DE102022126969B3 - Calibrating a rotation rate sensor - Google Patents

Abstract

The invention relates to a method for calibrating a rotation rate sensor (1): determining (S1) first test data by applying reference rotation rates in different orientations and amounts; Estimating (S2) an alignment error to the sensor axis, determining (S3) second test data by applying vibrations; Determining (S4) a respective first orientation transformation so that all reference rotation rates of the first test data are related to the coordinate system fixed to the sensor axis; Combining (S5) all test data into a combined test data set; Specifying (S6) a model structure; performing (S7) an all-encompassing regression analysis on the combined test data set to determine parameter values so that all parameters of the parametric model are obtained at once based on all test data; and calibrating (S8) the tested rotation rate sensor (1) based on the parametric model;

Description

The invention relates to a method for calibrating a rotation rate sensor.

Inertial measurement units (so-called IMUs) are used to record current kinematic variables in a reference coordinate system. Largely designed mechanically, they use known effects, such as the relationship between force and accelerated mass, the angular stability of rotating masses (gyroscopic stability) and the like, in order to determine further processable sensor signals of prevailing kinematic variables. Inertial measuring units based on optical or relativistic effects, for example for determining attitude angle rotation rates using laser gyros, are very expensive and are therefore rarely used; these can only be found in niche applications such as military missiles with extremely high expected accelerations. The focus in the following will therefore be on mechanical measuring units, although it cannot be ruled out that the solutions presented can also apply to optical systems.

The following information arises from professional considerations rather than necessarily arising from a specific prior art document: As with other sensors, the purpose of inertial measurement units is to detect a naturally occurring quantity and to generate a processable signal value from it. Naturally, every physical sensor has certain inaccuracies that can be influenced by environmental conditions and vary in particular over different ranges of measured quantities and/or frequencies of the measured quantity. A sensor therefore has a certain transfer function (in the algebraic and dynamic sense), which represents a mapping of the naturally occurring quantity onto the processable sensor signal.

Such a transfer function can be described using mathematical models. In principle, a variety of model types are possible for this, be it models in the form of algebraic equations or dynamic models, the latter describing a frequency-dependent transfer function and are therefore preferably formulated as differential equations or as equations in the frequency range (such as the Laplace range). A distinction must also be made between linear models and non-linear models. Statistical models allow many individual empirical data to be combined into one model. The more accurately a model reflects the reality of the sensor, the easier it is to correct the error introduced by the respective sensor in the transfer of the naturally occurring quantity to the processable sensor signal.

Gyroscopic instruments are often an essential component of an inertial measuring unit. Such gyroscopic instruments serve to be able to determine the orientation of a reference coordinate system in the form of position angles, particularly with respect to the earth. The effect exploited here is the stability of a rotating mass against changes in position angle. For example, if a gimbal-suspended, rapidly rotating mass is arranged in a housing on a moving body, when the body and thus the housing rotates, the housing rotates, so to speak, around the rotating mass, while the respective axis of the rotating mass maintains its alignment. From this relative change in orientation between the rotating mass and the housing, an orientation change in the form of differential angles can be determined. If a position angle is determined by temporal integration of a measured yaw rate, measurement errors in the measured yaw rate naturally integrate into larger errors in the position angles. To avoid this error integration, gyroscopic instruments are typically not used alone, but rather coupled with other types of sensors. Nevertheless, the measurement error in the measured rotation rate is included in the finally determined position angle.

With the aim of reducing sensor errors in gyroscopic instruments such as yaw rate sensors, such gyroscopic instruments were and are typically tested for characterization in the sense of system identification in order to eliminate errors in later operation by means of the characterization and the resulting knowledge about transmission errors of the gyroscopic instruments, i.e. H. to calibrate the gyroscopic instruments. A relatively simple model description of the transmission errors of a gyroscopic instrument consists in dividing the transmission errors into the components “scaling error” (also called “scale factor error”), “bias”, and “white noise” (so-called “white noise”). At least the scaling error and the bias can be calculated out and thus the original sensor signal can be corrected, especially if scaling errors and bias can be determined for each current situation depending on the acceleration to be measured and known environmental conditions.

A wide variety of methods for analyzing or calibrating inertial sensors are known in the prior art.

The WO 2013/ 033755 A1 relates to a method for determining an inertial sensor bias, the method comprising the following steps: determining the orientation of the inertial sensor relative to a housing; Obtaining a first measurement of the inertial sensor; Rotate the housing with the inertial sensor by approx. 180°; obtaining a second inertial sensor measurement; and determining the inertial sensor bias from the first inertial sensor measurement, the second inertial sensor measurement and the obtained orientation of the inertial sensor relative to the housing.

The DE 10 2021 004 103 A1 further relates to a method for detecting malfunctions in inertial measuring units that are used in a vehicle for measuring angular velocities and specific forces, wherein two inertial measuring units, each with several sensors, including accelerometers and gyroscopic sensors, are used, with a first inertial measuring unit as the master -Inertial measuring unit is used, with a second inertial measuring unit, the performance of which may be lower than that of the first inertial measuring unit, being used as a slave inertial measuring unit.

The WO 2013/ 112 230 A1 further relates to a method for calibrating an accelerometer in a portable device, comprising: receiving data from the accelerometer; providing acceleration measurements from the data based on one or more selection rules that adaptively select data that meet certain criteria; Adaptation of the acceleration measurements to a mathematical model; and providing a bias of the accelerometer based on a center point of the mathematical model.

The DE 10 2012 219 507 A1 also relates to a method for adjusting yaw rate sensors, wherein a yaw rate sensor has a substrate with a main extension plane, a seismic mass movable relative to the substrate, a drive arrangement for deflecting the seismic mass in a drive direction, a detection arrangement for detecting a force effect of the seismic mass in a drive direction vertical detection direction and a test arrangement for deflection of the seismic mass parallel to the detection direction.

The object of the invention is to improve the calibration of a rotation rate sensor by characterizing the rotation rate sensor as accurately as possible.

The invention results from the features of the independent claims. Advantageous further developments and refinements are the subject of the dependent claims.

• - Ermitteln von ersten Testdaten durch Aufbringen von Referenzdrehraten in unterschiedlichen Orientierungen relativ zum Gehäuse und mit unterschiedlichen Beträgen der Referenzdrehraten je Orientierung auf den Drehratensensor, wobei die ersten Testdaten die Beträge der Referenzdrehraten mit den zugehörigen Orientierungen relativ zum Gehäuse sowie die nominale Orientierung eines zum Gehäuse körperfesten Koordinatensystems relativ zu einem an einem Rotationstisch für das Aufbringen der Referenzdrehraten körperfesten Koordinatensystem sowie ein jeweiliges den Referenzdrehraten zugeordnetes verarbeitbares Sensorsignal des Drehratensensors umfassen;
• - Schätzen, für die ersten Testdaten, eines Ausrichtungsfehlers zwischen einem zur Sensorachse festen Koordinatensystem und dem zum Gehäuse körperfesten Koordinatensystem,
• - Ermitteln von zweiten Testdaten durch Aufbringen von Vibrationen in drei Orientierungen des Drehratensensors, insbesondere in genau drei Orientierungen, wobei die Orientierungen insbesondere an den Achsen des zum Gehäuse körperfesten Koordinatensystems ausgerichtet sind, wobei die zweiten Testdaten die aufgeprägten Vibrationen sowie die jeweilige Orientierung des Drehratensensors sowie das jeweilige verarbeitbare Sensorsignal des Drehratensensors umfassen;
• - Ermitteln einer jeweiligen ersten Orientierungstransformation auf Basis der nominalen Orientierungen und auf Basis des geschätzten Ausrichtungsfehlers jeweils der ersten Testdaten und Anwenden der jeweiligen ersten Orientierungstransformation auf die ersten Testdaten mit der Wirkung, dass alle Referenzdrehraten der ersten Testdaten auf das zur Sensorachse feste Koordinatensystem bezogen sind;
• - Zusammenfassen aller Testdaten in einen kombinierten Testdatensatz;
• - Vorgeben einer mathematischen Modellstruktur eines parametrischen Modells des getesteten Drehratensensors;
• - Ausführen einer allumfassenden Regressionsanalyse für den kombinierten Testdatensatz zur Ermittlung von Parameterwerten des parametrischen Modells, sodass mit der allumfassenden Regressionsanalyse sämtliche Parameter des parametrischen Modells auf Grundlage aller Testdaten aufeinmal erhalten werden; und
• - Kalibrieren des getesteten Drehratensensors auf Basis des parametrischen Modells mit den ermittelten Parameterwerten;

A first aspect of the invention relates to a method for calibrating a rotation rate sensor, wherein the rotation rate sensor has a sensor axis and a housing, comprising the steps:

• - Determining first test data by applying reference rotation rates in different orientations relative to the housing and with different amounts of reference rotation rates per orientation on the rotation rate sensor, the first test data being the amounts of the reference rotation rates with the associated orientations relative to the housing and the nominal orientation of a body fixed to the housing Coordinate system relative to a body-fixed coordinate system on a rotation table for applying the reference rotation rates and a respective processable sensor signal of the rotation rate sensor assigned to the reference rotation rates;
• - Estimating, for the first test data, an alignment error between a coordinate system fixed to the sensor axis and the coordinate system fixed to the housing,
• - Determining second test data by applying vibrations in three orientations of the yaw rate sensor, in particular in exactly three orientations, the orientations being aligned in particular with the axes of the coordinate system fixed to the housing, the second test data containing the imposed vibrations and the respective orientation of the yaw rate sensor as well include the respective processable sensor signal of the rotation rate sensor;
• - Determining a respective first orientation transformation based on the nominal orientations and based on the estimated alignment error of each of the first test data and applying the respective first orientation transformation to the first test data with the effect that all reference rotation rates of the first test data are related to the coordinate system fixed to the sensor axis;
• - Combining all test data into a combined test data set;
• - Specifying a mathematical model structure of a parametric model of the tested yaw rate sensor;
• - Perform an all-encompassing regression analysis on the combined test data set to determine parameter values of the parametric model so that the all-encompassing regression analysis obtains all parameters of the parametric model based on all test data at once; and
• - Calibration of the tested rotation rate sensor based on the parametric model with the determined parameter values;

The rotation rate sensor has a sensor axis. The measurement takes place along this sensor axis. By definition, the orientation of the measurement to the sensor axis is always known, but there is an alignment error to be determined in the sense of an unknown alignment difference between a coordinate system that is fixed to the sensor axis and the coordinate system that is fixed to the housing. The sensor housing and rotation table rotate in particular together with the reference rotation rates. The reference rotation rates are applied in different orientations relative to the housing, in particular three orientations, preferably around the axes of a Cartesian coordinate system. The nominal orientation of the coordinate system that is fixed to the body relative to a coordinate system that is fixed to the body on a rotation table for applying the reference rotation rates is in principle known in its standard value, hence “nominal”.

The yaw rate sensor is configured in such a way that a processable sensor signal is generated in response to an yaw rate or change in orientation that occurs. The processable sensor signal can include: an electrical voltage, preferably in the unit volts, or an electrical current intensity, preferably in the unit milliamperes. The processable sensor signal is thus generated by the rotation rate sensor in response to the respective reference rotation data. Accordingly, the rotation rate sensor has the function of mapping the input variable (at least the reference rotation data) to the output variable (the processable sensor signal). The input quantity can be scalar, but is typically vector, i.e. H. that it includes several components. Even if the input variable is vectorial, the rotation rate sensor can be used to map onto a scalar output variable for an axis under consideration (in which, in particular, exactly one change in orientation is measured).

This illustration is reproduced by the parametric model. The parametric model has a predetermined, in particular non-linear, mathematical structure, for example it is of a predetermined order (with which highest exponent variables of the input variable enter into the model); furthermore, the predetermined mathematical structure is preferably used to determine which components of the input variable enter into the model .

The parametric model can in principle be a static model or a dynamic model. The static model does not take into account the influence of the frequencies of the input variable and is therefore described by an algebraic equation. Such an algebraic equation assigns to each current input variable a corresponding current output variable. The output variable of the dynamic model, on the other hand, depends not only on the current input variable, but also on the past history of the input variables. Instead of an algebraic equation, a differential equation is therefore preferably used to describe a dynamic model, which can also be transformed into the frequency or Laplace range and can be represented as a transfer function. Furthermore, with a dynamic model it is possible to model the so-called “colored noise”, a frequency-dependent noise. Whether a static model or a dynamic model is more suitable for describing the transfer of the input variable to the output variable depends on the properties of the yaw rate sensor used and, in special cases, on which frequencies are expected in the accelerations that need to be recorded.

Regression analysis serves the purpose of identifying the parameters of the model in values and therefore, in a broader sense, the process of parameter identification. In general, regression analysis is a method based on statistical considerations to model statistical processes, i.e. H. to determine the relationship between one or more dependent variables and one or more independent variables, so that for a hypothetical value of an input variable, a corresponding value of an output variable of the model can be calculated. Which specific form of regression analysis is used depends in particular on the structure of the parametric model. Preferably, a form of regression analysis is used that takes a bias term into account.

In other words, according to the first aspect of the invention, the rotation rate sensor is placed on a typical rotation table. Here, the nominal sensor orientation is determined relative to the coordinate system, which is fixed to the rotation table. The rotation table is then controlled in such a way that different reference rotation rates are controlled, preferably over the entire detectable width of amounts of the rotation rate sensor, and under different orientations relative to a body-fixed axis of the rotation rate sensor, or at least in an orientation that deviates from the sensor axis under consideration. During this test, both the processable sensor signal and the reference rotation rate are determined and recorded. If the rotation table has sufficient accuracy, the reference acceleration of gravity can also be made dependent on the relation between the gravity vector and the current rotation table orientation. Otherwise, the reference acceleration from gravity can also be assumed to be zero, and this error is included in the model error. In the event that the rotation table is arranged above a seismically isolated platform, the observation of the measurement noise carried out in an embodiment mentioned below can be skipped and the measurement noise can be carried out together with the actual system identification (when applying the reference rotation rates). To do this, the rotation stage should be positioned statically to allow sufficient time to view the measurement noise, before, between, or after applying the reference rotation rates. The time course, ie the profile, of the reference rotation data should be selected so that all sensor axes are excited, and ideally with as many different reference rotation rate vectors as possible. As an additional option, a time-variant temperature ramp can be applied while the system identification is being carried out and the reference rotation rates are applied for this purpose.

Once this has been done, the rotation rate sensor is placed on a typical vibration table with a corresponding holder. The holder should be manufactured as precisely as possible, and the orientations of the rotation rate sensor relative to the holder should therefore be known as precisely as possible. The acceleration at the holder and with respect to a coordinate system that is body-fixed to the holder should be known as precisely as possible, in particular by measurement, and used as a term in the input variable. Again, care should be taken to provide sufficient excitation in frequency and amplitude, using various known profiles such as random vibrations, frequency responses (sine sweeps), shocks, and quasi-static loads. This is based in particular on which vibrations are expected during later operation of the yaw rate sensor. While the vibrations are being applied, the processable sensor signal and the known vibrations must be recorded and linked to each other in time. The nominal orientation of the rotation rate sensor must also be recorded with reference to the coordinate system of the holder. Optionally, if possible, the temperature can be changed during this vibration test, with the corresponding temperature being recorded and used as part of the input variable as part of the second test data. This vibration test is preferably repeated sequentially for all relevant sensor axes.

The vibrations are referenced to a nominal coordinate system with no alignment errors, while the reference rotation rates are initially referenced to an estimated coordinate system that has alignment errors, which in turn are estimated. After recording all alignment errors and nominal sensor orientations, all reference rotation rates are transformed into the coordinate system that is fixed to the sensor axis of the rotation rate sensor and all recorded accelerations are transformed into the coordinate system that is fixed to the housing. A special regression analysis is used to obtain all parameters of the parametric model in one go, which are precisely related to the coordinate systems mentioned above. With this preliminary model, the final model is preferably obtained by applying a further coordinate system transformation, so that the model is only related to the coordinate system that is body-fixed to the rotation rate sensor.

The invention has the following advantageous effects: Standard and not too expensive equipment can be used, such as a vibration table, a rotation table, and the like. Furthermore, different situations and different environmental conditions can be taken into account, such as different temperatures, vibrations and orientations. The most suitable, especially nonlinear, dynamic model with higher order terms and with nonlinear terms can be obtained and, if necessary, later reduced to components such as “scaling error” and “bias”, but with a better error model. The sensor error, or a sensor error component, can be modeled as a non-white random error, and a prevailing vibration as part of the environmental conditions is not limited to sinusoidal cases. The effort for the necessary tests is thereby advantageously kept low, so that one and the same vibration profiles are used for testing the sensor Environmental conditions were used, can also be used for the characterization of the sensor. In total, no more than three test runs need to be carried out regarding the external vibration influence. Temperature effects can also be taken into account in the characterization, whereby the bandwidth of the prevailing vibrations is only limited by the bandwidth of the test equipment, so that a very wide frequency range can usually be taken into account. Additionally, alignment can be estimated and taken into account, allowing this sensor characterization to occur independently of the test equipment.

