Vibration Noise Modeling for Measurement While Drilling ... - MDPI

sensors Article

Vibration Noise Modeling for Measurement While Drilling System Based on FOGs Chunxi Zhang, Lu Wang, Shuang Gao *, Tie Lin and Xianmu Li Key Laboratory of Inertial Technology, Institute of Opto-electronics Technology, School of Instrument Science and Opto-electronics Engineering, Beihang University, Beijing 100191, China; [email protected] (C.Z.);[email protected] (L.W.); [email protected] (T.L.); [email protected] (X.L.) * Correspondence: [email protected]; Tel.: +86-138-1002-5968; Fax: +86-10-8231-6547 (ext. 818) Received: 22 August 2017; Accepted: 12 October 2017; Published: 17 October 2017

Abstract: Aiming to improve survey accuracy of Measurement While Drilling (MWD) based on Fiber Optic Gyroscopes (FOGs) in the long period, the external aiding sources are fused into the inertial navigation by the Kalman filter (KF) method. The KF method needs to model the inertial sensors’ noise as the system noise model. The system noise is modeled as white Gaussian noise conventionally. However, because of the vibration while drilling, the noise in gyros isn’t white Gaussian noise any more. Moreover, an incorrect noise model will degrade the accuracy of KF. This paper developed a new approach for noise modeling on the basis of dynamic Allan variance (DAVAR). In contrast to conventional white noise models, the new noise model contains both the white noise and the color noise. With this new noise model, the KF for the MWD was designed. Finally, two vibration experiments have been performed. Experimental results showed that the proposed vibration noise modeling approach significantly improved the estimated accuracies of the inertial sensor drifts. Compared the navigation results based on different noise model, with the DAVAR noise model, the position error and the toolface angle error are reduced more than 90%. The velocity error is reduced more than 65%. The azimuth error is reduced more than 50%. Keywords: FOG-based MWD; system noise model; vibration noise; dynamic Allan variance

1. Introduction In the oil industry, horizontal drilling processes make use of Measurement While Drilling (MWD) instruments to monitor the position and the orientation of the bottom hole assembly (BHA). The traditional MWD instrument is comprised of three accelerometers and three magnetometers. The magnetometers determine the orientations by measuring the earth magic field. The major defect using the magnetometers is that their accuracy is influenced by the magnetic field existing in the oil hole. Consequently, the deteriorative measurements accuracy could lead to drilling failure [1]. For the sake of improving the survey accuracy, gyroscopic measure techniques based on fiber optic gyroscopes (FOGs) have been put forward to replace the magnetic instruments [2]. Though the FOG-based inertial navigation technique is a mature technique in the military domain, it encounters a lot of special problems because of the harsh work environment of drilling. It needs to work over 200 h with a high shock and strong vibration [1]. As we all know, the inertial errors accumulate with time, which leads to accuracy degradation. For long-term measuring and high accuracy surveying, the external aiding observations are fused into the FOG-based MWD by Kalman filter (KF), such as the drilling pipe length, the penetration rate and the zero velocity update (ZUPT) [3–5]. In KF, the inertial sensor’s noise should be modeled to be used as the system noise. They were often modeled as the stationary white Gaussian noise. However, during the vibration, the gyroscope noise is no longer stationary white Gaussian noise. What’s worse, an incorrect system noise model will influence the performance and Sensors 2017, 17, 2367; doi:10.3390/s17102367

www.mdpi.com/journal/sensors

Sensors 2017, 17, 2367

2 of 17

accuracy of KF greatly. Therefore, it is very critical to put forward an accurate modeling approach for the system noise. Various approaches were researched to model the random errors in the gyros and accelerometers. In Reference [6], the stochastic errors in the gyroscope were modeled with the autoregressive moving average model (ARMA). In Reference [7], the stochastic errors in gyros and accelerometers were simplified as an autoregressive (AR) model. But the ARMA model and the AR model were unsuitable to process higher-order stochastic processes, and high-dynamic ranges [8]. In Reference [9], the stochastic errors were assumed to be a stationary Gaussian-Markov process. Reference [10] pointed out that the stochastic noise of the inertial sensors were not stationary. Thus, the stationary Gaussian-Markov process was not an accurate model for the stochastic process. What’s worse, it is very difficult to determine the order, the coefficients, and the time constant of a Gaussian-Markov model for a particular stochastic noise. In Reference [11], the stochastic errors in the inertial sensors were assumed to be a white noise process. The user’s manual or data sheet of inertial sensors often disturb the power spectral density functions (PSDs) of this white noise process. Based on the PSD, the noises covariance matrix (Q) of the white noise which was needed in the Kalman Filter could be obtained. However, the real stochastic process in an inertial sensor may be much more complicated than white noise. For most applications this assumption is an approximation and a simple model; it sacrifices performances and precision. Reference [12] designed a colored-noise model for KF to diminish the effects of the vibration error. But this method was verified by the simulation. In reference [13], the random noise of the Micro-Electro-Mechanical System (MEMS) inertial sensor were identified and modeled by Allan variance. This model was applied into this low cost Inertial navigation system (INS) integrated system to improve the accuracy and performance of the system. But in order to simplify the model, some components of the stochastic noise were ignored, such as Quantization noise. In References [14,15], the stochastic errors in an inertial sensor were identified by Allan variance, and an equivalent differential equation representation for each kind of stochastic noise was established. The equivalent differential equation was augmented into the KF of GPS/INS integration. However, after augmenting, the KF estimation process will become much more complicated and take more time. In 2003, L. Galleani and P. Tavella developed the dynamic Allan variance (DAVAR) to track and reveal the anomy and non-stationary in the atom clock behavior [16]. In contrast to Allan variance, DAVAR can track and reveal the non-stationary characteristics of time series [17]. Li et al. [18], Wei et al. [19] and Zhang et al. [20] utilized the DAVAR to describe the non-stationary of the laser gyroscope. Wang et al. took advantage of DAVAR to identify and characterize the vibration noise of FOGs in the MWD system [21,22]. However, DAVAR hasn’t been applied to model the random noise to improve the performance of the KF. In this paper, the gyroscope vibration noise is identified by DAVAR, and then a noise model based on DAVAR was developed for the first time. With this accurate vibration noise model, the performance and the accuracy of the KF could be improved. The organization of this paper is as follows. In the Section 2, the Dynamic Allan variance method is introduced simply. Section 3, the KF for MWD is designed. Section 4, the new noise modeling method was developed and the KF based on the new model is designed. The experimental results are presented in the Section 5. Section 6 is the conclusion. 2. Dynamic Allan Variance The Allan variance for FOGs is defined as follows [23], σω2 (τ ) =

E 1D (ω (t + τ ) − ω (t))2 2

(1)

where τ is the observation interval, ω (t) is FOG output data. and hi is a symbol indicating a time averaging. The average of ω (t) is calculated by ω (t) =

1 τ

Z t+τ t

ω (u)du

(2)

Sensors 2017, 17, 2367

3 of 17

where u is the integral variable. The dynamic Allan variance (DAVAR) is developed based on Allan variance. It is a sliding Allan variance. Its computation process can be described as following. Firstly, at a given time epoch, we truncate the FOG data with a rectangular window. Secondly, the Allan variance of the truncated data could be calculated using Equation (1). As a result, we get the Allan variance at a given time epoch. Then repeating these two steps at every time epoch, the Allan variance at each time epoch can be obtained. In the end, plotting all the variances in a 3D graph, we can obtain the DAVAR figure. Please refer to reference [17] to get the detailed computation process. In this paper, we describe the definition of DAVAR as shown in Equation (3). σω2 (t, τ ) =

1

Z t+ Nw −τ 2

2τ 2 ( Nw − 2τ )

t− N2w +τ

(ω (u + τ ) − ω (u))2 du

(3)

where Nw is the length of the truncation window, t is the analysis time epoch, τ is the observation interval, σω2 (t, τ ) is the DAVAR. 3. Noise Modeling Based on DAVAR El-Sheimy et al. [24] showed that a unique relationship existing between Allan variance σω2 (τ ) and the power spectral density (PSD) of the random noise. It is, σω2 (τ ) = 4

Z ∞ 0

Sω ( f )

sin4 (π f τ )

( π f τ )2

du

(4)

where Sω ( f ) is the PSD of the random process ω (t). Substituting PSD of any physical meaningful random process into Equation (4,), we can obtain the Allan variance σω2 (τ ) of this random process [23]. As far as we all know, there are five kinds of noise existing in the random noise of the gyros. These are the quantization noise (Q), angular random walk (N), bias instability (B), rate random walk (R), and the rate slope (K). The quantization noise is one of the errors introduced into an analog signal by encoding it in digital form. The gyro angle random walk was contributed by the high-frequency noise terms that have correlation time much shorter than the sample time. They come from the light path noise of the gyro. The origin of bias instability noise is the electronics or other components that are susceptible to random flickering. Because of its low-frequency nature, it is indicated as the bias fluctuations in the data. Rate random walk noise is a random process of uncertain origin, possibly a limiting case of an exponentially correlated noise with a very long correlation time. The rate slope is considered to be a kind of deterministic error. It is caused by the slowly changing of the light source intensity or caused by the temperature of the environment. When the PSD of one random process passed through a filter with the transfer function

sin4 (π f τ ) , ( π f τ )2

we can get its corresponding Allan variance. That means that different types of random processes can appear in different region of τ and the different types of random noise can be examined by regulating τ [25]. With this particular characteristic, the various noise terms existing in the gyro could be identified. Table 1 summarized a relationship between noise terms and the Allan variance and observation τ.

