Investigation of the trade-off between the complexity of the accelerometer bias model and the state estimation accuracy in INS/GNSS integration

The integration of Inertial Navigation Systems and Global Navigation Satellite Systems (GNSS) represents the core navigation unit for mobile platforms in open sky environments. A realistic assessment of the accuracy of the navigation solution depends on the accurate modelling of inertial sensor errors. Sensor noise and biases contribute most to short-term navigation errors. For the latter, different models can be used, varying in complexity. This paper investigates how the use of two different models for the accelerometer bias affects the accuracy of the state estimate in an extended Kalman filter. For this purpose, the Allan variance technique is applied to a data sequence from a specific inertial sensor to identify and quantify the underlying noise processes. The estimated noise parameters are used to characterise a bias model for the accelerometers that in addition to the static bias model takes non-white noise processes of the inertial sensor under investigation into account. This detailed accelerometer biasmodel is compared to a classical modelling approach that only considers static biases. Both approaches are evaluated based on simulation studies for continuous and intermittent GNSS coverages. The results show no significant difference between the two modelling approaches in terms of horizontal position and attitude precision. Furthermore, the correctness of the accelerometer bias estimates is not significantly affected by the modelling approach. All in all, it can be concluded *Corresponding author: Gilles Teodori, Department of Geodesy and Geoinformation, Research Division Engineering Geodesy, TU Wien, Wiedner Hauptstraße 8, 1040 Vienna, Austria, E-mail: gilles.teodori@tuwien.ac.at Hans Neuner, Department of Geodesy and Geoinformation, Research Division Engineering Geodesy, TU Wien, Wiedner Hauptstraße 8, 1040 Vienna, Austria, E-mail: hans.neuner@tuwien.ac.at that a detailed bias model of the accelerometers does not outperform the classical modelling approach.


Introduction
For outdoor operations, the majority of autonomous mobile robot platforms is equipped with Inertial Navigation Systems (INS) and Global Navigation Satellite Systems (GNSS). The fusion of both navigation systems builds the core navigation unit. The integrated navigation solution, consisting of attitude, velocity and position solutions, is used as the basis for vehicle control or georeferencing of point clouds, for example.

Motivation
In a loose INS/GNSS integration architecture, the GNSS velocity and position observations have the purpose to correct the INS navigation solution and to estimate the systematic Inertial Measurement Unit (IMU) errors (known as calibration) [1][2][3], of which the biases are the dominant ones. Apart from the systematic IMU errors, the INS navigation solution is also corrupted by stochastic IMU errors, which in contrast to the systematic errors, cannot be calibrated. In the simplest case, only the white noise of the IMU is considered. However, depending on the IMU grade and the runtime, further non-white noise processes can emerge. Accurate handling of the complete stochastic properties of inertial sensors is particularly important when the INS has to bridge GNSS outages, as the navigation solution relies entirely on the INS during GNSS outages. It is also necessary for a realistic assessment of the accuracy of the navigation solution.
To identify all noise processes, the stochastic properties of inertial sensors have to be analysed. For this purpose, various techniques have been suggested. They all have in common that a sufficiently long data sequence of an IMU at rest must be available. A first technique consists of analysing the IMU data sequence in the frequency domain [4]. Here, the power spectrum is computed by the discrete Fourier transform of the autocorrelation function [5]. By plotting the power spectrum of the data sequence as a function of frequency on a log-log scale graph, the noise processes present in the data sequence can be identified by their specific slopes on the graph. The counterpart to frequency domain analysis is time domain analysis. Here, autocorrelation analysis is a possible technique that can describe the stochastic properties of inertial sensors in conjunction with stationary Gauss-Markov processes [6,7]. According to [8] the stability of the estimated correlation times strongly depends on the recorded length of the data sequence. In the same work and in [9,10], higher order autoregressive processes are introduced as an alternative. It is demonstrated that their characteristics are in many respects better suited than those of the autocorrelation function of a firstorder Gauss-Markov process, for example. The disadvantage of using higher order autoregressive processes is that the matrix dimensions in the Kalman filter considerably increase, as each sensor axis must be separately modelled using several parameters. All techniques presented above require some sort of de-noising of the inertial sensor data [4,8]. This is because the inertial sensor data is dominated by high frequency noise, which inhibits accurate estimation of the noise parameters. The analysis technique that has gained acceptance in the field of inertial navigation is the Allan variance (AVAR) technique [11]. The success of this technique is reflected in the developed standards [12,13] and in the fact that IMU manufacturers have adopted this technique to specify the noise in their products. The AVAR technique was originally designed for characterising noise and stability of oscillators. It can also be applied to inertial sensor data to identify and quantify the underlying noise processes [14]. In the classical AVAR technique, the AVAR is simply defined as the mean-square error. The errors are the differences between successive time averages of the inertial data sampled over the cluster time. The square root of the AVAR can be plotted as a function of different cluster times on a log-log scale graph. Similar to the analysis in the frequency domain, the different noise processes are represented by specific slopes in the log-log scale graph.