• - Ermitteln einer jeweiligen zweiten Orientierungstransformation auf Basis der zweiten Testdaten, und Anwenden der jeweiligen zweiten Orientierungstransformation auf die zweiten Testdaten mit der Wirkung, dass alle Vibrationen der zweiten Testdaten auf das zum Gehäuse feste Koordinatensystem bezogen sind; oder:
• - Ermitteln einer jeweiligen zweiten Orientierungstransformation auf Basis der zweiten Testdaten und der Schätzungen des Ausrichtungsfehlers aus den ersten Testdaten, und Anwenden der jeweiligen zweiten Orientierungstransformation auf die zweiten Testdaten mit der Wirkung, dass alle Vibrationen der zweiten Testdaten auf das zur Sensorachse feste Koordinatensystem bezogen sind;

According to a further advantageous embodiment, the method further comprises one of the steps:

• - Determining a respective second orientation transformation based on the second test data, and applying the respective second orientation transformation to the second test data with the effect that all vibrations of the second test data are related to the coordinate system fixed to the housing; or:
• - Determining a respective second orientation transformation based on the second test data and the estimates of the alignment error from the first test data, and applying the respective second orientation transformation to the second test data with the effect that all vibrations of the second test data are related to the coordinate system fixed to the sensor axis;

According to a further advantageous embodiment, the different orientations of the reference rotation rates are such that they each generate excitations of the rotation rate sensor in all axes of the coordinate system fixed to the housing.

According to a further advantageous embodiment, the parametric model comprises a NARMAX model. NARMAX stands for “nonlinear autoregressive moving average model with exogenous inputs”. NARMAX models can represent a broad class of nonlinear systems.

According to a further advantageous embodiment, a large number of possible parametric models of the tested yaw rate sensor are specified with different mathematical model structures, with a respective all-encompassing regression analysis being carried out for the combined test data set to determine all parameter values of the respective parametric model, the most suitable model according to a predetermined metric is selected and the tested rotation rate sensor is calibrated based on the selected parametric model with the associated determined parameter values.

According to a further advantageous embodiment, third test data is generated by statically arranging the yaw rate sensor on a seismically isolated platform, the third test data comprising a processable sensor signal of the yaw rate sensor comprising a respective measurement noise, and wherein for the third test data there is an alignment error between the to Sensor axis fixed coordinate system and the body-fixed coordinate system to the housing is determined. The third test data is then also added to the combined test data set.

In other words, according to this embodiment, the rotation rate sensor is mounted on a seismically isolated platform and the processed sensor signal and the vector of the earth's rotation rate are recorded relative to the holder. Furthermore, the nominal orientation of the rotation rate sensor to the holder is determined. Optionally, during these tests to determine the sensor noise, the temperature of the yaw rate sensor is changed, preferably with a variable temperature ramp, with the current temperature being recorded and assigned to the respective orientation of the yaw rate sensor. The alignment error is then determined without the rotation rate sensor being removed from the holder and repositioned. Additionally, optionally, the orientation of the rotation rate sensor relative to the holder and thus relative to the seismic isolated platform can be changed and the above-mentioned procedure can be repeated. However, if the rotation rate sensor is removed from the holder and reinstalled, the alignment error must be determined again each time it is reinstalled.

According to a further advantageous embodiment, the rotation rate sensor is brought into different static orientations relative to the seismically isolated platform, the third test data comprising the respective orientations of the rotation rate sensor.

According to a further advantageous embodiment, the earth's rotation rate is compensated for the first test data and/or for the second test data and/or for the third test data in that the vector of the earth's rotation rate is projected onto the axes of the coordinate system fixed to the housing, and the result of the Projections and the respective orientation of the rotation rate sensor are used as the respective component of the respective test data.

According to a further advantageous embodiment, different temperatures are applied to the rotation rate sensor within the first test data and/or the second test data and/or the test data, with the detected temperatures being used as a further component of the respective test data.

According to a further advantageous embodiment, a modified rational model estimator is used to determine a respective alignment error. The modified rational model estimator is also abbreviated as modified RME, where RME stands for “Rational Model Estimator”.

Further advantages, features and details emerge from the following description, in which at least one exemplary embodiment is described in detail - if necessary with reference to the drawing. Identical, similar and/or functionally identical parts are provided with the same reference numerals.

• 1: Ein Verfahren zum Kalibrieren eines Drehratensensors gemäß einem Ausführungsbeispiel der Erfindung.
• 2: Bezeichnungen und Definitionen für den Drehratensensor.
• 3: Ein Verfahren der erweiterten Methode der kleinsten Quadrate für ein Ausführungsbeispiel der Erfindung.
• 4: Ein Verfahren zur Ermittlung der Orientierungstransformation für ein Ausführungsbeispiel der Erfindung.

Show it:

• 1 : A method for calibrating a rotation rate sensor according to an exemplary embodiment of the invention.
• 2 : Names and definitions for the yaw rate sensor.
• 3 : An extended least squares method for an embodiment of the invention.
• 4 : A method for determining the orientation transformation for an embodiment of the invention.

The representations in the figures are schematic and not to scale.

• - Ermitteln S1 von ersten Testdaten durch Aufbringen von Referenzdrehraten in unterschiedlichen Orientierungen relativ zum Gehäuse und mit unterschiedlichen Beträgen der Referenzdrehraten je Orientierung auf den Drehratensensor 1, wobei die ersten Testdaten die Beträge der Referenzdrehraten mit den zugehörigen Orientierungen relativ zum Gehäuse sowie die nominale Orientierung eines zum Gehäuse körperfesten Koordinatensystems relativ zu einem an einem Rotationstisch für das Aufbringen der Referenzdrehraten körperfesten Koordinatensystem sowie ein jeweiliges den Referenzdrehraten zugeordnetes verarbeitbares Sensorsignal des Drehratensensors 1 umfassen;
• - Schätzen S2, für die ersten Testdaten, eines Ausrichtungsfehlers zwischen einem zur Sensorachse festen Koordinatensystem und dem zum Gehäuse körperfesten Koordinatensystem,
• - Ermitteln S3 von zweiten Testdaten durch Aufbringen von Vibrationen in drei Orientierungen des Drehratensensors 1, wobei die Orientierungen insbesondere an den Achsen des zum Gehäuse körperfesten Koordinatensystems ausgerichtet sind, wobei die zweiten Testdaten die aufgeprägten Vibrationen sowie die jeweilige Orientierung des Drehratensensors 1 sowie das jeweilige verarbeitbare Sensorsignal des Drehratensensors 1 umfassen;
• - Ermitteln S4 einer jeweiligen ersten Orientierungstransformation auf Basis der nominalen Orientierungen und auf Basis des geschätzten Ausrichtungsfehlers jeweils der ersten Testdaten und Anwenden der jeweiligen ersten Orientierungstransformation auf die ersten Testdaten mit der Wirkung, dass alle Referenzdrehraten der ersten Testdaten auf das zur Sensorachse feste Koordinatensystem bezogen sind;
• - Zusammenfassen S5 aller Testdaten in einen kombinierten Testdatensatz;
• - Vorgeben S6 einer mathematischen Modellstruktur eines parametrischen Modells des getesteten Drehratensensors 1;
• - Ausführen S7 einer allumfassenden Regressionsanalyse für den kombinierten Testdatensatz zur Ermittlung von Parameterwerten des parametrischen Modells, sodass mit der allumfassenden Regressionsanalyse sämtliche Parameter des parametrischen Modells auf Grundlage der Testdaten aller Testpunkte aufeinmal erhalten werden; und
• - Kalibrieren S8 des getesteten Drehratensensors 1 auf Basis des parametrischen Modells mit den ermittelten Parameterwerten;

1 shows a method for calibrating a rotation rate sensor 1, wherein the rotation rate sensor 1 has a sensor axis and a housing, comprising the steps:

• - Determining S1 of first test data by applying reference rotation rates in different orientations relative to the housing and with different amounts of the reference rotation rates per orientation on the rotation rate sensor 1, the first test data being the amounts of the reference rotation rates with the associated orientations relative to the housing as well as the nominal orientation of a to Housing body-fixed coordinate system relative to a body-fixed coordinate system on a rotation table for applying the reference rotation rates and a respective processable sensor signal of the rotation rate sensor 1 assigned to the reference rotation rates;
• - Estimate S2, for the first test data, of an alignment error between a coordinate system fixed to the sensor axis and the coordinate system fixed to the housing,
• - Determining S3 of second test data by applying vibrations in three orientations of the yaw rate sensor 1, the orientations being aligned in particular with the axes of the coordinate system fixed to the housing, the second test data containing the imposed vibrations and the respective orientation of the yaw rate sensor 1 as well as the respective processable Sensor signal of the rotation rate sensor 1 include;
• - Determining S4 a respective first orientation transformation based on the nominal orientations and based on the estimated alignment error of each of the first test data and applying the respective first orientation transformation to the first test data with the effect that all reference rotation rates of the first test data are related to the coordinate system fixed to the sensor axis ;
• - Combine S5 all test data into a combined test data set;
• - Specify S6 a mathematical model structure of a parametric model of the tested rotation rate sensor 1;
• - Execute S7 an all-encompassing regression analysis on the combined test data set to determine parameter values of the parametric model, so that the all-encompassing regression analysis obtains all parameters of the parametric model at once based on the test data of all test points; and
• - Calibrating S8 of the tested rotation rate sensor 1 based on the parametric model with the determined parameter values;

• 2 zeigt die Achsen eines skizzenhaften Drehratensensors 1 und die Winkel der Orientierungstransformation. Hierbei bezeichnen:
  • - IA: Die Eingangsachse, die gleich der Sensorachse des Drehratensensors gemäß der Beschreibung ist, dass der Drehratensensor eine Sensorachse und ein Gehäuse aufweist;
  • - IRA: Die Eingangsreferenzachse;
  • - XRA: Die X-Referenzachse;
  • - YRA: Die Y-Referenzachse; und
  • - α: den Rotationsvektor in der Ebene, die durch XRA und YRA definiert wird, und ist senkrecht zu IA wie auch zu IRA;

Further possible implementations of this method are set out below, where: 2 gives the basic designations for the other preferred embodiments:

• 2 shows the axes of a sketchy rotation rate sensor 1 and the angles of the orientation transformation. Here designate:
  • - IA: The input axis, which is equal to the sensor axis of the yaw rate sensor according to the description that the yaw rate sensor has a sensor axis and a housing;
  • - IRA: The input reference axis;
  • - XRA: The X reference axis;
  • - YRA: The Y reference axis; and
  • - α: the rotation vector in the plane defined by XRA and YRA and is perpendicular to both IA and IRA;

• - IEEE Standard Specification Format Guide and Test Procedure for SingleAxis Interferometric Fiber Optic Gyros. Standard IEEE 952:1997(R2008). The Institute of Electrical and Electronics Engineers (IEEE), Dec. 10, 2008. Sowie nach:
• - IEEE Standard Specification Format Guide and Test Procedure for SingleAxis Interferometric Fiber Optic Gyros, Corrigendum 1: Figure 1 and Subclauses 5.3.4, 8.3, 12.11.4.3.2, 12.11.4.3.3, 12.11.4.3.4, 12.12.3.1, and 12.12.4.1. Standard IEEE 952:1997(R2008), Cor 1-2016. The Institute of Electrical and Electronics Engineers (IEEE), Dec. 7, 2016.

The system equation (and therefore a candidate for a parametric model) for describing a uniaxial yaw rate sensor is standardized according to:

• - IEEE Standard Specification Format Guide and Test Procedure for SingleAxis Interferometric Fiber Optic Gyros. Standard IEEE 952:1997(R2008). The Institute of Electrical and Electronics Engineers (IEEE), Dec. 10, 2008. As well as after:
• - IEEE Standard Specification Format Guide and Test Procedure for SingleAxis Interferometric Fiber Optic Gyros, Corrigendum 1: Figure 1 and Subclauses 5.3.4, 8.3, 12.11.4.3.2, 12.11.4.3.3, 12.11.4.3.4, 12.12.3.1 , and 12.12.4.1. Standard IEEE 952:1997(R2008), Cor 1-2016. The Institute of Electrical and Electronics Engineers (IEEE), Dec. 7, 2016.

These sources provide a rational system equation that is dynamic and links the output of the yaw rate sensor with components of the reference rotations, temperatures, and drift errors along the reference and orthogonal axes. Based on this, the following system equation is used herein: $${\mathrm{<figure-callout\; id="0"\; label="S"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">S</figure-callout>}}_0=\left(\Delta N/\Delta t\right)=\frac{I+E+\mathrm{<figure-callout\; id="1"\; label="D"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">D</figure-callout>}}{1+{10}^{-6}{\epsilon }_K}$$

$$E=D_T\Delta T+D_{\dot{T}}\left(dT/dt\right)+{\overline{D}}_{\nabla \dot{\overline{T}}}\cdot \frac{d\nabla \overline{T}}{dt},\text{ und }D=D_F+D_R+D_Q,\text{mit:}$$

$$D_R=D_{RN}+D_{RB}+D_{RK}+D_{RR},\text{ und }{\epsilon }_K={\epsilon }_T\Delta T+f\left(I\right)$$

worin wiederum sind: ω_IRA, ω_XRA, ω_YRA Komponenten einer inertialen Eingangsdrehrate im Referenzkoordinatensystem des Drehratensensors 1; α der Ausrichtungsfehler von IA zu IRA; ${\alpha }^2={\alpha }_x^2+{\alpha }_y^2$

mit α_x der Ausrichtungsfehler von IA zu XRA; α_y der Ausrichtungsfehler von IA zu YRA; mit $\sqrt{1-{\left[\frac{\text{sin}\alpha }{\alpha }{\alpha }_x\right]}^2-{\left[\frac{\text{sin}\alpha }{\alpha }{\alpha }_y\right]}^2}=\text{cos }\alpha ;$

D_F der Bias; D_TΔT die Driftrate die zurückzuführen ist auf eine Temperaturänderung ΔT mit D_T dem Driftraten-Temperatur-Sensitivitätskoeffizienten; ε_TΔT der Skalenfaktor-Fehler der auf eine Temperaturänderung ΔT zurückzuführen ist mit ε_T dem Skalenfaktor-Temperatur-Sensitivitätskoeffizienten; D_Ṫ(ΔT/dt) die Driftrate die auf eine Temperaturrampe ΔT/dt zurückzuführen ist mit D_Ṫ dem Koeffizienten für die Temperaturrampe-Driftraten-Sensitivität; ${\overline{D}}_{\nabla \dot{\overline{T}}}\cdot \frac{d\nabla \overline{T}}{dt}$

die Driftrate die auf einen zeitvarianten Temperaturgradienten $\frac{d\nabla \overline{T}}{dt}$

zurückzuführen ist mit ${\overline{D}}_{\nabla \dot{\overline{T}}}$

dem Koeffizientenvektor der zeitabhängigen Temperaturgradient-Driftraten-Sensitivität; f (I) der Skalenfaktor-Fehler abhängig von der Eingangsrate; D_RN die zufällige Driftrate die auf einen Winkel-Random-Walk zurückzuführen ist mit N als dem Koeffizienten; D_RB die zufällige Driftrate die auf eine Bias Instabilität zurückzuführen ist mit B als dem Koeffizienten; D_RK die zufällige Driftrate die auf eine Winkelraten-Random-Walk zurückzuführen ist mit K als dem Koeffizienten; D_RR die zufällige Driftrate die auf eine Rampe zurückzuführen ist mit R als dem Koeffizienten; D_Q die äquivalente zufällige Driftrate die auf eine Winkelquantisierung zurückzuführen ist mit Q als dem Koeffizienten.Here are: S ₀ : The nominal scale factor ["/p]; (ΔN/Δt): The output pulse rate [p/s]; I: The inertial input variable [°/h]; E: The environmental influence-sensitive term [°/h] ; D: The drift term [°/h]; ε _K : The scale factor error term [ppm]; Furthermore: $$I={\omega }_{\text{<figure-callout id="1" label="IRA" filenames="DE102022126969B3\_0140.png" state="{{state}}">IRA</figure-callout>}}\sqrt{1-{\left[\frac{\text{sin}\alpha }{\alpha }{\alpha }_x\right]}^2-{\left[\frac{\text{sin}\alpha }{\alpha }{\alpha }_y\right]}^2}+{\omega }_{\text{XRA}}\left[\frac{\text{sin}\alpha }{\alpha }{\alpha }_y\right]-{\omega }_{\text{YRA}}\left[\frac{\text{sin}\alpha }{\alpha }{\alpha }_x\right]$$

$$E=D_T\Delta T+D_{\dot{T}}\left(dT/dt\right)+{\overline{D}}_{\nabla \dot{\overline{T}}}\cdot \frac{d\nabla \overline{T}}{dt},\text{and}D=D_F+D_R+D_Q,\text{with:}$$

$$D_R=D_{RN}+D_{Rb}+D_{RK}+D_{RR},\text{and}{\epsilon }_K={\epsilon }_T\Delta T+f\left(I\right)$$

where in turn: ω _IRA , ω _XRA , ω _YRA are components of an inertial input rotation rate in the reference coordinate system of the rotation rate sensor 1; α is the alignment error from IA to IRA; ${\alpha }^2={\alpha }_x^2+{\mathrm{<figure-callout\; id="2"\; label="\alpha \; y"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">\alpha </figure-callout>}}_y^{}$

with α _x the alignment error from IA to XRA; α _y is the alignment error from IA to YRA; with $\sqrt{1-{\left[\frac{\text{sin}\alpha }{\alpha }{\alpha }_x\right]}^2-{\left[\frac{\text{sin}\alpha }{\alpha }{\alpha }_y\right]}^2}=\text{cos}\alpha ;$

D _F the bias; D _T ΔT is the drift rate that can be attributed to a temperature temperature change ΔT with D _T the drift rate temperature sensitivity coefficient; ε _T ΔT the scale factor error due to a temperature change ΔT with ε _T the scale factor temperature sensitivity coefficient; D _Ṫ (ΔT/dt) is the drift rate due to a temperature ramp ΔT/dt with D _Ṫ the coefficient for the temperature ramp drift rate sensitivity; ${\overline{D}}_{\nabla \dot{\overline{T}}}\cdot \frac{d\nabla \overline{T}}{dt}$

the drift rate is due to a time-variant temperature gradient $\frac{d\nabla \overline{T}}{dt}$

is attributed to ${\overline{D}}_{\nabla \dot{\overline{T}}}$

the coefficient vector of the time-dependent temperature gradient drift rate sensitivity; f (I) the scale factor error depending on the input rate; D _RN is the random drift rate due to an angular random walk with N as the coefficient; D _RB is the random drift rate due to bias instability with B as the coefficient; D _RK is the random drift rate due to an angular rate random walk with K as the coefficient; D _RR is the random drift rate due to a ramp with R as the coefficient; D _Q is the equivalent random drift rate due to angle quantization with Q as the coefficient.