Sensors 2017, 17, 2367

4 of 17

Table 1. Relationship between noise terms, Allan variance and τ. Noise Coefficient

Noise Terms

SΩ (v) PSD of the Random Process (

the quantization noise

Q

angular random walk

N

4Q2 τ

sin2 (π f τ ) (2π f )2 τQ2

SΩ ( f ) =

B

rate random walk

K

the rate slope

1 2τ 1 2τ

SΩ ( f ) = N2 (

bias instability

f ≥ f
f0 0 f 0 cutoff frequency

SΩ ( f ) =

2B2 π

2

K SΩ ( f ) = ( 2π ) f12

SΩ ( f ) =

R

σ 2 (τ )

ln 2

K2 τ 3

2

R (2π f )3

2 2

R τ 2

0 0.5 1

Assuming that the noise terms existing in the gyro are statistically independent, the Allan variance could be rewritten as the sum of the Allan variances due to each random process at the different τ [25]. In other words, 2 ( τ ) + σ2 ( τ ) + σ2 ( τ ) + σ2 ( τ ) + σ2 ( τ ) σ2 (τ ) = σQ Q N N B B K K R R

=

3Q2 τQ 2

+

= C−2 τQ

N2 τN −2

+

2B2 π

ln 2 +

K2 τK 3

+

R2 τR 2 2

(5)

+ C−1 τN −1 + C0 τB 0 + C1 τK 1 + C2 τR 2

2 ( τ ), σ2 ( τ ), σ2 ( τ ), σ2 ( τ ), σ2 ( τ ) stand for Allan variance of each random noise term. where σQ Q N N B B K K R R τQ , τN , τB , τK , and τR are the observation times of individual random processes, respectively. C−2 , C−1 , C0 , C1 and C2 are the polynomial coefficients of the σ2 (τ ) − τ. With the line fitting approach, the relationship between the noise coefficients and the coefficients of the σ2 (τ ) could be established. If the unit of gyroscope signal is degree per hour (◦ /h), each noise coefficient could be obtained as follows: √ √ √ √ √ 1 3 C 106 π C−2 00 C0 ◦ √ ( ), R = 3600 2C2 (◦ /h2 ) ( /h), Q = N = 60−1 (◦ /h 2 ), K = 60 3C1 (◦ /h 2 ), B = 0.664 (6) 180×3600× 3

DAVAR is an assembling of the Allan variances at each the time point. Therefore, at any given time epoch, the DAVAR can be rewritten as Equation (7): 2 2 σ2 (t, τ ) = σQ (t, τQ ) + σN (t, τN ) + σB2 (t, τB ) + σK2 (t, τK ) + σR2 (t, τR )

(7)

Fitting the σ(t, τ ) − τ curve, the noise coefficients Q(t), N(t), B(t), K(t), and R(t) at that time t could be obtained. Therefore, coefficients for all the time t could be obtained. With all these coefficients, we can establish the dynamic model for the random error of the gyroscope. Institute of Electrical and Electronics Engineers (IEEE) 952 1997 [23] puts forward that the magnitude of the five noise term can be read off from the slope line of the log σ verse log τ curve, namely bi-logarithmic curves. The magnitude of Quantization noise can be read off from the logσ–logτ √ curve at τQ = 3, the magnitude of Angle random walk noise can be read off from the bi-logarithmic curves of slop −0.5 at τN = √1. The magnitude of Rate random walk can be obtained from the bi-logarithmic curves at τR = 2 and the magnitude of the Rate Slope noise can be read off at τK = 3. The numerical value of the bias instability has nothing with τ. Therefore, for a given time t, substituting observation time τQ , τN , τR , τK and their corresponding noise coefficients into Equation (5), we can obtain the accurate Allan variance including all kinds of noise. This accurate Allan variance which changes with time can be described as follows σ2 ( t )

2 ( t, τ ) + σ2 ( t, τ ) + σ2 ( t ) + σ2 ( t, τ ) + σ2 ( t, τ ) = σQ N K R Q N B K R

=

3Q2 2 τQ

+

N2 2 τN

+

2B2 π

ln 2 +

K2 τK 3

+

R2 τR2 2

(8)

Sensors 2017, 17, 2367

5 of 17

σ2 (t) in Equation (8) is the accurate noise model which could track and exhibit the non-stationary characteristics of the noise. What’s more, it not only contains the white noise, such as angler random walk, but also the color noise, such as Quantization noise, Bias instability noise, Rate random walk noise and Rate lamp noise. Applying this noise model to the Kalman filter may improve the performance of the Kalman filter greatly, which we will discuss in the next section. 4. DAVAR Aided Kalman Filter There are two types of continuous aiding observations which can be fused into the MWD surveying system [1]. The first one is the position information that can be obtained from the continuous measurement of the drill pipe length. The second one is the velocity which can be derived from pipe length and time. Therefore, position/velocity loose coupled navigation approach for MWDs is chosen in this paper. 4.1. Kalman Filter Model The state equation is expressed as follows: Xk = Fk,k−1 Xk−1 + Gk−1 Wk−1 Xk−1 = [ δL

δλ

δh

δV e

δV n

δV u

δI

δT

δA

aBx

Wk−1 = [01×3 waT w Tg 01×6 ]

(9)

aBy

aBz

gBx

gBy

gBz

]

T

(10)

T

(11)

where Xk is the error states vector. δL is the latitude error, δλ is longitude error, and δh are height error. δV e is the east velocity errors, δV n is the north velocity error, and δV u is up velocity error. δI is the inclination error, δT is the toolface angle error and δA is azimuth error. aBx , aBy and aBz are the bias errors of accelerometers along the X axis, Y axis and Z axis, respectively. gBx , gBy and gBz are the bias errors of gyroscope along with the X axis, Y axis and Z axis, respectively. Fk,k−1 is the dynamic matrix relating Xk−1 to Xk . Gk−1 is the noise coefficient matrix, and Wk−1 is the system noise vector which has the normal distribution with the variance matrix Qk−1 . wa is the noise model of accelerometers and w g is the noise model of gyroscopes. The measurement equation is Zk = Hk · Xk + Vk (12) Zk = [ Lins − Lupdate

λins − λupdate

hins − hupdate

e − Ve Vins update

n − Vn Vins update

u u Vins − Vupdate ]

T

(13)

Zk is external measurements or observations, Hk is the observation matrix. Vk is the random noise h iT e n u vector for the observations. Vk is random noise model of the observations. Vins is Vins Vins h iT e n u Vupdate Vupdate the velocity obtained from the INS. Vupdate is the drill bit rate of penetration. h iT h iT is the position calculated by the INS. is the Lupdate λupdate hupdate Lins λins hins position calculated by the drilling pipe length. Traditionally, the observation random noise vector Vk and the system measurement noise Wk are both assumed to be white sequence and not correlated with each other. The characteristics and relationship of Wk and Vk are expressed as h i   E[Wk ] = 0, Cov Wk , Wj = E Wk Wj T = Qk δkj

(14)

h i   E[Vk ] = 0, Cov Vk , Vj = E Vk Vj T = Rk δkj h i   Cov Wk , Vj = E Wk Vj T = 0

(15) (16)

Sensors 2017, 17, 2367

6 of 17

( 0, k 6= j δkj = 1, k = j

(17)

At time tk , the estimation Xˆ k of the error state vector Xk could be obtained by KF. Thus, the error covariance matrix of the estimation is described as Equation (19) E

h

Xˆ k − Xk




Xˆ k − Xk


T i

= Pk

(18)

On the diagonal of the error covariance matrix Pk , it is the mean square estimation error (MSEE) of each error state which represents the estimation accuracy. The estimation process starts by providing the prediction Xˆ k/k−1 of the state vector as follows Xˆ k/k−1 = Φk,k−1 Xˆ k−1

(19)

Secondly, we should predict the value of the error covariance matrix Pk/k−1 T T Pk/k−1 = Φk,k−1 Pk−1 Φk,k −1 + Γ k −1 Q k −1 Γ k −1

