Mathematical Problems in Engineering

Volume 2015, Article ID 168045, 9 pages

http://dx.doi.org/10.1155/2015/168045

## The Particle Filter Sample Impoverishment Problem in the Orbit Determination Application

^{1}University of São Paulo (USP)-EEL/LOB, Estrada Municipal do Campinho, s/n, 12602-810 Lorena, SP, Brazil^{2}National Institute for Space Research (INPE)-DMC, Avenida dos Astronautas, 1758 Jardim da Granja, 12227-010 São José dos Campos, SP, Brazil^{3}São Paulo Federal University (UNIFESP)-ICT/UNIFESP, Rua Talim, 330 Vila Nair, 12231-280 São José dos Campos, SP, Brazil

Received 11 February 2015; Accepted 6 May 2015

Academic Editor: Ruihua Liu

Copyright © 2015 Paula Cristiane Pinto Mesquita Pardal et al. 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

The paper aims at discussing techniques for administering one implementation issue that often arises in the application of particle filters: sample impoverishment. Dealing with such problem can significantly improve the performance of particle filters and can make the difference between success and failure. Sample impoverishment occurs because of the reduction in the number of truly distinct sample values. A simple solution can be to increase the number of particles, which can quickly lead to unreasonable computational demands, which only delays the inevitable sample impoverishment. There are more intelligent ways of dealing with this problem, such as roughening and prior editing, procedures to be discussed herein. The nonlinear particle filter is based on the bootstrap filter for implementing recursive Bayesian filters. The application consists of determining the orbit of an artificial satellite using real data from the GPS receivers. The standard differential equations describing the orbital motion and the GPS measurements equations are adapted for the nonlinear particle filter, so that the bootstrap algorithm is also used for estimating the orbital state. The evaluation will be done through convergence speed and computational implementation complexity, comparing the bootstrap algorithm results obtained for each technique that deals with sample impoverishment.

#### 1. Introduction

The orbit of an artificial satellite is determined using real data from the Global Positioning System (GPS) receivers. In the orbit determination process of artificial satellites, the nature of the dynamic system and the measurements equations are nonlinear. As a result, it is necessary to manage a fully nonlinear problem in which the disturbing forces as well as the measurements are not easily modelled. In this orbit determination problem, the variables that completely specify a satellite trajectory in the space are estimated, with the processing of a set of pseudorange measurements related to the body.

A spaceborne GPS receiver is a powerful resource to determine orbits of artificial Earth satellites by providing many redundant measurements, which ultimately yields high degree of the observability to the problem. The Jason satellite is a nice example of using GPS for space positioning. Through an on-board GPS receiver, the pseudoranges (error corrupted distance from satellite to each of the tracked GPS satellites) can be measured and used to estimate the full orbital state.

The bootstrap filter is a particle filter whose central idea is to express the required probability density function (PDF) as a set of random samples, instead of a function over state space [1–3].

Numerous strategies have been developed for solving the particles degeneracy (or sample impoverishment) problem that often arises in particle filter applications like introduction of a risk-sensitive particle filter as an alternative approach to mitigate sample impoverishment based on constructing explicit risk functions from a general class of factorizable functions [4]; incorporation of genetic algorithms into a particle filter [5, 6]; and many others [7–9]. All these strategies, although extremely interesting and suitable for the orbit determination problem, are not in the scope of this work. Here, the option was done for studying two classical methods to solve (or try to solve) the degeneracy problem: roughening and prior editing.

Herein, the main goal is to analyze the bootstrap filter behavior for the highly nonlinear orbit determination problem. Its simulation results are compared taking into account the sample impoverishment. A reference solution is a bootstrap particle filter (BPF) applied to orbit determination that has already been compared to the unscented Kalman filter solution for the same problem and works well for the analysis of the sample impoverishment issue [10].

#### 2. Particle Filter

The particle filter was designed to numerically implement the Bayesian estimator [2]. The Bayesian approach consists of constructing the PDF of the state based on all the available information, and, for nonlinear or non-Gaussian problem, the required PDF has no closed form. The bootstrap filter represents the required PDF as a set of random samples, which works as an alternative to the function over state space. This filter is a recursive algorithm for propagation and update of these samples for the discrete time problem. The Bayes rule, the key update stage of the method, is implemented as a weighted bootstrap [1].

