Algorithms, Models and Metrics for the Design of Workholding Using Part Concavities K. Gopalakrishnan, Ken Goldberg, IEOR, U.C. Berkeley.

Review Unilateral Fixtures - Experiments Deformation Space Optimal Deform Closure Grasps Assembly Line Simulation Conclusion Outline

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

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

C-Space and Form Closure y x  /3 (5,4) y x q 4 5  /3 (5,4,- p/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.

2D v-grips Expanding. Contracting.

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

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

Examples

Review Unilateral Fixtures - Experiments Deformation Space Optimal Deform Closure Grasps Assembly Line Simulation Conclusion Outline

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

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

Experimental Apparatus A1 A2 A3

Orientation error (degrees) Jaw separation (inches) Experiment Results

Ford D219 Door model ++

A4C A1C A2C A3R A5R A6C A7C A8R A9R B1C B2C B3C B4R B5R D219 Door: Contact set

Review Unilateral Fixtures - Experiments Deformation Space Optimal Deform Closure Grasps Assembly Line Simulation Conclusion Outline

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

Topology violating configuration Undeformed partAllowed deformation Topology Preservation Example for for system of parts

D-free: Examples Slice with 1-4 fixed Part and mesh x y Slice with 1,2,4,5 fixed x 3 y 3

For FEM with linear elasticity and linear interpolation, P.E. = (1/2) X T K X D-Space and Potential Energy qAqA qBqB UTUT Increase in potential energy U A needed to release part. Deform Closure if U A > 0.

Frame invariance. Form-closure  Deform-closure of equivalent deformable part. Results

Numerical Example Joules547 Joules

D-Obstacle symmetry - Prismatic extrusion of identical shape along multiple axes. - Point obstacles are identical but displaced. Symmetry of Topology preserving space (D T ). - Superset: Non-degenerate meshes. Symmetry in D-Space

Review Unilateral Fixtures - Experiments Deformation Space Optimal Deform Closure Grasps Assembly Line Simulation Conclusion Outline

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

Consider: - Threshold P.E. U A. - Additional P.E. needed for plastic deformation U L. Q = min { U A, U L } Quality metric n0n0 n1n1 LL

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

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

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

A B C E F G A B C D E F D G H Contact Graph: Edges Traversal with minimum increase in energy. Adjacent mesh nodes: Non-adjacent mesh nodes: 

 U ( v(n i, n j ),  ) Peak P.E.: Given release path

Peak P.E.: All release paths  U ( v *,  )

 U ( v o,  ), U ( v*,  ) Threshold P.E.

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

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 (  )  

Calculate U L. To determine U A : Algorithm inspired by Dijkstra’s algorithm for sparse graphs. Fixed  i Insert pic of contact graph drawn on 2D P.E. graph

 V -  Algorithm for U A (  i )

 V -  Algorithm for U A (  i )

Numerical Example Undeformed  = 10 mm. Optimal   = 5.6 mm.

Review Unilateral Fixtures - Experiments Deformation Space Optimal Deform Closure Grasps Assembly Line Simulation Conclusion Outline

Review Unilateral Fixtures - Experiments Deformation Space Optimal Deform Closure Grasps Assembly Line Simulation Conclusion Outline

2D v-grips. 3D v-grips. 3D vg-grips. Unilateral Fixtures. D-Space and Deform-Closure. Optimal Deform-Closure grasps. Assembly line simulation. Topics completed

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.

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

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