Bayesian filter based on grid filtration and its application to Multi-UAV tracking

Abstract

A filtering method called Grid Filtration Filter (GFF) is proposed based on Bayesian inference. First, we select the high-probability region of the current state according to the confidence parameter $\alpha$, and obtain samples uniformly in this region. Second, these samples are regarded as discretized “potential states” and their posterior weights are calculated based on the Bayesian inference. Third, the grid filtration method is used to choose these “potential states” with high weights, and these selected “potential states” and their normalized weights are used to represent the posterior distribution, and thus estimate the state. we finally verify the feasibility of the GFF algorithm in a typical two-dimensional linear non-Gaussian filtering scenario. Results show that the GFF has a slightly better accuracy than the particle filter (PF) algorithm, and the calculation speed is better by a factor of approximately 45 compared with the PF with 10,000 particles. We finally validate the GFF algorithm in a collaborative target tracking scenario. Results show that the accuracy of the GFF estimation is slightly better than that of the extended Kalman filtering and that of the unscented Kalman filtering, while the advantage of the estimation accuracy of velocity and acceleration is more obvious.

Introduction

In recent years, multi-sensor fusion has been paid much attention and has been widely applied to target tracking, localization, guidance and navigation, and fault detection [1], [2], [3], [4], [5], [6]. With the increase of the number and types of sensors, nonlinear control and nonlinear tracking have become major concerns in multi-sensor information fusion for researchers. For nonlinear control problems, [7] proposed a new command filter based on a second-order finite-time differentiator to generate command signals and their derivatives. [8] deal with the unknown nonlinear function using neural networks. [9] designed a modified FT command filter to ensures the output of the filter can faster approximate the derivatives of virtual signals, suppress chattering, and relax the input signal limit of the Levant differentiator.

For nonlinear tracking problems, Bayesian inference is a popular theory in the field of data fusion [10], [11], [12], [13], [14] study the application of Kalman filter (KF) class in target tracking. In the case of a linear Gaussian environment, the posterior PDF can always be expressed by means and covariances, and estimation can be implemented in terms of the famous updating equations of the KF [15], [16], [17]. In the case of a nonlinear environment, Bayesian inference is difficult to implement accurately. This is due to the inability to obtain the analytical solution of the prior distribution in the process of inference [18], [19], [20], [21]. The linearization of nonlinear functions is a method to solve this problem. Typical corresponding filter algorithms include extended Kalman filter (EKF) [22], unscented Kalman filter (UKF) [23], cubature Kalman filter (CKF) [24], Maximum correlation-entropy Kalman filtering (MCKF) [25] and so on. Although KF class algorithms can approximate solve the nonlinear problem, it is under the assumption of Gaussian noise model. It is difficult to achieve accurate filtering in non-Gaussian environment for KF class algorithms. Particle Filter (PF) is one of the most effective measures to address the non-Gaussian filters and it has been widely used to solve nonlinear non-Gaussian filter problems [15], [16], [17].

The main procedure of PF can be roughly summed up as: 1. obtain particles according to a proposed distribution; 2. calculate the prior weights and the likelihood weights according to the process noise model and measurement noise model respectively and mix them; 3. divide the mixed weights by the corresponding density of the proposed distribution and normalization. After that, the posterior distribution is reflected from these weighted particles [26]. From this procedure we can observe that it is more flexible in solving nonlinear non-Gaussian filtering problems. However, the sampling basis (i.e., the selection of the proposed distribution) and random disturbance introduced by the finite random samples often lead to instability of the PF performance, which restricts the development of the PF algorithm. Furthermore, the process of select particles is random, but the weights are calculated precisely according to the noise model. This phenomenon might cause random disturbances which could affect the filtering accuracy [27,28].

To overcome the aforementioned problem, a new grid filtration filter (GFF) is proposed based on the Bayesian inference that can avoid the random disturbances in a linear non-Gaussian Scenario. Firstly, the grids are used to partition the high probability region of the prior distribution, and setting particles in each position of the grid center, and each particle was considered as a“potential state”. Then, the Bayesian inference was used to calculate the posterior weights corresponding to “potential states”. Finally, these “potential states” were screened to avoid filtering divergency according to the corresponding size of weights and achieve the state estimation. The main contributions of this paper as follows:

• For the multi-dimensional PDF, we propose the concept of the probability box under the confidence condition. The box compresses the probability space of the multi-dimensional PDF into a bounded space. It is interesting that this bounded space contains a large amount of probability information, and the specific amount of information depends on the selection of confidence parameter α. This lays a foundation for the meshing of probability space, the inference of weights on grids and the final realization of the GFF algorithm.

• We realize the meshing of the space in the probability box and the discrete Bayesian inference process of the probability of each potential state in the grid. After meshing the probability box, we treat each grid as a potential state. The overall continuous Bayesian filtering process is then considered as a state transition process between potential states at different time steps. In this process, we use the discrete Bayesian theorem to infer the probability corresponding to these potential states. we finally adopt the discrete Bayesian filtering approximation to estimate the state of continuous Bayesian filtering.

• We propose a grid filtration method with which to correct the HPR of the state space. We screen the potential states according to the magnitude of the posterior weight; that is, the potential states with high posterior weight are retained, whereas the potential states with very low posterior weight are deleted. Adopting this grid filtration method, the potential state is compressed from the HPR of the prior distribution to that of the posterior distribution, moreover, this avoids the growth of the number of potential states in the process filtering so as to ensure the realizability of the GFF filtering method.

• The GFF algorithm is proposed. Simulation results show that the GFF performs better than the PF and other filtering algorithms in terms of both the filtering precision and calculation cost in linear non-Gaussian scenarios. Meanwhile, we verify the effectiveness of the proposed algorithm in a cooperative target tracking scenario involving multi-unmanned aerial vehicles (multi-UAVs).

The structure of this paper is as follows: In Section II, the problem model and the principle of the PF are introduced. In Section III, the main mechanism of the GFF is divided into three parts: the selection of the HPR in the state space, Bayesian inference-based weights update, grid filtration and state estimation, to be explained in detail. In Section IV, simulation results are provided that describe the performance of the GFF. In Section V this paper is concluded.