Multiple-Model Cardinality Balanced Multitarget Multi-Bernoulli Filter for Tracking Maneuvering Targets

Abstract

By integrating the cardinality balanced multitarget multi-Bernoulli (CBMeMBer) filter with the interacting multiple models (IMM) algorithm, an MM-CBMeMBer filter is proposed in this paper for tracking multiple maneuvering targets in clutter. The sequential Monte Carlo (SMC) method is used to implement the filter for generic multi-target models and the Gaussian mixture (GM) method is used to implement the filter for linear-Gaussian multi-target models. Then, the extended Kalman (EK) and unscented Kalman filtering approximations for the GM-MM-CBMeMBer filter to accommodate mildly nonlinear models are described briefly. Simulation results are presented to show the effectiveness of the proposed filter.

1. Introduction

Recently, the random-finite-set-(RFS-) based multitarget tracking approaches [1] have attracted extensive attention. Although theoretically solid, the RFS-based approaches usually are involved with intractable computations. By introducing the finite-set statistics (FISST) [2], Mahler developed the probability hypothesis density (PHD) [3] and cardinalized PHD (CPHD) [4] filters, which have been shown to be a computationally tractable alternative to full multitarget Bayes filters in the RFS framework. The sequential Monte Carlo (SMC) implementations for the PHD and CPHD filters were devised by Zajic and Mahler [5], Sidenbladh [6], and Vo et al. [7]. Vo et al. and Zhang et al. [8–10] devised the Gaussian mixture (GM) implementations for the PHD and CPHD filters under the linear-Gaussian assumption on target dynamics, birth process, and sensor model. The PHD-based approaches have been successfully used for many real-world problems [11–13]. However, the SMC-PHD and SMC-CPHD approaches require clustering to extract state estimates from the particle population, which is expensive and unreliable [14, 15].

In 2007, Mahler proposed the multitarget multi-Bernoulli (MeMBer) [2] recursion, which is an approximation to the full multitarget Bayes recursion using multi-Bernoulli RFSs under low clutter density scenarios. In 2009, Vo et al. showed that the MeMBer filter overestimates the number of targets and proposed a cardinality-balanced MeMBer (CBMeMBer) filter [16] to reduce the cardinality bias. Then, the SMC and GM implementations for the MeMBer and CBMeMBer filters were, respectively, proposed for generic and linear-Gaussian dynamic and measurement models. The MeMBer and CBMeMBer recursions propagate not the moments and cardinality distributions which are propagated by the PHD and CPHD filters but rather the approximate multitarget multi-Bernoulli posterior density. Therefore, the key advantage of the SMC-CBMeMBer filter over the SMC-PHD and SMC-CPHD filters is that the multi-Bernoulli representation of the posterior density allows reliable and inexpensive extraction of state estimates. The CBMeMBer filter has been applied for tracking multiple targets according to their audio and visual information [17].

The original CBMeMBer filter does not consider the target maneuvers. Maneuvering targets might switch between different models of operation, so tracking using a single-model CBMeMBer filter might fail since the filter does not match the actual system dynamics. It is well known that the interacting multiple models (IMM) approaches [18] have been proven to be very effective and have better performance than the single-model filters in tracking a single maneuvering target without clutter. In the IMM approaches, a bank of filters, each matched with a different target motion model, operate in parallel. In general, there are three key steps in the IMM estimators: (1) mixing the model-conditioned estimates; (2) model-conditioned base-state estimation; (3) deriving the overall state estimate by combining the estimates from each model-conditioned base-state filters.

By integrating the CBMeMBer filter with the IMM algorithm, an MM-CBMeMBer filter is proposed to address the problem of tracking multiple maneuvering targets in clutter, which is much more difficult than the problem of tracking a single maneuvering target without clutter since the association between the measurements and the targets is unknown. The SMC method is used to implement the filter for generic multitarget models while the GM method is used to implement the filter for linear-Gaussian multitarget models. Then, the extended Kalman (EK) [19] and unscented Kalman (UK) [20] filtering approximations for the GM-MM-CBMeMBer filter to accommodate mildly nonlinear models are described briefly. Nonlinear and linear-Gaussian examples of multiple maneuvering targets tracking are, respectively, presented for comparing the performance of the MM-CBMeMBer filter with that of the single-model CBMeMBer filters, MM-PHD filter [21–24], and MM-CPHD filter [25]. The simulation results show that (1) the proposed filter can estimate the number and states of multiple maneuvering targets effectively, whereas the performance of the single-model CBMeMBer filters is rather poor; (2) under relatively low clutter density, the SMC-MM-CBMeMBer filter outperforms the SMC-MM-PHD and SMC-MM-CPHD filters; (3) the performance of the GM-MM-CBMeMBer filter is similar to that of the GM-MM-PHD filter and hence is inferior to that of GM-MM-CPHD filter.

The rest of the paper is organized as follows. Section 2 describes the problem of multiple maneuvering targets tracking. In Section 3, the MM-CBMeMBer recursion is given. The generic SMC implementation of the MM-CBMeMBer filter is described in Section 4. The analytic GM implementation of the MM-CBMeMBer filter for linear-Gaussian multitarget models and its EK and UK extensions for nonlinear multitarget models are, respectively, given in Section 5. Numerical studies are shown in Section 6. The conclusions and the future work are given in Section 7.

2. Problem Statement for Multiple Maneuvering Targets Tracking

The multiple maneuvering targets appear and disappear randomly against time over an observation region. At time 𝑘, let 𝐱_𝑘∈ℝ^𝑛 denote the kinematical state of a target and 𝑛_𝑘∈ℕ the label of the model in effect, where ℕ is the discrete set of all model labels. The models follow a discrete Markov chain with transition probability ℎ_{𝑘∣𝑘−1}(𝑛_𝑘∣𝑛_𝑘−1). Let 𝐲_𝑘=(𝐱_𝑘,𝑛_𝑘)∈ℝ^𝑛×ℕ denote the augmented state vector, whose transition is governed by the density 𝑓_{𝑘∣𝑘−1}𝐲_𝑘∣𝐲_𝑘−1=𝑓_{𝑘∣𝑘−1}𝐱_𝑘∣𝐱_𝑘−1,𝑛_𝑘ℎ_{𝑘∣𝑘−1}𝑛_𝑘∣𝑛_𝑘−1, (1) where 𝑓_{𝑘∣𝑘−1}(𝐱_𝑘∣𝐱_𝑘−1,𝑛_𝑘) is the kinematical state transition density conditioned on model 𝑛_𝑘.

The measurement originates either from target or from random clutter (false alarm). Moreover, the target-generated measurements are indistinguishable from the clutter. At time 𝑘, let 𝐳_𝑘∈ℝ^𝑚 denote the measurement vector received by a sensor. The single-measurement single-target likelihood is described by the density conditioned on model 𝑛_𝑘𝑔_𝑘𝐳_𝑘∣𝐲_𝑘=𝑓_𝑘∣𝑘𝐳_𝑘∣𝐱_𝑘,𝑛_𝑘. (2)

At time 𝑘, let 𝑇_𝑘 denote the number of the existing targets and 𝑆_𝑘 the number of the measurements. Then, multiple augmented states and unlabelled sensor measurements can be represented as finite sets 𝑌_𝑘={(𝐱_𝑘^(𝑖),𝑛_𝑘^(𝑖))}^𝑇𝑘_𝑖=1 and 𝑍_𝑘={𝐳_𝑘^(𝑠)}^𝑆𝑘_𝑠=1, respectively. In addition, let 𝑍_1∶𝑘≜𝑍₁,…,𝑍_𝑘 denote a sequence of the measurement sets available up to and including time 𝑘.

3. MM-CBMeMBer Filter

A Bernoulli RFS 𝑌_𝑘 has probability 1−𝑟_𝑘 of being empty and probability 𝑟_𝑘 (0≤𝑟_𝑘≤1) of being a singleton whose only element is distributed according to a probability density 𝑝_𝑘. The probability density of 𝑌_𝑘 is 𝜋𝑌_𝑘=1−𝑟_𝑘,𝑌_𝑘𝑟=∅,_𝑘𝑝_𝑘𝐱_𝑘,𝑛_𝑘,𝑌_𝑘=𝐱_𝑘,𝑛_𝑘. (3)

A multi-Bernoulli RFS 𝑌_𝑘 is a union of a fixed number of independent Bernoulli RFSs 𝑌_𝑘^(𝑖), 𝑖=1,…,𝑀_𝑘, that is, 𝑌_𝑘=⋃^𝑀𝑘_𝑖=1𝑌_𝑘^(𝑖). 𝑌_𝑘 is thus completely described by the multi-Bernoulli parameter set {(𝑟_𝑘^(𝑖),𝑝_𝑘^(𝑖)(𝐱_𝑘,𝑛_𝑘))}^𝑀𝑘_𝑖=1 with the mean cardinality ∑^𝑀𝑘_𝑖=1𝑟_𝑘^(𝑖) and the probability density [2]𝜋𝑌_𝑘=^𝑀𝑘_𝑗=11−𝑟_𝑘^(𝑗)_{1≤𝑖1≠⋯≠𝑖𝑘||𝑌≤𝑀𝑘}^|𝑌𝑘|_𝑗=1𝑟^(𝑖𝑗)_𝑘𝑝^(𝑖𝑗)_𝑘𝐱_𝑘,𝑛_𝑘1−𝑟^(𝑖𝑗)_𝑘, (4) where |⋅| denotes the cardinality of a set.

Throughout this paper, we abbreviate a probability density of the form (4) by 𝜋(𝑌_𝑘)={(𝑟_𝑘^(𝑖),𝑝_𝑘^(𝑖)(𝐱_𝑘,𝑛_𝑘))}^𝑀𝑘_𝑖=1.

Let 𝑝_𝑆,𝑘(𝐲_𝑘−1) denote the probability that the maneuvering target with augmented state 𝐲_𝑘−1 survives at time 𝑘; let 𝑝_𝐷,𝑘(𝐲_𝑘) denote the probability that the maneuvering target with augmented state 𝐲_𝑘 generates an observation at time 𝑘. RFS modeling the multiple maneuvering targets state 𝑌_𝑘 and the sensor measurement 𝑍_𝑘 are, respectively, given by the union 𝑌_𝑘=_{𝐲𝑘−1∈𝑌𝑘−1}Ω_{𝑘∣𝑘−1}𝐲_𝑘−1∪Γ_𝑘,𝑍_𝑘=_{𝐲𝑘∈𝑌𝑘}Θ_𝑘𝐲_𝑘∪𝐾_𝑘, (5) where Γ_𝑘 denotes the multi-Bernoulli RFS of spontaneous births; the Bernoulli RFS Ω_{𝑘∣𝑘−1}(𝐲_𝑘−1) with 𝑟_𝑘=𝑝_𝑆,𝑘(𝐲_𝑘−1) and 𝑝_𝑘(𝐲_𝑘)=𝑓_{𝑘∣𝑘−1}(𝐲_𝑘∣𝐲_𝑘−1) is used to model the dynamic behavior of 𝐲_𝑘−1∈𝑌_𝑘−1; the Bernoulli RFS Θ_𝑘(𝐲_𝑘) with 𝑟_𝑘=𝑝_𝐷,𝑘(𝐲_𝑘) and 𝑝_𝑘(𝐳_𝑘)=𝑔_𝑘(𝐳_𝑘∣𝐲_𝑘) is used to model the observation behavior of 𝐲_𝑘∈𝑌_𝑘; the clutter is modeled as a Poisson RFS 𝐾_𝑘 with the intensity 𝜅_𝑘(𝐳_𝑘)=𝜆_𝑐,𝑘𝑓_𝑐,𝑘(𝐳_𝑘), where 𝜆_𝑐,𝑘 and 𝑓_𝑐,𝑘(⋅) are, respectively, the average clutter number and the probability density of clutter spatial distribution at time 𝑘.

