# Dynamic Indoor Localization Using Maximum Likelihood Particle Filtering
## 1 Introduction
In a particle filter, a set of new particles is drawn during each prediction step, whose weighs are determined during the subsequent update step. A drawback of this approach is that this weighting results in little use of those particles whose weights are very small. The number of wasted particles can be very large in the case where measurements already provide enough information to determine the target's location with a reasonable accuracy, without using the information provided by the motion model. This is often the case in localization setups, for otherwise, algorithms not using a motion model would not work with reasonable accuracy. Consequently, a very large number of particles are required to avoid particle depletion after the update step, which would result in an inaccurate result. The vast number of required particles renders the approach numerically intractable in many applications. In order to tackle the above shortcoming, a number of methods are available to reduce the estimation error resulting from using particle filtering with a restricted number of particles. In [21], a machine learning classifier is added to decide whether particles are in the correct room. In [22], a context variable, with values in the set {'free space', 'constrained space','static space'}, is added to the state to account for the movement constraints at the target's current position. Also, the authors of [23] divide the environment into reachable and unreachable areas, so as to discard particles falling into an unreachable area. Generally speaking, the above methods aim to reduce the particle depletion phenomenon described above by complementing the motion model with environment-dependent information to decide where to draw particles. The aim of this work is to make the most of this principle, i.e., ideally to totally avoid particle depletion. To this end, we propose a novel particle filtering scheme which combines the prediction and update steps into a single one. In other words, we avoid drawing unnecessary particles during the prediction step that will afterwards disappear during the subsequent update step. This drastically reduces the required number of particles, making the approach numerically feasible in virtually any application. In deriving our method, we exploit the principle underlying a Bayesian tracking method called maximum likelihood Kalman filtering (MLKF) [24]. We then call the proposed method the maximum likelihood particle filter (MLPF). The authors used the MLKF in [25] to propose a dynamic localization method which only applies to the case in which the motion model is linear and Gaussian. In this sense, the method proposed in this work can be thought of as the generalization of the method we proposed in [25] to the case of non-linear motion models. This kind of models are typically used in robotics [26], and as we show with experiments using real data, leads to great accuracy improvements. The rest of the paper is organized as follows: In Section 2, we state the dynamic localization problem. In Section 3, we describe the particle filtering approach for solving it. In Section 4, we derive the proposed MLPF method. In Section 5, we present experimental evidence, showing the numerical advantages offered by the proposed method. Concluding remarks are given in Section 6. For ease of readability, all proofs appear in Appendix C.