A novel distributed fusion algorithm for multisensor nonlinear tracking
 Jingxian Liu^{1},
 Zulin Wang†^{1, 2} and
 Mai Xu†^{1}Email author
https://doi.org/10.1186/s136340160362y
© Liu et al. 2016
Received: 30 December 2015
Accepted: 6 May 2016
Published: 18 May 2016
Abstract
The covariance intersection (CI), especially with feedback structure, can be easily combined with nonlinear filters to solve the distributed fusion problem of multisensor nonlinear tracking. However, this paper proves that the CI algorithm is suboptimal, thus degrading the fusion accuracy. To avoid such an issue, a novel distributed fusion algorithm, namely Monte Carlo Bayesian (MCB) algorithm, is proposed. First, it builds a distributed fusion architecture based on the Bayesian tracking framework. Then, the Monte Carlo sampling is incorporated into this architecture to form a feasible solution to nonlinear tracking. Finally, the simulation results verify that our MCB algorithm advances the stateoftheart distributed fusion of nonlinear tracking.
Keywords
1 Introduction
Distributed fusion [1] refers to combining the information of decentralized sensors [2, 3], in which the observation information of each sensor is processed independently. In comparison with centralized fusion, the distributed fusion can significantly save time and storage resources in the fusion center. As for distributed fusion, the conventional solution is BarShalomCampo algorithm [4]. Later, Chang et al. [5] proved that this algorithm is not optimal in terms of the rootmeansquareerror (RMSE), and they further presented an optimal fusion algorithm. Nevertheless, this algorithm can only be used in the scenario of two sensors fusion. Hence, Li et al. [6] proposed a unified fusion architecture, which can be used for two and more sensors. However, all aforementioned algorithms are designed with linear state space models (SSM), which results in the inferior fusion performance of nonlinear tracking.
Recently, some intelligent techniques, such as the adaptive fuzzy backstepping control technique [7] and the adaptive neural network technique [8] with nonlinear model predictive control, are proposed to advance the application of nonlinear models in multisensors or sensor network control [9]. Hence, the optimal distributed fusion with nonlinear model becomes one of the most important direction of signal processing. To our best knowledge, the most effective algorithms for fusion are based on covariance intersection (CI) [10, 11]. These CIbased algorithms can be easily combined with nonlinear filers to form the stateoftheart solutions to nonlinear problems, such as UKFSCI [12] and DPFICI [13]. The former is more accurate because the unscented Kalman filter (UKF) performs better in normally nonlinear tracking, while the latter has an advantage in nonGaussian scenarios benefitting from the particle filter (PF). However, these CIbased algorithms require the computation on additional fractional powers to calculate the fusion results [14], which causes an increment of estimation error. Although many applications utilize a feedback structure^{1} to improve the fusion accuracy [15], this paper proves that the suboptimality caused by the increment of estimation error still exists in CI. To our best knowledge, few algorithms are proposed to overcome this suboptimality. Hence, the present study is motivated to achieve a fusion algorithm which can outperform the CI algorithm. In summary, the novelty of this paper is that our algorithm overcomes the suboptimality problem in covariance intersection (CI) fusion, which has not been addressed in the previous work.

A novel distributed fusion architecture is proposed, which makes full use of the information of different sensors to produce robust fusion estimation.Table 1
Notation list
Notation
Meaning of the notation
x _{ k }
The state at time step k
P _{ k }
The variance of state at time step k
i
The ith sensor
I
The number of sensors
n
The nth particle
N
The number of particles
\(\boldsymbol {X}^{n}_{k}\)
The nth particle of state at time step k
\(\boldsymbol {z}^{i}_{k}\)
The observation of the ith sensor at time step k
\(\boldsymbol {z}^{f}_{1:k}\)
The observation set of all sensors from time step 1 to k
p(·)
The transition distribution of state
q(·)
The likelihood of state
h ^{ i }(·)
The observation function of the ith sensor
Q
The variance of observation noise
δ(·)
The Dirac delta function
C
The normalized constant
F
The transition function
n _{ k−1}
The transition noise at time k−1
d _{ x,k },d _{ y,k }
The distances along x and y direction at time step k
v _{ x,k },v _{ y,k }
The velocity along x and y direction at time step k
s _{ T }
The sampling interval
n _{ d },n _{ v }
The transition noise of distance and velocity with variance \({\sigma ^{2}_{d}}\) and \({\sigma ^{2}_{v}}\)
n _{ θ },n _{ r }
The observation noise of azimuth and distance with variance \(\sigma ^{2}_{\theta }\) and \({\sigma ^{2}_{r}}\)

A MCB algorithm is developed by means of the Monte Carlo sampling, which solves the nonlinear fusion problem based on the numerical approximation.
