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Robust Path-following for Keplerian Orbits

Rodolfo Batista Negri, Antônio Fernando Bertachini de Almeida Prado

TL;DR

A robust path following law is derived that is able to achieve any conic section, even for problems that do not involve inverse squared distance forces, using the framework of the sliding mode control theory.

Abstract

This work introduces a novel path-following control strategy inspired by the famous two-body problem, aiming to stabilize any Keplerian orbit. Utilizing insights from the mathematical structure of the two-body problem, we derive a robust path-following law adopting sliding mode control theory to achieve asymptotic convergence to bounded disturbances. The resulting control law is demonstrated to be asymptotically stable. Illustrative examples showcase its applicability, including orbiting an accelerated moving point, patching Keplerian trajectories for complex patterns, and orbital maintenance around the asteroid Itokawa. The proposed control law offers a significant advantage for the orbital station-keeping problem, as its sliding surface is formulated based on variables commonly used to define orbital dynamics. This inherent alignment facilitates easy application to orbital station-keeping scenarios.

Robust Path-following for Keplerian Orbits

TL;DR

A robust path following law is derived that is able to achieve any conic section, even for problems that do not involve inverse squared distance forces, using the framework of the sliding mode control theory.

Abstract

This work introduces a novel path-following control strategy inspired by the famous two-body problem, aiming to stabilize any Keplerian orbit. Utilizing insights from the mathematical structure of the two-body problem, we derive a robust path-following law adopting sliding mode control theory to achieve asymptotic convergence to bounded disturbances. The resulting control law is demonstrated to be asymptotically stable. Illustrative examples showcase its applicability, including orbiting an accelerated moving point, patching Keplerian trajectories for complex patterns, and orbital maintenance around the asteroid Itokawa. The proposed control law offers a significant advantage for the orbital station-keeping problem, as its sliding surface is formulated based on variables commonly used to define orbital dynamics. This inherent alignment facilitates easy application to orbital station-keeping scenarios.

Paper Structure

This paper contains 9 sections, 6 theorems, 66 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Two-body Problem Fundamentals
  4. Main Results
  5. Illustrative Examples
  6. Moving Path-following Example
  7. Patched-hyperboles Example
  8. Orbital Maintenance About Itokawa
  9. Conclusions

Key Result

Proposition 3.1

A sliding surface $s \in \mathbb{R}$, written as a linear combination of the radial and transverse components of an arbitrary vector $\vec{A} \in \mathbb{R}^{3 }$: $\lambda > 0$, will asymptotically converge to $A_T = A_R = 0$, if $\dot{\vec{A}}=0$.

Figures (6)

  • Figure 1: Controlled orbit for the MPF example.
  • Figure 2: The control components for the MPF example.
  • Figure 3: Patched hyperboles example in the inertial frame.
  • Figure 4: The control components for the patched hyperboles example.
  • Figure 5: Itokawa example in the inertial frame.
  • ...and 1 more figures

Theorems & Definitions (16)

  • Proposition 3.1
  • proof
  • Remark 1
  • Theorem 1
  • proof
  • Corollary 1.1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 2
  • ...and 6 more