1 Dissertation Workshop. Algorithms, Models and Metrics for the Design of Workholding using Part Concavities. K. Gopalakrishnan IEOR, U.C. Berkeley. Thesis Committee: Prof. Sheldon Ross (Chair) Prof. Alper Atamturk Prof. Ken Goldberg Prof. Paul Wright

2 Workholding GraspingFixturing

3 Review Unilateral Fixtures - Experiments Deformation Space Two Point Deform Closure Grasps Conclusion Outline

4 Inspiration. Related Work. C-Space. 2D v-grips. 3D vg-grips and unilateral fixtures. Review

5 Workholding: Rigid parts Contact Mechanics: Number of contacts –[Reuleaux, 1876], [Somoff, 1900] –[Mishra, Schwarz, Sharir, 1987], –[Nguyen, 1988] –[Markenscoff, Papadimitriou, 1990] –[Han, Trinkle, Li, 1999] Immobility, 2 nd Order Form Closure –[Rimon, Burdick, 1995] –[Ponce, Burdick, Rimon, 1995] [Mason, 2001]

6 Workholding: Rigid parts Caging Grasps –[Rimon, Blake, 1999] Summaries of results –[Bicchi, Kumar, 2000] –[Mason, 2001] C-Spaces for closed chains –[Milgram, Trinkle, 2002] Fixturing hinged parts –[van der Stappen et al, 2002] Antipodal Points for Curved Parts –[Jia 2002]

7 Bulky Complex Multilateral Dedicated, Expensive Long Lead time Designed by human intuition Conventional Fixtures [Toyota GBL, 2003]

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

9 C-Space and Form Closure C-Space (Configuration Space): [Perez, 1983] 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.

10 2D v-grips ExpandingContracting Contact at concavities. 2 points necessary and sufficient. Maxima/minima of distance.

11 vava I II III IV 2D v-grips: Geometric test Gripping Parts at Concave Vertices, K. "Gopal" Gopalakrishnan and K. Goldberg, IEEE International Conference on Robotics and Automation, May 2002.

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

13 3D vg-grips Use plane-cone contacts at concavities: –Jaws with conical grooves: Edge contacts. –Support Jaws with Surface Contacts.

14 3D vg-grips: Phase I Fast geometric tests.

15 3D vg-grips: Phase II

16 Examples

17 Review Unilateral Fixtures Deformation Space Two Point Deform Closure Grasps Conclusion Outline

18 Ford Motor Co. June-August Matthew Zaluzec, Rama Koganti. Scientific Research Laboratory. Vehicle Operations. Advanced Engineering Center.

19 Ford D219 Door model Ford Escape. Design in progress. Front right hand door. Multiple parts.

20 Ford D219 Door model + Door Inner Reinforcement Spot-welded Assembly

21 Ford D219 Door Assembly IGES to i-Deas conversion. FEM in i-Deas. 6.4mm dia welds. 0.3mm tolerances at welds. 1.5mm tolerance elsewhere. 2.2kN weld forces. 0.4kN clamping forces.

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

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

24 Tree of fixtures. Children have one more contact than parents. DFS by greedily choosing children. Fixture Tree and DFS...

25 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

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

27 Experimental Apparatus A1 A2 A3 A1-A A1-A

28 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 in March 2004 to the IEEE Transactions on Automation Sciences and Engineering.

29 Review Unilateral Fixtures - Experiments Deformation Space Two Point Deform Closure Grasps Conclusion Outline

30 Lack of definition of fixtures/grasps for deformable parts. Generalization of C-Space. D-Space

31 Related Work, C-Space D-Space: a “C-Space”-like framework for holding deformable parts. D free = D T  [  (DA i C )] Potential Energy Deform Closure Thm 1: Frame Invariance Thm 2: Form-Closure Equivalence Symmetry in D free. Outline

32 C-Space C-Space (Configuration Space): [Lozano-Perez, 1983] Dual representation Each degree of part freedom is one C-space dimension. y x  /3 (5,4) y x  (5,4,-  /3) Physical space C-Space

