Multitarget Tracking by Improved Particle Filter Based on 𝐻_∞ Unscented Transform

Abstract

This paper considers the problem of multitarget tracking in cluttered environment. To reduce the dependency on the noise priori knowledge, an improved particle filtering (PF) data association approach is presented based on the 𝐻_∞ filter (HF). This approach can achieve higher robustness in the condition that the measurement noise prior is unknown. Because of the limitations of the HF in nonlinear tracking, we first present the 𝐻_∞ unscented filter (HUF) by embedding the unscented transform (UT) into the 𝐻_∞ extended filter (HEF) structure. Then the HUF is incorporated into the Rao-Blackwellized particle filter (RBPF) framework to update the particles. Simulation results are provided to demonstrate the effectiveness of the proposed algorithms in linear and nonlinear multitarget tracking.

1. Introduction

Multitarget tracking is to estimate the targets’ current positions from a series of noise-corrupted measurements by filtering methods [1]. In cluttered environment, the foremost difficulty involves the problem of associating the correct measurements with the appropriate tracks. A number of strategies, including the joint probabilistic data association (JPDA) [2, 3], multiple hypothesis tracking (MHT) [4, 5], S-D assignment [6], and the probabilistic hypothesis density filer (PHDF) [7–9], are available to solve this problem.

Recently, the sequential Monte Carlo (SMC) data association approaches are also applied to the tracking and association problems [10, 11]. This paper tackles data association joint with state estimation via a SMC method called Rao-Blackwellized particle filter (RBPF) [12–14]. This particle method can be considered as a generalization of MHT, which represents the data association and state posteriors as a discrete set of hypotheses. Instead of the pure particle strategy, a mixture of Gaussian representation of the joint posterior distribution is used to reduce the estimation variance. In the application of RBPF for nonlinear target tracking, it can replace the Kalman filter (KF) in the data association algorithm by the extended Kalman filter (EKF) or the unscented Kalman filter (UKF) [15]. Although RBPF has shown higher tracking efficiency against the pure particle filtering schemes, there is still weakness lying in the strong assumptions on the system models and noise statistics [16]. That is, the KF or its variants can only be applied in the condition that a perfect system model is known and the process and measurement noise statistics are white and Gaussian with known covariance matrices. In many practical applications, these assumptions can hardly be satisfied, and the performance of the KF based methods may degrade severely when the measurement disturbances are not in definite Gaussian forms. Hence, we need a solution with more robustness against the uncertainties of the noise than the KF. The 𝐻_∞ filter (HF), which aims at minimizing the worst possible effects of the disturbances on the estimation errors, can provide an alternative to solve this problem [17–19]. Different from the KF, the only assumption made for HF is that the noise signals have a finite energy but without any statistic assumptions.

The HF techniques have been used in linear-model target tracking. Accordingly, the 𝐻_∞ extended filter (HEF) has also been proposed for nonlinear models by using the EKF structure. As another way to approximate the filtering distribution, the unscented transform (UT) can be more accurate compared to the EKF for it performs a higher order of the Taylor series expansion. By a Gaussian density instead of approximating the nonlinear functions as the EKF, the UT technique has shown its priority in handling nonlinear estimation problems and also been combined with the 𝐻_∞ filter [20]. In this paper, we aim to incorporate the HF technique into the framework of RBPF to reduce its dependency on a priori knowledge of the noise statistics in multitarget tracking. In RBPF, the continuous state is estimated using the KF or its variants, while the discrete state or the mode state is estimated using particle filters. Thus, the HF can be embedded into the RBPF structure directly. Because of the limitations of the HF in nonlinear tracking, we present the 𝐻_∞ unscented filter (HUF) by embedding the UT into the HEF structure. Since the HUF has the same observer structure as the UKF, it can also be embedded into the RBPF framework to update the continuous states of the particles.

