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A MRI-Based Toolbox for Neurosurgical Planning in Nonhuman Primates
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Reduction of placement problems using Minkowski decomposition.

L Dorst1, N Ahmed

  • 1Informatics Inst., Univ. of Amsterdam, Netherlands.

IEEE Transactions on Systems, Man, and Cybernetics. Part B, Cybernetics : a Publication of the IEEE Systems, Man, and Cybernetics Society
|February 2, 2008
PubMed
Summary
This summary is machine-generated.

This study explores the mathematical structure of collision-free object placement, introducing Minkowski decomposition to create intermediate reformulations. This offers a foundation for approximate placement methods and consistent rotation discretization, especially for simple shapes.

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Area of Science:

  • Robotics
  • Computational Geometry
  • Mathematical Optimization

Background:

  • Collision-free placement is crucial in robotics and automated systems.
  • Existing solutions include task space and configuration space approaches.
  • These methods often lack a unified mathematical foundation for approximations.

Purpose of the Study:

  • To analyze the mathematical structure of the object placement problem.
  • To establish a theoretical basis for approximate collision-free placement algorithms.
  • To provide a consistent method for discretizing rotations in placement problems.

Main Methods:

  • Utilizing Minkowski decomposition of the object.
  • Developing a hierarchy of intermediate reformulations of the placement problem.
  • Analyzing the mathematical properties of these reformulations.

Main Results:

  • Minkowski decomposition provides a structured hierarchy for placement problems.
  • This decomposition forms the mathematical basis for common approximate solution methods.
  • A consistent recipe for discretizing rotations is derived.

Conclusions:

  • The study offers a novel mathematical framework for collision-free object placement.
  • Minkowski decomposition unifies and explains existing approximate placement techniques.
  • The findings facilitate more efficient and consistent placement strategies, particularly for simple geometries.