Optimization and Gimbal Lock Control via Shifted Decomposition of Rotations

The present paper studies the effects of introducing a fourth rotation in the generalized Euler factorization. As a first immediate consequence, it is shown to weaken the well-known Davenport orthogonally condition on the decomposition axes, thus allowing much more freedom in the geometry of attitude control mechanisms. Furthermore, the additional parameter is used for avoiding gimbal lock singularities and optimization purposes. Some of these features have been pointed out by other authors and partially studied in the case of two axes. This text, however, provides exact closed form solutions based on recent results on the generalized Euler decomposition problem. The examples considered below clearly illustrate that the approach of four factor sequences can be significantly more efficient than standard Euler type decompositions. Finally, dynamical effects, such as the inertial anisotropy, are being modelled via introducing weight coefficients in the cost function.