Finding effective “force targets” for two-dimensional, multifinger frictional grips

In this paper we apply techniques from computational geometry to solve several problems in grasp planning and control in robotics. We consider the problem of calculating “force targets ” for a collection ofn fingers which grasp a two-dimensional object at known positions, at which the normals to the surface are also assumed to be known at least approximately. If the points at which the fingers touch the body do not allow apositive grip to be exerted (i.e., a grip in which the fingers hold the body in equilibrium by exerting friction-free forces in the directions of the corresponding inward-directed normals), it is appropriate to find the smallest coefficient of friction for which it is possible to assign a set of forces to be exerted by the fingers (so-calledfinger-force targets) which hold the object at equilibrium and such that each individual force lies within the corresponding cone of friction. We present an algorithm for this problem which runs in time0(n log2n log logn). We also present another algorithm for preprocessing the given data so as to allow fast computation of the desired coefficient of friction for the case in which one needs to balance any given “query” external force and torque. Finally, we discuss simpler variants of our techniques which are likely to be more efficient when the problem is solved for a small number of fingers.

[1]  Micha Sharir,et al.  Nonlinearity of davenport—Schinzel sequences and of generalized path compression schemes , 1986, FOCS.

[2]  Micha Sharir,et al.  On the Two-Dimensional Davenport Schinzel Problem , 2018, J. Symb. Comput..

[3]  Leonidas J. Guibas,et al.  Optimal Point Location in a Monotone Subdivision , 1986, SIAM J. Comput..

[4]  H. Hanafusa,et al.  Stable Prehension by a Robot Hand with Elastic Fingers , 1977 .

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

[6]  James Demmel,et al.  Theoretical and experimental studies using a multifinger planar manipulator , 1988, Proceedings. 1988 IEEE International Conference on Robotics and Automation.

[7]  Robert E. Tarjan,et al.  Planar Point Location Using Persistent Search Trees a , 1989 .

[8]  Richard Cole,et al.  A Parallel Median Algorithm , 1985, Inf. Process. Lett..

[9]  Richard Cokt.  Slowing Down Sorting Networks to Obtain Faster Sorting Algorithms , 1984 .

[10]  Nimrod Megiddo,et al.  Applying parallel computation algorithms in the design of serial algorithms , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[11]  M. Atallah.  Some dynamic computational geometry problems , 1985 .

[12]  János Komlós,et al.  Deterministic selection in O(loglog N) parallel time , 1986, STOC '86.

[13]  Nimrod Megiddo,et al.  Linear Programming in Linear Time When the Dimension Is Fixed , 1984, JACM.

[14]  E. Szemerédi,et al.  O(n LOG n) SORTING NETWORK. , 1983 .

[15]  David G. Kirkpatrick,et al.  Quantitative Steinitz's theorems with applications to multifingered grasping , 1990, STOC '90.

[16]  Van-Duc Nguyen,et al.  The synthesis of stable grasps in the plane , 1986, Proceedings. 1986 IEEE International Conference on Robotics and Automation.

[17]  Nimrod Megiddo,et al.  Linear-time algorithms for linear programming in R3 and related problems , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[18]  V.-D. Nguyen.  The Synthesis of Force Closure Grasps in the Plane. , 1985 .

[19]  Martin E. Dyer,et al.  Linear Time Algorithms for Two- and Three-Variable Linear Programs , 1984, SIAM J. Comput..

[20]  Leslie G. Valiant,et al.  Parallelism in Comparison Problems , 1975, SIAM J. Comput..

[21]  E. Szemerédi,et al.  Sorting inc logn parallel steps , 1983 .

[22]  Micha Sharir,et al.  Sharp upper and lower bounds on the length of general Davenport-Schinzel sequences , 2015, J. Comb. Theory, Ser. A.

[23]  J..  SOME DYNAMIC COMPUTATIONAL GEOMETRY PROBLEMS , 2009 .

[24]  Brenda S. Baker,et al.  Stable prehension with three fingers , 1985, STOC '85.

[25]  David G. Kirkpatrick,et al.  Optimal Search in Planar Subdivisions , 1983, SIAM J. Comput..

[26]  E. Galperin.  Linear time algorithms for linear programming , 1999 .