The remainder of this paper is organized as follows. In Section 2, we give a brief introduction of the generic RBPF target tracking approach and the basic 𝐻_∞ linear filter structure. The main work is given in Section 3 where the HUF is first presented. Then the HUF is incorporated into the RBPF framework to update the particles. Simulation results are provided in Section 4 to demonstrate the effectiveness of the proposed algorithms in linear and nonlinear multitarget tracking. In Section 5, conclusions are also given to summarize the main works of this paper.

2. Background

2.1. Rao-Blackwellized Particle Filter for Target Tracking

Consider the following time-varying state-space system: 𝐱_𝑘=𝐅_𝑘−1𝐱_𝑘−1+𝐰_𝑘−1,𝐳_𝑘=𝐇_𝑘𝐱_𝑘+𝐯_𝑘, (1) where 𝐱_𝑘∈ℝ^𝑛 and 𝐳_𝑘∈ℝ^𝑝 are the system state and measurement vectors at time step 𝑘, respectively. 𝐰_𝑘−1∼𝒩(0,𝐐_𝑘−1) and 𝐯_𝑘∼𝒩(0,𝐑_𝑘) are zero mean mutually independent Gaussian process noise, and 𝐅_𝑘−1 and 𝐇_𝑘 are matrices with compatible dimensions. Suppose that we are able to form another variable 𝑐_𝑘 to describe the matrices 𝐅_𝑘−1 and 𝐇_𝑘; then the RBPF algorithm can be applied to estimate the whole state {𝐱_𝑘;𝑐_𝑘} [21]. For space consideration, we omit the details of the RBPF algorithms which can be found in [22, 23]. In the application of RBPF for target tracking, the latent variable 𝑐_𝑘 is defined to be the data association event indicator [12]. That is, 𝑐_𝑘=0 when the measurement is from clutter, and 𝑐_𝑘=𝑗 when the measurement is from target 𝑗 (𝑗=1,…,𝑇, where 𝑇 is number of targets). The predictive probability 𝑝(𝑐_𝑘∣𝑐_1∶𝑘−1) gives the data association prior given the data association results {𝑐_1∶𝑘−1} in the previous 𝑘−1 time steps. The posterior distribution of 𝑐_𝑘 is 𝑝(𝑐_𝑘∣𝑐_1∶𝑘−1,𝐳_1∶𝑘) which can be calculated by 𝑝𝑐_𝑘∣𝑐_1∶𝑘−1,𝐳_1∶𝑘∝𝑝𝐳_𝑘∣𝑐_𝑘,𝑐_1∶𝑘−1,𝐳_1∶𝑘−1𝑝𝑐_𝑘∣𝑐_1∶𝑘−1, (2) where the data association prior 𝑝(𝑐_𝑘∣𝑐_1∶𝑘−1) is modeled as a recursive Markov chain, which guarantees the association assumption of one target per measurement in each time step. Accordingly, the 𝑝(𝑐_𝑘∣𝑐_1∶𝑘−1) in (2) has been replaced by 𝑝(𝑐_𝑘∣𝑐_{𝑘−𝑀∶𝑘−1}) [13], and the general form of the joint prior model is given by 𝑝𝑐_{𝑘+𝑀−1},…,𝑐_𝑘=^𝑀_𝑚=1𝑝𝑐_𝑘+𝑚∣𝑐_𝑘,…,𝑐_{𝑘+𝑚−1}. (3) That means if we obtain 𝑀 measurements in time step 𝑘, the 𝑚th (𝑚=2,…,𝑀) measurement’s association prior 𝑝(𝑐_{𝑘+𝑚−1}∣𝑐_{𝑘+𝑚−2},…,𝑐_𝑘) only depends on the previous association results {𝑐_{𝑘+𝑚−2},…,𝑐_𝑘} in the 𝑘th time step. Note that 𝑐_𝑘 has the prior 𝑝(𝑐_𝑘) if 𝑚=1. This RBPF multitarget tracking algorithm is also termed Rao-Blackwellized Monte Carlo data association (RBMCDA) [12]. In this paper, we aim to improve the robustness of the RBPF for target tracking by the 𝐻_∞ filter and unscented transform.