The main idea of the BPF is intuitive and direct. At the beginning, particles () are randomly generated, based on the known initial PDF . At each step of time , the particles are propagated to the next step using the dynamics equation [2]. After receiving the measurement at time , the PDF is evaluated. That is, the conditional relative likelihood of each particle is calculated. If an -dimensional measurement equation is given as and is a Gaussian random variable with a mean of zero and a variance of , , then a relative likelihood that the measurement is equal to a specific measurement , given the premise that is equal to the particle , can be computed as follows [2]:

In (1), the symbol means that the probability is directly proportional to the right side. So if the equation is used for all particles , then the relative likelihood that the state is equal to each particle is correct. The relative likelihood values are normalized to ensure that the sum of all likelihood values is equal to one. Next, a new set of randomly generated particles is computed from the relative likelihood . In the resampling step, roughening was used, in order to prevent sample impoverishment. At this point, there is a set of particles that are distributed according to the PDF , and any desired statistical measure of it can be computed [2].

The particle filter, adjusted to the orbit determination problem, can be summarized as follows. (1) The dynamic and the measurement equations are given as where and are independent white noise processes with known PDFs. (2) initial particles () are randomly generated on the basis of the known initial state PDF . is a parameter chosen as a trade-off between computational cost and estimation accuracy [2]. (3) For , (a) in the time propagation step, obtain a priori (predicted) particles , using the dynamics equation and the PDF of the process noise, both known: where each noise vector, , is randomly generated on the basis of the known PDF of ; (b) compute the relative likelihood of each particle , conditioned on the measurement , using the nonlinear measurement equation and the PDF of the measurement noise, as in (1); (c) normalize the relative likelihood values: (d) in the resampling step, generate a set of a posteriori (resampled) particles , on the basis of the relative likelihood ; (e) now, there is a set of particles distributed according to the PDF , and mean and covariance statistical measures can be computed.

In the implementation of the bootstrap filter, there is only a small overlap between the prior and the likelihood.

There are some procedures that may be implemented for combating the consequent reduction in the number of truly distinct sample values, such as increasing the number of particles, roughening, and prior editing [1]. Here, they were implemented: a bootstrap particle filter with resampling (PF); a PF with roughening (PFR); and a PFR with prior editing (PFPE), in order to evaluate roughening and prior editing strategies for dealing with sample impoverishment.

##### 2.1. Roughening

Roughening will be the first remedy for sample impoverishment to be discussed. It restrains the resampled particles spread (a posteriori particles) by adding random noise to them, which is similar to adding artificial process noise to the Kalman filter [2]. In roughening approach, the a posteriori particles are modified, after the resampling step, as follows: is a zero-mean random variable (usually Gaussian); is a constant tuning parameter; is the number of particles; is the state space dimension; and is a vector of the maximum difference between the particle elements before roughening. The th element of the vector is given as where is the step time and and are particle numbers.

The tuning parameter choice is a compromise. Being too large, a value would blur the distribution, but being too small, it would produce tight clusters of points around the original particles [1]. In this paper, .

##### 2.2. Prior Editing

Prior editing can be tried if roughening does not prevent sample impoverishment. Such approach edits the a posteriori particles from the prior time instant, (after roughening), if the a priori particle from actual instant, , does not satisfy a coarse, pragmatic acceptance test [1]. Therefore, this procedure artificially boosts the number of samples of the prior editing in the neighborhood of the likelihood, for if an a priori particle is in a region of state space with small , it is rejected. Then, the a priori rejected particle can be roughened as many times as required, according to (5), until it is in a region of significant [2]. The prior editing was implemented as follows [1]:(a)Pass the resampled sample from previous instant, , through roughening and system model to generate the predicted sample from current instant, .(b)Calculate , the residual between the true and the predicted measurements, for the th particle of the sample, considering that the actual instant observation is available.(c)If the magnitude of is higher than six standard deviations of the measurement noise, then it is highly unlikely that is chosen as an a posteriori particle. In this case, is rejected, and is roughened again and passes one more time through dynamic model to generate a new a priori particle . As has already passed through roughening and generated a rejected predicted particle, this procedure may be repeated while is in a region of no negligible probability.

Due to the high computational cost involving prior editing, such approach was done only once. It is important to make it clear that, here, the th particle is, in fact, an -dimensional vector, while a sample is a matrix, where the th state variable is represented by particles.

The accommodation of roughening and prior editing in the bootstrap particle filter algorithm can be schematized as Figure 1 shows.