Objectives
The investigations in this publication are based on a specific IMU that is currently used in combination with a mobile robot platform for research purposes at the Research Unit Engineering Geodesy of TU Wien. For this specific IMU, it is concretely investigated whether a detailed modelling of the stochastic properties of the accelerometers provides additional benefits in terms of a more realistic assessment of the accuracy of the state estimates. Here, the term accuracy is used to describe the correctness (i.e. deviation from the true value) and precision (i.e. standard deviation) of state estimates. Due to the IMU's properties (highgrade gyros and lower-grade accelerometers), the focus of the investigations is on the stochastic modelling of the accelerometers.
To derive the IMU's complete stochastic properties, an AVAR analysis is performed based on a recorded IMU data sequence, and the model parameters of the noise processes are estimated. This empirical approach is chosen because the available data sheet does not contain all the information needed for a complete stochastic modelling of the accelerometers.
Usually, the non-white noise processes are assigned to the biases (bias drift) and are thus taken into account by extending the bias model. For this purpose, a suitable state space model has to be introduced for the individual noise processes, which increases the complexity of the resulting INS/GNSS integration architecture. In order to investigate whether the added value of the more detailed accelerometer bias model justifies the increased model's complexity, a classical modelling approach that only considers static accelerometer biases is contrasted with a more detailed modelling approach that additionally takes non-white noise processes into account.
Thus, the following research questions are going to be investigated by means of simulated data with identical stochastic properties as the IMU under investigation: -How do the different bias models of the accelerometer translate into the state's precision depending on the GNSS observation availability? -How do the different bias models of the accelerometer affect the correctness of the accelerometer bias estimates?
This paper is organized as follows: In Section 2 the AVAR technique is used to analyse the stochastic properties of a tactical grade IMU. Furthermore, the model parameters of the individual noise processes are estimated. Section 3 presents the structure of the loose INS/GNSS integration architecture. The example of accelerometer biases is used to illustrate how the comprehensive stochastic properties of the accelerometers are incorporated into the loose INS/GNSS integration architecture. The simulation setup is outlined in Section 4 and Section 5 discusses the simulation results. Our conclusions are drawn in the final section and an outlook is given to future work.

Analysis of long-term IMU recordings
The aim of this section is to analyse the stochastic properties of the investigated IMU and to estimate the model parameters of the characterizing noise processes (noise parameters). A standard approach for analysing the stochastic properties of an IMU is to collect raw data over several hours, days or even weeks and apply the so-called AVAR technique to it. During the recording period, the IMU should not move and should be isolated from external influences (temperature fluctuations, vibrations, etc.) as best as possible. Under these conditions, the AVAR technique enables the identification and quantification of different noise processes in the IMU data.
This technique is also used in the present investigations. For this purpose, raw IMU data is recorded over 6 h at a frequency f = 200 Hz. The IMU used to collect the raw data consists of three fibre optic gyros and three servoaccelerometers. By using the AVAR technique (Sections 2.1 and 2.2), noise parameters are estimated (Section 2.3) and later used as basis for the simulation studies.

Overlapping Allan variance
In order to calculate the AVAR ( ) 2 from the recorded IMU data, the Python package AllanTools [15] is used. Specifically, the calculation was realized by the overlapping AVAR version, as this version achieves a higher confidence in the determination of the AVAR values than the classical AVAR calculation method. The overlapping AVAR takes into account all possible combinations of the dataset and not only the two adjacent samples as in the classical AVAR calculation method [16]. The overlapping AVAR is defined as [14] with L being the length of the dataset and the cluster time of duration mt 0 . m refers to the number of successive IMU observations within the cluster sampled at equidistant intervals t 0 . The input values i in Eq. (1) are obtained by integrating the originally recorded and detrended frequency data 1ũ (t) over time: 1 Given in units of m∕s 2 and rad∕s.
where T = Lt 0 refers to the total recording time. The Allan deviation (ADEV) is then given by the square root of the AVAR: ( ) = √ 2 ( ). Equation (1) is applied to a range of different cluster times i which are evenly spaced on a logscale (base 2). The resulting ADEV values can be plotted as a function of the cluster times on a log-log scale graph, where the different noise processes are represented by specific slopes in the graph.