(20)

Thirdly, with the error covariance matrix Pk/k−1 , the Kalman gain matrix Kk is computed by Equation (21). Kk = Pk/k−1 HkT ( Hk Pk/k−1 HkT + Rk )

−1

(21)

Next, the estimation Xˆ k could be obtained as follows: Xˆ k = Xˆ k/k−1 + Kk ( Zk − Hk Xˆ k/k−1 )

(22)

Finally, the error covariance matrix Pk of the estimate Xˆ k could be calculated by the following equation: Pk = ( I − Kk Hk ) Pk/k−1 (23) Based on the above equations, it can be seen that the Kalman gain matrix Kk is the major contributor to MSEE. Kk is directly proportional to the estimate error covariance Pk/k−1 and inversely proportional to the variance of the measurement noise Rk . According to Equation (20), Pk/k−1 is directly proportional to the system noise variance Qk . In conclusion, the Kk is also proportional to the system noise Qk . Therefore, the Kalman gain Kk presents a ratio of the uncertainty in the state estimate to the uncertainty in observations. Therefore, the accuracy of the measurement noise model Rk and the system noise model Qk could influence the estimation results of the KF. Because the measurement information is always the outer information, we don’t know it very accurately. The only way to improve the accuracy of the KF is to establish an accurate system noise model for the inertial sensors. 4.2. Noise Model Based on DAVAR The system noise vector Qk is always assumed to be white noise whose covariance given by the manufacturer or by the datasheet. However, the noise of the inertial sensors isn’t simple white noise. Especially in the vibration, it contains a lot of color noise. For the sake of providing an optimal estimation of the error states as mentioned above, we established a precise model for the sensors using the DAVAR. On the basis of Equation (6), the variance of the noise at each time in the X axis gyro,

Sensors 2017, 17, 2367

7 of 17

Y axis gyro and Z axis gyro could be obtained. The variance of the noise at time t can be expressed as follows:   03 × 3 03×3 03 × 3 03×6  0 03 × 3 03×6  Q3a ×3  3×3    σgx (t) 0 0   Qk ( t ) =  (24)   03 × 3 03×3 03×6  0 σgy (t) 0     0 0 σgz (t) 06 × 3 06×3 06 × 3 06×6 Q3a ×3 is the constant variance of the noise in the accelerometers which can be got from the manufacture. σgx (t), σgy (t) and σgz (t) is the DAVAR of the X axis gyro, Y axis gyro and Z axis gyro, respectively. Instead of a constant system noise model, the DAVAR noise model is dynamic model which changes with time and varies with different motion. Moreover, it contains not only the white Sensors but 2017,also 17, 2367 8 of 18 noise, the color noise. Applying the DAVAR noise model to the Kalman filter will improve the accurate and performance of the Kalman filter estimation. 5. Experiments 5. Experiments In order to validate that the proposed noise model based on DAVAR outperforms the conventional two vibration experiments done in on theDAVAR laboratory located in China In order toone, validate that the proposed noisewere model based outperforms theBeijing, conventional whose latitude is 39.9778° and longitude is 116.3448°. The FOGs-based MWD system was mounted one, two vibration experiments were done in the laboratory located in Beijing, China whose latitude is ◦ and longitude ◦ . The on the horizontal vibration platform as Figure 1 was shown. The vibration table was a 5Thorizontal vibrating 39.9778 is 116.3448 FOGs-based MWD system was mounted on the table produced by the American Ling Company, Model 1216VH, No. It is a linear vibration table. vibration platform as Figure 1 was shown. The vibration table was a 219. 5T vibrating table produced by 1 σ The accuracy of the FOG in the MWD under test is 0.5°/h ( ) and its total length of the optic fiber the American Ling Company, Model 1216VH, No. 219. It is a linear vibration table. The accuracy of the in theinfiber coil is 600 m.test Theisfirst was and a sine testofwith a fixed frequency 25 Hz fixed FOG the MWD under 0.5◦one /h (1σ) itsvibration total length the optic fiber in the fiber coiland is 600 m. vibration acceleration 5 g which imitated the low frequency vibration existing in the drilling process. The first one was a sine vibration test with a fixed frequency 25 Hz and fixed vibration acceleration Standards ofthe the low petroleum and vibration natural gas industry People’s Republic requires 5As g the which imitated frequency existing in of thethe drilling process. As of theChina Standards of [26],petroleum the MWDand should perform such a vibration for more than 1ofhour to requires test its reliability before the natural gas industry of the People’s Republic China [26], the MWD puttingperform the MWD instrument into The1second random vibration whosethe vibration should such a vibration forpractice. more than hour toone testwas its reliability before putting MWD PSD was reported in Figure Figureone 2a is therandom demand vibration PSD. vibration The Figure 2b was is the output in of instrument into practice. The2.second was vibration whose PSD reported the vibration table. Root-mean-square of PSD. the whole vibration is 13.12 While drilling, Figure 2. Figure 2a The is the demand vibration The Figure 2b magnitude is the output of theg.vibration table. a vibration sensor wasofmounted onvibration the MWD and collected theg.vibration information downsensor hole. The Root-mean-square the whole magnitude is 13.12 While drilling, a vibration Thismounted vibrationon PSD theand FFTcollected result ofthe thevibration vibrationinformation sensor data. Therefore, it can simulate strong was thewas MWD down hole. This vibration PSD was vibration while drilling. the FFT result of the vibration sensor data. Therefore, it can simulate strong vibration while drilling.

Figure Figure 1. 1. Vibration Vibration experiment experiment setup. setup.

Sensors 2017, 17, 2367

8 of 17

Figure 1. Vibration experiment setup.

Sensors 2017, 17, 2367

9 of 18

(a)

(b) Figure 2.2.power powerspectral spectraldensity density (PSD) of the random vibration. (a) of PSD the vibration; the Figure (PSD) of the random vibration. (a) PSD theof vibration; (b) the (b) output output of the vibration of the vibration table. table.

Because the MWD vibrated in the fixed place, its position didn’t change and was always the Because the MWD vibrated in the fixed place, its position didn’t change and was always the same same as the initial position. Its velocity could be assumed as zeros with random noise. Therefore the as the initial position. Its velocity could be assumed as zeros with random noise. Therefore the aiding aiding source for the vibration experiment is the initial position and the velocity. The measurement source for the vibration experiment is the initial position and the velocity. The measurement vector in vector in Equation (12) can be described as follows; Equation (12) can be described as follows; e n u T Z = [ Lins − Linitial λins − λinitial hins − hinitial Vins −e 0 Vins − 0n Vins − 0 ]Tu Zk = [ Linsk− Linitial −0] λins − λinitial hins − hinitial Vins − 0 Vins − 0 Vins

(25) (25)

5.1. Fixed-Frequency Vibration 5.1. Fixed-Frequency Vibration This vibration test was performed as the steps below. First, the vibration table was in static for vibration testvibration was performed as thetosteps below. First, the vibration table invibration static for aboutThis 5 min. Then the table started vibrate. After vibrating for about 1.5 was h, the aboutreturned 5 min. Then the vibration started to started vibrate.toAfter vibrating for vibrated about 1.5for h, about the vibration table to static. Next, the table vibration table vibrate again and 15 min. table returned to static. Next, the vibration table started to vibrate again and vibrated for about The vibration direction was along the X axis of the MWD. This experiment with different motions can 15 min. The vibration direction was along the X axis of the MWD. This experiment with different motivate the FOG noise. The raw date of the three FOGs was represented in Figure 3. It can be seen motions canthe motivate the the FOG noise. The raw dateFOG of the three FOGsvery waslarge. represented in Figurefrom 3. It that during vibration, noise existing in the data became It was different can be seen that during the vibration, the noise existing in the FOG data became very large. It was the noise in the static, so modeling the FOG noise during vibration as the stationary white noise was different from the noise in the static, so modeling the FOG noise during vibration as the stationary not corrected. whiteThen noise was notmodeling corrected.approach based on DAVAR has been used to analyze the vibration data. the noise FOGY output（°/S）FOGX output（°/S）

Its results are obtained with a truncation window whose length is NW = 50, 000 samples, and step FOGX output—time width is 5000 samples. 0.2 Figure 4 is the DAVAR results. The surface of the DAVAR is stationary both in the static and in 0 vibration. But when the motion status of the FOG is changing, the surface of the DAVAR appears to -0.2 0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

7000

8000

9000

10000

FOGY output—time

0.2 0 -0.2 0

1000

2000

3000

4000

5000

6000

(b) Sensors 2017, 17,Figure 2367 2. power spectral density (PSD) of the random vibration. (a) PSD of the vibration; (b) the output of the vibration table.