Based on the above RFS models of the multiple maneuvering targets and the method of Mahler’s FISST, the MM-CBMeMBer filter, which implicitly requires a finite number of single-model CBMeMBer filters operate in parallel, is derived by introducing the mixing and combination strategies in the IMM approaches [18]. As the multiple-model approaches, the MM-CBMeMBer filter does not need a maneuver detection decision and undergoes a soft switching between the models. One cycle of the recursive MM-CBMeMBer algorithm can be described as follows.

(1) The Mixing and Prediction Stage. If at time 𝑘−1, the posterior density is a multi-Bernoulli of the form 𝜋_𝑘−1(𝑌_𝑘−1∣𝑍_1∶𝑘−1)={(𝑟^(𝑖)_𝑘−1,𝑝^(𝑖)_𝑘−1(𝐱_𝑘−1,𝑛_𝑘−1))}^𝑀𝑘−1_𝑖=1, then the mixed multi-Bernoulli density is 𝜋𝑌^⌢_𝑘−1^⌢_𝑘−1∣𝑍_1∶𝑘−1=𝑟^(𝑖)_𝑘−1,𝑝^(𝑖)_𝑘−1𝐱_𝑘−1,𝑛_𝑘^𝑀𝑘−1_𝑖=1, (6) where 𝑝^(𝑖)_𝑘−1𝐱_𝑘−1,𝑛_𝑘=_𝑛𝑘−1𝑝^(𝑖)_𝑘−1𝐱_𝑘−1,𝑛_𝑘,𝑛_𝑘−1=_𝑛𝑘−1𝑝^(𝑖)_𝑘−1𝑛_𝑘∣𝐱_𝑘−1,𝑛_𝑘−1𝑝^(𝑖)_𝑘−1𝐱_𝑘−1,𝑛_𝑘−1. (7)

Since the models switching is only decided by the model transition probability and is independent of the target kinematical state: =_𝑛𝑘−1ℎ_{𝑘∣𝑘−1}𝑛_𝑘∣𝑛_𝑘−1𝑝^(𝑖)_𝑘−1𝐱_𝑘−1,𝑛_𝑘−1 (8) is a combination of the previous model-dependent densities. Finally, the mixed and predicted density is also a multi-Bernoulli and is given by 𝜋_{𝑘∣𝑘−1}𝑌_𝑘∣𝑍_1∶𝑘−1=𝑟^(𝑖)_{𝑃,𝑘∣𝑘−1},𝑝^(𝑖)_{𝑃,𝑘∣𝑘−1}𝐱_𝑘,𝑛_𝑘^𝑀𝑘−1_𝑖=1∪𝑟^(𝑖)_Γ,𝑘,𝑝^(𝑖)_Γ,𝑘𝐱_𝑘,𝑛_𝑘^𝑀Γ,𝑘_𝑖=1, (9) where {(𝑟^(𝑖)_Γ,𝑘,𝑝^(𝑖)_Γ,𝑘(𝐱_𝑘,𝑛_𝑘))}^𝑀Γ,𝑘_𝑖=1 are the parameters of the multi-Bernoulli RFS of births at time 𝑘: 𝑟^(𝑖)_{𝑃,𝑘∣𝑘−1}=𝑟^(𝑖)_{𝑘−1𝑛𝑘𝑛𝑘−1}ℎ_{𝑘∣𝑘−1}𝑛_𝑘∣𝑛_𝑘−1×𝑝^(𝑖)_𝑘−1⋅,𝑛_𝑘−1,𝑝_𝑆,𝑘⋅,𝑛_𝑘−1,𝑝^(𝑖)_{𝑃,𝑘∣𝑘−1}𝐱_𝑘,𝑛_𝑘=_𝑛𝑘−1ℎ_{𝑘∣𝑘−1}𝑛_𝑘∣𝑛_𝑘−1×𝑓_{𝑘∣𝑘−1}𝐱_𝑘∣⋅,𝑛_𝑘,𝑝^(𝑖)_𝑘−1⋅,𝑛_𝑘−1𝑝_𝑆,𝑘⋅,𝑛_𝑘−1×_{𝑛𝑘𝑛𝑘−1}ℎ_{𝑘∣𝑘−1}𝑛_𝑘∣𝑛_𝑘−1×𝑝^(𝑖)_𝑘−1⋅,𝑛_𝑘−1,𝑝_𝑆,𝑘⋅,𝑛_𝑘−1⁻¹, (10) where ⟨⋅,⋅⟩ defines the integral inner product, that is, 𝑝^(𝑖)_𝑘−1⋅,𝑛_𝑘−1,𝑝_𝑆,𝑘⋅,𝑛_𝑘−1=𝑝^(𝑖)_𝑘−1𝐱_𝑘−1,𝑛_𝑘−1𝑝_𝑆,𝑘𝐱_𝑘−1,𝑛_𝑘−1𝑑𝐱_𝑘−1. (11)

(2) The Update Stage. If at time 𝑘, the mixed and predicted density is a multi-Bernoulli of the form 𝜋_{𝑘∣𝑘−1}(𝑌_𝑘∣𝑍_1∶𝑘−1)={(𝑟^(𝑖)_{𝑘∣𝑘−1},𝑝^(𝑖)_{𝑘∣𝑘−1}(𝐱_𝑘,𝑛_𝑘))}^{𝑀𝑘∣𝑘−1}_𝑖=1, then the posterior density can be approximated by a multi-Bernoulli as follows:𝜋_𝑘𝑌_𝑘∣𝑍_𝑘≈𝑟^(𝑖)_𝐿,𝑘,𝑝^(𝑖)_𝐿,𝑘𝐱_𝑘,𝑛_𝑘^{𝑀𝑘∣𝑘−1}_𝑖=1∪𝑟_𝑈,𝑘𝐳_𝑘,𝑝_𝑈,𝑘𝐱_𝑘,𝑛_𝑘;𝐳_{𝑘𝐳𝑘∈𝑍𝑘}, (12) where𝑟^(𝑖)_𝐿,𝑘=𝑟^(𝑖)_{𝑘∣𝑘−1}1−∑_𝑛𝑘𝑝^(𝑖)_{𝑘∣𝑘−1}⋅,𝑛_𝑘,𝑝_𝐷,𝑘⋅,𝑛_𝑘1−𝑟^(𝑖)_{𝑘∣𝑘−1}∑_𝑛𝑘𝑝^(𝑖)_{𝑘∣𝑘−1}⋅,𝑛_𝑘,𝑝_𝐷,𝑘⋅,𝑛_𝑘,𝑝^(𝑖)_𝐿,𝑘𝐱_𝑘,𝑛_𝑘=1−𝑝_𝐷,𝑘𝐱_𝑘,𝑛_𝑘𝑝^(𝑖)_{𝑘∣𝑘−1}𝐱_𝑘,𝑛_𝑘1−∑_𝑛𝑘𝑝^(𝑖)_{𝑘∣𝑘−1}⋅,𝑛_𝑘,𝑝_𝐷,𝑘⋅,𝑛_𝑘,𝑟_𝑈,𝑘𝐳_𝑘=^{𝑀𝑘∣𝑘−1}_𝑖=11−𝑟^(𝑖)_{𝑘∣𝑘−1}𝑟^(𝑖)_{𝑘∣𝑘−1}∑_𝑛𝑘𝑝^(𝑖)_{𝑘∣𝑘−1}⋅,𝑛_𝑘,𝑔_𝑘𝐳_𝑘∣⋅,𝑛_𝑘𝑝_𝐷,𝑘⋅,𝑛_𝑘1−𝑟^(𝑖)_{𝑘∣𝑘−1}∑_𝑛𝑘𝑝^(𝑖)_{𝑘∣𝑘−1}⋅,𝑛_𝑘,𝑝_𝐷,𝑘⋅,𝑛_𝑘²×𝜅_𝑘(𝐳_𝑘)+^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^(𝑖)_{𝑘∣𝑘−1}∑_𝑛𝑘𝑝^(𝑖)_{𝑘∣𝑘−1}⋅,𝑛_𝑘,𝑔_𝑘𝐳_𝑘∣⋅,𝑛_𝑘𝑝_𝐷,𝑘⋅,𝑛_𝑘1−𝑟^(𝑖)_{𝑘∣𝑘−1}∑_𝑛𝑘𝑝^(𝑖)_{𝑘∣𝑘−1}⋅,𝑛_𝑘,𝑝_𝐷,𝑘⋅,𝑛_𝑘⁻¹,𝑝_𝑈,𝑘𝐱_𝑘,𝑛_𝑘;𝐳_𝑘=∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^(𝑖)_{𝑘∣𝑘−1}/1−𝑟^(𝑖)_{𝑘∣𝑘−1}𝑝^(𝑖)_{𝑘∣𝑘−1}𝐱_𝑘,𝑛_𝑘𝑔_𝑘𝐳_𝑘∣𝐱_𝑘,𝑛_𝑘𝑝_𝐷,𝑘𝐱_𝑘,𝑛_𝑘∑^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^(𝑖)_{𝑘∣𝑘−1}/1−𝑟^(𝑖)_{𝑘∣𝑘−1}∑_𝑛𝑘𝑝^(𝑖)_{𝑘∣𝑘−1}⋅,𝑛_𝑘,𝑔_𝑘𝐳_𝑘∣⋅,𝑛_𝑘𝑝_𝐷,𝑘⋅,𝑛_𝑘. (13)

(3) The Multitarget State Estimation. For the multi-Bernoulli representation 𝜋_𝑘(𝑌_𝑘∣𝑍_𝑘)={(𝑟_𝑘^(𝑖),𝑝_𝑘^(𝑖)(𝐱_𝑘,𝑛_𝑘))}^𝑀𝑘_𝑖=1, the extraction of multitarget number and state estimates are straightforward since the probability 𝑟_𝑘^(𝑖) indicates how likely the 𝑖th hypothesized track is a true track, and the posterior density 𝑝_𝑘^(𝑖)(𝐱_𝑘,𝑛_𝑘) describes the distribution of the estimated augmented state of the track. The state estimation procedure for the MM-CBMeMBer filter [8] is summarized in Algorithm 1.

given 𝜋𝑘(𝑌𝑘∣𝑍𝑘)={(𝑟𝑘(𝑖),𝑝𝑘(𝑖)(𝐱𝑘,𝑛𝑘))}𝑀𝑘𝑖=1; set 𝑇𝑘=0, 𝑋𝑘=Ø     for  𝑖=1,…,𝑀𝑘,      if  𝑟𝑘(𝑖)> a given threshold (i.e. 0.5);        𝑇𝑘=𝑇𝑘+1,  𝐱(𝑘𝑇)𝑘=∑𝑝𝑘(𝑖)(𝑛𝑘)𝑛𝑘probabilityofmodel𝑛𝑘⋅𝐱𝑘𝑝𝑘(𝑖)(𝐱𝑘|𝑛𝑘)𝑑𝐱stateestimationconditionedonmodel𝑛𝑘𝑘=∑𝑛𝑘∫𝐱𝑘𝑝𝑘(𝑖)(𝐱𝑘,𝑛𝑘)𝑑𝐱𝑘,        𝑋𝑘∶=𝑋𝑘,𝐱(𝑘𝑇)𝑘;      end;     end;     output: 𝑋𝑘={𝐱𝑘(𝑖)}𝑇𝑘𝑖=1

Algorithm 1 

Multitarget state estimation procedure for the MM-CBMeMBer filter.

