**1. Introduction**

During the last two decades, low cost, small size, accurate and reliable navigation system development is a hot research in the modern navigation technology. *The word navigation is a process of monitoring and controlling any moving object from one place to other. Inertial navigation system (INS) is a dead reckoning positioning method based on measurements and mathematical processing of the vehicle absolute acceleration and angular speed in order to estimate its attitude, speed and position related to difference* [1–5]. INS technology is categorized into (i) gimbal INS and (ii) strap-down INS. In the early 1940s, a gimbal INS system was developed based on the mechanical inertial sensor (i.e., accelerometers and gyroscopes) for providing the navigation information [5]. Its accuracy was limited by mechanical inertial sensor errors. The main drawbacks of the gimbal INS system are its designed complexity and it requires synchronous servo motors, slip rings, control electronics, etc., for acquiring the navigation information. Because of these factors, the gimbals INS systems are used in low grade navigation applications [6]. In the early 1950, strap-down INS (SINS) was developed based on solid state inertial sensor [5]. The SINS is a selfcontained navigation system that has been developed for providing the accurate navigation information (i.e., position, velocity and rotation information. It has three gyroscopes ad three accelerometers. In general, the operation principle of SINS follows the physical laws of motion equations. It is an emerging technology as compared to gimbal INS systems and it has significant features such as easy to design, lower cost of ownership, moderate manufacturing cost and also high reliability. SINS consist of an inertial measurement that includes 3-axis accelerometers and 3-axis gyroscopes, and a processing computer. IMU is a key device to the INS and has been widely used for measuring the rotation rate and acceleration of an object. In practice, SINS accuracy degrades due to internal and external errors of the inertial sensors. These errors are mainly caused due to fluctuation in temperature, pressure and internal electronics components of the sensor. Due to these factors, stochastic errors and drift errors are generated at the IMU output [7, 8].

wavelet transform (DWT), empirical mode decomposition (EMD) method and Forward linear prediction (FLP) methods have been developed and applied to MEMS sensors for filtering the high-frequency noise [28, 29]. These methods are not suitable when the sensor includes the correlated noise. Kalman filter (KF) is a most popular state estimation technique that has been used for minimizing the correlated noise of the MEMS sensor [30–34]. The priori knowledge of an initial values of the process and measurement noise covariance matrix are known exactly, when the KF become an optimal. However, in practice, these noise parameters may vary with time so that the performance of the KF can be degraded and then the

*Modeling of Inertial Rate Sensor Errors Using Autoregressive and Moving Average (ARMA)…*

To solve the divergence problems, Adaptive Kalman filter technique (AKF) is a better solution. The adaptation can be based on either (i) innovation based adaptive estimation AKF (IAE-AKF) or (ii) residual based estimation AKF (RAE-AKF) and also multiple model based AKF [34, 35]. Among the other methods, adaptive KF is developed using IAE. In general, an innovation sequence is defined as the difference between true and estimated values. In the IAE-AKF method, the measurement and process noise matrices are estimated based on innovation sequence and followed by sliding average window method. In real time, the selection of window size is a critical issue. Sage-Husa Adaptive KF is another version of adaptive KF that has been developed to improve the AKF performance by introducing a time varying estimator. In the SHAKF, using a time-varying noise estimator can be helpful in estimating the statistical characteristics of the uncertainty in the measurements in real time and mitigating the filter divergence. A further study on the SHAKF is developed based on adaptive factors for improving the filter practicability and

An adaptive fading Kalman filter (AFKF) was proposed for compensating the effect of the uncertainty in the measurements by transitive factor to the state error covariance (P). In AFKF, the state error covariance (P) is scaled with a single transitive factor for improving the filter variance and gain correction. When it is used for complex systems, the performance of AFKF degrades because of it may not be sufficient to use a single transitive factor for estimating the covariance matrix of the filter [24]. To overcome the difficulties of single transitive factor, multiple fading factors are used in AFKF. Because of that reason, authors are developed double transitive factor based SHAFKF that adapts both predicted state error covariance (P) and measurement noise covariance matrix (R) based on the innovation sequence. Although it has been successively applied to different domains, its performance for MEMS gyroscope sensor signal is not explored. The stochastic errors of MEMS gyroscope cannot be eliminated using calibration technique. It needs to be modeled before filtering the signal. Therefore, adaptive filtering techniques have been developed for minimizing the random noise from MEMS gyroscope system. In general, auto-regressive (AR), Moving Average (MA), and Auto-Regressive and Moving Average (ARMA) and Gauss-Markov model (GM) have been used for modeling stochastic signal [17]. Among these models, ARMA is a better choice for modeling MEMS gyroscope drift errors. In general, the ARMA modeling involved three steps as (i) randomness and stationary test (ii) selection of suitable time series model and (iii) estimation of model parameters. The unit root test and inverse sequence techniques have been used for checking the stationary of the signal. The model order is obtained by using auto correlation function (ACF) and partial auto correlation function (PACF). Moreover, Akaike Information Criterion also used to check the model order. The modified Yule-Walker method is used estimate the model parameters. Once an optimal ARMA model is defined, a suitable adaptive Kalman filter can be applied to minimize the drift of inertial sensors

