**4. Adaptive Kalman filtering**

#### **4.1 Conventional Kalman filter**

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

$$\mathbf{x}\_{k} = A\mathbf{x}\_{k-1} + Bu\_{k} + w\_{k} \tag{8}$$

$$z\_k = H\mathbf{x}\_k + v\_k \tag{9}$$

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 each other, following properties can be satisfied:

$$\to \{w\_k\} = \mathbf{0}, \mathbf{E}\left\{v\_k\right\} = \mathbf{0} \tag{10}$$

$$\mathbf{E}\left\{w\_{k}w\_{k}^{T}\right\}=\mathbf{Q}\_{k}\tag{11}$$

$$\mathbf{E}\left\{\boldsymbol{v}\_{k}\boldsymbol{v}\_{k}^{T}\right\}=\boldsymbol{R}\_{k}\tag{12}$$

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

Prediction equations can be expressed as

$$
\hat{\mathfrak{x}}\_k^- = A \hat{\mathfrak{x}}\_{k-1} \tag{13}
$$

$$P\_k^- = AP\_{k-1}A^T + Q\_k \tag{14}$$

In the above equations, *A* is the state transition matrix and *A<sup>T</sup>* denotes the transpose of *A*. *P*� *<sup>k</sup>* and *Qk* represents prediction state error covariance and process noise covariance matrix at epoch *k*.

In the linear Kalman filter, the measurement updated equations are

$$K\_k = P\_k^- H^T \left( H P\_k^- H^T + R \right)^{-1} \tag{15}$$

$$
\hat{\mathfrak{X}}\_{k} = \hat{\mathfrak{x}}\_{k}^{-} + K\_{k} \left( \mathbf{z}\_{k} - H \hat{\mathfrak{x}}\_{k}^{-} \right) \tag{16}
$$

$$P\_k = (I - K\_k H) P\_k^- \tag{17}$$

*<sup>R</sup>*^*<sup>k</sup>* <sup>¼</sup> *<sup>C</sup>*^*Vk* � *HP*�

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

vation matrix, *P*�

covariance matrix of innovation sequence.

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

covariance matrices, respectively.

where *dk* <sup>¼</sup> ð Þ <sup>1</sup> � *bk <sup>=</sup>* <sup>1</sup> � *<sup>b</sup><sup>k</sup>*þ<sup>1</sup>

Prediction equations as.

**45**

**4.3 Sage-Husa adaptive Kalman filter (SHAKF)**

where *R*^*<sup>k</sup>* is the estimated measurement noise covariance matrix, *H* is the obser-

Sage-Husa AKF (SHAKF) is another class of adaptive filtering that uses a timevarying noise statistical estimator to proceed recursively. It is also used to reduce the sensor noise in the presence of MEMS IMU signals [16]. The linear dynamical process and measurement model equations can be written in the Eqs. (4) and (5). The expectation and the covariance matrices of *wk* and *vk* are written as.

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

E *vkvT k*

The time-varying noise statistic recursive estimator is given by:

*<sup>R</sup>*^ *<sup>k</sup>*þ<sup>1</sup> <sup>¼</sup> ð Þ <sup>1</sup> � *dk <sup>R</sup>*^ *<sup>k</sup>* <sup>þ</sup> *dk VkV<sup>T</sup>*

*<sup>Q</sup>*^ *<sup>k</sup>*þ<sup>1</sup> <sup>¼</sup> ð Þ <sup>1</sup> � *dk <sup>Q</sup>*^ *<sup>k</sup>* <sup>þ</sup> *dk KkVkV<sup>T</sup>*

*k*

filtering output signal are expressed in terms of following equations.

*x*^�

*P*�

*Kk* ¼ *P*�

*x*^*<sup>k</sup>* ¼ *x*^�

Measurement updated equations are equations:

Here, an innovation sequence can be written as

where *Q*^ *<sup>k</sup>* and *R*^*<sup>k</sup>* are the initial estimated process and measurement noise

^*rk*þ<sup>1</sup> ¼ ð Þ 1 � *dk* ^*rk* þ *dk zk* � *Hx*^�

and 1. The Kalman filtering output signal and Sage-Husa self-adaptive Kalman

*<sup>k</sup> H<sup>T</sup> HP*�

*Pk* ¼ ð Þ *I* � *KkH P*�

*<sup>k</sup>* is the prediction state error covariance and *<sup>C</sup>*^*Vk* is the estimated

*<sup>k</sup> H<sup>T</sup>* (21)

E f g *wk* ¼ *q*^*<sup>k</sup>* (22) E f g *vk* ¼ ^*rk* (23)

<sup>¼</sup> *<sup>Q</sup>*^ *<sup>k</sup>* (24)

<sup>¼</sup> *<sup>R</sup>*^*<sup>k</sup>* (25)

*k*

*<sup>k</sup>* � *HP*�

*<sup>q</sup>*^*<sup>k</sup>*þ<sup>1</sup> <sup>¼</sup> ð Þ <sup>1</sup> � *dk <sup>q</sup>*^*<sup>k</sup>* <sup>þ</sup> *dk*ð Þ *xk* � *Ax*^*<sup>k</sup>* (28)

is the amnestic factor, value range between 0

*<sup>k</sup> K<sup>T</sup>*

*<sup>k</sup> <sup>H</sup><sup>T</sup>* <sup>þ</sup> *Rk*

*k*

*<sup>k</sup>* þ *Kk zk* � *Hx*^�

(26)

*<sup>k</sup> <sup>H</sup><sup>T</sup>* (27)

*<sup>k</sup>* <sup>þ</sup> *Pk* � *APk*�<sup>1</sup>*AT* (29)

*<sup>k</sup>*�<sup>1</sup> <sup>¼</sup> *Ax*^*<sup>k</sup>*�<sup>1</sup> <sup>þ</sup> *<sup>q</sup>*^*<sup>k</sup>* (30)

�<sup>1</sup> (32)

(33)

*<sup>k</sup>*�<sup>1</sup> (34)

*<sup>k</sup>*�<sup>1</sup> <sup>¼</sup> *APk*�<sup>1</sup>*A<sup>T</sup>* <sup>þ</sup> *<sup>Q</sup>*^ *<sup>k</sup>*�<sup>1</sup> (31)

*Vk* ¼ *zk* � *Hx*^*<sup>k</sup>* � ^*rk* (35)

where *x*^*<sup>k</sup>* is the estimated state, *Kk* is the gain matrix and *Pk* is the estimated of state vector. *R* and *I* are the measurement noise covariance matrix and identify matrix respectively.

