### Abstract

Given an object with n points on its boundary where fingers can be placed, we give algorithms to select a "strong" grasp with the least number c of fingers (up to a logarithmic factor) using several measures of goodness. Along similar lines, given an integer c, we find the "best" κc log c finger grasp for a small constant κ. In addition, we generalize existing measures for the case of frictionless assemblies of many objects in contact. We also give an approximation scheme which guarantees a grasp quality close to the overall optimal value where fingers are not restricted to preselected points. These problems translate into a collection of convex set covering problems where we either minimize the cover size or maximize the scaling factor of an inscribed geometric object L. We present an algorithmic framework which handles these problems in a uniform way and give approximation algorithms for specific instances of L including convex polytopes and balls. The framework generalizes an algorithm for polytope covering and approximation by Clarkson [Cla] in two different ways. Let γ = 1/[d/2], where d is the dimension of the Euclidean space containing L. For both types of problems, when L is a polytope, we get the same expected time bounds (with a minor improvement), and for a ball, the expected running time is O(n
^{1+δ} + (nc)
^{1/(1+γ/(1+δ))} + c log(n/c)(c log c)
^{[d/2]} for fixed d, and arbitrary positive δ. We improve this bound if we allow in addition a different kind of approximation for the optimal radius. We also give bounds when d is not a constant.

Original language | English (US) |
---|---|

Pages (from-to) | 345-363 |

Number of pages | 19 |

Journal | Algorithmica (New York) |

Volume | 26 |

Issue number | 3-4 |

State | Published - 2000 |

### Fingerprint

### Keywords

- Approximate geometric algorithms
- Closure grasps
- Fixturing
- Grasp metrics
- Grasping
- Multifinger robot hands
- Polytope covering

### ASJC Scopus subject areas

- Computer Graphics and Computer-Aided Design
- Software
- Safety, Risk, Reliability and Quality
- Applied Mathematics

### Cite this

*Algorithmica (New York)*,

*26*(3-4), 345-363.

**Probabilistic algorithms for efficient grasping and fixturing.** / Teichmann, M.; Mishra, Bhubaneswar.

Research output: Contribution to journal › Article

*Algorithmica (New York)*, vol. 26, no. 3-4, pp. 345-363.

}

TY - JOUR

T1 - Probabilistic algorithms for efficient grasping and fixturing

AU - Teichmann, M.

AU - Mishra, Bhubaneswar

PY - 2000

Y1 - 2000

N2 - Given an object with n points on its boundary where fingers can be placed, we give algorithms to select a "strong" grasp with the least number c of fingers (up to a logarithmic factor) using several measures of goodness. Along similar lines, given an integer c, we find the "best" κc log c finger grasp for a small constant κ. In addition, we generalize existing measures for the case of frictionless assemblies of many objects in contact. We also give an approximation scheme which guarantees a grasp quality close to the overall optimal value where fingers are not restricted to preselected points. These problems translate into a collection of convex set covering problems where we either minimize the cover size or maximize the scaling factor of an inscribed geometric object L. We present an algorithmic framework which handles these problems in a uniform way and give approximation algorithms for specific instances of L including convex polytopes and balls. The framework generalizes an algorithm for polytope covering and approximation by Clarkson [Cla] in two different ways. Let γ = 1/[d/2], where d is the dimension of the Euclidean space containing L. For both types of problems, when L is a polytope, we get the same expected time bounds (with a minor improvement), and for a ball, the expected running time is O(n 1+δ + (nc) 1/(1+γ/(1+δ)) + c log(n/c)(c log c) [d/2] for fixed d, and arbitrary positive δ. We improve this bound if we allow in addition a different kind of approximation for the optimal radius. We also give bounds when d is not a constant.

AB - Given an object with n points on its boundary where fingers can be placed, we give algorithms to select a "strong" grasp with the least number c of fingers (up to a logarithmic factor) using several measures of goodness. Along similar lines, given an integer c, we find the "best" κc log c finger grasp for a small constant κ. In addition, we generalize existing measures for the case of frictionless assemblies of many objects in contact. We also give an approximation scheme which guarantees a grasp quality close to the overall optimal value where fingers are not restricted to preselected points. These problems translate into a collection of convex set covering problems where we either minimize the cover size or maximize the scaling factor of an inscribed geometric object L. We present an algorithmic framework which handles these problems in a uniform way and give approximation algorithms for specific instances of L including convex polytopes and balls. The framework generalizes an algorithm for polytope covering and approximation by Clarkson [Cla] in two different ways. Let γ = 1/[d/2], where d is the dimension of the Euclidean space containing L. For both types of problems, when L is a polytope, we get the same expected time bounds (with a minor improvement), and for a ball, the expected running time is O(n 1+δ + (nc) 1/(1+γ/(1+δ)) + c log(n/c)(c log c) [d/2] for fixed d, and arbitrary positive δ. We improve this bound if we allow in addition a different kind of approximation for the optimal radius. We also give bounds when d is not a constant.

KW - Approximate geometric algorithms

KW - Closure grasps

KW - Fixturing

KW - Grasp metrics

KW - Grasping

KW - Multifinger robot hands

KW - Polytope covering

UR - http://www.scopus.com/inward/record.url?scp=0344128114&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0344128114&partnerID=8YFLogxK

M3 - Article

VL - 26

SP - 345

EP - 363

JO - Algorithmica

JF - Algorithmica

SN - 0178-4617

IS - 3-4

ER -