Analysis of the acquired data
When analysing IMU data with regard to their stochastic properties, different types of noise must be distinguished. The terminology used by IMU manufacturers sometimes differs from that used in textbooks on stochastic processes. To avoid confusion, the IMU manufacturer's terminology is added in brackets immediately after the introduction of a noise process. Of all conceivable processes, only those that are relevant for the following investigations are considered: Quantization noise originates from the digitization and therefore, the quantization error depends on the bit resolution of the analogue-to-digital converter. White noise (or velocity/angular random walk) is visible as short-term variation of the sensor readings and is a high frequency noise like quantization noise. The two noise types differ in that quantization noise is uniformly distributed, whereas white noise is usually assumed to follow a Gaussian distribution. Flicker noise (or bias instability) is noticeable as a low-frequency drift of the sensors' readings over a bounded period of time. Acceleration/rate random walk manifests as random drift of the accelerometer readings over a very long period of time. White noise, flicker noise and random walk processes are related to instabilities of the sensors' electrical, electronic and mechanical components. These components can be for example the power supply unit, signal conditioning circuitry and suspension mechanism of the pendulum in accelerometers [3, p. 157 ff]. External influences are mostly excluded because of the recording criteria described above.
According to the AVAR theory [12], these different noise processes are represented by specific slopes when their corresponding ADEV is plotted on a log-log scale graph. Figure 1 shows the ADEV ( ), versus the cluster time in a log-log scale plot for the recorded IMU data. By comparing Figure 1(a) and (b), it is easily recognisable that the accelerometers and gyros have different stochastic properties. In particular, it is noticeable that the gyros are characterized by a large time stability, whereas the noise characteristics of the accelerometers change with time. Having a closer look at the accelerometer data, it can be seen that for cluster times around 10 −2 s, the corresponding curves follow a slope of −1, indicating the presence of quantization noise. In the region of cluster times of 10 0 − 10 1 s, the curves flatten out and reach gradients of −1∕2, implying a white noise process. At the lowest points, the curves reach slopes close to 0. In this region, the flicker noise dominates the stochastic behaviour of the accelerometers. For cluster times close to 10 3 s and above, the curves achieve gradients of +1∕2, denoting the emergence of a random walk process.
The flicker noise component of the z-axis accelerometer is slightly lower than that of the xand y-axis accelerometers. In contrast to the accelerometers, the gyros are only affected by white noise. The three curves of the x-, yand z-axis gyros are also very similar, so that the extracted white noise parameters are of similar order of magnitude.
In addition to identifying different noise processes, Figure 1(a) gives an indication of when it might be useful to include these noise processes in the Kalman filter. This information can be read off by looking at which cluster time range the individual noise processes are most dominant. As already mentioned, the flicker noise of the accelerometers is most dominant for cluster times near 10 2 s. If the INS integrates the IMU outputs over these ranges, e.g. due to GNSS outages, then this noise process might be important for a realistic prediction of the INS navigation solution and covariance.
In order to include the different noise processes into a Kalman filter formulation, a state-space model is needed for each noise process. For the sake of clarity, the models are presented as scalars in this section. The extension to three dimensional space is presented in Section 3. Based on the similarity of the ADEV values for each sensor, we further assume that all three sensors of each sensor type can be identically modelled. The overall state-space model for the stochastic accelerometer error z a (t) has the form where the subscripts A, N, B and K refer to the quantization noise, white noise, flicker noise and random walk processes, respectively. Equation (3) assumes that the noise processes are mutually independent [17]. The overall statespace model for the stochastic gyro error z g (t) is z g (t) = z g,N (t). (4)

