The Cubli: Modeling Utilizing Quaternions

TL;DR

This paper models the Cubli, a 3D inverted pendulum with reaction wheels, using quaternions instead of Euler angles, enabling complex dynamic equations to be derived analytically and validated through simulations.

Contribution

It introduces a quaternion-based modeling approach for the Cubli, avoiding singularities and simplifying the derivation of dynamic equations.

Findings

Successful derivation of dynamic equations using quaternions

Validation through computer simulations and Poinsot trajectories

Enhanced modeling accuracy and simplicity

Abstract

This paper performs the modeling of a Cubli, a cube with three reaction wheels mounted on orthogonal faces that becomes a reaction wheel based 3D inverted pendulum when positioned in one of its vertices. The approach novelty is that quaternions are used instead of Euler angles. One nice advantage of quaternions, besides the usual arguments to avoid singularities and trigonometric functions, is that it allows working out quite complex dynamic equations completely by hand utilizing vector notation. Modeling is performed utilizing Lagrange equations and it is validated through computer simulations and Poinsot trajectories analysis.