filter become diverge.

*DOI: http://dx.doi.org/10.5772/intechopen.86735*

optimality [23].

[14, 30].

**39**

With the recent development of modern navigation technology, inertial sensor based SINS technology have been characterized into three categories, (i) low accuracy (tactical applications), (ii) medium accuracy (navigation applications) and (iii) high accuracy (strategic navigation applications) sensor technology. The performance improvements of inertial sensors are decided by the inertial sensor errors [9]. Currently, the strap-down INS use (i) low-cost MEMS and (ii) precision fiber optic gyroscope. MEMS sensor has more attractive to manufacturers of navigation systems because of their small size, low cost, light weight, low power consumption and ruggedness [10]. However, MEMS sensors give poor performance in the highly dynamic environment. Hence, the reliability of MEMS-based INS navigation accuracy is limited. Because of these features MEMS have only been used for low-end navigation applications (i.e., commercial domain) [11].

In the recent years, MEMS devices have been developed and tested successfully for low-end accuracy applications [12, 13]. MEMS sensor operates for a long time under poor condition and it generates the noise due to internal circuits and electronics interferences of the MEMS sensor [14–16]. As a result, noise and drift are generated at the MEMS output. In general, drift error is affected by ambient temperatures and magnetic field effect [17–19]. Many studies have been reported for temperature error model of MEMS sensor to capture the temperature variation affects [20]. According to the IEEE standard specification, MEMS errors can be characterized into two categories, such as (i) deterministic errors and (ii) stochastic errors. Deterministic errors are due to scale factor errors, bias and misalignment errors [18, 19]. Several calibration methods have been developed for eliminating the bias errors, scale factor errors in the lab environments. Stochastic errors are due to quantization effect, temperature effect (random bias), random drift, and additive noise of MEM sensor. In the case of stochastic errors analysis, calibration techniques cannot be suitable because of randomness [21–24]. This chapter concentrates on random errors modeling and random noise elimination techniques. The developments of random noise suppressing methods are helpful for improving the MEMS accuracy as well as SINS accuracy. In general stochastic error includes quantitation noise (QN), bias instability (BS), angle random walk (ARW), rate random walk (RRW) and rate ramp (RR) drift. With the extension of research, random noise and bias drift are the non-negligible errors in the MEMS sensor output. In this chapter, different signal processing techniques are developed to minimize the bias drift and random noise [25].

In time domain, Allan Variance (AV) is a popular technique has been widely used to identify and quantify different random noises present in the MEMS sensor [16, 26, 27]. In literature, several noise compensation techniques such as discrete

#### *Modeling of Inertial Rate Sensor Errors Using Autoregressive and Moving Average (ARMA)… DOI: http://dx.doi.org/10.5772/intechopen.86735*

wavelet transform (DWT), empirical mode decomposition (EMD) method and Forward linear prediction (FLP) methods have been developed and applied to MEMS sensors for filtering the high-frequency noise [28, 29]. These methods are not suitable when the sensor includes the correlated noise. Kalman filter (KF) is a most popular state estimation technique that has been used for minimizing the correlated noise of the MEMS sensor [30–34]. The priori knowledge of an initial values of the process and measurement noise covariance matrix are known exactly, when the KF become an optimal. However, in practice, these noise parameters may vary with time so that the performance of the KF can be degraded and then the filter become diverge.

To solve the divergence problems, Adaptive Kalman filter technique (AKF) is a better solution. The adaptation can be based on either (i) innovation based adaptive estimation AKF (IAE-AKF) or (ii) residual based estimation AKF (RAE-AKF) and also multiple model based AKF [34, 35]. Among the other methods, adaptive KF is developed using IAE. In general, an innovation sequence is defined as the difference between true and estimated values. In the IAE-AKF method, the measurement and process noise matrices are estimated based on innovation sequence and followed by sliding average window method. In real time, the selection of window size is a critical issue. Sage-Husa Adaptive KF is another version of adaptive KF that has been developed to improve the AKF performance by introducing a time varying estimator. In the SHAKF, using a time-varying noise estimator can be helpful in estimating the statistical characteristics of the uncertainty in the measurements in real time and mitigating the filter divergence. A further study on the SHAKF is developed based on adaptive factors for improving the filter practicability and optimality [23].

