Gunner Fritsch, Texas A&M University; Kyle DeMars, Texas A&M University
Keywords: Nonlinear, filtering, bayesian, probability of detection, estimation, tracking
Abstract:
The probability of detection is a term that quantifies the likeliness that a valid return will be generated by a given measurement model. It naturally arises in the update of Bayesian filters which model both false alarms and valid measurements separately, such as the probability hypothesis density filter. The inherent state dependency of the detection probability is a significant obstacle when attempting to seek closed-form filtering solutions, and thus is typically approximated as state-independent. However, the legitimacy of this approximation does not always hold, especially under large fluctuations in the detection probability, such as when objects are transitioning across a sensor’s field of view. Therefore, this work investigates possible alternatives to manage the state-dependent nature of the probability of detection using a proposed nonlinear filter inspired by earlier work in intrinsic fault resistance. The overall goal of the work is to survey different probability of detection models of varying fidelity, such that an appropriate model may be selected.
The proposed filter is constructed via a single-target, fault-cognizant model that is capable of processing multiple measurements simultaneously within the update, as well as account for both valid and faulty sensor returns. To achieve improved nonlinear, non-Gaussian operations, the Bayesian filtering solution is actualized via Gaussian mixture approximations of the state distribution. As false alarm modeling is not of primary interest to this work, the typical assumptions are taken wherein faulty returns are independent events temporally Poisson distributed and spatially uniform distributed.
It is found that certain models for state-dependent probability of detection can provide additional state information to the filter, directly improving the state estimate. However, such methods may require additional external information be provided, as the update requests a reported value of detection probability. To avoid this, additional approximations may be carefully applied, leading to different variations of the proposed filter. Thus, this paper includes three different filter updates, each representative of a different fidelity of detection probability modeling. The filter with the lowest fidelity model uses a method of residual editing to screen out false alarms, and essentially does no modeling of the detection probability at all. Next, an update that uses a zeroth-order approximation to treat the probability of detection as state-independent is used, which is considered standard practice in many situations and represesents a moderate fidelity model. Finally, the most promising variation presented by this work models the state-dependent probability of detection as a Gaussian distribution. This specific method is considered favorable as it not only aids in developing a closed-form solution, but it also allows for a level of uncertainty to be attributed with the probability of detection. Where many filters fallaciously operate upon the assumption that the probability of detection is known exactly, this method is able to account for some of the stochasticity produced by factors such as model inaccuracy.
The efficacy of the proposed filter is evaluated with several Monte-Carlo simulations, where it is tested against a Gaussian mixture extended Kalman filter, which is used as a performance baseline. To examine the filter’s behavior more clearly, a simulation with linear dynamics is included, as to eliminate any artifacts resulting from nonlinearities. The results are expected to indicate that including a state-dependent probability of detection improves the accuracy of the filtering solution and promotes increased robustness against large variations in the detection probability. To evaluate the filter in a more realistic system, an orbit determination simulation is presented where optical measurements are taken via a ground-based observer. A complex probability of detection model is created that is functioned on several state-dependent factors, including satellite visibility and illumination.
Date of Conference: September 14-17, 2021
Track: Dynamic Tasking