An Optimal Virtual Inertial Sensor Framework using Wavelet Cross Covariance
Yuming Zhang, Pennsylvania State University, USA; Haotian Xu, University of Geneva, Switzerland; Ahmed Radi, University of Calgary, Canada; Roberto Molinari, Stephane Guerrier, Pennsylvania State University, USA; Naser El-Sheimy, University of Calgary, Canada
Location: Big Sur
The utilization of inertial sensors in the design of Inertial Navigation Systems (INS) to provide short term position, velocity and attitude information is constantly growing for navigation purposes within different applications. Generally speaking, inertial sensors suffer from numerous errors that are accumulated through time and should be accurately modeled and compensated to achieve acceptable navigation results and to avoid navigation solution degradation. Hence, developing accurate and efficient models is highly needed to decrease the effect of inertial sensor errors. The modeling process of such inertial sensor errors is considered to be one of the most challenging tasks in the design of any navigation system, especially for low-cost ones.
Inertial sensors are typically corrupted by two categories of errors: deterministic and stochastic. A considerable amount of literature has contributed to the calibration of deterministic errors which can be found in Li et al. (2015) (and the references therein). However, the calibration of deterministic errors is beyond the scope of this study which deals with approaches used to calibrate stochastic errors that are still limited due to the complexity of this task. In fact, these stochastic errors are often very complex in nature and can only be represented through composite processes (i.e. the sum of multiple independent latent processes) which lead to considerable space for research to improve calibration in this setting.
In order to better understand and calibrate stochastic errors, approaches such as the regression-based approach using the Allan Variance (AV) proposed in El-Sheimy et al. (2008) and the Maximum Likelihood Estimators (MLE) have been considered and developed. To overcome some limitations of the latter methods and increase the generality and computational efficiency of available approaches, the Generalized Method of Wavelet Moments (GMWM) was proposed by Guerrier et al. (2013) as a statistically sound alternative to the MLE with a high computational efficiency and numerical stability. The standard GMWM is based on a quantity called Wavelet Variance (WV), which can be interpreted as the variance of a process after it has been subject to an approximate band-pass filter (i.e. the wavelet filter). The GMWM makes use of the fact that WV can identify many processes that underlie the observed error signals in order to provide a basis to estimate the parameters of the mentioned complex time series models.
However, the GMWM is usually used in settings where sensors, such as gyros and accelerometers, are modelled independently. Although this is still an open area of research under many aspects, it is now increasingly common to find settings with a redundant sensor architecture, which is composed of a redundant number of Inertial Measurement Unit (IMU) systems arranged against each other. This is a common approach used to improve the overall performance of low-cost based navigation system on several levels and for different applications such as car navigation, Unmanned Ground Vehicles (UGVs), sport applications and even military ones. Such an approach has become increasingly popular as they provide an alternative way to design stable INS using low-cost Micro-Electro-Mechanical Systems (MEMS) IMUs. In order to consider this common setting, a part of literature has started making progress in taking into account and combining the information coming from the error signals of the sensors that are part of the mentioned array. For example, Bayard and Ploen (2002, 2005) proposed a Kalman filter-based virtual gyro that can combine the results of an array of high-drift sensors to construct a virtual low-drift sensor. This approach highly depends on the correlations of the random drift components of the individual gyros. Inspired by this, Vaccaro and Zaki (2017) developed an algorithm to estimate the off-diagonal elements in the spectral density matrix for the drift components of an array of gyros, i.e. the covariance structure of the gyros. This algorithm is based on AV for a single gyro, and Allan covariance between gyros. Nevertheless, in their framework of study, this algorithm applies only to a multivariate process generated by a combination of a White Noise (WN) and a Random Walk (RW). Unfortunately, this is usually not the case in reality since the stochastic error models of low-cost MEMS-based inertial sensors are known to be extremely complicated and can rarely be fully characterized by WN and RW.
Given this new setting, the GMWM has already been extended to a setting that considers multiple signals. Indeed, a new calibration framework proposed in Bakalli et al. (2017) takes into account the parameter variation between independent replicates from the same sensor. However, in the proposed study we consider the previously described setting where multiple correlated signals are collected at the same time from different identical sensors, such as an array of gyros. Within this setting, we need to take into consideration the complex covariance structure that usually characterize the error signals of these sensors in order to deliver a virtual gyro that aggregates all the information we can obtain from the array of gyros.
Considering the above goal, in our proposed study we consider the quantity called Wavelet Cross-Covariance (WCCV) to characterize the covariance structure of the inertial sensors and improve the general modeling of inertial sensor errors. WCCV allows a more general and more flexible approach to estimate the covariance structure implied by the observed signals without limiting itself to the WN plus RW model thereby allowing to consider more realistic scenarios. We can show that this extended GMWM, namely Multivariate GMWM (MGMWM) which takes into account the WCCV, provides a better understanding of the multivariate process over time and allows to better characterize the covariance structure so as to find the linear combination of error signals that delivers the optimal virtual sensor. Preliminary simulation results suggest that this new method outperforms other currently available approaches, including the algorithm proposed by Vaccaro and Zaki (2017). Moreover, as mentioned earlier, this new method shows more flexibility as it can be applied to a more general setting where the latent processes are not just limited to WN and RW models. Finally, it also shows high computational efficiency and statistical consistency in the estimation procedure. Considering the above advantages, this new method is going to be applied to real data in which an array of low-cost gyros is considered to build an optimal virtual gyro in order to provide real world navigation performance improvements.
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