An adaptive fading Kalman filter (AFKF) was proposed for compensating the effect of the uncertainty in the measurements by transitive factor to the state error covariance (P). In AFKF, the state error covariance (P) is scaled with a single transitive factor for improving the filter variance and gain correction. When it is used for complex systems, the performance of AFKF degrades because of it may not be sufficient to use a single transitive factor for estimating the covariance matrix of the filter [24]. To overcome the difficulties of single transitive factor, multiple fading factors are used in AFKF. Because of that reason, authors are developed double transitive factor based SHAFKF that adapts both predicted state error covariance (P) and measurement noise covariance matrix (R) based on the innovation sequence. Although it has been successively applied to different domains, its performance for MEMS gyroscope sensor signal is not explored. The stochastic errors of MEMS gyroscope cannot be eliminated using calibration technique. It needs to be modeled before filtering the signal. Therefore, adaptive filtering techniques have been developed for minimizing the random noise from MEMS gyroscope system. In general, auto-regressive (AR), Moving Average (MA), and Auto-Regressive and Moving Average (ARMA) and Gauss-Markov model (GM) have been used for modeling stochastic signal [17]. Among these models, ARMA is a better choice for modeling MEMS gyroscope drift errors. In general, the ARMA modeling involved three steps as (i) randomness and stationary test (ii) selection of suitable time series model and (iii) estimation of model parameters. The unit root test and inverse sequence techniques have been used for checking the stationary of the signal. The model order is obtained by using auto correlation function (ACF) and partial auto correlation function (PACF). Moreover, Akaike Information Criterion also used to check the model order. The modified Yule-Walker method is used estimate the model parameters. Once an optimal ARMA model is defined, a suitable adaptive Kalman filter can be applied to minimize the drift of inertial sensors [14, 30].

the navigation information. Because of these factors, the gimbals INS systems are used in low grade navigation applications [6]. In the early 1950, strap-down INS (SINS) was developed based on solid state inertial sensor [5]. The SINS is a selfcontained navigation system that has been developed for providing the accurate navigation information (i.e., position, velocity and rotation information. It has three gyroscopes ad three accelerometers. In general, the operation principle of SINS follows the physical laws of motion equations. It is an emerging technology as compared to gimbal INS systems and it has significant features such as easy to design, lower cost of ownership, moderate manufacturing cost and also high reliability. SINS consist of an inertial measurement that includes 3-axis accelerometers and 3-axis gyroscopes, and a processing computer. IMU is a key device to the INS and has been widely used for measuring the rotation rate and acceleration of an object. In practice, SINS accuracy degrades due to internal and external errors of the inertial sensors. These errors are mainly caused due to fluctuation in temperature, pressure and internal electronics components of the sensor. Due to these factors,

*Gyroscopes - Principles and Applications*

stochastic errors and drift errors are generated at the IMU output [7, 8].

navigation applications (i.e., commercial domain) [11].

**38**

With the recent development of modern navigation technology, inertial sensor based SINS technology have been characterized into three categories, (i) low accuracy (tactical applications), (ii) medium accuracy (navigation applications) and (iii) high accuracy (strategic navigation applications) sensor technology. The performance improvements of inertial sensors are decided by the inertial sensor errors [9]. Currently, the strap-down INS use (i) low-cost MEMS and (ii) precision fiber optic gyroscope. MEMS sensor has more attractive to manufacturers of navigation systems because of their small size, low cost, light weight, low power consumption and ruggedness [10]. However, MEMS sensors give poor performance in the highly dynamic environment. Hence, the reliability of MEMS-based INS navigation accuracy is limited. Because of these features MEMS have only been used for low-end

