Hierarchical Segmentation of Automotive Surfaces and Fast Marching Methods

1. Hierarchical Segmentation of Automotive Surfaces and Fast Marching Methods David C. Conner Aaron Greenfield Howie Choset Alfred A. Rizzi • Prasad N. Atkar Microdynamic Systems Laboratory BioRobotics Lab

2. Complete Coverage Uniform Coverage Cycle time and Paint waste Programming Time 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

3. 35.08 0 Micr Spray Gun Challenges Deposition Pattern • Complex deposition patterns • Non-Euclidean surfaces • High dimensioned search-space for optimization Warping of the Deposition Pattern Target Surface

4. 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)

5. 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 Path Variables Output Characteristics Constraints Rule Based Planning System Parameters Output

6. y Q Q Q a(t) x Q 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

7. 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)

8. Average Normal 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)

9. Not a geodesic geodesic Effect of Surface Curvature • Offsets of geodesics are not geodesics in general!! • Geodesic curvature of passes depends on surface curvature • Gauss-Bonnet Theorem

10. Selecting position of Start Curve • Select start curve as a geodesic Gaussian curvature divider

11. 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

12. 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/levelset040401.pdf

13. 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

14. 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.

15. 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

16. FMM Contd. ∞ 1 ∞ 1 Dijkstra FMM First order approximation For 2-D grid In our example,

17. FMM on triangulated manifolds C 5 5 • Evaluate finite difference on a triangulated domain • Basis: two linearly independent vectors 4 2 B A Front grad. T(B)= 8 T(A)=10 Dijkstra: T(C)=min(T(A)+5, T(B)+5)=13 FMM: T(C)=8+4=12

18. 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

19. Geometrical Segmentation • To improve uniformity of paint deposition • Minimize Geodesic curvature of passes • Restrict the regions of high Gaussian curvature to boundaries

20. Geometrical Segmentation • Watershed Segmentation on RMS curvature of the surface • Maxima of RMS sqrt((k12+k22)/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

21. Symmetrized Gauss Map Medial Axis 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)

22. Pass Based Segmentation • Improves cycle time and paint waste associated with overspray • Segment out narrow regions • Generate slices at discrete intervals

23. 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

24. Region Merging Results Segmented Merged Segmented Merged Segmented Merged

25. 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

26. Future Work—Cell Stitching • Optimize ordering in which cells are painted • Optimize overspray to minimize the cross-boundary deposition • Optimize end effector velocity

27. Thank You!Questions?