Samuel Fedeler, University of Colorado Boulder; Marcus Holzinger, University of Colorado Boulder; William Whitacre, Draper Laborartory
Keywords: space situational awareness, sensor tasking, estimation, heuristic search
Abstract:
Whether the goal in consideration is near-Earth asteroid detection, debris mitigation, or collision avoidance, the ability to quickly make follow-up observations on space objects (SOs) is imperative. Often, these goals are further challenged by the prescence of underdetermined or uncertain target states. A newly detected SO may not have been fully observed. A target object may have maneuvered, introducing further uncertainty. If there are collision or operational concerns, urgency is the most critical component in the process of target recovery. This motivates extension of the sensor tasking problem to consider search over feasible regions of state space in an optimal manner, recovering targets as quickly as possible.
Generally, methodologies for driving sensor tasking can be categorized depending on the objective considered. First, one may wish to maintain existing estimates, informing knowledge on a catalog of SOs. A variety of strategies have been proposed assuming a priori knowledge on state estimates and uncertainties. Erwin et al apply linear optimization to form a tasking solution and propose useful quantities for interpreting the value of a tasking decision. This work is extended by Williams et al, using Lyapunov exponents to probe the stability of SO estimates. A variety of approaches have also taken inspiration from the machine learning literature, with techniques such as stochastic gradient ascent , asynchronous actor-critic methods, and Monte Carlo Tree Search. In each of these methods, the driving goal is determination of an optimal policy for decision making given a large set of candidate observations.
Alternately, one may wish to generate new state estimates, expanding the set of SOs studied by searching for natural objects, orbiting satellites, or debris. Wide-ranging techniques for this objective exist in literature. Often, long-period stares over an optical field are performed, acting as a sweep through orbital parameter space. Striping methodologies may also be formed in measurement space, and this strategy accomodates optimization.
It is also important to note that detections made with optical sensors generally do not fully observe the object state; as a result of this “Too Short Arc” problem, an admissible region (AR) of unobservable ranges and range rates may be formed. This admissible region is a two-dimensional manifold of feasible pairs that may be projected into the six-dimensional state space. Note that this region may be uniformly distributed or probabilistic if measurement uncertainty is incorporated. Gehly et al. leverage the AR methodology in tandem with Finite Set Statistics to approach the tracking problem, representing the admissible region as a Gaussian mixture to be ingested by a CPHD filter. Methodologies for generating Gaussian mixture representations of admissible regions are introduced by Demars and Jah. AR pairs over longer observation intervals may be used for initial orbit determination. These methodologies are not typically used in an online manner, but rather consider large populations of admissible regions generated from detected tracklets over several observation campaigns.
This literature illustrates the need for follow up observation to fully observe target states. Tasking in the context of this objective also becomes critical when considering maneuvering targets. Jaunzemis applies the Dempster-Shafer theory of evidence to this problem with success . Decision theoretic approaches have also been applied in tandem with multiple model filters. It is recognized, though, that in either scenario, it becomes challenging to locate a target object, because the projection of the admissible region or reachable set for a target may become quite large relative to the sensor field of view.
The problem of exhausting a feasible set has been explored by Hobson and Murphy. Murphy considers the direct follow-up tasking problem for an admissible region, representing the region as a feasible set that has grown over time in state space. The area of the admissible region is computed over time using high order Taylor series expansions, and the idea of minimizing the search set using the divergence of the set as an observational metric is considered. Simulated annealing in combination with this observational metric is found to achieve some success in minimizing the search space in a time-optimal manner. This research was extended by Fedeler, where further search heuristics were developed and leveraged in combination with Monte Carlo Tree Search.
The primary goal of this research is development of an efficient estimation scheme to be utilized in tandem with a prior tasking methodology. Of particular interest is whether the “spooky effect” is apparent, as may be seen in multi-target filters such as the Gaussian mixture Cardinalized Probability Hypothesis Density (CPHD) filter. This behavior describes the impact of a missed or null detection in one region of measurement space on the probability hypothesis density arbitrarility far away. A logical extension of this effect, then, is determining how a null detection in a subset of the projected feasible set may affect knowledge on other regions within that volume.
With this purpose in mind, the following developments are outlined. First, a brief overview of the process for generating a mixture representation of an admissible region is presented. A mixand weight update is then derived for null detections. Within this update, the expectation propagation methodology for efficient Gaussian integration is utilized. A novel methodology for splitting mixands is then outlined; this methodology was developed to ensure that updates avoid non-Gaussianity. Finally, a merging methodology is introduced to ensure the update process remains computationally efficient. The resultant filter is then fully outlined and applied to a geostationary follow-up tasking problem.
Date of Conference: September 14-17, 2021
Track: Dynamic Tasking