4. SMC-MM-CBMeMBer Filter

In this section, a generic SMC implementation of the proposed MM-CBMeMBer filter is presented for accommodating nonlinear dynamic and measurement models. In this implementation, the samples or particles, which are used to represent the multi-Bernoulli density of multiple maneuvering targets, consists of the kinematical state and model information with associated weights. One cycle of the recursive SMC-MM-CBMeMBer algorithm can be described as follows.

(1) The SMC Mixing and Prediction Stage. Suppose that at time 𝑘−1 the multi-Bernoulli posterior density 𝜋_𝑘−1(𝑌_𝑘−1∣𝑍_1∶𝑘−1)={(𝑟̃^(𝑖)_𝑘−1,𝑝^(𝑖)_𝑘−1(𝐱_𝑘−1,𝑛_𝑘−1))}^𝑀𝑘−1_𝑖=1 is given and each 𝑝^(𝑖)_𝑘−1(𝐱_𝑘−1,𝑛_𝑘−1), 𝑖=1,…,𝑀_𝑘−1, is composed of a set of weighted samples {𝜔^(𝑖,𝑙)_𝑘−1,𝐱^(𝑖,𝑙)_𝑘−1,𝑛^(𝑖,𝑙)_𝑘−1}^{𝐿(𝑖)𝑘−1}_𝑙=1, 𝑝^(𝑖)_𝑘−1𝐱_𝑘−1,𝑛_𝑘−1=^{𝐿(𝑖)𝑘−1}_𝑙=1𝜔^(𝑖,𝑙)_𝑘−1𝛿𝐱_𝑘−1−𝐱^(𝑖,𝑙)_𝑘−1,𝑛_𝑘−1−𝑛^(𝑖,𝑙)_𝑘−1, (14) where 𝛿(𝐱−𝐱^(𝑐)) is Dirac delta function centered at 𝐱^(𝑐). Then, the mixed and predicted multi-Bernoulli density 𝜋_{𝑘∣𝑘−1}(𝑌_𝑘∣𝑍_1∶𝑘−1)={(𝑟̃^(𝑖)_{𝑃,𝑘∣𝑘−1},𝑝^(𝑖)_{𝑃,𝑘∣𝑘−1}(𝐱_𝑘,𝑛_𝑘))}^𝑀𝑘−1_𝑖=1∪{(𝑟̃^(𝑖)_Γ,𝑘,𝑝^(𝑖)_Γ,𝑘(𝐱_𝑘,𝑛_𝑘))}^𝑀Γ,𝑘_𝑖=1 can be computed as follows: 𝑟̃^(𝑖)_{𝑃,𝑘∣𝑘−1}=𝑟̃^𝐿(𝑖)_𝑘−1^{(𝑖)𝑘−1}_𝑙=1𝜔^(𝑖,𝑙)_𝑘−1𝑝_𝑆,𝑘𝐱^(𝑖,𝑙)_𝑘−1,𝑛^(𝑖,𝑙)_𝑘−1ℎ_{𝑘∣𝑘−1}𝑛^(𝑖,𝑙)_{𝑃,𝑘∣𝑘−1}∣𝑛^(𝑖,𝑙)_𝑘−1𝛼_𝑘^(𝑖)𝑛^(𝑖,𝑙)_{𝑃,𝑘∣𝑘−1}∣𝑛^(𝑖,𝑙)_𝑘−1,𝑟̃^(𝑖)_Γ,𝑘𝑝=parametergivenbybirthmodel,_{(𝑖)𝑃,𝑘∣𝑘−1}𝐱_𝑘,𝑛_𝑘=^{𝐿(𝑖)𝑘−1}_𝑙=1𝜔^(𝑖,𝑙)_{𝑃,𝑘∣𝑘−1}𝛿𝐱_𝑘−𝐱^(𝑖,𝑙)_{𝑃,𝑘∣𝑘−1},𝑛_𝑘−𝑛^(𝑖,𝑙)_{𝑃,𝑘∣𝑘−1},𝑝^(𝑖)_Γ,𝑘𝐱_𝑘,𝑛_𝑘=^{𝐿(𝑖)Γ,𝑘}_𝑙=1𝜔^(𝑖,𝑙)_Γ,𝑘𝛿𝐱_𝑘−𝐱^(𝑖,𝑙)_Γ,𝑘,𝑛_𝑘−𝑛^(𝑖,𝑙)_Γ,𝑘, (15) where the particles 𝐱^(𝑖,𝑙)_{𝑃,𝑘∣𝑘−1}, 𝑛^(𝑖,𝑙)_{𝑃,𝑘∣𝑘−1} corresponding to the surviving maneuvering targets can be derived by sampling from the proposal densities 𝑞_𝑘^(𝑖)(⋅∣𝐱_𝑘−1,𝑛_𝑘,𝑍_𝑘) and 𝛼_𝑘^(𝑖)(⋅∣𝑛_𝑘−1)𝑛^(𝑖,𝑙)_{𝑃,𝑘∣𝑘−1}∼𝛼_𝑘^(𝑖)⋅∣𝑛^(𝑖,𝑙)_𝑘−1𝐱^(𝑖,𝑙)_{𝑃,𝑘∣𝑘−1}∼𝑞_𝑘^(𝑖)⋅∣𝐱^(𝑖,𝑙)_𝑘−1,𝑛^(𝑖,𝑙)_{𝑃,𝑘∣𝑘−1},𝑍_𝑘𝑙=1,…,𝐿^(𝑖)_𝑘−1 (16) with the associated weights 𝜔^(𝑖,𝑙)_{𝑃,𝑘∣𝑘−1}=𝜔̆^(𝑖,𝑙)_{𝑃,𝑘∣𝑘−1}∑^{𝐿(𝑖)𝑘−1}_𝑙=1𝜔̆^(𝑖,𝑙)_{𝑃,𝑘∣𝑘−1},𝜔̆^(𝑖,𝑙)_{𝑃,𝑘∣𝑘−1}=𝑓_{𝑘∣𝑘−1}𝐱^(𝑖,𝑙)_{𝑃,𝑘∣𝑘−1}∣𝐱^(𝑖,𝑙)_𝑘−1,𝑛^(𝑖,𝑙)_{𝑃,𝑘∣𝑘−1}𝑝_𝑆,𝑘𝐱^(𝑖,𝑙)_𝑘−1,𝑛^(𝑖,𝑙)_𝑘−1𝑞_𝑘^(𝑖)𝐱^(𝑖,𝑙)_{𝑃,𝑘∣𝑘−1}∣𝐱^(𝑖,𝑙)_𝑘−1,𝑛^(𝑖,𝑙)_{𝑃,𝑘∣𝑘−1},𝑍_𝑘⋅ℎ^(𝑖)_{𝑘∣𝑘−1}𝑛^(𝑖,𝑙)_{𝑃,𝑘∣𝑘−1}∣𝑛^(𝑖,𝑙)_𝑘−1𝛼_𝑘^(𝑖)𝑛^(𝑖,𝑙)_{𝑃,𝑘∣𝑘−1}∣𝑛^(𝑖,𝑙)_𝑘−1⋅𝜔^(𝑖,𝑙)_𝑘−1 (17) and the particles 𝐱^(𝑖,𝑙)_Γ,𝑘, 𝑛^(𝑖,𝑙)_Γ,𝑘 corresponding to the new born maneuvering targets can be derived by sampling from the proposal densities 𝑏_𝑘^(𝑖)(⋅∣𝑛_𝑘,𝑍_𝑘) and 𝛽_𝑘^(𝑖)(⋅)𝑛^(𝑖,𝑙)_Γ,𝑘∼𝛽_𝑘^(𝑖)(⋅)𝐱^(𝑖,𝑙)_Γ,𝑘∼𝑏_𝑘^(𝑖)⋅∣𝑛^(𝑖,𝑙)_Γ,𝑘,𝑍_𝑘𝑙=1,…,𝐿^(𝑖)_Γ,𝑘 (18) with the associated weights 𝜔^(𝑖,𝑙)_Γ,𝑘=𝜔̆^(𝑖,𝑙)_Γ,𝑘∑^{𝐿(𝑖)Γ,𝑘}_𝑙=1𝜔̆^(𝑖,𝑙)_Γ,𝑘,𝜔̆^(𝑖,𝑙)_Γ,𝑘=𝑝^(𝑖)_Γ,𝑘𝐱^(𝑖,𝑙)_Γ,𝑘,𝑛^(𝑖,𝑙)_Γ,𝑘𝑏_𝑘^(𝑖)𝐱^(𝑖,𝑙)_Γ,𝑘∣𝑛^(𝑖,𝑙)_Γ,𝑘,𝑍_𝑘𝛽_𝑘^(𝑖)𝑛^(𝑖,𝑙)_Γ,𝑘. (19)

(2) The SMC Update Stage. Suppose that at time 𝑘 the mixed and predicted multi-Bernoulli density 𝜋_{𝑘∣𝑘−1}(𝑌_𝑘∣𝑍_1∶𝑘−1)={(𝑟̃^(𝑖)_{𝑘∣𝑘−1},𝑝^(𝑖)_{𝑘∣𝑘−1}(𝐱_𝑘,𝑛_𝑘))}^{𝑀𝑘∣𝑘−1}_𝑖=1 is given and each 𝑝^(𝑖)_{𝑘∣𝑘−1}(𝐱_𝑘,𝑛_𝑘), 𝑖=1,…,𝑀_{𝑘∣𝑘−1}, is composed of a set of weighted samples {𝜔^(𝑖,𝑙)_{𝑘∣𝑘−1},𝐱^(𝑖,𝑙)_{𝑘∣𝑘−1},𝑛^(𝑖,𝑙)_{𝑘∣𝑘−1}}^{𝐿(𝑖)𝑘∣𝑘−1}_𝑙=1, 𝑝^(𝑖)_{𝑘∣𝑘−1}𝐱_𝑘,𝑛_𝑘=^{𝐿(𝑖)𝑘∣𝑘−1}_𝑙=1𝜔^(𝑖,𝑙)_{𝑘∣𝑘−1}𝛿𝐱_𝑘−𝐱^(𝑖,𝑙)_{𝑘∣𝑘−1},𝑛_𝑘−𝑛^(𝑖,𝑙)_{𝑘∣𝑘−1}. (20)