2.2. 𝐻_∞ Linear Filter

Consider the model given by (1), where the process noise 𝐰_𝑘−1 and the measurement noise 𝐯_𝑘 are assumed to be uncorrelated zero-mean white noise processes with unknown statistical properties. Note that they are also the energy bounded 𝑙₂[0,+∞] signals. Let 𝐱_𝑘∣𝑘≜ℒ{𝐳₀,𝐳₁,…,𝐳_𝑘} denote the estimation of 𝐱_𝑘 given measurements 𝐳_0∶𝑘. We can define the estimation error as ℰ_𝑘≜𝐱_𝑘|𝑘−𝐱_𝑘 and denote 𝒯_𝑘(ℒ) as the transfer operator that maps the unknown disturbances 𝐱₀−𝐱_0∣0, 𝐰_0∶𝑘−1, and 𝐯_1∶𝑘 to the estimation errors ℰ_1∶𝑘, where 𝐱_0|0 is a priori estimate of 𝐱₀ and 𝐱₀−𝐱_0∣0 represents unknown initial estimation error.

In optimal 𝐻_∞ filter, it is operated to minimize the possible worst effects of the unknown disturbances on the estimation errors. That is, the estimation strategy ℒ should be designed so as to minimize the 𝐻_∞ norm of the operator 𝒯_𝑘(ℒ). Actually, it is hard to obtain the closed-form solution of the optimal 𝐻_∞ filtering except in some specific cases. For example, the desired accuracy of the optimal 𝐻_∞ filter can be obtained by iterating the 𝛾² of the suboptimal solution [16]. This paper considers the suboptimal solutions that can bound the maximum energy gain from the disturbance to the estimation errors under the prescribed disturbance tolerance level. Given a scalar 𝛾²>0, find the estimation strategies ℒ such that the 𝐻_∞ norm of 𝒯_𝑘(ℒ) satisfies [24, 25] 𝒯_𝑘(ℒ)_∞≜sup_{𝐱0,𝐰𝑡,𝐯𝑡∈𝑙2}∑^𝑘_𝑡=1ℰ_𝑡²₂𝐱₀−𝐱_0∣0²_{𝐏−10∣0}+∑^𝑘_𝑗=0𝐰_𝑡²_𝐐𝑡−1+∑^𝑘_𝑡=1𝐯_𝑡²_𝐑𝑡−1<𝛾², (4) where the notation ||𝑎||²_𝛽 is defined as the square of the weighted 𝑙₂ norm of 𝑎, that is, ||𝑎||²_𝛽=𝑎^𝑇𝛽𝑎. The matrix 𝐏_0∣0>0 reflects a priori knowledge of how close 𝐱₀ is to the initial estimate 𝐱_0∣0. 𝐐_𝑡>0 and 𝐑_𝑡>0 are weighting matrices, by which the designer can make appropriate choice to satisfy the performance requirements. The solution to the 𝐻_∞ filtering algorithm as shown in [26] is given by 𝐱_{𝑘∣𝑘−1}=𝐅_𝑘−1𝐱_{𝑘−1∣𝑘−1},𝐏_{𝑘∣𝑘−1}=𝐅_𝑘−1𝐏_{𝑘−1∣𝑘−1}𝐅^𝑇_𝑘−1+𝐐_𝑘−1,𝐱_𝑘∣𝑘=𝐱_{𝑘∣𝑘−1}+𝐊_𝑘𝐳_𝑘−𝐇_𝑘𝐱_{𝑘∣𝑘−1},𝐊_𝑘=𝐏_{𝑘∣𝑘−1}𝐇^𝑇_𝑘𝐑_𝑘+𝐇_𝑘𝐏_{𝑘∣𝑘−1}𝐇^𝑇_𝑘⁻¹,𝐏_𝑘∣𝑘=𝐏_{𝑘∣𝑘−1}−𝐏_{𝑘∣𝑘−1}𝐇^𝑇_𝑘𝐼𝑅⁻¹_ℰ,𝑘𝐇^𝑇_𝑘𝐼^𝑇𝐏_{𝑘∣𝑘−1}, (5)(6)(7)(8)(9) where 𝐼 is an identity matrix with compatible dimension. The matrix 𝑅_ℰ,𝑘 is given by 𝑅_ℰ,𝑘=𝐑_𝑘00−𝛾²𝐼+𝐇_𝑘𝐼𝐏_{𝑘∣𝑘−1}𝐇^𝑇_𝑘𝐼. (10)

