function of zF(x,t)
• Solve the PDE (H-J eqn)

given the initial front F(x,t0)
http//www.imm.dtu.dk/mbs/downloads/levelset04040
1.pdf
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Fast Marching Method Sethian

• F(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

?
T0
given
T3

• Using efficient sorting and causality, compute
  T(x) at all x quickly.

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FMM Similarity with Dijkstra

• Similar to Dijkstras algorithm
• Wavefront expansion
• O(N logN) for N grid points
• Improves accuracy by first order approximation to
  distance

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FMM Contd.
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1
8
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
5
4
2
B
A
Front grad.
T(A)10
T(B) 8
Dijkstra T(C)min(T(A)5, T(B)5)13
FMM T(C)8412
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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((k12k22)/2) Maxima of
  Gaussian curvature k1k2
• Four Steps
• Minima detection
• Minima expansion
• Descent to minima
• Merging based on Watershed Height

http//cmm.ensmp.fr/beucher/wtshed.html
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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)

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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 ill-effects 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
Merged
Segmented
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 WorkCell 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?