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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

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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?