33 Avoiding Collisions: C-obstacles Blue part collides with point obstacle A at a set of configurations. In C-space, the set is the C-obstacle, CA. In C-space, the part shrinks to a point and the obstacle grows accordlingly CA c = C free. Physical space C-Space x y

34 C-space applied to Workholding “Finger bodies” are obstacles. For planar rigid part, C-space has 3 dimensions. Surfaces of C-obstacles. Physical space C-Space x  y

35 Form Closure A part is grasped in Form Closure if any infinitesimal motion results in collision. Form Closure is equivalent to an isolated point in C-free. (Force Closure = ability to resist any wrench). Physical space C-Space x y

36 Elastic Fingers, Soft Contacts [Hanafusa and Asada, 1982] [Salisbury and Mason, 1985] [Kim, Hirai, Inoue, 2003] Physical Models [Joukhadar, Bard, Laugier, 1994] Bounded Force Closure [Wakamatsu, Hirai, Iwata, 1996] Learned Grasps of Deformable Objects [Howard, Bekey, 1999] Holding Deformable Parts

37 Holding Deformable Parts Path Planning for Elastic Sheets and Bodies [Kosuge et al, 1995] [Kavraki et al, 1998, 2000] Robot manipulation [Henrich and Worn, 2000] Robust manipulation with Vision [Hirai, Tsuboi, Wada 2001] Modified spring-mass modeling [Yasuda, Yokoi, 2001]

38 Holding sheet metal parts with dexterous parameter estimation [Ceglarek et al, 2002] Haptic rendering of deformations [Laycock, Day, 2003] Simulation of human hand deformation [Latombe et al, 2003] Gripper design: reducing deformation [Causey, 2003] Holding Deformable Parts

39 Deformable parts “Form closure” insufficient: The part can always escape if deformed.

40 Part Model Planar Polygonal Boundary Triangular Mesh Nodes (like hinges) Edges (like struts) Elements (deformable)

41 Deformation Space (D-Space) Each node has 2 DOF Analogous to configurations in C-Space D-Space: 2N-dimensional space of mesh node positions. point q in D-Space is a “deformation” q 0 is initial (undeformed) point 30-dimensional D-space

42 D-Space: Example Example part: 3-node mesh, 2 fixed. D-Space: 2D x y Physical space D-Space q0q0

43 Topology violating deformation Undeformed part Allowed deformation Self Collisions

44 D T : Topology Preserving Subspace x y Physical space D-Space D T  D-Space. DTDT DTC:DTC:

45 D-Obstacles x y Physical space D-Space Collision of any mesh element with obstacle. A Physical obstacle A i defines a deformation- obstacle DA i in D-Space. A1A1 DA 1

46 D-Space: Example Physical space x y D-Space Like C free, we define D free. D free = D T  [  (DA i C )]

47 With more nodes: D free Slice with only node 5 moving. Part and mesh x y Slice with only node 3 moving. x 3 y 3 x 5 y 5 x 5 y 5 x 5 y 5 Physical space D-Space D free = D T  [  (DA i C )]

48 Displacement between a pair of deformations q 0 : part’s nominal shape. X = q - q 0 : vector of nodal translations. Equivalent to moving origin in D-Space to q 0 and aligning nodal reference frames. q0q0 q X

49 Potential Energy Assume Linear Elasticity, Zero Friction K = FEM stiffness matrix. (2N  2N matrix for N nodes.) Forces at nodes: F = K X. Potential Energy: U(q) = (1/2) X T K X

50 Potential Energy “Surface” U : D free  R  0 Equilibrium: q where U is at a local minimum. In absence of friction or inertia, part will come to rest at an equilibrium. Stable Equilibrium: q where U is at a strict local minimum. = “Deform Closure Grasp” q U(q)

51 Potential Energy Needed to Escape from a Stable Equilibrium Consider: Stable equilibrium q A, Equilibrium q B. “Capture Region”: K(q A )  D free, such that any configuration in K(q A ) returns to q A. Saddlepoints [Rimon, Blake, 1995] q A qBqB q U(q) K( q A )

