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Optimal Sampling-Based Motion Planning under Differential Constraints: the Drift Case with Linear Affine Dynamics.

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Area of Science:

  • Robotics
  • Control Theory
  • Computer Science

Background:

  • Drift control systems, characterized by momentum preventing instantaneous stops, pose challenges for optimal control.
  • Existing sampling-based algorithms often lack rigorous optimality guarantees.
  • Linear affine dynamical systems, including double-integrators, are crucial models for understanding complex control behaviors.

Purpose of the Study:

  • To establish a theoretical framework for assessing optimality guarantees of sampling-based algorithms in drift control systems.
  • To design and analyze a novel sampling-based algorithm with guaranteed asymptotic optimality.
  • To derive concrete bounds on the convergence rate of the proposed algorithm.

Main Methods:

  • Development of a rigorous theoretical framework for optimality assessment.
  • Design and analysis of the Differential Fast Marching Tree (DFMT) algorithm.
  • Application of perturbation analysis for two-point boundary value problems.

Main Results:

  • The proposed framework rigorously assesses optimality guarantees for sampling-based algorithms.
  • The DFMT algorithm is shown to be asymptotically optimal, converging to the optimal solution as sample count increases.
  • Concrete bounds on the convergence rate of the DFMT algorithm are provided.

Conclusions:

  • The theoretical framework enables provable correctness for sampling-based algorithms in drift control systems.
  • The DFMT algorithm offers a robust solution for optimal control of linear affine drift systems.
  • This work lays the foundation for extending provably correct sampling-based methods to non-linear drift control systems.