The above-mentioned standards from the “Institute of Electrical and Electronics Engineers (IEEE)” specify a model that is both deterministic and stochastic. The deterministic part is represented by the factors S ₀ , I, E, ε _K and D _F , which are assigned to the basic sensor principle and relate temperature and sensitivity to environmental influences and deterministic errors. On the other hand, the stochastic part is represented by the factors D _R and D _Q , which can be summarized under the term measurement noise, but not necessarily just white noise, also called “white noise”.

Measurement noise is a highly undesirable but very relevant influence in inertial measurement units. In the following, the stress-strain theory is examined and then analyzed in the frequency range with experimental data in order to verify the following vibration model: $$S_{\mathrm{\Omega }}={\left|H\text{'}\left(f\right)f\right|}^2{S}_{vib}$$

worin c die Koeffizienten sind, die bei der Charakterisierung erhalten werden und k das jeweilige Sample für eine Variable ist. Die Driftterme D_RN, D_RB, D_RK und D_RR können ferner durch die folgende Gleichung modelliert werden: $$D_R\left(s\right)=\left(n_{\dot{\Omega }}\left(s\right)+\frac{1}{s}+n_{F\dot{\Omega }}\left(s\right)\frac{1}{\sqrt{s}}\right)\frac{1}{s}+n_{\mathrm{\Omega }}\left(s\right)+n_{F\mathrm{\Omega }}\left(s\right)\frac{1}{\sqrt{s}}$$

worin n_Ω(s), n_FΩ(s), n_Ω̇(s) und n_FΩ̇(s) die den zufälligen White-Noise beschreibenden Terme sind mit einem Mittelwert von Null und einer Varianz von jeweils $\frac{N}{{\tau }_0},\frac{2B^2}{\pi },\frac{K^2{\tau }_0}{3}$

und $\frac{R^2{\tau }_0^2}{2},$

während τ₀ die Samplingzeit ist. Nach Auswertung der obigen Gleichung können die Terme für das Messrauschen klassifiziert werden als weißes Rauschen (White-Noise), pinkes Rauschen, rotes Rauschen und schwarzes Rauschen. In praktischen Anwendungen sind nur die Terme des Messrauschens n_Ω(s) und n_FΩ(s) in der Regel relevant. Deshalb kann die obige Gleichung vereinfacht werden zu: $$D_R\left(s\right)=n_{\mathrm{\Omega }}\left(s\right)+n_{F\mathrm{\Omega }}\left(s\right)\frac{1}{\sqrt{s}}$$

Here S _Ω is the power spectral density of the output rate and S _vib is the power spectral density of the random vibration. Based on the system equation shown above, it is necessary to design the parametric model for use in both simulation and estimation scenarios. The temperature gradient is an effect that is too complex to model; It is better to leave it as a model error. The environment sensitive terms depend on temperature, but can be approximated by the following difference equation: $$E_T\left(k\right)=c_{T1}T\left(k\right)+c_{T2}T\left(k-1\right)$$

where c are the coefficients obtained during the characterization and k is the respective sample for a variable. The drift terms D _RN , D _RB , D _RK and D _RR can be further modeled by the following equation: $$D_R\left(s\right)=\left(n_{\dot{\Omega }}\left(s\right)+\frac{1}{s}+n_{F\dot{\Omega }}\left(s\right)\frac{1}{\sqrt{s}}\right)\frac{1}{s}+n_{\mathrm{\Omega }}\left(s\right)+n_{F\mathrm{\Omega }}\left(s\right)\frac{1}{\sqrt{s}}$$

where n _Ω (s), n _FΩ (s), n _Ω̇ (s) and n _FΩ̇ (s) are the terms describing the random white noise with a mean of zero and a variance of respectively $\frac{\mathrm{<figure-callout\; id="0"\; label="N\; \tau "\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">N</figure-callout>}}{{\tau }_{}},\frac{2{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}{\pi },\frac{{\mathrm{<figure-callout\; id="2"\; label="K"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">K</figure-callout>}}^2{\mathrm{<figure-callout\; id="0"\; label="\tau "\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">\tau </figure-callout>}}_0}{3}$

and $\frac{R^2{\tau }_0^2}{2},$

while τ ₀ is the sampling time. After evaluating the above equation, the measurement noise terms can be classified as white noise, pink noise, red noise and black noise. In practical applications, only the measurement noise terms n _Ω (s) and n _FΩ (s) are usually relevant. Therefore the above equation can be simplified to: $$D_R\left(s\right)=n_{\mathrm{\Omega }}\left(s\right)+n_{F\mathrm{\Omega }}\left(s\right)\frac{1}{\sqrt{s}}$$

ist pinkes Messrauschen proportional zu B². Die Übertragungsfunktion $\frac{1}{\sqrt{s}}$

kann nicht genau als ein lineares System modelliert werden. Sie kann jedoch durch eine lineare Übertragungsfunktion der dritten Ordnung angenähert werden, die weißes Rauschen filtert. Es seien ferner A(q) und B(q) Polynome aus q, wobei q den Vorwärtsoperator im Zeitbereich angibt, d. h. q^-1x(k) = x(k - 1). Das pinke Messrauschen kann daher angenähert werden durch: $$d_{F\mathrm{\Omega }}\left(k\right)=\frac{b{\text{}}_0^{\text{'}}+b_1^{\text{'}}{q}^{-1}+b_2^{\text{'}}{q}^{-2}+b_3^{\text{'}}{q}^{-3}}{1+a_1^{\text{'}}{q}^{-1}+a_2^{\text{'}}{q}^{-2}+a_3^{\text{'}}{q}^{-3}}{n}_{F\mathrm{\Omega }}\left(k\right)=\frac{B\text{'}\left(q\right)}{A\text{'}\left(q\right)}{n}_{F\mathrm{\Omega }}\left(k\right)$$

Here n _Ω (s) white measurement noise is proportional to N ² , and $n_{F\mathrm{\Omega }}\left(s\right)\frac{1}{\sqrt{s}}$

pink measurement noise is proportional to B ² . The transfer function $\frac{1}{\sqrt{s}}$

cannot be accurately modeled as a linear system. However, it can be approximated by a third-order linear transfer function that filters white noise. Furthermore, let A(q) and B(q) be polynomials of q, where q indicates the forward operator in the time domain, ie q ^-1 x(k) = x(k - 1). The pink measurement noise can therefore be approximated by: $$d_{F\mathrm{\Omega }}\left(k\right)=\frac{b{}_0^{\text{'}}+b_1^{\text{'}}{q}^{-1}+b_2^{\text{'}}{q}^{-2}+b_3^{\text{'}}{q}^{-3}}{1+a_1^{\text{'}}{q}^{-1}+a_2^{\text{'}}{q}^{-2}+a_3^{\text{'}}{q}^{-3}}{n}_{F\mathrm{\Omega }}\left(k\right)=\frac{b\text{'}\left(q\right)}{A\text{'}\left(q\right)}{n}_{F\mathrm{\Omega }}\left(k\right)$$

The drift terms, on the other hand, are summarized by the difference equation: $$D_R\left(k\right)=\frac{b\text{'}\left(q\right)}{A\text{'}\left(q\right)}{n}_{F\mathrm{\Omega }}\left(k\right)+n_{\mathrm{\Omega }}\left(k\right)$$

For the purpose of characterizing the system behavior, it will be difficult in practice to separate the two sources of white noise n _FΩ (k) and n _Ω (k). To remedy this, a noise model of the resulting measurement noise as a result of filtering only one source of white measurement noise with adequate combined variance can be used: $$D_R\left(k\right)=\frac{b\left(q\right)}{A\left(q\right)}{n}_{\mathrm{\Omega }}^{\text{'}}\left(k\right)$$

The scale factor error f (I) mentioned in the context of the system equation as a function of the input rotation rate I, which in turn is a function of ω _IRA , ω _XRA and ω _YRA , is used to reduce the order of the model as follows, where assumed is that it is only a function of ω _IRA based on the prior knowledge that the largest influences on the signal come from ω _IRA . This results in: $$f\left(I\right)\left(k\right)={\mathrm{<figure-callout\; id="1"\; label="c\; f"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">c</figure-callout>}}_f{\omega }_{IRA}\left(k\right)+{\mathrm{<figure-callout\; id="2"\; label="c\; f"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">c</figure-callout>}}_f{\omega }_{IRA}^2\left(k\right)+c_{f\text{3}}\left|{\omega }_{IRA}\right|\left(k\right)$$

Here c _f1 , c _f2 , and c _f3 are arbitrary coefficients that need to be characterized. It is likely that during characterization one or more of these parameters will turn out to be irrelevant. However, it is safer to take all of these parameters into account first and only remove them later when you are certain. For simplicity, factor I can also be reduced to: $$I\left(k\right)=c_{I1}{\omega }_{IRA}\left(k\right)+c_{I2}{\omega }_{XRA}\left(k\right)+c_{I3}{\omega }_{YRA}\left(k\right)$$

From equation (A) it follows that the transfer function from a mechanical acceleration to the output variable is described by H' (f) f. This is a transfer function that is multiplied by a violet noise filter. A violet noise filter is a derivative of the white noise signal. The transfer function of the closed chain can thus be determined as: $$H\text{'}\left(ƒ\right)=\frac{1}{Ts+1}{e}^{-\tau s}$$

$$E_a\left(k\right)=\frac{D\left(q\right)}{C\left(q\right)}{a}_{IRA}\left(k\right)+\frac{F\left(q\right)}{E\left(q\right)}{a}_{XRA}\left(k\right)+\frac{H\left(q\right)}{G\left(q\right)}{a}_{YRA}\left(k\right)$$

worin a_IRA(k), a_XRA(k) und a_YRA(k) externe Beschleunigungen beschreiben, die auf den Drehratensensor in jeder Achse zur Instanz k ausgeübt werden. Aus sämtlichen vorhergehenden Gleichungen kann damit die vorgeschlagene Ausgangsgröße Ω(k) des Drehratensensors zur Instanz k modelliert werden durch folgende Gleichung: $$\begin{array}{l}\mathrm{\Omega }\left(k\right)S_0\left[1+{10}^{-6}{\epsilon }_{T}T\left(k\right)+c_{ƒ1}{\omega }_{IRA}\left(k\right)+c_{f\text{2}}{\omega }_{IRA}^2\left(k\right)+c_{f\text{3}}\left|{\omega }_{IRA}\right|\left(k\right)\right]=\\ =c_{I1}{\omega }_{IRA}\left(k\right)+c_{I\text{2}}{\omega }_{XRA}\left(k\right)+c_{I3}{\omega }_{YRA}\left(k\right)+c_{T1}T\left(k\right)+c_{T2}T\left(k-1\right)+\\ +\frac{B\left(q\right)}{A\left(q\right)}{n}_{\mathrm{\Omega }}^{\text{'}}\left(k\right)+\frac{D\left(q\right)}{C\left(q\right)}{a}_{IRA}\left(k\right)+\frac{F\left(q\right)}{E\left(q\right)}{a}_{XRA}\left(k\right)+\frac{H\left(q\right)}{G\left(q\right)}{a}_{YRA}\left(k\right)+D_F\end{array}$$

Here T and τ are time constants. It therefore makes sense to approximate the transfer function of the vibration influence using a second-order difference equation such as: $$\begin{array}{l}{E}_a\left(k\right)=\frac{{\mathrm{<figure-callout\; id="0"\; label="d"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">d</figure-callout>}}_0+d_1{q}^{-1}+d_2{q}^{-2}}{1+c_1{q}^{-1}+c_2{q}^{-2}}{a}_{IRA}\left(k\right)+\frac{{\mathrm{<figure-callout\; id="0"\; label="ƒ"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">ƒ</figure-callout>}}_0+{ƒ}_1{q}^{-1}+{ƒ}_2{q}^{-2}}{1+e_1{q}^{-1}+e_2{q}^{-2}}{a}_{XRA}\left(k\right)+\\ \frac{{\mathrm{<figure-callout\; id="0"\; label="H"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">H</figure-callout>}}_0+H_1{q}^{-1}+H_2{q}^{-2}}{1+G_1{q}^{-1}+G_2{q}^{-2}}{a}_{YRA}\left(k\right)\end{array}$$

$$E_a\left(k\right)=\frac{D\left(q\right)}{C\left(q\right)}{a}_{IRA}\left(k\right)+\frac{F\left(q\right)}{E\left(q\right)}{a}_{XRA}\left(k\right)+\frac{H\left(q\right)}{G\left(q\right)}{a}_{YRA}\left(k\right)$$

where a _IRA (k), a _XRA (k) and a _YRA (k) describe external accelerations exerted on the yaw rate sensor in each axis to instance k. From all the previous equations, the proposed output variable Ω(k) of the rotation rate sensor can be modeled for instance k using the following equation: $$\begin{array}{l}\mathrm{\Omega }\left(k\right)S_0\left[1+{10}^{-6}{\epsilon }_{T}T\left(k\right)+{\mathrm{<figure-callout\; id="1"\; label="c\; ƒ"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">c</figure-callout>}}_{ƒ}{\omega }_{IRA}\left(k\right)+{\mathrm{<figure-callout\; id="2"\; label="c\; f"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">c</figure-callout>}}_f{\omega }_{IRA}^2\left(k\right)+c_{f\text{3}}\left|{\omega }_{IRA}\right|\left(k\right)\right]=\\ =c_{I1}{\omega }_{IRA}\left(k\right)+c_{I\text{2}}{\omega }_{XRA}\left(k\right)+c_{I3}{\omega }_{YRA}\left(k\right)+c_{T1}T\left(k\right)+c_{T2}T\left(k-1\right)+\\ +\frac{b\left(q\right)}{A\left(q\right)}{n}_{\mathrm{\Omega }}^{\text{'}}\left(k\right)+\frac{D\left(q\right)}{C\left(q\right)}{a}_{IRA}\left(k\right)+\frac{F\left(q\right)}{E\left(q\right)}{a}_{XRA}\left(k\right)+\frac{H\left(q\right)}{G\left(q\right)}{a}_{YRA}\left(k\right)+D_F\end{array}$$

Let F _S = {n _i , n _p , n _o } be the reference coordinate system within the yaw rate sensor 1, where n _i is aligned with the axis IA of the yaw rate sensor 1. Furthermore, let F _C = {n _ira , n _xra , n _yra } be the reference coordinate system on the rotation rate sensor 1 as in 2 shown, where the transformation from F _C to F _S is known. Based on this, the above equation can be rewritten in terms of F _S : $$\begin{array}{l}\mathrm{\Omega }\left(k\right)S_0\left[1+{10}^{-6}{\epsilon }_{T}T\left(k\right)+{\mathrm{<figure-callout\; id="1"\; label="c\; ƒ"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">c</figure-callout>}}_{ƒ}{\omega }_i\left(k\right)+{\mathrm{<figure-callout\; id="2"\; label="c\; ƒ"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">c</figure-callout>}}_{ƒ}{\omega }_i^2\left(k\right)+c_{ƒ\text{3}}\left|{\omega }_i\right|\left(k\right)\right]=\\ c_{I1}{\omega }_i\left(k\right)+c_{T1}T\left(k\right)+c_{T2}T\left(k-1\right)+\\ +\frac{b\left(q\right)}{A\left(q\right)}{n}_{\mathrm{\Omega }}^{\text{'}}\left(k\right)+\frac{D\left(q\right)}{C\left(q\right)}{a}_i\left(k\right)+\frac{F\left(q\right)}{E\left(q\right)}{a}_p\left(k\right)+\frac{H\left(q\right)}{G\left(q\right)}{a}_O\left(k\right)+D_F\end{array}$$

It should be noted that other rotation terms do not appear in this equation since by definition F _S is exactly aligned with the sensor axis and therefore no alignment error occurs.