In the recent years, MEMS devices have been developed and tested successfully for low-end accuracy applications [12, 13]. MEMS sensor operates for a long time under poor condition and it generates the noise due to internal circuits and electronics interferences of the MEMS sensor [14–16]. As a result, noise and drift are generated at the MEMS output. In general, drift error is affected by ambient temperatures and magnetic field effect [17–19]. Many studies have been reported for temperature error model of MEMS sensor to capture the temperature variation affects [20]. According to the IEEE standard specification, MEMS errors can be characterized into two categories, such as (i) deterministic errors and (ii) stochastic errors. Deterministic errors are due to scale factor errors, bias and misalignment errors [18, 19]. Several calibration methods have been developed for eliminating the bias errors, scale factor errors in the lab environments. Stochastic errors are due to quantization effect, temperature effect (random bias), random drift, and additive noise of MEM sensor. In the case of stochastic errors analysis, calibration techniques cannot be suitable because of randomness [21–24]. This chapter concentrates on random errors modeling and random noise elimination techniques. The developments of random noise suppressing methods are helpful for improving the MEMS accuracy as well as SINS accuracy. In general stochastic error includes quantitation noise (QN), bias instability (BS), angle random walk (ARW), rate random walk (RRW) and rate ramp (RR) drift. With the extension of research, random noise and bias drift are the non-negligible errors in the MEMS sensor output. In this chapter, different signal processing techniques are developed to minimize the bias drift and random noise [25].

In time domain, Allan Variance (AV) is a popular technique has been widely used to identify and quantify different random noises present in the MEMS sensor [16, 26, 27]. In literature, several noise compensation techniques such as discrete

In this chapter, we developed double transitive factors based on Sage-Husa adaptive fading Kalman filter (SHAFKF), namely SHAFKF-P Adaption and SHAFKF-R adaption. In addition, ARMA model is used to model the random drift errors of MEMS sensor. ARMA model based SHAFKF algorithm is developed and applied for minimizing the bias drift and random noise in the presence of MEMS gyroscope signal. The suggested algorithm is analyzed in two stages. In the first stage, the predicted state error covariance is adapted by a transitive factor, whereas, in the second stage, another transitive factor is scaled to the measurement noise covariance matrix (R). The efficiency of the algorithm is analyzed using Allan Variance technique.

The remainder of the paper is organized as follows. Section 2, explains the theory of ARMA models for MEMS gyroscope random noise analysis. The Allan Variance method is explained in Section 3. In Section 4, Conventional and adaptive Kalman filters are discussed based on innovation sequence. Section 5 explains the proposed algorithm based on double transitive factors. Designing state space model for ARMA (2, 1) model is presented in Section 6. Experimental results and static test analysis are explained in Section 7 and also followed by conclusions in Section 8.

#### **2. Auto regressive and moving average (ARMA) model**

In literature, several time series models have been widely used in many fields such as industry, science and engineering. Among the other model, auto regressive (AR) and moving average (MA) models have been most popular and since then widely used for forecasting [14–16]. The combination of AR and MA models has been used for inertial sensors error modeling. In this chapter, stationary ARMA model is proposed for characterizing the stochastic errors of the MEMS gyroscope signals. In general, the ARMA model is a combination of weighted sum of AR and MA model. The expression for the ARMA model with an order (*p*, *q*) is defined as

$$Y\_n = \sum\_{i=1}^p \mathcal{Q}\_i Y\_{n-i} \sum\_{j=1}^q \theta\_j \varepsilon\_{n-j} + \varepsilon\_n \tag{1}$$

The samples autocorrelation function (ACF) is defined as

1 *N* P*N*�*<sup>k</sup>*

*<sup>n</sup>*¼<sup>1</sup> *Yn* � *<sup>μ</sup><sup>y</sup>*

1 *N* P*<sup>N</sup>*

**Model order ACF PACF**

MA(q) Cut of at order q Tail off ARMA(p, q) Tail off Tail off

AR(p) Tail off Cut of at order P

*Modeling of Inertial Rate Sensor Errors Using Autoregressive and Moving Average (ARMA)…*

*<sup>j</sup>*¼<sup>1</sup> *gk*�<sup>1</sup> � � *gk*

where *k* is the lag and *gk* is the sample autocorrelation. The *μ<sup>y</sup>* and *gkk* are the

This can also cross checked using Akaike Information Criterion (AIC) method. In this work, AIC values of the time series data are evaluated using **Table 1**. The

where Θ denotes the estimated residual of the model. *daic* and *Naic* are the model

Allan variance (AV) is a popular time domain method has been widely used for identifying and quantifying random errors in the presence of inertial sensor [14]. Cluster based analysis is used to develop the AV technique. In the AV analysis, the IMU raw data can be divided into clusters with specified length, "*m*." Let us take "*n*" measurements of gyroscope ð Þ *<sup>ω</sup>* , denote it by *<sup>ω</sup><sup>x</sup>*½ � <sup>1</sup> *, <sup>ω</sup><sup>x</sup>*½ � <sup>2</sup> *, <sup>ω</sup><sup>x</sup>*½ � <sup>3</sup> *,* ……*ωx n*½ �. The collected MEMS sensor data is sampled at rate of *fs* (samples per seconds). The set of samples called as cluster and denoted as "*kc*". The total number of clusters can be

