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The calibration of low-cost inertial sensors has become increasingly important over the last couple of decades, especially when dealing with sensor stochastic errors. This procedure is commonly performed on a single error measurement from an inertial sensor taken over a certain amount of time, although it is extremely frequent for different replicates to be taken for the same sensor, thereby delivering important information which is often left unused. In order to address the latter problem, this paper presents a general wavelet variance-based framework for multisignal inertial sensor calibration, which can improve the modeling and model selection procedures of sensor stochastic errors using all replicates from a calibration procedure and allows to understand the properties, such as stationarity, of these stochastic errors. The applications using microelectromechanical system inertial measurement units confirm the importance of this new framework, and a new graphical user interface makes these tools available to the general user. The latter is developed based on an R package called mgmwm and allows the user to select a type of sensor for which different replicates are available and to easily make use of the approaches presented in this paper in order to carry out the appropriate calibration procedure.

Different inertial sensor calibration techniques have been proposed to consider the sources of measurement error from inertial sensors. There has been a significant amount of literature which studies the stochastic errors calibration of such devices. The recent results of [1] have proved that among all possible methods the (Generalized Method of Wavelet Moments) (GMWM) presents various optimality and is computationally reliable. However, the GMWM estimators depend on weight matrix which considerably impact the quality of the estimated stochastic error models. In addition, such models are made of different components (typically high-frequency and low-frequency components) whose impacts on navigation vary depending on the context. For example, the high-frequency component of the error model may be more important when considering low-cost IMUs mounted on small size drones used for short-term missions. On the other hand, the situation may be reversed when considering navigational grade IMUs used, often autonomously, for long-term missions. With these differences, one may wish to select a GMWM estimator whose weight matrix has been tailored to estimate more reliably the elements of an error model believed to have the greatest impacts on navigation accuracy. In this article, we provide a formal answer to this question by proposing an optimally weighted GMWM estimator. Our results show that the proposed estimator is optimal for all parameters of the sensor error model we wish to estimate with the smallest possible uncertainty of the estimation. Therefore, regardless of the application, and independently of the context, the same optimally weighted estimator can be employed.

Complex time series models such as (the sum of) ARMA$(p,q)$ models with additional noise, random walks, rounding errors and/or drifts are increasingly used for data analysis in fields such as biology, ecology, engineering and economics where the length of the observed signals can be extremely large. Performing inference on and/or prediction from these models can be highly challenging for several reasons: (i) the data may contain outliers that can adversely affect the estimation procedure; (ii) the computational complexity can become prohibitive when models include more than just a few parameters and/or the time series are large; (iii) model building and/or selection adds another layer of (computational) complexity to the previous task; and (iv) solutions that address (i), (ii) and (iii) simultaneously do not exist in practice. For this reason, this paper aims at jointly addressing these challenges by proposing a general framework for robust two-step estimation based on a bounded influence M-estimator of the wavelet variance. In this perspective, we first develop the conditions for the joint asymptotic normality of the latter estimator thereby providing the necessary tools to perform (direct) inference for scale-based analysis of signals. Taking advantage of the model-independent weights of this first-step estimator that are computed only once, we then develop the asymptotic properties of two-step robust estimators using the framework of the Generalized Method of Wavelet Moments (GMWM), hence defining the Robust GMWM (RGMWM) that we then use for robust model estimation and inference in a computationally efficient manner even for large time series. Simulation studies illustrate the good finite sample performance of the RGMWM estimator and applied examples highlight the practical relevance of the proposed approach.

The task of inertial sensor calibration has required the development of various techniques to take into account the sources of measurement error coming from such devices. The calibration of the stochastic errors of these sensors has been the focus of increasing amount of research in which the method of reference has been the so-called “Allan variance (AV) slope method” which, in addition to not having appropriate statistical properties, requires a subjective input which makes it prone to mistakes. To overcome this, recent research has started proposing “automatic” approaches where the parameters of the probabilistic models underlying the error signals are estimated by matching functions of the AV or wavelet variance with their model-implied counterparts. However, given the increased use of such techniques, there has been no study or clear direction for practitioners on which approach is optimal for the purpose of sensor calibration. This article, for the first time, formally defines the class of estimators based on this technique and puts forward theoretical and applied results that, comparing with estimators in this class, suggest the use of the Generalized method of Wavelet moments (GMWM) as an optimal choice. In addition to analytical proofs, experiment-driven Monte Carlo simulations demonstrated the superior performance of this estimator. Further analysis of the error signal from a gyroscope was also provided to further motivate performing such analyses, as real-world observed error signals may show significant deviation from manufacturer-provided error models.