The latter equation is a complex nonlinear autoregressive model with moving average components and an external input (the so-called “Non-linear Autoregressive Model with Moving Average and Exogenous Input” for short, NARMAX); A NARMAX model is used below, which is described in more detail in the publication “ QM Zhu and SA Billings. “Identification of Polynomial and Rational Narmax Models.” In: IFAC Proceedings Volumes 27.8 (July 1994), pp. 259-264. issn: 14746670. doi: 10. 1016 / S1474 - 6670(17) 47725 - 1. http://dx.doi.org/10.1016/S1474-6670(17)47725-1 “is explained and derived. This generalized NARMAX model can be described by: $$y\left(k\right)={\displaystyle \sum _{j=1}^m{\varphi }_j\left(k\right){\theta }_j+e\left(k\right)}$$

Die Abhängigkeit von y(k) in u(k - d) wird hierin angenommen als d = 1, 2, ...; hierzu kann gezeigt werden, dass diese Annahme keinen Einfluss auf die Tatsache hat, dass der im Folgenden diskutierte Schätzer „unbiased“ ist. Zum Zwecke der Schätzung wird die obige Gleichung (B) des generalisierten NARMAX Modells in Matrix Notation umgeschrieben: $$Y=\mathrm{\Phi \Theta +}e,$$

wobei gilt: $$Y={\left[y\left(1\right),\dots ,y\left(N\right)\right]}^T$$

$$\mathrm{\Phi }=\left(\mathbf{PV}+\mathbf{PE}\right)$$

$$\mathbf{PV}=\left[\begin{array}{ccc}{p}_1\left(1\right)v_1\left(1\right)& \ddots & p_m\left(1\right)v_{m1}\left(1\right)\\ ⋮& ⋮& ⋮\\ p_1\left(N\right)v_1\left(N\right)& \ddots & p_m\left(N\right)v_m\left(N\right)\end{array}\right]$$

$$\mathbf{P}\mathbf{E}=\left[\begin{array}{ccc}{p}_1\left(1\right)e_1\left(1\right)& \ddots & p_m\left(1\right)e_{m1}\left(1\right)\\ ⋮& ⋮& ⋮\\ p_1\left(N\right)e_1\left(N\right)& \ddots & p_m\left(N\right)e_m\left(N\right)\end{array}\right]$$

$$\mathrm{\Theta =}{\left[{\theta }_1,\dots ,{\theta }_m\left(N\right)\right]}^T,\mathbf{e}={\left[e\left(1\right),\dots ,e\left(N\right)\right]}^T$$

und N ist die Datenlänge.Therein ϕ _j (k) = p _j (k) (v _j (k) + e _j (k)), and u(k) represents the input variable and y(k) the output variable at a time k(k = 1, 2, 3, ...). Furthermore, p _j (k) = p _j (y(k - 1), ..., y(k - r), u(k), ..., u(k - r), e(k - 1 ), ..., e(k - r)) and v _j (k) = v _j (y(k - 1), ..., y(k - r), u(k), ..., u(k - r), e(k - 1), ..., e(k - r)) are free of actual measurement noise terms and can be expressed in any nonlinear ways. Furthermore, r denotes the order of the model and e(k) and e _j (k) are independent current measurement noise terms with a mean of zero and a variance of ${\sigma }_{e\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">j</figure-callout>}}^2.$

The dependence of y(k) in u(k - d) is assumed herein as d = 1, 2, ...; It can be shown that this assumption has no influence on the fact that the estimator discussed below is “unbiased”. For the purpose of estimation, the above equation (B) of the generalized NARMAX model is rewritten in matrix notation: $$Y=\mathrm{\Phi \Theta +}e,$$

where: $$Y={\left[y\left(1\right),\dots ,y\left(N\right)\right]}^T$$

$$\mathrm{\Phi }=\left(\mathbf{PV}+\mathbf{P.E}\right)$$

$$\mathbf{PV}=\left[\begin{array}{ccc}{p}_1\left(1\right)v_1\left(1\right)& \ddots & p_m\left(1\right)v_{m1}\left(1\right)\\ ⋮& ⋮& ⋮\\ p_1\left(N\right)v_1\left(N\right)& \ddots & p_m\left(N\right)v_m\left(N\right)\end{array}\right]$$

$$\mathbf{P}\mathbf{E}=\left[\begin{array}{ccc}{p}_1\left(1\right)e_1\left(1\right)& \ddots & p_m\left(1\right)e_{m1}\left(1\right)\\ ⋮& ⋮& ⋮\\ p_1\left(N\right)e_1\left(N\right)& \ddots & p_m\left(N\right)e_m\left(N\right)\end{array}\right]$$

$$\mathrm{\Theta =}{\left[{\mathrm{<figure-callout\; id="1"\; label="\theta "\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">\theta </figure-callout>}}_1,\dots ,{\theta }_m\left(N\right)\right]}^T,\mathbf{e}={\left[e\left(1\right),\dots ,e\left(N\right)\right]}^T$$

and N is the data length.

für die gilt: $$a_{sys}\left(k\right)={\displaystyle \sum {}_{j=1}^{num}{p}_{sys|nj}\left(k\right)}{\theta }_{sys|nj}$$

$$b_{sys}\left(k\right)={\displaystyle \sum {}_{j=1}^{den}{p}_{sys|dj}\left(k\right)}{\theta }_{sys|dj}$$

jeweils zur Parametrisierung des Zählers bzw. des Nenners des Systemmodells (C). Die Terme p_sys|nj (k) und p_sys|dj (k) bestehen aus den Kombinationen y_sys (k - 1), ..., y_sys (k - r), u_sys (k), ..., u_sys (k - r), e_sys1 (k - 1), ..., e_sys1 (k - r), und e_sys2 (k - 1), ..., e_sys2 (k - r); ferner sind θ_sys|nj und θ_sys|dj die Parameter für jeden Term. Außerdem gilt θ_sys|d1 = 1 und num + den (symbolisch für Zähler plus Nenner) steht für die vollständige Zahl von unbekannten Parametern. Zur Angleichung des Formats des obig eingeführten generalisierten NARMAX Modells in Gleichung (B) kann erhalten werden: $$\begin{array}{l}y\left(k\right)\leftarrow y_{sys}\left(k\right)p_{sys|d1}\left(k\right)\\ p_j\left(k\right)\leftarrow \{\begin{array}{cc}{p}_{sys|nj}\left(k\right)& \text{for 1}\le j\le num\\ p_{sys|d\left(j-num+1\right)}\left(k\right)& \text{for 1}+num\le j\le num+den-1\end{array}\\ v_j\left(k\right)\leftarrow \{\begin{array}{cc}1& \text{for 1}\le j\le num\\ \frac{a_{sys}\left(k\right)}{b_{sys}\left(k\right)}& \text{for 1}+num\le j\le num+den-1\end{array}\\ {\theta }_j\left(k\right)\leftarrow \{\begin{array}{cc}{\theta }_{sys|nj}& \text{for 1}\le j\le num\\ {\theta }_{sys|d\left(j-num+1\right)}& \text{for 1}+num\le j\le num+den-1\end{array}\\ e_j\left(k\right)\leftarrow \{\begin{array}{cc}0& \text{for 1}\le j\le num\\ \frac{e_{sys1}\left(k\right)}{b_{sys}\left(k\right)}+e_{sys2}\left(k\right)& \text{for 1}+num\le j\le num+den-1\end{array}\\ e\left(k\right)\leftarrow e_{sys1}\left(k\right)+b_{sys}\left(k\right)e_{sys2}\left(k\right)\end{array}$$

Let us now introduce a parametric (system) model using the following equation: $$y_{sys}\left(k\right)=\frac{a_{sys}\left(k\right)+e_{sys1}\left(k\right)}{b_{sys}\left(k\right)}+e_{sys2}\left(k\right)$$

for which applies: $$a_{sys}\left(k\right)={\displaystyle \sum {}_{j=1}^{num}{p}_{sys|nj}\left(k\right)}{\theta }_{sys|nj}$$

$$b_{sys}\left(k\right)={\displaystyle \sum {}_{j=1}^{den}{p}_{sys|dj}\left(k\right)}{\theta }_{sys|dj}$$

each for the parameterization of the numerator or the denominator of the system model (C). The terms p _sys|nj (k) and p _sys|dj (k) consist of the combinations y _sys (k - 1), ..., y _sys (k - r), u _sys (k), ... , u _sys (k - r), e _sys1 (k - 1), ..., e _sys1 (k - r), and e _sys2 (k - 1), ..., e _sys2 (k - r); further, θ _sys|nj and θ _sys|dj are the parameters for each term. Furthermore, θ _sys|d1 = 1 and num + den (symbolic of numerator plus denominator) represents the complete number of unknown parameters. To approximate the format of the generalized NARMAX model introduced above in equation (B), one can obtain: $$\begin{array}{l}y\left(k\right)\leftarrow y_{sys}\left(k\right)p_{sys|d1}\left(k\right)\\ p_j\left(k\right)\leftarrow \{\begin{array}{cc}{p}_{sys|nj}\left(k\right)& \text{for 1}\le j\le num\\ p_{sys|d\left(j-num+1\right)}\left(k\right)& \text{for 1}+num\le j\le num+den-1\end{array}\\ v_j\left(k\right)\leftarrow \{\begin{array}{cc}1& \text{for 1}\le j\le num\\ \frac{a_{sys}\left(k\right)}{b_{sys}\left(k\right)}& \text{for 1}+num\le j\le num+den-1\end{array}\\ {\theta }_j\left(k\right)\leftarrow \{\begin{array}{cc}{\theta }_{sys|nj}& \text{for 1}\le j\le num\\ {\theta }_{sys|d\left(j-num+1\right)}& \text{for 1}+num\le j\le num+den-1\end{array}\\ e_j\left(k\right)\leftarrow \{\begin{array}{cc}0& \text{for 1}\le j\le num\\ \frac{e_{sys1}\left(k\right)}{b_{sys}\left(k\right)}+e_{sys2}\left(k\right)& \text{for 1}+num\le j\le num+den-1\end{array}\\ e\left(k\right)\leftarrow e_{sys1}\left(k\right)+b_{sys}\left(k\right)e_{sys2}\left(k\right)\end{array}$$

The unbiased least squares estimate of the parameters Θ for equation (B) is: $$\widehat{\Theta }={\left[{\mathrm{\Phi }}^T\mathrm{\Phi }-{\left[\mathbf{P.E}\right]}^T\mathbf{P.E}\right]}^{-1}\left[{\overline{\Phi }}^{T}Y-{\left[\mathbf{P.E}\right]}^{T}Y\right]$$

wobei gilt: ${\sigma }_e^2=\frac{1}{N-md}{\displaystyle \sum _{k=md+1}^N{e}^2\left(k\right)},$

und wobei md der maximale Verzug in den Termen ist.The covariance of the estimator is also given by ${\displaystyle \sum {}_{\widehat{\Theta }}}={\sigma }_e^2{\left[{\mathrm{\Phi }}^T\mathrm{\Phi }\right]}^{-1},$

where: ${\sigma }_e^2=\frac{1}{N-md}{\displaystyle \sum _{k=md+1}^N{e}^2\left(k\right)},$

and where md is the maximum delay in the terms.

For the system in Equation (C), the error variance of the white measurement noise can be estimated by: $${\sigma }_{e_{sys1}}^2=\frac{1}{N-md}{\displaystyle \sum _{k=md+1}^N{e}_{sys1}^2\left(k\right)},\text{and}{\sigma }_{e_{sys2}}^2=\frac{1}{N-md}{\displaystyle \sum _{k=md+1}^N{e}_{sys2}^2\left(k\right)}$$

The variable e(k) is obviously not measurable and therefore the estimator is not sufficient on its own. The rational model estimator (RME for short). 3 can be used to remedy the situation.

However, for very complex structures, this method could be inefficient and computationally intensive. Should the in 3 If the algorithm shown turns out to be computationally intensive, the algorithm described in the publication “ Yan Pu, Jing Chen, Yongqing Yang, and Quanmin Zhu. Accelerated identification algorithms for rational models based on the vector transformation. In: Optimal Control Applications and Methods December 2021 (Jan. 2022), pp. 1-17. issn: 0143-2087 . doi: 10.1002/oca.2849. url: https://onlinelibrary.wiley.com/doi/10. 1002/oca.2849”, which may have faster convergence and is robust to estimates of initial parameters. This algorithm is “unbiased” and can be applied to high-scale systems and nonlinear systems.

wobei p_sys|n(k) und p_sys|d(k) die vektoriellen die Äquivalente von p_sys|nj(k) und P_sys|dj(k) sind. In diesem Fall ist ein e_sys2(k) Ausgangs-additiver Messfehler.If equation (G) is written in the form of equation (C), it will be possible using the algorithm of 3 to characterize the rotation rate sensor 1, that is, in particular to identify the parameters of the parametric model. By means of a few intermediate steps and transformations so that terms with Ω(k) are on the right-hand side of the equation, equation (C) can be obtained: $$\begin{array}{l}{\mathbf{p}}_{sys|n}\left(k\right)\leftarrow [\mathrm{\Omega }\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right),\mathrm{\Omega }\left(k-1\right)T\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right)T\left(k-9\right),\\ \mathrm{\Omega }\left(k-1\right){\omega }_i\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right){\omega }_i\left(k-9\right),\\ \mathrm{\Omega }\left(k-1\right){\omega }_i^2\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right){\omega }_i^2\left(k-9\right),\\ \mathrm{\Omega }\left(k-1\right)\left|{\omega }_i\left(k-1\right)\right|,\dots ,\mathrm{\Omega }\left(k-9\right)\left|{\omega }_i^2\left(k-9\right)\right|,\\ {\omega }_i\left(k\right),\dots ,{\omega }_i\left(k-9\right),T\left(k\right),\dots ,T\left(k-10\right),n_{\mathrm{\Omega }}^{\text{'}}\left(k-1\right),\dots ,n_{\mathrm{\Omega }}^{\text{'}}\left(k-9\right),\\ a_i\left(k\right),\dots ,a_i\left(k-9\right),a_i\left(k-9\right),a_i\left(k\right),\dots ,n_{\mathrm{\Omega }}^{\text{'}}\left(k-9\right)\mathrm{,1}]\\ {\mathbf{p}}_{sys|d}\left(k\right)\leftarrow \left[\mathrm{1,}T\left(k\right),{\omega }_i\left(k\right),{\omega }_i^2\left(k\right),\left|{\omega }_i\left(k\right)\right|\right]\\ e_{sys1}\left(k\right)=n_{\mathrm{\Omega }}^{\text{'}}\left(k\right)\\ md=10\end{array}$$

where p _sys|n (k) and p _sys|d (k) are the vectorial equivalents of p _sys|nj (k) and P _sys|dj (k). In this case there is an e _sys2 (k) output additive measurement error.

$$\mathrm{\Phi }\leftarrow {\left[{\mathrm{\Phi }}_0^T\cdots {\mathrm{\Phi }}_M^T\right]}^T$$

Regarding the all-encompassing regression analysis: With the method described above for carrying out a least squares procedure, a set of parameters of the given parametric model can be determined for a test point. In order to combine the respective advantageous characteristics of the different test points, the test data of the different test points are combined into a combined test data set and the parameters of the parametric model are finally determined on the basis of the combined test data set. For example, one test point has higher accuracy due to a better controlled environment, while other test points have broader variations in the reference quantities. For the purpose of combining these advantageous properties, the test data from the different test points are combined into a combined test data set. The regression analysis is carried out in one piece, i.e. a single regression analysis, on the combined test data set. For this purpose: $$Y\leftarrow {\left[Y_0^T\cdots Y_M^T\right]}^T$$

$$\mathrm{\Phi }\leftarrow {\left[{\mathrm{\Phi }}_0^T\cdots {\mathrm{\Phi }}_M^T\right]}^T$$

Where M indicates the number of different test points, for Y and Φ see above.

aus den Testpunkten s gezeigt: $$\text{par}=\frac{{\displaystyle \sum {}_{s=1}^M\frac{1}{{\sigma }_s^2}{\text{par}}_s}}{{\displaystyle \sum {}_{s=1}^M\frac{1}{{\sigma }_s^2}}},{\sigma }^2=M\frac{{\displaystyle \prod {}_{s=1}^M{\sigma }_s^2}}{{\displaystyle \sum {}_{s=1}^M{\sigma }_s^2}}$$

It can happen that the size of the combined test data set becomes very large, ie the combined test data set requires a considerable file storage size and leads to corresponding computational effort in the regression analysis. This means that the number of regressors becomes very large, which in turn leads to large matrices. Even if the regression analysis is carried out offline, memory can fill up and the numerical limits of the computer used can be reached. To avoid these problems, the combined test data set can be prepared by pre-processing in order to reduce the amount of data in the combined test data set, while the information of the data points from different tests and experiments is generally retained in the combined test data set. If this strategy is not possible, the following can be used: The ultimately obtained parameters of the parametric model can be determined individually for each test point and finally averaged. The averaging is preferably carried out by weighting with the inverse of the respective determined estimate of the variances, as explained above. In the following equation is an example of the final parameter estimate par and their respective variances σ ² for all parameters par _s and variances ${\sigma }_{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">s</figure-callout>}}^2$

shown from the test points s: $$\text{par}=\frac{{\displaystyle \sum {}_{s=1}^{\mathrm{<figure-callout\; id="1"\; label="M"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">M</figure-callout>}}\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="\sigma \; s"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">\sigma </figure-callout>}}_s^{}}{\text{par}}_s}}{{\displaystyle \sum {}_{s=1}^{\mathrm{<figure-callout\; id="1"\; label="M"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">M</figure-callout>}}\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="\sigma \; s"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">\sigma </figure-callout>}}_s^{}}}},{\mathrm{<figure-callout\; id="2"\; label="\sigma "\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">\sigma </figure-callout>}}^2=M\frac{{\displaystyle \prod {}_{s=1}^M{\sigma }_s^2}}{{\displaystyle \sum {}_{s=1}^{\mathrm{<figure-callout\; id="2"\; label="M\; \sigma \; s"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">M</figure-callout>}}{\sigma }_s^{}{}_2^{}}}$$

The parametric model with its parameters that will obtain these estimated parameters can then be compared with other models and their respective estimated parameters to determine the quality of the respective model.

