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Polynomial Regression on Lie Groups and Application to SE(3).

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This study introduces a novel method for estimating mobile robot trajectories using Lie group theory. The technique accurately models drone movement by finding the best-fit geodesic path from noisy position data.

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Area of Science:

  • Robotics and Control Systems
  • Geometric Mechanics
  • Data Analysis

Background:

  • Estimating the precise trajectory of mobile robots, like drones, from sensor data is crucial for navigation and control.
  • Traditional methods can struggle with the complex, non-linear motion of rigid bodies in 3D space.
  • Lie group theory provides a mathematically robust framework for modeling such motions.

Purpose of the Study:

  • To implement geodesic regression on the Special Euclidean group SE(3) for accurate mobile robot pose estimation.
  • To adapt existing Riemannian manifold regression techniques to the SE(3) Lie group context.
  • To validate the proposed method using simulated trajectory data.

Main Methods:

  • Utilizing the Special Euclidean group SE(3) to model the 3D motion of rigid bodies.
  • Applying geodesic regression on Riemannian manifolds to fit noisy trajectory data.
  • Employing a Riemannian least squares approach to find the optimal geodesic path.
  • Implementing parametric frameworks for regression on manifolds.

Main Results:

  • Successfully demonstrated the application of geodesic regression in the SE(3) Lie group for trajectory estimation.
  • The method effectively fits noisy position measurements to determine the best geodesic path.
  • Simulated data applications illustrate the practical utility of the technique.

Conclusions:

  • The Lie group framework, specifically SE(3) with geodesic regression, offers a powerful approach for mobile robot pose estimation from noisy data.
  • This method provides a robust way to model complex trajectories.
  • Further research can explore limitations and expand to real-world applications.