Relative Spacecraft Position and Attitude in the Circular Restricted Three-Body Problem: TSE(3) vs. Dual Quaternions

Abstract

Relative motion has been a crucial component in crewed space missions dating back as far as the mid 1960s. Such operations exist in a dynamic environment where rotational and translational motion can be highly coupled. Traditional, minimal attitude parameterization sets (such as Euler angles) cannot be used for maneuvers with large ranges of rotational motion without encountering singularities, so modern treatment of attitude maneuvers favors the use of unit quaternions and dual quaternions for orbit-attitude coupling due to the balance between lesser storage requirements when compared with rotation matrices and their nonsingularity when compared with minimal sets. However, quaternions and dual quaternions are susceptible to the unwinding phenomenon and have some non-uniqueness associated with them, as a single attitude corresponds to two antipodal quaternions on the three-sphere S^3. Hence, an accurate dynamics formalism is required in such scenarios. Describing rigid body motion on the special Euclidean group SE(3), which leverages rotation matrices to represent attitude, and its tangent bundle is particularly well-suited to handle these cases. In this work, the relative motion problem is presented on the special Euclidean group and its tangent bundle during a case study that involves simulated relative motion maneuvers with the Gateway within the near rectilinear halo orbit, or NRHO. To illustrate the validity of the formalism introduced, results are compared in terms of both accuracy and computational load with those obtained using dual quaternions.