Regarding model selection: If, instead of specifying a single parametric model with its special mathematical structure, a large number of possible parametric models of the tested yaw rate sensor with different mathematical model structures are specified, an all-encompassing regression analysis is carried out with the combined test data set to determine parameter values for all of the structurally specified parametric ones Models executed.

• - Error Reduction Ratio (ERR),
• - t-statistic,
• - Akaike Information Criterion (AIC), und
• - eine angepasste Schätzung der Kovarianz.

According to a predetermined metric, the most suitable model is selected and the tested yaw rate sensor is calibrated based on the selected parametric model with the associated determined parameter values. For this purpose, a combination of the following four metrics is applied:

• - Error Reduction Ratio (ERR),
• - t-statistic,
• - Akaike Information Criterion (AIC), and
• - an adjusted estimate of covariance.

wobei gilt: $$\widehat{g}=\left[{\widehat{g}}_1,\dots ,{\widehat{g}}_p\right]={\left(Q^{T}Q\right)}^{-1}{Q}^T\mathbf{Y}$$

$$Q=\left[{\mathbf{q}}_1,\dots ,{\mathbf{q}}_p\right]$$

$$\mathrm{\Phi }=QR$$

The ERR is a measure for each model parameter, which indicates the contribution or relevance of the respective model parameter to the output variable. Accordingly, a model parameter with a higher ERR is considered more important than a model parameter with a lower ERR. The ERR for a parameter j is: $${\left[\text{ERR}\right]}_j={\widehat{G}}_j^2\frac{{\mathbf{q}}_j^T{\mathbf{q}}_j}{{\mathbf{Y}}^T\mathbf{Y}},$$

where: $$\widehat{G}=\left[{\widehat{G}}_1,\dots ,{\widehat{G}}_p\right]={\left(Q^{T}Q\right)}^{-1}{Q}^T\mathbf{Y}$$

$$Q=\left[{\mathbf{<figure-callout\; id="1"\; label="q"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">q</figure-callout>}}_1,\dots ,{\mathbf{q}}_p\right]$$

$$\mathrm{\Phi }=QR$$

Furthermore, QR is the QR decomposition of the regression matrix Φ, while R indicates the upper triangular matrix with zeros on the diagonal, Q has the orthogonal columns q _j and Q ^T Q = D, where D is diagonal.

The AIC, on the other hand, attempts to address the so-called “bias-variance dilemma” by including the error variance, but penalizing the inclusion of additional parameters. Therefore, a model with a lower AIC value is statistically better than a model with a higher AIC value. Therefore: $$\text{AIC}\left(n_{\theta }\right)=N\text{ln}\left[{\sigma }_e^2\left(p\right)\right]+2p$$

ist die Residuenvarianz der Identifikationsdaten für das parametrische Modell mit p Parametern.Herein indicate: N the number of data, p the number of model parameters, and ${\sigma }_e^2\left(p\right)$

is the residual variance of the identification data for the parametric model with p parameters.

The “t-statistic” serves the purpose of identifying and removing irrelevant parameters of the parametric model. If a parameter has an influence on the output variable, its value must statistically deviate from zero. If this is not the case, the parametric model can be simplified by removing this parameter. Assuming that Θ̂ is normally distributed and has a covariance of Σ _Θ̂ calculated as above, the so-called Student's t-test can be carried out to test the null hypothesis as to whether θ̂ _j = 0 for j=1,...,p. The t-statistic for this hypothesis is: $$t_j=\frac{{\widehat{\theta }}_j}{\sqrt{{\sum }_{\widehat{\Theta },j}}}\sim t\left(N-p-1\right)$$

Here ∑ _Θ̂,j is the jth element on the diagonal of Σ _Θ̂ . Using t _j in the two-tailed cumulative Student t distribution with N - p - 1 degrees of freedom, it is possible to determine the confidence level necessary to reject the null hypothesis (θ̂ _j = 0).

Finally, if different parametric models with a given structure are compared, a data set is used specifically for verification. The combined test data set is therefore used in two parts, on the one hand for parameter identification using regression analysis and on the other hand for verification of the respective candidate parametric model. The goal is to select the parametric model with identified parameters that best fits the data set for verification. The covariance σ _θ̂ is the exact expected value of the difference between the actual parameters and the estimated ones. This metric is therefore particularly suitable for comparing the various fully identified models. For this purpose, the following metric is defined to compare the parametric models with identified parameters: $${\mathrm{\Delta }}_i={\displaystyle \sum _{s=1}^M\Vert {}_i^s{\sum }_{\widehat{\Theta }}\Vert 1.1}$$

erhalten wurde, M beschreibt die Zahl der verschiedenen Testpunkte um die Testdaten zu erhalten, und || • ||_1,1 ist die 1-Norm für eine Matrix, welche durch Summenbildung der Komponenten der Matrix ermittelt wird. Mit diesen Grundlagen wird folgender Algorithmus (Q) der Modellauswahl angewendet:

• Stelle bereit: ^sΩ(k), ^sp_sys|n(k), ^sp_sys|d(k);
  • 1) Trenne ^sΩ(k), ^sp_sys|n(k), ^sp_sys|d(k) in einen Identifikations-Datensatz und einen Verfikations-Datensatz;
  • 2) Für den Identifikations-Datensatz: Fasse alle ${}_{ID}^s\mathrm{\Omega }\left(k\right)$
    
    in einen Term _IDΩ(k), alle ${}_{ID}^s{\mathbf{p}}_{sys|n}\left(k\right)$
    
    in einen Term _IDp_sys|n(k), und alle ${}_{ID}^s{\mathbf{p}}_{sys|d}\left(k\right)$
    
    in einen Term _IDp_sys|d(k) zusammen, wenn möglich;
  • 3) wenn die allumfassende Regressionsanalyse möglich ist, dann:
  • 4) Schätze die Parameter Θ̂ des parametrischen Modells mithilfe der erweiterten Methode der kleinsten Quadrate aus 3 und dem Datensatz _IDΩ(k), _IDp_sys|n(k), _IDp_sys|d(k);
  • 5) Sortiere die Parameter θ̂_j nach ${\left[\text{ERR}\right]}_j={\widehat{g}}_j^2\frac{{\text{q}}_j^T{\text{q}}_j}{{\text{Y}}^T\text{Y}};$
    
  • 6) Beginne Schleife: Für alle j sortiert nach absteigender Reihenfolge in [ERR]_j tue bis zum Schleifenende:
  • 7) Modifiziere ${}_{ID}^s{\mathbf{p}}_{sys|n}\left(k\right){\text{ und }}_{ID}^s{\mathbf{p}}_{sys|d}\left(k\right),$
    
    sodass es nur Spalten mit Bezug zu den Parametern θ̂₁, ..., θ̂_j umfasst;
  • 8) Schätze die reduzierte Menge der Modellparameter θ̂_1,...,j durch die erweiterte Methode der kleinsten Quadrate nach 3 und dem Datensatz _IDΩ(k), _IDp_sys|n(k), _IDp_sys|d(k).
  • 9) Entferne alle störenden Parameter mittels des Studentschen t-Tests und aktualisiere Modell;
  • 10) Berechne AIC_j mittels $\text{AIC}\left(n_{\theta }\right)=N\text{ ln}\left[{\sigma }_e^2\left(p\right)\right]+2p;$
    
  • 11) Berechne Σ_Θ̂ mittels _IDΩ(k), _IDp_sys|n(k), _IDp_sys|d(k), dem identifizierten θ̂_1,...j und der erweiterten Methode der kleinsten Quadrate nach 3 zur Schätzung des zufälligen Fehlers (random error);
  • 12) Beende Schleife;
  • 13) Wähle das bestes Modell abhängig von AIC_j und ||∑_Θ̂||_1,1
  • 14) Andernfalls zu Schritt 3):
  • 15) Beginne Schleife: Für alle Testpunkte s tue:
  • 16) Schätze Modellparameter Θ̂^is mit den Schritten 4) und 13) und ${}_{ID}^s\mathrm{\Omega }\left(k\right),{}_{ID}^s{\text{p}}_{sys|n}\left(k\right),{}_{ID}^s{\text{p}}_{sys|d}\left(k\right);$
    
  • 17) Beende Schleife;
  • 18) Schätze Modellparameter Θ̂^iM+1 durch $\text{par}=\frac{{\displaystyle {\sum }_{s=1}^m\frac{1}{{\sigma }_s^2}{\text{par}}_s}}{{\displaystyle {\sum }_{s=1}^M\frac{1}{{\sigma }_s^2}}},$
    
    und ${\sigma }^2=M\frac{{\displaystyle {\prod }_{s=1}^M{\sigma }_s^2}}{{\displaystyle {\sum }_{s=1}^M{\sigma }_s^2}};$
    
  • 19) Beginne Schleife: Für alle Modelle i_s für s = 1, ..., M + 1, tue bis zum Schleifenende:
  • 20) Berechne ${}_{i_s}^s{\sum }_{\widehat{\Theta }}$
    
    mittels ${{}_V}_F^s\mathrm{\Omega }\left(k\right){,}_V{}_F^s{\text{p}}_{sys|n}\left(k\right){,}_V{}_F^s{\text{p}}_{sys|d}\left(k\right),$
    
    dem identifizierten θ̂_1,....j und der erweiterten Methode der kleinsten Quadrate nach 3 zur Schätzung des zufälligen Fehlers (random error);
  • 21) Berechne Δ_is ;
  • 22) Beende Schleife;
  • 23) Wähle bestes Modell aus ← argmin_x∈{i,is} (Δ_x);

Here Δ _i is the verification metric for model i, and s denotes the respective test for which ${}_i^s{\sum }_{\widehat{\Theta }}$

was obtained, M describes the number of different test points to obtain the test data, and || • || _1.1 is the 1-norm for a matrix, which is determined by summing the components of the matrix. With these basics, the following algorithm (Q) of model selection is applied:

• Provide: ^s Ω(k), ^s p _sys|n (k), ^s p _sys|d (k);
  • 1) Separate ^s Ω(k), ^s p _sys|n (k), ^s p _sys|d (k) into an identification data set and a verification data set;
  • 2) For the identification data set: Collect all ${}_{ID}^s\mathrm{\Omega }\left(k\right)$
    
    into a term _ID Ω(k), all ${}_{ID}^s{\mathbf{p}}_{sys|n}\left(k\right)$
    
    into a term _ID p _sys|n (k), and all ${}_{ID}^s{\mathbf{p}}_{sys|d}\left(k\right)$
    
    into a term _ID p _sys|d (k), if possible;
  • 3) if the all-encompassing regression analysis is possible, then:
  • 4) Estimate the parameters Θ̂ of the parametric model using the extended least squares method 3 and the data set _ID Ω(k), _ID p _sys|n (k), _ID p _sys|d (k);
  • 5) Sort the parameters θ̂ _j by ${\left[\text{ERR}\right]}_j={\widehat{G}}_j^2\frac{{\text{q}}_j^T{\text{q}}_j}{{\text{Y}}^T\text{Y}};$
    
  • 6) Start loop: For all j sorted in descending order in [ERR] _j do until the end of the loop:
  • 7) Modify ${}_{ID}^s{\mathbf{p}}_{sys|n}\left(k\right){\text{and}}_{ID}^s{\mathbf{p}}_{sys|d}\left(k\right),$
    
    so that it only includes columns related to the parameters θ̂ ₁ , ..., θ̂ _j ;
  • 8) Estimate the reduced set of model parameters θ̂ _1,...,j by the extended least squares method 3 and the data set _ID Ω(k), _ID p _sys|n (k), _ID p _sys|d (k).
  • 9) Remove all confounding parameters using Student's t-test and update model;
  • 10) Calculate AIC _j using $\text{AIC}\left(n_{\theta }\right)=N\text{ln}\left[{\sigma }_e^2\left(p\right)\right]+2p;$
    
  • 11) Calculate Σ _Θ̂ using _ID Ω(k), _ID p _sys|n (k), _ID p _sys|d (k), the identified θ̂ _1,...j and the extended least squares method 3 to estimate random error;
  • 12) End loop;
  • 13) Choose the best model depending on AIC _j and ||∑ _Θ̂ || _1.1
  • 14) Otherwise go to step 3):
  • 15) Start loop: For all test points do:
  • 16) Estimate model parameters Θ̂ ^{i s} with steps 4) and 13) and ${}_{ID}^s\mathrm{\Omega }\left(k\right),{}_{ID}^s{\text{p}}_{sys|n}\left(k\right),{}_{ID}^s{\text{p}}_{sys|d}\left(k\right);$
    
  • 17) End loop;
  • 18) Estimate model parameters Θ̂ ^{i M+1} through $\text{par}=\frac{{\displaystyle {\sum }_{s=1}^m\frac{1}{{\sigma }_s^2}{\text{par}}_s}}{{\displaystyle {\sum }_{s=1}^M\frac{1}{{\sigma }_s^2}}},$
    
    and ${\sigma }^2=M\frac{{\displaystyle {\prod }_{s=1}^M{\sigma }_s^2}}{{\displaystyle {\sum }_{s=1}^M{\sigma }_s^2}};$
    
  • 19) Start loop: For all models i _s for s = 1, ..., M + 1, do until the end of the loop:
  • 20) Calculate ${}_{i_s}^s{\sum }_{\widehat{\Theta }}$
    
    by means of ${{}_v}_F^s\mathrm{\Omega }\left(k\right){,}_v{}_F^s{\text{p}}_{sys|n}\left(k\right){,}_v{}_F^s{\text{p}}_{sys|d}\left(k\right),$
    
    the identified θ̂ _1,....j and the extended least squares method 3 to estimate random error;
  • 21) Calculate Δ _{i s} ;
  • 22) End loop;
  • 23) Select best model ← argmin _{x∈{i,i s }} ( _Δx );

• - Es wird für jede Messung des Drehratensensors die aktuell vorliegende Orientierung in Form von Lagewinkeln mit aufgezeichnet;
• - Die bekannte, durchschnittliche Rotationsrate ω_e der Erde wird auf die Achsen des Drehratensensors projiziert, sodass ω_e|i, ω_e|p und ω_e|o erhalten werden; und
• - Die Ergebnisse dieser Projektionen werden als bekannte Variablen zur Charakterisierung und letztendlich damit zur Kalibrierung des Drehratensensors verwendet, im Detail:

$${\omega }_i\left(k\right)\leftarrow {\omega }_i\left(k\right)+{\omega }_{e|i}\left(k\right)$$

$${\omega }_p\left(k\right)\leftarrow {\omega }_p\left(k\right)+{\omega }_{e|p}\left(k\right)$$

$${\omega }_o\left(k\right)\leftarrow {\omega }_o\left(k\right)+{\omega }_{e|o}\left(k\right)$$

To compensate for the rotation of the earth: The rotation rate sensor is fundamentally subject to the rotation rate of the earth compared to an inertial system. If this influence is not taken into account, the characterization of the yaw rate sensor will be incorrect. To compensate for the Earth's rotation rate, the following procedure is followed:

• - For each measurement of the yaw rate sensor, the current orientation is recorded in the form of position angles;
• - The known average rotation rate ω _e of the Earth is projected onto the axes of the rotation rate sensor, so that ω _e|i, ω _e|p and ω _e|o are obtained; and
• - The results of these projections are used as known variables to characterize and ultimately calibrate the yaw rate sensor, in detail:

$${\omega }_i\left(k\right)\leftarrow {\omega }_i\left(k\right)+{\omega }_{e|i}\left(k\right)$$

$${\omega }_p\left(k\right)\leftarrow {\omega }_p\left(k\right)+{\omega }_{e|p}\left(k\right)$$

$${\omega }_O\left(k\right)\leftarrow {\omega }_O\left(k\right)+{\omega }_{e|O}\left(k\right)$$

The steps described above remove the bias caused by the Earth's rotation rate including all nonlinearities and model effects.