Because the MWD vibrated in the fixed place, its position didn’t change and was always the

9 of 17

have bigsame crests which can be seen clearly at t = 300 s, t = 6000 s and 7800s. The DAVAR results are as the initial position. Its velocity could be assumed as zeros with random noise. Therefore the consistent with the raw data of theexperiment FOGs in isFigure 3. position Fitting and the the bi-logarithm σ2 (t, τ ) − τ at aiding source for the vibration the initial velocity. Thegraph measurement vector time in Equation can be described as noise follows;terms can be acquired as Figure 4b,d,f is showing. each analysis t, the (12) coefficients of each At time points t = 300 s, t = 6000 s and 7800s, all the noise terms are very large. During the vibration e n u = [ Lins − Linitialofλthe −λ hcoefficients − hinitial Vins − 0as Vsmall − 0 as Vins − 0 ]Tin the static, (25)except for k ins initial ins ins (t = 300 s to 6000 s), theZmagnitude noise is that the Quantization noise. Though the Quantization noise is larger, its value doesn’t change greatly in vibration.5.1.SoFixed-Frequency we can conclude that the noise terms excepting Quantization noise were not motivated Vibration by the fixed frequency vibration. The reason is that the fixed-frequency vibration is a stable and This vibration test was performed as the steps below. First, the vibration table was in static for disciplinary motion without extreme changing. The Quantization noise has been motivated because aboutSensors 5 min. Then the vibration table started to vibrate. After vibrating for about 1.5 h, the10vibration 2017, 17, 2367 of 18 any dynamic motion will make thetheQuantization larger thanagain that and in the stationary. But when table returned to static. Next, vibration tableerror started to vibrate vibrated for about Figure 4 is the DAVAR results. The surface of the DAVAR is stationary both in the static and in 15 min. vibration along theto X static, axis of the This items experiment with different MWD began toThe vibrate anddirection when itwas was back all MWD. the noise of MWD became bigger. vibration. But when theFOG motion status of raw the FOG changing, surface ofrepresented the DAVAR appears to 3. It motions can motivate the noise. The date the threethe FOGs was Figure Therefore, the DAVAR could track and reveal theisofinstability of the noise itemsinin the FOGs data. have bigthat crests whichthe canvibration, be seen clearly at t = existing 300 s, t = in 6000 and 7800s. DAVAR can be seen during the noise thes FOG data The became veryresults large.are It was 2 With the DAVAR results, we can establish a precise model for the vibration noise. The noise model consistent withnoise the raw datastatic, of the so FOGs in Figure Fitting the bi-logarithm graph as σ the (t ,τ )τ at different from the in the modeling the3.FOG noise during vibration stationary based onwhite DAVAR iswas showed 5. of each noise terms can be acquired as Figure 4b,d,f is showing. each analysis time t, in the Figure coefficients noise not corrected. FOGZ output（°/S）FOGY output（°/S）FOGX output（°/S）

At time points t = 300 s, t = 6000 s and 7800s, all the noise terms are very large. During the vibration (t = 300 s to 6000 s), the magnitude of the noise coefficients is as small as that in the static, except for FOGX output—time 0.2 Though the Quantization noise is larger, its value doesn’t change greatly in the Quantization noise. vibration. So we can conclude that the noise terms excepting Quantization noise were not motivated 0 by the fixed frequency vibration. The reason is that the fixed-frequency vibration is a stable and -0.2 disciplinary motion without extreme noise9000 has been 0 1000 2000 changing. 3000 4000 The 5000Quantization 6000 7000 8000 10000 motivated because FOGY output—time any dynamic motion will make the Quantization error larger than that in the stationary. But when 0.2 MWD began to vibrate and when it was back to static, all the noise items of MWD became bigger. Therefore, the DAVAR0could track and reveal the instability of the noise items in the FOGs data. With the DAVAR results, we can establish a precise model for the vibration noise. The noise model based -0.2 0 Figure 1000 5. 2000 3000 4000 5000 6000 7000 8000 9000 10000 on DAVAR is showed in FOGZ output—time Figure 5 shows the variance of FOGs noise derived from DAVAR. It can be seen clearly that the 0.2 noise in different motions is different. While the motion of the gyro is changing from static to 0 vibration, the variance of the noise is the largest in the whole experiment. During the vibration, the variance of the noise-0.2 is bigger than that in the static. Table 2 lists the variance of the noise at different 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 time epochs. In static, the noise is the smallest.time(s) In vibration, the noise becomes much bigger. When the state of the motion is changing, the noise in the FOG is the biggest. Therefore, In the KF, if the system noise modeled stationary white noise(FOGs) with ainconstant must result in Figure 3. is Raw data of as theaFiber Optic Gyroscopes the fixedvariance, frequencyit vibration. Figure 3. Raw data of the Fiber Optic Gyroscopes (FOGs) in the fixed frequency vibration. degrading the performance and the accuracy of the KF.

step width is 5000 samples.

R(°/h2) K(°/h(3/2) B(°/h) N (°/sqrt(h) Q(＇＇)

Then the noise modeling approach based on DAVAR has been used to analyze the vibration data. Its results are obtained with a truncation window 5whose length is NW = 50,000 samples, and 0 0 0.2 0.1 0 0 4 2 0 0 40

20 0 0 100 50 0 0

2000

4000

6000

8000

10000

2000

4000

6000

8000

10000

2000

4000

6000

8000

10000

2000

4000

6000

8000

10000

2000

4000

6000 time(s)

8000

10000

(b) R(°/h2) K(°/h(3/2) B(°/h) N (°/sqrt(h) Q(＇＇)

(a) 5

0 0 0.1 0.05 0 40 2 0 0 20

2000

4000

6000

8000

10000

2000

4000

6000

8000

10000

2000

4000

6000

8000

10000

2000

4000

6000

8000

10000

2000

4000 6000 time(s)

8000

10000

10 0 0 100 50 0 0

(c)

(d)

Figure 4. Cont.

Sensors 2017, 17, 2367

11 of 18 R(°/h2) K(°/h(3/2) B(°/h) N (°/sqrt(h) Q(＇＇)

Sensors 2017, 17, 2367

R(°/h2) K(°/h(3/2) B(°/h) N (°/sqrt(h) Q(＇＇)

Sensors 2017, 17, 2367

10 of 17

5

0 0 0.2 5 0.1 0 0 0 0 0.2 0.1 5 0 0 5

11 of 18

2000

2000

4000

6000

8000

10000

4000

6000

8000

10000

2000 2000

2000

2000

6000

4000

4000

6000

6000

2000

8000

4000

4000

6000

6000

4000 6000 time(s)4000

2000

(e)

6000

4000

2000

2000

0 0 40 0 0 40 20 20 0 0 1000 1000 50 50 0 0 0 0

4000

10000

8000

10000

8000

10000

8000

10000

8000

10000

10000

8000

10000

8000

10000

6000 time(s)

8000

(f)

(e)

(f)

Figure 4. Dynamic Allan variance (DAVAR) and the noise term of the fixed-frequency vibration test.

5 0 0

5 0 0

σz(t)(°/h)

10

2000

3000

2000 1000

3000 2000

4000 5000 6000 time(s) variance of FOGY

7000

8000

4000 5000 6000 7000 time(s) 4000 5000 6000 7000 8000 variance of FOGY time(s)

3000

9000

8000

9000

9000

variance of FOGZ

10 σz(t)(°/h)

σy(t)(°/h)

10

0 0

1000

5

1000 0 0

5

variance of FOGX

10 σy(t)(°/h)

σx(t)(°/h)

10

σx(t)(°/h)

Figure 4. Dynamic Allan of variance (DAVAR) and noise term the infixed-frequency vibration test. (a) The DAVAR the FOG along X axis; (b) Thethe coefficients noiseof terms the FOG along X axis; Figure 4. Dynamic Allan variance (DAVAR) and the noise of term of the fixed-frequency vibration test. (c) The of the FOG X along Y axis; Thecoefficients coefficients of noise terms terms in the FOG alongFOG Y axis;along X axis; (a) The DAVAR of DAVAR the FOG along axis; (b)(d) The of noise in the (a) The DAVAR ofDAVAR the FOG along X axis; (b)(f)The coefficients of noise inalong the ZFOG (e) The of the FOG along Z axis; The coefficients of noise terms interms the FOG axis. along X axis; (c) The DAVAR of the FOG along Y axis; (d) The coefficients of noise terms in the FOG along Y axis; (c) The DAVAR of the FOG along Y axis; (d) The coefficients of noise terms in the FOG along Y axis; variance of FOGX (e) The DAVAR of the FOG Z axis; (f) The coefficients of noise terms in the FOG along Z axis. 10 along (e) The DAVAR of the FOG along Z axis; (f) The coefficients of noise terms in the FOG along Z axis.

