# Almost-global tracking for a rigid body with internal rotors

## TL;DR

This paper introduces a new PID control law for a rigid body with three internal rotors, enabling almost-global orientation tracking from nearly all initial states, addressing a less-studied problem in geometric control.

## Contribution

It presents the first geometric PID control law for internally actuated rigid bodies, expanding the scope of trajectory tracking control methods.

## Key findings

- Achieves almost-global tracking from nearly all initial conditions.
- Demonstrates effectiveness through theoretical analysis and simulations.
- Addresses a gap in geometric control for internally actuated systems.

## Abstract

Almost-global orientation trajectory tracking for a rigid body with external actuation has been well studied in the literature, and in the geometric setting as well. The tracking control law relies on the fact that a rigid body is a simple mechanical system (SMS) on the $3-$dimensional group of special orthogonal matrices. However, the problem of designing feedback control laws for tracking using internal actuation mechanisms, like rotors or control moment gyros, has received lesser attention from a geometric point of view. An internally actuated rigid body is not a simple mechanical system, and the phase-space here evolves on the level set of a momentum map. In this note, we propose a novel proportional integral derivative (PID) control law for a rigid body with $3$ internal rotors, that achieves tracking of feasible trajectories from almost all initial conditions.