The Cubli: Modeling and Nonlinear Attitude Control Utilizing Quaternions

This paper covers the modeling and nonlinear attitude control of the 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 proposed approach utilizes quaternions instead of Euler angles as feedback control states. A 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. The derived nonlinear control law is based on feedback linearization technique, thus being time-invariant and equivalent to a linear one dynamically linearized at the given reference. Moreover, it is characterized by only three straightforward tuning parameters. Experimental results are presented.