Journal of Electrical and Computer Engineering

Volume 2018 (2018), Article ID 4761601, 15 pages

https://doi.org/10.1155/2018/4761601

## Rao-Blackwellized Gaussian Sum Particle Filtering for Multipath Assisted Positioning

German Aerospace Center (DLR), Institute of Communications and Navigation, Muenchner Str. 20, 82334 Wessling, Germany

Correspondence should be addressed to Markus Ulmschneider; ed.rld@redienhcsmlU.sukraM

Received 25 August 2017; Revised 27 November 2017; Accepted 18 January 2018; Published 23 April 2018

Academic Editor: Huimin Chen

Copyright © 2018 Markus Ulmschneider 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

In multipath assisted positioning, multipath components arriving at a receiver are regarded as being transmitted by a virtual transmitter in a line-of-sight condition. As the locations and clock offsets of the virtual and physical transmitters are in general unknown, simultaneous localization and mapping (SLAM) schemes can be applied to simultaneously localize a user and estimate the states of physical and virtual transmitters as landmarks. Hence, multipath assisted positioning enables localizing a user with only one physical transmitter depending on the scenario. In this paper, we present and derive a novel filtering approach for our multipath assisted positioning algorithm called Channel-SLAM. Making use of Rao-Blackwellization, the location of a user is tracked by a particle filter, and each landmark is represented by a sum of Gaussian probability density functions, whose parameters are estimated by unscented Kalman filters. Since data association, that is, finding correspondences among landmarks, is essential for robust long-term SLAM, we also derive a data association scheme. We evaluate our filtering approach for multipath assisted positioning by simulations in an urban scenario and by outdoor measurements.

#### 1. Introduction

The amount of available and potential services requiring precise localization of a user has steadily increased over the recent years. Global navigation satellite systems (GNSSs) can often satisfy the demands for localization in scenarios where the receiver has a clear view of the sky. However, if the view of the sky is obstructed, such as indoors, in urban canyons, or in tunnels, the positioning performance of GNSSs may be drastically decreased, or no positioning solution may be obtained at all [1]. Reasons for this include a low received signal power due to signal blocking or shadowing and multipath propagation.

In contrast to GNSS signals, many kinds of terrestrial signals are likely to have a good coverage in GNSS denied places. In particular, cellular radio frequency (RF) signals are designed to be reliably available at least in populated areas, and they may be used as signals of opportunity (SoOs) for positioning. However, also terrestrial signals experience multipath propagation. Multipath propagation biases range estimates if standard correlator based methods are used. Various approaches to handle the multipath problem have been addressed in the literature, for example, in [2]. Advanced methods such as maximum likelihood (ML) mitigation algorithms try to estimate the channel impulse response (CIR) and to mitigate the influence of multipath components (MPCs) on the line-of-sight (LoS) path [3].

The idea of multipath assisted positioning is contrary, though. Instead of regarding multipath propagation as ill, the spatial information of MPCs on the receiver position is exploited. In [4], the information of MPCs is used in a fingerprinting scheme. Going one step further, each MPC can be regarded as being transmitted by a virtual transmitter in a pure LoS condition, and the virtual transmitters can be used to locate the user. Such an approach is called multipath assisted positioning.

The authors of [5, 6] derived some theoretical bounds for multipath assisted positioning. Multipath assisted positioning schemes have, for example, been applied in radar applications [7], using ultrawideband (UWB) [8, 9] or 5G [10] systems and in cooperative systems [11].

If the locations of physical transmitters and reflecting and scattering objects are known, the locations of virtual transmitters can be calculated based on geometrical considerations. The authors of [12] assume the room layout to be known and focus on the association among virtual transmitters and reflecting walls. In a general setting, however, the scenario is unknown to the user.

The authors of [13, 14] have presented a multipath assisted positioning scheme named Channel-SLAM that does not rely on prior information on the scenario. Instead, the locations of the physical and virtual transmitters are estimated simultaneously with the user position in a simultaneous localization and mapping (SLAM) [15, 16] approach. In general, SLAM describes the simultaneous estimation of a user position and the locations of landmarks. In Channel-SLAM, the landmarks are the physical and virtual transmitters. Previous extensions to Channel-SLAM include mapping of the user positions [17], the consideration of vehicular applications [18], and data association methods [19, 20], for example.

Nonlinearities in the prediction and update equations of the Bayesian recursive estimation framework prohibit the use of optimal algorithms such as the Kalman filter, since the integrals involved in the estimation process cannot be solved in closed form or become intractable. A popular alternative is the extended Kalman filter (EKF) [21], which linearizes the nonlinear terms using a first-order Taylor series expansion. However, such a linearization can introduce large errors in the estimation process [22]. The unscented Kalman filter (UKF) [23, 24] uses a nonlinear transformation to deal with nonlinearities and outperforms the EKF in a wide range of applications [22, 25].

UKF methods have found their way into localization problems, for example, in [27, 28]. The authors of [29] propose Gaussian sum cubature filters. In [30, 31], the authors consider a Rao-Blackwellization scheme for SLAM with a particle filter for the user state and UKFs for the landmark states, where the measurement model is based on linearization, though.

The current Channel-SLAM algorithm uses a Rao-Blackwellized particle filter to estimate the user state and the location of transmitters simultaneously. Hence, both the user state probability density function (PDF) and the transmitter state PDFs are represented by a large set of particles, tending to result in a high memory occupation. This paper is an extension of [32], where we proposed a novel estimation approach for Channel-SLAM scheme based on Rao-Blackwellization and performed first simulations. We refer to this new estimation method as Rao-Blackwellized Gaussian sum particle filter (RBGSPF). In the RBGSPF, the user position is tracked by a sequential importance resampling (SIR) particle filter, while the physical and virtual transmitter state PDFs are represented by Gaussian mixture models estimated by UKFs. This parametrized representation of the transmitter states is a key enabler for exchanging maps of transmitters among users, since the amount of data that has to be communicated among users can be decreased drastically compared to the nonparametric representation with particles. Such an exchange of maps may be performed directly among users or via a central entity, for example, in form of local dynamic maps (LDMs) in an intelligent transportation system (ITS) context. In this paper, we provide a full and detailed derivation of our novel algorithm. In particular, we derive the calculation of the particle weights in the user particle filter given the representation of the transmitters in the UKF framework. Since data association is an essential feature for the accuracy in long-term SLAM, we also derive a data association method based on [33]. We evaluate our algorithm by both simulations in an urban scenario and outdoor measurements.

The remainder of this paper is structured as follows. Section 2 describes the fundamental idea behind multipath assisted positioning and Channel-SLAM. In Section 3, we briefly summarize some concepts of nonlinear Kalman filtering. The derivation of the RBGSPF is presented in Section 4, and a solution to data association is presented in Section 5. After the experimental results in Section 6, Section 7 concludes the paper.

Throughout the paper, we use the following notation:(i)As indices, stands for a user particle, denotes a transmitter or a signal component, is a component in a Gaussian mixture model, and stands for a sigma point.(ii) denotes the transpose of a matrix or vector.(iii) denotes the identity matrix of dimension .(iv) and denote the zero matrices of dimensions and , respectively.(v) denotes the PDF of a normal distribution in with mean and covariance .(vi) denotes the speed of light.(vii) denotes the Euclidean norm of a vector.

#### 2. Multipath Assisted Positioning

##### 2.1. Virtual Transmitters

The idea of virtual transmitters is illustrated in Figure 1. The physical transmitter Tx transmits an RF signal. A mobile user equipped with an RF receiver receives the transmitted signal via three different propagation paths.