5 0 0

1000

1000

2000

2000

3000

3000

4000

5000

6000

7000

8000

time(s)5000 4000 6000 7000 time(s) Figure 5. The noise model based on DAVAR. variance of FOGZ

9000

8000

9000

Table 2. The variance of the noise.

5

Motion FOGX Noise FOGY Noise FOGY Noise During vibration (3000 s) 3.404 2.689 3.379 0 Static (6500 s) 0.023 0.012 0.009 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Static to vibration (6800) 6.248 6.058 8.959 time(s)

We applied the proposed model to the KF, namely KF. In order to prove that the Figurenoise 5. The noise model basedDAVAR on DAVAR. Figure 5. The noise model based on DAVAR. proposed noise model based on DAVAR can improve the performance of the KF, it was compared with the classic KF with a white noise model. We labelled it as the classic KF. The variance of the Table 2. The variance of the noise. noise model was set to 0.02°/h since the nominal accuracy of the FOG is 0.02°/h. While the white Figure 5white shows the variance of FOGs noise derived from DAVAR. It can be seen clearly that the noise model only includes white noise, the noise model based on DAVAR has taken all the possible Motion FOGX Noise FOGY Noise FOGY Noise noise in different motions isinertial different. the motion of the is model, changing static to vibration, stochastic noise in sensorsWhile into consideration. Thus, withgyro this new the KFfrom can estimate During vibration (3000 s) 3.404 2.689 3.379 the inertial sensors drifts more accurately. Figure 6 represented the estimate value of FOGs drifts andthe variance the variance of the noise is the largest in the whole experiment. During the vibration, Accelerometers drifts. Static (6500 s) 0.023 0.012 0.009

of the noise is bigger than that in the static. Table 2 lists the variance of the noise at different time Static vibration 6.058becomes much 8.959bigger. When the epochs. In static, theto noise is the (6800) smallest. In 6.248 vibration, the noise state of the motion is changing, the noise in the FOG is the biggest. Therefore, In the KF, if the system applied as theaproposed theaKF, namely DAVARit KF. Inresult order in todegrading prove thatthe the noise We is modeled stationarynoise whitemodel noise to with constant variance, must proposed noise model based on DAVAR can improve the performance of the KF, it was compared performance and the accuracy of the KF. with the classic KF with a white noise model. We labelled it as the classic KF. The variance of the white noise model was set to 0.02°/h since of the FOG is 0.02°/h. While the white Table 2. the Thenominal variance accuracy of the noise. noise model only includes white noise, the noise model based on DAVAR has taken all the possible Motion FOGX Noise Thus, FOGY Noise stochastic noise in inertial sensors into consideration. with this new FOGY model,Noise the KF can estimate the inertial sensors more accurately. Figure the estimate value Duringdrifts vibration (3000 s) 3.404 6 represented 2.689 3.379 of FOGs drifts and Static (6500 s) 0.023 0.012 0.009 Accelerometers drifts. Static to vibration (6800)

6.248

6.058

8.959

Sensors 2017, 17, 2367

11 of 17

We applied the proposed noise model to the KF, namely DAVAR KF. In order to prove that the proposed noise model based on DAVAR can improve the performance of the KF, it was compared with the classic KF with a white noise model. We labelled it as the classic KF. The variance of the white noise model was set to 0.02◦ /h since the nominal accuracy of the FOG is 0.02◦ /h. While the white noise model only includes white noise, the noise model based on DAVAR has taken all the possible stochastic noise in inertial sensors into consideration. Thus, with this new model, the KF can estimate the inertial sensors drifts more accurately. Figure 6 represented the estimate value of FOGs drifts and Sensors 2017, 17, 2367 12 of 18 Accelerometers drifts.

0.2 Sensors 2017, 17, 2367

10

0.2

0 -0.2 0.4

ACCY drift—time

0.4

ACCZ ） drift（ug） ACCZ drift（ug） FOGZ drift （°/h） FOGZ drift （°/h

ACCX drift—time

ACCY drift （ug） ACCY drift （ug） FOGY drift（°/h） FOGY drift（°/h）

°/h） drift （ug） ACCX drift （ug） FOGX drift（°/h） FOGX drift（ACCX

0.4

ACCX drift—time

-0.4 0.20 0 x 10-3 2

-0.2 0.4

5000 10000 time(s) FOGX drift—time

-0.2 1

0 ACCY drift—time

-0.4 0.20

5000 time(s)

10000

0x 10-4 FOGY drift—time 2

-0.2 1

-0.4 00 -3

-1 x 10 2 -2 10

5000 10000 time(s) FOGX drift—time 5000 time(s)

10000

0

-0.4 00

5000 time(s) -4

-1 x 10 2 -2 10

10000

FOGY drift—time 5000 time(s)

10000

0

ACCZ drift—time

0

12 of 18

-10 -20 -30 10

Classic KF ACCZ drift—time DAVAR KF

-400 0

5000 10000 time(s) -10 -3 FOGZ drift—time x 10 1 -20 Classic KF -30 DAVAR KF 0.5 -40 00

5000 time(s) -3

-0.5 x 10 1 -1 0.50

10000

FOGZ drift—time 5000 time(s)

10000

0

-1 -1FOGsdrifts -0.5 Figure 6. Estimate drifts and drifts.drifts. Figure 6. Estimate of of FOGs andAccelerometers Accelerometers -2

-2

-1

5000 10000 0 5000 10000 0 5000 10000 In the Kalman0Filtering, the drifts of the inertial sensors (gyros and accelerometers) are estimated time(s) time(s) time(s) In theand Kalman Filtering, the drifts of is, the sensors (gyros and accelerometers) compensated consecutively. That in inertial every filtering cycle, the residual drifts are estimated are and estimated compensated. Hence, ifFigure the KF works normally, afterand theAccelerometers consecutive the are estimated and compensated consecutively. That is, in filtering cycle, thecompensation, residual drifts estimated and 6. Estimate ofevery FOGs drifts drifts. drifts should converge to zero gradually. In Figure 6, it can be noted that the maximum absolute compensated. Hence, if the KF works normally, after the consecutive compensation, the estimated amplitude the sensor driftsthe estimated the classical noise model is much larger than using In theof Kalman drifts ofusing the inertial (gyros accelerometers) arethat estimated drifts should converge toFiltering, zero In Figuresensors 6, it can beand noted that the maximum absolute the noise model.gradually. WithThat the is, proposed estimated value for are the estimated sensor drifts andproposed compensated consecutively. in every model, filteringthe cycle, the residual drifts and amplitudeconverged of the sensor drifts estimated using the classical noise model is much larger than that to smaller values which were almost zero, not only for the bias error of the gyros, but also compensated. Hence, if the KF works normally, after the consecutive compensation, the estimated for the bias error of the accelerometers. With this developed method, the whole performance of the using the proposed noise model. With the proposed model, the estimated value for the sensor drifts drifts should converge to zero gradually. In Figure 6, it can be noted that the maximum absolute become better and the estimated sensor’s drift canthe beclassical estimated andmodel compensated with amplitude of the sensor drifts using is much larger than thatgyros, using convergedKF tohad smaller values which were almost zero, not noise only for the biasmore erroraccurately of the but also the noise modeling the DAVAR proposed noise model. approach. With the proposed model, the estimated value for the sensor drifts for the bias error of the accelerometers. With this developed method, the whole performance of the KF Before to vibration, we have 5 minwere to alignment. Thenot alignment areerror that of thethe inclination angle converged smaller values which almost zero, only forresults the bias gyros, but also had become better and sensor’s drift be this estimated and compensated more accurately is toolface angle is −0.028° andcan the Azimuth is 186.780°. The navigation results based on of these for−0.089°, the bias errorthe of the accelerometers. With developed method, the whole performance the with the two noised models areand compared in Figure DAVAR noise modeling approach. KF had become better the sensor’s drift7.can be estimated and compensated more accurately with

-0.08 0

-0.1 -0.02 0 0

Ve(m/s)

0.02 0

5000 10000 5000 10000 time(s) time(s) vs time east velocity

0.5 0 0 -0.02 -0.5 -0.04 0 0 0 0.2

height(m)

5000 time(s) longitude vs time toolface vs time

5000 10000 5000 10000 time(s) time(s) toolface vs time north velocity vs time

-0.02 0 -0.04 -0.2 0 0 0.2

0 0

10000

0.4 186.85

Azimuth(deg) height(m)

longitude(m)

toolface(deg) longitude(m)

5000 10000 time(s) 5000 10000 time(s) Inclination vs time east velocity vs time

-0.5 0

5000 10000 5000 10000 time(s) time(s) vs time north velocity

5000 time(s) height vs time Azimuth vs time

10000

5000 5000 time(s) time(s) Azimuth vs time up velocity vs time

10000 10000

Classic KF DAVAR KF

0.2 186.8

0 186.75 0 0

Vup(m/s) Azimuth(deg)

-0.06 0.02

10000

186.85 0.05 186.8 0 186.75 -0.05 0 0 0.05

Vup(m/s)

Inclination(deg) Ve(m/s)

Inclination(deg) Latitude(m)

0.5 -0.06 0 -0.08 -0.5 -0.1 0 0

5000 time(s) Latitude vs time Inclination vs time

Vn(m/s) toolface(deg)

-0.5 0

Vn(m/s)

Latitude(m)

DAVAR noise modeling approach. Beforethe vibration, we have 5 min to alignment. The alignment results are that the inclination angle Latitude vs time longitude vs time height vs time Classic KF Before vibration, we have 5◦min0.5to alignment. The alignment0.4 results are that the inclination angle ◦ ◦ 0.5 DAVAR KF is −0.089 , toolface angle is −0.028 and the Azimuth is 186.780 . The navigation results based on is −0.089°, toolface angle is −0.028° and the Azimuth is 186.780°. The navigation results based on these 0 0 Figure 7. 0.2 these two noised models are compared in two noised models are compared in Figure 7.