The parametric estimation of stochastic error signals is a common task in many engineering applications, such as inertial sensor calibration. In the latter case, the error signals are often of complex nature, and very few approaches are available to estimate the parameters of these processes. A frequently used approach for this purpose is the maximum likelihood (ML), which is usually implemented through a Kalman filter and found via the expectation-maximization algorithm. Although the ML is a statistically sound and efficient estimator, its numerical instability has brought to the use of alternative methods, the main one being the generalized method of wavelet moments (GMWM). The latter is a straightforward, consistent, and computationally efficient approach, which nevertheless loses statistical efficiency compared with the ML method. To narrow this gap, in this paper, we show that the performance of the GMWM estimator can be enhanced by making use of model moments in addition to those provided by the vector of wavelet variances. The theoretical findings are supported by simulations that highlight how the new estimator not only improves the finite sample performance of the GMWM but also allows it to approach the statistical efficiency of the ML. Finally, a case study with an inertial sensor demonstrates how useful this development is for the purposes of sensor calibration.

The task of inertial sensor calibration has always been challenging, especially when dealing with stochastic errors that remain after the deterministic errors have been filtered out. Among others, the number of observations is becoming increasingly high since sensor measurements are taken at high frequencies over longer periods of time, thereby placing considerable limitations on the estimation of the complex models that characterize stochastic errors (without considering testing and selection procedures). Moreover, before estimating these models, there is a need for tests that determine whether the error signals are characterized by a model that remains constant over time and, if so, which model best predicts these errors. Considering these needs, this paper presents an open-source software platform that allows practitioners to carry out these procedures by making use of two recent proposals which stem from the Generalized Method of Wavelet Moments framework. These proposals make use of the growing amount of signal replicates issued during sensor calibration procedures and the proposed platform allows users to easily employ various functions that implement these methods in a user-friendly and computationally efficient manner.

The Global Navigation Satellite System (GNSS) is currently used in many fields, such as autonomous driving, robotics application, and Unmanned Aerial Vehicles (UAVs), where accurate position information is required. These applications require high positioning accuracy which, in turn, require precise analysis of the residual noise characteristics of the GNSS positioning solutions and their quantitative models. This paper investigates the Generalized Method of Wavelet Moments (GMWM) method for stochastic modelling of low-cost GNSS receiver signal. The paper also compares the results of GMWM to the Allan Variance (AV) which is currently the most common method to study the stochastic characteristics of different time series. Different datasets were collected using two low-cost GNSS receivers at different frequencies and were processed in Single Point Positioning (SPP) mode where position errors are expressed in the Local-Level Frame (LLF) of reference. Both techniques were used in identifying and characterizing the different latent stochastic process and their related coefficients for GNSS position residual signals where precise models of the latter have been built. The test results showed that for low-cost GNSS receivers, a white noise process alone is not sufficient for accurate position residual signals' modeling. The results also stressed out that the GNSS error signal models are complicated where the corresponding error model structures were represented as a sum of white noise and one or more 1st order Gauss-Markov (GM) processes which indicates the existence of short and relatively long correlation between consecutive observations, especially for observations collected at higher sampling rates. Moreover, the results showed that the GMWM approach in general outperforms the AV method in terms of correlated noise identification and characterization.

