Sam Wishnek, University of Colorado Boulder; Marcus Holzinger, University of Colorado Boulder; Patrick Handley, Ball Aerospace
Keywords: Cislunar, IOD, Orbit Determination
Abstract:
As the population of near-Earth objects has grown, more scientific, commercial, and military operators have expanded their utilization of the domain. One area of growing interest to these operators is exploration of the opportunities provided by the cislunar domain. Methods for maintaining space domain awareness in the cislunar domain are essential to support these current and future missions. One of the essential tools of space domain awareness is initial orbit determination for characterizing the orbits using a minimal set of observations. However, existing initial orbit determination algorithms rely on the two-body assumption for estimating the state. For cislunar missions that operate where moon gravitational effects are significant, new methods for initial orbit determination are required in order to estimate orbit state under multi-body dynamics.
In order to perform initial orbit determination in the cislunar domain, the proposed method applies an optimization-based approach to find the state or set of states that can feasibly match the observations. The approach leverages admissible regions to constrain the search space and improve convergence characteristics. The optimization for the cislunar domain is run over four dimensions. These are range and three velocity components. The optimized cost function is the Mahalanobis distance between predicted and observed measurements based on a trial value.
The approach implements JPL’s SPICE Toolkit to accurately predict the relative positions of the Earth, Sun, and Moon. The gravitational influence of these three bodies dominates the dynamics of orbits in cislunar space and beyond. The influence of the Sun is essential for identifying if a target object is in a cislunar, geocentric, or heliocentric orbit as targets sufficiently far from the observer can be difficult to differentiate in terms of orbit domains.
The proposed method uses angles only measurements with three observations. This is necessary for the cislunar domain as the angular velocity of a target in cislunar orbit typically has a much slower inertial velocity than an object in a near-Earth orbit and the significant increase in separation distance between target and observer makes the observed angle rates much smaller. The relative drop in angle rate observability compared to the near-Earth orbits significantly restricts the types of measurements that can accurately capture angle rates. Accordingly, angles-only measurements are far more applicable for the cislunar domain. The cost function itself is the sum of six Mahalanobis distances. From a single trial value, these are the distances between the predicted and observed measurements for the orbit propagated to each measurement time. The measurement treated as truth is then changed and the process repeated until each measurement is given equal weighting in the estimate. For perfect measurements and an accurate state guess, the cost function returns zero. We find that small measurement errors can still yield accurate estimated states.
The Mahalanobis-based cost function alone tends to form long curves through the solution space that some optimization methods may struggle with. In order to improve the geometry of the underlying contours for optimization, modifications to the solution space and cost function are investigated. This includes investigating the influence of the velocity coordinate system. Another method used to improve convergence for the cislunar domain is the modification of the admissible regions from the two-body implementation. The admissible regions for cislunar space differ substantially from the near-Earth orbits, and these limits are shifted accordingly to penalize the cost function in a way that improves convergence and limits the search domain. Investigated penalty function parameters include geocentric periapse and apoapse limits, heliocentric periapse and apoapse, obstruction of line of sight due to the Moon and Earth, occultation of the Sun on the target due to the Moon and Earth, and absolute limits on maximum and minimum velocities and ranges. Well targeted application of these admissible regions can significantly restrict the search space and improve the convergence properties of the optimization.
The results of this cislunar initial orbit determination method are investigated in terms of convergence accuracy with respect to measurement error and the relative orbit geometry of the observer and target using a series of Monte Carlo simulations. There will also be an attempt to collect observational data of cislunar objects for empirical verification. Successfully collecting this data depends on the installation of a telescope system as currently planned in April.
The proposed method is an approach for performing initial orbit determination in cislunar space. The efficacy of this method is demonstrated through a series of Monte Carlo simulations investigating the response of the algorithm across cislunar space and in response to measurement error.
Date of Conference: September 14-17, 2021
Track: Cislunar SSA