Yannick Stebler

Wavelet-Variance-Based Estimation for Composite Stochastic Processes

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 errors 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.

Generalized method of wavelet moments for inertial navigation filter design

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 calibration 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.

Implementation and performance of a GPS/INS tightly coupled assisted PLL architecture using MEMS inertial sensors.

The use of global navigation satellite system receivers for navigation still presents many challenges in urban canyon and indoor environments, where satellite availability is typically reduced and received signals are attenuated. To improve the navigation performance in such environments, several enhancement methods can be implemented. For instance, external aid provided through coupling with other sensors has proven to contribute substantially to enhancing navigation performance and robustness. Within this context, coupling a very simple GPS receiver with an Inertial Navigation System (INS) based on low-cost micro-electro-mechanical systems (MEMS) inertial sensors is considered in this paper. In particular, we propose a GPS/INS Tightly Coupled Assisted PLL (TCAPLL) architecture, and present most of the associated challenges that need to be addressed when dealing with very-low-performance MEMS inertial sensors. In addition, we propose a data monitoring system in charge of checking the quality of the measurement flow in the architecture. The implementation of the TCAPLL is discussed in detail, and its performance under different scenarios is assessed. Finally, the architecture is evaluated through a test campaign using a vehicle that is driven in urban environments, with the purpose of highlighting the pros and cons of combining MEMS inertial sensors with GPS over GPS alone.

A framework for inertial sensor calibration using complex stochastic error models

Modeling and estimation of gyroscope and accelerometer errors is generally a very challenging task, especially for low-cost inertial MEMS sensors whose systematic errors have complex spectral structures. Consequently, identifying correct error-state parameters in a INS/GNSS Kalman filter/smoother becomes difficult when several processes are superimposed. In such situations, the classical identification approach via Allan Variance (AV) analyses fails due to the difficulty of separating the error-processes in the spectral domain. For this purpose we propose applying a recently developed estimation method, called the Generalized Method of Wavelet Moments (GMWM), that is excepted from such inconveniences. This method uses indirect inference on the parameters using the wavelet variances associated to the observed process. In this article, the GMWM estimator is applied in the context of modeling the behavior of low-cost inertial sensors. Its capability to estimate the parameters of models such as mixtures of GM processes for which no other estimation method succeeds is first demonstrated through simulation studies. The GMWM estimator is also applied on signals issued from a MEMS-based inertial measurement unit, using sums of GM processes as stochastic models. Finally, the benefits of using such models is highlighted by analyzing the quality of the determined trajectory provided by the INS/GNSS Kalman filter, in which artificial GNSS gaps were introduced. During these epochs, inertial navigation operates in coasting mode while GNSS-supported trajectory acts as a reference. As the overall performance of inertial navigation is strongly dependent on the errors corrupting its observations, the benefits of using the more appropriate error models (with respect to simpler ones estimated using classical AV graphical identification technique) are demonstrated by a significant improvement in the trajectory accuracy.

Theoretical Limitations of Allan Variance-based Regression for Time Series Model Estimation

This letter formally proves the statistical inconsistency of the Allan variance-based estimation of latent (composite) model parameters. This issue has not been sufficiently investigated and highlighted since it is a technique that is still being widely used in practice, especially within the engineering domain. Indeed, among others, this method is frequently used for inertial sensor calibration, which often deals with latent time series models and practitioners in these domains are often unaware of its limitations. To prove the inconsistency of this method, we first provide a formal definition and subsequently deliver its theoretical properties, highlighting its limitations by comparing it with another statistically sound method.

An Approach for Observing and Modeling Errors in MEMS-Based Inertial Sensors Under Vehicle Dynamic

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.

Wavelet-Based Improvements for Inertial Sensor Error Modeling

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.

Study of MEMS-based inertial sensors operating in dynamic conditions

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.