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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Advances in the development of micro-electromechanical systems (MEMS) have made possible the fabrication of cheap and small dimension accelerometers and gyroscopes, which are being used in many applications where the global positioning system (GPS) and the inertial navigation system (INS) integration is carried out,

Currently, many land vehicles are equipped with global positioning systems (GPS), which can give an acceptable positioning. However, GPS is affected by several errors (

Recently, new low-cost and small sized inertial sensors have been built, increasing the demand of the low-cost INS and expanding its use in several applications, where GPS/INS are blended. Nonetheless, low-cost inertial sensors are characterized by high noise and large uncertainties in their outputs, such as bias, scale factor and non-orthogonalities [

This paper is focused on the identification and modeling of the bias-drift stochastic error, applying the most used techniques currently available to analyze these random processes. Additionally, since these noises have both high-frequency noise (short-term) and low-frequency noise (long-term), it is necessary to minimize both of them in order to improve the accuracy of the INS. Wavelet de-noising has been used in similar works, because of its great effectiveness removing high-frequency noises, as is shown in [

This paper is organized as follows. Firstly, we begin with an introduction to the noises that are involved in a low-cost INS (micro-electromechanical systems (MEMS) grade) (Section 2). Secondly, the architecture employed to integrate INS and GPS data is described, as well as the state-space form of different error models (Section 3). Thirdly, the analysis of the underlying random processes that affect the inertial sensors is achieved by different techniques: autocorrelation, Allan variance (AV), power spectral density (PSD) and autoregressive processes (Section 4). Subsequently, the parameters of various stochastic models are obtained from the methods presented in the previous section by using experimental data collected in the laboratory (Section 5). This section also explains the combination between wavelet de-noising and autoregressive (AR) models with different orders, and the combination between AV and wavelet de-noising techniques using different levels of decomposition. Finally, the models that are identified using AV and PSD, wavelet de-noising/AR models and the proposed method based on wavelet de-nosing/AV are adapted to the loosely-coupled integration, assessed using real data collected in urban roadways and compared (Sections 6 and 7).

The strapdown inertial navigation system (INS) involves mechanization equations, which are the numerical tool to implement the physical phenomenon that relates the inertial sensor measurements to the navigation state (

The inertial measurement unit (IMU), which is part of the INS, is the device where the inertial sensors are mounted; it provides the accelerations and angular rotations along three orthogonal directions with respect to the body frame (

Deterministic errors are due to manufacturing and mounting defects and can be calibrated out from the data; on the other hand, the stochastic errors are the random errors that occur due to random variations of bias or scale factor over time [

An additional factor that also affects the inertial sensors based on MEMS technology is the temperature. However, it will not be covered in this work. For further details of temperature dependence of the stochastic error and the different errors that affect the MEMS sensors, refer to [

The deterministic errors can be minimized before implementing the mechanization equations by following different procedures through laboratory calibrations (see [

The next section will present some of the stochastic processes that are usually used to model the bias-drift that affects the INS and their state-space representation. We will also explain the loosely-coupled architecture that is addressed to integrate INS and GPS data.

It is common to blend GPS and INS using different integration approaches (

On the other hand, the feed-back includes a close-loop that allows us to correct the INS error, where in the case of a GPS outage, the navigation solution will depend only on the INS, which will be corrected by its correspondent inertial sensor error model. The block diagram of the GPS/INS integration with feedback is shown in

In this strategy, the position and velocity obtained from the mechanization
^{n}^{n}

The system model for loosely-coupled approach is given by position error, velocity error and attitude error, which represent the navigation error states,

The next section will describe the stochastic processes that will augment the state-space model with the IMU error states associated to the inertial sensor bias-drift. The corresponding stochastic model for each error state will be selected in Section 6, after having analyzed the inertial sensors data with the techniques that will be explained in Section 4. For further details about the navigation error states for the loosely-coupled integration, refer to [

Various stochastic processes are well detailed in [

The bias-drift models adapted into the LC approach are the first order Gauss-Markov, random walk and autoregressive processes, which are a generalized representation of the first mentioned one.

