Construction of Dynamically-Dependent Stochastic Error Models
Philipp Clausen, Swiss Federal Institute of Technology in Lausanne, Switzerland; Samuel Orso, University of Geneva, Switzerland; Jan Skaloud, Swiss Federal Institute of Technology in Lausanne, Switzerland; Stéphane Guerrier, Pennsylvania State University, USA
Location: Big Sur
The current manufacturing and technological advances in the domain of Micro-Electro-Mechanical Systems (MEMS) allows for a large expansion of the use of inertial sensing technologies. Low-cost, low-power, and low-weight Inertial Measurement Units (IMUs) are easily producible in high quantities. A large volume of production allows then for extremely low prices which are the reasons for their attractiveness and implementation in all kind of applications [4, 5, 9]. Nevertheless, a successful employment of these sensors in positioning and navigation applications requires a correct characterization of the error signals coming from these devices.
For this purpose the error signals are considered containing a deterministic as well as a stochastic part. The components of the former can be related, for example, to the physical misalignment of sensor axes, to the constant offsets in their readings (biases) and their variations depending on the environment (e.g. changes in the temperature). Whereas a significant part of the deterministic errors can be compensated and accounted for by a calibration procedure , a proper dealing with the stochastic components is in general more challenging, especially for those with complex spectral structure. Moreover, the stochastic properties typically change from sensor to sensor, even when considering the sensor of the same type, the individual axes may exhibit different amounts of noise. Therefore, each sensor and each axis requires an individual stochastic calibration in order to fully grasp its potential.
A typical stochastic calibration session consists of the acquisition of a long dataset where the noise of the sensor is recorded in static conditions. Such a dataset typically contains millions of observations in order to properly analyze the signal. Ideally, sensors should be isolated from environmental conditions to eliminate any additional perturbations (e.g. vibrations, temperature variations). Other external variations like for instance the temperature-changes are minimized apart their planned values (e.g. within a temperature chamber). This static data is then analyzed with different techniques like the maximum likelihood estimator [6, 10], the Allan variance [1, 2, 8], or the Generalized Method of Wavelet Moments (GMWM) . The latter technique has become lately freely accessible via an open-piece of software implemented using the statistical tool ``R''. This method analyses the signal and estimates the model parameters with their confidence levels for a user-defined set of noise models. Such models can include many different types and combinations of the following noise characteristics: quantization noise, white noise, random walk and other autoregressive processes.
However, the determined parameters of the different stochastic noise properties of the sensors may evolve in time due to external/environmental influences and do not correspond to the ones found in the controlled environment. One of these influences could be for instance their dependence on the temperature or the physical motion of the system. Although part of these conditions are adequately addressed when dealing with the deterministic errors, this is unfortunately not the case for the stochastic characterization (e.g. a sensor in a heating chamber observes a constant signal and the temperature-dependent model then forces the signal back to such a constant).
In this work we focus on the dependency of the stochastic properties of a sensor and the dynamics (rotation and acceleration) it undergoes. In other words we examine the stochastic noise variations as a function of external movement under the expanded framework of the GMWM. The resulting stochastic noise calibration is able to analyze its variation over time at known ``dynamic'' that is provided by a reference. Particularly, we study the impact on the gyro output of predefined angular speeds (and jerks) on a rotation table over certain time periods. Similarly, by mounting the sensors eccentrically from the rotation axis we excite also the accelerometers to different extents.
In the paper, we first present the theoretical background that extents the GMWM methodology so that it is capable to describe the stochastic noise properties as a function of the dynamics provided by the reference. This dependence can then be put as function to the rotational speed or to the acceleration of the platform. A profound analysis is carried out first on a synthetic signal demonstrating this new capability of the adapted GMWM. We then present a practical calibration scenario with different MEMS-IMUs. Two types of sensors are employed from the lower and higher price scales of the MEMS-IMUs used in different applications in robotics and navigation. The gathered data at constant temperature on a rotation table are used to determine the functional relationship of stochastic parameters with their encountered dynamics.
Finally, we conclude with a short discussion on the importance of considering the varying stochastic parameters within an estimator for applications where a MEMS-IMU of this type is the backbone of integrated navigation. Indeed, the closer the model corresponds to the reality, the better will be the predicted uncertainties of the IMU coasting as well as the fusion with other navigation sensors. Platforms which are subject to a wide range of dynamics, such as micro-aerial vehicles, will thus profit most from this improved modeling. Other applications of the adapted GMWM methodology could then include additional environmental influences (e.g. take the air pressure into account) and open the doors to future research.
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