� � *Yn*þ*<sup>k</sup>* � *<sup>μ</sup><sup>y</sup>*

*<sup>n</sup>*¼<sup>1</sup> *Yn* � *<sup>μ</sup><sup>y</sup>*

*g*<sup>1</sup> *if k* ¼ 1

2*daic*

� �

� � *if k* <sup>¼</sup> <sup>2</sup>*,* <sup>3</sup>*,* …*, n,*

*Naic* � � � � (4)

� �<sup>2</sup> (2)

(3)

*ACF* ¼ *gk* ¼

*Determining the model and order of the MEMS gyro signal.*

*DOI: http://dx.doi.org/10.5772/intechopen.86735*

and the partial autocorrelation is expressed as

8 >><

>>:

*gk* P*<sup>k</sup>*�*<sup>j</sup> <sup>j</sup>*¼<sup>1</sup> *gk*�<sup>1</sup> � � *gk*�*<sup>j</sup>* � �

model order is selected based on the minimum value of AIC.

order and the number of time series observation respectively.

<sup>1</sup> � <sup>P</sup>*<sup>k</sup>*�*<sup>j</sup>*

The general expression of Akaike information criterion (AIC) is

*AIC* ¼ log Θ 1 þ

Suitable model parameters are estimated by using Yule-Walker, Burg, Unconstrained Least-Squares method and Levinson-Durbin methods. In general, for large data-set analysis, Yule-Walker and Unconstrained Least-Squares method

*PACF* ¼ *gkk* ¼

**Table 1.**

**2.2 Model parameter estimation**

are the better estimators.

**41**

**3. Allan variance analysis**

samples mean and partial correlation at lag *k*.

where *p* and *q* are the AR and MA model orders, receptively. *ε<sup>n</sup>* is a sequence of independent and identical distributed random variable. *Yn* is the measured time series data of MEMS gyroscope signal. ∅1, ∅2, ∅3, …, ∅*<sup>p</sup>* and *θ*1, *θ*2, *θ*3, …, *θ<sup>q</sup>* are the auto regressive (AR) and moving average (MA) coefficients, respectively. The MEME gyroscope sensor raw data is used to test the normality and zero mean of the time series data of MEMS gyroscope. In general, the skewness and Kurtosis should be 0 and 1 that tells that checking the zero mean and normal distributed data of the time series data of the sensors.

#### **2.1 Time series model selection**

In the time series analysis, several methods have been developed for selecting the order of the AR, MA and ARMA order. In general, auto-correlation function (ACF) and partial ACF (PACF) are the basic methods to select the model based on the characteristics of the ACF and PCF graphs as shown in **Table 1**. From **Table 1**, we observed that both ACF and PACF are tail off. In this chapter, ARMA (p, q) is suitable for modeling the MEMS Gyroscope data.

*Modeling of Inertial Rate Sensor Errors Using Autoregressive and Moving Average (ARMA)… DOI: http://dx.doi.org/10.5772/intechopen.86735*


**Table 1.**

In this chapter, we developed double transitive factors based on Sage-Husa adaptive fading Kalman filter (SHAFKF), namely SHAFKF-P Adaption and SHAFKF-R adaption. In addition, ARMA model is used to model the random drift errors of MEMS sensor. ARMA model based SHAFKF algorithm is developed and applied for minimizing the bias drift and random noise in the presence of MEMS gyroscope signal. The suggested algorithm is analyzed in two stages. In the first stage, the predicted state error covariance is adapted by a transitive factor, whereas, in the second stage, another transitive factor is scaled to the measurement noise covariance matrix (R). The efficiency of the algorithm is analyzed using Allan

The remainder of the paper is organized as follows. Section 2, explains the theory of ARMA models for MEMS gyroscope random noise analysis. The Allan Variance method is explained in Section 3. In Section 4, Conventional and adaptive Kalman filters are discussed based on innovation sequence. Section 5 explains the proposed algorithm based on double transitive factors. Designing state space model for ARMA (2, 1) model is presented in Section 6. Experimental results and static test analysis are explained in Section 7 and also followed by conclusions in

