Geometric Solution to Probabilistic Admissible Region (PAR)

In the Initial Orbit Determination (IOD) context, Admissible Regions are subsets of parameter space of orbital elements that are reckoned as functions of the tracking measurement variables. Physically acceptable orbits, e.g., orbits with negative energies, constitute an Admissible Region. This paper defines sets of orbits that satisfy constraints imposed by the measurement variables. Using a probabilistic representation of constraints on some of the orbital parameters the Admissible Region can be further constrained to give a Probabilistic Admissible Region (PAR). PAR gives a particle cloud representation of the initial probability density function (pdf) of the state of a Resident Space Object (RSO). This paper presents a geometric solution to the Probabilistic Admissible Region (G-PAR). G-PAR is a set of algorithms sharing the same underlying template that geometrically maps postulated statistics on some orbital elements and statistics of the measurement process to the uncertainty in the states. The proposed scheme gives a simple closed-form solution for mapping particles to get the PAR pdf for the first time. This speeds up the PAR initial orbit determination with a single partial state measurement. The effectiveness of the proposed G-PAR is shown on diverse combinations of sensors and prior knowledge. [ABSTRACT FROM AUTHOR]