Curvature-Based Computation of Antipodal Grasps

1. Curvature-Based Computation of Antipodal Grasps Yan-Bin Jia Department of Computer Science Iowa State University Ames, IA 50011-1040

2. Antipodal Grasp

3. s* t* N(s*) N(t*) Antipodal Points Curve:closed, simple, and twice continuously differentiable. Two pointss*andt*onareantipodal if their inward normals are collinear and pointing at each other. N(s*) + N(t*) = 0 N(s*)  ((t*) – (s*)) = 0 N(s*)  ((t*) – (s*)) > 0

4. Antipodal Grasp & Points Hong et al. 1990; Chen & Burdick 1992; Ponce et al. 1993; Blake & Taylor 1993 Grasping & Fixturing Salisbury & Roth 1983; Mishra et al. 1987; Nguyen 1988; Markenscoff et al. 1992; Trinkle 1992;Brost & Goldberg 1994; Bicchi 1995 Curve Processing Goodman 1991; Manocha & Canny 1992; Sakai 1999; Jia 2001 Computational Geometry Preparata & Hong 1977; Yao 1982; Chazelle et al. 1993; Matousek & Schwarzkopf 1996; Ramos 1997; Bespamyatnikh 1998 Previous Work

5.  Incomplete  (local methods) do not guarantee to find any if exist.  do not guarantee to find all that exist.  Inefficient  many guesses of antipodal positions are needed. Problems with Existing Methods geometric formulation nonlinear programming + heuristics Why not exploit geometry (global & differential)?

6. t In the neighborhoods of antipodal points define opposite points by N(t ) N(s) + N(t) = 0 N(s) : antipodal angle s Antipodal Angle s and t are antipodal if and only if (s) = 0. t* r(s) s* simple antipodal pointss* and t* if (s*) = 0 but (s*)  0.

7. s* t* osculating circle t* s* (k–1) (k) kth order if (s*) = … =  (s*) = 0 but  (s*)  0. Higher Order Antipodal Points 2nd order antipodal points if (s*) = (s*) = 0 but (s*)  0. 1/(s*) + 1/(t*) equals the distance between s* and t*. curvature at s* radius of curvature at s* center of curvature We are primarily interested in finding simple antipodal points.

8. t s s b a s T(s ) T(s ) T(s ) T(s ) a a b b t t a b The Case of Two Segments T Assumptions for clarity (removable): i) No intersection ii) Curvature everywhere positive (convex) or everywhere negative (concave) iii) Opposite normals at corresponding endpoints (which are not antipodal) S iv) Total curvatures  [–, ] (and with equal absolute value) total curvature v) Normals at every pair of opposite points (s, t) pointing towards each other

9.  uniquepair of antipodal points if (s )< 0 and (s )> 0. a b s s b a (s ) (s ) b a s* Bisection over [s , s ]. a b t* t t a b Concave – Concave antipodal angleincreases monotonicallyas s increases.  no antipodal points otherwise. S T

10. Case 1: (s )and (s )have the same sign. a b t t 1 2 t = 0 0 1 t t b a monotonic sequences defined by s ,s , … N(s )  r(s ) = 0 i+1i t , t , … 0 1 N(t ) = – N(s ) i+1i+1 s s a a s = s 0 2 s 1 Convex – Convex : Case 1 t* Marching  If antipodal points exist, the procedure will converge to the closest pair at linear rate. s*  If no antipodal points exist, the procedure will terminate at the other endpoints.

11. t* t t a b s s b a s* Geometry at Antipodal Points (1) curvatures at the antipodal points (s*), (t*) > 0. 1/(t*) 1/(s*) + 1/(t*) < ||r(s*)|| center of curvature r(s*) osculating circle Geometrically, the two centers of curvature do not cross each other on r(s*).

12. Case 2: (s )and (s )have different signs. a b s* 1/(s*) + 1/(t*) < ||r(s*)|| or 1/(s*) + 1/(t*) > ||r(s*)|| t* Convex – Convex : Case 2  At least one pair of antipodal points exists. Bisection finds the first pair.

13. s* 2 t* s* 1 1 Recursion tree: s* , t* 1 1 t* Case 1 (march) 2 march s* , t* bisection Case 2 (bisection) 2 2 Case 1 Case 1 Recursively Finding All pairs

14. s s t t s s a a b b a a t t b b T ray of N(t ) ray of N(t ) S b a a b Convex – Concave Segments Suppose segment S is convex and segment T is concave.  No antipodal points exist if neither the ray of N(t ) nor the ray of N(t ) intersects segment S.

15. t t a b s s a b monotonic sequences: s ,s , … 0 1 t , t , … t 0 1 t = t* 1 0 t 2 s* s 2 s 1 s = 0 Marching - the ray of N(t ) or N(t ) intersects segment S -the two endpoint antipodal angles have the same sign. a b  When antipodal points exist, the procedure converges to the closest pair at linear rate.  If no antipodal points exist, the procedure will terminate at the other endpoints.

16. Geometry at Antipodal Points (2) 1/(s*) +1/(t*) > ||r(s*)|| 1/(s*) where (s*) > 0 and (t*) < 0 t* -1/(t*) t* and its center of curvature lie between s* and its center of curvature. s* Bisection is used when the antipodal angles have different signs at the two endpoints of segment S. Interleave marching with bisection recursively to find all pairs of antipodal points on S and T.

17. simple Inflection:  = 0 but   0 Preprocessing – Inflection Points How to generate segment pairs that satisfy conditions i)-v)?  > 0  > 0  < 0  < 0 Step 1Compute all simple inflection points. Step 2Split every segment whose total curvature  [–, ].

18. s s b a t b t a Preprocessing – Common Tangents a pair of normals pointing away from each other Step 3 Intersect normal cones Step 4Compute common tangent lines (quadratic convergence)

19. limacon  = 4 + 2.5 cos elliptic lemniscate 2 2 2 2 1/2  = (6 cos + 3 sin ) Experiments closed cubic spline (10 pairs)  = 3/(1+0.5 cos(3))

20. 2  Efficiency – O(n + m) two-level calls of numerical primitives Conclusion  New insights into differential geometry of antipodal points.  Completeness – finding all antipodal points  Dissecting the curve into monotone segments (preprocessing of global geometry)  Interleaving marching with numerical bisection (based on local/differential geometry)  Provable convergence rate (independent of parameterization) (about 300 iterations per call) #inflections #antipodals

21. Future Work  Computation of special substructures on curves  Optimization over parameterized curves  Extension of the scheme to finding antipodal points on 3-D surfaces