Robust target motion analysis using the possibility particle filter

IET Radar, Sonar & NavigationVolume 13, Issue 1 p. 18-22 Research ArticleFree Access Robust target motion analysis using the possibility particle filter Branko Ristic, Corresponding Author Branko Ristic branko.ristic@rmit.edu.au orcid.org/0000-0001-8561-4412 RMIT University, School of Engineering, 124 La Trobe St, Melbourne, VIC, 3000 AustraliaSearch for more papers by this authorJeremie Houssineau, Jeremie Houssineau National University of Singapore, SingaporeSearch for more papers by this authorSanjeev Arulampalam, Sanjeev Arulampalam Defence Science and Technology Group, Edinburgh, SA, 5111 AustraliaSearch for more papers by this author Branko Ristic, Corresponding Author Branko Ristic branko.ristic@rmit.edu.au orcid.org/0000-0001-8561-4412 RMIT University, School of Engineering, 124 La Trobe St, Melbourne, VIC, 3000 AustraliaSearch for more papers by this authorJeremie Houssineau, Jeremie Houssineau National University of Singapore, SingaporeSearch for more papers by this authorSanjeev Arulampalam, Sanjeev Arulampalam Defence Science and Technology Group, Edinburgh, SA, 5111 AustraliaSearch for more papers by this author First published: 01 January 2019 https://doi.org/10.1049/iet-rsn.2018.5144Citations: 5AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Bearings-only target motion analysis (TMA) is the process of estimating the state of a moving emitting target from noisy measurements collected by a single passive observer. The focus of this study is on recursive TMA, traditionally solved using the Bayesian filters (e.g. extended or unscented Kalman filters, particle filters). The TMA is a difficult problem and may result in track divergence, especially when the assumed probabilistic models are imperfect or mismatched. As a robust alternative to Bayesian filters for TMA, the authors present a recently proposed stochastic filter referred to as the possibility filter. The filter is implemented in the sequential Monte Carlo framework, and named the possibility particle filter. This study demonstrates its superior performance against the standard (Bayesian) particle filter in the presence of a model mismatch, while in the case of the exact model match, its performance equals that of the standard particle filter. 1 Introduction Target motion analysis (TMA) is a collective term for tracking algorithms developed for passive sensors (e.g. passive sonar, radar or infrared). TMA algorithms estimate the state of a moving target (e.g. its position and velocity vector), from noise-corrupted passive measurements, such as the bearings to the target, collected by a single moving observer [1]. Development of bearings-only TMA algorithms has a long history, particularly in the context of submarine combat systems for underwater surveillance [2, 3]. The modern emphasis is on stochastic filters, such as the recursive Bayesian estimators [4] [e.g. the extended/unscented Kalman filter, or particle filters (PFs)], because they provide the entire posterior probability density function (PDF) of the state and thereby provide a measure of uncertainty to any derived point estimate. Assuming a plane geometry and the state vector which includes target position and velocity, a Bayesian TMA algorithm estimates recursively the time-evolving posterior PDF over the four-dimensional state space. TMA is a peculiar non-linear filtering problem, because the target range remains unobservable until the observer performs an appropriate manoeuver [5]. Consequently, Bayesian TMA algorithms can occasionally fail and cause track divergence. This is particularly true in situations where the non-linearity is high or the assumed target dynamic model or the measurement noise model are imperfect or mismatched. In this paper, we propose a novel stochastic filtering scheme, referred to as the possibility filter, for bearings-only TMA. The possibility filter has recently been introduced as a special instance of a class of outer measure based stochastic filters [6]. Bayesian filtering style analytic expressions for prediction and update of outer measures are available and their numerical implementation is discussed in [6-8]. The motivation for using outer measures instead of the probabilistic framework is to provide a more generalised representation of uncertainty, capable of handling in a rigorous mathematical manner the situations of ignorance or partial knowledge. For example, in situations where the precise knowledge of probability distributions is unavailable, the outer measure can be seen as defining the upper bound for probability distributions, which captures uncertainty both due to randomness and the lack of knowledge [9]. As there is no closed-form analytic solution to the prediction and