#### **4.2 Innovation based adaptive estimation adaptive Kalman filter (IAE-AKF)**

CKF requires a prior knowledge of the measurement and dynamic process models of MEMS IMU. In practice, statistical noise models of the process and measurement models are varying with time because of that the CKF would deprive optimality. To address this divergence, an adaptive KF (AKF) is a better solution. In the AKF, the adaptation can be carried out using three ways: (a) varying *Q* by trial and error until a stable state is estimated with fixed *R* [20]; (b) varying *R* by keeping *Q* fixed; (c) varying *Q* and *R* simultaneously [21]. In the IAE-AKF algorithm, we selected the second adaption method is that varying the measurement noise covariance matrix (R) by keeping Q fixed based on innovation sequence *Vk*.

The innovation sequence is defined as the difference between true measurements and predicated measurements that can assume to be additional information to the filter. The innovation sequence is a zero-mean white Gaussian noise sequence, defined as

$$V\_k = \mathbf{z}\_k - H\hat{\mathbf{x}}\_k^- \tag{18}$$

The weighted innovation *Kk zk* � *Hx*^� *k* � � acts as a correction to the predicted estimation *x*^� *<sup>k</sup>* to form *x*^*k*. By substituting the measurement model (5) in (14), we get *Vk* ¼ *H xk* � *x*^� *k* � ). By taking variance on both sides of this, the theoretical covariance matrix of *Vk* is

$$\mathbf{C}\_{V\_k} = H\mathbf{P}\_k^-\mathbf{H}^T + \mathbf{R}\_k \tag{19}$$

The optimal estimation of covariance matrix of innovation sequence using average window method can be expressed as

$$\hat{\mathbf{C}}\_{V\_k} = \frac{1}{D\_s} \sum\_{j=j0}^{D\_s} V\_j V\_j^T \tag{20}$$

where *Vj* is the innovation sequence, *Ds* is the window size, *j*0 ¼ *k* � *Ds* þ 1 is the first epoch. If the window size is too small, the measurement estimation covariance can be noisy; on the other hand, the estimation of measurement covariance will be smoother. Usually, window size is chosen empirically for statistical smoothing.

The estimated measurement noise covariance based on innovation sequence is

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

$$
\hat{R}\_k = \hat{\mathbf{C}}\_{V\_k} - H \mathbf{P}\_k^- H^T \tag{21}
$$

where *R*^*<sup>k</sup>* is the estimated measurement noise covariance matrix, *H* is the observation matrix, *P*� *<sup>k</sup>* is the prediction state error covariance and *<sup>C</sup>*^*Vk* is the estimated covariance matrix of innovation sequence.

#### **4.3 Sage-Husa adaptive Kalman filter (SHAKF)**

In the above equations, *A* is the state transition matrix and *A<sup>T</sup>* denotes the

*<sup>k</sup> H<sup>T</sup> HP*�

*Pk* ¼ ð Þ *I* � *KkH P*�

**4.2 Innovation based adaptive estimation adaptive Kalman filter (IAE-AKF)**

CKF requires a prior knowledge of the measurement and dynamic process models of MEMS IMU. In practice, statistical noise models of the process and measurement models are varying with time because of that the CKF would deprive optimality. To address this divergence, an adaptive KF (AKF) is a better solution. In the AKF, the adaptation can be carried out using three ways: (a) varying *Q* by trial and error until a stable state is estimated with fixed *R* [20]; (b) varying *R* by keeping *Q* fixed; (c) varying *Q* and *R* simultaneously [21]. In the IAE-AKF algorithm, we selected the second adaption method is that varying the measurement noise covariance matrix (R) by keeping Q fixed based on innovation sequence *Vk*. The innovation sequence is defined as the difference between true measurements and predicated measurements that can assume to be additional information

to the filter. The innovation sequence is a zero-mean white Gaussian noise

*CVk* ¼ *HP*�

*<sup>C</sup>*^*Vk* <sup>¼</sup> <sup>1</sup> *Ds* X *Ds*

*Vk* ¼ *zk* � *Hx*^�

*k*

*<sup>k</sup>* to form *x*^*k*. By substituting the measurement model (5) in (14), we

� ). By taking variance on both sides of this, the theoretical

The optimal estimation of covariance matrix of innovation sequence using aver-

*j*¼*j*0

where *Vj* is the innovation sequence, *Ds* is the window size, *j*0 ¼ *k* � *Ds* þ 1 is the first epoch. If the window size is too small, the measurement estimation covariance can be noisy; on the other hand, the estimation of measurement covariance will be smoother. Usually, window size is chosen empirically for statistical

The estimated measurement noise covariance based on innovation sequence is

*VjV<sup>T</sup>*

where *x*^*<sup>k</sup>* is the estimated state, *Kk* is the gain matrix and *Pk* is the estimated of state vector. *R* and *I* are the measurement noise covariance matrix and identify

*<sup>k</sup>* þ *Kk zk* � *Hx*^�

In the linear Kalman filter, the measurement updated equations are

*Kk* ¼ *P*�

*x*^*<sup>k</sup>* ¼ *x*^�

*<sup>k</sup>* and *Qk* represents prediction state error covariance and process

*k*

*<sup>k</sup> <sup>H</sup><sup>T</sup>* <sup>þ</sup> *<sup>R</sup>* � ��<sup>1</sup> (15)

� � (16)

*<sup>k</sup>* (17)

*<sup>k</sup>* (18)

*<sup>k</sup> <sup>H</sup><sup>T</sup>* <sup>þ</sup> *Rk* (19)

*<sup>j</sup>* (20)

� � acts as a correction to the predicted

transpose of *A*. *P*�

matrix respectively.

sequence, defined as

estimation *x*^�

smoothing.

