Modern smartphones have all the sensing capabilities required for accurate and robust navigation and tracking. In specific environments some data streams may be absent, less reliable, or flat out wrong. In particular, the GNSS signal can become flawed or silent inside buildings or in streets with tall buildings. In this application paper, we aim to advance the current state-of-the-art in motion estimation using inertial measurements in combination with partial GNSS data on standard smartphones. We show how iterative estimation methods help refine the positioning path estimates in retrospective use cases that can cover both fixed-interval and fixed-lag scenarios. We compare estimation results provided by global iterated Kalman filtering methods to those of a visual-inertial tracking scheme (Apple ARKit). The practical applicability is demonstrated on real-world use cases on empirical data acquired from both smartphones and tablet devices.
On the more technical side, we apply iterative filtering methods in this paper. Kalman filters and smoothers (see, , @cite_19 for an excellent overview of non-linear filtering) are recursive estimation schemes and thus iterative already per definition. Iterated filtering often refers to local ( inner-loop') iterations (over a single sample period). They are used together with extended Kalman filtering as a kind of fixed-point iteration to work the extended Kalman update towards a better linearization point (see, , @cite_17 ). The resulting iterated extended Kalman filter and iterated linearized filter-smoother can provide better performance if the system non-linearities are suitable. We however, are interested in iterative re-linearization of the dynamics and passing information over the state history for extended periods. Thus we focus on so-called global ( outer-loop') schemes, which are based on iteratively re-running of the entire forward--backward pass in the filter smoother. These methods relate directly to other iterative global linearization schemes like the so-called Laplace approximation in statistics machine learning (see, , @cite_16 ) or Newton iteration based methods (see, , @cite_29 and references therein).
@cite_19 @cite_29 @cite_16 @cite_17
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This paper points out the flaws in using the extended Kalman filter (EKE) and introduces an improvement, the unscented Kalman filter (UKF), proposed by Julier and Uhlman (1997). A central and vital operation performed in the Kalman filter is the propagation of a Gaussian random variable (GRV) through the system dynamics. In the EKF the state distribution is approximated by a GRV, which is then propagated analytically through the first-order linearization of the nonlinear system. This can introduce large errors in the true posterior mean and covariance of the transformed GRV, which may lead to sub-optimal performance and sometimes divergence of the filter. The UKF addresses this problem by using a deterministic sampling approach. The state distribution is again approximated by a GRV, but is now represented using a minimal set of carefully chosen sample points. These sample points completely capture the true mean and covariance of the GRV, and when propagated through the true nonlinear system, captures the posterior mean and covariance accurately to the 3rd order (Taylor series expansion) for any nonlinearity. The EKF in contrast, only achieves first-order accuracy. Remarkably, the computational complexity of the UKF is the same order as that of the EKF. Julier and Uhlman demonstrated the substantial performance gains of the UKF in the context of state-estimation for nonlinear control. Machine learning problems were not considered. We extend the use of the UKF to a broader class of nonlinear estimation problems, including nonlinear system identification, training of neural networks, and dual estimation problems. In this paper, the algorithms are further developed and illustrated with a number of additional examples. Filtering and smoothing methods are used to produce an accurate estimate of the state of a time-varying system based on multiple observational inputs (data). Interest in these methods has exploded in recent years, with numerous applications emerging in fields such as navigation, aerospace engineering, telecommunications and medicine. This compact, informal introduction for graduate students and advanced undergraduates presents the current state-of-the-art filtering and smoothing methods in a unified Bayesian framework. Readers learn what non-linear Kalman filters and particle filters are, how they are related, and their relative advantages and disadvantages. They also discover how state-of-the-art Bayesian parameter estimation methods can be combined with state-of-the-art filtering and smoothing algorithms. The book's practical and algorithmic approach assumes only modest mathematical prerequisites. Examples include MATLAB computations, and the numerous end-of-chapter exercises include computational assignments. MATLAB GNU Octave source code is available for download at www.cambridge.org sarkka, promoting hands-on work with the methods. Increasingly, for many application areas, it is becoming important to include elements of nonlinearity and non-Gaussianity in order to model accurately the underlying dynamics of a physical system. Moreover, it is typically crucial to process data on-line as it arrives, both from the point of view of storage costs as well as for rapid adaptation to changing signal characteristics. In this paper, we review both optimal and suboptimal Bayesian algorithms for nonlinear non-Gaussian tracking problems, with a focus on particle filters. Particle filters are sequential Monte Carlo methods based on point mass (or "particle") representations of probability densities, which can be applied to any state-space model and which generalize the traditional Kalman filtering methods. Several variants of the particle filter such as SIR, ASIR, and RPF are introduced within a generic framework of the sequential importance sampling (SIS) algorithm. These are discussed and compared with the standard EKF through an illustrative example. We establish a full relationship between Kalman filtering and Amari's natural gradient in statistical learning. Namely, using an online natural gradient descent on data log-likelihood to evaluate the parameter of a probabilistic model from a series of observations, is exactly equivalent to using an extended Kalman filter to estimate the parameter (assumed to have constant dynamics). In the i.i.d. case, this relation is a consequence of the "information filter" phrasing of the extended Kalman filter. In the recurrent (state space, non-i.i.d.) case, we prove that the joint Kalman filter over states and parameters is a natural gradient on top of real-time recurrent learning (RTRL), a classical algorithm to train recurrent models. This exact algebraic correspondence provides relevant settings for natural gradient hyperparameters such as learning rates or initialization and regularization of the Fisher information matrix.