Alternative set of nonsingular quaternionic orbital elements

Quaternionic elements in orbital mechanics are usually related to the Kustaanheimo-Stiefel transformation or to the definition of the orbital plane. The new set of regular elements presented in this paper stems from the form of the equations of motion of a rotating solid, which model the evolution of a quaternion defining the orientation of a bodyfixed frame and the change in the angular velocity of such frame. By replacing the body-fixed frame with a special orbital frame and accounting for the radial motion separately, an equivalent solution to orbital motion can be constructed. The variation of parameters technique furnishes a new set of elements that is independent from the orbital plane. A second-order Sundman transformation introduces a fictitious time that replaces the physical time as the independent variable. This technique improves the numerical performance of the method and simplifies the derivation. The use of a time element yields an even smoother evolution of the orbital elements under perturbations. Once the Lagrange and Poisson brackets are obtained, the most general nonosculating version of the set of elements is presented. Regarding the performance, numerical experiments show that the method is comparable to other formulations involving similar stabilization and regularization techniques.