First order Gauss-Markov (GM): This process has been widely used for modeling random errors, not only because it is able to represent a large number of physical processes, but also because it has a relatively simple mathematical description [_{c}_{k}_{c}_{c}

Random walk (RW): This process results when uncorrelated signals are integrated, e.g., when white noise is integrated during the mechanization stage. The continuous and discrete time of the RW are represented by:
_{k}_{k}_{+1} – _{k}_{RW}^{2}Δ

The noise covariance of the RW process can be obtained from power spectral density or Allan variance analysis, which will be described in Sections 4.3 and 4.4, respectively. This process is used to represent rate/acceleration random walk (K).

A typical bias-drift of a inertial sensor can be represented by a combination of different random processes, such as white noise (WN), RW and first order GM processes. These processes can be added into the KF by writing them in a state-space model. According to the previous definitions, a random process that combines WN, RW and first order GM can be generated using the following discrete time-invariant state-space model:
_{WN}_{k}^{st}GM

Autoregressive (AR) process: An AR process is a time series produced by linear combination of past values, which can be described by the following linear equation [_{0}; _{k}

In order to include the AR process in the EKF transition matrix, it is necessary to express

This represents the AR model in state-space for one of the inertial sensors. It should be noted that if the order of the AR model increases by one, the variables in the state vector of the Kalman filter will increase by six, since this model is applied to each axis of inertial sensors.

The stochastic processes that are used to model the inertial sensors bias-drift are augmented into the Kalman filter, as was explained in this section. In order to obtain the parameters of each stochastic process, an analysis of the sensors data needs to be done. The methods addressed to get these parameters are discussed in Section 4, and the experimental analysis of each method is presented in Section 5.

The stochastic modeling of the inertial sensors is a challenging task that in most practical cases, is performed by tuning the GPS/INS Extended Kalman Filter, which is often sensitive and difficult, by using sensors available specifications, but low-cost sensors do not provide enough information to develop this sort of models, or by experience [

The autocorrelation function has been used in previous works to analyze the stochastic error of the inertial sensors [

For a random process, _{c}_{c}^{2} is the variance of the process at zero time lag (_{xx}_{xx}

One of the limitations of this method is that an accurate autocorrelation curve from experimental data is rarely done, due to the fact that the data collected is limited and finite. As it is discussed in [

In [

To avoid the problem of inaccurate modeling of inertial sensor random errors, as in the case with the low-precise autocorrelation function, described in Section 4.1, another method, which was introduced in [

Although first order Gauss-Markov (GM) process has been very useful for modeling random errors of inertial sensors, better stochastic modeling can be achieved by modeling these errors as higher order AR models [

According to _{0}, _{k}_{1} = −1, it becomes a random walk (RW), and if _{1} = 0, it would be a white noise (WN). The coefficients of this process are estimated by Burg's method, since it overcomes some of the drawbacks of other methods by providing more stable models and improved estimates with shorter data records [

In this paper, we focus on AR models up to the third order, since a higher order would increase the computational load and might result in unstable solutions [

Power spectral density (PSD) is an important descriptor of a random process, because it provides information of the signal that is not easy to extract from the time domain.

The PSD is related to the autocorrelation function with:
_{x}_{xx}

Basically, the PSD is used to identify the stochastic errors of the inertial sensors from the frequency components, and the parameters obtained from the PSD are eventually used in the stochastic model of the INS.

Although it is not covered in this paper, it should be mentioned that recently, a more effective analysis in the frequency-domain has been presented by El-Diasty and Pagiatakis, where a GPS/INS impulse response model that is applied in the bridging GPS outages using as input the INS-only navigation solution is developed; for further details about this method, refer to [

So far, we have presented the autocorrelation, where the stochastic model parameters are extracted from the autocorrelation curve, the autoregressive processes that estimates the coefficients of an AR model applying Burg's method over the de-noised sensor data and the power spectral density that identifies the noise terms based on the slopes in a log-log PSD curve. The following section will describe the Allan variance technique, which is similar to the PSD, but in the time domain.

