1 Hierarchical Segmentation of Automotive Surfaces and Fast Marching Methods David C. Conner Aaron Greenfield Howie Choset Alfred A. Rizzi BioRobotics Lab Microdynamic Systems Laboratory Prasad N. Atkar
Jan 07, 2016
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Hierarchical Segmentation of Automotive Surfaces and Fast Marching Methods
David C. Conner
Aaron Greenfield
Howie Choset
Alfred A. Rizzi
BioRoboticsLab
Microdynamic SystemsLaboratory
Prasad N. Atkar
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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
Bone Shaving CNC Milling
Complete Coverage
Uniform Coverage
Cycle time and Paint waste
Programming Time
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Challenges
• Complex deposition patterns
• Non-Euclidean surfaces
• High dimensioned search-space for optimization
0 Micr
35.08
Deposition
Pattern
Spray Gun
Target Surface
Warping of the
Deposition Pattern
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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
Constraints Path Variables SimulationOutput
Characteristics
Rule Based Planning System
Parameters
Model Based Planning
Output
Dimensionality Reduction
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Our Approach: Decomposition
• Segment surface into cells– Topologically
simple/monotonic – Low surface curvature
y
x(t)
• Generate passes in each cell
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
Select passes with minimal geodesic curvature (uniformity)
Avoid painting holes (cycle time, paint waste)
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)
Average Normal
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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
geodesic
Not ageodesic
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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 changesInitial front
Front at a later instance Marker pt. soln.Images from http://www.imm.dtu.dk/~mbs/downloads/levelset040401.pdf
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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
given
• Using efficient sorting and causality, compute T(x) at all x quickly.
T=0
Г
T=3
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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.
In our example,
For 2-D grid
Dijkstra FMM
First order approximation
1
1
∞
∞
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FMM on triangulated manifolds
• Evaluate finite difference on a triangulated domain– Basis: two linearly
independent vectorsT(A)=10
C
T(B)= 8
5 5
Dijkstra: T(C)=min(T(A)+5, T(B)+5)=13
FMM: T(C)=8+4=12
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A BFront
grad.
2
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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+k2
2)/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)SymmetrizedGauss 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 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 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?