Johnathan Tucker, University of Colorado Boulder; Daniel Aguilar Marsillach, University of Colorado Boulder; Anna Montgomery, University of Colorado Boulder; Marcus Holzinger, University of Colorado Boulder
Keywords: Custody, Reachability, Catalog Maintenance
Abstract:
Using optimal control formulations of reachability analysis a bounding sphere of possible locations of a space object (SO), after some time horizon, can be created. Projecting this sphere onto the focal plane of a telescope allows for the determination of a degree of custody. Here custody represents the knowledge that, at any time, the observing sensor could look at and acquire the space object in question. The contribution made in this paper is an efficient algorithm that computes the time horizon when the SO will no longer be within a sensor field of view – when sensor custody of the SO may be lost. This work has impacts particularly for Department of Defense (DoD) agencies that are involved with telescope tasking and SO tracking. Given the increase in space objects and limited sensing resources, computationally efficient algorithms that allow for the determination of both custody and the time until custody is lost are highly desirable. This type of analysis can inform both sensor tasking as well as tasking CONOPS for terrestrial, space-based, and cislunar sensor platforms.
The United States Space Force defines space domain awareness (SDA) as “the effective identification, characterization, and understanding of any factor associated with the space domain.” It should be recognized that, with the ever-increasing population of cislunar space objects, SDA is a big data problem. Here each data point has unstable dynamics that make them difficult to track and therefore also difficult to maintain custody. This problem demands a combination of solutions such as ground and space-based observers as well as efficient custody determination and telescope tasking algorithms. This paper aims to provide the ability to determine custody and task telescopes efficiently for both space and ground-based observers. More specifically, the work fundamentally aims to make ground and space-based observers more robust against SO maneuvers. There is a particular need for this as there is a gap in the literature when it comes to cislunar space.
Past work has shown that by applying optimal control methods and reachability analysis, one can create a sphere that over-approximates the set of reachable positions of a space object after a specified time horizon. This is generally achieved by computing the position reachable set, which requires solving an optimal control problem for every state on the boundary of an initial set. Furthermore, the assumption of minimum-time optimal controllers is made because this provides the largest possible reachable set given assumed thrust constraints. The literature has also shown that there are many approaches to computing position reachable sets. These are based on Hamilton-Jacobi-Bellman (HJB) Partial Differential Equation (PDE) solutions for a set, non-optimal control formulations, ellipsoidal/polytopic over- and under-approximations, or sampling-based methods using optimal control. The latter provides a tractable solution to approximate the reach set in a computationally efficient manner. Finally, previous work has shown that the position reachable set space can be projected onto the focal plane of a telescope for custody determination.
This paper conceptually extends the above work by using sampling-based optimal control methods to calculate the position reachable space, which is then projected onto the focal plane of a telescope at varying time horizons. Ultimately this allows operators to compute how long the telescope does not have to actively observe the SO but may still maintain custody (i.e., the object cannot maneuver out of the sensor field of view in the computed time horizon). Furthermore, this paper will extend the above capabilities of reachability analysis by implementing it on both ground and space-based observers for custody determination. This will be accomplished through simulations based on real-world scenarios to ultimately show these methods can be used for efficient telescope tasking and custody determination. The first simulation will include a ground-based observer with a space object in a Geosynchronous Earth Orbit (GEO). This simulation has direct applications to DoD projects such as the custody maintenance of Echostar 15. The next simulation will be with a space-based observer and a space object in a highly elliptical near-rectilinear halo orbit (NRHO), such as the Lunar Gateway.
In summary, this paper contributes the following to the field of cislunar space domain awareness:
Extension of past work that includes simulation with a variety of real-world applications.
A computationally efficient method for the determination of custody as well as a method for determining how long the telescope can not directly observe an SO before losing custody.
Date of Conference: September 14-17, 2021
Track: Dynamic Tasking