The Allan variance (AV) is a time domain analysis technique originally developed to study the frequency stability of oscillators [

The Allan variance is estimated as follows:

The basic idea to estimate the AV is to take a long sequence of data (

In AV, the uncertainty in the data is assumed to be generated by noise sources of specific character, as for instance, rate random walk, angle random walk, bias instability,

The AV obtained from _{x}

An interpretation of ^{4}(^{2}. This filter depends on

Computation of AV needs a finite number of clusters that can be generated from the raw data measurements of the sensors. Depending on the size of these clusters, AV can identify any noise term that is affecting the data sensor. It is important to mention that the estimation accuracy of the AV for a given

For example, if 360,000 data points are collected from an inertial sensor and if we want to compute the estimation accuracy of the AV for a bias instability (

The following section presents wavelet de-noising technique, which will be combined with autoregressive processes, as well as Allan variance.

The Discrete Wavelet Transform (DWT) is a widely used technique in digital signal processing, and one of its characteristics is that allows us to do a multiresolution analysis. Basically, when DWT is applied to a signal, _{0}(_{1}(_{0}(_{k}_{k}

Moreover, wavelet de-noising takes advantage of the sub-band decomposition performed by the DWT and removes the noise by eliminating the frequency components that are less relevant; in general, this procedure is called wavelet de-nosing and is well described in [

This technique is the current state-of-the-art technique used in the accuracy enhancement of inertial sensors [

Wavelet de-noising might be used to remove long-term noises (low-frequency) by increasing the level of decomposition that at the same time, increases the number of frequency bands that can be de-noised. However, in land-vehicle applications, these low-frequency components consist not only of long-term noises, but also vehicle motion dynamics. Since wavelet de-noising can be used to remove the high-frequency components and the AV method can be used to model the long-term noises without removing the vehicles motion, these two methods are combined in order to enhance the INS accuracy. The mixture between these two techniques is addressed in the following section, as well as the experimental analysis for each method explained.

In order to evaluate and compare the previous methods, the static data for analysis was obtained from the IMU 3DM-GX3-25 MEMS grade of MicroStrain (

The IMU was configured with a sampling frequency of 100 Hz, and the second moving average filter stage implemented in the microcontroller was adjusted with a filter width of 15; this means an attenuation of 14.16% at 20 Hz; for further details of this digital filter, which is embedded on the IMU, see [

The characteristics provided by the manufacturer can be seen in

After the seven hour-length data collecting, we used the autocorrelation method to achieve the analysis of the random errors that affect the accelerometers and gyroscopes of the IMU. Nevertheless, before processing the raw samples, we removed the turn-on bias for each sensor. Then, the high-frequency terms were attenuated by applying the wavelet de-noising technique. The idea in this step is to minimize the uncorrelated noise that is present in the sensors. Subsequently, the autocorrelation is calculated (_{c}

The same wavelet de-noising procedure was repeated to analyze the gyroscope's characteristics. The results are depicted in

In the case of inertial sensors based on MEMS technology, the assumption that the stochastic error follows a first order Gauss-Markov process is not valid in most of the situations. This can be visible by comparing

It is worth mentioning that the uncorrelated noise could be minimized by applying more levels of decomposition during the wavelet de-noising procedure, or a very high order autoregressive model could be used to create the model. However, the use of such a complex AR model in the integration filter would drastically increase the matrices sizes, as well as the computational burden. In addition, due to the fact that the autocorrelation has some other limitations (see Section 4.1), the method that will be analyzed in the following section is more appropriate to model higher order autoregressive processes.