**44**

get *Vk* ¼ *H xk* � *x*^�

covariance matrix of *Vk* is

The weighted innovation *Kk zk* � *Hx*^�

*k*

age window method can be expressed as

noise covariance matrix at epoch *k*.

*Gyroscopes - Principles and Applications*

Sage-Husa AKF (SHAKF) is another class of adaptive filtering that uses a timevarying noise statistical estimator to proceed recursively. It is also used to reduce the sensor noise in the presence of MEMS IMU signals [16]. The linear dynamical process and measurement model equations can be written in the Eqs. (4) and (5).

The expectation and the covariance matrices of *wk* and *vk* are written as.

$$\mathcal{E}\left\{w\_{k}\right\} = \hat{q}\_{k} \tag{22}$$

$$\mathbf{E}\left\{\boldsymbol{v}\_{k}\right\}=\hat{\mathbf{r}}\_{k}\tag{23}$$

$$\mathbf{E}\left\{w\_{k}w\_{k}^{T}\right\}=\hat{Q}\_{k}\tag{24}$$

$$\mathbf{E}\left\{\boldsymbol{v}\_{k}\boldsymbol{v}\_{k}^{T}\right\}=\hat{\mathbf{R}}\_{k}\tag{25}$$

where *Q*^ *<sup>k</sup>* and *R*^*<sup>k</sup>* are the initial estimated process and measurement noise covariance matrices, respectively.

The time-varying noise statistic recursive estimator is given by:

$$
\hat{r}\_{k+1} = (1 - d\_k)\hat{r}\_k + d\_k(z\_k - H\hat{x}\_k^-) \tag{26}
$$

$$\hat{R}\_{k+1} = (\mathbf{1} - d\_k)\hat{R}\_k + d\_k \left(V\_k V\_k^T - H P\_k^- H^T\right) \tag{27}$$

$$
\hat{q}\_{k+1} = (\mathbf{1} - d\_k)\hat{q}\_k + d\_k(\mathbf{x}\_k - A\hat{\mathbf{x}}\_k) \tag{28}
$$

$$
\hat{Q}\_{k+1} = (1 - d\_k)\hat{Q}\_k + d\_k \left( K\_k V\_k V\_k^T K\_k^T + P\_k - A P\_{k-1} A^T \right) \tag{29}
$$

where *dk* <sup>¼</sup> ð Þ <sup>1</sup> � *bk <sup>=</sup>* <sup>1</sup> � *<sup>b</sup><sup>k</sup>*þ<sup>1</sup> *k* is the amnestic factor, value range between 0 and 1. The Kalman filtering output signal and Sage-Husa self-adaptive Kalman filtering output signal are expressed in terms of following equations.

Prediction equations as.

$$
\hat{\boldsymbol{x}}\_{k-1}^{-} = A\hat{\boldsymbol{x}}\_{k-1} + \hat{\boldsymbol{q}}\_{k} \tag{30}
$$

$$P\_{k-1}^{-} = AP\_{k-1}A^T + \hat{Q}\_{k-1} \tag{31}$$

Measurement updated equations are equations:

$$K\_k = P\_k^- H^T \left( H P\_k^- H^T + R\_k \right)^{-1} \tag{32}$$

$$
\hat{\mathfrak{x}}\_{k} = \hat{\mathfrak{x}}\_{k}^{-} + K\_{k} \left( \mathbf{z}\_{k} - H\hat{\mathfrak{x}}\_{k}^{-} \right) \tag{33}
$$

$$P\_k = (I - K\_k H) P\_{k-1}^- \tag{34}$$

Here, an innovation sequence can be written as

$$V\_k = \mathbf{z}\_k - H\hat{\mathbf{x}}\_k - \hat{r}\_k \tag{35}$$

The *Kk* is the Kalman updated gain. *Rk* and I are the measurement noise covariance matrix and identity matrix respectively.

The suboptimal state and update the measurement equations as

*<sup>k</sup> H<sup>T</sup> HP*�

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

*Pk* ¼ ð Þ *I* � *KkH P*�

**5.2 Stage two: adaptation of measurement noise covariance matrix (R)**

Eq. (45). Thus the modified residual sequence can be defined as

termed as AUFKF-R adaptation.

*Flow chart of the SHAFKF-P adaptation algorithm.*

**Figure 2.**

**47**

The stage one algorithm requires prior knowledge of the state error vector and kinematic of model errors. To overcome this drawback and to eliminate the influence of the measurement noise disturbances, another transitive factor is introduced for updating the measurement noise covariance matrix (R). This stage is also

In this stage, modified residual sequence is evaluated as the difference between measurement vector **zk** and the suboptimal estimated state (*x*^*k*) evaluated using

where *x*^*<sup>k</sup>* indicates the suboptimal estimated state vector and *CVk* denotes the suboptimal covariance matrix of innovation the state vector. For optimal filter purpose, *x*^*<sup>k</sup>* and *Pk* are further passed to the next stage. The flowchart of the stage

*<sup>k</sup> <sup>H</sup><sup>T</sup>* <sup>þ</sup> *Rk*

*k*

*<sup>k</sup>* þ *Kk zk* � *Hx*^�

�<sup>1</sup> (41)

(42)

*<sup>k</sup>*�<sup>1</sup> (43)

*Kk* ¼ *P*�

one of the algorithm is shown in **Figure 2**.

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

*x*^*<sup>k</sup>* ¼ *x*^�

## **5. Proposed: Sage-Husa adaptive fading Kalman filter (SHAFKF) based on double transitive factors**

Adaptive estimation methods have been developed for improving the CKF performance [22, 23]. In the AKF, covariance matching techniques is used to estimate the covariance matrix of the innovation or residual by fixing the values of *Q*. By using a scale factor in the AKF hence the performance filter was improved for estimating the state error covariance and also it improves the variance of the predicted state. Further, adaptive fading Kalman filters have been developed for improving the filter performance by introducing multiple adaptive scaling factors [24]. In the proposed algorithms, adaptive transitive factors based linear Adaptive Kalman filter algorithm is proposed also used for improving the MEMS gyroscope performance [25, 30]. However, a limited work has been reported the use of transitive factors in ARMA model based Sage-Husa KF. The proposed algorithm is explained in two cascaded stages. The predicted state error covariance *P* is adapted in the stage one, whereas in the second stage, the measurement noise covariance *R* is adapted by another transitive factor. The proposed scheme is shown in Sections 5.1 and 5.2, respectively.