• Es wird der Ausrichtungsfehler der Halterung zusammen mit dem Ausrichtungsfehler des Drehratensensors geschätzt. Dies führt zu einer entsprechend höheren Zahl von zu identifizierenden Parametern und zu zusätzlich benötigten Rechenschritten, um die Charakterisierung des Drehratensensors unabhängig von Ausrichtungsfehlern nach Gleichung (G) durchzuführen. Es sei F_E = {n_x, n_y, n_z} ein externes Referenzkoordinatensystem, in dem die Rotationsraten und Beschleunigungen bekannt sind. Die nominale Koordinatensystem- Transformation (auch genannt Richtungs-Kosinus-Matrix, abgekürzt DCM), ohne irgendwelche Ausrichtungsfehler, gibt die Koordinatensystem-Transformation aus dem System F_E nach F_C an und ist eine 3x3 Matrix ${\text{M}}_E^C,$
  
  sodass gilt:

$$\left[\begin{array}{c}{\text{n}}_{ira}\\ {\text{n}}_{xra}\\ {\text{n}}_{yra}\end{array}\right]={\text{M}}_E^C\left[\begin{array}{c}{\text{n}}_x\\ {\text{n}}_y\\ {\text{n}}_z\end{array}\right]$$

For orientation transformation: The algorithm of 3 carried out, the transformation from F _C to F _S is not known. In principle it is possible to assume the identity transformation, according to which n _i is exactly n _ira , and to accept the resulting errors. On the other hand, this transformation from an external reference system to F _C is known. However, alignment errors remain in any real test setup, adding uncertainties to this transformation matrix. If these uncertainties are appropriately controlled and thereby kept low, they can basically be ignored. If greater accuracy is required, each test operation in the device should be performed at least twice, in a first upright orientation and in a second orientation opposite to the first. This can lead to shortening or at least reducing the orientation transformation in the parameter identification. However, the number of test points naturally doubles, and depending on the holder, it can be difficult to test the yaw rate sensor in two opposite directions. The procedure is therefore suggested as follows:

• The alignment error of the holder is estimated together with the alignment error of the yaw rate sensor. This leads to a correspondingly higher number of parameters to be identified and to additional calculation steps required in order to carry out the characterization of the yaw rate sensor independently of alignment errors according to equation (G). Let F _E = {n _x , n _y , n _z } be an external reference coordinate system in which the rotation rates and accelerations are known. The nominal coordinate system transformation (also called direction cosine matrix, abbreviated DCM), without any alignment errors, gives the coordinate system transformation from the system F _E to F _C and is a 3x3 matrix ${\text{M}}_E^C,$
  
  so that:

$$\left[\begin{array}{c}{\text{n}}_{ira}\\ {\text{n}}_{xra}\\ {\text{n}}_{yra}\end{array}\right]={\text{M}}_E^C\left[\begin{array}{c}{\text{n}}_x\\ {\text{n}}_y\\ {\text{n}}_{\mathrm{e.g}}\end{array}\right]$$

von F_E nach F_S, wenn alle Ausrichtungsfehler berücksichtigt werden, gilt: $$\left[\begin{array}{c}{\text{n}}_i\\ {\text{n}}_p\\ {\text{n}}_o\end{array}\right]={\text{M}}_E^S\left[\begin{array}{c}{\text{n}}_x\\ {\text{n}}_y\\ {\text{n}}_z\end{array}\right]=\left[\begin{array}{ccc}{\alpha }_{ix}& {\alpha }_{iy}& {\alpha }_{iz}\\ {\alpha }_{px}& {\alpha }_{py}& {\alpha }_{pz}\\ {\alpha }_{ox}& {\alpha }_{oy}& {\alpha }_{oz}\end{array}\right]\left[\begin{array}{c}{\text{n}}_x\\ {\text{n}}_y\\ {\text{n}}_z\end{array}\right]$$

With the DCM Matrix ${\text{M}}_E^S$

from F _E to F _S , if all alignment errors are taken into account, then: $$\left[\begin{array}{c}{\text{n}}_i\\ {\text{n}}_p\\ {\text{n}}_O\end{array}\right]={\text{M}}_E^S\left[\begin{array}{c}{\text{n}}_x\\ {\text{n}}_y\\ {\text{n}}_{\mathrm{e.g}}\end{array}\right]=\left[\begin{array}{ccc}{\alpha }_{ix}& {\alpha }_{iy}& {\alpha }_{i\mathrm{e.g}}\\ {\alpha }_{px}& {\alpha }_{py}& {\alpha }_{p\mathrm{e.g}}\\ {\alpha }_{Ox}& {\alpha }_{Oy}& {\alpha }_{O\mathrm{e.g}}\end{array}\right]\left[\begin{array}{c}{\text{n}}_x\\ {\text{n}}_y\\ {\text{n}}_{\mathrm{e.g}}\end{array}\right]$$

worin ω_E = [ω_x, ω_y, ω_z]^T die Messungen der Rotationsraten im Koordinatensystem F_E sind. Die Beschleunigungen, die entlang dieser Achsen des Drehratensensors wirken, werden für die Achsen im Einzelnen erhalten zu: $$a_i\left(k\right)={\alpha }_{ix}{a}_x\left(k\right)+{\alpha }_{iy}{a}_y\left(k\right)+{\alpha }_{iz}{a}_z\left(k\right)$$

$$a_p\left(k\right)={\alpha }_{px}{a}_x\left(k\right)+{\alpha }_{py}{a}_y\left(k\right)+{\alpha }_{pz}{a}_z\left(k\right)$$

$$a_o\left(k\right)={\alpha }_{ox}{a}_x\left(k\right)+{\alpha }_{oy}{a}_y\left(k\right)+{\alpha }_{oz}{a}_z\left(k\right)$$

worin a_S = [a_i, a_p, a_o]^T die Beschleunigungsmessungen im Koordinatensystem F_S sind, a_E = [a_x, a_y, a_z]^T die Beschleunigungsmessungen im Koordinatensystem F_E; die Untermengen p_ωa|n(k) aus p_sys|n(k) und p_ωa|d(k) aus p_sys|d(k) sammeln lediglich die Regressionen mit Bezug zu den Rotationsraten und Beschleunigungen mit der Ausnahme der Regressoren für die absoluten Werte, sodass gilt: $$\begin{array}{l}{\text{p}}_{\omega a|n}\left(k\right)=[\mathrm{\Omega }\left(k-1\right){\omega }_i\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right){\omega }_i\left(k-9\right),\mathrm{\Omega }\left(k-1\right){\omega }_i^2\left(k-1\right),\dots ,\\ \mathrm{\Omega }\left(k-9\right){\omega }_i^2\left(k-9\right),{\omega }_i\left(k\right),\dots ,{\omega }_i\left(k-9\right),a_i\left(k\right),\dots ,\\ a_i\left(k-9\right),a_p\left(k\right),\dots ,a_p\left(k-9\right),a_o\left(k\right),\dots ,a_o\left(k-9\right)]\end{array}$$

$${\text{p}}_{\omega a|d}\left(k\right)=\left[{\omega }_i\left(k\right),{\omega }_i^2\left(k\right)\right]$$

The rotation rates that are detected in a respective axis of the rotation rate sensor can therefore be determined by: $${\omega }_i\left(k\right)={\alpha }_{ix}{\omega }_x\left(k\right)+{\alpha }_{iy}{\omega }_y\left(k\right)+{\alpha }_{i\mathrm{e.g}}{\omega }_{\mathrm{e.g}}\left(k\right)$$

where ω _E = [ω _x , ω _y , ω _z ] ^T are the measurements of the rotation rates in the coordinate system F _E. The accelerations that act along these axes of the yaw rate sensor are obtained for the axes in detail as: $$a_i\left(k\right)={\alpha }_{ix}{a}_x\left(k\right)+{\alpha }_{iy}{a}_y\left(k\right)+{\alpha }_{i\mathrm{e.g}}{a}_{\mathrm{e.g}}\left(k\right)$$

$$a_p\left(k\right)={\alpha }_{px}{a}_x\left(k\right)+{\alpha }_{py}{a}_y\left(k\right)+{\alpha }_{p\mathrm{e.g}}{a}_{\mathrm{e.g}}\left(k\right)$$

$$a_O\left(k\right)={\alpha }_{Ox}{a}_x\left(k\right)+{\alpha }_{Oy}{a}_y\left(k\right)+{\alpha }_{O\mathrm{e.g}}{a}_{\mathrm{e.g}}\left(k\right)$$

where a _S = [a _i , a _p , a _o ] ^T are the acceleration measurements in the coordinate system F _S , a _E = [a _x , a _y , a _z ] ^T are the acceleration measurements in the coordinate system F _E ; the subsets p _ωa|n (k) from p _sys|n (k) and p _ωa|d (k) from p _sys|d (k) collect only the regressions related to the rotation rates and accelerations with the exception of the regressors for the absolute values, so that: $$\begin{array}{l}{\text{p}}_{\omega a|n}\left(k\right)=[\mathrm{\Omega }\left(k-1\right){\omega }_i\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right){\omega }_i\left(k-9\right),\mathrm{\Omega }\left(k-1\right){\omega }_i^2\left(k-1\right),\dots ,\\ \mathrm{\Omega }\left(k-9\right){\omega }_i^2\left(k-9\right),{\omega }_i\left(k\right),\dots ,{\omega }_i\left(k-9\right),a_i\left(k\right),\dots ,\\ a_i\left(k-9\right),a_p\left(k\right),\dots ,a_p\left(k-9\right),a_O\left(k\right),\dots ,a_O\left(k-9\right)]\end{array}$$

$${\text{p}}_{\omega a|d}\left(k\right)=\left[{\omega }_i\left(k\right),{\omega }_i^2\left(k\right)\right]$$

$${\mathrm{\Theta }}_{\omega a|d}=\left[{\theta }_{\omega }^d,{\theta }_{\omega \omega }^d\right]$$

To simplify the notation, we also consider the parameter subsets Θ _ωa|n and Θ _ωa|d from Θ, which are directly related to the regressors in p _ωa|n (k) and p _ωa|d (k), so that: $$\begin{array}{l}{\mathrm{\Theta }}_{\omega a|n}=[{\theta }_{\mathrm{\Omega }\mathrm{<figure-callout\; id="1"\; label="\omega "\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">\omega </figure-callout>}}^1,\dots ,{\theta }_{\mathrm{\Omega }\omega }^9,{\theta }_{\mathrm{\Omega }\omega \omega }^1,\dots ,{\theta }_{\mathrm{\Omega }\omega \omega }^9,{\theta }_{{\omega }_i}^0,\dots ,{\theta }_{{\omega }_i}^9\\ {\theta }_{a_{\mathrm{<figure-callout\; id="0"\; label="i"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">i</figure-callout>}}}^0,\dots ,{\theta }_{a_i}^9,{\theta }_{a_{\mathrm{<figure-callout\; id="0"\; label="p"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">p</figure-callout>}}}^0,\dots ,{\theta }_{a_p}^9,{\theta }_{a_{\mathrm{<figure-callout\; id="0"\; label="O"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">O</figure-callout>}}}^0,\dots ,{\theta }_{a_O}^9]\end{array}$$

$${\mathrm{\Theta }}_{\omega a|d}=\left[{\theta }_{\omega }^d,{\theta }_{\omega \omega }^d\right]$$

dann können p_sys|n(k) und p_sys|d(k) modifiziert werden, sodass folgende Gleichung (L) erhalten wird: $$\begin{array}{c}{\text{p}}_{sys|n}^{EC}\left(k\right)\leftarrow [\mathrm{\Omega }\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right),\mathrm{\Omega }\left(k-1\right)T\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right)T\left(k-9\right),\\ \mathrm{\Omega }\left(k-1\right){\omega }_x\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right){\omega }_x\left(k-9\right),\\ \mathrm{\Omega }\left(k-1\right){\omega }_y\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right){\omega }_y\left(k-9\right),\\ \mathrm{\Omega }\left(k-1\right){\omega }_z\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right){\omega }_z\left(k-9\right),\\ \mathrm{\Omega }\left(k-1\right){\omega }_x^2\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right){\omega }_x^2\left(k-9\right),\\ \mathrm{\Omega }\left(k-1\right){\omega }_y^2\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right){\omega }_y^2\left(k-9\right),\\ \mathrm{\Omega }\left(k-1\right){\omega }_z^2\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right){\omega }_z^2\left(k-9\right),\\ \mathrm{\Omega }\left(k-1\right){\omega }_x\left(k-1\right),{\omega }_y\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right){\omega }_x\left(k-9\right){\omega }_y\left(k-9\right),\\ \mathrm{\Omega }\left(k-1\right){\omega }_x\left(k-1\right),{\omega }_z\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right){\omega }_x\left(k-9\right){\omega }_z\left(k-9\right),\\ \mathrm{\Omega }\left(k-1\right){\omega }_y\left(k-1\right),{\omega }_z\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right){\omega }_y\left(k-9\right){\omega }_z\left(k-9\right),\\ \mathrm{\Omega }\left(k-1\right)\left|{\omega }_i\left(k-1\right)\right|,\dots ,\mathrm{\Omega }\left(k-9\right)\left|{\omega }_i\left(k-9\right)\right|,\\ {\omega }_x\left(k\right),\dots ,{\omega }_x\left(k-9\right),{\omega }_y\left(k\right),\dots ,{\omega }_y\left(k-9\right),{\omega }_z\left(k\right),\dots ,{\omega }_z\left(k-9\right),\\ T\left(k\right),\dots ,T\left(k-10\right),n_{\mathrm{\Omega }}^{\text{'}}\left(k-1\right),\dots ,n_{\mathrm{\Omega }\left(k-9\right),}^{\text{'}}\\ a_{IRA}\left(k\right),\dots ,a_{IRA}\left(k-9\right),a_{XRA}\left(k\right),\dots ,a_{XRA}\left(k-9\right),a_{YRA}\left(k\right),\dots ,\\ a_{YRA}\left(k-9\right)\mathrm{,1}]\end{array}$$

$$\begin{array}{l}{\text{p}}_{sys|d}^{EC}\left(k\right)\leftarrow [\mathrm{1,}T\left(k\right),{\omega }_x\left(k\right),{\omega }_y\left(k\right),{\omega }_z\left(k\right),{\omega }_x^2\left(k\right),{\omega }_y^2\left(k\right),{\omega }_z^2\left(k\right),\\ {\omega }_x\left(k\right){\omega }_y\left(k\right)\mathrm{,,}{\omega }_x\left(k\right){\omega }_z\left(k\right),{\omega }_y\left(k\right){\omega }_z\left(k\right)\left|{\omega }_i\left(k\right)\right|]\end{array}$$

