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This study defines an "ideal" interpolating planar motion for bounded objects by minimizing distances between positions. The resulting motion couples translation and rotation using sinusoidal functions, differing from standard linear interpolation.

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Area of Science:

  • Robotics
  • Computational Geometry
  • Motion Planning

Background:

  • Characterizing spatial separation between object positions is crucial for motion planning.
  • Existing methods often simplify object geometry or motion interpolation.
  • Defining appropriate metrics for bounded objects remains a challenge.

Purpose of the Study:

  • To develop a novel method for constructing interpolating planar motion for bounded objects.
  • To define an "ideal" motion, termed "motion sweep", by minimizing spatial distances.
  • To investigate the mathematical properties of this optimized motion.

Main Methods:

  • Utilized shape-dependent object norms to compute average distances between object positions.
  • Defined the "motion sweep" as the path minimizing the sum of average distances to end positions.
  • Derived the mathematical formulation for the resulting translational and rotational components.

Main Results:

  • The optimal interpolating motion ("motion sweep") is not a simple linear interpolation.
  • The translational and rotational components are coupled through sinusoidal functions.
  • This coupling differs from standard independent linear interpolation of translation and rotation.

Conclusions:

  • The proposed "motion sweep" offers a more accurate interpolating motion for bounded objects.
  • The findings challenge conventional approaches to motion interpolation in robotics.
  • This work provides a new perspective on defining optimal paths in geometric spaces.