Conference Proceedings

Modeling a Global Navigation Satellite System (GNSS) receiver clock instability is crucial to advancing GNSS-based navigation systems, especially regarding low-cost devices. This work proposes a way to obtain an insight to the drift that characterizes Local Oscillators (LOs) of GNSS receivers by using a raw GNSS baseband signal, namely the instantaneous code phases and then model its random behaviour using a state-of-the-art stochastic modeling framework, the Generalized Method of Wavelet Moments (GMWM). First, a GNSS Software-Defined Radio (SDR) is used to process three types of IntermediateFrequency (IF) GPS signals (i.e., L1 C/A, L5 data, and L5 pilot) and generate the respective code phase measurements, which are synthesized by using the LO from a GNSS frontend. Then, a pre-processor computes clock bias errors by subtracting the code phases from their references. Finally, the single-difference results of the pre-processor outputs, meaning the first order change increments of the previously computed clock bias errors, are inputted into the Robust GMWM (RGMWM) framework, which offers a stochastic modeling solution that is partially protected by the influence of outliers in the data at hand. The suggested methodology was tested by carrying out real-world experiments under open sky conditions, which validated the effectiveness of the RGMWM in stochastically modeling the LO instability using the instantaneous code phase of GNSS signals. In addition, a new type of sinusoidal noise has been identified in the input data computed from the GPS L1 C/A tracking and quantified via the RGMWM, while the L5 signals appeared to be unaffected by such noise type. This finding has the potential to contribute in the development of more advanced GNSS receivers by leveraging this more reliable modeling in either the baseband processors or navigators in the future.

The use of low-cost inertial sensors is nowadays wide-spread in many different mass-market applications, especially for navigation, for example in drones and smartphones. However, ensuring their performance is challenging since it is highly dependent on the available knowledge for the inertial sensor random error behavior, and which is hard to obtain accurately in practice. The main reason is that in many cases, the inertial sensor measurements collected during calibration contain outliers caused by either external (e.g., vibrations) or internal factors (e.g., ageing of the IMU). Therefore, it is essential that the modeling of that behavior is conducted by an estimator that has the capability to effectively reduce the influence of potential outliers from its estimation product (robustness), such that the estimated model parameters, when supplied to the chosen navigation algorithm, lead to optimal performances. The current state-of-the-art to handle the intricate nature of the stochastic errors, typically of low-cost inertial sensors, is the Generalized Method of Wavelet Moments (GMWM) and its Multi Signal extension (MS-GMWM). Although the GMWM possesses such a robustness feature thanks to an M-estimator for its fundamental quantity, the wavelet variance, it can be difficult to detect whether the analyzed data contain outliers or not, while this feature has not been extended to the multi signal approach. In this paper, it is demonstrated through simulations, that the utilization of the robust version of the GMWM in every scenario is a worthwhile trade-off between reduction of the outlier impact and reduction of the estimator’s efficiency. Furthermore, two new robust estimators in the context of the MS-GMWM are proposed and their performance is evaluated: the first one is able to decrease the influence of outliers in the analyzed data, while the second reinforces protection against calibration signal replicate(s) that present(s) significantly different stochastic error behavior compared to the others.

The following topics are dealt with: control system synthesis

Proper modeling of stochastic errors in inertial sensors plays a crucial role in the achievable quality of GNSS-INS integration especially with low-cost inertial sensors. Generalized Method of Wavelet Moments (GMWM) can model the underlying process for such errors with arbitrarily complex structure to obtain close match to the observed errors in terms of wavelet variance. In comparison to the widely used and IEEE adopted error modeling using Allan Variance, this method provides consistent estimation, ability to estimate parameters of composite stochastic models of much higher complexities, and considerably easier usage. However, the level of improvement in navigation quality does not necessarily grow proportionally to the fidelity of the error models. Therefore, opting for unnecessarily complex models may only increase computational load with no tangible gain in navigation quality. On the other hand, converging estimation of higher complexity models, generally speaking, requires longer estimation periods and higher dynamics to improve observability of error states. This implies yet another motivation to find an optimal sensor error model avoiding unnecessary complexities. In this paper, we employ two methods to investigate the effect of model complexity on integrated navigation performance. Firstly, a covariance propagation is performed on static conditions, as is the standard scenario in error analysis of inertial navigation systems. Afterwards, an emulation study is performed based on a real Unmanned Aerial Vehicle (UAV) flight and error signals of a Navchip V2 Inertial Measurement Unit (IMU). Results of both methods are in general agreement, and suggest that more complex models in general provide higher accuracy for the navigation system and a more consistent covariance prediction by the navigation filter. This difference is, however, more noticable only in GNSS outages of longer duration (tens of seconds). However, the benefits of more complex models may be only marginal in other applications, depending on the duration of inertial coasting and availability of other sensory data such as GNSS observations.