#### **5.1 Stage one: adaptation of predicted state error covariance (P)**

In this stage, the predicted state error covariance is modified using an adaptive transitive factor. This stage is also termed as SHAFKF-P adaptation. The transitive factor is used to reduce the process noise of kinematic model based on the residual sequence.

The transitive factor *a*1ð Þ k is evaluated as

$$a\_1(\mathbf{k}) = \begin{cases} 1, & tr\left(\mathbf{C}\_{V\_k}\right) > tr\left(\hat{P}\_{\overline{v}k}\right) \\\\ \frac{tr\left(\hat{\mathbf{C}}\_{V\_k} - R\_k\right)}{tr\left(\mathbf{C}\_{V\_k} - R\_k\right)}, & \text{Otherwise} \end{cases} \tag{36}$$

where *tr* is the trace function and *P*^*vk* is the estimated covariance matrix of the residual sequence expressed as

$$
\hat{P}\_{\overline{v}k} = \overline{V}\_k \overline{V}\_k^T \tag{37}
$$

The predicted state covariance *P*^� *<sup>k</sup>* is updated as

$$
\hat{P}\_k^- = \frac{\mathbf{1}}{a\_1(k)} \hat{P}\_{k-1}^- \tag{38}
$$

The SHAFKF-P adaptation algorithm, the predicted and estimated state error covariance are updated based on the SHAKF algorithm.

$$
\hat{P}\_k^- = \frac{1}{a\_1(k)} \hat{P}\_{k-1}^- \tag{39}
$$

$$\mathbf{C}\_{V\_k} = H\mathbf{P}\_k^-\mathbf{H}^T + \mathbf{R}\_k \tag{40}$$

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

The suboptimal state and update the measurement equations as

$$K\_k = P\_k^- H^T \left( H P\_k^- H^T + R\_k \right)^{-1} \tag{41}$$

$$
\hat{\mathfrak{x}}\_{k} = \hat{\mathfrak{x}}\_{k}^{-} + K\_{k} \left( \mathbf{z}\_{k} - H \hat{\mathfrak{x}}\_{k}^{-} \right) \tag{42}
$$

$$P\_k = (I - K\_k H) P\_{k-1}^- \tag{43}$$

where *x*^*<sup>k</sup>* indicates the suboptimal estimated state vector and *CVk* denotes the suboptimal covariance matrix of innovation the state vector. For optimal filter purpose, *x*^*<sup>k</sup>* and *Pk* are further passed to the next stage. The flowchart of the stage one of the algorithm is shown in **Figure 2**.

**Figure 2.**

The *Kk* is the Kalman updated gain. *Rk* and I are the measurement noise covari-

**5. Proposed: Sage-Husa adaptive fading Kalman filter (SHAFKF) based**

Adaptive estimation methods have been developed for improving the CKF performance [22, 23]. In the AKF, covariance matching techniques is used to estimate the covariance matrix of the innovation or residual by fixing the values of *Q*. By using a scale factor in the AKF hence the performance filter was improved for estimating the state error covariance and also it improves the variance of the predicted state. Further, adaptive fading Kalman filters have been developed for improving the filter performance by introducing multiple adaptive scaling factors [24]. In the proposed algorithms, adaptive transitive factors based linear Adaptive Kalman filter algorithm is proposed also used for improving the MEMS gyroscope performance [25, 30]. However, a limited work has been reported the use of transitive factors in ARMA model based Sage-Husa KF. The proposed algorithm is explained in two cascaded stages. The predicted state error covariance *P* is adapted in the stage one, whereas in the second stage, the measurement noise covariance *R* is adapted by another transitive factor. The proposed scheme is shown in Sections 5.1 and 5.2, respectively.

In this stage, the predicted state error covariance is modified using an adaptive transitive factor. This stage is also termed as SHAFKF-P adaptation. The transitive factor is used to reduce the process noise of kinematic model based on the residual

1*, tr CVk*

� � *, Otherwise*

where *tr* is the trace function and *P*^*vk* is the estimated covariance matrix of the

*<sup>P</sup>*^*vk* <sup>¼</sup> *VkV<sup>T</sup>*

*<sup>k</sup>* is updated as

*<sup>a</sup>*1ð Þ*<sup>k</sup> <sup>P</sup>*^�

*<sup>a</sup>*1ð Þ*<sup>k</sup> <sup>P</sup>*^�

The SHAFKF-P adaptation algorithm, the predicted and estimated state error

*tr <sup>C</sup>*^*Vk* � *Rk* � �

*tr CVk* � *Rk*

*P*^� *<sup>k</sup>* <sup>¼</sup> <sup>1</sup>

*P*^� *<sup>k</sup>* <sup>¼</sup> <sup>1</sup>

*CVk* ¼ *HP*�

� �> *tr P*^*vk*

� �

*<sup>k</sup>* (37)

*<sup>k</sup>*�<sup>1</sup> (38)

*<sup>k</sup>*�<sup>1</sup> (39)

*<sup>k</sup> <sup>H</sup><sup>T</sup>* <sup>þ</sup> *Rk* (40)

(36)

**5.1 Stage one: adaptation of predicted state error covariance (P)**

The transitive factor *a*1ð Þ k is evaluated as

8 >>><

>>>:

covariance are updated based on the SHAKF algorithm.

*a*1ð Þ¼ k

residual sequence expressed as

The predicted state covariance *P*^�

ance matrix and identity matrix respectively.

**on double transitive factors**

*Gyroscopes - Principles and Applications*

sequence.

**46**

*Flow chart of the SHAFKF-P adaptation algorithm.*

#### **5.2 Stage two: adaptation of measurement noise covariance matrix (R)**

The stage one algorithm requires prior knowledge of the state error vector and kinematic of model errors. To overcome this drawback and to eliminate the influence of the measurement noise disturbances, another transitive factor is introduced for updating the measurement noise covariance matrix (R). This stage is also termed as AUFKF-R adaptation.