3. Improved Rao-Blackwellized Particle Filter

3.1. 𝐻_∞ Nonlinear Filter with Unscented Transform

Consider the following discrete-time nonlinear state-space model: 𝐱_𝑘=𝑓𝐱_𝑘−1+𝐰_𝑘−1,𝐳_𝑘=ℎ𝐱_𝑘+𝐯_𝑘. (11) The HEF just replaces 𝐅_𝑘−1 and 𝐇_𝑘 in (5) and (7) by 𝑓(𝐱_{𝑘−1∣𝑘−1}) and ℎ(𝐱_{𝑘∣𝑘−1}), respectively. Here, we present the HUF by embedding the UT technique into the HF structure, which can be used to update the continuous-state particles in the RBPF framework. Suppose that (2𝑛+1) sigma points are generated based on the state estimates at time 𝑘−1, 𝜁⁰_{𝑘−1∣𝑘−1}=𝐱_{𝑘−1∣𝑘−1},𝜁^𝑠_{𝑘−1∣𝑘−1}=𝐱_{𝑘−1∣𝑘−1}+(𝑛+𝒟)𝐏_{𝑘−1∣𝑘−1}^𝑠,𝜁^𝑠+𝑛_{𝑘−1∣𝑘−1}=𝐱_{𝑘−1∣𝑘−1}−(𝑛+𝒟)𝐏_{𝑘−1|𝑘−1}^𝑠,𝑠=1,…,𝑛, (12) where the state estimate 𝐱_{𝑘−1∣𝑘−1} and its covariance 𝐏_{𝑘−1|𝑘−1} have been obtained at time 𝑘−1. 𝒟∈ℝ is a scaling factor, and ((𝑛+𝒟)𝐏_{𝑘−1|𝑘−1})^𝑠 is the 𝑠th row or column of the matrix squares root of (𝑛+𝒟)𝐏_{𝑘−1∣𝑘−1}. By implementing the UT into (5) and (6), the predicted mean and covariance can be obtained as follows: 𝜁^𝑠_{𝑘∣𝑘−1}=𝑓𝜁^𝑠_{𝑘−1∣𝑘−1},𝐱_{𝑘∣𝑘−1}=^2𝑛_𝑠=0𝜔^𝑠𝜁^𝑠_{𝑘∣𝑘−1},𝐏_{𝑘∣𝑘−1}=^2𝑛_𝑠=0𝜔^𝑠𝜁^𝑠_{𝑘∣𝑘−1}−𝐱_{𝑘∣𝑘−1}×𝜁^𝑠_{𝑘∣𝑘−1}−𝐱_{𝑘∣𝑘−1}^𝑇+𝐐_𝑘−1, (13) where 𝜔^𝑠 is the normalized weight associated with the 𝑠th sigma points (see [27, 28]). Since the linearized measurement function 𝐇_𝑘 does not exist explicitly under the unscented transform framework, the statistical linear error propagation method [29] is used to reformulate the updated equations. Approximately, the measurement covariance and its cross-correlation covariance can be given by 𝐏^𝑧𝑧_{𝑘∣𝑘−1}≈𝐇_𝑘𝐏_{𝑘∣𝑘−1}𝐇^𝑇_𝑘,𝐏^𝑥𝑧_{𝑘∣𝑘−1}≈𝐏_{𝑘∣𝑘−1}𝐇^𝑇_𝑘. (14) By using the predicted sigma points, they can be calculated by 𝐏^𝑧𝑧_{𝑘∣𝑘−1}=^2𝑛_𝑠=0𝜔^𝑠ℎ𝜁^𝑠_{𝑘∣𝑘−1}−𝐳̂_{𝑘∣𝑘−1}×ℎ𝜁^𝑠_{𝑘∣𝑘−1}−𝐳̂_{𝑘∣𝑘−1}^𝑇,𝐏^𝑥𝑧_{𝑘∣𝑘−1}=^2𝑛_𝑠=0𝜔^𝑠𝜁^𝑠_{𝑘∣𝑘−1}−𝐱_{𝑘∣𝑘−1}×ℎ𝜁^𝑠_{𝑘∣𝑘−1}−𝐳̂_{𝑘∣𝑘−1}^𝑇, (15) where 𝐳̂_{𝑘∣𝑘−1}=^2𝑛_𝑠=0𝜔^𝑠ℎ𝜁^𝑠_{𝑘∣𝑘−1}. (16) By substituting (14) into (7)–(10), the filtered estimates can be obtained by 𝐱_𝑘∣𝑘=𝐱_{𝑘∣𝑘−1}+𝐏^𝑥𝑧_{𝑘∣𝑘−1}𝐑_𝑘+𝐏^𝑧𝑧_{𝑘∣𝑘−1}⁻¹×𝐳_𝑘−𝐳̂_{𝑘∣𝑘−1},𝐏_𝑘∣𝑘=𝐏_{𝑘∣𝑘−1}−𝐏^𝑥𝑧_{𝑘∣𝑘−1}𝐏_{𝑘∣𝑘−1}×𝐑⁻¹_ℰ,𝑘𝐏^𝑥𝑧_{𝑘∣𝑘−1}𝐏_{𝑘∣𝑘−1}^𝑇, (17) where 𝑅_ℰ,𝑘=𝐑_𝑘+𝐏^𝑧𝑧_{𝑘∣𝑘−1}𝐏^𝑥𝑧_{𝑘∣𝑘−1}^𝑇𝐏^𝑥𝑧_{𝑘∣𝑘−1}−𝛾²𝐼+𝐏_{𝑘∣𝑘−1}. (18) It should be pointed out that the HUF has the same observer structure as that of the UKF, and 𝐐_𝑘−1 and 𝐑_𝑘 play the same role as the covariance matrices of the process noise and the measurement noise when using the UKF. Hence, the weighting matrices can be adjusted with no conflictions to the framework of KF. For the HUF, it not only outperforms the HEF in accuracy, but also achieves more robustness than the UKF for unknown noise statistics [16].

