Rotation Representations and Their Conversions

A rigid body motion, which can be decomposed into rotation and translation, is essential for engineers and scientists who deal with moving systems in a space. While translation is as simple as vector addition, rotation is hard to understand because rotations are non-Euclidean, and there are many ways to represent them. Additionally, each representation comes with complex operations, and the conversions between different representations are not unique. Therefore, in this tutorial we review rotation representations which are widely used in industry and academia such as rotation matrices, Euler angles, rotation axis-angles, unit complex numbers, and unit quaternions. In particular, for better understanding we begin with rotations in a two dimensional space and extend them to a three dimensional space. In that context, we learn how to represent rotations in a two dimensional space with rotation angles and unit complex numbers, and extend them respectively to Euler angles and unit quaternions for rotations in a three dimensional space. The definitions and properties of mathematical entities used for representing rotations as well as the conversions between various rotation representations are summarized in tables for the reader’s later convenience.

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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.

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