Journal of Computational Engineering

Volume 2014, Article ID 175820, 6 pages

http://dx.doi.org/10.1155/2014/175820

## An Improved Unscented Particle Filter with Global Sampling Strategy

Hefei Electronic and Engineering Institute, Hefei 230000, China

Received 13 July 2014; Accepted 22 November 2014; Published 10 December 2014

Academic Editor: Hongli Dong

Copyright © 2014 Yi-zheng Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Particle filter (PF) has many variations and one of the most popular is the unscented particle filter (UPF). UPF uses the unscented Kalman filter (UKF) to generate particles in the PF framework and has a better performance than the standard PF. However, UPF suffers from its high computation complexity because it has to execute UKF to each particle to obtain proposal distribution. This paper gives an improved UPF aiming at reducing the computation complexity of the algorithm. In comparison to the standard UPF, the new strategy generates proposal distribution from the mean and covariance value of the whole particles instead of from each particle. Thus the improved algorithm utilizes the characteristics of the whole particles and only needs to perform UKF algorithm once to get the proposal distribution at each time step. Experimental results show that, compared to standard UPF, the improved algorithm reduces the time consumption greatly almost without performance degradation.

#### 1. Introduction

Nonlinear and non-Gaussian filtering has a wide range of applications in many fields [1–3]. Among the many methods that have been proposed in the literature for these applications, particle filter (PF) has become one of the most popular. For decades PF has been applied to a variety of problems, such as computer vision, signal processing [4], target tracking [5], and financial pricing [6].

However when designing a PF a major problem is to choose a proper proposal distribution of the particles. Due to the fact that the particles are drawn from this distribution, and the weight values of particles are also related to this distribution, the performance of a PF is strongly influenced by the choice of the proposal distribution.

To design better proposal distributions, several techniques based on linearization have been proposed. In one method, an extended Kalman filter (EKF) is used to generate the proposal distribution, and this is known as extended Kalman PF (EKPF). However, the linearization operation in EKPF introduces modeling errors, which can yield large estimation errors if the system is highly nonlinear [7]. To overcome this problem, a more accurate PF was proposed using unscented Kalman filter (UKF) to generate the proposal distribution [8]. The UKF can accurately compute the mean and covariance of nonlinear systems up to the second order of the Taylor series expansions. This type of particle filter using UKF to generate proposal distribution is known as the unscented particle filter (UPF). The UKF produces proposal distributions that exhibit a larger support overlap with the true posterior probability than the EKF proposal distributions; thus it is more suitable for the proposal distribution generation, and consequently the UPF performs much better than EKPF [9].

However, in UPF framework, an unscented Kalman filtering operation which is often more complex to compute needs to be applied to each particle to generate and propagate a Gaussian proposal distribution. Generally there are many particles in particle filter; therefore the time complexity of UPF is often tens of times larger than general PF; therefore the application of UPF is limited in many occasions.

In order to reduce the time cost of the UPF, Fasheng and Yuejin [10] proposed an IUPF method that divides the particles into two parts, one part generated from the transition prior and another part from the UKF. Basically this method is a combination of the general PF and the UPF. Because only part of the particles needs to be generated using UKF, this method has a less time cost than UPF. For the same reason, however, with less particles generated from UKF, the filter accuracy is also decreased compared to the UPF. Thus IUPF method can hardly keep high accuracy and low computation simultaneously.

To reduce the computation complexity of the UPF and maintain the advantage of its high precision, this paper focuses on an improved UPF which generates proposal distribution from the mean and covariance of the whole particles instead of from each particle. Stimulation experiments show that, by this strategy, the time cost of UPF can be reduced greatly and almost without loss of accuracy.

The rest of this paper is organized as follows. In Section 2, we briefly reviewed the general particle filter algorithm and the UPF. Section 3 gives the details of the proposed particle filter. Experiments and analysis are given in Section 4. The last section draws the conclusion.

#### 2. Particle Filter

##### 2.1. General Particle Filter

The nonlinear, non-Gaussian filtering problem considered here consists of recursively computing the posterior probability density function of the state vector in a general discrete-time state-space model, given the observed measurements. Such a general model can be formulated as where denotes the output observations, denotes the input observations, denotes the state of the system, denotes the process noise, and denotes the measurement noise. The mappings and represent the deterministic process and measurement models. The prior distribution at is represented by .

The main idea underlying the particle filter is to approximate the posterior probability density distribution by using a set of random samples with associated weights, which are called “particles.” The weight assigned to each particle is proportional to the probability. Let denote a random measure that characterizes the posterior probability density distribution , and is a set of particles with their associated weights . The weights are normalized as , where is the number of samples used in the approximation. The posterior density at time can be approximated as

The normalized importance weights are chosen by using the principle of importance sampling. If the samples are drawn from a so-called proposal distribution with the principle of Sequential Importance Sampling, can be given by where

Equation (4) provides a mechanism to sequentially update the importance weights, given an appropriate choice of proposal distribution, . The choice of proposal distribution is one of the most important issues for particle filter [6]. The standard particle filter uses the transition prior as proposal density [7], which means

By substitution of (5) into (4), the weights can be recursively computed and updated as

It is worth to notice that if the samples are generated from a suboptimal proposal such as in (5), the weights of most of them will approach zero after a few iterations. Therefore the filter spends a considerable amount of computational time in updating unlikely samples. Generally this can be avoided by the resampling step, which consists in replacing the unlikely samples with the more likely ones [8].

A better proposal distribution can help solve this problem with no need to resample particles, which is discussed in the next section.

##### 2.2. UPF

In order to overcome weights degeneracy of general particle filter, the unscented Kalman filter is adopted as the proposal density function in general particle filter, which is called UPF. It has been proven that the UKF is a better proposal than the transition prior in that it incorporates the most recent observations which usually contain much valuable information for estimating states [9].

In comparison to other linearization strategies, UPF tends to generate more accurate estimates of the true covariance of the state. Unlike the EKFP, this method does not linearize the nonlinear equations of the system, and consequently it can accurately capture the nonlinearity of the system. Details of the UPF algorithm can be found in [10]. Figure 1 gives the main steps of UPF in a flow chart.