Since the autocorrelation is a low-accurate technique to identify the noises affecting a low-cost INS, a method based on AR models have been used to overcome this issue (see [

For static drift data of the inertial sensors, the approximation part of the DWT includes the earth gravity, the earth rotation rate frequency components and the long-term error, while the detail part of the DWT contains the high-frequency noise and other disturbances [

By working with inertial data collected in a stationary condition, we first applied the wavelet de-noising technique, and then, the AR model coefficients were estimated with Burg's method. This procedure is executed for each sensor and for two AR models: first and third order. In this work, the attention is focused on these two models, because the first order AR models is one of the most used in the navigation field, and also up to the third order, because as it is explained by Nassar

The power spectral density was implemented using Welch's method, since this has been found to have the widest application in engineering and experimental physics [^{20} data points for the seven hours of the data collection. The results for the PSD are shown in

The values for each noise parameter (B,N,K) were extracted drawing straight lines for each frequency band influenced by the noise. The interception of each line with a specific point was taken into account. For instance, the PSD curve for the ^{−4} Hz and 2.29 × 10^{−3} Hz. This parameter is obtained by fitting a straight line with a slope of −2, starting from 1 × 10^{−4} Hz, until it meets the vertical line of

For details of the intercepts to determine the noise parameters, see [^{−3} Hz and 7.1 × 10^{−2} Hz, with a slope of −1, while the velocity random walk (N) is present between 0.1248 Hz and 20 Hz. After 20 Hz, there is an attenuation, because of the digital moving average filter, which is used to minimize high-frequency spectral noise produced by the MEMS sensors.

Regarding the gyroscopes,

For Allan variance analysis, the acceleration and the angular rate were integrated to obtain the instantaneous velocity and angle. Subsequently, the log-log plot of Allan variance standard deviation versus cluster times (

The values for each noise parameter were extracted as in the PSD, drawing straight lines for each error with its corresponding slope, but in this case, the interceptions are different. To clarify,

The Allan variance standard deviation

For the

For further details of the intercepts of each noise term in the log-log AV curve, see [

_{c}_{c}

This verifies the results that were obtained with PSD analysis, where velocity random walk (N), bias instability (B) and acceleration random walk (K) for accelerometers data and angle random walk (N) and bias instability (B) for gyro data were also identified. It can be seen that most of the estimated values in PSD (see

The next section presents the inertial sensor error model that mixtures of AV and wavelet de-noising techniques.

In order to combine wavelet de-noising (WD) and Allan variance under dynamic conditions, it is necessary to process the inertial sensors measures with wavelet de-noising before computing the mechanization (see _{s}_{1}) and a spectrum between 25–50 Hz for the details coefficients (_{1}), considering perfect filters. Therefore, the frequency band of the wavelet de-nosing output will be limited to _{s}^{k}_{k}_{k}

Thus, the test was achieved using the Matlab Wavelet Toolbox from three LOD, where the band of approximation coefficients is limited to 6.25 Hz (100/(2 × 2^{3}) = 6.25), up to eight levels of decomposition, where the output band is limited to 0.1953 Hz (100/(2 × 2^{8}) = 0.1953). This is taking into account that we use a sampling frequency of _{s}

If we consider these two cases—the first one applying wavelet de-noising with three LOD and the second one applying eight LOD (

Given that these motions of the vehicle are mixed with the long-term noises, a suitable LOD should be selected with the purpose of not removing relevant components that would compromise the performance of the navigation system. Therefore, to analyze the effect of wavelet de-nosing, we evaluated the enhancement accuracy of the GPS/INS solution with two vehicle tests, where a total of seven GPS outages were introduced under different dynamic conditions with a duration of 30 s and 60 s (see

The wavelet de-nosing parameters that provided the most significant enhance accuracy of the GPS/INS solution are summarized in

The use of Stein's Unbiased Risk Estimate (SURE) as a threshold rule helps us not to loose coefficients associated with the vehicle, since it is a conservative threshold that is usually used when small details of the signal lie in the noise range [

Having selected the LOD for wavelet de-noising, the long-term noises are modeled and compensated by the AV parameters obtained in Section 5.5. Overall, under dynamic conditions, wavelet de-noising will be computed for inertial sensor measurements prior to the INS mechanization (see

Having identified the random errors using AV and PSD, the parameters obtained with AV were used in the loosely-coupled GPS/INS integration scheme (_{se}

Regarding the 3DM-GX3-25 gyro stochastic error, _{se}_{c}

On the other hand, the AR coefficients obtained from Burg's method are adapted into the KF taking parameters that were shown in

As explained in Section 3, we use loosely-coupled integration with feed-back, which corrects the INS error through a close-loop. The INS error dynamics equations are built in the KF, having initially nine states for position, velocity and attitude error plus additional states to estimate the bias of each sensor of the IMU.