Then, the multi-Bernoulli approximation of the updated density 𝜋_𝑘(𝑌_𝑘∣𝑍_1∶𝑘)≈{(𝑟̃^(𝑖)_𝐿,𝑘,𝑝^(𝑖)_𝐿,𝑘(𝐱_𝑘,𝑛_𝑘))}^{𝑀𝑘∣𝑘−1}_𝑖=1∪{(𝑟̃_𝑈,𝑘(𝐳_𝑘),𝑝_𝑈,𝑘(𝐱_𝑘,𝑛_𝑘;𝐳_𝑘))}_{𝐳𝑘∈𝑍𝑘} can be computed as follows:𝑟̃^(𝑖)_𝐿,𝑘=𝑟̃^(𝑖)_{𝑘∣𝑘−1}1−∑^{𝐿(𝑖)𝑘∣𝑘−1}_𝑙=1𝜔^(𝑖,𝑙)_{𝑘∣𝑘−1}𝑝_𝐷,𝑘𝐱^(𝑖,𝑙)_{𝑘∣𝑘−1},𝑛^(𝑖,𝑙)_{𝑘∣𝑘−1}1−𝑟̃^(𝑖)_{𝑘∣𝑘−1}∑^{𝐿(𝑖)𝑘∣𝑘−1}_𝑙=1𝜔^(𝑖,𝑙)_{𝑘∣𝑘−1}𝑝_𝐷,𝑘𝐱^(𝑖,𝑙)_{𝑘∣𝑘−1},𝑛^(𝑖,𝑙)_{𝑘∣𝑘−1},𝑝^(𝑖)_𝐿,𝑘𝐱_𝑘,𝑛_𝑘=^{𝐿(𝑖)𝑘∣𝑘−1}_𝑙=1𝜔^(𝑖,𝑙)_𝐿,𝑘𝛿𝐱_𝑘−𝐱^(𝑖,𝑙)_{𝑘∣𝑘−1},𝑛_𝑘−𝑛^(𝑖,𝑙)_{𝑘∣𝑘−1},𝑟̃_𝑈,𝑘𝐳_𝑘=^{𝑀𝑘∣𝑘−1}_𝑖=11−𝑟̃^(𝑖)_{𝑘∣𝑘−1}𝑟̃^(𝑖)_{𝑘∣𝑘−1}∑^{𝐿(𝑖)𝑘∣𝑘−1}_𝑙=1𝜔^(𝑖,𝑙)_{𝑘∣𝑘−1}𝑔_𝑘𝐳_𝑘∣𝐱^(𝑖,𝑙)_{𝑘∣𝑘−1},𝑛^(𝑖,𝑙)_{𝑘∣𝑘−1}𝑝_𝐷,𝑘𝐱^(𝑖,𝑙)_{𝑘∣𝑘−1},𝑛^(𝑖,𝑙)_{𝑘∣𝑘−1}1−𝑟̃^(𝑖)_{𝑘∣𝑘−1}∑^{𝐿(𝑖)𝑘∣𝑘−1}_𝑙=1𝜔^(𝑖,𝑙)_{𝑘∣𝑘−1}𝑝_𝐷,𝑘𝐱^(𝑖,𝑙)_{𝑘∣𝑘−1},𝑛^(𝑖,𝑙)_{𝑘∣𝑘−1}²×𝜅_𝑘𝐳_𝑘+^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟̃^(𝑖)_{𝑘∣𝑘−1}∑^{𝐿(𝑖)𝑘∣𝑘−1}_𝑙=1𝜔^(𝑖,𝑙)_{𝑘∣𝑘−1}𝑔_𝑘𝐳_𝑘∣𝐱^(𝑖,𝑙)_{𝑘∣𝑘−1},𝑛^(𝑖,𝑙)_{𝑘∣𝑘−1}𝑝_𝐷,𝑘𝐱^(𝑖,𝑙)_{𝑘∣𝑘−1},𝑛^(𝑖,𝑙)_{𝑘∣𝑘−1}1−𝑟̃^(𝑖)_{𝑘∣𝑘−1}∑^{𝐿(𝑖)𝑘∣𝑘−1}_𝑙=1𝜔^(𝑖,𝑙)_{𝑘∣𝑘−1}𝑝_𝐷,𝑘𝐱^(𝑖,𝑙)_{𝑘∣𝑘−1},𝑛^(𝑖,𝑙)_{𝑘∣𝑘−1}⁻¹,𝑝_𝑈,𝑘𝐱_𝑘,𝑛_𝑘;𝐳_𝑘=^{𝑀𝑘∣𝑘−1𝐿}_𝑖=1^{(𝑖)𝑘∣𝑘−1}_𝑙=1𝜔^(𝑖,𝑙)_𝑈,𝑘𝐳_𝑘𝛿𝐱_𝑘−𝐱^(𝑖,𝑙)_{𝑘∣𝑘−1},𝑛_𝑘−𝑛^(𝑖,𝑙)_{𝑘∣𝑘−1}, (21)

where 𝜔^(𝑖,𝑙)_𝐿,𝑘=𝜔̆^(𝑖,𝑙)_𝐿,𝑘∑^{𝐿(𝑖)𝑘∣𝑘−1}_𝑙=1𝜔̆^(𝑖,𝑙)_𝐿,𝑘,𝜔̆^(𝑖,𝑙)_𝐿,𝑘=𝜔^(𝑖,𝑙)_{𝑘∣𝑘−1}1−𝑝_𝐷,𝑘𝐱^(𝑖,𝑙)_{𝑘∣𝑘−1},𝑛^(𝑖,𝑙)_{𝑘∣𝑘−1}1−∑^{𝐿(𝑖)𝑘∣𝑘−1}_𝑙=1𝜔^(𝑖,𝑙)_{𝑘∣𝑘−1}𝑝_𝐷,𝑘𝐱^(𝑖,𝑙)_{𝑘∣𝑘−1},𝑛^(𝑖,𝑙)_{𝑘∣𝑘−1},𝜔^(𝑖,𝑙)_𝑈,𝑘𝐳_𝑘=𝜔̆^(𝑖,𝑙)_𝑈,𝑘𝐳_𝑘∑^{𝑀𝑘∣𝑘−1}_𝑖=1∑^{𝐿(𝑖)𝑘∣𝑘−1}_𝑙=1𝜔̆^(𝑖,𝑙)_𝑈,𝑘𝐳_𝑘,𝜔̆^(𝑖,𝑙)_𝑈,𝑘𝐳_𝑘=𝑟̃^(𝑖)_{𝑘∣𝑘−1}1−𝑟̃^(𝑖)_{𝑘∣𝑘−1}𝜔^(𝑖,𝑙)_{𝑘∣𝑘−1}𝑔_𝑘×𝐳_𝑘∣𝐱^(𝑖,𝑙)_{𝑘∣𝑘−1},𝑛^(𝑖,𝑙)_{𝑘∣𝑘−1}𝑝_𝐷,𝑘𝐱^(𝑖,𝑙)_{𝑘∣𝑘−1},𝑛^(𝑖,𝑙)_{𝑘∣𝑘−1}. (22)

(3) The Resampling and Pruning Stage. It is the same as the resampling and pruning stage of the SMC-CBMeMBer filter [16].

(4) The SMC Multitarget State Estimation. Given the SMC multi-Bernoulli posterior density 𝜋_𝑘𝑌_𝑘∣𝑍_1∶𝑘=𝑟̃_𝑘^(𝑖),𝑝_𝑘^(𝑖)𝐱_𝑘,𝑛_𝑘^𝑀𝑘_𝑖=1with𝑝_𝑘^(𝑖)𝐱_𝑘,𝑛_𝑘=^{𝐿𝑘(𝑖)}_𝑙=1𝜔_𝑘^(𝑖,𝑙)𝛿𝐱_𝑘−𝐱_𝑘^(𝑖,𝑙),𝑛_𝑘−𝑛_𝑘^(𝑖,𝑙), (23) from the method described in Algorithm 1, the SMC multitarget state estimation can be easily obtained as 𝑋_𝑘=𝐱_𝑘^(𝑖)𝑇_𝑘𝑖=1with𝐱_𝑘^(𝑖)=^{𝐿𝑘(𝑖)}_𝑙=1𝐱_𝑘^(𝑖,𝑙)𝜔_𝑘^(𝑖,𝑙),𝑖=1,…,𝑇_𝑘. (24)

Note that the MCMC move step [26] can be introduced for increasing the particle variety after the resample step without affecting the validity of the SMC approximation.

5. GM-MM-CBMeMBer Filter and Its EK and UK Extensions

An analytic solution to the MM-CBMeMBer recursion for linear-Gaussian multiple maneuvering targets models is presented in this section. The resulting filter propagates the GM multi-Bernoulli density against time. Some certain assumptions about the linear-Gaussian multiple maneuvering targets models are firstly summarized below.

(A) The dynamic and measurement models for the augmented state of each maneuvering target have the form 𝑓_{𝑘∣𝑘−1}𝐱_𝑘,𝑛_𝑘∣𝐱_𝑘−1,𝑛_𝑘−1=𝒩𝐱_𝑘;𝐹_𝑘𝑛_𝑘𝐱_𝑘−1,Α_𝑘𝑛_𝑘𝑄_𝑘𝑛_𝑘Α_𝑘𝑛_𝑘^𝑇×ℎ_{𝑘∣𝑘−1}𝑛_𝑘∣𝑛_𝑘−1,𝑔_𝑘𝐳_𝑘∣𝐱_𝑘,𝑛_𝑘=𝒩𝐳_𝑘;𝐻_𝑘𝑛_𝑘𝐱_𝑘,Β_𝑘𝑛_𝑘𝑅_𝑘𝑛_𝑘Β_𝑘𝑛_𝑘^𝑇, (25) where 𝒩(⋅;𝐦,𝑃) denotes the density of Gaussian distribution with the mean 𝐦 and covariance 𝑃; 𝐹_𝑘(𝑛_𝑘), 𝑄_𝑘(𝑛_𝑘), and Α_𝑘(𝑛_𝑘) are, respectively, the kinematical state transition, process noise covariance, and process noise coefficient matrixes conditioned on model 𝑛_𝑘; 𝐻_𝑘(𝑛_𝑘), 𝑅_𝑘(𝑛_𝑘), and Β_𝑘(𝑛_𝑘) are, respectively, the observation, observation noise covariance, and observation noise coefficient matrixes conditioned on model 𝑛_𝑘.

(B) The probabilities of maneuvering target survival and maneuvering target detection are independent of the kinematical state: 𝑝_𝑆,𝑘𝐱_𝑘−1,𝑛_𝑘−1=𝑝_𝑆,𝑘𝑛_𝑘−1,𝑝_𝐷,𝑘𝐱_𝑘,𝑛_𝑘=𝑝_𝐷,𝑘𝑛_𝑘. (26)

(C) The birth model for the maneuvering targets is a multi-Bernoulli with parameter set {(𝑟^(𝑖)_Γ,𝑘,𝑝^(𝑖)_Γ,𝑘(𝐱_𝑘,𝑛_𝑘))}^𝑀Γ,𝑘_𝑖=1, where 𝑝^(𝑖)_Γ,𝑘(𝐱_𝑘,𝑛_𝑘), 𝑖=1,…,𝑀_Γ,𝑘, are GM of the form 𝑝^(𝑖)_Γ,𝑘𝐱_𝑘,𝑛_𝑘=𝑝^(𝑖)_Γ,𝑘𝐱_𝑘∣𝑛_𝑘ℎ^(𝑖)_Γ,𝑘𝑛_𝑘=ℎ^(𝑖)_Γ,𝑘𝑛_𝑘^{𝐽(𝑖)Γ,𝑘(𝑛𝑘)}_𝑗=1𝜛^(𝑖,𝑗)_Γ,𝑘𝑛_𝑘×𝒩𝐱_𝑘;𝐦^(𝑖,𝑗)_Γ,𝑘𝑛_𝑘,𝑃^(𝑖,𝑗)_Γ,𝑘𝑛_𝑘, (27) where ℎ^(𝑖)_Γ,𝑘(𝑛_𝑘) is the distribution of model births and 𝑝^(𝑖)_Γ,𝑘(𝐱_𝑘∣𝑛_𝑘) is the distribution of the birth kinematical state given model 𝑛_𝑘.𝑝^(𝑖)_Γ,𝑘(𝐱_𝑘∣𝑛_𝑘) is GM of the form with the parameter set {𝜛^(𝑖,𝑗)_Γ,𝑘(𝑛_𝑘),𝐦^(𝑖,𝑗)_Γ,𝑘(𝑛_𝑘),𝑃^(𝑖,𝑗)_Γ,𝑘(𝑛_𝑘)}^{𝐽(𝑖)Γ,𝑘(𝑛𝑘)}_𝑗=1.

