Uncertainty models for physically realizable Inertia Dyadics

Abstract

This paper proposes new probability-distribution functions (PDFs) for the inertia dyadic. Representing uncertainty in rigid-body rotational dynamics with these PDFs preserves constraints on the moments and products of inertia that arise from the underlying physics but which conventional approaches violate. Specifically, we propose representing uncertainty in terms of parameters closely related to the radii of gyration and in terms of Euler parameters (for the orientation of the principal axes with respect to a set of body-fixed reference coordinates). The physical constraints on the inertia matrix are shown to be satisfied for any distribution in these special parameters and if the Euler parameters are given a radially constrained Gaussian distribution. Although unconventional, this uncertainty structure is shown to be broadly applicable. We consider the example of spacecraft/launch vehicle separation analysis for an early-stage spacecraft program, when mass properties are ill-defined. Here, misplaced conservatism in requirements can be costly. The proposed PDFs are shown to guarantee physically realizable results, while more naïve approaches yield non-physical behaviors.