In this stage, modified residual sequence is evaluated as the difference between measurement vector **zk** and the suboptimal estimated state (*x*^*k*) evaluated using Eq. (45). Thus the modified residual sequence can be defined as

*Gyroscopes - Principles and Applications*

$$
\overline{V}\_k = \mathbf{z}\_k - H\hat{\mathbf{x}}\_k - \hat{r}\_k \tag{44}
$$

The Kalman gain and state equations are updated as Eqs. (41)–(46). In this algorithm, measurement noise covariance matrix is multiplied by the adaptive transitive factor, *a*2ð Þ k . If *a*2ð Þ k large, *Rk* becomes larger, this helps to reduce the influence of uncertain measurement noise [23, 24]. The flow chart of the stage two,

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

The ARMA (p, q) model order is obtained using AIC method as in **Table 3**. The minimum values of AIC can be decided the optimal order of the ARMA (2, 1) is chosen. The ARMA (2, 1) model parameters such as **Φ1** = �**0.5422**, **Φ2** = �**0.1204** and *θ***1** = **0.1382** are estimated based on the minimum AIC value, i.e., �5.7612. The

The ARMA (2, 1) model is used to approximate the MEMS Gyro sensor as:

where **Φ** is the AR coefficients and *θ* is the MA model parameter, *ε<sup>n</sup>* is the system

*Yn*�<sup>2</sup> � �

þ

. In the CKF, the process and measurement noise covariance matrices

*Yn Yn*�<sup>1</sup> � �

where *Wk* <sup>=</sup> *<sup>ε</sup><sup>k</sup> <sup>ε</sup><sup>k</sup>*�<sup>1</sup> ½ �*<sup>T</sup>* is the process noise. The initialize the state estimate *<sup>x</sup>*^<sup>0</sup> <sup>=</sup> ½ � 0 0 *<sup>T</sup>* and state error covariance, *<sup>P</sup>*^<sup>0</sup> <sup>=</sup> *<sup>I</sup>* are selected. In practice, the process noise covariance matrix and the measurement noise covariance matrix are assumed as

are constant whereas in the adaptive proposed algorithms, these parameters are

**Model** *φ***<sup>1</sup>** *φ***<sup>2</sup>** *θ***<sup>1</sup> AIC** AR(1) �0.5422 �5.1769 AR(2) �0.5422 �0.1204 �5.1832 MA(1) �0.1382 �5.1860 ARMA(1, 1) �0.5422 �0.1382 �5.5726 ARMA(2, 1) �0.5422 �0.1382 �5.7612

The experimental setup consists of a single axis a prototype Xsens MTi 10 series MEMS sensor, turn table control unit, data acquisition board, and data processing computer. The MEMS gyroscope specification and test conditions of three single axis

*Yn* ¼ *φ*1*Yn*�<sup>1</sup> þ *φ*2*Yn*�<sup>2</sup> þ *θ*1*ε<sup>n</sup>*�<sup>1</sup> þ *ε<sup>n</sup>* (49)

1 *θ*<sup>1</sup>

*n*. State-space represen-

0 0 � �*Wk* (50)

þ *Vk* (51)

i.e., SHAFKF-R adaptation, is shown in **Figure 3**.

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

parameters are tabulated in **Table 3**.

*σ*2 *Wk* 0 0 *σ*<sup>2</sup> *Vk*

**Table 3.**

**49**

" #

changed iteratively.

**7. Test results and discussion**

*Parameters estimation results with AIC values.*

**6. Designing state space model for ARMA (2, 1) model**

input Gaussian white noise with zero mean and variance *σ*<sup>2</sup>

*Xk* <sup>¼</sup> *<sup>φ</sup>*<sup>1</sup> *<sup>φ</sup>*<sup>2</sup>

1 0 � � *Yn*�<sup>1</sup>

*Zk* ¼ ½ � 1 0

tation of the optimal ARMA (2, 1) model is described as

Furthermore, using the suboptimal state error covariance *P*^*vk* similar to Eq. (41), the estimated covariance matrix of the residual sequence can be written as

$$\mathbf{C}\_{\overline{v}k} = H\mathbf{P}\_k^-H^T + \mathbf{R}\_k \tag{45}$$

The suboptimal estimation of covariance matrix of residual sequence using the average window method is

$$
\hat{\mathbf{C}}\_{\overline{v}k} = \frac{1}{D\_s} \sum\_{j=j\mathbf{0}}^k \overline{V}\_k(j) \overline{V}\_k^T(j) \tag{46}
$$

The transitive factor *a*2ð Þ k for the stage two is evaluated as

$$a\_2(\mathbf{k}) = \begin{cases} 1, & tr(\mathbf{C}\_{\overline{v}k}) > tr\left(\hat{\mathbf{C}}\_{\overline{v}k}\right) \\ \frac{tr(\mathbf{C}\_{\overline{v}k})}{tr\left(\hat{\mathbf{C}}\_{\overline{v}k}\right)}, & O(therwise \end{cases} \tag{47}$$

In this algorithm, the measurement noise covariance matrix is scaled by a factor *a*2ð Þ k . Thus Eq. (48) can be rewritten as

$$\mathbf{C\_{\overline{vk}}} = HP\_k^- H^T + a\_2(\mathbf{k}) R\_k \tag{48}$$

**Figure 3.**

*Flow chart of the SHAFKF-R adaptation algorithm.*

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

The Kalman gain and state equations are updated as Eqs. (41)–(46). In this algorithm, measurement noise covariance matrix is multiplied by the adaptive transitive factor, *a*2ð Þ k . If *a*2ð Þ k large, *Rk* becomes larger, this helps to reduce the influence of uncertain measurement noise [23, 24]. The flow chart of the stage two, i.e., SHAFKF-R adaptation, is shown in **Figure 3**.

#### **6. Designing state space model for ARMA (2, 1) model**

The ARMA (p, q) model order is obtained using AIC method as in **Table 3**. The minimum values of AIC can be decided the optimal order of the ARMA (2, 1) is chosen. The ARMA (2, 1) model parameters such as **Φ1** = �**0.5422**, **Φ2** = �**0.1204** and *θ***1** = **0.1382** are estimated based on the minimum AIC value, i.e., �5.7612. The parameters are tabulated in **Table 3**.

