The Cubli: Modeling and Nonlinear Control Utilizing Unit Complex Numbers

TL;DR

This paper introduces a novel modeling and nonlinear control approach for the Cubli using unit complex numbers and quaternions, simplifying the control design and enabling effective stabilization of the inverted pendulum in 1D and 3D configurations.

Contribution

It presents a new control framework using complex numbers and quaternions for the Cubli, reducing complexity and improving understanding over traditional methods.

Findings

Control law is equivalent to a linear controller with three tuning parameters.

Experimental validation confirms the effectiveness of the proposed control approach.

Modeling using complex numbers simplifies the nonlinear control design.

Abstract

This paper covers the modeling and nonlinear control of the Cubli, a cube with three reaction wheels mounted on orthogonal faces that becomes a reaction wheel-based 1D/3D inverted pendulum when positioned in one of its edges (1D) or vertices (3D). Instead of angles, unit complex numbers are used as control states for the 1D configuration. This approach is useful not only to get rid of trigonometric functions, but mainly because it is a specific case of the 3D configuration, that utilizes unit ultra-complex numbers (quaternions) as system states, and therefore facilitates its understanding. The derived nonlinear control law is equivalent to a linear one and is characterized by only three straightforward tuning parameters. Experiment results are presented to validate modeling and control.