Probabilistic Methods for Initial Orbit Determination and Orbit Determination in Cislunar Space

Abstract

In orbital mechanics, Gauss's method for orbit determination (OD) is a popular, minimal assumption solution for obtaining the initial state estimate of a passing resident space object (RSO). Since much of the cislunar domain relies on three-body dynamics, a key assumption of Gauss's method is rendered incompatible, creating a need for a new, minimal assumption method for initial orbit determination (IOD). In this work, we present a framework for short and long term probabilistic target tracking in cislunar space which produces an initial state estimate with as few assumptions as possible. Specifically, we propose an IOD method involving the kinematic fitting of several series of noisy, consecutive ground-based observations. Once a probabilistic initial state estimate in the form of a particle cloud is formed, we apply the powerful Particle Gaussian Mixture (PGM) Filter to reduce the uncertainty of our state estimate over time. This combined IOD/OD framework is demonstrated for several classes of trajectories in cislunar space and compared to better-known filtering frameworks.

Figures (24)

• Figure 1: Kinematic polynomial fitting illustration over the first (50%) of an RSO's pass, totaling 9 measurements (illustrated as blue circles) over roughly 5 hours. We fit fourth-order polynomials (in red, green, and cyan) as functions of $t$ through range, azimuth, and elevation measurements, converted to $x$, $y$, and $z$ in the observer-centered topocentric reference frame using Eq. \ref{['eq8:AZ-EL']}. When we take the temporal derivatives for $x$, $y$, and $z$, an initial state estimate is formed at the last timestep in these plots (indicated by a star symbol). Because a single estimate will not accurately capture the target state, we simulate thousands of time-series sets of angular observations to obtain an initial state estimate. For comparison purposes, the target truth (denoted by a black cross) shows how closely our curve fitting approach matches the true state of the object over time.
• Figure 2: To obtain the initial state estimate in particle cloud form, we use the polynomial fitting method shown in Figure \ref{['fig:1polyIOD']} several hundreds or thousands of times, drawing each measurement and range information from the noise statistics or using a priori knowledge of our target's orbit. Each point that would be indicated by a star from Figure \ref{['fig:1polyIOD']} would correspond to a single blue particle in this figure. To visually verify that our initial state estimate is consistent with the truth, it is clear that the target truth, given as a black cross, lies within the particle cloud -- abstracted into a probability density function (PDF) -- that makes a probabilistic initial state estimate.
• Figure 3: Using the polynomial fitting method shown in Figure \ref{['fig:1polyIOD']} several hundreds or thousands of times, we develop an initial state estimate. The choice of trajectory for the target (denoted by a black X) is one which is propagated by a CR3BP patch point as described by Williams, et al. 2017williams2017. Using k-means clustering, we partition our propagated estimate into six clusters, representing a priori GMM components. Based on the number of particles in each component, the cluster means, and cluster covariances, an a priori PDF is generated for our orbit.
• Figure 4: Component-wise update and resampling steps to the estimate shown in Figure \ref{['fig:3bAprioriEx_Williams']}. As expected, the cluster defined in green is expected to retain a large fraction of the weight update. Due to low measurement likelihoods, the purple and yellow clusters will vanish.
• Figure 5: Initial state estimate derived from kinematically fitting polynomials through several series of possible measurements whose range information is derived from a large uniform PDF. Particles in the position space are pruned if they lie outside of the bounds of cislunar space. Although the resulting PDF is large and very uncertain in the range direction, it is localized in terms of AZ and EL, and we may utilize the PGM Filter to reduce the uncertainty.