We present an algorithm for determining the nature of stochastic processes and their parameters based on the analysis of time series of inertial errors. The algorithm is suitable mainly (but not only) for situations where several stochastic processes are superposed. The proposed approach is based on a recently developed method called the Generalized Method of Wavelet Moments (GMWM), whose estimator was proven to be consistent and asymptotically normally distributed. This method delivers a global selection criterion based on the wavelet variance that can be used to determine the suitability of a candidate model (compared to other models) and apply it to low-cost inertial sensors. By allowing candidate model ranking, this approach enables us to construct an algorithm for automatic model identification and determination. The benefits of this methodology are highlighted by providing practical examples of model selection for two types of MEMS IMUs. Copyright (C) 2015 Institute of Navigation.

This paper studies the error behavior of low-cost inertial sensors in dynamic conditions. After proposing a method for error observations per sensor (i.e., gyroscope or accelerometer) and axes, their properties are estimated via the methodology of generalized method of wavelet moments. The developed model parameters are compared with those obtained under static conditions. Then, an attempt is presented to link the parameters of the established model to the dynamic of the vehicle. It is found that a linear relation explains a large portion of the exhibited variability. These findings suggest that the static methods employed for the calibration of inertial sensors could be improved when exploiting such a relationship.

The use of Inertial Measurement Units (IMU) for navigation purposes is constantly growing and they are increasingly being considered as the core dynamic sensing device for Inertial Navigation Systems (INS). However, these systems are characterized by sensor errors that can affect the navigation precision of these devices and consequently a proper calibration of the sensors is required. The first step in this direction is usually taken by evaluating the deterministic type of errors, such as bias and scale factor, which can be taken into account through known physical models. The second step consists in finding an appropriate model to describe the stochastic nature of the sensor errors. The focus of this paper is related to the second of such calibration procedures. Indeed, we propose an automatic model selection approach which is particularly appropriate when we observe/collect several independent replicates of the error signal of interest. In short, the proposed approach relies on the Generalized Methods of Wavelet Moments (GMWM) and the Wavelet Variance Information Criterion (WVIC), where we proposed a procedure to compute a Cross-Validation (CV) like estimator of the goodness-of-fit of a candidate model. This estimator provides by construction a tradeoff between model fit and model complexity, therefore allowing rank all candidate models and select the one (or the ones) that appears to be the most appropriate for the task of stochastic sensor calibration.

Inertial sensors are increasingly being employed in different types of applications. The reduced cost and the extremely small size makes them the number-one-choice in miniature embedded devices like phones, watches, and small unmanned aerial vehicles. The more complex the application, the more it is necessary to understand the structure of the error signal coming from these sensors. Indeed, their error signals are composed of deterministic and stochastic parts. The deterministic errors or faults can be compensated by proper calibration while the stochastic signal is usually ignored since its modeling is relatively difficult due to computational or statistical reasons, especially due to its complex spectral structure. However, a recently proposed approach called the Generalized Method of Wavelet Moments overcomes these limitations and this paper presents the software platform that implements this method for the analysis of the stochastic errors. As an example throughout the paper we will consider an inertial measurement unit, but the platform can be used for the stochastic calibration of any kind of sensor. The software is developed in the widely used statistical tool R using C++ language. The tools enable the user to study with ease any signal by the means of a vast range of predefined models and tools.

This article presents a new estimation method for the parameters of a time series model. We consider here composite Gaussian processes that are the sum of independent Gaussian processes which, in turn, explain an important aspect of the time series, as is the case in engineering and natural sciences. The proposed estimation method offers an alternative to classical estimation based on the likelihood, that is straightforward to implement and often the only feasible estimation method with complex models. The estimator furnishes results as the optimization of a criterion based on a standardized distance between the sample wavelet variances (WV) estimates and the model-based WV. Indeed, the WV provides a decomposition of the variance process through different scales, so that they contain the information about different features of the stochastic model. We derive the asymptotic properties of the proposed estimator for inference and perform a simulation study to compare our estimator to the MLE and the LSE with different models. We also set sufficient conditions on composite models for our estimator to be consistent, that are easy to verify. We use the new estimator to estimate the stochastic error's parameters of the sum of three first order Gauss–Markov processes by means of a sample of over 800, 000 issued from gyroscopes that compose inertial navigation systems. Supplementary materials for this article are available online.