The ARMA (2, 1) model is used to approximate the MEMS Gyro sensor as:

$$Y\_n = \rho\_1 Y\_{n-1} + \rho\_2 Y\_{n-2} + \theta\_1 \varepsilon\_{n-1} + \varepsilon\_n \tag{49}$$

where **Φ** is the AR coefficients and *θ* is the MA model parameter, *ε<sup>n</sup>* is the system input Gaussian white noise with zero mean and variance *σ*<sup>2</sup> *n*. State-space representation of the optimal ARMA (2, 1) model is described as

$$X\_k = \begin{bmatrix} \rho\_1 & \rho\_2 \\ \mathbf{1} & \mathbf{0} \end{bmatrix} \begin{bmatrix} Y\_{n-1} \\ Y\_{n-2} \end{bmatrix} + \begin{bmatrix} \mathbf{1} & \theta\_1 \\ \mathbf{0} & \mathbf{0} \end{bmatrix} W\_k \tag{50}$$

$$Z\_k = \begin{bmatrix} \mathbf{1} & \mathbf{0} \end{bmatrix} \begin{bmatrix} Y\_n \\ Y\_{n-1} \end{bmatrix} + V\_k \tag{51}$$

where *Wk* <sup>=</sup> *<sup>ε</sup><sup>k</sup> <sup>ε</sup><sup>k</sup>*�<sup>1</sup> ½ �*<sup>T</sup>* is the process noise. The initialize the state estimate *<sup>x</sup>*^<sup>0</sup> <sup>=</sup> ½ � 0 0 *<sup>T</sup>* and state error covariance, *<sup>P</sup>*^<sup>0</sup> <sup>=</sup> *<sup>I</sup>* are selected. In practice, the process noise covariance matrix and the measurement noise covariance matrix are assumed as *σ*2 *Wk* 0 " #

0 *σ*<sup>2</sup> *Vk* . In the CKF, the process and measurement noise covariance matrices

are constant whereas in the adaptive proposed algorithms, these parameters are changed iteratively.


#### **Table 3.**

*Vk* ¼ *zk* � *Hx*^*<sup>k</sup>* � ^*rk* (44)

*<sup>k</sup> <sup>H</sup><sup>T</sup>* <sup>þ</sup> *Rk* (45)

� �

*<sup>k</sup> <sup>H</sup><sup>T</sup>* <sup>þ</sup> *<sup>a</sup>*2ð Þ <sup>k</sup> *Rk* (48)

*<sup>k</sup>* ð Þ*j* (46)

(47)

Furthermore, using the suboptimal state error covariance *P*^*vk* similar to Eq. (41),

The suboptimal estimation of covariance matrix of residual sequence using the

*Vk*ð Þ*<sup>j</sup> <sup>V</sup><sup>T</sup>*

<sup>1</sup>*, tr C*ð Þ *vk* <sup>&</sup>gt;*tr <sup>C</sup>*^*vk*

� � *, Otherwise*

In this algorithm, the measurement noise covariance matrix is scaled by a factor

*j*¼*j*0

the estimated covariance matrix of the residual sequence can be written as

*Cvk* ¼ *HP*�

*<sup>C</sup>*^*vk* <sup>¼</sup> <sup>1</sup> *Ds* X *k*

The transitive factor *a*2ð Þ k for the stage two is evaluated as

8 >>><

>>>:

*tr C*ð Þ *vk tr C*^*vk*

*Cvk* ¼ *HP*�

*a*2ð Þ¼ k

*a*2ð Þ k . Thus Eq. (48) can be rewritten as

**Figure 3.**

**48**

*Flow chart of the SHAFKF-R adaptation algorithm.*

average window method is

*Gyroscopes - Principles and Applications*

*Parameters estimation results with AIC values.*

#### **7. Test results and discussion**

The experimental setup consists of a single axis a prototype Xsens MTi 10 series MEMS sensor, turn table control unit, data acquisition board, and data processing computer. The MEMS gyroscope specification and test conditions of three single axis gyro sensor detailed results are reported in [36]. The experimental raw data is collected for 1 hour duration with sampling frequency at 100 Hz at room temperature. In the static condition, MEMS gyro is in zero rotation under the room temperature, for a more detailed specification of the Xsens MTi 100 series MEMS please refer to [36, 37].

test the zero mean values for the sensor raw data before analyzing the Allan variance (AV) results [16]. Three single-axes of the MEMS Gyro sensor signals and corresponding AV results are plotted in **Figure 4a** and **b** respectively. From these figures, we see that the 1/2 slope indicates the angle random walk (ARW), which is a white noise characteristics. Bias instability (Bs) is due to internal and external electronic components of the sensor and is indicated at zero slope in log-log AV plot [16]. The three axes of MEMS IMU sensors are identified and quantified using AV

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

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

*(a) X-axis MEMS gyro signal and de-noised results using the SHAFKF algorithm and (b) corresponding Allan*

**Figure 5.**

**51**

*variance plot.*

### **7.1 Static performance test analysis**

Three single-axis MEMS gyro sensor raw data are collected for 1 hour duration with sampling frequency at 100 Hz. The pre-processing methods are required to

**Figure 4.** *(a) Three single axes of MEMS gyroscope raw signals and (b) corresponding Allan variance plot.*

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

test the zero mean values for the sensor raw data before analyzing the Allan variance (AV) results [16]. Three single-axes of the MEMS Gyro sensor signals and corresponding AV results are plotted in **Figure 4a** and **b** respectively. From these figures, we see that the 1/2 slope indicates the angle random walk (ARW), which is a white noise characteristics. Bias instability (Bs) is due to internal and external electronic components of the sensor and is indicated at zero slope in log-log AV plot [16]. The three axes of MEMS IMU sensors are identified and quantified using AV

**Figure 5.** *(a) X-axis MEMS gyro signal and de-noised results using the SHAFKF algorithm and (b) corresponding Allan variance plot.*

gyro sensor detailed results are reported in [36]. The experimental raw data is collected for 1 hour duration with sampling frequency at 100 Hz at room temperature. In the static condition, MEMS gyro is in zero rotation under the room temperature, for a more detailed specification of the Xsens MTi 100 series MEMS please refer to [36, 37].