5000 10000 5000 10000 time(s) time(s)vs time up velocity

Figure 7. Navigation results of the fixed-frequency vibration. 0

0

-0.02 -0.2in this experiment, the position -0.05 results and velocity results are Because the 5000 MWD didn’t move 0 10000 0 5000 10000 0 5000 10000 time(s) time(s) time(s) also the navigation errors. It can be noted that the longitude is 0.42 m using the conventional noise model, while it becomes 0.0697.m using theresults proposed model. The longitude error is decreased Figure Navigation of thenoise fixed-frequency vibration. Figureis 7. Navigation results oftothe fixed-frequency vibration. by 83.6%. The height also reduced from 0.4 m 0.13 m. The fluctuation range of the inclination

Because the MWD didn’t move in this experiment, the position results and velocity results are also the navigation errors. It can be noted that the longitude is 0.42 m using the conventional noise model, while it becomes 0.069 m using the proposed noise model. The longitude error is decreased by 83.6%. The height is also reduced from 0.4 m to 0.13 m. The fluctuation range of the inclination

Sensors 2017, 17, 2367

12 of 17

Sensors 2017, 17, 2367

13 of 18

Because the MWD didn’t move in this experiment, the position results and velocity results are errorthe is navigation repressed by the DAVAR toolfaceiserror computed the classic KF is also errors. It can benoise notedmodel. that theThe longitude 0.42 m using theby conventional noise divergent with time. While using the DAVAR KF, it is converged to 0.02°. Because the FOG based model, while it becomes 0.069 m using the proposed noise model. The longitude error is decreased by MWD is vibrating, velocity is fluctuating zero. can be seenrange that the fluctuation range of 83.6%. The height isthe also reduced from 0.4 m around to 0.13 m. TheItfluctuation of the inclination error therepressed velocity by calculated by DAVAR KF isThe smaller thanerror the classical KF. the KF accuracy of the is the DAVAR noise model. toolface computed byHence, the classic is divergent DAVAR KFWhile is higher than classic KF, KF. it is converged to 0.02◦ . Because the FOG based MWD is with time. using thethe DAVAR vibrating, the velocity is fluctuating around zero. It can be seen that the fluctuation range of the 5.2. Random-Frequency VibrationKF is smaller than the classical KF. Hence, the accuracy of the DAVAR velocity calculated by DAVAR KF isThis higher than the classic test KF. was implemented as the steps below. Firstly, the vibration table was random vibration

FOGZ output（°/S）FOGY output（°/S）FOGX output（°/S）

stationary for about 10 min. Then the vibration platform began to vibrate and kept vibrating for about 5.2. Random-Frequency Vibration 10 min. Finally the vibration platform returned to static and kept static for about 10 min. The vibration direction the MWD system was along the X axis.asThe data of the three the FOGs is represented in This of random vibration test was implemented theraw steps below. Firstly, vibration table was Figure 8. During the10vibration, the noise existing in the began FOG data is veryand big. Thevibrating magnitude the stationary for about min. Then the vibration platform to vibrate kept for of about noise is Finally about 2°/s the vibration. 10 min. the in vibration platform returned to static and kept static for about 10 min. The vibration Thenofthe noise modeling hasThe been utilized model vibration error. The direction theDAVAR MWD system was alongapproach the X axis. raw data oftothe threethis FOGs is represented in DAVAR computed by a truncation window samples, a step of width Figure 8. was During the vibration, the noise existingofinlength the FOG is very big. Theand magnitude of the NWdata = 50,000 noise is about 2◦ /s in the vibration. 5000 samples. FOGX output—time

2 0 -2 0

200

400

600

800

1000

1200

1400

1600

1800

1200

1400

1600

1800

1400

1600

1800

FOGY output—time

2 0 -2 0

200

400

600

800

1000

FOGZ output—time 2 0 -2 0

200

400

600

800 1000 time(s)

1200

Figure 8. 8. Raw in the the random random vibration. vibration. Figure Raw data data of of the the FOGs FOGs in

Figure 9 shows the DAVAR results. DAVAR surface was stationary when the vibration table Then the DAVAR noise modeling approach has been utilized to model this vibration error. was static. Then a big crest which started at t = 600 s and stopped at t = 1200 s was appearing. At the The DAVAR was computed by a truncation window of length NW = 50, 000 samples, and 2a step of end of5000 the test, the DAVAR came back to the stationary. Fitting the bi-logarithmic curve σ (t ,τ )-τ , width samples. the noise canthe be DAVAR obtained as Figure 9b,d,f was was shown. Beforewhen vibration (t < 600table s), each Figureitems 9 shows results. DAVAR surface stationary the vibration was coefficient was small and they didn’t have obvious change. While the vibration table began to static. Then a big crest which started at t = 600 s and stopped at t = 1200 s was appearing. At thevibrate end of (t =test, 600 the s), each noise term changed sharply. During vibration the noise items the DAVAR came back to the stationary. Fitting the bi-logarithmic curve σ2 (change t, τ ) − τ,obviously, the noise which is different to the noise items in the fixed frequency vibration. After vibration, coefficients of items can be obtained as Figure 9b,d,f was shown. Before vibration (t < 600 s), each coefficient was noise terms were all back to the small value, similar to the value in static. In conclusion, the noise in small and they didn’t have obvious change. While the vibration table began to vibrate (t = 600 s), each the FOGs not always unchangeable and the DAVAR could identify reveal the highly dynamic noise termischanged sharply. During vibration the noise items changeand obviously, which is different instability in the FOG’s data. to the noise items in the fixed frequency vibration. After vibration, coefficients of noise terms were all back to the small value, similar to the value in static. In conclusion, the noise in the FOGs is not always unchangeable and the DAVAR could identify and reveal the highly dynamic instability in the FOG’s data.

Sensors 2017, 17, 2367

13 of 17

14 of 18

R (°/h 2 ) K (°/h ( 3/2) B (°/h) N (°/s qrt(h) Q ( ＇＇)

Sensors 2017, 17, 2367 4 2 0 0 1

500

1000

1500

500

1000

1500

500

1000

1500

500

1000

1500

1000

1500

0.5 0 0 0.5

0 0 0.2 0.1 0 0 0.02 0.01 0 0

500

(b) R (° /h 2 ) K (° /h ( 3 /2 ) B (°/h ) N (° /s q rt(h ) Q ( ＇＇)

(a) 4 2 0 0 0.5

0 0 0.2 0.1 0 0 0.04 0.02 0 0x 10-3 4 2 0 0

500

1000

1500

500

1000

1500

500

1000

1500

500

1000

1500

1000

1500

500

time(s)

(d) R (°/h 2 ) K (° /h ( 3 /2 ) B (° /h ) N (° /s q rt(h ) Q ( ＇＇ )

(c) 5

0 0 1 0.5 0 0 0.4 0.2 0 0 0.1 0.05 0 0 0.01 0.005 0 0

(e)

time(s)

500

1000

1500

500

1000

1500

500

1000

1500

500

1000

1500

1000

1500

500

time(s)

(f)

Figure 9. DAVAR and the noise terms in the random vibration test. (a) The DAVAR of the FOGX; (b) Figure 9. DAVAR and the noise terms in the random vibration test. (a) The DAVAR of the FOGX; The coefficients of noise terms in FOGX; (c) The DAVAR of the FOGY; (d) The coefficients of noise (b) The coefficients of noise terms in FOGX; (c) The DAVAR of the FOGY; (d) The coefficients of noise terms in FOGY; (e) The DAVAR of the FOGZ; (f) The coefficients of noise terms in FOGZ. terms in FOGY; (e) The DAVAR of the FOGZ; (f) The coefficients of noise terms in FOGZ.

