Mobile Information Systems

Volume 2016 (2016), Article ID 9595306, 15 pages

http://dx.doi.org/10.1155/2016/9595306

## Measuring the Uncertainty of Probabilistic Maps Representing Human Motion for Indoor Navigation

German Aerospace Center (DLR), Institute for Communications and Navigation, Oberpfaffenhofen, Germany

Received 17 February 2016; Accepted 7 August 2016

Academic Editor: Yuwei Chen

Copyright © 2016 Susanna Kaiser 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

Indoor navigation and mapping have recently become an important field of interest for researchers because global navigation satellite systems (GNSS) are very often unavailable inside buildings. FootSLAM, a SLAM (Simultaneous Localization and Mapping) algorithm for pedestrians based on step measurements, addresses the indoor mapping and positioning problem and can provide accurate positioning in many structured indoor environments. In this paper, we investigate how to compare FootSLAM maps via two entropy metrics. Since collaborative FootSLAM requires the alignment and combination of several individual FootSLAM maps, we also investigate measures that help to align maps that partially overlap. We distinguish between the map entropy conditioned on the sequence of pedestrian’s poses, which is a measure of the uncertainty of the estimated map, and the entropy rate of the pedestrian’s steps conditioned on the history of poses and conditioned on the estimated map. Because FootSLAM maps are built on a hexagon grid, the entropy and relative entropy metrics are derived for the special case of hexagonal transition maps. The entropy gives us a new insight on the performance of FootSLAM’s map estimation process.

#### 1. Introduction

Over the last few years professional and consumer location based applications have grown supported by digital cartography and GNSS (Global Navigation Satellite System), where real-time location estimation is combined with accurate maps. Motivated by the success of outdoor navigation the demand for indoor or underground navigation has increased. However, in indoors GNSS is often unavailable [1].

Different techniques address the particular challenges in indoor navigation. An overview of existing techniques is given in [2] and more recently in [3]. One approach is to install additional transmitters (GNSS Pseudolites) or receivers (e.g., RFIDs (Radio Frequency Identification), UWBs (Ultra Wide Band transmitters)). Another possibility is to use existing radio base stations originally installed for other purposes (such as UMTS or WLAN) or to enhance the quality of the GNSS signals (high sensitivity receivers and/or assisted GNSS services). The main drawback of these techniques is that new infrastructure needs to be installed and maintained to achieve high positioning accuracy. Another promising approach that deals with the problems of indoor navigation is the use of foot mounted inertial sensors [4] or other suitable forms of pedestrian dead reckoning (PDR) with alternate sensor placements or including additional sensors like cameras or magnetic field sensors [5]. In this case, no extra infrastructure is needed and the sensors can usually be implemented at low cost. In this paper, we will focus on positioning estimation for pedestrians based purely on PDR.

Foot mounted inertial measurement units (IMUs) can be used very effectively with the supply of zero velocity updates (ZUPTs) to a PDR estimation algorithm [4], reducing the nonlinear error growth over time to a reasonably small linear drift. The drift experienced by all PDR approaches can be addressed by using available floor plans to constrain the estimate [6–8]. However, maps of indoor scenarios may be only partially available, completely unavailable, outdated, or proprietary. To address the problem of unknown maps, the principle of Simultaneous Localization and Mapping (SLAM) using PDR can be called into play [9, 10]. SLAM estimates the location and a suitable presentation of the environment conditioned on a sequence of noisy measurements.

In this paper, we focus on FootSLAM [11] to estimate useful maps of probabilistic human motion (in the following referred to as probabilistic maps). FootSLAM is a form of SLAM for pedestrians. Rooted in a Bayesian estimation formulation, FootSLAM allows for the generation of a rich map for the visited areas, in our case a* probabilistic transition map* built on a hexagon raster, while simultaneously making accurate estimates of the pedestrian’s pose, her position and orientation, within the map. FootSLAM builds on a Rao-Blackwellized particle filter (RBPF) [12] based on the FastSLAM factorization [13] to process human odometry, that is, human step measurements obtained, for example, with an IMU located on the pedestrian’s shoe or in the pocket [14]. These odometry measurements are computed by an unscented Kalman Filter [15–17] using the raw IMU data and are then used as input to the FootSLAM particle filter. Each particle in the particle filter [18] randomly draws odometry errors and effectively represents one hypothesis for the path followed by the pedestrian. Those particles that revisit similar transitions are rewarded and are more likely to represent the actual path that the pedestrian followed. More precisely, each particle learns the FootSLAM map by counting each transition it makes across the edges of the hexagons and stores its whole path through the hexagonal grid.

The FootSLAM* probabilistic transition maps* are uncertain because they are estimated from human odometry. A suitable metric of the uncertainty of transition maps has not yet been addressed but having one would be useful for performance evaluation, theoretical analysis, map comparison, map combination, and map selection. Map combination and selection are, for instance, needed by the so-called “Turbo” FeetSLAM algorithm [19] whereby several maps resulting from several datasets are processed and combined in an iterative way to obtain a more accurate and extensive map of an environment.

The paper is organized as follows: First, we will discuss the term uncertainty and give an overview of different uncertainty metrics related to the probabilistic map. After that, we will give an overview of the related work (Section 1.2). In Section 2, we will describe FootSLAM’s Bayesian formulation and how the map is generated during the FootSLAM estimation process. Next, Section 3 will introduce the representation of the pedestrian’s history of poses as a random walk on a weighted graph, and Section 4 presents the derivation of the entropy and relative entropy metrics. Finally, we will show experimental results and outline the main conclusions.

##### 1.1. Uncertainty

In the context of localization we can find different map definitions (Figure 1). For instance, we can differentiate between roadmaps, floor plans, topological maps, or images like satellite images, and so forth. We can calculate the entropy of any of these maps to obtain a measure of the* information content* of that map. However, the nature of the maps estimated by a SLAM algorithm is completely different. If we use human odometry and process it using FootSLAM, we obtain a human motion map that is inherently a probabilistic representation of human motion. This map is uncertain because it is being estimated from odometry, which in turn is also uncertain due to uncertainty in the sensor measurements. Our goal is to derive a metric that reflects the uncertainty of a map.