According to the above Assumptions A, B, and C, a closed form solution to the MM-CBMeMBer recursion, namely, the GM-MM-CBMeMBer filter, can be derived by applying the following two standard results for Gaussian functions: 𝒩𝐱;𝐹𝐱^′,𝑄𝒩𝐱^′;𝐦,𝑃𝑑𝐱^′=𝒩𝐱;𝐹𝐦,𝑄+𝐹𝑃𝐹^𝑇,𝒩(𝐳;𝐻𝐱,𝑅)𝒩(𝐱;𝐦,𝑃)=𝒩𝐳;𝐻𝐦,𝑅+𝐻𝑃𝐻^𝑇𝒩𝐱;𝐦^⌢,𝑃^⌢, (28) where 𝐾=𝑃𝐻^𝑇𝐻𝑃𝐻^𝑇+𝑅⁻¹,_⌢𝐦=𝐦+𝐾(𝐳−𝐻𝐦_⌢),𝑃=(𝐼−𝐾𝐻)𝑃. (29)

One cycle of the recursive GM-MM-CBMeMBer algorithm can be described as follows.

(1) The GM Mixing and Prediction Stage. Suppose that at time 𝑘−1 the multi-Bernoulli posterior density 𝜋_𝑘−1(𝑌_𝑘−1∣𝑍_1∶𝑘−1)={(𝑟^(𝑖)_𝑘−1,𝑝^(𝑖)_𝑘−1(𝐱_𝑘−1,𝑛_𝑘−1))}^𝑀𝑘−1_𝑖=1 is given and each 𝑝^(𝑖)_𝑘−1(𝐱_𝑘−1,𝑛_𝑘−1), 𝑖=1,…,𝑀_𝑘−1, is composed of GM of the form 𝑝^(𝑖)_𝑘−1𝐱_𝑘−1,𝑛_𝑘−1=^{𝐽(𝑖)𝑘−1(𝑛𝑘−1)}_𝑗=1𝜛^(𝑖,𝑗)_𝑘−1𝑛_𝑘−1×𝒩𝐱_𝑘−1;𝐦^(𝑖,𝑗)_𝑘−1𝑛_𝑘−1,𝑃^(𝑖,𝑗)_𝑘−1𝑛_𝑘−1. (30) Then, the mixed and predicted multi-Bernoulli density 𝜋_{𝑘∣𝑘−1}(𝑌_𝑘∣𝑍_1∶𝑘−1)={(𝑟^(𝑖)_{𝑃,𝑘∣𝑘−1},𝑝^(𝑖)_{𝑃,𝑘∣𝑘−1}(𝐱_𝑘,𝑛_𝑘))}^𝑀𝑘−1_𝑖=1∪{(𝑟^(𝑖)_Γ,𝑘,𝑝^(𝑖)_Γ,𝑘(𝐱_𝑘,𝑛_𝑘))}^𝑀Γ,𝑘_𝑖=1 can be computed as follows: 𝑟^(𝑖)_{𝑃,𝑘∣𝑘−1}=𝑟^(𝑖)_{𝑘−1𝑛𝑘𝑛𝑘−1}^{𝐽(𝑖)𝑘−1(𝑛𝑘−1)}_𝑗=1ℎ_{𝑘∣𝑘−1}𝑛_𝑘∣𝑛_𝑘−1×𝑝_𝑆,𝑘𝑛_𝑘−1𝜛^(𝑖,𝑗)_𝑘−1𝑛_𝑘−1,𝑝^(𝑖)_{𝑃,𝑘∣𝑘−1}𝐱_𝑘,𝑛_𝑘=_𝑛𝑘−1^{𝐽(𝑖)𝑘−1(𝑛𝑘−1)}_𝑗=1𝜛^(𝑖,𝑗)_{𝑃,𝑘∣𝑘−1}𝑛_𝑘,𝑛_𝑘−1×𝒩𝐱_𝑘;𝐦^(𝑖,𝑗)_{𝑃,𝑘∣𝑘−1}𝑛_𝑘,𝑛_𝑘−1,𝑃^(𝑖,𝑗)_{𝑃,𝑘∣𝑘−1}𝑛_𝑘,𝑛_𝑘−1,𝑟^(𝑖)_Γ,𝑘,𝑝^(𝑖)_Γ,𝑘𝐱_𝑘,𝑛_𝑘^𝑀Γ,𝑘_𝑖=1=givenbythebirthmodel(27), (31) where 𝐦^(𝑖,𝑗)_{𝑃,𝑘∣𝑘−1}𝑛_𝑘,𝑛_𝑘−1=𝐹_𝑘𝑛_𝑘𝐦^(𝑖,𝑗)_𝑘−1𝑛_𝑘−1,𝑃^(𝑖,𝑗)_{𝑃,𝑘∣𝑘−1}𝑛_𝑘,𝑛_𝑘−1=𝐹_𝑘𝑛_𝑘𝑃^(𝑖,𝑗)_𝑘−1𝑛_𝑘−1𝐹_𝑘𝑛_𝑘^𝑇+Α_𝑘𝑛_𝑘𝑄_𝑘𝑛_𝑘Α_𝑘𝑛_𝑘^𝑇,𝜛^(𝑖,𝑗)_{𝑃,𝑘∣𝑘−1}𝑛_𝑘,𝑛_𝑘−1=ℎ_{𝑘∣𝑘−1}𝑛_𝑘∣𝑛_𝑘−1𝑝_𝑆,𝑘𝑛_𝑘−1𝜛^(𝑖,𝑗)_𝑘−1𝑛_𝑘−1×_{𝑛𝑘𝑛𝑘−1}^{𝐽(𝑖)𝑘−1(𝑛𝑘−1)}_𝑗=1ℎ_{𝑘∣𝑘−1}𝑛_𝑘∣𝑛_𝑘−1×𝑝_𝑆,𝑘𝑛_𝑘−1𝜛^(𝑖,𝑗)_𝑘−1𝑛_𝑘−1⁻¹. (32)

(2) The GM Update Stage. Suppose that at time 𝑘 the mixed and predicted multi-Bernoulli density 𝜋_{𝑘∣𝑘−1}(𝑌_𝑘∣𝑍_1∶𝑘−1)={(𝑟^(𝑖)_{𝑘∣𝑘−1},𝑝^(𝑖)_{𝑘∣𝑘−1}(𝐱_𝑘,𝑛_𝑘))}^{𝑀𝑘∣𝑘−1}_𝑖=1 is given and each 𝑝^(𝑖)_{𝑘∣𝑘−1}(𝐱_𝑘,𝑛_𝑘), 𝑖=1,…,𝑀_{𝑘∣𝑘−1}, is composed of GM of the form 𝑝^(𝑖)_{𝑘∣𝑘−1}𝐱_𝑘,𝑛_𝑘=^{𝐽(𝑖)𝑘∣𝑘−1(𝑛𝑘)}_𝑗=1𝜛^(𝑖,𝑗)_{𝑘∣𝑘−1}𝑛_𝑘×𝒩𝐱_𝑘;𝐦^(𝑖,𝑗)_{𝑘∣𝑘−1}𝑛_𝑘,𝑃^(𝑖,𝑗)_{𝑘∣𝑘−1}𝑛_𝑘. (33) Then, the multi-Bernoulli approximation of the updated density 𝜋_𝑘(𝑌_𝑘∣𝑍_1∶𝑘)≈{(𝑟^(𝑖)_𝐿,𝑘,𝑝^(𝑖)_𝐿,𝑘(𝐱_𝑘,𝑛_𝑘))}^{𝑀𝑘∣𝑘−1}_𝑖=1∪{(𝑟_𝑈,𝑘(𝐳_𝑘),𝑝_𝑈,𝑘(𝐱_𝑘,𝑛_𝑘;𝐳_𝑘))}_{𝐳𝑘∈𝑍𝑘} can be computed as follows:𝑟^(𝑖)_𝐿,𝑘=𝑟^(𝑖)_{𝑘∣𝑘−1}1−∑_𝑛𝑘∑^{𝐽(𝑖)𝑘∣𝑘−1(𝑛𝑘)}_𝑗=1𝜛^(𝑖,𝑗)_{𝑘∣𝑘−1}𝑛_𝑘𝑝_𝐷,𝑘𝑛_𝑘1−𝑟^(𝑖)_{𝑘∣𝑘−1}∑_𝑛𝑘∑^{𝐽(𝑖)𝑘∣𝑘−1(𝑛𝑘)}_𝑗=1𝜛^(𝑖,𝑗)_{𝑘∣𝑘−1}𝑛_𝑘𝑝_𝐷,𝑘𝑛_𝑘,𝑟_𝑈,𝑘𝐳_𝑘=^{𝑀𝑘∣𝑘−1}_𝑖=11−𝑟^(𝑖)_{𝑘∣𝑘−1}𝑟^(𝑖)_{𝑘∣𝑘−1}∑_𝑛𝑘∑^{𝐽(𝑖)𝑘∣𝑘−1(𝑛𝑘)}_𝑗=1𝜛^(𝑖,𝑗)_{𝑘∣𝑘−1}𝑛_𝑘𝑝_𝐷,𝑘𝑛_𝑘𝒬_𝑘^(𝑖,𝑗)𝑛_𝑘;𝐳_𝑘1−𝑟^(𝑖)_{𝑘∣𝑘−1}∑_𝑛𝑘∑^{𝐽(𝑖)𝑘∣𝑘−1(𝑛𝑘)}_𝑗=1𝜛^(𝑖,𝑗)_{𝑘∣𝑘−1}𝑛_𝑘𝑝_𝐷,𝑘(𝑛_𝑘)²×𝜅_𝑘𝐳_𝑘+^{𝑀𝑘∣𝑘−1}_𝑖=1𝑟^(𝑖)_{𝑘∣𝑘−1}∑_𝑛𝑘∑^{𝐽(𝑖)𝑘∣𝑘−1(𝑛𝑘)}_𝑗=1𝜛^(𝑖,𝑗)_{𝑘∣𝑘−1}𝑛_𝑘𝑝_𝐷,𝑘𝑛_𝑘𝒬_𝑘^(𝑖,𝑗)𝑛_𝑘;𝐳_𝑘1−𝑟^(𝑖)_{𝑘∣𝑘−1}∑_𝑛𝑘∑^{𝐽(𝑖)𝑘∣𝑘−1(𝑛𝑘)}_𝑗=1𝜛^(𝑖,𝑗)_{𝑘∣𝑘−1}𝑛_𝑘𝑝_𝐷,𝑘𝑛_𝑘⁻¹𝑝^(𝑖)_𝐿,𝑘𝐱_𝑘,𝑛_𝑘=∑^{𝐽(𝑖)𝑘∣𝑘−1(𝑛𝑘)}_𝑗=11−𝑝_𝐷,𝑘𝑛_𝑘𝜛^(𝑖,𝑗)_{𝑘∣𝑘−1}𝑛_𝑘𝒩𝐱_𝑘;𝐦^(𝑖,𝑗)_{𝑘∣𝑘−1}𝑛_𝑘,𝑃^(𝑖,𝑗)_{𝑘∣𝑘−1}𝑛_𝑘1−∑_𝑛𝑘∑^{𝐽(𝑖)𝑘∣𝑘−1(𝑛𝑘)}_𝑗=1𝜛^(𝑖,𝑗)_{𝑘∣𝑘−1}𝑛_𝑘𝑝_𝐷,𝑘𝑛_𝑘,𝑝_𝑈,𝑘𝐱_𝑘,𝑛_𝑘;𝐳_𝑘=^{𝑀𝑘∣𝑘−1𝐽}_𝑖=1^{(𝑖)𝑘∣𝑘−1(𝑛𝑘)}_𝑗=1𝜛^(𝑖,𝑗)_𝑈,𝑘𝑛_𝑘;𝐳_𝑘𝒩𝐱_𝑘;𝐦^(𝑖,𝑗)_𝑈,𝑘𝑛_𝑘;𝐳_𝑘,𝑃^(𝑖,𝑗)_𝑈,𝑘𝑛_𝑘, (34)