Three single-axis MEMS gyro sensor raw data are collected for 1 hour duration with sampling frequency at 100 Hz. The pre-processing methods are required to

*(a) Three single axes of MEMS gyroscope raw signals and (b) corresponding Allan variance plot.*

**7.1 Static performance test analysis**

*Gyroscopes - Principles and Applications*

**Figure 4.**

**50**

**Figure 6.** *(a) Y-axis MEMS gyro signal and de-noised results using the SHAFKF algorithm and (b) corresponding Allan variance plot.*

analysis, which are presented in **Table 2**. From this table, we can observe that ARW and BI are the two that dominate noises in the presence of the MEMS sensor.

algorithm, an innovation sequence is used to adjust the noise parameters of process and measurement noise matrices ad it is followed by covariance matching principle. In the IAE-AKF algorithm, the window width selection is critical and can decide the filter optimality. In general, the window width is varied between 5 and 30. In this analysis, we observed that 15 samples of the window width is the optimal choice for

*(a) Z-axis MEMS gyro signal and de-noised results using the SHAFKF algorithm and (b) corresponding Allan*

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

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

statistical smoothing.

**Figure 7.**

**53**

*variance plot.*

Conventional Kalman filter (CKF) algorithm is applied for minimizing the all three axis MEMS gyro static signal. In this experiment, the initial values of measurement and process noise covariance matrix are chosen as 0.098 and 0.0001 respectively. In practical application, these noise covariance matrices vary with time. In real-time, by adjusting the noise parameters are critical. The adaptive KF *Modeling of Inertial Rate Sensor Errors Using Autoregressive and Moving Average (ARMA)… DOI: http://dx.doi.org/10.5772/intechopen.86735*

**Figure 7.**

analysis, which are presented in **Table 2**. From this table, we can observe that ARW and BI are the two that dominate noises in the presence of the MEMS sensor. Conventional Kalman filter (CKF) algorithm is applied for minimizing the all three axis MEMS gyro static signal. In this experiment, the initial values of measurement and process noise covariance matrix are chosen as 0.098 and 0.0001 respectively. In practical application, these noise covariance matrices vary with time. In real-time, by adjusting the noise parameters are critical. The adaptive KF

*(a) Y-axis MEMS gyro signal and de-noised results using the SHAFKF algorithm and (b) corresponding Allan*

**Figure 6.**

**52**

*variance plot.*

*Gyroscopes - Principles and Applications*

*(a) Z-axis MEMS gyro signal and de-noised results using the SHAFKF algorithm and (b) corresponding Allan variance plot.*

algorithm, an innovation sequence is used to adjust the noise parameters of process and measurement noise matrices ad it is followed by covariance matching principle. In the IAE-AKF algorithm, the window width selection is critical and can decide the filter optimality. In general, the window width is varied between 5 and 30. In this analysis, we observed that 15 samples of the window width is the optimal choice for statistical smoothing.

In the SHAKF algorithm, the innovation sequence is used to estimate the measurement noise covariance matrix and followed by sliding window average method. In addition, statistical noise estimator is used in the AKF frame work for updating the noise coefficients in each iteration recursively. The window width is 15 samples for statistical smoothing. The SHAKF algorithm results are plotted in **Figures 5a**–**7a**, respectively.

In the proposed approach, the predicted state error covariance is updated by one transitive factor whereas the measurement noise covariance matrix is updated using another transitive factors based on the residual sequence. The covariance matrix of residual sequence is estimated using sliding average window method. In this method, window width is chosen empirically as 15. In the first stage of the proposed algorithm (SHAFKF-P adaption), the transitive factor (*a*1) is calculated in stage one. The measurement noise covariance matrix is scaled by an adaptive transitive factor (*a*2) is in the second stage. The transitive factors are used to scale *Rk* and reciprocal to *P*^� *<sup>k</sup>* for reducing the variance of uncertainty in the process model and measurements, respectively. The developed algorithm is also applied to X, Y and Zaxis MEMS gyroscope static signal. The test results of the proposed algorithm for X, Y and Z-axis data are shown in **Figures 5a**–**7a**, respectively. From these figures, it is observed that the angle random walk (ARW) and bias instability (Bs) noise are the dominated noise sources. The quantified noise coefficients are tabulated in the **Tables 4**–**6**, respectively. All the random noise and drift are quantified before and after applying the de-noising algorithm. The drift is also calculated before and after de-noising MEMS signal and tabulated in **Tables 4**–**6**, respectively. From these tables, it is observed that the ARW is reduced by 1000 and also Bs random noise is minimalized by order of 100 compared to the original value.

measurement noise covariance is scaled by the transitive factor. It ensures the variance is inversely proportional to the uncertainty of measurement. Due to this,

*Allan variance and drift results of Z-axis MEMS gyro using proposed scheme in static condition.*

MEMS raw data 38.9222 9.8105 1.758 CKF 24.3805 8.3228 1.216 SHKF 7.0068 3.502 0.597 IAE-SHAKF 3.3229 3.071 0.0013 SHAFKF-P adaption 1.2516 0.832 0.00028 SHAFKF-R adaption 0.914 0.542 0.00014

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

In addition, we observed the Drift error for the MEMS gyroscope signals. Drift error is considered as one of the performance indicator of all the proposed algorithms. From **Tables 4**–**6**, it is observed that the proposed SHAFKF-R adaptation filter performs better than CKF, IAE-AKF SHAKF, and SHAFKF-P adaptation filters because of that the measurement noise covariance tunes by the adaptive transitive factor *a*2ð Þ k to reduce the influence of uncertainty in measurement noise of

*hr* <sup>p</sup> **) BS (°***=hr***) Drift (°***=hr***)**

In this chapter, the MEMS gyroscope drift is modeled by using ARMA (2, 1) for characterizing the MEMS gyro noise behavior. Moreover, ARMA-based linear Sage-Husa adaptive fading Kalman filter with double transitive factors is proposed. In the proposed algorithm, double adaptive transitive factors are used to update in the predicted state vector and measurement noise covariance matrix. The suggested algorithm is used to reduce the drift and random noise in the presence of MEMS gyroscope. From the AV analysis, the noise terms of ARW and Bs are reduced by order of 100. The proposed SHAFKF outperforms the CKF, IAE-AKF, and SHAKF algorithms in static case. It concludes that the SHAFKF algorithm is suitable for

SHAFK-R adaptation algorithm outperforms other algorithms.