The EKF was adapted for each designed bias model in order to evaluate the accuracy of the stochastic processes that were obtained from the previous analysis. Firstly, the two models extracted from AV\PSD were implemented, so the vector error states of the Extended Kalman Filter were augmented with six and nine states, respectively. The latter error model was combined with wavelet de-noising in order to evaluate the enhancement accuracy when Allan variance parameters and wavelet de-noising techniques are blended together. Finally, two autoregressive models were assessed augmenting EKF with six and 18 states.

The EKF for the loosely-coupled integration has 15 states for two models: one is the model obtained with AV\PSD, where the bias instability (B) of both accelerometers and gyro are modeled with a first order Gauss-Markov process (GM) plus velocity\angle random walk (N), which is modeled as white noise (WN) for accelerometers and gyros, respectively. The second model with 15 states is a first order AR model. Although it is not depicted in

In order to assess the performance of the inertial sensor error models, a car was equipped with the 3DM-GX3-25 MEMS grade IMU, which was integrated with the Sat-Surf platform with a u-blox LEA-5X receiver [

Two data sets were collected in urban roadways inside the city of Turin, Italy. After the data collection campaign, the loosely-coupled integration architecture with the error models presented in this paper were evaluated. Although there were no GPS outages during the campaigns, we intentionally introduced several GPS outages off line, lasting 30 s and 60 s. During an outage, the system works in prediction mode only, and the accuracy of the loosely-coupled's performance relies entirely on the INS error model and, in particular, on the INS bias model. Therefore, it is straightforward to consider different outage lengths and different vehicle's dynamic conditions in order to have a clearer answer on the accuracy of the bias models under investigation. It is really worth mentioning that since these results are based on the loosely-coupled strategy, the simulated outages have complete GPS signal blockages. The GPS/INS solution without any outages was used as a reference to compare the performance of the different error models during the simulated GPS signal blockages.

The first trajectory that was used to asses the different sensor error models is shown in Google Earth map (

The easting and northing position for two of the three outages (outage 1 and outage 2) are presented in

During the first GPS outage (

To further validate the performance of the different stochastic error models, a second road test trajectory was collected in some urban roadways in the city of Turin; there is also a part of the path on a highway in the outskirts of the city. The road-test trajectory is 15.05 min long and is depicted in

Regarding GPS outage 5 (

In order to summarize the maximum and mean position error for both trajectories and each GPS outage performed,

Overall, the model based on AV and wavelet de-noising is the one that has provided the best accuracy in most of the cases under investigation. For instance, taking into account the results that are depicted in

We can also clearly appreciate how 18AV shows better results compared with 15AV in most situations where the GPS signal is not available (see

As far as the AR technique is concerned, the main objective of using AR models and wavelet de-noising is to remove the uncorrelated noise of the inertial sensors as much as possible. In fact, if we are able to remove the main quantity of the uncorrelated noise, we can then obtain a smooth autocorrelations curve, and the noise can be modeled with an higher order Gauss-Markov process (e.g., third order AR model), with a consequent benefit on the accuracy and performance of the GPS/INS system. Unfortunately, this is not the case of the low-cost inertial sensors (MEMS IMUs) we have used in this work, since, as is shown in Section 5.2, the autocorrelation function of some of the inertial sensors after processing the data with the de-noising technique does not have a smooth autocorrelation curve, which makes the estimation of the parameters less accurate compared to the parameters obtained with AV (

At last, the mixture between AV and wavelet de-noising has shown much better enhancement accuracy of the INS than the others methods presented in this work compensating for the short-term and long-term noises that affect the inertial sensors.

