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This study introduces a time-optimal scaling for constant linear velocity (CLV) spirals, significantly reducing trajectory tracking errors and image distortion in applications like medical imaging and robot exploration.

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Area of Science:

  • Robotics
  • Medical Imaging
  • Computational Geometry

Background:

  • The Archimedean spiral is a space-filling curve utilized in path planning and scan generation.
  • Constant linear velocity (CLV) parameterization offers fixed path velocity and isotropic sampling.
  • High accelerations near the origin of CLV spirals cause trajectory tracking issues.

Purpose of the Study:

  • To derive a time-optimal time scaling for CLV spirals to address singularity-induced tracking errors.
  • To improve the utility of CLV spirals in applications requiring precise trajectory following.

Main Methods:

  • Derivation of a closed-form time-optimal time scaling for CLV spirals.
  • Analysis of the CLV spiral's acceleration envelope to mitigate origin singularity.
  • Application and evaluation of the time scaling on a two-degree-of-freedom Cartesian scanner.

Main Results:

  • Trajectory tracking error reduced by up to 97.1% compared to naive CLV spirals.
  • Demonstrated low computational overhead for the proposed time scaling method.
  • Elimination of central image distortion near the origin in scanning applications.

Conclusions:

  • The derived time-optimal time scaling effectively mitigates CLV spiral singularity issues.
  • This method enhances the practical application of CLV spirals in robotics and medical imaging.
  • The approach offers significant improvements in accuracy and image quality with minimal computational cost.