**Methods ARW (°***=* ffiffiffiffiffi

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

the sensor.

**55**

**Table 6.**

**8. Conclusions**

MEMS gyroscope signal drift minimization.

**Methods ARW (°***=* ffiffiffiffiffi *hr* <sup>p</sup> **) BS (°***=hr***) Drift (°***=hr***)** MEMS raw data 165.115 8.775 1.758 CKF 103.235 7.459 1.362 IAE-AKF 24.858 3.496 0.859 SHAKF 4.228 2.296 0.0014 SHAFKF-P Adaption 1.279 0.690 0.00038 SHAFKF-R Adaption 0.331 0.421 0.00012

From these tables, it is evident that SHAFKF of R adaptation using transitive factor improves the performance of the algorithm. In this proposed algorithm,

#### **Table 4.**

*Allan variance and drift results of X-axis MEMS gyro using proposed scheme in static condition.*


#### **Table 5.**

*Allan variance and drift results of Y-axis MEMS gyro using proposed scheme in static condition.*

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


**Table 6.**

In the SHAKF algorithm, the innovation sequence is used to estimate the measurement noise covariance matrix and followed by sliding window average method. In addition, statistical noise estimator is used in the AKF frame work for updating the noise coefficients in each iteration recursively. The window width is 15 samples for statistical smoothing. The SHAKF algorithm results are plotted in **Figures 5a**–**7a**,

In the proposed approach, the predicted state error covariance is updated by one transitive factor whereas the measurement noise covariance matrix is updated using another transitive factors based on the residual sequence. The covariance matrix of residual sequence is estimated using sliding average window method. In this method, window width is chosen empirically as 15. In the first stage of the proposed algorithm (SHAFKF-P adaption), the transitive factor (*a*1) is calculated in stage one. The measurement noise covariance matrix is scaled by an adaptive transitive factor (*a*2) is in the second stage. The transitive factors are used to scale *Rk* and

measurements, respectively. The developed algorithm is also applied to X, Y and Zaxis MEMS gyroscope static signal. The test results of the proposed algorithm for X, Y and Z-axis data are shown in **Figures 5a**–**7a**, respectively. From these figures, it is observed that the angle random walk (ARW) and bias instability (Bs) noise are the dominated noise sources. The quantified noise coefficients are tabulated in the **Tables 4**–**6**, respectively. All the random noise and drift are quantified before and after applying the de-noising algorithm. The drift is also calculated before and after de-noising MEMS signal and tabulated in **Tables 4**–**6**, respectively. From these tables, it is observed that the ARW is reduced by 1000 and also Bs random noise is

From these tables, it is evident that SHAFKF of R adaptation using transitive factor improves the performance of the algorithm. In this proposed algorithm,

*hr*

MEMS raw data 165.115 8.775 1.758 CKF 103.235 7.459 1.362 IAE-AKF 24.858 3.496 0.859 SHAKF 4.228 2.296 0.0014 SHAFKF-P Adaption 1.279 0.690 0.00038 SHAFKF-R Adaption 0.331 0.421 0.00012

*Allan variance and drift results of X-axis MEMS gyro using proposed scheme in static condition.*

*hr*

MEMS raw data 33.0437 4.7297 1.7587 CKF 30.6510 1.5762 1.3624 SHKF 4.0098 1.2862 0.0859 IAE-SHAKF 3.4075 0.4371 0.00386 SHAFKF-P adaption 0.6570 0.2653 0.000562 SHAFKF-R adaption 0.442 0.150 0.000312

*Allan variance and drift results of Y-axis MEMS gyro using proposed scheme in static condition.*

<sup>p</sup> **) BS (°***=hr***) Drift (°***=hr***)**

<sup>p</sup> **) BS (°***=hr***) Drift (°***=hr***)**

minimalized by order of 100 compared to the original value.

**Methods ARW (°***=* ffiffiffiffiffi

**Methods ARW (°***=* ffiffiffiffiffi

*<sup>k</sup>* for reducing the variance of uncertainty in the process model and

respectively.

*Gyroscopes - Principles and Applications*

reciprocal to *P*^�

**Table 4.**

**Table 5.**

**54**

*Allan variance and drift results of Z-axis MEMS gyro using proposed scheme in static condition.*

measurement noise covariance is scaled by the transitive factor. It ensures the variance is inversely proportional to the uncertainty of measurement. Due to this, SHAFK-R adaptation algorithm outperforms other algorithms.

In addition, we observed the Drift error for the MEMS gyroscope signals. Drift error is considered as one of the performance indicator of all the proposed algorithms. From **Tables 4**–**6**, it is observed that the proposed SHAFKF-R adaptation filter performs better than CKF, IAE-AKF SHAKF, and SHAFKF-P adaptation filters because of that the measurement noise covariance tunes by the adaptive transitive factor *a*2ð Þ k to reduce the influence of uncertainty in measurement noise of the sensor.

## **8. Conclusions**

In this chapter, the MEMS gyroscope drift is modeled by using ARMA (2, 1) for characterizing the MEMS gyro noise behavior. Moreover, ARMA-based linear Sage-Husa adaptive fading Kalman filter with double transitive factors is proposed. In the proposed algorithm, double adaptive transitive factors are used to update in the predicted state vector and measurement noise covariance matrix. The suggested algorithm is used to reduce the drift and random noise in the presence of MEMS gyroscope. From the AV analysis, the noise terms of ARW and Bs are reduced by order of 100. The proposed SHAFKF outperforms the CKF, IAE-AKF, and SHAKF algorithms in static case. It concludes that the SHAFKF algorithm is suitable for MEMS gyroscope signal drift minimization.

*Gyroscopes - Principles and Applications*