In this work, different stochastic error models for the measurement noise components of a MEMS-based IMU have been derived from experimental data and compared, specifically, autoregressive/wavelet de-noising models, Allan variance and Allan variance/wavelet de-noising. These stochastic models obtained from several techniques were adapted to the loosely-coupled strategy integration. Additionally, their performance was assessed in a low-cost navigation application by means of intentionally introducing several GPS outages in two trajectories collected in real urban roadways. The artificial GPS blockages were introduced in straight and curved portions of the trajectories comprising conditions of real GPS signal degradation inside a city.

Although AR processes combined with wavelet de-noising are commonly used for modeling INS stochastic errors, due to the fact that they have more modeling flexibility than first order Gauss-Markov, random walk and white noise processes, it is necessary to consider that the autocorrelation function of the IMU's raw measurements in static condition is expected to be a smooth curve (after de-noising), to use a low-order AR model, but this desired situation does not always apply for a low-cost inertial sensors (MEMs grade)

As was mentioned, the inertial sensors (MEMS grade) are affected not only by short-term noises, but also by long-term noises. Minimizing the latter is not an easy task, since these are combined with vehicle motion dynamics. In this work, we evaluated a error model that is a mixture of AV parameters and wavelet de-noising techniques (18AVWD); this model showed better performance than the other traditional methods based on AV and AR models during different GPS outages; specifically, with the 18AVWD model, we got a maximum horizontal error of 53.61 m with respect to 92.51 m (15AV), 96.32 m (18AV), 181.38 m (15AR) and 82.74 m (27AR). The 18AVWD stochastic error model uses the parameters obtained from AV to compensate for the long-term noises, while wavelet de-noising is employed to minimize the short-term noises that affect the inertial sensor of the IMU. Therefore, the wavelet de-noising technique has once again demonstrated its utility for removing the short-term noises of the inertial sensors. Nevertheless, other adaptive filtering techniques based on wavelet packet could be used in the future to get even better results, as the structure of decomposition of the sensor signal could be adapted according to the vehicle motion dynamics. It is also important to mention that, depending on the application, the selection of the decomposition level has to be carefully analyzed, due to the fact that frequency components that are associated with, e.g., vehicle motion dynamics may be eliminated after performing a de-noising technique.

It is well known that the AV technique presents drawbacks, such as: uncertainty of large clusters, so it requires large data sets to generate consistent AV curves [

We combined AV/Wavelet de-nosing and evaluated different levels of decomposition showing that although some vehicle motion components might be attenuated; we verified by simulation that the selected LOD provide more benefits concerning position accuracy.

Moreover, since we were dealing with a low-cost IMU, we noticed that it required many levels of decomposition to attenuate part of the uncorrelated noise and observe an enhancement in the position accuracy using wavelet de-nosing, which is not the case for high-end IMUs.

In the future, the error models analyzed in this paper could be adapted in more complex GPS/INS integration strategies, such as tightly-coupled, in order to enhance the position accuracy by using GPS estimates of pseudoranges and Doppler.

Support funded by the Spanish Ministry of Economy and Competitiveness under DELPHIS project: TEC 2009-09712.

The equations that represent the error dynamics in the n-frame for the loosely-coupled approach are given by position error, (^{n}), velocity error, (^{n}^{n}). The description of the transition matrix for these nine states is detailed in [^{n}^{n}^{n}

To include the bias of the inertial sensors (_{a}_{g}_{c}_{a}

The complete error states after adapting _{a}_{,}_{bi}_{g}_{,}_{bi}_{a}_{,}_{k}

The design matrix,

The noise covariance matrix Q of the model is:
_{a,n}_{a,bi}_{a}_{,}_{k}_{g}_{,}_{n}_{g}_{,}_{bi}_{ax}_{,}_{b}_{i}_{,} in discrete time for the first order GM process is given by [_{c}_{,}_{ax}_{ax}

The spectral density, _{ax}_{,}_{bi}_{ax}

The transition matrix in the discrete time used to augment the KF with a bias-drift modeled as a third order AR process for each inertial sensors can be described by

The complete error states of the KF will have 27 states, which are given by:
_{a}_{,}_{b}_{1}, _{a,b}_{2} and _{a,b}_{3} are the nine states associated to the third order AR models of the three accelerometer, while _{g,b}_{1}, _{g,b}_{2} and _{g,b}_{3} are the nine states associated to the third order AR models of the three gyros.