3.2. 𝐻_∞ Unscented Rao-Blackwellized Particle Filter

In this section, the HUF based RBPF multiple target tracking algorithm (HURBPF) is provided. It can be found that the 𝑝(𝐱_𝑘∣𝑐_{𝑖,1∶𝑘},𝐳_1∶𝑘) will not be strictly Gaussian if the system dynamic and (or) measurement function are (is) nonlinear. In this case, the HUF can be a better candidate than HEF or UKF for updating the continuous state within the particle filtering framework. The conditional distribution mentioned above can be chosen as 𝑝𝐱_𝑘∣𝑐_{𝑖,1∶𝑘},𝐳_1∶𝑘=𝒩𝐱_𝑘;𝐱_{𝑖,𝑘∣𝑘},𝐏_{𝑖,𝑘∣𝑘}, (19) where 𝐱_{𝑖,𝑘∣𝑘} and 𝐏_{𝑖,𝑘∣𝑘} are the mean and covariance of 𝐱_𝑘 computed by the HUF (17). The main procedure for the HURBPF algorithm is presented as follows (see Algorithm 1). In this algorithm, the target state priors can be represented as a weighted importance sample set 𝑝𝐱₀^(𝑗)=^𝑁𝑝_𝑖=1𝐰_𝑖,0𝒩𝐱₀^(𝑗)∣𝐱^(𝑗)_𝑖,0,𝐏^(𝑗)_𝑖,0, (20) where 𝑖 (𝑖=1,…,𝑁_𝑝) is the identifier of particle.