update equations of the possibility filter for TMA, an approximate solution based on the sequential Monte Carlo (SMC) estimation method has been implemented [7] and referred to as the possibility particle filter (PPF). The paper is organised as follows. Section 2 introduces the mathematical formulation of the problem and reviews the standard Bayes filter. The concept of stochastic filtering using possibility functions, including the prediction and update equations of the possibility filter, is presented in Section 3. A SMC implementation of the possibility filter for bearings-only TMA is described in Section 4. A numerical study comparing the performance of the standard Bayes PF and the PPF is carried out in Section 5, followed by conclusions in Section 6. 2 Background 2.1 Bearings-only TMA formulation The problem formulation is standard, as used in numerous publications [4, 10]. The state of a moving target at time (where is a discrete-time index) is fully characterised by the vector , where is the state space. Let us adopt the two-dimensional Cartesian coordinate system and define the state vector as (1) Here and represent the target position and velocity, respectively, and T denotes the matrix transpose. The observer (ownship) is also moving in the same coordinate system. Its state vector is assumed known and correspondingly defined as . The target motion model and the observation model will be expressed in terms of the relative state vector . The target motion model is adopted as the nearly constant velocity (CV) model [11] (2) where is the transition matrix and is a known deterministic matrix taking into account the effect of observer accelerations. The product in (2) is introduced to model small deviations from CV motion. Matrices , and are defined as (3) where is the Kroneker product, is the identity matrix and is the sampling interval. For the rest of the paper we adopt a constant sampling interval , with simplified notation , . Process noise in (2) is modelled as being zero-mean white Gaussian, with covariance matrix [11] (4) The covariance matrix of the product in (2) is then . Bearing measurement is related to the target state at time as follows: (5) where measurement noise is independent of process noise and modelled by zero-mean white Gaussian noise whose variance is . The non-linear function in (5) is the four-quadrant inverse tangent function (6) resulting in the true target bearing at time . The problem is to estimate sequentially the posterior distribution of the state, that is , where . The initial density, at , is typically constructed using the first bearings measurement [4]. 2.2 Standard Bayes filter This section briefly summarises the standard Bayes filter, which is the basis of recursive TMA. In this framework, the current knowledge about the state of the target is fully specified by the posterior PDF , where is the sequence of all bearings measurements up to the discrete time k. The Bayes filter estimates sequentially , assuming that the initial PDF is given. This is carried out in two stages, the prediction stage and the update stage. Suppose the posterior PDF at , i.e. , is known. The prediction stage involves using the target motion model (2) to obtain the predicted PDF of the state via the Chapman–Kolmogorov equation [12] (7) The transition density, , which features in (7), according to the specification in Section 2.1 is a Gaussian distribution with the mean and the covariance matrix . The update stage applies the Bayes theorem to compute the posterior PDF at k as follows: (8) The likelihood function , which features in (8), according to the measurement model specification in Section 2.1 is a Gaussian distribution with the mean and the variance . A point estimate is typically computed from the posterior as its mean or its maximum [4]. 3 Possibility filter The concept of an uncertain variable on the state space was introduced in [6] in order to model jointly the uncertainty due to randomness (aleatory uncertainty) and to incomplete knowledge of the probabilistic model (epistemic uncertainty) [9]. An uncertain variable will be represented by possibility functions, introduced in the seminal paper [13]. Let be a subset of the state space and let be a possibility measure associated with . Then the possibility measure of takes a value in the interval representing the possibility that , and is defined as , where is the possibility function (or distribution [14]) corresponding to . The possibility function is the primitive object of possibility theory [14, 15], which assigns to each a degree of possibility of being the true value of the state. It can be seen as a membership function determining the fuzzy restriction of minimal specificity (In the sense that any hypothesis not known to be impossible