Based on the Equation (6), the noise model based on DAVAR could be obtained. Figure 10 shows Based on of theFOGs Equation (6),derived the noise model on DAVAR could obtained. 10 shows the variance noise from the based DAVAR. It can be seenbeclearly thatFigure the noise in the the variance of FOGs noise derived from the DAVAR. It can be seen clearly that the noise in the random vibration is very big. The variance of the noise in the vibration is 4°/h, while itrandom is only ◦ /h in the vibration very big.state. The variance the noisenoise in the vibration is 4◦ /h, to while it is Figure only 0.03 0.03°/h inisthe static Then thisofproposal model was applied the KF. 11 represents static state. Then this model was applied to the KF. Figure 11 represents the estimate the estimate drifts of proposal the FOGsnoise and ACCs. drifts of the FOGs and ACCs.

Sensors 2017, 17, 2367 Sensors2017, 2017,17, 17,2367 2367 Sensors

14 of 17 15ofof18 18 15

VarianceofofFOGX FOGX Variance

σ x(t)(°/h) σ x(t)(°/h)

44 22 00 00

200 200

400 400

600 800 600 800 time(s) time(s) VarianceofofFOGY FOGY Variance

1000 1000

1200 1200

1400 1400

200 200

400 400

600 800 600 800 time(s) time(s) VarianceofofFOGZ FOGZ Variance

1000 1000

1200 1200

1400 1400

200 200

400 400

600 800 600 800 time(s) time(s)

1000 1000

1200 1200

1400 1400

σ y(t)(°/h) σ y(t)(°/h)

44 22 00 00

σ z(t)(°/h) σ z(t)(°/h)

55

00 00

Figure 10. Noise model based on DAVAR. Figure10. 10.Noise Noisemodel modelbased basedon onDAVAR. DAVAR. Figure

40 40

00

-50 -50

ACCYdrift—time drift—time ACCY

ACCZdrift—time drift—time ACCZ

100 100

20 20

ClassicKF KF Classic DAVARKF KF DAVAR

200 200 ACCZ ACCZ drift（ug） drift（ug）

ACCXdrift—time drift—time ACCX

ACCY ACCY drift（ug） drift（ug）

ACCX ACCX drift（ug） drift（ug）

50 50

00

-100 -100

00

-200 -200

00

-1 -1 -2 -2 -3 -3 00

FOGZ FOGZ drift（°/h） drift（°/h）

FOGY FOGY drift（°/h） drift（°/h）

-20 -300 -20 -300 500 1000 1500 00 500 1000 1500 1500 500 1000 1500 1500 500 1000 1500 500 1000 00 500 1000 time(s) time(s) time(s) time(s) time(s) time(s) -3-3 -3-3 FOGY drift—time -3-3 FOGZ drift—time drift—time FOGXdrift—time drift—time drift—time 10 FOGX 10 FOGY 10 FOGZ xx10 xx10 xx10 11 44 44

FOGX FOGX drift（°/h） drift（°/h）

-100 -100 00

22 00

1500 1500

-4 -4 00

00

-2 -2

-2 -2

500 1000 500 1000 time(s) time(s)

22

-4 -4

500 1000 500 1000 time(s) time(s)

1500 1500

-6 -6 00

500 1000 500 1000 time(s) time(s)

1500 1500

Figure 11. Estimate drifts of FOGs and ACCs. Figure Figure11. 11.Estimate Estimatedrifts driftsof ofFOGs FOGsand andACCs. ACCs.

Usingthe theproposed proposedmodel, model,the themaximum maximumabsolute absoluteamplitude amplitudeof ofestimation estimationvalue valueof ofthe thegyros’ gyros’ Using Using the proposed model, the maximum absolute amplitude of estimation value of the gyros’ driftswas wasmuch muchsmaller smallerthan thanthat thatby bythe theconventional conventionalmethod. method.Though Thoughthe thefluctuation fluctuationamplitude amplitudefor for drifts drifts was much smaller than that by the conventional method. Though the fluctuation amplitude the Y axis FOG and Z axis FOG is a littler bigger, they converge to a smaller value after 1000 s. the Y axis FOG and Z axis FOG is a littler bigger, they converge to a smaller value after 1000 s. ItIt for the Y axis FOG and Z axis FOG is a littler bigger, they converge to a smaller value after 1000 s. provedthat, that,compared comparedto tothe theclassical classicalKF, KF,the thesensor’s sensor’sdrift driftcan canbe beestimated estimatedand andcompensated compensatedmore more proved It proved that, compared to the classical KF, the sensor’s drift can be estimated and compensated more accuratelywith withthe theproposed proposedapproach. approach. accurately accurately with the proposed approach. Thenavigation navigationresults resultsbased basedon onthe theDAVAR DAVARnoise noisemodel modeland andthe theclassical classicalwhite whitenoise noisemodel model The The navigation results based on the DAVAR noise model and the classical white noise model are arereported reportedin inFigure Figure12. 12. are reported in Figure 12.

500 1000 time(s) Inclination vs time

-0.6

1500

-0.7 -0.8 0

500 1000 time(s) east velocity vs time

1500

500 1000 time(s)

1500

height(m)

-10 0 -89

500 1000 time(s) toolface vs time

500 1000 time(s) north velocity vs time

1500

0 0 184

500 1000 time(s)

1500

500 1000 time(s) Azimuth vs time

1500

500 1000 time(s) up velocity vs time

1500

500 1000 time(s)

1500

183 182 0 2

0 -0.5 0

10

1500

-90 -91 0

height vs time Classic KF DAVAR KF

20

0

0.5

0 -0.5 0

toolface(deg)

-10 0

longitude vs time

Azimuth(deg)

longitude(m)

0

10

Vup(m/s)

Latitude vs time

10

0.5 Ve(m/s)

15 of 17 16 of 18

Vn(m/s)

Inclination(deg)

Latitude(m)

Sensors 2017, 17, 2367 Sensors 2017, 17, 2367

0 -2 0

Figure 12. Navigation results of the random vibration.

Using theclassic classicKF, KF, position diverges min random vibration, the Using the thethe position diverges withwith time.time. After After 20 min20random vibration, the latitude, latitude, the longitude and the are all to increased 10 contrary, m. On the contrary, using noise the DAVAR the longitude and the height areheight all increased 10 m. Ontothe using the DAVAR model, noise model, the position is limited to 0.27 m, 0.697 m and 0.741 m, respectively. So the position error the position is limited to 0.27 m, 0.697 m and 0.741 m, respectively. So the position error is reduced is reduced more 90% byKF. the DAVAR the KF. attitude Comparing attituderesult to the alignmentangle result more than 90% by than the DAVAR Comparing to thethe alignment (inclination is (inclination angle is −0.735°, the toolface angle is −90.278° and the azimuth is 182.633°), the attitude ◦ ◦ ◦ −0.735 , the toolface angle is −90.278 and the azimuth is 182.633 ), the attitude error from the DAVAR error from the DAVAR model displays a smaller drift than the classical one. We can see that the model displays a smaller drift than the classical one. We can see that the toolface angle obtained toolface angle obtained by the classic KF drifts to −0.78°while the DAVAR KF drifts only 0.02°. Using by the classic KF drifts to −0.78◦ while the DAVAR KF drifts only 0.02◦ . Using the classical KF, the the classical KF, the maximum fluctuation range of the azimuth is 1°. While using the DAVAR KF, maximum fluctuation range of the azimuth is 1◦ . While using the DAVAR KF, the maximum fluctuation the maximum fluctuation range of the azimuth is 0.5°. For the velocity, when using the classical noise range of the azimuth is 0.5◦ . For the velocity, when using the classical noise model, the maximum model, the maximum absolute amplitude error of the velocity is 0.5 m/s. However, when using the absolute amplitude error of the velocity is 0.5 m/s. However, when using the DAVAR noise model, the DAVAR noise model, the maximum fluctuation range of velocity is only 0.2 m/s. Table 3 listed the maximum fluctuation range of velocity is only 0.2 m/s. Table 3 listed the navigation error obtained navigation error obtained from both of the classic KF and the DAVAR KF. It can be seen that each from both of the classic KF and the DAVAR KF. It can be seen that each kind of navigation error is kind of navigation error is decreased dramatically. The position error was reduced more than 90%. decreased dramatically. The position error was reduced more than 90%. The velocity error was reduced The velocity error was reduced more than 60%. The attitude error was reduced by 30% at least. more than 60%. The attitude error was reduced by 30% at least. Therefore, the navigation results Therefore, the navigation results obtained using the DAVAR noise model are much more accurate obtained using the DAVAR noise model are much more accurate than using the conventional one. than using the conventional one. Table 3. The navigation error. Table 3. The navigation error.

