Algorithms, Models and Metrics for Workholding using Part Concavities. K. Gopalakrishnan IEOR, U.C. Berkeley.

Workholding GraspingFixturing

Conventional Fixtures Bulky Complex Multilateral Dedicated, Expensive Long Lead time, Designed by human intuition Ideal Fixtures Compact Simplified Unilateral Modular, Amortizable Rapid Setup, Designed by CAD/CAM software

Inspiration GBL (Global Body Line) (Toyota, 1998-) –Multiple models. –Fewer Jigs/Fixtures.

Workholding: Basic concepts Immobility –Any part motion causes collision Force Closure –Any external Wrench resisted by applying suitable forces

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

Avoiding Collisions: C-obstacles Obstacles prevent parts from moving freely. Images in C-space are called C-obstacles. Rest is C free. Physical space C-Space x y

Workholding and C-space Multiple contacts. 1 Contact = 1 C-obstacle. C free = Collision with no obstacle. Surface of C-obstacle: Contact, not collision. Physical space C-Space x  y

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

First order Immobility Consider escape path. Distance to C-obstacles. Truncate to First order.

First order Immobility Physical space C-Space

First order Immobility In n dimensions there are n(n+1)/2 DOF: n translations n(n-1)/2 rotations For first order immobility, n(n+1)/2+1 are necessary and sufficient

Fast Test for First Order Immobility Any infinitesimal motion on the plane is a rotation. No center of rotation possible for a part in Form-Closure. Try to identify possible centers

Workholding: Rigid parts Number of contacts –[Reuleaux, 1876], [Somoff, 1900] –[Mishra, Schwarz, Sharir, 1987], [Markenscoff, 1990] Nguyen regions –[Nguyen, 1988] Form and Force Closure –[Rimon, Burdick, 1995] Immobilizing three finger grasps –[Ponce, Burdick, Rimon, 1995] [Mason, 2001]

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 –[Cheong, Goldberg, Overmars, van der Stappen, 2002] Contact force prediction –[Wang, Pelinescu, 2003]

2D v-grips Expanding. Contracting.

Algorithm Step1:We list all concave vertices. Step2:At these vertices, we draw normals to the edges through the jaw’s center. Step3:We label the 4 regions as shown: I II IV III Theorem: Both jaws lie strictly in the other’s Region I means it is an expanding v-grip or Both jaws lie in the other’s Region IV, at least one strictly, means it is a contracting v-grip

Maximum change in orientation occurs with one jaw at a vertex. The metric is given by |d  /dl|. Using sine rule and neglecting 2 nd order terms, |d  /dl| = |tan(  )/l| Ranking Grips

3D v-grips 3D v-grip: –Start from a stable initial orientation. –Close jaws monotonically. –Deterministic Quasi-static process. –Final configuration is a 3D v-grip if only vertical translation is possible. Input: A CAD model of the part and the position of its center of mass. Output: A list (possibly empty) of all 3D v-grips.

Phase I A candidate 2D v-grip occurs at end of phase I This is because a minimum height of COM occurs at minimum jaw distance

Phase II All configurations in Phase II are candidate 2D v-grips.

Gear & Shaft We assume that the gear is a cylinder (no teeth) This part is symmetric about the axis (one redundant degree of freedom). Search is thus reduced to 0 dimensions! to allow gripping.

Gear & Shaft: Solution Part OrientationShaft Trajectory

Example without Symmetry Orthogonal views:

Unilateral Fixtures “Unilateral” loading of body panels. Fixture lies on interior of assembled body. Reconfigurable fixtures.

Proposed Modular Components Use plane-cone contacts: –Jaws with conical grooves: Edge contacts. –Support Jaws with Surface Contacts.

Definition: Vg-grips Rigid approximation. is a vg-grip if: –Jaws engage part at v a, v b. –Achieves form closure. Not easy to check.

Notation: Coordinate Axes x: line joining verticesProjection perpendicular to x x

Sufficient Test Form-closure is achieved if: 1.2D v-grip in x-y plane. 2.2D v-grip in x-z plane (same nature as 1) 3.  q ij, i=a,b; j=1,2; penetrate cone (angle with axis less than half-cone angle) q ij r ij exex q ij = e x x r ij.