The design matrix,

In this case, the noise covariance matrix, _{a,n}_{g,n}_{a,b}_{g,b}_{0} is the standard deviation of the AR process.

The authors declare no conflict of interest.

Navigation frame inertial navigation system (INS) mechanization; figure kindly taken from [

Inertial sensor error modeling; figure kindly taken from [

Loosely-coupled Kalman Filter (KF) integration with feedback; figure kindly taken from [

The autocorrelation function of the first order Gauss-Markov process.

Hypothetical power spectral density (PSD) in single-sided form of an inertial sensor; PSD plot from the IEEE Std 1293-1998 [

Hypothetical Allan variance (AV) of an inertial sensor; AV plot from the IEEE Std 952-1997 [

Filter banks of the discrete wavelet transform.

Band frequency distribution after applying four levels of decomposition.

3DM-GX3-25 inertial measurement unit (IMU).

Power spectral density accelerometer Z IMU 3DM-GX-25.

IMU 3DM-GX3-25 Allan variance for three gyro axes.

Allan variance accelerometer Z IMU 3DM-GX-25 after applying wavelet de-noising with three and eight levels of decomposition.

Experimental setup mounted inside the test vehicle.

(

(

(

(

3DM-GX3-25 IMU characteristics. Acc, accelerometer.

Acc bias stability | ±0.5 g for ±5 g |

Acc nonlinearity | 0.2% |

Gyro bias stability | ±0.2°/s for ±360°/s |

Gyro repeatability | 0.2° |

Gyro nonlinearity | 0.2% |

Autoregressive process coefficients for each inertial sensor obtained with Burg's method after wavelet de-noising with six LOD.

_{1} |
_{2} |
_{3} |
_{0}^{2}(^{2})^{2} | |
---|---|---|---|---|

Acc X | −1 | 2.129 * 10^{−10} | ||

−2.582 | 2.166 | −0.585 | 4.973 * 10^{−13} | |

Acc Y | −1 | 1.148 * 10^{−9} | ||

−2.564 | 2.136 | −0.572 | 8.399 * 10^{−12} | |

Acc Z | −1 | 1.014 * 10^{−9} | ||

−2.564 | 2.136 | −0.572 | 7.964 * 10^{−12} | |

_{1} |
_{2} |
_{3} |
_{0}^{2}(^{2} | |

| ||||

Gyro X | −0.999 | 1.014 * 10^{−10} | ||

−2.564 | 2.131 | −0.567 | 2.739 * 10^{−13} | |

Gyro Y | −0.999 | 7.04 * 10^{−11} | ||

−2.565 | 2.133 | −0.568 | 1.905 * 10^{−13} | |

Gyro Z | −0.999 | 8.066 * 10^{−11} | ||

−2.562 | 2.128 | −0.566 | 2.181 * 10^{−13} |

Identified error coefficients for accelerometers and gyro of the 3DM-GX3 IMU with PSD.

^{3/2}) | |||
---|---|---|---|

Acc X | 0.045 | 4.6447 | 168.60 |

Acc Y | 0.044 | 4.6700 | 26.66 |

Acc Z | 0.047 | 1.7733 | 14.60 |

1–4 | ^{3/2}) | ||

| |||

Gyro X | 2.297 | 43.438 | |

Gyro Y | 1.937 | 39.614 | |

Gyro Z | 2.058 | 30.705 |

Identified error coefficients for accelerometers and gyro of the 3DM-GX3 IMU with AV.