6. Conclusions

Parameter Parameter Latitudeerror error(m) (m) Latitude Longitude error (m) Longitude error (m) Height error (m) Height error Inclination error(m) (deg) Inclination error(deg) (deg) toolface errer Azimuth error(deg) (deg) toolface errer East velocity error (m/s) Azimuth error (deg) North velocity error (m/s) East error(m/s) (m/s) upvelocity velocity error North velocity error (m/s) up velocity error (m/s)

Classic KF DAVAR KF Classic KF DAVAR KF 8.562 0.257 8.562 0.257 7.962 0.697 7.962 0.697 10.129 0.741 10.129 0.741 0.06 0.04 0.06 0.04 0.78 0.02 0.5 0.781 0.02 0.3415 0.05 1 0.5 0.452 0.160 0.3415 0.05 1.191 0.399 0.452 0.160 1.191 0.399

Optimized Optimized 96.99% 96.99% 91.25% 91.25% 92.68% 92.68% 33.33% 33.33% 97..45% 50% 97..45% 85.36% 50% 64.60% 85.36% 66.50% 64.60% 66.50%

This paper proposed a new noise modeling approach for vibration noise and applied this new 6. Conclusions noise model to the Kalman filter. Firstly, the random noise items were identified and separated using This paper proposed a new noise modeling approach for vibration noise and applied this new dynamic Allan variance. Then, the noise model including white noise and color noise was established noise model to the Kalman filter. Firstly, the random noise items were identified and separated using based on the results of DAVAR. Finally, this new noise model was applied to the Kalman filter to dynamic Allan variance. Then, the noise model including white noise and color noise was established based on the results of DAVAR. Finally, this new noise model was applied to the Kalman filter to

Sensors 2017, 17, 2367

16 of 17

provide an accuracy model for the system noise. Two vibration experiments have been performed to validate the new noise modeling approach. One was with fixed frequency vibration and another one was with random vibration. Both the experiments’ results demonstrated that the proposed approach could improve the performance and accuracy of the KF greatly, especially for the random vibration experiment. For the random vibration experiment, using the DAVAR KF, the position error was reduced more than 90%. The velocity error was reduced more than 60%. The attitude error was reduced by 30% at least. Acknowledgments: Here and now, I would like to extend my sincere thanks to Huipeng Li who has helped me make this thesis better. He checked through my thesis with patience and has given me instructive suggestions. This research is supported by the Ministry of Science and Technology of the People’s Republic of China. The name of the project is development and application of ultra-high temperature drilling trajectory measuring instrument. (Project number 2013YQ050791). Author Contributions: Lu Wang, Chunxi Zhang conceived the study idea. Lu Wang and Shuang Gao designed the experiments. Lu Wang, Tie Lin and Xianmu Li performed the experiments. Lu wang and Tie Lin processed the experimental data. Tie Lin and Xianmu Li draw the figures and tables in this paper. Lu Wang wrote the whole paper. Chunxi Zhang and Shuang Gao checked the theory part and guarantee the innovation of this paper. Tie Lin and Xianmu Li checked the grammar errors in this paper. Conflicts of Interest: The authors declare no conflict of interest.

References 1. 2.

3. 4. 5. 6.

7. 8. 9. 10. 11. 12.

13.

14.

Noureldin, A. New Measurement-While-Drilling Surveying Technique Utilizing Sets of Fiber Optic Rotation Sensors. Ph.D. Thesis, University of Calgary, Calgary, AB, Canada, 2002; pp. 12–26. Noureldin, A.; Irvine-Halliday, D.; Mintchev, M.P. Measurement-while-drilling surveying of highly inclined and horizontal well sections utilizing single-axis gyro sensing system. Meas. Sci. Technol. 2004, 15, 2426–2434. [CrossRef] ElGizawy, M. Continuous Measurement-While-Drilling Surveying System Utilizing MEMS Inertial Sensors. Ph.D. Thesis, University of Calgary, Calgary, AB, Canada, 2009; pp. 14–21. Ben, Y.Y.; Sun, F.; Gao, W.; Chen, M. Study of zero velocity update for inertial navigation. J. Syst. Simul. 2008, 20, 4639–4642. Zhang, C.; Lin, T. A long-term performance enhancement method for FOG-based measurement while drilling. Sensors 2016, 16, 1186. [CrossRef] [PubMed] Nassar, S.; Schwarz, K.P.; Noureldin, A.; El-Sheimy, N. Modeling inertial sensor errors using autoregressive (AR) models. In Proceedings of the 2003 National Technical Meeting of The Institute of Navigation, Anaheim, CA, USA, 22–24 January 2003. Yi, Y. On Improving the Accuracy and Reliability of GPS/INS-Based Direct Sensor Georeferencing. Ph.D. Thesis, Ohio State University, Columbus, OH, USA, 2007; p. 102. Schwarz, K.P.; Wei, M. INS/GPS Integration for Geomatics; University of Calgary: Calgary, AB, Canada, 2001; pp. 34–42. Wang, J.; Lee, H.K.; Hewitson, S.; Lee, H.K. Influence of dynamics and trajectory on integrated GPS/INS navigation performance. J. Glob. Position Syst. 2003, 2, 109–116. [CrossRef] Grejner-Brzezinska, D.; Toth, C.; Yi, Y. On improving navigation accuracy of GPS/INS systems. Photogram. Eng. Remote Sens. 2005, 71, 377–389. [CrossRef] Han, S.; Wang, J. Monitoring the degree of observability in integrated GPS/INS systems. In Proceedings of the International Symposium on GPS/GNSS 2008, Tokyo, Japan, 11–14 November 2008. Kumar, A. Colored-noise Kalman filter for vibration mitigation of position/attitude estimation systems. In Proceedings of the AIAA Guidance, Navigation and Control Conference and Exhibit, Hilton, SC, USA, 20–23 August 2007. Kim, H.; Lee, J.G.; Park, C.G. Performance improvement of GPS/INS integrated system using Allan variance analysis. In Proceedings of the 2004 International Symposium on GNSS/GPS, Sydney, Australia, 6–8 December 2004. Han, S.; Wang, J. Quantization and colored noises error modeling for inertial sensors for GPS/INS integration. IEEE Sens. J. 2011, 11, 1493–1503. [CrossRef]

Sensors 2017, 17, 2367

15. 16. 17. 18. 19. 20. 21. 22. 23.

24. 25. 26.

17 of 17

Gu, S.; Liu, J.; Zeng, Q.; Lv, P. A Kalman filter algorithm based on exact modeling for FOG GPS/SINS integration. Optik 2014, 125, 3476–3481. [CrossRef] Galleani, L.; Tavella, P. The dynamic Allan variance. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 2009, 56, 450–464. [CrossRef] [PubMed] Sesia, I.; Galleani, L.; Tavella, P. Application of the dynamic Allan varinacne for the characterization of space clock behavior. IEEE Trans. Aerosp. Electron. Syst. 2011, 47, 884–895. [CrossRef] Li, Y.; Chen, X.; Song, S. Dynamic Allan variance analysis for the drift error of fiber optical gyroscope. J. Optoelectron. Laser 2008, 19, 183–186. Wei, G.; Long, X. Research on stochastic errors of dithered ring laser gyroscope based on dynamic Allan cariance. Chin. J. Lasers 2010, 37, 2975–2979. Li, X.; Zhang, N. Analysis of dynamic characteristics of a fiber optic gyroscope based on dynamic Allan variance. J. Harbin Eng. Univ. 2011, 32, 183–187. Wang, L.; Zhang, C.; Lin, T.; Li, X.; Wang, T. Characterization of a fiber optic gyroscope in a measurement while drilling system with the dynamic Allan variance. Measurement 2015, 75, 263–272. [CrossRef] Wang, L.; Zhang, C. Fast algorithm of the dynamic Allan variance for FOG. Optik 2016, 127, 2413–2418. [CrossRef] Institute of Electrical and Electronics Engineers (IEEE). IEEE 952-1997: IEEE Standard Specification Format Guide and Test Procedure for Single-Axis Interferometric Fiber Optic Gyros; Institute of Electrical and Electronics Engineers (IEEE): Piscataway, NJ, USA, 1997. El-Sheimy, N.; Hou, H.; Niu, X. Analysis and modeling of inertial sensors using Allan variance. IEEE Trans. Instrum. Meas. 2008, 57, 140–149. [CrossRef] Hou, H. Modeling Inertial Sensors Errors Using Allan Variance; University of Calgary: Calgary, AB, Canada, 2004. Xi’an Petroleum Exploration Instrument Factory; Shengli Oilfield Drilling Technology Research Institute. Technical Specifications of Measurement While Drilling, Standards of the Petroleum and Natural Gas Industry of the People’s Republic of China; SY/T 6702-2007; Petroleum Industry Press: Beijing, China, 2007. © 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).