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

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.