The prediction of storm surge is an important part of risk analysis for hurricanes because storm surge is the cause of a significant amount of hurricane damage. Although still debated, the effects of climate change on hurricanes may lead to an increase in storm surge occurrences and in the related damages. Consequently, there is the need to analyze the possible consequences of climate change for several possible scenarios. However, the available models for storm surge analyses are either too computationally expensive or incapable of accounting for climate change effects. This paper proposes a random field model for storm surge predictions based on the Improved Latent Space Approach. Contrary to models available in the literature, the presented metamodel can be trained with both data coming from high-fidelity simulations and observations from historical records.

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 practice of inertial sensor calibration has commonly been carried out by taking into account the deterministic and stochastic components of the error measurements issued from a calibration session. Once the deterministic components have been taken into account through physical models, the remaining stochastic component has always been dealt with for each sensor separately. The latter process involves estimating complex probabilistic models for each sensor which has been proven to be extremely complicated over the past years, although recent proposals have allowed to overcome most of the limitations that have characterized this task. However, the separate stochastic calibration of the individual sensors composing an inertial measurement unit may not be wise in many cases since there can be considerable degrees of dependence between the sensors, especially between the gyroscopes. For this reason, there has been growing attention towards this issue in order to consider the influence of the stochastic behaviour of the sensors on each other, with few proposals that address this problem. Among these proposals there has been the idea of integrating the information coming from the different gyroscopes so as to build a virtual gyroscope. In this paper we build on this idea and, using a recently proposed method for multivariate signal modelling, we deliver a general and flexible framework that allows to consider many different modelling options which provide the basis to construct a virtual sensor that optimally combines the information from the individual sensors and considerably improves navigation accuracy.

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.

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.

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.

The task of inertial sensor calibration has become increasingly important due to the growing use of low-cost inertial measurement units which are however characterized by measurement errors. Being widely employed in a variety of mass-market applications, there is considerable focus on compensating for these errors by taking into account the deterministic and stochastic factors that characterize them. In this paper we focus on the stochastic part of the error signal where it is customary to register the latter and use the observed error signal to identify and estimate the stochastic models, often complex in nature, that underlie this process. However, it is often the case that these error signals are observed through a series of replicates for the same inertial sensor and equally often it can be noticed that these replicates have the same model structure but their parameters appear to be different between replicates. This phenomenon has not been taken into account by current stochastic calibration procedures which therefore can be conditioned by flawed parameter estimation. For this reason, this paper aims at delivering an approach that takes into account the parameter variation between replicates by delivering an estimator that minimizes a loss function that considers each replicate, thereby improving measurement precision on the long run, and allows to build a statistical test to determine the presence of parameter variation between replicates.

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 presents the new open-source statistical software package for inertial sensor calibration. This platform is based on the Generalized Method of Wavelet Moments that was recently proposed to estimate simple and composite stochastic models that are typically used in sensor calibration. As opposed to existing techniques, this new package allows to easily and efficiently visualize, estimate and test a wide range of stochastic models that are used to describe and predict the error signals coming from accelerometers and gyroscopes that characterize inertial sensors. The availability of this new tool is of considerable importance since, given the growing use of low-cost IMUs, correctly identifying and precisely estimating the models for the error signals of these sensors allows to greatly improve their navigation accuracy.

The identification and selection of a small set of models that are able to well describe and predict the error signals coming from inertial sensors is of utmost importance to improve the navigation precision of these devices. For this reason, in this paper we describe the implementation of the WVIC model selection criterion by improving the computational efficiency of its associated algorithm. This criterion is based on the Generalized Method of Wavelet Moments that was recently proposed to estimate the parameters of inertial sensor error models. Using this approach, the model selection procedure is included within an algorithm that allows it to be executed more efficiently and is implemented within a new package in the opensource statistical platform R. The efficient implementation of this model selection procedure enables engineers and researchers to rapidly identify a restricted set of models for inertial sensor calibration.

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.