cannot be ruled out.) on [13]. Any PDF can be transformed into a possibility function (p.f.) , and conversely, any p.f. can be transformed into a PDF, using for example the following relationships: (9) (10) Other types of transformations have been discussed in [15]. Considering that in the formulation of the TMA problem in Section 2.1 both process noise and measurement noise were modelled by Gaussian distributions, in this paper we focus on the Gaussian p.f. defined as (11) for some and for some positive definite matrix with real coefficients. With abuse of language, we will refer to and as to the mean (The possibilistic mean value has been defined as a closed interval [16], although other interpretations exist.) and variance of the Gaussian p.f. . Referring to the TMA problem, the state vector is now treated as an uncertain variable. Let us introduce the concept of a posterior p.f. , as the conditional p.f. of given (the sequence of measurements up to time ). The problem of recursive TMA is to estimate sequentially , assuming that the initial p.f. is given. The prediction and update equations of the possibility filter are presented next. Suppose the posterior p.f. at time , that is , is available. The prediction equation explains how to compute the p.f. at time k based on the target dynamic model only. It is given by [6] (12) where is the conditional p.f. describing the transition from to . This transition was described by (2) for the TMA problem. Process noise in (2) is now considered to be an uncertain variable characterised by the Gaussian p.f. (11), with zero-mean and covariance matrix given by (4). The conditional p.f. in this case can be expressed by a Gaussian p.f. (11) with the mean and covariance matrix . Note that the prediction equation (12) is an analogue of the Chapman–Kolmogorov equation in the standard Bayes filter (7), except that: (i) the integral is replaced by the supremum and (ii) the PDFs are replaced by possibility functions. The update step of the possibility filter 'corrects' the predicted p.f. using the information contained in the new measurement . The update equation is given by [6] (13) Equation (13) is an analogue of the Bayes' theorem (8), again with the exception that: (i) the supremum replaces the integral, and (ii) PDFs are replaced by possibility functions. The term in (13) represents the likelihood function. Considering that in (5) is assumed to be Gaussian, we treat it now as an uncertain variable characterised by a Gaussian p.f. (11) whose mean is zero and variance . Then, the likelihood function is also a Gaussian p.f. (11), with the mean and variance . The question is how to compute a point estimate from the posterior p.f. . One option is to adopt a point estimate as the supremum of [6]. An alternative, which we adopt in this paper, is to compute a point estimate as the conditional expectation (14) where is obtained from using for example (10). Finally, the initial possibility function (at ) can be computed from the initial PDF, constructed using the first bearings measurement , followed by the transformation (9). In the presence of non-linearities, practical computation of prediction and update steps of the possibility filter are non-trivial. There is no analytic closed-form solution to (12) and (13) and hence we need to apply a numerical approximation. One such approximation would be a grid-based method, used in [8]. Grid methods, however, suffer from the curse of dimensionality and instead we choose to implement the possibility filter using the SMC method. SMC algorithms have become widespread in Bayesian estimation and their properties, extensions and applications have been studied in numerous publications [4, 17-19]. 4 Monte Carlo approximation of possibility filter A practical implementation of the possibility filter for stochastic filtering is based on an adaptation of the SMC method for the sake of propagating the support points of the posterior p.f. over time. These points are known as particles and hence the resulting filter is referred to as the PPF. The first problem is that one cannot sample directly from possibility functions [7]. Instead, for a given possibility distribution function , samples must be drawn from a PDF p which is induced by . While there is an infinite number of ways one can construct p from (one being (10)), the natural solution is the one that results in the least informative p. This can be achieved by application of the maximum entropy principle, formally stated as follows: given a possibility distribution function , we are after the PDF that will maximise the differential entropy (defined as ), subject to constraints that: (i) integrates to and (ii) is lower or equal to for every . The described constrained optimisation is numerically difficult to