The calibration of (low-cost) inertial sensors has become increasingly important over the past years since their use has grown exponentially in many applications going from unmanned aerial vehicle navigation to 3D-animation. However, this calibration procedure is often quite problematic since the signals issued from these sensors have a complex spectral structure and the methods available to estimate the parameters of these models are either unstable, computationally intensive and/or statistically inconsistent. This paper presents a new software platform for inertial sensor calibration based on the Generalized Method of Wavelet Moments which provides a computationally efficient, flexible, user-friendly and statistically sound tool to estimate and select from a wide range of complex models. The software is developed within the open-source statistical software R and is based on C++ language allowing it to achieve high computational performance., 20 pages, 6 figures

A new open-source software platform that, among others, allows to select models for inertial sensor stochastic calibration is presented in this paper. This platform consists in a package included in the statistical software R. The identification of stochastic models and estimation of model parameters is based on the method of Generalized Method of Wavelet Moments. This approach provides an extremely general framework for the identification, estimation and testing of models to describe and predict the error signals coming from inertial sensors. With the possibility of estimating complex models made of the sum of different underlying processes, this paper also presents the method with which a model, or a restrict set of models, can be selected that best describes and predicts the error signal.

This paper aims at studying the behaviour of the errors coming from inertial sensors when measured in dynamic conditions. After proposing a method for constructing the error process, the properties of these errors are estimated via the Generalized Method of Wavelets Moments methodology. The developed model parameters are compared to those obtained under static conditions. Finally an attempted is presented to find the link between the encountered dynamic of the vehicle and error-model parameters.

A robust approach to the estimation of time series models is proposed. Taking froma new estimation method called the Generalized Method of Wavelet Moments (GMWM)which is an indirect method based on the Wavelet Variance (WV), we replace the classicalestimator of the WV with a recently proposed robust M-estimator to obtain a robustversion of the GMWM. The simulation results show that the proposed approach can beconsidered as a valid robust approach to the estimation of time series and state-spacemodels.

Stochastic behavior of an instrument is often analyzed by constructing the Allan (or wavelet) variance signatures from an error signal. For inertial sensors, such a signature is conveniently obtained by recording data at rest. The analysis of this signal will result in noise-parameters adequate to such situation. Nonetheless, the value of the noise parameters may change under dynamics or other kind of external influences like for instance the temperature. In this research we study first the influence of the rotational dynamics on the signal of MEMS gyroscopes and then we show how to link this property to the noise-parameter estimation in a rigorous way by a modified version of the Generalized Method of Wavelet Moments (GMWM) estimator. The results of such analysis can then for instance be used in a Kalman filter, where the noise parameters are adapted according to such predetermined functional relationship between sensor noise and the encountered dynamics of the platform/sensor.

The integration of observations issued from a satellite-based system (GNSS) with an inertial navigation system (INS) is usually performed through a Bayesian filter such as the extended Kalman filter (EKF). The task of designing the navigation EKF is strongly related to the inertial sensor error modeling problem. Accelerometers and gyroscopes may be corrupted by random errors of complex spectral structure. Consequently, identifying correct error-state parameters in the INS/GNSS EKF becomes difficult when several stochastic processes are superposed. In such situations, classical approaches like the Allan variance (AV) or power spectral density (PSD) analysis fail due to the difficulty of separating the error processes in the spectral domain. For this purpose, we propose applying a recently developed estimator based on the generalized method of wavelet moments (GMWM), which was proven to be consistent and asymptotically normally distributed. The GMWM estimator matches theoretical and sample-based wavelet variances (WVs), and can be computed using the method of indirect inference. This article mainly focuses on the implementation aspects related to the GMWM, and its integration within a general navigation filter alibration procedure. Regarding this, we apply the GMWM on error signals issued from MEMS-based inertial sensors by building and estimating composite stochastic processes for which classical methods cannot be used. In a first stage, we validate the resulting models using AV and PSD analyses and then, in a second stage, we study the impact of the resulting stochastic models design in terms of positioning accuracy using an emulated scenario with statically observed error signatures. We demonstrate that the GMWM-based calibration framework enables to estimate complex stochastic models in terms of the resulting navigation accuracy that are relevant for the observed structure of errors.