Algorithm 1. 𝐻_∞ unscented Rao-Blackwellized particle filter.
For 𝑘=1 to 𝑛 do.
Step 1. Do prediction step. For 𝑖=1,2,…,𝑁_𝑝 and 𝑗=1,2,…,𝑇, perform HUF prediction from the mean 𝐱_{𝑖,𝑘−1∣𝑘−1} and covariances 𝐏_{𝑖,𝑘−1∣𝑘−1} to generate the predicted estimates 𝐱^(𝑗)_{𝑖,𝑘∣𝑘−1} and 𝐏^(𝑗)_{𝑖,𝑘∣𝑘−1}.
Step 2. Calculate the data association priors 𝑝(𝑐_𝑖,𝑘∣𝑐_{𝑖,1∶𝑘−1}) based on the recursive Markov chain described in [13].
Step 3. For 𝑖=1,2,…,𝑁_𝑝, perform HUF update for each particle to get contemporary filtered estimates 𝐱^(𝑗)_{𝑖,𝑘∣𝑘} and 𝐏^(𝑗)_{𝑖,𝑘∣𝑘}.
Step 4. Calculate the posterior distribution of 𝑐_𝑖,𝑘: 𝑝𝑐_𝑖,𝑘∣𝑐_{𝑖,1∶𝑘−1},𝐳_1∶𝑘=𝑝𝐳_𝑘∣𝑐_𝑖,𝑘,𝑐_{𝑖,1∶𝑘−1},𝐳_1∶𝑘−1𝑝𝑐_𝑖,𝑘∣𝑐_{𝑖,1∶𝑘−1}. (21)
Step 5. Sample a new association 𝑐_𝑖,𝑘=𝑗 with probability 𝜌_𝑖^(𝑗): 𝜌_𝑖^(𝑗)=𝜌_𝑖^(𝑗)∑^𝑇_𝑗′=0𝜌^(𝑗′)_𝑖,𝑗=0,…,𝑇, (22) where 𝜌_𝑖⁽⁰⁾=𝑝𝑐_𝑖,𝑘=0∣𝑐_{𝑖,1∶𝑘−1},𝐳_1∶𝑘,𝜌_𝑖^(𝑗)=𝑝𝑐_𝑖,𝑘=𝑗∣𝑐_{𝑖,1∶𝑘−1},𝐳_1∶𝑘. (23)
Step 6. Calculate the new weights 𝐰_𝑖,𝑘∝𝐰_{𝑖,𝑘−1}𝑝𝐳_𝑘∣𝑐_𝑖,𝑘,𝑐_{𝑖,1∶𝑘−1},𝐳_1∶𝑘−1𝑝𝑐_𝑖,𝑘∣𝑐_{𝑖,1∶𝑘−1}𝑝𝑐_𝑖,𝑘∣𝑐_{𝑖,1∶𝑘−1},𝐳_1∶𝑘. (24)
Step 7. Update the 𝑗th target according to the new 𝑐_𝑖,𝑘 (𝑖=1,2,…,𝑁_𝑝). If 𝑐_𝑖,𝑘=𝑗 (𝑗≠0), then 𝐱^(𝑗)_{𝑖,𝑘∣𝑘}=𝐱^(𝑗)_{𝑖,𝑘∣𝑘},𝐏^(𝑗)_{𝑖,𝑘∣𝑘}=𝐏^(𝑗)_{𝑖,𝑘∣𝑘}. (25)
Step 8. Calculate the target state vectors 𝐱^(𝑗)_𝑘|𝑘=^𝑁𝑝_𝑖=1𝐰_𝑖,𝑘𝐱^(𝑗)_{𝑖,𝑘|𝑘}. (26)
End for. (Resample if needed [30–32].)