We present an algorithm for determining the nature of stochastic processes together with its parameters based on the analysis of time series of inertial errors. The algorithm is suitable mainly (but not only) for situations when several stochastic processes are superposed. In such cases, classical approaches based on the analysis of Allan variance or PSD are likely to fail due to the difficulty of separating the underlying error-processes in the spectral domain. The developed alternative is based on the recently proposed method called the Generalized Method of Wavelet Moments (GMWM), which estimator was proven to be consistent and asymptotically normally distributed. The principle of this method is to match the empirical and model-based wavelet variances (WV). In this study we propose a goodness-of-fit criterion which can be used to determine the suitability of a model candidate and apply it to low-cost inertial sensors. The suggested approach of model selection relies on an unbiased estimate of the distance between the theoretical WV and the empirical WV which would be obtained on an independent sample issued from the stochastic process of interest. Such goodness-of-fit criterion is however “penalized” by the complexity of the model. In some sense, the proposed methodology is a generalization of Mallow’s Cp applied to models estimated by the GMWM. By allowing to rank candidate models, this approach permits 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, the latter of higher quality.

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.

Stochastic modeling is a challenging task for lowcost inertial sensors whose errors can have complex spectral structures. This makes the tuning process of the INS/GNSS Kalman filter often sensitive and difficult. We are currently investigating two approaches for bounding the errors in the mechanization. The first is an improved modeling of stochastic errors through the superposition of several Auto-Regressive (AR) processes. A new algorithm is presented based on the Expectation-Maximization (EM) principle that is able to estimate such complex models. The second approach focuses on redundancy through the use of multiple IMUs which don’t need to be calibrated a priori. We present a synthetic IMU computation in which the residuals are modeled by a single ARMA model. The noise power issued from the residuals is then continuously estimated by a GARCH model, which enables a proper weighting of the individual devices in the synthetic IMU.

This research presents methods for detecting and isolating faults in multiple Micro-Electro-Mechanical System (MEMS) Inertial Measurement Unit (IMU) configurations. Traditionally, in the inertial technology, the task Fault Detection and Isolation (FDI) is realized by the parity space method. However, this approach performs poorly with low-cost MEMS-IMUs, although, it provides satisfactory results when applied to tactical or navigation grade IMUs. In this article, we propose a more complex approach to detect outliers that takes into account the shape and size of multivariate data. The proposed method is based on Mahalanobis distances. Such approach has already been successfully applied in other fields of applied multivariate statistics, however, it has never been tested with inertial sensors. As Mahalanobis distances (as well as the parity space method) is very sensitive to the presence of the same outliers this method aims to detect, we propose using its robust version. The performances of the proposed algorithm are evaluated using dynamical experiments with several MEMS-IMUs and a reference signal provided by a tactical-grade IMU run in parallel. The conducted experiment shows that, for example, the percentage of false alarms is approximately ten times lower when using a method based on Mahalanobis distances as compared to that based on the parity space approach.

Although experimental results have demonstrated that redundant MEMS-IMUs integrated with GPS are an efficient way to improve navigation performances, the precise relationship between the number of sensors employed and the accuracy enhancement remains unclear. This article aims at demonstrating, with the help of simulations, that multiple MEMS-IMU systems can be designed according to specifications. This enables to better define the relationship between the number of sensors employed and the accuracy improvement as well as to ascertain the precise number of sensors needed to fulfill the system’s requirements. This proves to be highly helpful in designing navigation systems for applications that require a specific precision. This article also aims at demonstrating the impact of sensors’ orientation on the system performances. To achieve this, a new method based on partial redundancies is introduced to formalize the determination of optimal geometry of multi-IMU systems. It shows that, when dealing with IMU triads, the optimality of such systems is independent of the geometry between them. This result, moreover, presents important practical implications since it demonstrates that complicated geometries, traditionally employed in such systems, can be avoided. Additionally, it also proves that navigation performances obtained by simulations with a certain number of sensors are valid independently from orientation amongst these sensors.

In this article, we investigate two different algorithms for the integration of GPS with redundant MEMS-IMUs. Firstly, the inertial measurements are combined in the observation space to generate a synthetic set of data which is then integrated with GPS by the standard algorithms. In the second approach, the method of strapdown navigation needs to be adapted in order to account for the redundant measurements. Both methods are evaluated in experiments where redundant MEMS-IMUs are fixed in different geometries: orthogonally-redundant and skew-redundant IMUs. For the latter configuration, the performance improvement using a synthetic IMU is shown to be 30% on the average. The extended mechanization approach provides slightly better results (about 45% improvement) as the systematic errors of the individual sensors are considered separately rather than their fusion when forming compound measurements. The maximum errors are shown to be reduced even by a factor of 2.