2 Distributed fusion architecture based on BTF
where p(·) is the transition distribution, and q(·) is the likelihood of x _{ k } in the ith sensor. According to the BTF, the fusion process is to obtain the posterior distribution \(p\left (\boldsymbol {x}_{k}\boldsymbol {z}^{f}_{1:k}\right)\) based on the SSM, where \(\boldsymbol {z}^{f}_{1:k} = \left \{\boldsymbol {z}^{i}_{1:k}\right \}^{I}_{i=1}\) is the integrated observation set combining all the sensors information from step 1 to k. Notice that, in the feedback structure, we normally have \(\boldsymbol {z}^{i}_{1:k}=\left \{\boldsymbol {z}^{i}_{k},\boldsymbol {z}^{f}_{1:k1}\right \}\) because the prior observation information has been shared between sensors. Then, the fusion is calculated iteratively with two steps: the prediction and update steps.
is the marginal likelihood.
Figure 1 shows that, in the fusion process of time step k, the fusion center makes a prediction with (3). Then, the information of prediction is sent to each distributed sensor, in which the likelihood is calculated. Afterwards, each sensor sends the likelihood back to the fusion center to yield the final result with (6). Finally, the fusion result feeds back to the fusion center as the prior information of the next step.
3 Suboptimality in CI algorithm with feedback structure
Lemma 1.
where w _{ i }∈(0,1) and \(\sum _{i=1}^{I}w_{i}=1\).
Proof.
This completes the proof of Lemma 1.
In the feedback structure, Lemma 1 provides a generalized form of posterior distribution \(p\left (\boldsymbol {x}_{k}\boldsymbol {z}^{f}_{1:k}\right)\) calculated in CI fusion. As seen from (8) and (6), the CI algorithm adds a fractional power ω _{ i } to each likelihood.
Lemma 2.
Assuming that likelihoods obey Gaussian distribution, i.e., \(q\left (\boldsymbol {z}^{i}_{k}\boldsymbol {x}_{k}\right) = \mathcal {N}\left (\boldsymbol {z}^{i}_{k}; h^{i}(\boldsymbol {x}_{k}),{\sigma ^{2}_{i}}\right)\) for each i={1,2,…I}, where h ^{ i }(·) is the mapping from \(\mathbb {R}^{d_{x}}\) to \(\mathbb {R}^{d_{z^{i}}}\), and \({\sigma ^{2}_{i}}\) is the corresponding variance. Then, the variance of likelihood in each sensor becomes \({\sigma ^{2}_{i}}/\omega _{i}\) in CI algorithm.
Proof.
Hence, the variance of each likelihood in CI fusion has turned into \({\sigma ^{2}_{i}}/\omega _{i}\) in the process of calculating the posterior distribution. This completes the proof of Lemma 2.
ω _{ i } increases the variance of likelihood in CI fusion. Hence, compared with (6) in the distributed fusion of Section 2, the estimation (8) calculated in CI contains more uncertainty under the same observations. Such uncertainty leads to the increment of estimation error. As a result, the suboptimality still exists in CI fusion with feedback structure.
4 Monte Carlo Bayesian algorithm
To avoid the increment of estimation error, this paper utilizes the distributed fusion architecture in Section 2 to develop the fusion algorithm. In the most common applications, the observation is nonlinear [20]. Therefore, the posterior distributed in (6) cannot be simply calculated by joint Gaussian method with the procedure of squares completion [21]. To solve this problem, our MCB algorithm is proposed. This algorithm incorporates the Monte Carlo sampling into the distributed fusion architecture, by which the posterior distribution can be directly approximated with particles.
