Abstract A Bayesian solution and its approximations to out-of-sequence measurement problems

Keywords: Time delayed measurement; Networked nsor; Smoothing; Augmented state; Target tracking
1. Introduction
In a multi-sensor centralized tracking system, sensors produce observations that are sent to a fusion center over communication networks which can introduce random delays. This problem has appeared in the literature under various names such as the out-of-sequence measurements (OOSM) problem [1,2], the problem of tracking with random sampling and delays [3–5], and the problem of incorporating random time delayed measurements [6]. In this paper, we refer to this problem as the OOSM problem. The basic problem can be formulated using a single sensor with measurements received in incorrect time order [1].
q This work is supported by SSD division, Defence Science Technology Organization (DSTO), Australia.
* Corresponding author. Tel.: +61-3-8344-7974; fax: +61-3-83446678.
In contrast, existing solutions in the literature modify the sensor measurement equation to account for the time delay and explicitly deal with the resulting correlations that arise in the process noise and current target state. In the Bayesian solution proposed in this paper, such cross correlations are treated implicitly. Under linear Gaussian assumptions, the Bayesian solution reduces to an augmented state Kalman filter (AS-KF) for scenarios devoid of clutter and an augmented state probabilistic data association filter (AS-PDA) for scenarios involving clutter. Computationally efficient versions of AS-KF and AS-PDA are considered in this paper. Simulations are presented to evaluate the performance of these solutions. Ó 2003 Published by Elsevier B.V.
yðsÞ ¼ HsxðsÞ þ vðsÞ; s < tk
ð5Þ
arrives at time tk after (1) and (2) have been computed, one faces the problem of updating the state estimate and its covariance with the delayed measurement (5), i.e., to compute
E-mail addresses: s.challa@ee.mu.oz.au (S. Challa), r.evans@ee. mu.oz.au (R.J. Evans), xu.wang@ee.mu.oz.au (X. Wang).
1566-2535/$ - see front matter Ó 2003 Published by Elsevier B.V. doi:10.1016/S1566-2535(03)00037-X
x^ðts j kÞ , EðxðtkÞ j Y k; yðsÞÞ
ð6Þ
and
Ps j k , E½ðxðtkÞ À x^ðtk j kÞÞðxðtkÞ À x^ðtk j kÞÞ0 j Y k; yðsÞ ð7Þ
For the scenario where the delay corresponds to the time between the last two updates (tkÀ1 6 s < tk), an optimal solution has been proposed in the filtering framework by Bar-Shalom in [1] where it is also stated that extension to longer time delays involves some kind of nonstandard smoothing. Such an approach was later considered by Mallick et al. in [7]. Prior to this, the OOSM problem was solved, approximately, in [4]. The key difference between the algorithms in [1] and [4] is the effect of the process noise w on the filtering scheme. Although both consider the effect of process noise, the optimal solution incorporates the non-zero conditional mean of the process noise into the filter update equations while the approximate filter does not. A solution that completely ignores the process noise appeared in [8] and a numerical comparison between these various approaches was carried out on a simulated example in [1]. Based on these simulations and the complexity issues in tracker implementations, it was concluded in [1] that the sub-optimal solution that assumes the conditional mean of the process noise to be zero provides a good balance between accuracy and computational complexity especially when data association issues (e.g., tracking in clutter) are involved. The OOSM problem in clutter was first considered by the authors in [9]. It was almost simultaneously considered by Orton and Marrs using particle filters in [10]. In this work, Orton and Marrs, correctly recognize the importance of considering target states from both the current time and delayed time.
Denoting a ‘‘standard’’ measurement sequence Y k ¼ fyðt1Þ; yðt2Þ; . . . ; yðtkÞg, the ‘‘standard’’ target tracking problem reduces to the problem of computing the conditional mean estimate of the target state
Department of Electrical and Electronic Engineering, Center for Sensor Signal and Information Processing (CSSIP), University of Melbourne, Victoria 3010, Australia
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S. Challa et al. / Information Fusion 4 (2003) 185–199
estimate x^ðtk j kÞ of the target state, which is optimal in the minimum variance sense, can be computed recursively using the Kalman filter (KF). KF based solutions treat this problem as a filtering problem where all the measurements, including the current measurement are used to obtain the best estimate at the current time. The basic KF algorithm can be easily extended to multisensor systems where the data is assumed to arrive at known times and in correct time sequence. A key problem arising when dealing with multiple interconnected sensors with communication links is time delay between the sensor and tracking computer. This problem can be defined as follows. When a measurement corresponding to time, s, expressed as