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Efficient Stabilization of Hybrid Coulomb Spacecraft Formations using Control Lyapunov Functions

Adam M Tahir

TL;DR

This work tackles stabilizing a multi-spacecraft formation equipped with both Coulomb actuation and thrusters. It introduces a control allocation framework based on a control Lyapunov function $V(\Xi)$, partitioning the CLF decrease between Coulomb inputs $qq^\top$ and thrusts $T$ via a design parameter $\eta$, and solving a sequence of tractable problems to compute $q^\star$ and $T^\star$. The method leverages the structure of $\dot{V}$ as a quadratic form in $q$ and affine in $T$, enabling a QCQP formulation; feasibility is guaranteed by the system’s over-actuated nature. Numerical results on a four-spacecraft formation show an $85\%$ reduction in propellant relative to thruster-only maneuvers, with careful tuning of $\eta$ and switching strategies further reducing fuel use while maintaining acceptable steady-state error. The approach offers a practical, decomposition-based way to realize stable, propellant-efficient formation control for high-dimensional HCSFs with potential applicability to mission planning and autonomous swarm operations.

Abstract

A control allocation algorithm using control Lyapunov functions to determine stabilizing charges and thrusts of hybrid Coulomb spacecraft formations (HCSFs) is presented. The goal is to stabilize a desired configuration while minimizing the thruster actuation and maximizing Coulomb actuation to minimize propellant usage. A proportion of the decrease of the control Lyapunov function is designated for Coulomb actuation and the rest is performed by thrusters. Simulations show that an 85% reduction of propellant compared to using solely thrusters is attainable using the proposed algorithm. It is shown that the best role for thrusters in a HCSF is to provide small corrections that cannot be provided by Coulomb actuation.

Efficient Stabilization of Hybrid Coulomb Spacecraft Formations using Control Lyapunov Functions

TL;DR

This work tackles stabilizing a multi-spacecraft formation equipped with both Coulomb actuation and thrusters. It introduces a control allocation framework based on a control Lyapunov function , partitioning the CLF decrease between Coulomb inputs and thrusts via a design parameter , and solving a sequence of tractable problems to compute and . The method leverages the structure of as a quadratic form in and affine in , enabling a QCQP formulation; feasibility is guaranteed by the system’s over-actuated nature. Numerical results on a four-spacecraft formation show an reduction in propellant relative to thruster-only maneuvers, with careful tuning of and switching strategies further reducing fuel use while maintaining acceptable steady-state error. The approach offers a practical, decomposition-based way to realize stable, propellant-efficient formation control for high-dimensional HCSFs with potential applicability to mission planning and autonomous swarm operations.

Abstract

A control allocation algorithm using control Lyapunov functions to determine stabilizing charges and thrusts of hybrid Coulomb spacecraft formations (HCSFs) is presented. The goal is to stabilize a desired configuration while minimizing the thruster actuation and maximizing Coulomb actuation to minimize propellant usage. A proportion of the decrease of the control Lyapunov function is designated for Coulomb actuation and the rest is performed by thrusters. Simulations show that an 85% reduction of propellant compared to using solely thrusters is attainable using the proposed algorithm. It is shown that the best role for thrusters in a HCSF is to provide small corrections that cannot be provided by Coulomb actuation.

Paper Structure

This paper contains 17 sections, 3 theorems, 40 equations, 8 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Problem Setup
  4. Formation Definition
  5. Relative Dynamics and Objective
  6. Control Lyapunov Function
  7. Useful Lemma
  8. Control Allocation Algorithm
  9. Overview of the Algorithm
  10. Finding $q^\star$
  11. Summary and Implementation
  12. Relationship between Algorithm 1 and Pointwise Optimal Allocation Approach
  13. The Parameter $\eta$
  14. Numerical Example
  15. Computational Burden
  16. ...and 2 more sections

Key Result

Lemma 1

Given a desired relative acceleration $u_\text{acc}^\text{des}\in\mathbb{R}^{d(\mathcal{N}-1)}$ and any charge $q^\star\in\mathbb{R}^\mathcal{N}$ and configuration $\tilde{\xi}\in\mathbb{R}^{d(\mathcal{N}-1)}$, there exists a thrust $T^\star\in\mathbb{R}^{d\mathcal{N}}$ such that

Figures (8)

  • Figure 1: Diagram of a HCSF with $\mathcal{N}=3$ in $\mathbb{R}^2$. The positions of each spacecraft are $x_1,x_2,x_3\in\mathbb{R}^2$, and their respective charges are $q_1,q_2,q_3\in\mathbb{R}$. Each spacecraft is depicted with a set of thrusters that can generate thrusts $T_1,T_2,T_3\in\mathbb{R}^2$.
  • Figure 2: Trajectory when $\eta =0.99$.
  • Figure 3: Charges when $\eta =0.99$.
  • Figure 4: Thrust magnitude when $\eta =0.99$.
  • Figure 5: Zoom in on a portion of the charges when $\eta=0.99$.
  • ...and 3 more figures

Theorems & Definitions (6)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition 1
  • proof