4.1 Monte Carlo sampling in MCB algorithm
where C is the normalization constant.
4.2 Fusion with MCB algorithm
According to the distributed fusion architecture in section 2, the estimation of state x _{ k } is calculated iteratively by (3) and (6).
Second, the particles are sent to each sensor node. After achieving the observation \(\boldsymbol {z}^{i}_{k}\) in the ith sensor, the likelihood is approximated by the set: \(\left \{q\left (\boldsymbol {z}^{i}_{k}\boldsymbol {X}^{n}_{k}\right)\right \}^{N}_{n=1}\). Then, each sensor sends its own approximated likelihood back to the fusion center.
5 Simulation results
In this section, the simulation results are provided to validate the performance of our MCB algorithm. In our simulation, a classic twodimensional (2D) fusion scenario with the nonlinear observation of one active and two passive radars is considered. In such a scenario, the fusion process is simulated by tracking a single target under several cases. For each case, different noise and kinematic models of transition equation are applied. Finally, 100 Monte Carlo simulations have been run for each case, and the fusion performance of nonlinear tracking is compared between our MCB algorithm and the UKFSCI algorithm [12] with feedback structure. Note that, in this paper, we concentrate on Gaussian tracking scenarios, in which the UKFSCI outperforms the DPFICI [13].
5.1 Simulation setup
In addition, n _{ k }=[n _{ d }, n _{ d }, n _{ v }, n _{ v }]^{T} represents the state transition noise in distance and velocity. For fair comparison, this paper utilizes the same noise for different models which obeys the Gaussian distribution, i.e., \(n_{d} \sim \mathcal {N}(n_{d}; 0, {\sigma ^{2}_{d}})\) and \(n_{v} \sim \mathcal {N}(n_{v}; 0, {\sigma ^{2}_{v}})\). Here, σ _{ d } and σ _{ v }=2σ _{ d }/s _{ T } are standard deviations of transition noise of distance and velocity, respectively.
where \(n_{\theta } \sim \mathcal {N}(n_{\theta }; 0,\sigma ^{2}_{\theta })\), \(n_{r} \sim \mathcal {N}(n_{r}; 0,{\sigma ^{2}_{r}})\); σ _{ θ } and σ _{ r } are standard deviations for tracking azimuth and distance. Then, we evaluate the performance of fusion with three radars, i.e., one active and two passive radars. Following [25], the aforementioned parameters are set as s _{ T }=0.1(s), σ _{ θ }=0.0001(r a d), σ _{ r }=0.5(m), and σ _{ d }={0.25,0.3,0.35,0.4}(m).
The simulation setting may be applied to a composite guidance scenario [26], in which there are one active and two passive radars [27, 28]. Although we simplify the observation equation in a 2D scenario, the simulation scenario is still suitable for the practical application of tracking and surveillance in network centric warfare (NCW) [29].
5.2 Evaluation
In our simulation, the single target tracking is performed with the aforementioned kinematic models. For each model, both MCB and UKFSCI algorithms are utilized to fuse the information of three radars (one active and two passive) together, and obtain the final tracking result. Furthermore, in the tracking process of each model, the fusion performance is evaluated with four standard deviations of transition noise (i.e., {0.25,0.3,0.35,0.4}(m)). Note that 100 Monte Carlo runs are applied in our simulation for each scenario, and the particle number N in our MCB algorithm is set to be 200.
