Kevin Vanslette, Raytheon BBN; Roddy Taing, University of Wisconsin-Madison
Keywords: Space Situational Awareness, SSA, Conjunction, Space Domain Awareness, SDA, Sensor Tasking, Information, Inferential Moments, Value of Information, Sensor Planning, Algorithms
Abstract:
Information Gain and Fisher Information are often used as objective functions for optimizing sensor tasking schedules to maintain custody of assets for Space Domain Awareness (SDA). Because sensing resources are finite and imperfect, not every space object can be tracked with certainty at all times. By choosing optimal schedules according to information-based objective functions, one seeks solutions that reduce track uncertainty in an informational sense.
A recent article Inferential Moments of Uncertain Multivariable Systems [1] defines a new set of quantities, inferential moments, for reasoning with joint probability distributions in inference settings. Similar to an Information Gain or Fisher Information, these quantities are functionals of a joint probability distribution and in principle exist for any suitably behaved distribution. The first inferential moment corresponds to the equation for a marginal probability distribution p(a), which in this context is understood to be the expected value of the conditional probability E_B[p(a|b)]. Inferential moments are higher order conditional probability moments E_B[p(a|b)^n] that describe how a distribution is expected to respond to new information.
Of particular interest is the second order central inferential moment, which is called the inferential variance. It is a measure of the mean square error of using a marginal probability distribution p(a) to estimate the value of the conditional probability distribution p(a|b) when b is an uncertain but definite value. Thus, for every marginal probability distribution, one can use the square root of the inferential variance to quantify its expected error analogous to the use of standard deviations for quantifying the expected error of a mean value.
Preliminary comparisons between the Mutual Information and Information Gain and inferential moments are made in [1]. Similar to a Mutual Information or Information Gain, when variables are independent, central inferential moments are zero. Further, it is found that the Mutual Information can be expanded in terms of central inferential moments, where its leading order term is proportional to the inferential variance. For optimal sensor tasking, however, this begs the question, Why would it be preferable to task a sensor according to a maximum Mutual Information if we could instead more directly task a sensor to eliminate the maximum amount of error in terms of inferential variance? The article employs a simple greedy algorithm using inferential variance for sensor tasking over a static discrete Bayesian Network and demonstrates that this approach generally outperforms a similar greedy Mutual Information algorithm in terms of its ability to reduce state prediction error, as is expected.
In this article, we develop an inferential moment approach for LEO multi-object SDA. We explore a few sensor tasking algorithms using inferential moments in the continuous domain for space object tracking and compare them against similar approaches using Information Gain, Fisher Information, and rule-of-thumb tasking. Because our algorithms will be aimed are reducing probabilistic error through the utilization of inferential variances we expect our algorithms to outperform these approaches, which are instead not directly optimizing for error. Thus, we expect this inferential moment based approach will offer a competitive alternative to traditional SDA methods in terms of probabilistic prediction error.
We consider two optimal sensor tasking objectives the first objective is the standard objective of maintaining accurate custody of assets. Recently [2] demonstrated that the Mutual Information improves upon the Fisher Information in terms of custody-based sensor tasking effectiveness due to its ability to properly capture Kalman Filter paradigm nonlinearities in dynamics and measurement. Because the exhaustive inferential variance is monotonic in the same probabilistic decision variables (determinant ratios) as the Mutual Information, both lead to the same sensor tasking ranking scheme, and thus the inferential moment approach equally improves over the Fisher Information in terms of custody-based sensor tasking.
The second sensor tasking objective we consider optimizes sensor tasking toward maintaining and more accurately tracking conjunction probability estimates, which we call conjunction-based sensor tasking. While custody-based sensor tasking has been studied extensively in the literature, the authors are not aware of formalized approaches for studying something like conjunction-based sensor tasking outside of rule-of-thumb approaches. For space objects that are labelled as a conjunction risk, it is typical to prioritize sensor tasking based on one of the following qualities: maximum conjunction probability, largest current covariance, or by largest asset value. We demonstrate how using an inferential moment approach for sensor tasking improves over these rule-of-thumb approaches in terms of predicted conjunction probability error.
[1] Vanslette, Kevin. “Inferential Moments of Uncertain Multivariable Systems.” arXiv preprint arXiv:2305.01841 (2023).
[2] Adurthi, Nagavenkat, Puneet Singla, and Manoranjan Majji. “Mutual information based sensor tasking with applications to space situational awareness.” Journal of Guidance, Control, and Dynamics 43.4 (2020): 767-789.
Date of Conference: September 19-22, 2023
Track: Space Domain Awareness