solve and instead we will adopt its approximation, achieved by what we refer to as the 'water pouring operation' [7]. First note that if we were to ignore conditions (i) and (ii) above, the uniform probability distribution over the support of p would be the solution. By the water-pouring operation, we construct as the density closest to the uniform, which satisfies constraints (i) and (ii) above. This operation is denoted and is illustrated in Fig. 1 for one-dimensional continuous uncertain variable. The same principle can be applied to a discrete-valued uncertain variable. Fig. 1Open in figure viewerPowerPoint Water-pouring operation for a continuous distribution in one dimension A sample drawn from the probability distribution has to be weighted by . The set of weighted samples is then a Monte Carlo approximation of the p.f. . Note that the density of samples has no impact on the supremum, as opposed to the integral in the standard formulation. The pseudo-code of the kth cycle of the PPF is given in Algorithm 1 (see Fig. 2). The input to the algorithm is the bearings measurement and the weighted sample , with the property , which approximates the posterior p.f. at , i.e. . The code in Algorithm 1 (Fig. 2) is self-explanatory, except for line 11; here we apply the water-pouring operation to discrete p.f. in order to induce the probability mass function. The loop in lines 10–14 performs resampling for the sake of focusing the computational effort on the volume of the state space with non-negligible likelihood. The point estimate according to (14) is computed in line 16 for reporting only. Fig. 2Open in figure viewerPowerPoint Algorithm 1: The kth cycle of the PPF 5 Numerical results The observer–target scenario used in simulations is inspired by sonar underwater surveillance. Its top-down view is plotted in Fig. 3, with the start of each trajectory marked with 'white square'. The observer performs two manoeuvres while the target is moving with CV. The sampling interval is s; the target initial range and bearing are km and 80° (clockwise from north), respectively; the number of measurements is . We compare two stochastic filters for bearings-only TMA: the standard [4]PF and the PPF. Fig. 3Open in figure viewerPowerPoint Top-down view of the TMA scenario used in simulations The initial PDF used in the standard PF is adopted as being Gaussian, with the mean and covariance matrix computed using the first bearings measurement [4]. Thus, , with (15) (16) where [4]: , and . Here is a parameter of the PF indicating the (initial) target range and is adopted as km (recall, the true range is km); is the standard deviation assigned to the value of and is adopted as D km; is the standard deviation of measurement noise introduced in Section 2.1, and its value is . Finally, and which feature in (16) are m/s. The initial set of particles in the standard PF is a random draw from , with equal weights. The initial set of particles in the PPF is also a random draw from , however the weights are computed using (11) followed by normalisation , for . Standard deviation of process noise is adopted as in both stochastic filters. The output of the standard PF is the target state estimate computed as the weighted mean of the particles. Next we conduct simulation-based performance evaluation in three different setups, detailed below. Note that in all simulations, the two stochastic filters for bearings-only TMA are unchanged. 5.1 Matched models In this setup for performance evaluation, the stochastic models (the motion model and the measurement model) are matched to the models described in Section 2.1 and used in the standard PF. The PPF is using the same Gaussian models, however, in the form of Gaussian possibility functions (11). The number of particles in both the standard PF and the PPF was set to . Fig. 4a shows the RMS positional errors, obtained from 500 independent Monte Carlo runs, against the theoretical Cramer–Rao lower bound (CRLB) [4]. None of the two contesting filters diverged in 500 runs. Fig. 4b displays the result of a statistical consistency test based on the 50-run-average normalised estimation error squared (NEES) [11]. The horizontal dashed lines indicate the 95% acceptance interval. The results in Fig. 4 are shown from onwards, that is after the first manoeuvre, when the target state becomes fully observable (Note that prior to the first manoeuvre, the target range estimate is biased.). Fig. 4Open in figure viewerPowerPoint Matched models (the number of particles ) (a) RMS positional error against the theoretical CRLB, (b) Statistical consistency test result The conclusion from this first performance evaluation is that in the case where the models are matched, the PPF performs equally good as the standard PF. Using a large number of particles, the RMS errors of both TMA filters are very close to the best achievable error performance indicated by the CRLB. In addition, the average NEES of both contesting TMA filters is within the acceptance interval, indicating statistical consistency. Finally, we point out that the computational cost of the two TMA filters is similar. 