Methods and Procedure for the nonlinear dynamic characterization of Inertial Sensors under diverse operational conditions
Authors:   Deutsches Zentrum für Luft- und Raumfahrt (DLR), Bremen, Germany, Borges Farconi Leonardo , Bestard Körner Matias 
Published: January 2024

Gyroscopes and accelerometers are sensors of ultimate importance in Inertial Navigation Systems (INS), where the rate and acceleration readings are cumulated to estimate the current attitude and position of a system. The errors in those readings therefore degrade the attitude and position estimates with time. With the purpose of reducing those errors inertial sensors have been characterized in laboratories to better model and compensate the errors by means of calibrations in the respective navigation filters or to simulate them for performance analysis. Historically, these errors have been reduced to scale, bias and random white noise. The scale and bias are corrected from the sensor readings and any further variations are estimated online as non-repeatable figures. With a bit more of effort, some would characterize the sensor under different temperature conditions and consider the offline scale and bias as a function of temperature.

Even though for some applications this process may be sufficient, one important question is whether this can be improved for applications where the operational conditions are more extreme, as in the case of reusable launchers with lower grade sensors, for example. Here environmental conditions like vibration, shocks, different temperature ranges, and the combination of the effects, all cause influences which are usually unknown or, if known, usually not accounted for in the calibrations.

The Institute of Electrical and Electronics Engineers (IEEE) provides some standards for the characterization of inertial sensors in which nonlinear multiple-input models are presented. Different methods are introduced to characterize the model parameters. The methods are not necessarily correlated and may generate parameters in different conditions that, when put together back in a sensor model, may actually be an inaccurate representation of the sensor. Some of these tests also provide figures that are usable only for requirement verification but are not directly related to a time or frequency domain model. When trying to characterize higher-order parameters in accelerometers, like the ones dependent on vibration, the proposed methods turn out to be extremely long and over-stressing to the sensors, which in a real application scenario is undesired. In summary, obtaining a single model for different environmental and testing conditions become unpractical.

Other previous works that addressed these issues were limited in some sense or not providing a complete solution to the question of facilitating a more complete characterization of inertial sensors. In essence, they have either limited frequency scopes, limited model orders, prolonged test steps, no consideration of environmental conditions, use of specific equipment usually not available in industry, or no consideration for misalignments.

The procedure proposed here makes use of standard equipment available in industry, namely electrodynamic shaker and rotation table or equivalent, associated with a special proposed fixture of easy manufacturing. The data is collected in the time domain at the different operational conditions and, in the case of shaker tests, a sensor fusion between position and acceleration sensors is used for obtaining the ground truth with better accuracy when compared to the standard available data. With the use of ARMAX and NARMAX model structures, the data between the different data sets is then brought to a same reference frame by removing the misalignments for each test condition. With a bundled data set, the models are identified by using recursive Rational Model Estimator (RME) and Extended Least Squares (ELS) methods. With adaptations to these methods and the use of a proposed model selection strategy, based on the Error Reduction Ratio (ERR), the t-statistic, the Akaike Information Criterion (AIC) and a customized covariance estimator, the final model can then be quantitatively chosen.

The proposed procedure and methods reduce the characterization effort to the same effort employed for usual environmental acceptance or qualification procedures for products in the space industry, with added information about the sensor behaviour beyond only scale factor and bias at nominal testing conditions. In an even better scenario, the characterization procedure can be embedded in the engineering, acceptance and/or qualification tests if properly implemented.

The result of such procedure provides nonlinear with high-orders, dynamic, and verified models to be included in Model, Processor and Hardware in-the-loop setups. On the other side, it also provides simpler models regularly used for navigation filters but verified against and with the added information about diverse and potentially extreme operational conditions.

Due to the ongoing development of the projects where this strategy is being applied, numerical results will be limited to simulations to illustrate and prove the concept.