1 Dissertation Workshop: Algorithms, Models and Metrics for the Design of Workholding Using Part Concavities K. Gopalakrishnan IEOR, U.C. Berkeley.

2 Review Unilateral Fixtures - Experiments Deformation Space Two Point Deform Closure Grasps Assembly Line Simulation Conclusion Outline

3 Bulky Complex Multilateral Dedicated, Expensive Long Lead time Designed by human intuition Conventional Fixtures

4 Modular Fixturing Existence and algorithm: Brost and Goldberg, 1996.

5 C-Space and Form Closure y x  /3 (5,4) y x q 4 5  /3 (5,4,-  /3) C-Space (Configuration Space): Describes position and orientation. Each degree of freedom of a part is a C-space axis. Form Closure occurs when all adjacent configurations represent collisions.

6 2D v-grips Expanding. Contracting.

7 N-2-1 approach Cai et al, Decoupling beam elements Shiu et al, Manipulation of sheet metal part Kavraki et al, Deformable parts

8 3D vg-grips Use plane-cone contacts: –Jaws with conical grooves: Edge contacts. –Support Jaws with Surface Contacts.

9 vava I II III IV 3D vg-grips: Phase I Fast geometric tests.

10 3D vg-grips: Phase II

11 Examples

12 Review Unilateral Fixtures Deformation Space Two Point Deform Closure Grasps Assembly Line Simulation Conclusion Outline

13 Ford Motor Co. ++

14 Ford D219 Door model Datum points. Spot welding access. Variation in tolerances. Multiple parts. Clamping mechanism.

15 Ford D219 Door model WELDING A4C A1C A2C A3R A5R A6C A7C A8R A9R B1C B2C B3C B4R B5R

16 Complete algorithm. BFS. Scale independent quality metric. New Experiments. Stay-in and stay-out regions (for datum points). Rigorous algorithm and clarification of concepts. Unilateral Fixtures: Improvements

17 Quality Metric Sensitivity of orientation to infinitesimal jaw relaxation. Maximum of R x, R y, R z. R y, R z : Approximated to v-grip. R x : Derived from grip of jaws by part. Jaw Part

18 Apparatus: Schematic Baseplate Track Slider Pitch- Screw Mirror Dial Gauge

19 Experimental Apparatus A1 A2 A3

20 Orientation error (degrees) Jaw relaxation (inches) Experiment Results "Unilateral Fixtures for Sheet Metal Parts with Holes" K. Gopalakrishnan, Ken Goldberg, Gary M. Bone, Matthew, Zaluzec, Rama Koganti, Rich Pearson, Patricia Deneszczuk. Accepted to the IEEE Transactions on Automation Sciences and Engineering.

21 Review Unilateral Fixtures - Experiments Deformation Space Two Point Deform Closure Grasps Assembly Line Simulation Conclusion Outline

22 Lack of definition of fixtures/grasps for deformable parts. Generalization of C-Space. Based on FEM model. D-Space

23 C-Space C-Space: describes position and orientation. Each DOF is a coordinate axis. y x  /3 (5,4) y x q 4 5  /3 (5,4,-  /3)

24 Obstacles Obstacles prevent parts from moving freely. Images in C- space are called C-obstacles. Rest is Free Space.

25 Mesh M Part E Deformable parts: FEM Part represented as Mesh. Stiffness properties assigned. F = K X. X = nodal displacement vector.

26 Topology violating configuration Undeformed partAllowed deformation Avoiding mesh collisions: D T Example for for system of parts

27 Avoiding collisions: D-obstacles No collision Collision No collision (with obstacle) Slice of complement of D-obstacle. Nodes 1,2,3 fixed.

28 Free Space: D free Slice with nodes 1-4 fixed Part and mesh x y Slice with nodes 1,2,4,5 fixed x 3 y 3

29 Nominal configuration Deformed configuration D-Space and Potential Energy Nodal displacement: Distance preserving transformation. X = T (q - q 0 ) q0q0 q For FEM with linear elasticity and linear interpolation, U = (1/2) X T K X

30 Deform Closure qAqA qBqB Equilibrium configuration: Local minimum of U. Increase in potential energy U A needed to release part. Deform Closure if U A > 0. q U(q)

31 Frame invariance. Form-closure  Deform-closure of equivalent deformable part. Theorems M E x1x1 y1y1 x1x1 y1y1 

32 Numerical Example 4 Joules547 Joules

33 D-Obstacle symmetry - Prismatic extrusion of identical shape along multiple axes. Symmetry of Topology preserving space (D T ). Symmetry in D-Space

34 Review Unilateral Fixtures - Experiments Deformation Space Two Point Deform Closure Grasps Assembly Line Simulation Conclusion Outline

35 Given: Deformable polygonal part. FEM model. Pair of contact nodes. Determine: Optimal jaw separation. Optimal? Problem Description M E n0n0 n1n1 

36 If Q = U A : Quality metric

37 Q = min { U A, U L } Stress Strain Plastic Deformation LL Quality metric

38 Given: Deformable polygonal part. FEM model. Pair of contact mesh nodes. Assume: Sufficiently dense mesh. Linear Elasticity. Problem Description M, K E n0n0 n1n1 

39 Points of interest: contact at mesh nodes. Construct a graph: Each graph vertex = 1 pair of perimeter mesh nodes. p perimeter mesh nodes. O(p 2 ) graph vertices. Contact Graph

