1. 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

2. 2 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

3. 3 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

4. 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. 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 Constraints Path Variables Simulation Output Characteristics Rule Based Planning System Parameters Model Based Planning Output Dimensionality Reduction

6. 6 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

7. 7 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)

8. 8 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

9. 9 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 a geodesic

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

11. 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. 12 Offset Pass Generation (Implementation) Marker points Self-intersections difficulty Topological changes Initial front Front at a later instanceMarker pt. soln. Images from http://www.imm.dtu.dk/~mbs/downloads/levelset040401.pdf

13. 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. 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 given Using efficient sorting and causality, compute T(x) at all x quickly. T=0 Г T=3

15. 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. 16 FMM Contd. In our example, For 2-D grid DijkstraFMM First order approximation 1 1 ∞ ∞

17. 17 FMM on triangulated manifolds Evaluate finite difference on a triangulated domain –Basis: two linearly independent vectors T(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 4 A B Front grad. 2

18. 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. 19 Geometrical Segmentation To improve uniformity of paint deposition –Minimize Geodesic curvature of passes –Restrict the regions of high Gaussian curvature to boundaries

20. 20 Geometrical Segmentation Watershed Segmentation on RMS curvature of the surface –Maxima of RMS sqrt((k 1 2 +k 2 2 )/2) ≈ Maxima of Gaussian curvature k 1 k 2 Four Steps –Minima detection –Minima expansion –Descent to minima –Merging based on Watershed Height http://cmm.ensmp.fr/~beucher/wtshed.html

21. 21 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

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

23. 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. 24 Region Merging Results Segmented Merged SegmentedMerged Segmented Merged

25. 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. 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. 27 Thank You! Questions?