where 𝒬_𝑘^(𝑖,𝑗)𝑛_𝑘;𝐳_𝑘=𝒩𝐳_𝑘;𝐻_𝑘𝑛_𝑘𝐦^(𝑖,𝑗)_{𝑘∣𝑘−1}𝑛_𝑘,Β_𝑘𝑛_𝑘𝑅_𝑘𝑛_𝑘Β_𝑘𝑛_𝑘^𝑇+𝐻_𝑘𝑛_𝑘𝑃^(𝑖,𝑗)_{𝑘∣𝑘−1}𝑛_𝑘𝐻_𝑘𝑛_𝑘^𝑇𝜛^(𝑖,𝑗)_𝑈,𝑘𝑛_𝑘;𝐳_𝑘=𝑟^(𝑖)_{𝑘∣𝑘−1}1−𝑟^(𝑖)_{𝑘∣𝑘−1}𝜛^(𝑖,𝑗)_{𝑘∣𝑘−1}𝑛_𝑘𝑝_𝐷,𝑘𝑛_𝑘𝒬_𝑘^(𝑖,𝑗)𝑛_𝑘;𝐳_𝑘×^{𝑀𝑘∣𝑘−1}_{𝑖=1𝑛𝑘}^{𝐽(𝑖)𝑘∣𝑘−1(𝑛𝑘)}_𝑗=1𝑟^(𝑖)_{𝑘∣𝑘−1}1−𝑟^(𝑖)_{𝑘∣𝑘−1}𝜛^(𝑖,𝑗)_{𝑘∣𝑘−1}𝑛_𝑘𝑝_𝐷,𝑘×𝑛_𝑘𝒬_𝑘^(𝑖,𝑗)𝑛_𝑘;𝐳_𝑘⁻¹,𝐾^(𝑖,𝑗)_𝑈,𝑘𝑛_𝑘=𝑃^(𝑖,𝑗)_{𝑘∣𝑘−1}𝑛_𝑘𝐻_𝑘𝑛_𝑘^𝑇×𝐻_𝑘𝑛_𝑘𝑃^(𝑖,𝑗)_{𝑘∣𝑘−1}𝑛_𝑘𝐻_𝑘𝑛_𝑘^𝑇+Β_𝑘𝑛_𝑘𝑅_𝑘𝑛_𝑘Β_𝑘𝑛_𝑘^𝑇−1,𝐦^(𝑖,𝑗)_𝑈,𝑘𝑛_𝑘;𝐳_𝑘=𝐦^(𝑖,𝑗)_{𝑘∣𝑘−1}𝑛_𝑘+𝐾^(𝑖,𝑗)_𝑈,𝑘𝑛_𝑘×𝐳_𝑘−𝐻_𝑘𝑛_𝑘𝐦^(𝑖,𝑗)_{𝑘∣𝑘−1}𝑛_𝑘,𝑃^(𝑖,𝑗)_𝑈,𝑘𝑛_𝑘=𝐼−𝐾^(𝑖,𝑗)_𝑈,𝑘𝑛_𝑘𝐻_𝑘𝑛_𝑘𝑃^(𝑖,𝑗)_{𝑘∣𝑘−1}𝑛_𝑘. (35)

(3) The Pruning and Merging Stage. It is the same as the pruning and merging stage of the GM-CBMeMBer filter [16].

(4) The GM Multitarget State Estimation. Given the GM multi-Bernoulli posterior density 𝜋_𝑘𝑌_𝑘∣𝑍_1∶𝑘=𝑟_𝑘^(𝑖),𝑝_𝑘^(𝑖)𝐱_𝑘,𝑛_𝑘^𝑀𝑘_𝑖=1with𝑝_𝑘^(𝑖)𝐱_𝑘,𝑛_𝑘=^{𝐽𝑘(𝑖)(𝑛𝑘)}_𝑗=1𝜛_𝑘^(𝑖,𝑗)𝑛_𝑘𝒩𝐱_𝑘;𝐦_𝑘^(𝑖,𝑗)𝑛_𝑘,𝑃_𝑘^(𝑖,𝑗)𝑛_𝑘 (36) from the method described in Algorithm 1, the GM multitarget state estimation can be easily obtained as 𝑋_𝑘=𝐱_𝑘^(𝑖)𝑇_𝑘𝑖=1with𝐱_𝑘^(𝑖)=_𝑛𝑘^{𝐽𝑘(𝑖)(𝑛𝑘)}_𝑗=1𝜛_𝑘^(𝑖,𝑗)𝑛_𝑘𝐦_𝑘^(𝑖,𝑗)𝑛_𝑘,𝑖=1,…,𝑇_𝑘. (37)

Now turn to considering the extension of the GM-MM-CBMeMBer filter to nonlinear dynamical and observation models using the EK filtering approximation. Assumptions B and C are still required, but the dynamic and observation processes can be relaxed to the nonlinear models 𝐱_𝑘=𝑎_𝑘𝐱_𝑘−1,𝐰_𝑘𝑛_𝑘,𝑛_𝑘,𝐳_𝑘=𝑢_𝑘𝐱_𝑘,𝐯_𝑘𝑛_𝑘,𝑛_𝑘, (38) where 𝑎_𝑘(⋅,⋅,𝑛_𝑘) and 𝑢_𝑘(⋅,⋅,𝑛_𝑘) are known model-dependent nonlinear functions, and 𝐰_𝑘(𝑛_𝑘) and 𝐯_𝑘(𝑛_𝑘) are model-dependent process and observation noise vectors of known statistics.

For the EK-GM-MM-CBMeMBer filter, the closed form expressions for the mixing, prediction, and update of individual Gaussian components are approximated by replacing 𝐹_𝑘(𝑛_𝑘), Α_𝑘(𝑛_𝑘), 𝐻_𝑘(𝑛_𝑘), Β_𝑘(𝑛_𝑘) in the corresponding recursive equations (30)–(35) of the GM-MM-CBMeMBer filter with the corresponding local linearization of the nonlinear dynamical and observation models 𝐹^EK_𝑘𝑛_𝑘=𝜕𝑎_𝑘𝐱_𝑘−1,𝐰_𝑘𝑛_𝑘,𝑛_𝑘𝜕𝐱_{𝑘−1𝐱𝑘−1=}𝐱_{𝑘−1𝐰𝑘(𝑛𝑘)=𝟎},Α^EK_𝑘𝑛_𝑘=𝜕𝑎_𝑘𝐱_𝑘−1,𝐰_𝑘𝑛_𝑘,𝑛_𝑘𝜕𝐰_𝑘𝑛_𝑘^𝐱_𝑘−1⁼𝐱_{𝑘−1𝐰𝑘(𝑛𝑘)=𝟎},𝐻^EK_𝑘𝑛_𝑘=𝜕𝑢_𝑘𝐱_𝑘,𝐯_𝑘𝑛_𝑘,𝑛_𝑘𝜕𝐱_𝑘^𝐱_𝑘−1⁼𝐱_{𝑘∣𝑘−1𝐯𝑘(𝑛𝑘)=𝟎},Β^EK_𝑘𝑛_𝑘=𝜕𝑢_𝑘𝐱_𝑘,𝐯_𝑘𝑛_𝑘,𝑛_𝑘𝜕𝐯_𝑘(𝑛_𝑘)^𝐱_𝑘−1⁼𝐱_{𝑘∣𝑘−1𝐯𝑘(𝑛𝑘)=𝟎}. (39)

Note that the unscented Kalman version for the GM-MM-CBMeMBer filter can be derived by approximating the mean and covariance of individual Gaussian components with a set of sigma points and the unscented transform [20]. Because of the space limitation, the details of the UK-GM-MM-CBMeMBer filter are not presented here.

6. Simulations

6.1. Nonlinear Example Using SMC Implementations

In this nonlinear example, we evaluate the performance of the proposed MM-CBMeMBer filter by benchmarking it against the single-model CBMeMBer filters, the MM-PHD filter, and the MM-CPHD filter using the SMC implementations.

Consider a two-dimensional scenario with an unknown and time varying number of the maneuvering targets observed over the region [−1000,1000]×[−1000,1000] (m) for a period of 𝑁=50 time steps. The sampling interval is Δ𝑡=1 (s). Each of the targets may move at a nearly constant velocity or execute a coordinated turn in the surveillance period. Therefore, the model set designed for this example can be composed of a constant velocity (CV) model and a coordinated turn (CT) model with varying turn rate [27]. The target kinematical state is 𝐱_𝑘=𝑥_𝑘𝑥̇_𝑘𝑦_𝑘𝑦̇_𝑘𝜗_𝑘^𝑇, where 𝑥_𝑘𝑦_𝑘^𝑇 and 𝑥̇_𝑘𝑦̇_𝑘^𝑇, respectively, represent the position and the velocity in 𝑥 and 𝑦 coordinates and 𝜗_𝑘 represents the turn rate. For the turn rate 𝜗_𝑘, let the anticlockwise direction be positive and the clockwise direction be negative.

The model-dependent dynamics for the individual maneuvering target is given by the linear-Gaussian model 𝑓_{𝑘∣𝑘−1}𝐱_𝑘∣𝐱_𝑘−1,𝑛_𝑘=𝒩𝐱_𝑘;𝐹_𝑘𝑛_𝑘𝐱_𝑘−1,𝑄_𝑘𝑛_𝑘. (40)

Let 𝑛_𝑘=1 denote the CV model and 𝑛_𝑘=2 the CT model; then 𝐹_𝑘𝑛_𝑘=𝐹=1^CV0,𝐹_𝑘𝑛_𝑘=𝐹=2^CT𝜗_𝑘−11,𝑄_𝑘𝑛_𝑘=1=𝜎²_𝑤𝑛_𝑘𝑄0,𝑄=1_𝑘𝑛_𝑘=𝜎=2^21,𝑤𝑛_𝑘=2𝑄Δ𝑡²𝜎²_2,𝑤𝑛_𝑘=2 (41) with 𝐹_CV=,𝐹1Δ𝑡000100001Δ𝑡0001_CT𝜗_𝑘−1=1Δ𝑡sin𝜗_𝑘−1𝜗_𝑘−10−1−Δ𝑡cos𝜗_𝑘−1𝜗_𝑘−10Δ𝑡cos𝜗_𝑘−10−Δ𝑡sin𝜗_𝑘−101−Δ𝑡cos𝜗_𝑘−1𝜗_𝑘−11Δ𝑡sin𝜗_𝑘−1𝜗_𝑘−10Δ𝑡sin𝜗_𝑘−10Δ𝑡cos𝜗_𝑘−1,𝑄=Δ𝑡⁴4Δ𝑡³200Δ𝑡³2Δ𝑡²0000Δ𝑡⁴4Δ𝑡³200Δ𝑡³2Δ𝑡², (42) where 𝜎_𝑤(𝑛_𝑘) is the level of the power spectral density of the process noise for model 𝑛_𝑘. In this example, they are given by 𝜎_𝑤(𝑛_𝑘=1)=0.1 (m/s²), 𝜎_1,𝑤(𝑛_𝑘=2)=0.2 (m/s²), 𝜎_2,𝑤(𝑛_𝑘=2)=1×10⁻³ (rad/s²).