The measurement results of the rotation rates are used in the coordinate system F _E , and the measurement results of the accelerations are used in the coordinate system F _C , where applies ${\text{a}}_C=\left[a_{IRA},a_{XRA},a_{YRA}\right]={\text{M}}_E^C{\text{a}}_E,$

then p _sys|n (k) and p _sys|d (k) can be modified so that the following equation (L) is obtained: $$\begin{array}{c}{\text{p}}_{sys|n}^{EC}\left(k\right)\leftarrow [\mathrm{\Omega }\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right),\mathrm{\Omega }\left(k-1\right)T\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right)T\left(k-9\right),\\ \mathrm{\Omega }\left(k-1\right){\omega }_x\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right){\omega }_x\left(k-9\right),\\ \mathrm{\Omega }\left(k-1\right){\omega }_y\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right){\omega }_y\left(k-9\right),\\ \mathrm{\Omega }\left(k-1\right){\omega }_{\mathrm{e.g}}\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right){\omega }_{\mathrm{e.g}}\left(k-9\right),\\ \mathrm{\Omega }\left(k-1\right){\omega }_x^2\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right){\omega }_x^2\left(k-9\right),\\ \mathrm{\Omega }\left(k-1\right){\omega }_y^2\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right){\omega }_y^2\left(k-9\right),\\ \mathrm{\Omega }\left(k-1\right){\omega }_{\mathrm{e.g}}^2\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right){\omega }_{\mathrm{e.g}}^2\left(k-9\right),\\ \mathrm{\Omega }\left(k-1\right){\omega }_x\left(k-1\right),{\omega }_y\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right){\omega }_x\left(k-9\right){\omega }_y\left(k-9\right),\\ \mathrm{\Omega }\left(k-1\right){\omega }_x\left(k-1\right),{\omega }_{\mathrm{e.g}}\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right){\omega }_x\left(k-9\right){\omega }_{\mathrm{e.g}}\left(k-9\right),\\ \mathrm{\Omega }\left(k-1\right){\omega }_y\left(k-1\right),{\omega }_{\mathrm{e.g}}\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right){\omega }_y\left(k-9\right){\omega }_{\mathrm{e.g}}\left(k-9\right),\\ \mathrm{\Omega }\left(k-1\right)\left|{\omega }_i\left(k-1\right)\right|,\dots ,\mathrm{\Omega }\left(k-9\right)\left|{\omega }_i\left(k-9\right)\right|,\\ {\omega }_x\left(k\right),\dots ,{\omega }_x\left(k-9\right),{\omega }_y\left(k\right),\dots ,{\omega }_y\left(k-9\right),{\omega }_{\mathrm{e.g}}\left(k\right),\dots ,{\omega }_{\mathrm{e.g}}\left(k-9\right),\\ T\left(k\right),\dots ,T\left(k-10\right),n_{\mathrm{\Omega }}^{\text{'}}\left(k-1\right),\dots ,n_{\mathrm{\Omega }\left(k-9\right),}^{\text{'}}\\ a_{IRA}\left(k\right),\dots ,a_{IRA}\left(k-9\right),a_{XRA}\left(k\right),\dots ,a_{XRA}\left(k-9\right),a_{YRA}\left(k\right),\dots ,\\ a_{YRA}\left(k-9\right)\mathrm{,1}]\end{array}$$

$$\begin{array}{l}{\text{p}}_{sys|d}^{EC}\left(k\right)\leftarrow [\mathrm{1,}T\left(k\right),{\omega }_x\left(k\right),{\omega }_y\left(k\right),{\omega }_{\mathrm{e.g}}\left(k\right),{\omega }_x^2\left(k\right),{\omega }_y^2\left(k\right),{\omega }_{\mathrm{e.g}}^2\left(k\right),\\ {\omega }_x\left(k\right){\omega }_y\left(k\right)\mathrm{,,}{\omega }_x\left(k\right){\omega }_{\mathrm{e.g}}\left(k\right),{\omega }_y\left(k\right){\omega }_{\mathrm{e.g}}\left(k\right)\left|{\omega }_i\left(k\right)\right|]\end{array}$$

$${\widehat{\Theta }}_{\omega a|n}^{EC}=\left[{\widehat{K}}_{\omega x}^d,{\widehat{K}}_{\omega y}^d,{\widehat{K}}_{\omega z}^d,{\widehat{K}}_{\omega \omega x}^d,{\widehat{K}}_{\omega \omega y}^d,{\widehat{K}}_{\omega \omega z}^d,{\widehat{K}}_{\omega \omega xy}^d,{\widehat{K}}_{\omega \omega xz}^d,{\widehat{K}}_{\omega \omega yz}^d\right]$$

This shows that the number of regressors has increased. This results in the following equivalent parameter subsets after characterizing the above system: $$\begin{array}{c}{\widehat{\Theta }}_{\omega a|n}^{EC}=[{\widehat{K}}_{\mathrm{\Omega }\omega x}^1,\dots ,{\widehat{K}}_{\mathrm{\Omega }\omega x}^9,{\widehat{K}}_{\mathrm{\Omega }\omega y}^1,\dots ,{\widehat{K}}_{\mathrm{\Omega }\omega y}^9,{\widehat{K}}_{\mathrm{\Omega }\omega \mathrm{e.g}}^1,\dots ,{\widehat{K}}_{\mathrm{\Omega }\omega \mathrm{e.g}}^1,\\ {\widehat{K}}_{\mathrm{\Omega }\omega \omega x}^1,\dots ,{\widehat{K}}_{\mathrm{\Omega }\omega \omega x}^9,{\widehat{K}}_{\mathrm{\Omega }\omega \omega y}^1,\dots ,{\widehat{K}}_{\mathrm{\Omega }\omega \omega y}^9,{\widehat{K}}_{\mathrm{\Omega }\omega \omega \mathrm{e.g}}^1,\dots ,{\widehat{K}}_{\mathrm{\Omega }\omega \omega \mathrm{e.g}}^9\\ {\widehat{K}}_{\mathrm{\Omega }\omega \omega x\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">y</figure-callout>}}^1,\dots ,{\widehat{K}}_{\mathrm{\Omega }\omega \omega xy}^9,{\widehat{K}}_{\mathrm{\Omega }\omega \omega x\mathrm{e.g}}^1,\dots ,{\widehat{K}}_{\mathrm{\Omega }\omega \omega x\mathrm{e.g}}^9,{\widehat{K}}_{\mathrm{\Omega }\omega \omega y\mathrm{e.g}}^1,\dots ,{\widehat{K}}_{\mathrm{\Omega }\omega \omega y\mathrm{e.g}}^9\\ {\widehat{K}}_{\omega x}^0,\dots ,{\widehat{K}}_{\omega x}^9,{\widehat{K}}_{\omega y}^0,\dots ,{\widehat{K}}_{\omega y}^9,{\widehat{K}}_{\omega \mathrm{e.g}}^0,\dots ,{\widehat{K}}_{\omega \mathrm{e.g}}^0\\ {\widehat{K}}_{a\mathrm{<figure-callout\; id="0"\; label="I\; R\; A"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">I</figure-callout>}RA}^0,\dots ,{\widehat{K}}_{aIRA}^9,{\widehat{K}}_{a\mathrm{<figure-callout\; id="0"\; label="X\; R\; A"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">X</figure-callout>}RA}^0,\dots ,{\widehat{K}}_{aXRA}^9,{\widehat{K}}_{a\mathrm{<figure-callout\; id="0"\; label="Y\; R\; A"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">Y</figure-callout>}RA}^0,\dots ,{\widehat{K}}_{aYRA}^9\end{array}$$

$${\widehat{\Theta }}_{\omega a|n}^{EC}=\left[{\widehat{K}}_{\omega x}^d,{\widehat{K}}_{\omega y}^d,{\widehat{K}}_{\omega \mathrm{e.g}}^d,{\widehat{K}}_{\omega \omega x}^d,{\widehat{K}}_{\omega \omega y}^d,{\widehat{K}}_{\omega \omega \mathrm{e.g}}^d,{\widehat{K}}_{\omega \omega xy}^d,{\widehat{K}}_{\omega \omega x\mathrm{e.g}}^d,{\widehat{K}}_{\omega \omega y\mathrm{e.g}}^d\right]$$

$${\widehat{K}}_{\mathrm{\Omega }\omega y}^{l1}={\theta }_{\mathrm{\Omega }\omega }^{l1}{\alpha }_{iy}$$

$${\widehat{K}}_{\mathrm{\Omega }\omega z}^{l1}={\theta }_{\mathrm{\Omega }\omega }^{l1}{\alpha }_{iz}$$

$${\widehat{K}}_{\mathrm{\Omega }\omega \omega x}^{l1}={\theta }_{\mathrm{\Omega }\omega \omega }^{l1}{\alpha }_{ix}^2$$

$${\widehat{K}}_{\mathrm{\Omega }\omega \omega y}^{l1}={\theta }_{\mathrm{\Omega }\omega \omega }^{l1}{\alpha }_{iy}^2$$

$${\widehat{K}}_{\mathrm{\Omega }\omega \omega z}^{l1}={\theta }_{\mathrm{\Omega }\omega \omega }^{l1}{\alpha }_{iz}^2$$

$${\widehat{K}}_{\mathrm{\Omega }\omega \omega xy}^{l1}=2{\theta }_{\mathrm{\Omega }\omega \omega }^{l1}{\alpha }_{ix}{\alpha }_{iy}$$

$${\widehat{K}}_{\mathrm{\Omega }\omega \omega xz}^{l1}=2{\theta }_{\mathrm{\Omega }\omega \omega }^{l1}{\alpha }_{ix}{\alpha }_{iz}$$

$${\widehat{K}}_{\mathrm{\Omega }\omega \omega yz}^{l1}=2{\theta }_{\mathrm{\Omega }\omega \omega }^{l1}{\alpha }_{iy}{\alpha }_{iz}$$

$${\widehat{K}}_{\omega x}^{l0}={\theta }_{\omega i}^{l0}{\alpha }_{ix}$$

$${\widehat{K}}_{\omega y}^{l0}={\theta }_{\omega i}^{l0}{\alpha }_{iy}$$

$${\widehat{K}}_{\omega z}^{l0}={\theta }_{\omega i}^{l0}{\alpha }_{iz}$$

$${\widehat{K}}_{\omega x}^d={\theta }_{\omega }^d{\alpha }_{ix}$$

$${\widehat{K}}_{\omega y}^d={\theta }_{\omega }^d{\alpha }_{iy}$$

$${\widehat{K}}_{\omega z}^d={\theta }_{\omega }^d{\alpha }_{iz}$$

$${\widehat{K}}_{\omega \omega x}^d={\theta }_{\omega \omega }^d{\alpha }_{ix}^2$$

$${\widehat{K}}_{\omega \omega y}^d={\theta }_{\omega \omega }^d{\alpha }_{iy}^2$$

$${\widehat{K}}_{\omega \omega z}^d={\theta }_{\omega \omega }^d{\alpha }_{iz}^2$$

$${\widehat{K}}_{\omega \omega xy}^d={\theta }_{\omega \omega }^d{\alpha }_{ix}{\alpha }_{iy}$$

$${\widehat{K}}_{\omega \omega xz}^d={\theta }_{\omega \omega }^d{\alpha }_{ix}{\alpha }_{iz}$$

$${\widehat{K}}_{\omega \omega yz}^d={\theta }_{\omega \omega }^d{\alpha }_{iy}{\alpha }_{iz}$$

für l1 = 1, ..., 9 und l0 = 0, ..., 9. Selbst wenn berücksichtigt wird, dass n_i, n_p, n_o und n_x, n_y, n_z jeweils orthonormale Rechtshand-Koordinatensysteme bilden, ist das unmittelbar oben gezeigte System aus der Vielzahl der Gleichungen immer noch unterbestimmt und es gibt einen Unterraum von möglichen Orientierungen, die dem Gleichungssystem genügen. Eine der Möglichkeiten, dieses Problem zu lösen, ist im Algorithmus der 4 zur Ermittlung der Orientierungstransformation gezeigt, wobei die nominale DCM Matrix ${\text{M}}_E^C$

von F_E nach F_C wie oben beschrieben ist. Ferner ist in 4 im Schritt ‚9‘ R(n̂_R, δ_R) die Transformationsmatrix der Rotation um den Winkel δ_R um die n̂_R Achse.From equation (H) we get: $${\widehat{K}}_{\mathrm{\Omega }\omega x}^{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">l</figure-callout>}1}={\theta }_{\mathrm{\Omega }\omega }^{l1}{\alpha }_{ix}$$

$${\widehat{K}}_{\mathrm{\Omega }\omega y}^{l1}={\theta }_{\mathrm{\Omega }\omega }^{l1}{\alpha }_{iy}$$

$${\widehat{K}}_{\mathrm{\Omega }\omega \mathrm{e.g}}^{l1}={\theta }_{\mathrm{\Omega }\omega }^{l1}{\alpha }_{i\mathrm{e.g}}$$

$${\widehat{K}}_{\mathrm{\Omega }\omega \omega x}^{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">l</figure-callout>}1}={\theta }_{\mathrm{\Omega }\omega \omega }^{l1}{\alpha }_{ix}^2$$

$${\widehat{K}}_{\mathrm{\Omega }\omega \omega y}^{l1}={\theta }_{\mathrm{\Omega }\omega \omega }^{l1}{\alpha }_{iy}^2$$

$${\widehat{K}}_{\mathrm{\Omega }\omega \omega \mathrm{e.g}}^{l1}={\theta }_{\mathrm{\Omega }\omega \omega }^{l1}{\alpha }_{i\mathrm{e.g}}^2$$

$${\widehat{K}}_{\mathrm{\Omega }\omega \omega x\mathrm{<figure-callout\; id="1"\; label="y\; l"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">y</figure-callout>}}^l=2{\theta }_{\mathrm{\Omega }\omega \omega }^{l1}{\alpha }_{ix}{\alpha }_{iy}$$

$${\widehat{K}}_{\mathrm{\Omega }\omega \omega x\mathrm{e.g}}^{l1}=2{\theta }_{\mathrm{\Omega }\omega \omega }^{l1}{\alpha }_{ix}{\alpha }_{i\mathrm{e.g}}$$

$${\widehat{K}}_{\mathrm{\Omega }\omega \omega y\mathrm{e.g}}^{l1}=2{\theta }_{\mathrm{\Omega }\omega \omega }^{l1}{\alpha }_{iy}{\alpha }_{i\mathrm{e.g}}$$

$${\widehat{K}}_{\omega x}^{\mathrm{<figure-callout\; id="0"\; label="l"\; filenames="DE102022126969B3\_0140.png"\; state="\{\{state\}\}">l</figure-callout>}0}={\theta }_{\omega i}^{l0}{\alpha }_{ix}$$

$${\widehat{K}}_{\omega y}^{l0}={\theta }_{\omega i}^{l0}{\alpha }_{iy}$$

$${\widehat{K}}_{\omega \mathrm{e.g}}^{l0}={\theta }_{\omega i}^{l0}{\alpha }_{i\mathrm{e.g}}$$

$${\widehat{K}}_{\omega x}^d={\theta }_{\omega }^d{\alpha }_{ix}$$

$${\widehat{K}}_{\omega y}^d={\theta }_{\omega }^d{\alpha }_{iy}$$

$${\widehat{K}}_{\omega \mathrm{e.g}}^d={\theta }_{\omega }^d{\alpha }_{i\mathrm{e.g}}$$

$${\widehat{K}}_{\omega \omega x}^d={\theta }_{\omega \omega }^d{\alpha }_{ix}^2$$

$${\widehat{K}}_{\omega \omega y}^d={\theta }_{\omega \omega }^d{\alpha }_{iy}^2$$

$${\widehat{K}}_{\omega \omega \mathrm{e.g}}^d={\theta }_{\omega \omega }^d{\alpha }_{i\mathrm{e.g}}^2$$

$${\widehat{K}}_{\omega \omega xy}^d={\theta }_{\omega \omega }^d{\alpha }_{ix}{\alpha }_{iy}$$

$${\widehat{K}}_{\omega \omega x\mathrm{e.g}}^d={\theta }_{\omega \omega }^d{\alpha }_{ix}{\alpha }_{i\mathrm{e.g}}$$

$${\widehat{K}}_{\omega \omega y\mathrm{e.g}}^d={\theta }_{\omega \omega }^d{\alpha }_{iy}{\alpha }_{i\mathrm{e.g}}$$

for l1 = 1, ..., 9 and l0 = 0, ..., 9. Even if it is taken into account that n _i , n _p , n _o and n _x , n _y , n _z each form orthonormal right-hand coordinate systems , the system of the multitude of equations shown immediately above is still underdetermined and there is a subspace of possible orientations that satisfy the system of equations. One of the ways to solve this problem is in the algorithm 4 to determine the orientation transformation, where the nominal DCM matrix ${\text{M}}_E^C$

from F _E to F _C as described above. Furthermore, in 4 in step '9' R(n̂ _R , δ _R ) the transformation matrix of the rotation by the angle δ _R around the n̂ _R axis.

sodass gilt: $$\left[\begin{array}{c}{\text{n}}_{IRA}\\ {\text{n}}_{XRA}\\ {\text{n}}_{YRA}\end{array}\right]={\text{M}}_E^C\left[\begin{array}{c}{\text{n}}_i\\ {\text{n}}_p\\ {\text{n}}_o\end{array}\right]=\left[\begin{array}{ccc}{\alpha }_{Ii}& {\alpha }_{Ip}& {\alpha }_{Io}\\ {\alpha }_{Xi}& {\alpha }_{Xp}& {\alpha }_{Xo}\\ {\alpha }_{Yi}& {\alpha }_{Yp}& {\alpha }_{Yo}\end{array}\right]\left[\begin{array}{c}{\text{n}}_i\\ {\text{n}}_p\\ {\text{n}}_o\end{array}\right]$$

The DCM matrix for the coordinate system transformation from F _S to F _C is ${\text{M}}_S^C={\text{M}}_E^C{\left({\text{M}}_E^S\right)}^T,$

so that: $$\left[\begin{array}{c}{\text{n}}_{IRA}\\ {\text{n}}_{XRA}\\ {\text{n}}_{YRA}\end{array}\right]={\text{M}}_E^C\left[\begin{array}{c}{\text{n}}_i\\ {\text{n}}_p\\ {\text{n}}_O\end{array}\right]=\left[\begin{array}{ccc}{\alpha }_{Ii}& {\alpha }_{Ip}& {\alpha }_{IO}\\ {\alpha }_{Xi}& {\alpha }_{Xp}& {\alpha }_{XO}\\ {\alpha }_{Yi}& {\alpha }_{Yp}& {\alpha }_{YO}\end{array}\right]\left[\begin{array}{c}{\text{n}}_i\\ {\text{n}}_p\\ {\text{n}}_O\end{array}\right]$$

$$a_{XRA}\left(k\right)={\alpha }_{Xi}{a}_i\left(k\right)+{\alpha }_{Xp}{a}_p\left(k\right)+{\alpha }_{Xo}{a}_o\left(k\right)$$

$$a_{YRA}\left(k\right)={\alpha }_{Yi}{a}_i\left(k\right)+{\alpha }_{Yp}{a}_p\left(k\right)+{\alpha }_{Yo}{a}_o\left(k\right)$$

