Singularidades em rotações de corpos rígidos: uma comparação entre Ângulos de Euler e Quatérnions

Abstract

The determination of three-dimensional movements allows not only the prediction of objects trajectories, but also their attitude control. The rotational movement of rigid bodies, focus of this work, can be parameterized through means of several tools. The technique formulated by Leonhard Euler and that bears his name, is widely used to specify the attitude of objects in tridimensional space. However, for some occasions, Euler's Theorem of Rotations makes this parameterization unfeasible because it presents mathematical singularities that result in the loss of control of the body. As not all singular cases are due to physical factors, it is possible to deal with these cases taking hold of another mathematical artifice for analysis, known as Quaternions, and that despite being a less intuitive way, it does not present analytical singularities. In this work, we will present an exploration for both methods and consequent comparison, in order to find the forms of conversion from one tool to another, and how each one behaves when singularities arise. Finally, three situations with different rotations of a symmetric gyroscope will be simulated, being the first one an indeterminacy-free situation, another close to singular condition and one exactly where the sigularity happens, in order to signal the differences in the behaviour of Euler Angles and Quaternions.