Optimal Transport Based Tracking of Space Objects in Cylindrical Manifolds

Abstract

In this paper, we examine the performance of satellite state estimation algorithms in the modified equinoctial coordinate system, defined on the cylindrical manifold of $\mathbb {R}^{5}\times \mathbb {S}$, where $\mathbb {R}$ is the space of reals and $\mathbb {S}$ denotes circular space. A comparison is made between an optimal transport based filter and ensemble Kalman filter algorithm in the context of satellite state estimation. The initial state joint probability density function is modeled in $\mathbb {R}^{5}\times \mathbb {S}$ using the Gauss von Mises distribution. The sensor noise for optimal transport filtering is modeled in the same manifold. The ensemble Kalman filter, by definition, requires the sensor noise to be Gaussian and is modeled in $\mathbb {R}^{6}$ for this problem. We observe that there is a clear advantage in using an optimal transport based filtering algorithm where we represent the initial condition uncertainty and sensor noise, in the cylindrical manifold. These two filtering algorithms are implemented on a simulated International Space Station orbit, with measurement at equal intervals. We compare two distinct scenarios in this simulation based study. In the first one, sensor noise characteristics are assumed to be known. For the second one, we present a new algorithm within the optimal transport framework with unknown sensor noise characteristics, a practical and relevant issue in the space-tracking problems. The optimal transport based algorithm provides more consistent and robust estimates compared to that of ensemble Kalman filter, in each of these scenarios.