It should be pointed out that in 𝐻_∞ filter the level 𝛾 must be selected carefully to guarantee the existence of the HUF, or else the filtering program will fail in the applications of HURBPF approach. To adaptively adjust 𝛾 to its minimum at each iteration, we can choose the value of 𝛾_𝑘 as [16] 𝛾²_𝑘=𝛼maxeig𝐏⁻¹_{𝑘∣𝑘−1}+𝐏⁻¹_{𝑘∣𝑘−1}𝐏^𝑥𝑧_{𝑘∣𝑘−1}𝐑_𝐤⁻¹×𝐏⁻¹_{𝑘∣𝑘−1}𝐏^𝑥𝑧_{𝑘∣𝑘−1}^𝑇−1, (27) where 𝛼 is a scalar larger than one and max{eig(𝐴)⁻¹} denotes the maximum eigenvalue of the matrix 𝐴⁻¹.

4. Simulation Results

This section presents the two-dimensional (2D) target tracking examples to demonstrate the performance of the proposed tracking algorithms.

Example 1. The targets are modeled with near constant velocity model in Cartesian coordinates. The discrete-time dynamic and measurement models of the 𝑗th target have the following form: 𝐱_𝑗,𝑘=𝐅_{𝑗,𝑘−1}𝐱_{𝑗,𝑘−1}+𝐰_𝑘−1,𝐳_𝑗,𝑘=𝐇_𝑗,𝑘𝐱_𝑗,𝑘+𝐯_𝑘, (28) where 𝐅_{𝑗,𝑘−1}=10𝛿_𝑡0010𝛿_𝑡,𝐇00100001_𝑗,𝑘=,10000100 (29)𝐰_𝑘−1 and 𝐯_𝑘 are zero mean Gaussian process noises. The standard process and measurement noise variances are selected as 𝜎_𝑥=0.001 km and 𝜎_𝑧=20 km, respectively. The sample interval 𝛿_𝑡=1 and the correct measurements return with a known detection probability 𝑃_𝑑=1 (detection missing will not happen). At each time step 𝑘, the target is located at coordinates (𝑥_𝑘,𝑦_𝑘) and moves with constant velocity vector (𝑥̇_𝑘,𝑦̇_𝑘), which are combined with the target’s state vector 𝐱_𝑘=(𝑥_𝑘,𝑦_𝑘,𝑥̇_𝑘,𝑦̇_𝑘)^𝑇. In this two-target-crossing scenario (see Figure 1), target 1 and target 2 begin at 𝑘=0 with position-velocity coordinates 𝐱₀⁽¹⁾=(50km,50km,10km/s,2.5km/s)^𝑇 and 𝐱₀⁽²⁾=(50km,400km,10km/s,−9km/s)^𝑇, respectively. The total number of tracking time steps is 40.

Figure 1 

Example of two targets tracking (a successful track by HRBPF).

The KRBPF serves as the baseline algorithm, and the proposed HRBPF algorithm is compared with it. Both algorithms are designed based on the same assumptions, and the performance of the two algorithms is evaluated by the average results over Monte Carlo runs. The initial state estimates of the two targets are set to 𝐱⁽¹⁾_0∣0=(30km,50km,10km/s,2.5km/s)^𝑇 and 𝐱⁽²⁾_0∣0=(30km,400km,10km/s, −9km/s)^𝑇, respectively. The clutter is modeled as independent and identically distributed with uniform spatial distribution in a detection region of the coordinate plane [0 km, 500 km] × [0 km, 500 km], and the number of clutter measurements obeys a Poisson distribution with the Poisson random number 𝜆=10 (clutter rate). The number of particles used in the simulation is 100. The standard measurement variance used in both algorithms, KRBPF (Kalman filter based RBPF) and HRBPF (𝐻_∞ filter based RBPF), is 𝜎^𝑎_𝑧=30km. The position root mean square errors (RMSEs) of the KRBPF and HRBPF algorithms are illustrated in Figure 2 (target 1) and Figure 3 (target 2). We can see that the overall performance of the proposed HRBPF is significantly better than that of the KRBPF as expected. This is due to the fact that the HF outperforms the KF when the statistics of the noise processes are not known by the filtering algorithms.