Estimation of noise parameters
In order to estimate the noise parameters, suitable noise models must first be defined. The models presented in the following are applicable to both sensor types, which is why the subscripts a and g are omitted in these definitions.
In this work, the white noise term refers to a white Gaussian noise process [17,Eq. (21)] with a time-constant and frequency independent power spectral density (PSD) S z N ( f ) = S N = N 2 . The white noise coefficient N is usually specified by IMU manufacturers under the denotation of velocity/angular random walk. The reason is that white noise is integrated in the INS and thus leads to a random walk of the inertial velocity and attitude solution. However, this is not to be confused with the random walk process that is present in the raw accelerometer data. A random walk model is given by [17,Eq. (24)] where K (t) is assumed to be a white Gaussian noise process with time-constant and frequency independent PSD S K . Flicker noise cannot be represented by a finite-dimensional state-space model, because the corresponding transfer function is an irrational function [18]. Hence, it is common practice to approximate the flicker noise process z B (t) via a first-order Gauss-Markov process, which is defined by [ where B (t) is assumed to be a white Gaussian noise process with time-constant and frequency independent PSD S B , and T B denotes the correlation time. Quantization noise is uniformly distributed and therefore per definition not directly integrable into a Kalman filter formulation. Since a detailed discussion on how to incorporate this noise component is beyond the context of this paper, we restrict ourself to the estimation of the corresponding PSD S A .
In order to establish a relationship between the determined AVAR and the noise parameters to be estimated, the superposition principle is used to write the overall PSD of the stochastic accelerometer error specified in Eq. (3) by means of the sum of the PSD of the individual noise processes [17]: The interested reader finds the descriptions of the PSDs in the Appendix A. According to [12, Eq. (C.1)], it can be shown that the AVAR is related to the PSD as follows Thus, by integrating the PSDs of the individual noise processes according to Eq. (9), the overall AVAR model for the accelerometers can be derived [12, Eq. (C.21)], [17]: A detailed description of the integration of the individual noise processes can be found in [19, p. 84 ff]. The AVAR model of the gyros can be analogously derived and equals Having established the AVAR models, the noise parameters can be estimated. For this purpose, the parameter vec- is defined, allowing to rewrite Eq. (10) according to 2 a,z ( , ) = Based on Eqs.
where M represents the number of cluster times for which the AVAR was computed. Equation (13) slightly differs from the proposed cost function in [17] due to the logarithmic transformation of the AVAR values. Without this transformation, i.e. an adjustment in linear space, the estimated ADEV curve z a ( ,̂) strongly fits to the ADEV values at short cluster times and ignores much of the ADEV values for large cluster times. This is caused by the fact that for short cluster times, the ADEV values have a much higher order of magnitude than for large cluster times (approximately 10 −3 to 10 −5 ). As a result, the ADEV values at smaller cluster times have a larger weight in minimising the sum of squared deviations than the ADEV values at large cluster times. Even though this could be improved by introducing weights to the cost functions, the effort to manually adjust these weights is considered to be inefficient compared to the adjustment in the logarithmic space.
As mentioned above, one AVAR model is used each for the three accelerometers and for the three gyros. Due to the relatively simple AVAR model of the gyros compared to the accelerometers and for the sake of brevity, Figure 2 shows the result of the least squares adjustment using the x-axis accelerometer as an example. From a glance at Figure 2, we can clearly note that the chosen model structure for the accelerometers describes the ADEV values of the x-axis accelerometer sufficiently well. Including the ADEV models of each noise process in the same figure shows how at specific cluster times each noise process contributes to the overall ADEV. It is worth noting that the inclusion of the first-order Gauss-Markov process in the accelerometer model structure has the advantage that the wider flat part of the ADEV can be better represented than without it. Table 1 summarises the estimated noise parameters for each noise process of the x-axis accelerometer and gyro. Also listed are the manufacturer's specifications for white noise. From the values in Table 1 three conclusions can be drawn: At first, it can be seen that the estimated white noise parameters are within the range specified by the manufacturer. However, it also becomes clear that in addition to the white noise, three other noise processes are present in the accelerometer data. Finally, the magnitude of the sensors' noise parameters allow for making statements regarding the sensors' grade: Thus, it can be seen that the gyroscopes are high-grade sensors, whereas the accelerometers are of a lower grade. Accelerometers of a comparable grade can be e.g. found in IMUs costing less than 1000 euros.
The values given in Table 1 are used in Section 4 to replicate an IMU, the stochastic properties of which are identical to those of the IMU studied in this section, except for quantisation noise. Further details on the dashed orange and dotted red line in Figure 2 are also given in Section 5. The following section demonstrates how the identified noise processes, except for quantisation noise, are integrated into the loose INS/GNSS integration architecture.