5.2 Mismatched measurement model The second simulation setup investigates the performance of the two TMA filters when the measurement noise models are mismatched and the number of particles N is reduced. We continue to use the same two PFs (the standard PF and the PPF), but now the actual measurement noise samples in (5) are drawn from a zero-mean t location-scale (a.k.a. generalised Student-t) distribution [20, p. 102] with and the degrees-of-freedom parameter varied (rather than from a Gaussian distribution). The results obtained from 500 Monte Carlo runs are shown in Table 1. Due to the measurement model mismatch and a smaller number of particles, the two contesting TMA filters occasionally can diverge. A divergence is declared if the positional error at the end of the scenario is larger than 1 km. Table 1 presents the percentage of divergent runs for different values of N and . Note that as is increased, the tails of the t location-scale distribution are reduced, and in the limit , this distribution equals the Gaussian with the same mean and standard deviation. Table 1. Percentage of divergent runs: measurement model mismatch (a) 3 5 8 standard PF 43.6 21.2 12.4 5.4 PPF 21.0 6.2 1.8 0.6 (b) 3 5 8 standard PF 34.0 11.6 6.2 0.6 PPF 17.8 2.8 0.4 0.0 From Table 1 one can observe that the PPF is more robust than the standard PF. The most dramatic improvement can be noted for , when the model mismatch is relatively mild. For example, at , the percentage of divergent runs is reduced 7 and 15 times using and , respectively. The PPF is also more robust against the reduction in the number of particles N; at , which corresponds to the matched model case, the probability of divergence is significantly lower. 5.3 Mismatched motion model The third simulation setup investigates the performance when the actual target motion model is not matched to the motion model used by filters. In particular, at , the target performs a coordinated turn [11] with the turn rate of , lasting D sampling intervals. For a given D, a deviation from the initial target course can be easily computed as a product . We use the same two TMA filters (the standard PF and the PPF) tested earlier and estimate the probability of track divergence from 500 Monte Carlo runs, in the same manner described in Section 5.2. Table 2 shows the percentage of divergent runs as a function of the duration of the turn. The number of particles in both TMA filters is (i) and (ii) . Table 2. Percentage of divergent runs: motion model mismatch (a) Duration of the turn D 2 4 6 8 standard PF 5.0 9.0 20.2 42.8 PPF 0.2 0.4 10.4 34.8 (b) Duration of the turn D 2 4 6 8 standard PF 1.0 2.4 8.0 38.8 PPF 0.0 0.0 3.2 29.0 Table 2 demonstrates that the PPF is more robust than the standard PF to the mismatch in target dynamics. The most dramatic improvement again is achieved when the model mismatch is relatively mild, that is for short duration turns (e.g. turn duration of 2 and 4 sampling intervals). 6 Conclusions Stochastic filtering in the Bayesian framework relies heavily on mathematical models, in particular on the target motion model and the sensor observation model. The paper demonstrated, in the context of bearings-only TMA, that sensitivity to model mismatch can be greatly reduced by performing stochastic filtering using the possibility functions. A comparison between the standard PF and the PPF reveals the following. In the absence of a model mismatch, the two filters perform identically. In the presence of a measurement or a motion model mismatch, the PPF is consistently more robust, resulting in a lower probability of divergence, than the standard PF developed in the Bayesian framework. The improvement is particulary prominent when the model mismatch is relatively mild. Robustness of the PPF in the presence of a model mismatch intuitively can be explained by the fact that it is not assuming the precise probabilistic models. Instead, it is using only an upper bound for probability distributions, which is more likely to be correct. 7 Acknowledgments This research was partly funded by the Collaborative Research Agreement 'Robust Bayesian TMA using a class of outer measures' between DST Group and RMIT University. 8 References 1Becker, K.: ' Target motion analysis (TMA)', in S. Stergiopoulos, (Ed.): ' Advanced signal processing handbook' ( CRC Press, 2001), ch. 9, pp. 385– 408 2Lingren, A. 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