Figure 2 

Comparison of the average position estimation errors of target 1.

Figure 3 

Comparison of the average position estimation errors of target 2.

Example 2. Consider a scenario of tracking two targets using bearings-only measurements received by two static sensors which are located at (𝑠^𝑖_𝑥,𝑘,𝑠^𝑖_𝑦,𝑘), 𝑖=1,2 (see Figure 4, where “□" represents the location of the sensor). The dynamic of discrete-time velocity model is the same as (28), and the measurement function is given by 𝑧^𝑖_𝑗,𝑘𝑦=arctan^𝑗,𝑘−𝑠^𝑖_𝑦,𝑘𝑥_𝑗,𝑘−𝑠^𝑖_𝑥,𝑘+𝑟^𝑖_𝑘, (30) where 𝑟^𝑖_𝑘∼𝑁(0,𝜎²_𝑧) with 𝜎_𝑧=0.01 rad. Since this measurement model is nonlinear, the URBPF (UKF based RBPF) and HURBPF are employed for tracking. The clutter originated measurements obey Poisson distribution with clutter rate 𝜆=5 in a single time step. The detection region of the coordinated plane is set to be [0 km, 50 km] × [0 km, 50 km]. The standard bearings-only measurement variance used in both algorithms is 𝜎^𝑎_𝑧=0.03 rad.

Figure 4 

Example of two targets bearings-only tracking (a successful track by HURBPF).

The performance of the HURBPF is compared with that of the URBPF using 50 particles. The two targets begin at 𝑘=0 with position-velocity coordinates 𝐱₀⁽¹⁾=(15km,5km,0.2km/s,1km/s)^𝑇 and 𝐱₀⁽²⁾=(45km,5km,−0.8km/s,1km/s)^𝑇, respectively. The total number of tracking time steps is 40. In this example, the initial state estimates of the two targets are set to 𝐱⁽¹⁾_0∣0=(13km,3km, 0.05km/s,0km/s)^𝑇 and 𝐱⁽²⁾_0∣0=(43km,3km,−0.2km/s,0km/s)^𝑇.

The tracking performance of the HURBPF and the URBPF in terms of RMSE in position is shown in Figures 5 and 6. It can be seen that the HURBPF also outperforms the URBPF for almost the entire simulation interval since the HURBPF deals with the unknown measurement noises variance priors. By the typical examples demonstrated above, a conclusion can be drawn that the 𝐻_∞ based RBPF algorithms show good robustness against unknown noise statistics.

Figure 5 

Comparison of the average position estimation errors of target 1.

Figure 6 

Comparison of the average position estimation errors of target 2.

5. Conclusions

In this paper, we present an improved Rao-Blackwellized particle filtering algorithm by using the 𝐻_∞ unscented transform. The main benefit lies in it requiring no priori knowledge of the statistical properties of the measurement noise. By decomposing the RBPF filtering distribution, the HRBPF and HURBPF algorithms are developed based on the 𝐻_∞ filter for solving the multitarget tracking problems with unknown noise statistics. The proposed algorithms are tested by both the linear and nonlinear tracking experiments. Simulation results show that they can achieve better tracking performance than the standard KF and UKF based RBPF algorithms.

Acknowledgment

The author would like to thank the anonymous reviewers for their helpful comments on this paper.