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

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.