The relationship between the detected accelerations can therefore be given by: $$a_{IRA}\left(k\right)={\alpha }_{Ii}{a}_i\left(k\right)+{\alpha }_{Ip}{a}_p\left(k\right)+{\alpha }_{IO}{a}_O\left(k\right)$$

$$a_{XRA}\left(k\right)={\alpha }_{Xi}{a}_i\left(k\right)+{\alpha }_{Xp}{a}_p\left(k\right)+{\alpha }_{XO}{a}_O\left(k\right)$$

$$a_{YRA}\left(k\right)={\alpha }_{Yi}{a}_i\left(k\right)+{\alpha }_{Yp}{a}_p\left(k\right)+{\alpha }_{YO}{a}_O\left(k\right)$$

This allows the original acceleration-sensitive parameters to be determined with respect to the coordinate system F _S as described below for l = 0, ..., 9: $$\begin{array}{l}{\theta }_{a_i}^l={\widehat{K}}_{aIRA}^l{\alpha }_{Ii}+{\widehat{K}}_{aXRA}^l{\alpha }_{Xi}+{\widehat{K}}_{aYRA}^l{\alpha }_{Yi}\hfill \\ {\theta }_{a_p}^l={\widehat{K}}_{aIRA}^l{\alpha }_{Ip}+{\widehat{K}}_{aXRA}^l{\alpha }_{Xp}+{\widehat{K}}_{aYRA}^l{\alpha }_{Yp}\hfill \\ {\theta }_{a_O}^l={\widehat{K}}_{aIRA}^l{\alpha }_{IO}+{\widehat{K}}_{aXRA}^l{\alpha }_{XO}+{\widehat{K}}_{aYRA}^l{\alpha }_{YO}\hfill \end{array}$$

$${\text{p}}_{sys|d}^{SC}\left(k\right)\leftarrow \left[\mathrm{1,}T\left(k\right),{\omega }_i\left(k\right),{\omega }_i^2\left(k\right),\left|{\omega }_i\left(k\right)\right|\right]$$

For future reference, we hereby also introduce the case in which the rotation rate measurements are from the coordinate system F _S and the acceleration measurements are from the coordinate system F _C. Then p _sys|n (k) and p _sys|d (k) can be modified so that in the following equation (M): $$\begin{array}{c}{\text{p}}_{sys|n}^{SC}\left(k\right)\leftarrow [\mathrm{\Omega }\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right),\mathrm{\Omega }\left(k-1\right)T\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right)T\left(k-9\right),\\ \mathrm{\Omega }\left(k-1\right){\omega }_i\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right){\omega }_i\left(k-9\right),\\ \mathrm{\Omega }\left(k-1\right){\omega }_i^2\left(k-1\right),\dots ,\mathrm{\Omega }\left(k-9\right){\omega }_i^2\left(k-9\right),\\ \mathrm{\Omega }\left(k-1\right)\left|{\omega }_i\left(k-1\right)\right|,\dots ,\mathrm{\Omega }\left(k-9\right)\left|{\omega }_i\left(k-9\right)\right|\\ {\omega }_i\left(k\right),\dots ,{\omega }_i\left(k-9\right),T\left(k\right),\dots ,T\left(k-10\right),n_{\mathrm{\Omega }}^{\text{'}}\left(k-1\right),\dots ,n_{\mathrm{\Omega }}^{\text{'}}\left(k-9\right),\\ a_{IRA}\left(k\right),\dots ,a_{IRA}\left(k-9\right),a_{XRA}\left(k\right),\dots ,a_{XRA}\left(k-9\right),a_{YRA}\left(k\right),\dots ,\\ a_{YRA}\left(k-9\right)\mathrm{,1}]\end{array}$$

$${\text{p}}_{sys|d}^{SC}\left(k\right)\leftarrow \left[\mathrm{1,}T\left(k\right),{\omega }_i\left(k\right),{\omega }_i^2\left(k\right),\left|{\omega }_i\left(k\right)\right|\right]$$

The relationship to equation (J) can be easily established using transformation (K).

die nominale DCM Matrix von F_E nach F_C ohne Berücksichtigung eines Orientierungstransformation ist. Algorithmus (X) zur Ausführung des modifizierten rationalen Modell Schätzers (modifizierte RME Methode):

• Ausgehend von $\text{Y},{\omega }_b,{\text{a}}_b,{\text{M}}_E^C,\eta >\mathrm{0,}$
  
  führe aus:

The parameters of equation (C) and the alignment errors of equation (H) and the equations of the accelerations derived therefrom, which act along the axes of the yaw rate sensor, can be determined using the following algorithm, where ${\text{M}}_E^C$

is the nominal DCM matrix from F _E to F _C without considering an orientation transformation. Algorithm (X) for executing the modified rational model estimator (modified RME method):

• Starting from $\text{Y},{\omega }_b,{\text{a}}_b,{\text{M}}_E^C,\eta >\mathrm{0,}$
  
  execute:

Analogous to above, the covariance can be calculated by ${\sum }_{\widehat{\Theta }}={\sigma }_e^2{\left[{\mathrm{\Phi }}^T\mathrm{\Phi }\right]}^{-1}.$

• 1. Trenne Ω(k), ${\text{p}}_{sys|n}^E\left(k\right),{\text{p}}_{sys|d}^E\left(k\right)$
  
  in einen Identifikations-Datensatz und einen Verfikations-Datensatz;
• 2. Schätze die Sensor Parameter Θ̂^SC und den Ausrichtungsfehler
  
  mit Algorithmus (X) und dem Datensatz _IDΩ(k), ${}_{ID}{\text{p}}_{sys|n}^E\left(k\right){\text{ und }}_{ID}{\text{p}}_{sys|d}^E\left(k\right);$
  
• 3. Sortiere die Sensorparameter ${\widehat{\theta }}_j^{EC}$
  
  nach ${\left[\text{ERR}\right]}_j={\widehat{g}}_j^2\frac{{\text{q}}_j^T{\text{q}}_j}{{\text{Y}}^T\text{Y}}$
  
  und Φ^EC der vorhergehenden Schritte;
• 4. Beginne Schleife: Für alle j sortiert nach absteigender Reihenfolge in [ERR]_j tue bis zum Schleifenende:
• 5. Modifiziere ${}^s{\text{p}}_{sys|n}^{SC}\left(k\right){\text{ und }}^s{\text{p}}_{sys|d}^{SC}\left(k\right),$
  
  sodass sie nur Spalten mit Bezug zu den Parametern ${\widehat{\theta }}_1^{SC},\dots ,{\widehat{\theta }}_j^{SC}$
  
  umfassen. Wenn alle Sensorparameter in einer der Gleichungen (H) Null sind, entferne die äquivalente Spalte dieses Parameters von ${\text{P}}_{sys|n}^E\left(k\right)$
  
  und ${\text{P}}_{sys|d}^E\left(k\right)$
  
  in Algorithmus (X) im nächsten Schritt;
• 6. Schätze die reduzierte Menge der Modellparameter ${\widehat{\theta }}_{\mathrm{1,}\dots j}^{SC}$
  
  mit Algorithmus (X) und dem Datensatz _IDΩ(k), ${}_{ID}{\text{p}}_{sys|n}^{SC}\left(k\right){,}_{ID}{\text{p}}_{sys|d}^{SC}\left(k\right){,}_{ID}{\text{p}}_{sys|n}^E\left(k\right){\text{ und }}_{ID}{\text{p}}_{sys|d}^E\left(k\right)$
  
  mit den Modifikationen des vorhergehenden Schritts;
• 7. Entferne alle störenden Parameter mittels des Studentschen t-Tests und Φ^SC der vorhergehenden Schritte, und aktualisiere Modell;
• 8. Berechne AIC_j mit $\text{AIC}\left(n_{\theta }\right)=N\text{ ln}\left[{\sigma }_e^2\left(p\right)\right]+2p;$
  
• 9. Berechne Σ_Θ̂ mit _VFΩ(k), ${}_{VF}{\text{p}}_{sys|n}^{SC}\left(k\right){,}_{VF}{\text{p}}_{sys|n}^{SC}\left(k\right),$
  
  dem identifizierten ${\widehat{\theta }}_{\mathrm{1,}\dots j,}^{SC}$
  
  den geschätzten Beschleunigungen der Sensorachsen und Rotationsraten â_C(k) und ω̂_S(k), und 3 zur Schätzung des zufälligen Fehlers;
• 10. Beende Schleife;
• 11. Wähle bestes Modell abhängig von AIC_j und ||∑_Θ̂||_1,1;

Regarding the modified model selection strategy taking the orientation transformation into account: When estimating the orientation transformation and the sensor parameters at isolated test points (without the all-encompassing regression analysis), steps 4) to 13) of algorithm (Q) must be applied. However, minor modifications are necessary and Algorithm (Y) should be used to select the best model for this test while determining the alignment error. Algorithm (Y) for model selection strategy for isolated test points taking into account the estimation of the Orientie transformation:
Starting from Ω(k), ${\text{p}}_{sys|n}^E\left(k\right){\text{and p}}_{sys|d}^E\left(k\right):$

• 1. Separate Ω(k), ${\text{p}}_{sys|n}^E\left(k\right),{\text{p}}_{sys|d}^E\left(k\right)$
  
  into an identification data set and a verification data set;
• 2. Estimate the sensor parameters Θ̂ ^SC and the alignment error
  
  with algorithm (X) and the data set _ID Ω(k), ${}_{ID}{\text{p}}_{sys|n}^E\left(k\right){\text{and}}_{ID}{\text{p}}_{sys|d}^E\left(k\right);$
  
• 3. Sort the sensor parameters ${\widehat{\theta }}_j^{EC}$
  
  after ${\left[\text{ERR}\right]}_j={\widehat{G}}_j^2\frac{{\text{q}}_j^T{\text{q}}_j}{{\text{Y}}^T\text{Y}}$
  
  and Φ ^EC of the previous steps;
• 4. Start loop: For all j sorted in descending order in [ERR] _j do until the end of the loop:
• 5. Modify ${}^s{\text{p}}_{sys|n}^{SC}\left(k\right){\text{and}}^s{\text{p}}_{sys|d}^{SC}\left(k\right),$
  
  so that they only contain columns related to the parameters ${\widehat{\theta }}_1^{SC},\dots ,{\widehat{\theta }}_j^{SC}$
  
  include. If all sensor parameters in any of equations (H) are zero, remove the equivalent column of that parameter from ${\text{P}}_{sys|n}^E\left(k\right)$
  
  and ${\text{P}}_{sys|d}^E\left(k\right)$
  
  in Algorithm (X) in the next step;
• 6. Estimate the reduced set of model parameters ${\widehat{\theta }}_{\mathrm{1,}\dots j}^{SC}$
  
  with algorithm (X) and the data set _ID Ω(k), ${}_{ID}{\text{p}}_{sys|n}^{SC}\left(k\right){,}_{ID}{\text{p}}_{sys|d}^{SC}\left(k\right){,}_{ID}{\text{p}}_{sys|n}^E\left(k\right){\text{and}}_{ID}{\text{p}}_{sys|d}^E\left(k\right)$
  
  with the modifications of the previous step;
• 7. Remove all confounding parameters using Student's t-test and Φ ^SC from previous steps and update model;
• 8. Calculate AIC _j $\text{AIC}\left(n_{\theta }\right)=N\text{ln}\left[{\sigma }_e^2\left(p\right)\right]+2p;$
  
• 9. Calculate Σ _Θ̂ with _VF Ω(k), ${}_{vF}{\text{p}}_{sys|n}^{SC}\left(k\right){,}_{vF}{\text{p}}_{sys|n}^{SC}\left(k\right),$
  
  the identified one ${\widehat{\theta }}_{\mathrm{1,}\dots j,}^{SC}$
  
  the estimated accelerations of the sensor axes and rotation rates â _C (k) and ω̂ _S (k), and 3 to estimate the random error;
• 10. End loop;
• 11. Choose best model depending on AIC _j and ||∑ _Θ̂ || _1.1 ;

Although the invention has been illustrated and explained in detail by preferred embodiments, the invention is not limited by the examples disclosed and other variations may be derived therefrom by those skilled in the art without departing from the scope of the invention. It is therefore clear that a large number of possible variations exist. It is also to be understood that exemplary embodiments are truly examples only and should not be construed in any way as limiting the scope, application, or configuration of the invention. Rather, the preceding description and the description of the figures enable the person skilled in the art to concretely implement the exemplary embodiments, whereby the person skilled in the art can make a variety of changes with knowledge of the disclosed inventive concept, for example with regard to the function or the arrangement of individual elements mentioned in an exemplary embodiment, without departing from the scope of protection defined by the claims.

Reference symbol list

1

Rotation rate sensor

S1

Determine

S2

Treasure

S3

Determine

S4

Determine

S5

Summarize

S6

Pretend

S7

Carry out

S8

Calibrate

Claims (10)

Method for calibrating a rotation rate sensor (1), wherein the rotation rate sensor (1) has a sensor axis and a housing, comprising the steps: - Determining (S1) first test data by applying reference rotation rates in different orientations relative to the housing and with different amounts of the reference rotation rates per orientation to the rotation rate sensor (1), the first test data being the amounts of the reference rotation rates with the associated orientations relative to the housing and the nominal orientation of a coordinate system fixed to the housing relative to a coordinate system fixed to the body on a rotation table for applying the reference rotation rates, as well as a respective processable sensor signal of the rotation rate sensor (1) assigned to the reference rotation rates; - Estimating (S2), for the first test data, an alignment error between a coordinate system fixed to the sensor axis and the coordinate system fixed to the housing, - Determining (S3) second test data by applying vibrations in three orientations of the rotation rate sensor (1), the orientations being aligned in particular with the axes of the coordinate system fixed to the housing, the second test data containing the imposed vibrations and the respective orientation of the rotation rate sensor ( 1) and the respective processable sensor signal of the rotation rate sensor (1); - Determining (S4) a respective first orientation transformation based on the nominal orientations and based on the estimated alignment error of each of the first test data and applying the respective first orientation transformation to the first test data with the effect that all reference rotation rates of the first test data are based on the coordinate system fixed to the sensor axis are related; - Combining (S5) all test data into a combined test data set; - Specifying (S6) a mathematical model structure of a parametric model of the tested rotation rate sensor (1); - Executing (S7) an all-encompassing regression analysis on the combined test data set to determine parameter values of the parametric model so that the all-encompassing regression analysis obtains all parameters of the parametric model based on all test data at once; and - Calibration (S8) of the tested rotation rate sensor (1) based on the parametric model with the determined parameter values; Procedure according to Claim 1 , further comprising one of the steps: - determining a respective second orientation transformation based on the second test data, and applying the respective second orientation transformation to the second test data with the effect that all vibrations of the second test data are related to the coordinate system fixed to the housing; or: - Determining a respective second orientation transformation based on the second test data and the estimates of the alignment error from the first test data, and applying the respective second orientation transformation to the second test data with the effect that all vibrations of the second test data are related to the coordinate system fixed to the sensor axis are; Method according to one of the preceding claims, wherein the different orientations of the reference rotation rates are such that they each generate excitations of the rotation rate sensor (1) in all axes of the coordinate system fixed to the housing. Method according to one of the preceding claims, wherein the parametric model comprises a NARMAX model. Method according to one of the preceding claims, wherein a plurality of possible parametric models of the tested rotation rate sensor (1) are specified with different mathematical model structures, a respective all-encompassing regression analysis being carried out for the combined test data set to determine all parameter values of the respective parametric model, according to The most suitable model is selected from a given metric and the tested rotation rate sensor (1) is calibrated on the basis of the selected parametric model with the associated determined parameter values. Method according to one of the preceding claims, wherein third test data is generated by the yaw rate sensor (1) being statically arranged on a seismically isolated platform, the third test data comprising a processable sensor signal of the yaw rate sensor (1) comprising a respective measurement noise, and wherein For the third test data, an alignment error is determined between the coordinate system fixed to the sensor axis and the coordinate system fixed to the housing. Procedure according to Claim 6 , wherein the rotation rate sensor (1) is brought into different static orientations relative to the seismically isolated platform, the third test data comprising the respective orientations of the rotation rate sensor (1). Method according to one of the preceding claims, wherein for the respective test data, in particular for the first test data and / or for the second test data and / or for the third test data, a compensation of the earth's rotation rate takes place in that the vector of the earth's rotation rate is applied to the axes of the Housing body-fixed coordinate system is projected, and the result of the projections and the respective orientation of the rotation rate sensor (1) is used as the respective component of the respective test data. Method according to one of the preceding claims, wherein different temperatures are applied to the rotation rate sensor (1) within the respective test data, in particular within the first test data and/or the second test data and/or the third test data, the detected temperatures being a further component of the respective test data can be used. Method according to one of the preceding claims, wherein a modified rational model estimator is used to determine a respective alignment error.

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