Modelling of IMU errors in a loose INS/GNSS integration architecture
The loose INS/GNSS integration is the simplest integration architecture to combine both navigation systems. It consists of comparing the velocity and position solutions of whereb a andb g denote the estimated IMU biases. The superscript b denotes that the IMU observations are resolved about the body frame. The tilde quantities,̃, represent the raw IMU observations, which are modelled as follows [2, Eqs. (11.46) & (11.47) p. 391] where the quantities without any accents indicate the true counterparts of the quantities labelled witĥor̃. The additive terms z a,N and z g,N represent the white noise processes of the inertial sensors. The analysis of the accelerometer data in Section 2 showed that additional terms can be added in order to incorporate the complete stochastic properties (except of quantization noise). This is done by extending the accelerometer bias model by the first-order Gauss-Markov z a,B and acceleration random walk process z a,K : (16) with b a,0 and b g,0 describing constant, but unknown biases which vary from run-to-run. This variation is generally represented by a zero-mean Gaussian distribution with known initial variances 2 b g,0 and 2 b a,0 . In the classical modelling approach reported in many field specific textbooks, the IMU biases in Eq. (15) are modelled as static biases. 2 This model is referred to as the N model in Sections 4 and 5, because only accelerometer white noise ("N") is considered. The more detailed model is referred to as NBK model, as accelerometer white noise ("N"), flicker noise ("B") and random walk ("K") are considered. The differences between the classical and the detailed accelerometer bias model are indicated in 2 b a = b a,0 andḃ a = 0; b g = b g,0 andḃ g = 0.
Eq. (16) and in the succeeding equations of this section by the red colored terms, which represent the extensions of the detailed model compared to the classical one. These extensions only apply to the accelerometer biases, as the gyros are only affected by white noise as shown in Section 2.2. Apart from the temporal evolution of navigation errors, the system model of the EKF has to account for the uncalibrated portion of the IMU biases b = b −b that remain in the corrected IMU observations (see Eq. (14)). Thus, the system model has the following structure [1, p. 586 ff] (17) where n represents the attitude error defined as small angle error between the true and the estimated navigation frame n, which corresponds to the local level frame. The velocity and position errors are denoted by v n and r n , respectively. The rotation matrix C n b depicts the transformation from the body to the navigation frame. As the presentation of the navigation errors' dynamics (first three rows and columns of F) is very space consuming and well documented in [1, p. 586 ff], we limit ourselves here to the presentation of the system model structure. The adaptation of the system noise covariance matrix is done by extending the diagonal matrix containing the PSD of the system noise vector w [1, p. 99 ff] × × (18) where Δt is equal to the propagation interval and the transition matrix may be approximated by T k−1 ≈ e FΔt . The first order approximation used to derive the system noise covariance matrix shown in Eq. (18) mainly depends on the magnitude of the elements in the system matrix F and the length of the propagation interval. The system matrix is calculated using the navigation solution and the IMU observations. Therefore, the elements of the system matrix are dependent on the magnitude of the navigation solution, e.g. velocity and orientation of the platform, as well as the platform dynamics, e.g. acceleration and rotational motions. As a consequence, the use of Eq. (18) is reasonable and the approximation errors are negligible, as long as no high platform dynamics are involved and the propagation interval is sufficiently small. According to [1, p. 591], the system noise matrix could be further simplified if the propagation interval Δt is less than 0.2 s. It is further worth mentioning that in general the system noise covariance matrix Q k−1 is filled with non-zero elements due to the multiplication with the transition matrix T k−1 . The noise distribution matrix G k−1 is assumed to be constant over the propagation interval. The difference between GNSS and INS position and velocity solutions serves as input into the EKF [1, Eq. (14.101) p. 598]: , (19) where the subscripts I and G denote the INS and GNSS quantities, respectively. The vectors and r represent the GNSS observation noise. Equation (19) does not include a lever arm between both sensors, hence the measurement model yields and the corresponding measurement noise covariance matrix is

Simulation setup
In order to investigate the trade-off between the modelling of the stochastic accelerometer errors and the accuracy of the state estimation, a simulation study is conducted. A simulation study serves two purposes: Firstly, it can be used to make statements about the correctness of the estimates. Secondly, within the controlled environment of a simulation study, common influence encountered in real world data, such as vibrations caused by surface conditions, tyre treads, etc. or time-correlated GNSS observations, can initially be neglected. The simulation study consists of two related parts: the design of a motion scenario and the replication of sensor errors.

Motion scenario and replication of sensor errors
The motion scenario shown in Figure 3 is common in geodetic applications, but also in other disciplines like precise farming for example. The motion scenario includes a 5 min non-moving phase at the beginning, followed by a constant acceleration of 1 s towards the north to reach a speed of 1 m∕s. After an eight-section serpentine line, the simulated mobile platform decelerates within 1 s to come to a standstill. Immediately afterwards, the platform rotates 180 • and returns to the starting position along the same path in reverse order. Throughout the motion scenario, the platform is assumed to be levelled and to travel at a constant speed. In addition, the x-axis of the body frame coincides with the direction of travel (red arrows in Figure 3). The mobile platform is considered to be equipped with an IMU and a GNSS receiver, both located in the platform's centre of mass for simplicity. The true IMU and GNSS observations are determined from the simulated motion scenario. Sensor errors are added to these quantities to replicate a real-world application. For this purpose, the GNSS observation errors r and are drawn from a zero-mean Gaussian distribution with the following standard deviations: The stochastic IMU errors are generated based on the estimated noise parameters for the x-axis of the IMU investigated in Section 2 (see Table 1). Quantization errors are not generated as this type of error is not included in the EKF presented in Section 3. For convenience, the same parameters are used for each sensor axis. To verify whether the stochastic errors of the simulated accelerometers match the stochastic errors of the real x-axis accelerometer, the ADEV of the simulated errors is calculated and plotted against the ADEV of the real x-axis accelerometer (dashed orange and black line in Figure 2). The comparison of the two lines clearly shows that the simulation reproduces the real stochastic errors of the x-axis accelerometer well. The runto-run IMU biases b a,0 and b g,0 are drawn from zero-mean Gaussian distributions with standard deviations 1.5 mg and 0.75 deg ∕h, respectively. These values are taken from the data sheet of the IMU studied. After all IMU errors have been generated, they are added to the true IMU observations in order to obtain the final simulated observations. To ensure comparability, all of the simulation runs, with the exception of the Monte Carlo (MC) simulation, use the same simulation set-up. The MC simulation of this work includes 250 simulation runs, where each simulated data set was generated using the identical configuration as described above.