^{3/2}) | |||
---|---|---|---|

Acc X | 0.045 ± 0.00023 | 5.1581 ± 0.0370 | 166.30 ± 4.6398 |

Acc Y | 0.045 ± 0.00022 | 4.5507 ± 0.0506 | 24.95 ± 2.8368 |

Acc Z | 0.047 ± 0.00050 | 1.8336 ± 0.0524 | 13.53 ± 1.8685 |

^{3/2}) | |||

| |||

Gyro X | 2.420 ± 0.0974 | 44.533 ± 5.14 | |

Gyro Y | 1.988 ± 0.0565 | 38.810 ± 2.51 | |

Gyro Z | 2.164 ± 0.0599 | 31.717 ± 2.29 |

Identified correlation time, (

Correlation time _{c} |
4.56 | 7.26 | 20.74 | 490.89 | 623.25 | 735.17 |

Standard deviation (^{2} − |
0.0068 | 0.0065 | 0.0063 | 0.0055 | 0.0045 | 0.0048 |

Maximum horizontal position error during GPS outages before and after applying wavelet de-noising.

| |||||
---|---|---|---|---|---|

| |||||

1* | 30 | 25.16 | 85.12 | 85.13 | 76.08 |

2* | 30 | 18.86 | 162.98 | 162.67 | 157.73 |

3* | 30 | 42.53 | 189.67 | 189.69 | 188.82 |

4* | 30 | 23.56 | 52.20 | 52.20 | 51.62 |

5* | 30 | 39.32 | 54.03 | 54.01 | 43.99 |

6* | 60 | 103.36 | 232.54 | 232.74 | 217.28 |

7* | 60 | 122.73 | 279.22 | 279.03 | 274.74 |

Wavelet de-noising parameters for each sensor under kinematic conditions.

_{k} |
|||
---|---|---|---|

Acc X | 7 | 0.39 | soft, SURE |

Acc Y | 6 | 0.78 | soft, SURE |

Acc Z | 6 | 0.78 | soft, SURE |

Gyro X | 7 | 0.39 | soft, SURE |

Gyro Y | 7 | 0.39 | soft, SURE |

Gyro Z | 7 | 0.39 | soft, SURE |

Number of states in the loosely coupled integration architecture for different error models.

AV\PSD _{se} |
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AV\PSD _{se} |
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AR _{se}_{se} |
1^{st} |
3^{rd} |

Maximum and mean horizontal position error during GPS outages for trajectory 1. 15AR, 15 state AR; 27AR, 27 state AR; 15AV, 15 state AV; 18AV, 18 state AV; 18AVWD, 18 state AV with wavelet de-noising

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1 | 30 | 23.58 | 17.11 | 21.84 | 16.82 | 21.60 | 12.56 | 18.42 | 20.74 | 38.72 | 19.62 | 23.94 |

2 | 60 | 42.31 | 14.41 | 44.84 | 14.10 | 38.04 | 13.16 | 34.81 | 42.82 | 122.07 | 29.70 | 71.49 |

3 | 30 | 47.60 | 13.34 | 49.97 | 11.86 | 40.41 | 11.80 | 39.69 | 17.48 | 64.26 | 19.02 | 70.21 |

Maximum and mean horizontal position error during GPS outages for trajectory 2.

4 | 30 | 33.60 | 10.85 | 22.70 | 10.44 | 21.13 | 7.29 | 10.81 | 6.72 | 14.72 | 9.99 | 29.26 |

5 | 30 | 52.25 | 44.57 | 64.46 | 44.36 | 64.17 | 36.50 | 53.61 | 44.34 | 65.28 | 44.43 | 64.65 |

6 | 30 | 20.44 | 2.74 | 6.19 | 2.71 | 5.91 | 2.25 | 5.32 | 4.12 | 13.17 | 4.18 | 13.41 |

7 | 60 | 112.42 | 39.40 | 92.51 | 40.85 | 96.32 | 8.74 | 17.20 | 49.62 | 181.38 | 32.27 | 82.74 |