In literature, several time series models have been widely used in many fields such as industry, science and engineering. Among the other model, auto regressive (AR) and moving average (MA) models have been most popular and since then widely used for forecasting [14–16]. The combination of AR and MA models has been used for inertial sensors error modeling. In this chapter, stationary ARMA model is proposed for characterizing the stochastic errors of the MEMS gyroscope signals. In general, the ARMA model is a combination of weighted sum of AR and MA model. The expression for the ARMA model with an order (*p*, *q*) is

**2. Auto regressive and moving average (ARMA) model**

*Yn* <sup>¼</sup> <sup>X</sup> *p*

*i*¼1

∅*iYn*�*<sup>i</sup>*

X *q*

*θjε<sup>n</sup>*�*<sup>j</sup>* þ *ε<sup>n</sup>* (1)

*j*¼1

where *p* and *q* are the AR and MA model orders, receptively. *ε<sup>n</sup>* is a sequence of independent and identical distributed random variable. *Yn* is the measured time series data of MEMS gyroscope signal. ∅1, ∅2, ∅3, …, ∅*<sup>p</sup>* and *θ*1, *θ*2, *θ*3, …, *θ<sup>q</sup>* are the auto regressive (AR) and moving average (MA) coefficients, respectively. The MEME gyroscope sensor raw data is used to test the normality and zero mean of the time series data of MEMS gyroscope. In general, the skewness and Kurtosis should be 0 and 1 that tells that checking the zero mean and normal distributed data of the

In the time series analysis, several methods have been developed for selecting the order of the AR, MA and ARMA order. In general, auto-correlation function (ACF) and partial ACF (PACF) are the basic methods to select the model based on the characteristics of the ACF and PCF graphs as shown in **Table 1**. From **Table 1**, we observed that both ACF and PACF are tail off. In this chapter, ARMA (p, q) is

Variance technique.

*Gyroscopes - Principles and Applications*

Section 8.

defined as

**40**

time series data of the sensors.

**2.1 Time series model selection**

suitable for modeling the MEMS Gyroscope data.

*Determining the model and order of the MEMS gyro signal.*

The samples autocorrelation function (ACF) is defined as

$$\text{ACF} = \mathbf{g}\_k = \frac{\frac{1}{N} \sum\_{n=1}^{N-k} \left( Y\_n - \mu\_\mathbf{y} \right) \left( Y\_{n+k} - \mu\_\mathbf{y} \right)}{\frac{1}{N} \sum\_{n=1}^{N} \left( Y\_n - \mu\_\mathbf{y} \right)^2} \tag{2}$$

and the partial autocorrelation is expressed as

$$PACF = \mathbf{g}\_{kk} = \begin{cases} \mathbf{g}\_1 & \text{if } k = 1\\ \mathbf{g}\_k \sum\_{j=1}^{k-j} \left( \mathbf{g}\_{k-1} \right) \left( \mathbf{g}\_{k-j} \right) \\ \hline 1 - \sum\_{j=1}^{k-j} \left( \mathbf{g}\_{k-1} \right) \left( \mathbf{g}\_k \right) \end{cases} \qquad \text{if } k = 2, 3, \dots, n,\tag{3}$$

where *k* is the lag and *gk* is the sample autocorrelation. The *μ<sup>y</sup>* and *gkk* are the samples mean and partial correlation at lag *k*.

This can also cross checked using Akaike Information Criterion (AIC) method. In this work, AIC values of the time series data are evaluated using **Table 1**. The model order is selected based on the minimum value of AIC.

The general expression of Akaike information criterion (AIC) is

$$AIC = \log\left(\Theta\left[\mathbf{1} + \frac{2d\_{\rm{air}}}{N\_{\rm{air}}}\right]\right) \tag{4}$$

where Θ denotes the estimated residual of the model. *daic* and *Naic* are the model order and the number of time series observation respectively.

#### **2.2 Model parameter estimation**

Suitable model parameters are estimated by using Yule-Walker, Burg, Unconstrained Least-Squares method and Levinson-Durbin methods. In general, for large data-set analysis, Yule-Walker and Unconstrained Least-Squares method are the better estimators.