GNSS observation availability
The non-white processes become dominant only for longer cluster times (cf. Figure 2). Therefore, it is assumed that a modelling of these processes becomes meaningful, when GNSS outages occur and, consequently, the EKF and the INS have to predict the covariance matrix and navigation solution, respectively, over longer time spans. For this reason, the comparison between the N and NBK model (cf. Section 3) is conducted for two different cases of GNSS observation availability.
The first case includes a continuous GNSS observation coverage over the whole motion scenario with GNSS position and velocity solution every second. Over this time period, the dominant stochastic IMU errors are the white noises. Both, the N and the NBK model include a white noise term of similar PSD as will be detailed in Section 5. Therefore, no significant differences in the accuracy of the state estimates are expected between the results of the two models.
In the second case, a GNSS observation outage is introduced over a period of 300 s between the exit of turn 4 and shortly before turn 7 (see Figure 3). The outage period therefore includes straight lines and turns. Apart from the outage period, full GNSS coverage is assumed for the rest of the motion scenario. For outage periods of 300 s, the contribution of the first-order Gauss-Markov and random walk process to the stochastic accelerometer errors predominates, as shown in Figure 1(a). For this scenario, benefits in terms of state estimation accuracy are expected when using the NBK rather than the N model approach.

Simulation results
The simulated data is the basis for the comparison of the filtering results obtained with the different stochastic accelerometer models. As obvious from Section 3, the NBK model differs from the N model in the way the accelerometer bias model is extended to incorporate the non-white noise processes. Furthermore, the two approaches differ in the way to set up the white noise parameters: The detailed modelling approach uses the estimated white noise parameters, whereas in the classical approach, the manufacturer's specification of the accelerometer white noise is used to set up the system noise covariance matrix. Figure 2 shows that the ADEV of the manufacturer's white noise specification (red dotted line) is above the ADEV of the simulated stochastic accelerometer error (dashed orange line) for a large part of the cluster times. This reverses for cluster times of a few hundred seconds or more. In terms of stochastic accelerometer modelling, this means that the N model evaluates the simulated accelerometer data too pessimistically over a wide range of cluster times and too optimistically for cluster times over a few hundred seconds. The gyro noises for both approaches are kept identically.