The proportion of azimuth RMSE reduced by MCB algorithm over UKFSCI, with four standard deviations of transition noise
0.25(m)  0.3(m)  0.35(m)  0.4(m)  

CV  −2.51 %  1.29 %  4.90 %  7.78 % 
CT with turn rate of 2.5°/s  −3.19 %  0.88 %  4.10 %  7.43 % 
CT with turn rate of 5°/s  −0.53 %  3.47 %  7.47 %  10.36 % 
The proportion of distance RMSE reduced by MCB algorithm over UKFSCI, with four standard deviations of transition noise
0.25(m)  0.3(m)  0.35(m)  0.4(m)  

CV  3.55 %  5.80 %  8.02 %  10.37 % 
CT with turn rate of 2.5°/s  3.51 %  5.57 %  8.06 %  10.25 % 
CT with turn rate of 5°/s  4.54 %  7.06 %  9.42 %  12.24 % 
Tables 2 and 3 depict the proportions of reduced RMSE for different kinematic models and noises. As seen in these two tables, the reduction of RMSE becomes larger when the transition noise increases. This is consistent with the results shown in aforementioned figures. Moreover, the reduction of RMSE with CT model of turn rate 5°/s is largest among all three kinematic models. That means, our MCB algorithm has more advantage for high maneuvering targets.
In summary, based on the BTF, our MCB algorithm makes full use of the information of all observations, and fuses it to obtain more accurate estimation on target tracking. Hence, compared with the CI algorithm, there are two advantages in our MCB algorithm: (1) When the transition noise is large, the fusion RMSE of azimuth and distance is still small; (2) When the turn rate is large, the fusion RMSE of azimuth and distance is small as well. In other words, in the high maneuvering cases such as large transition noise and turn rate, our MCB outperforms the stateoftheart CI algorithm, in terms of the fusion RMSE.
5.3 Computational complexity
In this section, the computational complexity of our MCB algorithm is analysed. According to Section 4.2, in the fusion center, the algorithm contains two steps for each iteration: the prediction and update steps. In the prediction step, according to (18) and (19), the mean and variance are calculated with a summed form, in which the computational complexity is proportional to the dimension of tracking state. Moreover, N particles are drawn by sampling precess whose complexity is O(N). In the update step, firstly, we need to multiple all I sensors information of each particle to form a fusion likelihood. The computational complexity of this process is O(I). Then, the likelihood is used to compute the fusion results with the complexity being O(N), according to (20) and (21). In summary, the computational complexity in the update step is O(I·N).
To further evaluate the computational complexity of our MCB algorithm, we have recorded the computational time of the prediction and update steps, respectively, for one iteration in the simulation. Specially, the computer used for the test is with Intel Core i73770 CPU at 3.4 GHz and 4 GB RAM. In the aforementioned tracking case in Sections 5.1 and 5.2, the dimension of tracking state is 4, the particle number is 200 and the sensor number is 3. Through the simulation, we found out that the prediction and update steps take around 0.563 and 2.3 ms for one iteration. In other word, we only need 2.863 ms to compute the tracking results in the fusion center at each iteration. Hence, this algorithm is fast enough to utilize in the fields of radar tracking and fusion.
6 Conclusions
In this paper, we have proposed a novel MCB algorithm to achieve the distributed fusion estimation of nonlinear tracking. First, the distributed fusion architecture is set up based on BTF. Second, the suboptimality in CI algorithms is proved. Then, to solve the estimation problem of nonlinear tracking, the Monte Carlo sampling method is incorporated into the distributed architecture. Benefiting from this sampling method, the approximation of fusion results is obtained through random particles. Simulation results verify that our MCB algorithm outperforms the stateoftheart CI algorithm.
In summary, there are three directions of the future work in our paper. (1) Our MCB algorithm only offers a distributed calculation on the update step. Hence, a total distributed fusion structure is needed to further reduce the computation and communication overhead. (2) The timediscrete SSM used in our paper is actually a special case of continuoustime SSM. Therefore, we can extend our method to the exponential tracking scenario, in which the filtering can be processed with partially unknown and uncertain transition probabilities [30–33] with Markovian jump system. (3) Unknown inputs which represent the faults can be added into SSM and the residuals are calculated [34]. Hence, the fault detection algorithms [35] and the fuzzy model [36] can also be incorporated into our fusion system to strengthen the reliability of fusion process in the future work.
7 Endnote
^{1} The feedback structure means a structure which feeds the tracking result back to each sensor as the prior knowledge for the next step.
Declarations
Acknowledgements
This work was supported by the National Nature Science Foundation of China projects under Grants 61471022 and 61573037.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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