The Markovian model transition probability matrix is taken as ℎ_{𝑘∣𝑘−1}𝑛_𝑘∣𝑛_𝑘−1=0.80.20.20.8. (43)

At time 𝑘, the range 𝜌_𝑘 and bearing 𝜑_𝑘 measurements of the targets are generated by a sensor located at 00^𝑇. The measurement noise is independent and identically distributed (IID) zero-mean Gaussian white noise with covariance matrix 𝑅_𝑘=diag𝜎²_𝜌𝜎²_𝜑, where diag(⋅) denotes the diagonal matrix, and 𝜎_𝜌 and 𝜎_𝜑 are, respectively, standard deviations (STDs) of the range and bearing measurements. In this example, they are taken as 𝜎_𝜌=10 (m) and 𝜎_𝜑=0.01 (rad). The single-measurement single-target likelihood density is 𝑔_𝑘𝐳_𝑘∣𝐲_𝑘=𝒩𝐳_𝑘;𝑥²_𝑘+𝑦²_𝑘𝑦arctan^𝑘𝑥_𝑘,𝑅_𝑘. (44)

The detection probability and the survival probability are, respectively, taken as 𝑝_𝐷,𝑘(𝐱_𝑘,𝑛_𝑘)=𝑝_𝐷=0.95 and 𝑝_𝑆,𝑘(𝐱_𝑘−1,𝑛_𝑘−1)=𝑝_𝑆=0.95 in this example.

The clutter is modeled as a Poisson RFS with the intensity 𝜅_𝑘(𝐳_𝑘)=𝜆_𝑐,𝑘𝑓_𝑐,𝑘(𝐳_𝑘). In this example, we take 𝜆_𝑐,𝑘=20 and 𝑓_𝑐,𝑘(⋅)=𝒰(⋅), where 𝒰(⋅) denotes the density of the uniform distribution over the observation region.

Figure 1 shows the true trajectories for the maneuvering targets and sensor location.

Figure 1 

The true trajectories for the maneuvering targets and sensor location.

In Figure 1, “∘” denotes the locations at which targets are born and “□” denotes the locations at which targets die. Target 1 is born at 1 s and dies at 30 s. It first moves at a nearly constant velocity from the first second to the 15th second and then executes a coordinated turn in the anticlockwise direction from the 16th second to the 30th second. Target 2 is born at 1 s and dies at 35 s. It first executes a coordinated turn in the anticlockwise direction from the first second to the 20th second and then moves at a nearly constant velocity from the 21st second to the 35th second. Target 3 is born at 10 s and dies at 42 s. It first executes a coordinated turn in the anticlockwise direction from the 10th second to the 30th second and then moves at a nearly constant velocity from the 31st second to the 42nd second. Target 4 is born at 20 s and dies at 50 s. It first moves at a nearly constant velocity from the 20th second to the 30th second and then executes a coordinated turn in the clockwise direction from the 31st second to the 50th second. The motions of the targets are summarized in Table 1.

Table 1 

The motions of the targets.

The birth process is a multi-Bernoulli RFS with density 𝜋_Γ,𝑘(𝑌_𝑘)={(𝑟^(𝑖)_Γ,𝑘,𝑝^(𝑖)_Γ,𝑘(𝐱_𝑘,𝑛_𝑘))}³_𝑖=1, where 𝑟⁽¹⁾_Γ,𝑘=0.04, 𝑟⁽²⁾_Γ,𝑘=𝑟⁽³⁾_Γ,𝑘=0.02, 𝑝^(𝑖)_Γ,𝑘(𝐱_𝑘,𝑛_𝑘)=ℎ^(𝑖)_Γ,𝑘(𝑛_𝑘)𝒩(𝐱_𝑘;𝐦^(𝑖)_Γ,𝑘,𝑃^(𝑖)_Γ,𝑘) with 𝐦⁽¹⁾_Γ,𝑘=−600080000^𝑇,𝐦⁽²⁾_Γ,𝑘=−6500−80000^𝑇,𝐦⁽³⁾_Γ,𝑘=4000−40000^𝑇,𝑃⁽¹⁾_Γ,𝑘=𝑃⁽²⁾_Γ,𝑘=𝑃⁽³⁾_Γ,𝑘=diag4004004004000.01 (45) and the distribution of the model births ℎ^(𝑖)_Γ,𝑘𝑛_𝑘=0.50.5. (46)

For the purpose of comparison, we estimate the number and states of the maneuvering targets using the proposed SMC-MM-CBMeMBer filter, the CV model SMC-CBMeMBer filter, the CT model SMC-CBMeMBer filter, the SMC-MM-PHD filter, and the SMC-MM-CPHD filter, respectively. At each time step in the SMC implementations of the CBMeMBer-based filters, a maximum of 𝐿_max=1000 and minimum of 𝐿_min=300 particles per hypothesized track are imposed, and pruning of hypothesized tracks is performed with a threshold of 𝑟̃_threshold=0.001. At each time step in the SMC implementations of the PHD-based filters, 1000 particles are used to represent one target and 𝐾-means method [14] is used to cluster the resampled particles to extract the multitarget states. The proposal densities 𝛼_𝑘^(𝑖)(𝑛_𝑘∣𝑛_𝑘−1), 𝛽_𝑘^(𝑖)(𝑛_𝑘), 𝑞_𝑘^(𝑖)(𝐱_𝑘∣𝐱_𝑘−1,𝑛_𝑘,𝑍_𝑘), and 𝑏_𝑘^(𝑖)(𝐱_𝑘∣𝑛_𝑘,𝑍_𝑘) in (16) and (18) are, respectively, taken as ℎ_{𝑘∣𝑘−1}(𝑛_𝑘∣𝑛_𝑘−1), ℎ^(𝑖)_Γ,𝑘(𝑛_𝑘),  𝑓_{𝑘∣𝑘−1}(𝐱_𝑘∣𝐱_𝑘−1,𝑛_𝑘) and 𝒩(𝐱_𝑘;𝐦^(𝑖)_Γ,𝑘,𝑃^(𝑖)_Γ,𝑘). We now conduct 500 Monte Carlo (MC) simulation experiments on the same clutter intensity and target trajectories, but with independently generated clutter and target-generated measurements in each trial.

The MC averages of the mean and STD of the cardinality distribution for the five methods at each time step are shown along with the true target number in Figure 2, respectively.

(a)

(b)

(c)

(d)

(e)

Figure 2 

The 500 MC run averages of cardinality statistics versus time for the (a) CV model SMC-CBMeMBer filter, (b) CT model SMC-CBMeMBer filter, (c) SMC-MM-PHD filter, (d) SMC-MM-CPHD filter, and (e) SMC-MM-CBMeMBer filter.

Figures 2(a)–2(e) demonstrate that the target number estimates from the SMC-MM-PHD, SMC-MM-CPHD, and SMC-MM-CBMeMBer filters converge to the ground truth, whereas the CV model SMC-CBMeMBer and CT model SMC-CBMeMBer filters produce significant bias in estimating the target number. This is because the SMC-MM-PHD, SMC-MM-CPHD, and SMC-MM-CBMeMBer filters can effectively capture the model switching property of the maneuvering targets, so their performance is significantly better than that of the two single-model SMC-CBMeMBer filters, which show poor adaptation to target maneuvers and yield larger estimation errors.

Moreover, as plotted in Figures 2(c)–2(e), the STD of the cardinality distribution from the SMC-MM-CBMeMBer filter is lower than that of the SMC-MM-PHD filter, but larger than that of SMC-MM-CPHD filter. In addition, the STDs of the cardinality distributions from the three MM-based filters increase in different degrees at the instances when the maneuver occurs (i.e., 16 (s), 21 (s), and 31 (s)). The STD plots of the SMC-MM-PHD and SMC-MM-CPHD filters seem to fluctuate more obviously than the SMC-MM-CBMeMBer filter. This phenomenon indicates that the performance of the SMC-MM-CBMeMBer filter may be more stable and robust at the maneuver instances than that of the SMC-MM-PHD and SMC-MM-CPHD filters.

The optimal subpattern assignment (OSPA) metric [28], which can jointly capture differences in cardinality and individual elements between two finite sets, is used to evaluate the performance of the five methods. Given the actual and estimated multitarget state sets 𝑋_𝑘={𝐱_𝑘^(𝑖)}^𝑇𝑘_𝑖=1 and 𝑋_𝑘={𝐱_𝑘^(𝑖)}𝑇_𝑘𝑖=1, the OSPA metric of order 𝑝=2 with cut-off 𝑐 between the two sets is defined by OSPA^(𝑐)_2,𝑘𝑋_𝑘,𝑋_𝑘=1𝑇_𝑘min_𝜋∈Π𝑇_𝑘^{𝑇𝑘𝑖=1}min𝑐,𝐱_𝑘^(𝑖)−𝐱_𝑘^(𝜋(𝑖))₂²+𝑐²𝑇_𝑘−𝑇_𝑘^1/2 (47) if 𝑇_𝑘≤𝑇_𝑘 and OSPA^(𝑐)_2,𝑘(𝑋_𝑘,𝑋_𝑘)=OSPA^(𝑐)_2,𝑘(𝑋_𝑘,𝑋_𝑘) if 𝑇_𝑘>𝑇_𝑘. Π𝑇_𝑘 denotes the set of permutations on {1,2,…,𝑇_𝑘}. ‖⋅‖₂ denotes the 2-norm. In this example, we take 𝑐=100 (m).

The MC averages of the OSPA metric for the target position estimates, derived by the five methods, are shown in Figure 3.

Figure 3 

The 500 MC run averages of OSPA against time.

The OSPA metric is composed of two components each separately accounting for “localization” and “cardinality” errors. This results in high peaks in OSPA metric at the instances where the estimated number is incorrect. Figure 3 shows that (1) both the single-model SMC-CBMeMBer filters perform significantly worse than the other MM-based filters because of the large cardinality errors produced by the two filters as seen in Figures 2(a) and 2(b); (2) although the SMC-MM-CPHD filter can estimate the target number most accurately, the OSPA metric of the SMC-MM-CBMeMBer filter is smaller than that of the SMC-MM-CPHD filter, which is in turn smaller than that of the SMC-MM-PHD filter. This phenomenon indicates that the SMC-MM-CBMeMBer filter outperforms the SMC-MM-CPHD (and hence SMC-MM-PHD) filter in jointly estimating the multitarget number and states. A reason for this is that the additional errors could be introduced in the clustering processes of the SMC-MM-PHD and SMC-MM-CPHD filters to extract state estimates from the particle population; (3) the OSPA plots of the three MM-based filters in Figure 3 fluctuate against time due to the varying target number, the target maneuvers, and clutter. However, increase of the OSPA from the SMC-MM-CBMeMBer filter seems to be smallest at the maneuver instances (i.e., 16 (s), 21 (s), and 31 (s)) among the three MM-based methods. This phenomenon also indicates that the performance of the SMC-MM-CBMeMBer filter may be more stable and robust at the maneuver instances than that of the SMC-MM-PHD and SMC-MM-CPHD filters.