Case 1: continuous availability of GNSS observations -precision analysis
First, we consider case 1, where GNSS observations are available at a rate of 1 Hz. Figure 4(a)-(d) presents the standard deviations of the estimated error states as a function of time. In an error state-space EKF, the state vector x describes a vector of error terms. Terms such as "position error, pitch angle error, etc.", which are used in the following sections, therefore refer here to the error states and are not to be confused with deviations from true values. In order to establish a relationship between the 2D trajectory in Figure 3 and the time representations, the rotational motions (e.g. turns) are indicated by grey hatching pattern. The translational motions (acceleration and deceleration phases) are indicated in the time representation by means of vertical black lines. As shown in Figure 4(a)-(d), there is no significant difference in the standard deviations of most error states between the N and NBK model. Smaller differences however, can be observed between the two stochastic accelerometer models in the standard deviations of the vertical position error, the pitch angle errors and the uncalibrated bias of the x-axis accelerometer. The differences in the pitch angle error and the uncalibrated bias of the x-axis accelerometer are locally limited to a short period of time. The differences in the vertical position error, in contrast, extend globally over the whole trajectory. As can be seen in the smaller zoomed windows of Figure 4(a), the standard deviation of the horizontal position errors increases faster than the standard deviation of the vertical position error during the prediction interval of the EKF. This is because more error states are coupled into the horizontal position errors than into the vertical component when the platform is leveled and not accelerating. Also, the errors coupled into the horizontal position errors are more difficult to observe than those coupled into the vertical position error. For example, the error that contributes most to the vertical position error is the vertical uncalibrated accelerometer bias [1, p. 574 ff]. The vertical position error, like all position and velocity errors, is by definition observable as it is measured by the GNSS sensor. Consequently, the vertical uncalibrated accelerometer bias is strongly observable and its standard deviation decreases rapidly during the initial non-moving phase of the simulated motion scenario (see Figure 4(c)). The estimated IMU biases are used to correct the raw IMU observations before the INS integration steps. Since the vertical accelerometer bias is well estimated and removed from the accelerometer observations, the vertical position error now increases mainly due to the integration of the accelerometer white noise. The lower accelerometer noise of the NBK model compared to the N model (8.1 μg∕ √ Hz to 50 μg∕ √ Hz) results in the NBK model achieving a lower standard deviation of the vertical position error than the N model.
In contrast to the vertical position error, several errors are simultaneously coupled into the horizontal position errors. The pitch and roll angle errors (so-called tilt angles) are constantly coupled with the horizontal position errors. Although the horizontal position errors are observable, the contribution from the tilt angle errors cannot be separated from the uncalibrated biases of the xy accelerometer axes. [1, p. 574 ff]. As depicted in Figure 4(b) and (c), the standard deviation of these four error states do not converge during the initial non-moving phase of the simulated trajectory (first 5 min). When the platform accelerates or rotates about the vertical axis (turns), significant improvements in the standard deviation of the tilt angle errors and the uncalibrated biases of the xy accelerometer axes are noticeable. Interestingly, these improvements have no impact on the rate of increase in the standard deviations of the horizontal position errors between GNSS observations, which remains unchanged throughout the scenario. Thus, there is evidence that accelerometer errors generally do not contribute much to horizontal position errors.
The uncalibrated biases of the xy gyro axes are coupled into the horizontal position errors, when the platform is levelled and not accelerating. Figure 4(d) shows slowly decreasing standard deviations for the uncalibrated biases of the xy gyro axes. However, this does not affect the rate of increase in the standard deviations of the horizontal position errors. Consequently, the increase in horizontal position errors is due to the remaining white noise of the gyros, which is modeled equally by the NBK and N models. Figure 4(b) and (d) indicate that the standard deviation of the yaw angle error and the uncalibrated bias of the z-axis gyro improve when the platform performs horizontal accelerations.
All in all, the results of case 1 confirm the expectations: Since only periods in which white noise dominates have to be bridged by the INS, the inclusion of further non-white noise processes has no significant influence on the precision of the states.

Case 2: outage of GNSS observations -precision analysis
In case 2, an outage of GNSS observations is simulated for a total duration of 300 s, during which the platform navigates through several turns and straight lines. For outage periods of 300 s, the contribution of the first-order Gauss-Markov and the random walk process to the stochastic accelerometer errors predominates, as shown in Figure 1(a).
For cluster times of several hundred seconds, the dashed orange line overtakes the dotted red line, which refer to the NBK and N model, respectively. From Figure 2, it can be concluded that the standard deviations of the error states for the NBK model would overtake those of the N model after a few hundred seconds if the stochastic errors of accelerometers significantly contribute to the states' precision.
This behaviour can be observed from the standard deviation of the vertical position error (see zoomed window in Figure 5(b)). However, this is neither observed for the standard deviation of the horizontal position nor the standard deviation of attitude errors (see Figure 5(a)). This is further evidence that the accelerometer errors do not contribute much to the horizontal position and attitude errors.
The analysis of Figure 5(c) corresponds with the two modelling approaches of the accelerometer biases. Unlike in the NBK model, the biases are modelled to be static and without a noise component in the N model. As a consequence, the standard deviation of the uncalibrated accelerometer biases for the N model remains unchanged during the GNSS outage period.
Furthermore, it is noticeable that no dependencies between the vehicle motion and the diagonal of the error covariance matrix are recognisable in the figures. Although there exist such a dependency in principle (cf. Eq. (18)), the simulated platform dynamics in this scenario (maximum speed of 1 m/s) are assumed to be too low.
From the results of case 2, it can be concluded that although the GNSS outage period is in the order of magnitude where the non-white noise processes dominate, the precision of the majority of the states is not considerably affected by the different stochastic accelerometer models. The only exception is the standard deviation of the position error in downwards direction. Here, a discrepancy of 0.4 m -which is large in relation to the overall magnitude of the total drift in height -is visible, with the N model estimating the precision too optimistic.

