Computing All Immobilizing Grasps of a Simple Polygon with Few Contacts

AbstractWe study the output-sensitive computation of all the combinations of edges and concave vertices of a simple polygon that allow an immobilizing grasp with less than four frictionless point contacts. More specifically, if n is the number of edges, and m is the number of concave vertices of the polygon, we show how to compute: in O(m4/3 log1/3 m + K) time, all K combinations that allow a form-closure grasp with two contacts; in O(n2 log4 m + K) time, all K combinations that allow a form-closure grasp with three contacts; in O(n log4 m + (nm)2/3 log2+ε m + K) time (for any constant ε > 0), all K combinations of one concave vertex and one edge that allow a grasp with one contact on the vertex and one contact on the interior of the edge, satisfying Czyzowicz's weaker conditions for immobilization; in O(n2 log3 n + K) time, all K combinations of three edges that allow a grasp with one contact on the interior of each edge, satisfying Czyzowicz's weaker conditions for immobilization.

[1]  Pankaj K. Agarwal,et al.  Geometric Range Searching and Its Relatives , 2007 .

[2]  Joel W. Burdick,et al.  Mobility of bodies in contact. I. A 2nd-order mobility index for multiple-finger grasps , 1994, IEEE Trans. Robotics Autom..

[3]  J. Burdick,et al.  Mobility of Bodies in Contact — Part II : How Forces are Generated by Curvature Effects , 1998 .

[4]  Mark H. Overmars,et al.  Computing Immobilizing Grasps of Polygonal Parts , 2000, Int. J. Robotics Res..

[5]  Kenneth Y. Goldberg,et al.  Gripping parts at concave vertices , 2002, Proceedings 2002 IEEE International Conference on Robotics and Automation (Cat. No.02CH37292).

[6]  Chandler Davis.  THEORY OF POSITIVE LINEAR DEPENDENCE. , 1954 .

[7]  Bernard Chazelle,et al.  Cutting hyperplanes for divide-and-conquer , 1993, Discret. Comput. Geom..

[8]  Mark de Berg,et al.  Computational geometry: algorithms and applications , 1997 .

[9]  Jirí Matousek,et al.  Range searching with efficient hierarchical cuttings , 1992, SCG '92.

[10]  Jean Ponce,et al.  On Computing Two-Finger Force-Closure Grasps of Curved 2D Objects , 1993, Int. J. Robotics Res..

[11]  Kenneth Y. Goldberg,et al.  A complete algorithm for designing planar fixtures using modular components , 1996, IEEE Trans. Robotics Autom..

[12]  Joel W. Burdick,et al.  New bounds on the number of frictionless fingers requied to immobilize , 1995, J. Field Robotics.

[13]  Pankaj K. Agarwal,et al.  Partitioning arrangements of lines II: Applications , 2011, Discret. Comput. Geom..

[14]  Forrest R. Jones,et al.  Kinematics of machinery , 2022 .

[15]  Joel W. Burdick,et al.  New bounds on the number of frictionless fingers required to immobilize 2D objects , 1995, Proceedings of 1995 IEEE International Conference on Robotics and Automation.

[16]  Joel W. Burdick,et al.  Finding antipodal point grasps on irregularly shaped objects , 1992, IEEE Trans. Robotics Autom..

[17]  A. Frank van der Stappen,et al.  Output-Sensitive Computation of All Form-Closure Grasps of a Semi-Algebraic Set , 2005, Proceedings of the 2005 IEEE International Conference on Robotics and Automation.

[18]  Leonidas J. Guibas,et al.  The power of geometric duality , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[19]  Christos H. Papadimitriou,et al.  The Geometry of Grasping , 1990, Int. J. Robotics Res..

[20]  A. J. Goldman,et al.  Polyhedral Convex Cones , 1956 .

[21]  Van-Duc Nguyen,et al.  Constructing force-closure grasps , 1986, Proceedings. 1986 IEEE International Conference on Robotics and Automation.

[22]  B. Faverjon,et al.  On computing three-finger force-closure grasps of polygonal objects , 1991 .

[23]  V. Torczon,et al.  RANK ORDERING AND POSITIVE BASES IN PATTERN SEARCH ALGORITHMS , 1996 .

[24]  Kenneth Y. Goldberg,et al.  Friction and part curvature in parallel- jaw grasping , 1995, J. Field Robotics.

[25]  Hong Liu,et al.  A new algorithm for three-finger force-closure grasp of polygonal objects , 2003, 2003 IEEE International Conference on Robotics and Automation (Cat. No.03CH37422).

[26]  Micha Sharir,et al.  On the existence and synthesis of multifinger positive grips , 2015, Algorithmica.

[27]  J. Kenneth Salisbury,et al.  Articulated Hands , 1982 .

[28]  Ivan Stojmenovic,et al.  Immobilizing a Shape , 1999, Int. J. Comput. Geom. Appl..

[29]  Joel W. Burdick,et al.  Mobility of Bodies in Contact{ii: How Forces Are Generated by Curvature Eeects? , 1994 .

[30]  Jr. J. Kenneth Salisbury,et al.  Kinematic and force analysis of articulated hands , 1982 .

[31]  Yan-Bin Jia.  Curvature-based computation of antipodal grasps , 2002, Proceedings 2002 IEEE International Conference on Robotics and Automation (Cat. No.02CH37292).