# The Principal Fiber Bundle Structure of the Gimbal-Spacecraft System

## TL;DR

This paper models the gimbal-spacecraft system within a geometric mechanics framework, providing insights into its structure and potential control strategies, and serves as an accessible tutorial for applying differential geometry to aerospace systems.

## Contribution

It introduces a geometric mechanics formulation of the gimbal-spacecraft system, highlighting the principal fiber bundle structure and connection form for control design.

## Key findings

- Highlights the principal fiber bundle structure of the system
- Connects geometric mechanics tools to spacecraft attitude control
- Provides a tutorial introduction to differential geometry in this context

## Abstract

The gimbal-spacecraft system, that consists of a variable speed control moment gyro (VSCMG) mounted inside a spacecraft, has been employed as an actuator for the attitude control of a spacecraft and has been much studied in the aerospace control community. Employing a Newtonian approach, the equations of motion are derived, and further study focusses on singularity issues and control law synthesis. While the geometric mechanics community has studied many mechanical systems of engineering interest, including spinning rotors (or momentum wheels) that are used as actuators, there has not been a particular effort to model and control the gimbal-spacecraft system in a geometric framework. This article serves two purposes: it presents the gimbal-spacecraft system in a geometric mechanics framework, and in particular, highlights the connection form, that could form the basis for future control design, and secondly, the exposition is of a tutorial nature whereby the willing reader, with minimal prerequisites, is introduced to the tools of differential geometry in this context.