#### **3. Allan variance analysis**

Allan variance (AV) is a popular time domain method has been widely used for identifying and quantifying random errors in the presence of inertial sensor [14]. Cluster based analysis is used to develop the AV technique. In the AV analysis, the IMU raw data can be divided into clusters with specified length, "*m*." Let us take "*n*" measurements of gyroscope ð Þ *<sup>ω</sup>* , denote it by *<sup>ω</sup><sup>x</sup>*½ � <sup>1</sup> *, <sup>ω</sup><sup>x</sup>*½ � <sup>2</sup> *, <sup>ω</sup><sup>x</sup>*½ � <sup>3</sup> *,* ……*ωx n*½ �. The collected MEMS sensor data is sampled at rate of *fs* (samples per seconds). The set of samples called as cluster and denoted as "*kc*". The total number of clusters can be represented by "K," (i.e., *<sup>K</sup>* <sup>¼</sup> *<sup>n</sup> <sup>m</sup>*). The measured date of the gyroscope can be written as

$$\alpha^{\mathbf{x}[1]}, \alpha^{\mathbf{x}[2]}, \alpha^{\mathbf{x}[3]}, \dots \\
\alpha^{\mathbf{x}[m]}, \alpha^{\mathbf{x}[m+1]}, \dots \\
\alpha^{\mathbf{x}[2m]}, \dots \\
\alpha^{\mathbf{x}[n-m]} \dots \\
\alpha^{\mathbf{x}[n]}$$

To calculate each clusters average is

$$
\overline{w}^{\kappa[k\_{\epsilon}]} = \sum\_{i=1}^{m} \overline{w}^{\kappa[k\_{\epsilon}-1]m+i} \tag{5}
$$

The different random noise processes are characterized at various frequencies that are fitted by the AV method. The root Allan variance with each correlation time

**Noisy type Units Slope Root Allan variance**

*Modeling of Inertial Rate Sensor Errors Using Autoregressive and Moving Average (ARMA)…*

Bias instability (BS) °*=hr* 0 *σBS*ð Þ¼ *τ* 0*:*668 *Bs*

Rate ramp (RR) °*=hr*<sup>2</sup> <sup>1</sup> *<sup>σ</sup>RR*ð Þ¼ *<sup>τ</sup> RRW <sup>τ</sup>*ffiffi

Quantization noise (QN) °*<sup>=</sup>* sec �<sup>1</sup> *<sup>σ</sup>QN*ð Þ¼ *<sup>τ</sup>*

The application of conventional Kalman filter (CKF) for the MEMS gyroscope requires a prior knowledge of dynamic process and measurement models. In addition, the process and measurement noise of the MEMS gyroscope. Considering a linear dynamic system, the state and measurement equations can be written as

where *xk* is the state vector at epoch k; *A* is the state transition matrix; *wk* is the system (process) noise; *zk* is the observation (measurement) at epoch k; *H* represents the observation matrix; and *vk* is the measurement noise. Let us assume that the process *wk* and measurement noises ð Þ *vk* are the Gaussian white noise with zero mean and finite variance that means that these are statistically independent from

> E *wkw<sup>T</sup> k*

E *vkvT k*

*x*^�

*P*�

Basically, the Kalman Filtering estimation algorithm comprises two steps, namely prediction and updating equations. The main Kalman Filtering equations

*xk* ¼ *Axk*�<sup>1</sup> þ *Buk* þ *wk* (8)

*hr* <sup>p</sup> �1/2 *<sup>σ</sup>ARW* ð Þ¼ *<sup>τ</sup> ARW*ffiffi

*hr* <sup>p</sup><sup>3</sup> 1/2 *<sup>σ</sup>RRW*ð Þ¼ *<sup>τ</sup> RRW* ffiffi

E f g *wk* ¼ 0*,* E f g *vk* ¼ 0 (10)

� � <sup>¼</sup> *Qk* (11)

� � <sup>¼</sup> *Rk* (12)

*<sup>k</sup>* ¼ *Ax*^*<sup>k</sup>*�<sup>1</sup> (13)

*<sup>k</sup>* <sup>¼</sup> *APk*�<sup>1</sup>*AT* <sup>þ</sup> *Qk* (14)

*zk* ¼ *Hxk* þ *vk* (9)

ffiffi 3 <sup>p</sup> *QN τ*

*τ* p

*τ* 3 p

2 p

and slope are computed and presented in **Table 2**.

Angle random walk (ARW) °*=* ffiffiffiffiffi

*DOI: http://dx.doi.org/10.5772/intechopen.86735*

Rate random walk (RRW) °*=* ffiffiffiffiffi

each other, following properties can be satisfied:

Prediction equations can be expressed as

are given below.

**43**

**4. Adaptive Kalman filtering**

**Table 2.**

*Allan variance analysis results.*

**4.1 Conventional Kalman filter**

Here, *kc* =1, 2, 3, …, K is the number of clusters.