For comparing the overall performance of the three MM-based filters, the 500 MC trial averages of the OSPA distance (time-averaged over the duration of the scenario) for the three MM-based filters are shown in Table 2 against the clutter rate from 𝜆_𝑐,𝑘=20 to 𝜆_𝑐,𝑘=100. The result of time-averaging can be viewed as a broad indication of filter performance, although the average is likely to be scenario dependent.

Table 2 

Time-averaged OSPA distance (𝑚) in various 𝜆_𝑐,𝑘.

Table 2 shows that the OSPA distances of the three MM-based filters increase with higher 𝜆_𝑐,𝑘. It reflects that the performance of the three MM-based algorithms degrades by different degrees as the 𝜆_𝑐,𝑘 increases. Among the three MM-based algorithms, the SMC-MM-PHD filter always works the worst. The SMC-MM-CBMeMBer filter outperforms the SMC-MM-CPHD filter when 𝜆_𝑐,𝑘 is relatively lower (𝜆_𝑐,𝑘≤60). However, as the 𝜆_𝑐,𝑘 increases, the OSPA distance of SMC-MM-CBMeMBer filter increases more rapidly than that of the SMC-MM-CPHD filter. Therefore, as 𝜆_𝑐,𝑘 continues to increase until it reaches 𝜆_𝑐,𝑘=80, the OSPA distance of SMC-MM-CBMeMBer filter is very close to that of the SMC-MM-CPHD filter. When 𝜆_𝑐,𝑘 is relatively higher (i.e., 𝜆_𝑐,𝑘=80), the SMC-MM-CPHD filter outperforms the SMC-MM-CBMeMBer filter. A possible reason for this is that, compared with the SMC-MM-CBMeMBer filter, the advantage of the target number estimate for the SMC-MM-CPHD filter is more obvious as the 𝜆_𝑐,𝑘 increases and it finally leads that the OSPA distance of the SMC-MM-CPHD filter is smaller than that of the SMC-MM-CBMeMBer filter when 𝜆_𝑐,𝑘 is relatively higher (i.e., 𝜆_𝑐,𝑘=80).

6.2. Linear-Gaussian Example Using GM Implementations

In this linear-Gaussian example, we evaluate the performance of the proposed MM-CBMeMBer filter by benchmarking it against the single-model CBMeMBer filters, the MM-PHD filter, and the MM-CPHD filter using the GM implementations.

The simulation scenario and true trajectories for the maneuvering targets are the same as those of Example 1. The target kinematical state now turns into 𝐱_𝑘=𝑥_𝑘𝑥̇_𝑘𝑦_𝑘𝑦̇_𝑘^𝑇. The model set for this example is designed as follows. Model 𝑛_𝑘=1 is a CV model with linear-Gaussian dynamics given by 𝒩(𝐱_𝑘;𝐹_CV𝐱_𝑘−1,𝜎²_CV,𝑤𝑄); models 𝑛_𝑘=2,3,4,5 are, respectively, CT models with turn rates of 𝜗=𝜋/30,−𝜋/30,𝜋/20,𝜋/15 (rad/s) with linear-Gaussian dynamics given by 𝒩(𝐱_𝑘;𝐹_CT(𝜗)𝐱_𝑘−1,𝜎²_CT,𝑤𝑄). In this example, 𝜎_CV,𝑤 and 𝜎_CT,𝑤 are given by 𝜎_CV,𝑤=0.1 (m/s²), 𝜎_CT,𝑤=0.2 (m/s²).

The Markovian model transition probability matrix now turns into ℎ_{𝑘∣𝑘−1}𝑛_𝑘∣𝑛_𝑘−1=0.60.10.10.10.10.10.60.10.10.10.10.10.60.10.10.10.10.10.60.10.10.10.10.10.6. (48)

The 𝑥-position and 𝑦-position measurements 𝐳_𝑘=𝑥_𝑘𝑦_𝑘^𝑇 of the maneuvering targets are generated by the linear-Gaussian single-measurement single-target likelihood density given by 𝒩(𝐳_𝑘;𝐻_𝑘𝐱_𝑘,𝑅_𝑘) with 𝐻_𝑘=10000010 (49) and 𝑅_𝑘=diag𝜎²_𝑥𝜎²_𝑦. In this example, they are taken as 𝜎_𝑥=𝜎_𝑦=8 (m), and the kinematical state independent survival and detection probabilities are taken as 𝑝_𝐷,𝑘(𝑛_𝑘)=𝑝_𝐷=0.95 and 𝑝_𝑆,𝑘(𝑛_𝑘−1)=𝑝_𝑆=0.95.

The experiment settings of the clutter and birth model are also the same as those of Example 1 except that the 𝐦^(𝑖)_Γ,𝑘,𝑃^(𝑖)_Γ,𝑘, 𝑖=1, 2, 3, and [ℎ^(𝑖)_Γ,𝑘(𝑛_𝑘)] turn into 𝐦⁽¹⁾_Γ,𝑘=−60008000^𝑇,𝐦⁽²⁾_Γ,𝑘=−6500−8000^𝑇,𝐦⁽³⁾_Γ,𝑘=4000−4000^𝑇,𝑃⁽¹⁾_Γ,𝑘=𝑃⁽²⁾_Γ,𝑘=𝑃⁽³⁾_Γ,𝑘,ℎ=diag400400400400_{(𝑖)Γ,𝑘}𝑛_𝑘=0.20.20.20.20.2. (50)

For the purpose of comparison, we estimate the number and states of the maneuvering targets using the proposed GM-MM-CBMeMBer filter, the CV model GM-CBMeMBer filter, the CT model GM-CBMeMBer filter with turn rate of 𝜗=𝜋/20 (rad/s), (this turn rate seems to be most suitable for the scenario among the above four turn rates), the GM-MM-PHD filter, and the GM-MM-CPHD filter, respectively. At each time step in the GM implementations of the CBMeMBer-based filters, pruning of hypothesized tracks is performed with a threshold of 𝑟̃_threshold=0.001. In addition, the pruning and merging of Gaussian components are performed for each hypothesized track using a weight threshold of 10⁻⁵, a merging threshold of 4 (m), and a maximum of 𝐽_max=100 components, which are also used in the GM implementations of the PHD-based filters.

The MC averages of the mean and STD of the cardinality distribution for the five methods at each time step are shown along with the true target number in Figure 4, respectively.

(a)

(b)

(c)

(d)

(e)

Figure 4 

The 500 MC run averages of cardinality statistics versus time for the (a) CV model GM-CBMeMBer filter, (b) CT model GM-CBMeMBer filter, (c) GM-MM-PHD filter, (d) GM-MM-CPHD filter, and (e) GM-MM-CBMeMBer filter.

Similar to the SMC implementations, Figures 4(a)–4(e) demonstrate that the GM implementations of the three MM-based filters are unbiased in the target number estimates, whereas the GM implementations of the two single-model GM-CBMeMBer filters are significantly biased. Moreover, the GM-MM-CBMeMBer filter has a lower STD of the estimated cardinality than the GM-MM-PHD filter but has a larger STD than the GM-MM-CPHD filter. Again, The STD plots of the GM-MM-PHD and GM-MM-CPHD filters seem to fluctuate more obviously than the GM-MM-CBMeMBer filter at the maneuver instances (i.e., 16 (s), 21 (s), and 31 (s)).

The MC averages of the OSPA metric for the target position estimates, derived by the five methods, are shown in Figure 5.

Figure 5 

The 500 MC run averages of OSPA against time.

In contrast to the SMC case, Figure 5 shows that (1) the rather poor performance of the two single-model GM-CBMeMBer filters can be expected as the direct results of their significant cardinality biase as seen in Figures 4(a) and 4(b); (2) the OSPA metric of the GM-MM-CBMeMBer filter is similar to that of the GM-MM-PHD filter but is larger than that of the GM-MM-CPHD filter. This is because that, like the MM-CBMeMBer filter, the GM implementations of the MM-PHD and MM-CPHD filters also allow state estimates to be extracted from the posterior intensity in a much more efficient and reliable manner than particle clustering in the SMC-based approach. As a result, the GM-MM-CPHD filter, which has the lowest STD of the estimated cardinality, performs best among the three MM-based filters. Although the GM-MM-CBMeMBer filter has a lower STD of the estimated cardinality than the GM-MM-PHD filter, the performance of the two filters is similar. A reason for this is that the GM-MM-PHD filter may have more of an advantage than the GM-MM-CBMeMBer filter in the relatively high signal to noise ratio (SNR) of this scenario.

Although the GM-MM-CPHD filter outperforms the proposed GM-MM-CBMeMBer filter, it can be only used in the linear-Gaussian condition. In the nonlinear non-Gaussian conditions, both the MM-CPHD filter and MM-CBMeMBer filter must be implemented by the SMC method. In this case, the GM-MM-CBMeMBer filter outperforms the GM-MM-CPHD filter significantly, which is shown in Section 6.1.

The 500 MC trial averages of the OSPA distance (time-averaged over the duration of the scenario) for the three MM-based filters are shown in Table 3 against the clutter rate from 𝜆_𝑐,𝑘=20 to 𝜆_𝑐,𝑘=100.

Table 3 

Time-averaged OSPA distance (𝑚) in various 𝜆_𝑐,𝑘.

Similar to the SMC implementations, Table 3 shows that the OSPA distances of the GM implementations of the three MM-based filters increase with higher 𝜆_𝑐,𝑘. However, in various 𝜆_𝑐,𝑘, the GM-MM-CPHD filter always has the best performance among the three MM-based algorithms, and the GM-MM-CBMeMBer filter has the similar performance with the GM-MM-PHD filter.

7. Conclusions and Future Work

An MM-CBMeMBer filter, which is a multiple-model extension to the CBMeMBer filter, is proposed for tracking multiple maneuvering targets in clutter. The SMC and GM implementations of the proposed filter are, respectively, presented for generic models and for linear-Gaussian models. Then, the EK and UK filtering approximations for the GM-MM-CBMeMBer filter in nonlinear condition are described briefly. Simulation results show that (1) the proposed MM-CBMeMBer filter significantly outperforms the single-model CBMeMBer filters in tracking multiple maneuvering targets; (2) under relatively low clutter density, the SMC-MM-CBMeMBer filter outperforms the SMC-MM-PHD and SMC-MM-CPHD filters; (3) the performance of the GM-MM-CBMeMBer filter is similar to that of the GM-MM-PHD filter and hence is inferior to that of GM-MM-CPHD filter.

The future work is focused on the following three aspects. First, the track labeling problem in the proposed approach needs to be considered. Second, practical data need to be used for the performance evaluation of the proposed approaches. Third, the multiple-sensor versions of the CBMeMBer and MM-CBMeMBer filters need to be proposed for improving the performance of the single-sensor CBMeMBer and MM-CBMeMBer filters.

Acknowledgments

This research work was supported by the National Key Fundamental Research & Development Programs (973) of China (2013CB329405), Foundation for Innovative Research Groups of the National Natural Science Foundation of China (61221063), Natural Science Foundation of China (61203221, 61004087), Ph.D. Programs Foundation of Ministry of Education of China (20100201120036), China Postdoctoral Science Foundation (2011M501442, 20100481338), and Fundamental Research Funds for the Central University.