Gripping Parts at Concave Vertices K. “Gopal” Gopalakrishnan Ken Goldberg U.C. Berkeley

Inspiration Related Work 2D v-grips 3D v-grips Conclusion Outline

Inspiration

Part’s position and orientation are fixed.

Advantages Inexpensive Lightweight Small footprint Self-Aligning Multiple grips

Inspiration Related Work 2D v-grips 3D v-grips Conclusion Outline

Basics of Grasping Summaries of results in grasping –[Mason, 2001] –[Bicchi, Kumar, 2000] Rigorous definitions of Form and Force Closure –[Rimon, Burdick, 1996] –[Mason, 2001] [Mason, 2001]

Orders of Form-Closure First & second order form-closure –[Rimon, Burdick, 1995] For first order form-closure, n(n+1)/2+1 contacts are necessary and sufficient –[Realeaux, 1963] –[Somoff, 1900] –[Mishra, Schwarz, Sharir, 1987] –[Markenscoff, 1990]

Caging Grasps [Rimon, Blake, 1999] Efficient Computation of Nguyen regions [Van der Stappen, Wentink, Overmars, 1999] Multi-DOF Grips for Robotic Fixtureless Assembly [Plut, Bone, 1996 & 1997] Other Related Work

Inspiration Related Work 2D v-grips 3D v-grips Conclusion Outline

2D v-grips Expanding. Contracting.

2D Problem Definition We first analyze two-dimensional parts on the horizontal plane. Assumptions: Rigid Part. No out-of-plane rotation. Polygonal perimeter and Polygonal holes. Frictionless contacts. Zero Jaw radii.

2D Problem Definition Let v a and v b be two concave vertices. We call the unordered pair a v-grip if jaws placed at these vertices will provide frictionless form- closure of the part. vava vbvb

2D Problem Definition Input: Vertices of polygons representing the part’s boundary and/or holes, in counter-clockwise order, and jaw radius. Output: A list (possibly empty) of all v-grips sorted by quality measure.

2D 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

Conditions for V-grip Configurations like this are also contracting v-grips:

The Distance Function:  (s 1,s 2 ) Represents the distance between any 2 points on the part’s perimeter. The points are represented by an arclength parameter s. [Blake, Taylor, 1993] & [Rimon, Blake, 1998] O s

Proving the Theorem The proof lies in proving the equivalence of these 4 statements: For any pair of concave vertices, A: v 1 and v 2 both lie strictly in the other’s region I. B:  (v 1,s 2 ) and  (s 1,v 2 ) are local maxima at  (v 1,v 2 ). C:  (v 1,v 2 ) is a strict local maximum of  (s 1,s 2 ). D: The grasp at v1,v2 is an expanding v-grip. And similarly for contracting grasps (with region IV and minima)

Proof: Sketch A  B: The shortest distance from a point to a line is along the perpendicular. B  C: v2v2 v1v1 v' 1 v' 2 I P Q R v

Proof: Sketch (Contd.) C  D: Worst case analysis. – Any motion results in a collision. D  C: Assume form-closure but not C. – Case I: Contracting v-grip. – Form-Closure at non-extremum: Slide part along constant  contour. v1v1 v2v2

2D Algorithm Thus, If 2 vertices lie in region I of each other (A is true), an expanding v-grip is achieved (D is true). We enumerate all pairs of Concave vertices and apply theorems 1 and 2 for each pair to check for v-grips to generate an unranked list of v-grips.

Ranking Grips Based on sensitivity to small disturbances. Relax the jaws slightly. (Change the distance between them.) Consider maximum error in orientation due to this.

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

Metric evaluates grasp AC as better than BD Ranking Grips: Example D A C B

Computational Complexity O(n) to identify k concave vertices. O(k 2 ) to list v-grips and evaluate metrics. O(k 2 log k) to sort list. Total: O(n + k 2 log k) for 0 radius

Jaws with non-zero radii Jaw has a radius r The part is transformed with a Minkowsky addition, offsetting the polygons with a disk of radius r. Apply 2D algorithm to transformed part. O(n log n) time required.

Inspiration Related Work 2D v-grips 3D v-grips Conclusion Outline

3D v-grips Initial orientationFinal orientation after v-grip

3D v-grips Initial orientationFinal orientation after v-grip

3D Problem Definition In 3D, v-grips can be achieved with a pair of frictionless vertical cylinders and a planar work- surface. Assumptions: Rigid part Part is defined by a polyhedron. Frictionless contacts Jaws have zero radii.

3D Problem Definition 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.

3D Algorithm We describe a numerical algorithm for computing all 3D v-grips. The grasp occurs in 2 phases: –Rotation in plane –Rotation out of plane We find part trajectory during the second phase. We describe the algorithm for contracting 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.

3D Algorithm Enumerate starting positions. Identify 2D v-grips of projections. Compute Phase II trajectory: –Incrementally close jaws. –Find local minimum of COM height among candidate 2D v-grips. –Check termination criteria.

1.3D v-grip. 3D Algorithm: Termination. 3.The part falls away. All termination conditions checked in wrench-space. 2.3D equilibrium grip. Part can move but Gripper cannot close.

Example: Gear & Shaft Orthogonal views:

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

3D Example without Symmetry Orthogonal views:

3D Example Part Trajectory

Inspiration Related work 2D v-grips 3D v-grips Conclusion Outline

Conclusions: 2D Fast algorithm to find all 2D v-grips Quality Metric that is fast to compute and is consistent with intuition in most cases. Extended to non-zero jaw radii. Implemented in Java applet available online.

Conclusions: 3D 3D algorithm determines all 3D v-grips. The algorithm reduces a 6D search to a 1D search. Critical part parameters for Design for Mfg