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Lagrange-Poincaré-Kepler Equations of Disturbed Space-Manipulator Systems in Orbit

Borna Monazzah Moghaddam, Robin Chhabra

TL;DR

This work extends geometric reduction methods by embedding Keplerian orbital dynamics directly into Lagrange-Poincaré reductions for free-floating space manipulators. The LPKE framework combines Euler-Poincaré base dynamics, closed-form orbital evolution, and reduced manipulator dynamics on the shape space, expressed with exponential coordinates to avoid singularities. Key contributions include closed-form orbital forcing terms, a robust reduced mass metric, and mechanical connections that preserve momentum structure while enabling seamless integration with planning and control pipelines. Numerical validation against a high-fidelity benchmark demonstrates that LPKE Mode I achieves machine-precision agreement with substantially lower computation time than fully coupled alternatives, validating its suitability for real-time onboard planning and control in orbital environments.

Abstract

This article presents an extension of the Lagrange-Poincare Equations (LPE) to model the dynamics of spacecraft-manipulator systems operating within a non-inertial orbital reference frame. Building upon prior formulations of LPE for vehicle-manipulator systems, the proposed framework, termed the Lagrange-Poincare-Kepler Equations (LPKE), incorporates the coupling between spacecraft attitude dynamics, orbital motion, and manipulator kinematics. The formalism combines the Euler-Poincare equations for the base spacecraft, Keplerian orbital dynamics for the reference frame, and reduced Euler-Lagrange equations for the manipulator's shape space, using an exponential joint parametrization. Leveraging the Lagrange-d'Alembert principle on principal bundles, we derive novel closed-form structural matrices that explicitly capture the effects of orbital disturbances and their dynamic coupling with the manipulator system. The LPKE framework also systematically includes externally applied, symmetry-breaking wrenches, allowing for immediate integration into hardware-in-the-loop simulations and model-based control architectures for autonomous robotic operations in the orbital environment. To illustrate the effectiveness of the proposed model and its numerical superiority, we present a simulation study analyzing orbital effects on a 7-degree-of-freedom manipulator mounted on a spacecraft.

Lagrange-Poincaré-Kepler Equations of Disturbed Space-Manipulator Systems in Orbit

TL;DR

This work extends geometric reduction methods by embedding Keplerian orbital dynamics directly into Lagrange-Poincaré reductions for free-floating space manipulators. The LPKE framework combines Euler-Poincaré base dynamics, closed-form orbital evolution, and reduced manipulator dynamics on the shape space, expressed with exponential coordinates to avoid singularities. Key contributions include closed-form orbital forcing terms, a robust reduced mass metric, and mechanical connections that preserve momentum structure while enabling seamless integration with planning and control pipelines. Numerical validation against a high-fidelity benchmark demonstrates that LPKE Mode I achieves machine-precision agreement with substantially lower computation time than fully coupled alternatives, validating its suitability for real-time onboard planning and control in orbital environments.

Abstract

This article presents an extension of the Lagrange-Poincare Equations (LPE) to model the dynamics of spacecraft-manipulator systems operating within a non-inertial orbital reference frame. Building upon prior formulations of LPE for vehicle-manipulator systems, the proposed framework, termed the Lagrange-Poincare-Kepler Equations (LPKE), incorporates the coupling between spacecraft attitude dynamics, orbital motion, and manipulator kinematics. The formalism combines the Euler-Poincare equations for the base spacecraft, Keplerian orbital dynamics for the reference frame, and reduced Euler-Lagrange equations for the manipulator's shape space, using an exponential joint parametrization. Leveraging the Lagrange-d'Alembert principle on principal bundles, we derive novel closed-form structural matrices that explicitly capture the effects of orbital disturbances and their dynamic coupling with the manipulator system. The LPKE framework also systematically includes externally applied, symmetry-breaking wrenches, allowing for immediate integration into hardware-in-the-loop simulations and model-based control architectures for autonomous robotic operations in the orbital environment. To illustrate the effectiveness of the proposed model and its numerical superiority, we present a simulation study analyzing orbital effects on a 7-degree-of-freedom manipulator mounted on a spacecraft.
Paper Structure (17 sections, 9 theorems, 58 equations, 10 figures, 4 tables)

This paper contains 17 sections, 9 theorems, 58 equations, 10 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries of Relative Motion
  3. Kinematics of Spacecraft-Manipulator Systems in Orbit
  4. Orbital Motion
  5. Space Manipulator Kinematics
  6. Space Manipulator Dynamics Relative to Orbital Frame
  7. Numerical Validation
  8. Simulation Platform
  9. Simulation Results
  10. Computational Efficiency
  11. Discussion
  12. Conclusion
  13. Proof of Lemma \ref{['orbitdyn']}
  14. Proof of Lemma \ref{['orbitkinlemma']}
  15. Proof of Lemma \ref{['noninertial:lemmaPorbit']}
  16. ...and 2 more sections

Key Result

Lemma 1

(Evolution of the orbital parameter $\theta$) The True Anomaly $\theta$ of an undisturbed elliptic orbit in the perifocal frame with the eccentricity $e_\odot$ and constant orbital angular momentum $\mu_\odot$ as a function of time is found from the well-known Kepler's equations as curtis2013orbital where the Mean Anomaly function $\textbf{MA}(E):=E-e_\odot \text{sin}(E)$ is the fraction of orbita

Figures (10)

  • Figure 1: Space Manipulator in Orbit
  • Figure 2: Quasi-Inertial Frame $I$, Perifocal frame $\oplus$, and Orbital frame $\odot$
  • Figure 3: Simscape model and the arm frame assignment
  • Figure 4: 3D representation of the orbit and the quasi-inertial frame's paths around Earth
  • Figure 5: Path of the Orbital and Quasi-Inertial Frame and the relative position of orbit to the quasi-inertial frame
  • ...and 5 more figures

Theorems & Definitions (25)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof : Proof
  • Lemma 4
  • proof : Proof
  • Definition 1
  • Lemma 5
  • ...and 15 more