52 U A (q A ) = Increase in Potential Energy needed to escape from q A. = minimum external work needed to escape from q A. U A : Quality Measure q A qBqB q U(q) UAUA Potential Energy Needed to Escape from a Stable Equilibrium K( q A )

53 Quality Measure Example U A = 4 JoulesU A = 547 Joules

54 Definition of Deform closure is frame-invariant. Proof: Consider D-spaces D 1 and D 2. - Consider q 1  D 1, q 2  D 2. such that physical meshes are identical. Theorem 1: Frame Invariance x y x y D1:D1: D2:D2:

55 There exists distance preserving linear transformation T such that q 2 = T q 1. and U A2 (q 2 ) = U A1 (q 1 ) U A does not depend on frame. Theorem 1: Frame Invariance x y x y D1:D1: D2:D2:

56 Form-closure of rigid part Theorem 2: Form Closure and Deform Closure  Deform-closure of equivalent deformable part.  If in form-closure, strict local minimum: work needed to escape. If in deform-closure, and no deformation allowed: form closure.

57 High Dimensional. Computing D T and DA i. Exploit symmetry. Computing D free DA i D free DTCDTC

58 Consider obstacle A and one triangular element. Consider the slice D e of D, corresponding to the 6 DOF of this element. Along all other axes of D, D e is constant. Extruded cross-section is a 6D prism. The shape of DA e is same for all elements. Computing DA i

59 Thus, DA is the union of identical prisms with orthogonal axes. Center of DA is the deformation where the part has been shrunk to a point inside A. Similar approach for D T. Computing DA i 

60 Review Unilateral Fixtures - Experiments Deformation Space Two Point Deform Closure Grasps Conclusion Outline

61 Description. Quality metrics. Points of Interest. Contact Graph. Peak Potential Energy. Algorithm. Examples. Two-Point Deform Closure Grasps

62 Given: Pair of contact nodes. Determine: Optimal jaw separation. Optimal? Problem Description M E n0n0 n1n1 

63 Naïve Quality metric If Quality metric Q = U A. Maximum U A trivially at   = 0

64 Quality metric Plastic deformation. Occurs when strain exceeds e L.

65 New Quality Metric Additional work U L done by jaws for plastic deformation. New Q = min { U A, U L } Maximize min { U A, U L } Stress Strain Plastic Deformation A B C eLeL A B C

66 Additional input: e L : Elastic limit strain.  : allowed error in quality metric. Additional assumptions: Sufficiently dense mesh. Linear Elasticity. No collisions. No Friction. Monotonicity of U. Problem Description M, K E n0n0 n1n1 

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

68 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

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

70 Contact Graph Potential Energy Jaw 1 position Jaw 2 position

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

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

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

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

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

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

77 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 (  )  

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

79 Algorithm for U A (  i )

80 Algorithm for U A (  i )

81 U Vertex v (traversed on path of minimum work) U(v) U(v*) Estimates for maximum U

82 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

83 Review Unilateral Fixtures - Experiments Deformation Space Two Point Deform Closure Grasps Conclusion Outline

84 2D v-grips - Grasping at concavities. - New Quality metric. - Fast necessary and sufficient conditions. 3D v-grips: - Gripping at projection concavities. - Fast path planning. Contributions

85 Unilateral Fixtures: - New type of fixture: concavities at concavities - New Quality metric. - Combination of fast geometric and numeric approaches. D-Space and Deform-Closure: - Defined workholding for deformable parts. - Frame invariance. - Symmetry in D-Space. Contributions

86 Two Jaw Deform-Closure grasps: - Quality metric. - Fast algorithm for given jaw separation. - Error bounded optimal separation. Contributions

87 Publications Computing Deform Closure Grasps K. "Gopal" Gopalakrishnan and Ken Goldberg, accepted 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, 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.

88 Optimal node selection for two- point deform closure grasps. Convergence with denser meshes. Deform closure fixturing using internal nodes. Future work

89 2D v-grips. 3D v-grips. 3D vg-grips and unilateral fixtures. D-Space. Deform closure. Optimal two-jaw deform closure grasps. Summary