40 A B C E F G D Contact Graph: Edges Adjacent mesh nodes: A B C D E F G H H

41 Potential Energy vs.  nini njnj k ij Potential Energy (U) Distance between FEM nodes Undeformed distance Expanding Contracting

42 Contact Graph REDO

43 Contact Graph: Edges Non-adjacent mesh nodes: 

44 Traversal with minimum increase in energy. FEM solution with two mesh nodes fixed. nini njnj Deformation at Points of Interest

45  U ( v(n i, n j ),  ) Peak Potential Energy Given release path

46 Peak Potential Energy: All release paths  U ( v *,  )

47  U ( v o,  ), U ( v*,  ) Threshold Potential Energy U ( v*,  ) U ( v o,  ) U A (  ) U A (  ) = U ( v*,  ) - U ( v o,  )

48  U A (  ), U L (  )  Quality Metric U A (  ) U L (  ) Q (  )

49 Possibly exponential number of pieces. Sample in intervals of . Error bound on max. Q =  * max {  0 (n i, n j ) * k ij } Numerical Sampling  Q (  )  

50 Calculate U L. To determine U A : Algorithm inspired by Dijkstra’s algorithm for sparse graphs. Fixed  i

51  V -  Algorithm for U A (  i ) List of known least-work nodes: . List of estimated least work for vertices adjacent to .

52  V -  Algorithm for U A (  i )

53 Numerical Example Undeformed  = 10 mm. Optimal   = 5.6 mm. Rubber foam. FEM performed using ANSYS. Computing Deform Closure Grasps, K. "Gopal" Gopalakrishnan and Ken Goldberg, submitted to Workshop on Algorithmic Foundations of Robotics (WAFR), Oct

54 Review Unilateral Fixtures - Experiments Deformation Space Two Point Deform Closure Grasps Assembly Line Simulation Conclusion Outline

55 Review Unilateral Fixtures - Experiments Deformation Space Two Point Deform Closure Grasps Assembly Line Simulation Conclusion Outline

56 2D v-grips: Fast necessary and sufficient algorithm. 3D v-grips: Fast path planning. Unilateral Fixtures: - Combination of fast geometric and numeric approaches. - Quality metric. Contributions

57 D-Space and Deform-Closure: - Defined workholding for deformable parts. - Frame invariance. - Symmetry in D-Space. Two Jaw Deform-Closure grasps: - Fast algorithm for given jaw separation. - Error bounded optimal separation. Assembly line simulation: Cost analysis for modular tooling. Contributions

58 Publications Computing Deform Closure Grasps K. "Gopal" Gopalakrishnan and Ken Goldberg, submitted to Workshop on Algorithmic Foundations of Robotics (WAFR), Oct D-Space and Deform Closure A Framework for Holding Deformable Parts, K. "Gopal" Gopalakrishnan and Ken Goldberg. IEEE International Conference on Robotics and Automation, May Unilateral Fixtures for Sheet Metal Parts with Holes K. "Gopal" Gopalakrishnan, Ken Goldberg, Gary M. Bone, Matthew Zaluzec, Rama Koganti, Rich Pearson, Patricia Deneszczuk, tentatively accepted for IEEE Transactions on Automation Science and Engineering. Revised version December “Unilateral” Fixturing of Sheet Metal Parts Using Modular Jaws with Plane- Cone Contacts K. “Gopal” Gopalakrishnan, Matthew Zaluzec, Rama Koganti, Patricia Deneszczuk and Ken Goldberg, IEEE International Conference on Robotics and Automation, September Gripping Parts at Concave Vertices K. "Gopal" Gopalakrishnan and K. Goldberg, IEEE International Conference on Robotics and Automation, May 2002.

59 Optimal node selection. Given a deformable part and FEM model. - Determine optimal position of a pair of jaws. - Optimal: Minimize deformation-based metric over all FEM nodes. Future work

60 1 “Unilateral” Fixturing of Sheet Metal Parts Using Modular Jaws with Plane-Cone Contacts, K. “Gopal” Gopalakrishnan, Matthew Zaluzec, Rama Koganti, Patricia Deneszczuk and Ken Goldberg, IEEE International Conference on Robotics and Automation (ICRA), Sep D-Space and Deform Closure: A Framework for Holding Deformable Parts, K. "Gopal" Gopalakrishnan and Ken Goldberg, IEEE International Conference on Robotics and Automation (ICRA), May Computing Deform Closure Grasps, K. "Gopal" Gopalakrishnan and Ken Goldberg, submitted to Workshop on Algorithmic Foundations of Robotics (WAFR), Oct JunJulAugSepOctNovDecJanFebMarAprMay Qualifying Exam Ford Research Laboratory: Designed fixture prototype. D-Space: Finalized definitions and derived initial results. Submitted ICRA '04 paper 2. ICRA '03 paper presented 1. Revised T-ASE paper 3 and performed new experiments. Optimizing deform closure grasps. Optimal node selection for deform-closure. Dissertation workshop. Write Thesis. Submitted WAFR’04 paper Revise WAFR ’04 paper. Ford Research Laboratory: Finish prototype and experiments with new modules and mating parts. D-Space: Formalize basic definitions. Submit ICRA '04 paper. Improve locator optimization algorithm Complete mating parts algorithm. Submit IROS’04 paper Locator strategy for multiple parts. Cutting planes/heuristics for MIP formulation. Proposed timeline (in May ’03) Current Timeline (in March ’04) Assembly line simulation for cost effectiveness. Timeline

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