Proof: Outline Any displacement of part guarantees jaw displacement. Jaws are rigid. Thus Form-closure is achieved.

Quality Metric Maximum sensitivity 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 Mirror Dial Gauge

Experimental Apparatus A1 A2 A3

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.

Secondary Jaws

Grasp planning: Combining Geometric and Physical models - [Joukhadar, Bard, Laugier, 1994] Bounded force-closure -[Wakamatsu, Hirai, Iwata, 1996] Minimum Lifting Force - [Howard, Bekey, 1999] Holding Deformable Parts

Manipulation of flexible sheets -[Kavraki et al, 1998] Quasi-static path planning. - [Anshelevich et al, 2000] Robust manipulation - [Wada, Hirai, Mori, Kawamura, 2001]

Deformable parts “Form closure” does not apply: Can always avoid collisions by deforming the part.

Deformation Space: A Generalization of Configuration Space. Based on Finite Element Mesh. D-Space

Deformable Polygonal parts: Mesh Planar Part represented as Planar Mesh. Mesh = nodes + edges + Triangular elements. N nodes Polygonal boundary.

D-Space A Deformation: Position of each mesh node. D-space: Space of all mesh deformations. Each node has 2 DOF. D-Space: 2N-dimensional Euclidean Space. 30-dimensional D-space

Nominal mesh configuration Deformed mesh configuration Deformations Deformations (mesh configurations) specified as list of translational DOFs of each mesh node. Mesh rotation also represented by node displacements. Nominal mesh configuration (undeformed mesh): q 0. General mesh configuration: q. q0q0 q

D-Space: Example Simple example: 3-noded mesh, 2 fixed. D-Space: 2-dimensional Euclidean Space. D-Space shows moving node’s position. x y Physical space D-Space q0q0

Topological Constraints: D T x y Physical space D-Space Mesh topology maintained. Non-degenerate triangles only. DTDT

Topology violating deformation Undeformed part Allowed deformation Self Collisions and D T

D-Obstacles x y Physical space D-Space Collision of any mesh triangle with an object. Physical obstacle A i has an image DA i in D-Space. A1A1 DA 1

D-Space: Example Physical space x y D-Space D free = D T  [  (DA i C )]

Free Space: 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

Nodal displacement X = q - q 0 : vector of nodal translations. Equivalent to moving origin in D-Space to q 0. D- space q0q0 q

Potential Energy Linear Elasticity. 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

Potential Energy “Surface” U : D free  R  0 Equilibrium: q where U is at a local minimum. Stable Equilibrium: q where U is at a strict local minimum. Stable Equilibrium = “Deform Closure Grasp” q U(q)

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. q A qBqB q U(q) K( q A )

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 : Measure of “Deform Closure Grasp Quality” q A qBqB q U(q) UAUA Potential Energy Needed to Escape from a Stable Equilibrium K( q A )

Deform Closure Stable equilibrium = Deform Closure where U A > 0. qAqA qBqB q U(q)

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

There exists distance preserving linear transformation T such that q 2 = T q 1. It can be shown that U A2 (q 2 ) = U A1 (q 1 ) Theorem 1: Frame Invariance x y x y

Form-closure of rigid part Theorem 2: Form Closure and Deform Closure  Deform-closure of equivalent deformable part. 

Numerical Example 4 Joules547 Joules

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

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 prism. The shape of DA e is same for all elements. Computing DA i

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

Given: Pair of contact nodes. Determine: Optimal jaw separation. Optimal? Two Point Deform Closure Grasps M E n0n0 n1n1 

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

New Quality Metric Plastic deformation. Occurs when strain exceeds e L.

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

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

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

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

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

Contact Graph

Contact Graph: Edges Non-adjacent mesh nodes: 

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

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

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

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

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

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

Algorithm for U A (  i )

U Vertex v (traversed on path of minimum work) U(v) U(v*)

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

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

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. Summary

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