Case 2: outage of GNSS observations -MC simulations
As mentioned in the beginning of Section 5, the N model differs from the NBK model in that the accelerometer biases are modelled as static biases with no driving noise components. However, the simulated accelerometer biases are generated using the noise parameter of a real IMU. In the worst case, this circumstance can lead to a systematic deviation in the estimates. This eventuality is investigated by an MC simulation (see Section 4). Figure 6(a) and (b) shows the results of the MC simulation for the N and NBK models, respectively. The individual grey transparent lines represent the time progression of the deviation between the estimated and the true value of the accelerometer bias on the x-axis for each simulated run. The collection of the grey transparent lines is referred to as the ensemble. The time progression of the ensemble mean is shown as the green line. The red dotted line represents the threefold standard deviation of the (uncalibrated) accelerometer bias on the x-axis given by the EKF.
From a glance at Figure 6(a) and (b), the following conclusions can be drawn: First, the ensemble is within the 3 range of the EKF, indicating that the EKF provides a realistic assessment of the accelerometer bias precision. Second, the ensemble mean is close to zero over the whole scenario and indicates the correctness of the accelerometer bias estimates. Smaller deviations of the ensemble mean with respect to zero can be observed in the zoomed windows for example. However, as the discrepancies' magnitude (≈ 0.01mg) is small compared to the magnitude of the run-to-run bias variation (1 = 1.5 mg), these discrepancies are negligible. It is worth noting that the ensemble mean is zero in the initial non-moving phase of the platform (first 5 min), although the x-axis accelerometer bias is not observable in such phases. This effect can be explained by the fact that the run-to-run accelerometer biases are drawn from a zero-mean Gaussian distribution and the initial biases are set to zero. For this reason, the ensemble mean only becomes meaningful after the accelerometer bias becomes observable due to e.g. rotations around the vertical axis of the platform. Third, there are no significant differences between Figure 6(a) and (b) and, thus, between the results of the two modelling approaches across the entire scenario, including the GNSS outage period.
The results of the MC simulation for the other two accelerometer bias axes were examined analogously. The same statements could be derived from these results, which is why these results are omitted here. Thus, the MC simulations reveal that the correctness of the accelerometer bias estimates is not considerably affected by the choice of the stochastic accelerometer model or GNSS outages.

Conclusions and future work
The aim of this research was to investigate whether it is necessary to model in detail the stochastic accelerometer errors of a specific IMU, or whether it is sufficient to consider accelerometer white noise. For this purpose, the Allan variance technique was used on recorded IMU data to identify and quantify the underlying noise processes. The observed non-white noise processes were considered within a detailed bias model and included into the loose INS/GNSS integration architecture. By means of simulation studies the effect of different bias models of the accelerometer with respect to precision and correctness of state estimates were investigated. The simulation results reveal that the largest contribution to the precision of the navigation solution comes from the gyros, rather than from the accelerometers. Consequently, the first research question posed in Section 1.2 can be answered: For the precision of most error states in the EKF the modelling of accelerometer white noise is sufficient. Only for the height component and extended GNSS outage periods, detailed modelling leads to a more realistic precision assessment. The conduction of MC simulations additionally allows for answering the second research question posed in Section 1.2: The use of different bias models does not considerably affect the correctness of the accelerometer bias estimates.
One drawback of the AVAR technique used in this paper is that environmentally induced errors, such as vibrations of the platform, temperature changes, etc., are not taken into account. However, these types of error are frequently encountered in real-world applications. Research is planned to investigate the suitability of the noise parameters estimated with the AVAR technique for real-world applications.
The results also show that platform motions improve the estimation of some error states. Rotational motions about the vertical platform axis allow the tilt angle errors to be separated from the uncalibrated biases of the xy accelerometer axes. Another example is the yaw angle error that can be observed with horizontal acceleration changes. The dynamic dependence of observability of error states is known in the field of INS/GNSS integration [20,21]. Research is already in progress to solve the issue of observability of some error states in the preparation phase of a robotic mission.
Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission. Research funding: This work was carried out within the framework of the ZAP-ALS project funded by the Austrian Research Promotion Agency (FFG). ZAP-ALS is short for Zuverlässiger, Automatischer und Präziser: integrierte Schätzung von Trajektorien und Punktwolken aus GNSS, INS und ALS. The authors acknowledge TU Wien Bibliothek for financial support through its Open Access Funding Programme.

Conflict of interest statement:
The authors declare no conflicts of interest regarding this article.

Appendix A: Power spectral densities of noise processes
The power spectral density (PSD) of a random walk process defined by Eq. (6) is given by [2, Eq. (4.49) p. 127] where H(⋅) denotes the transfer function of Eq. (6) [17] and j refers to the imaginary unit. S K refers to the PSD of the driving noise K (t) of the random walk process z K (t).
Using the first line of Eq. (22) and the transfer function of Eq. (7) [17, p. 13], the PSD of a first-order Gauss-Markov process z B (t) results in S B refers to the PSD of the driving noise B (t) of the firstorder Gauss-Markov process z B (t).
The rate PSD of quantization noise' z A (t) is [12, Eq. (C.13)] where S A = A 2 and A denotes the quantization noise coefficient. t 0 refers to the fixed and uniform sampling interval of the IMU.