The Allan variance is computed from two successive cluster averages for the specified correlation time which is defined as:

$$\sigma^2(\tau\_m) = \frac{1}{2(K-1)} \sum\_{k\_\epsilon=1}^{K-1} \left( \left( \overline{w}^{\mathbf{x}[k\_\epsilon+1]}(m) - \overline{w}^{\mathbf{x}[k\_\epsilon]}(m) \right)^2 \right) \tag{6}$$

where *kc* =1, 2, 3, …, K, and *<sup>τ</sup><sup>m</sup>* <sup>¼</sup> *<sup>m</sup> <sup>f</sup> <sup>=</sup> <sup>s</sup>* is averaged period (or specified correlation time). The AV can be computed as a function of correlation times versus the Allan deviation plot are shown in **Figure 1**. The different contribution error sources are carried out simply by examining the slope of the AV plot. To extract the information on a specific source of error from the AV plot.

There is a unique relationship between the Allan Variance (time domain) and the PSD (frequency domain) of the random process as:

$$\sigma^2(\tau\_m) = 4 \int\_0^\infty \mathcal{S}\_\Omega(f) \frac{\sin^4(\pi fT)}{\left(\pi fT\right)^2} \tag{7}$$

where *S*<sup>Ω</sup> *fs* � � is the power spectral density (PSD) of the random process and sin <sup>4</sup>ð Þ *<sup>π</sup>fT* ð Þ *<sup>π</sup>fT* <sup>2</sup> is the transfer function of PSD.

**Figure 1.** *Allan variance log-log plot.*

*Modeling of Inertial Rate Sensor Errors Using Autoregressive and Moving Average (ARMA)… DOI: http://dx.doi.org/10.5772/intechopen.86735*


**Table 2.**

represented by "K," (i.e., *<sup>K</sup>* <sup>¼</sup> *<sup>n</sup>*

*Gyroscopes - Principles and Applications*

To calculate each clusters average is

Here, *kc* =1, 2, 3, …, K is the number of clusters.

1 2ð Þ *K* � 1

specified correlation time which is defined as:

tion on a specific source of error from the AV plot.

ð Þ *<sup>π</sup>fT* <sup>2</sup> is the transfer function of PSD.

the PSD (frequency domain) of the random process as:

*σ*2

ð Þ¼ *τ<sup>m</sup>* 4

*σ*2 ð Þ¼ *τ<sup>m</sup>*

where *S*<sup>Ω</sup> *fs*

sin <sup>4</sup>ð Þ *<sup>π</sup>fT*

**Figure 1.**

**42**

*Allan variance log-log plot.*

written as

*<sup>m</sup>*). The measured date of the gyroscope can be

*<sup>ω</sup>x k*½ � *<sup>c</sup>*þ<sup>1</sup> ð Þ� *<sup>m</sup> <sup>ω</sup>x kc* ½ �ð Þ *<sup>m</sup>* � �<sup>2</sup> � �

*wx k*½ � *<sup>c</sup>*�<sup>1</sup> *<sup>m</sup>*þ*<sup>i</sup>* (5)

ð Þ *<sup>π</sup>fT* <sup>2</sup> (7)

(6)

*ωx*½ � <sup>1</sup> *, ωx*½ � <sup>2</sup> *,ωx*½ � <sup>3</sup> *,* ……*ωx m*½ �*,ωx m*½ � <sup>þ</sup><sup>1</sup> *,* ……*ωx*½ � <sup>2</sup>*<sup>m</sup> ,* ……*ωx n*½ � �*<sup>m</sup>* …*ωx n*½ �

*i*¼1

The Allan variance is computed from two successive cluster averages for the

where *kc* =1, 2, 3, …, K, and *<sup>τ</sup><sup>m</sup>* <sup>¼</sup> *<sup>m</sup> <sup>f</sup> <sup>=</sup> <sup>s</sup>* is averaged period (or specified correlation time). The AV can be computed as a function of correlation times versus the Allan deviation plot are shown in **Figure 1**. The different contribution error sources are carried out simply by examining the slope of the AV plot. To extract the informa-

There is a unique relationship between the Allan Variance (time domain) and

*<sup>S</sup>*Ωð Þ*<sup>f</sup>* sin <sup>4</sup>ð Þ *<sup>π</sup>fT*

� � is the power spectral density (PSD) of the random process and

∞ð

0

*<sup>ω</sup>x k*d e*<sup>c</sup>* <sup>¼</sup> <sup>X</sup>*<sup>m</sup>*

X *K*�1

*kc*¼1

*Allan variance analysis results.*

The different random noise processes are characterized at various frequencies that are fitted by the AV method. The root Allan variance with each correlation time and slope are computed and presented in **Table 2**.
