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Accurate calibration for drift of fiber optic gyroscope in multi-position north-seeking phase
Optik 125 (2014) 7244–7246
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Accurate calibration for drift of fiber optic gyroscope in multi-position north-seeking phase Jianye Pan ∗ , Chunxi Zhang, Yanxiong Niu, Zhe Fan School of Instrumentation Science and Optoelectronics Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
a r t i c l e
i n f o
Article history: Received 13 December 2013 Accepted 20 July 2014 Keywords: Inertial measurement unit Fiber optic gyroscope Accelerometer bias North-seeking Calibration
a b s t r a c t This paper presents an accurate calibration method for drift of fiber optic gyroscope (FOG) in multiposition north-seeking phase. To avoid the effect of the rate of change of accelerometer bias on the drift estimate of FOG, north-seeking by coarse-alignment is performed in multi positions and the northseeking results are used to compute the drift of FOG. The laboratory test results prove that the proposed method can not only achieve multi-position north-seeking, but also accurately calibrate the drift of FOG. The comparison with the traditional method highlights the superior performance of the proposed method. © 2014 Elsevier GmbH. All rights reserved.
1. Introduction Inertial navigation is entirely self-contained and can provide information including position, velocity and attitude at a high rate. It is now widely used as the main navigation means in missiles, aircraft, robots and other autonomous vehicles [1]. Calibration for gyroscope drift is very important in inertial navigation technologies. A 1 nm/h class strapdown inertial navigation system (SINS) for fighter aircraft typically requires a stability and a calibration accuracy of 0.005◦ /h for gyroscope drifts [2]. The calibration techniques for gyroscope drifts are divided into two categories according to their measurement data types. The first category is direct calibration methods, which compare raw sensor data measured on the turntable with projections of reference gravity and earth rate [3,4]. This calibration approach requires not only rotating around inside-axis of turntable, but also rotating around outside-axis of turntable. The other category of calibration methods is using two-position alignment based on Kalman filter [5,6]. It uses the velocity and attitude indications of SINS to estimate calibration coefficient. Such methods are relatively robust as to instrumentation errors because they have the advantage of using a navigation algorithm that can determine the initial attitude and the amount of change
∗ Corresponding author. E-mail address: [email protected] (J. Pan). http://dx.doi.org/10.1016/j.ijleo.2014.07.126 0030-4026/© 2014 Elsevier GmbH. All rights reserved.
in angular position. However, in these methods gyroscope drift estimate is affected by the rate of change of accelerometer bias. Before the SINS being used in actual, to validate the accuracy of north-seeking in different azimuth directions, multi-position north-seeking test is usually performed. Multi-position northseeking test can be described as: for example, an IMU mounted on a turntable, with 7 times continuous 45◦ rotations around the inside-axis of turntable, north-seeking result (azimuth) in each position is recorded. The accuracy of north-seeking directly depends on the accuracy of gyroscope drift. In this current work, an accurate calibration method for drift of FOG in multi-position north-seeking phase is proposed, not only achieving multi-position north-seeking, but also calibrating the drift of FOG. To avoid the effect of the rate of change of accelerometer bias on drift estimate of FOG, north-seeking by coarse-alignment is performed in multi positions. After that, the drifts of FOGs are computed by the north-seeking results. The paper is organized as follows. In Section 2, the effect of the rate of change of accelerometer bias on the drift estimate of FOG is discussed. In Section 3, north-seeking by coarse-alignment is performed in multi positions and the north-seeking results are used to estimate the drift of FOG. In Section 4, the laboratory test results of the calibration as well as accuracy verification are reported. In Section 5, the conclusion is discussed.
J. Pan et al. / Optik 125 (2014) 7244–7246 2000
Table 1 North-seeking results in four positions.
1900
Test number Azimuth (0◦ ) Azimuth (90◦ ) Azimuth (180◦ ) Azimuth (270◦ )
Accelerometer outputs (μg)
1800
Test 1 Test 2 . . . Test n
1700 1600 1500
1100
0
2000
4000
6000 Time (s)
8000
10000
12000
Fig. 1. Static data output of an accelerometer.
2. Effect of the rate of change of accelerometer bias on the drift estimate of FOG In fine-alignment based on Kalman filter, azimuth alignment error is [7]
∇ nE +
n
∇E
g
tan L −
εnE +
ωN
ε nE
n
−
∇ N /g ωN
,
(1)
T
where ∇ n = [ ∇ nE ∇ nN ∇ nU ] is the accelerometer bias vector in navigation frame (the subscripts stand for north, upward and east n
n
n
n
T
velocity components, respectively), ∇ = [ ∇ E ∇ N ∇ U ] is the rate of change of accelerometer bias vector in navigation frame (the subscripts stand for north, upward and east velocity components, respectively), εnE and ε nE are the east gyroscope drift and the rate of change of east gyroscope drift in navigation frame, respectively, g is the gravity, L is the latitude and ωN is the earth rotation rate of north component. This result shows that azimuth alignment error depends on not only the gyroscope drift and accelerometer bias, but also on the rate of change of accelerometer bias. From Eq. (1), rate of change of accelerometer bias of 1 mg/h corresponds to gyroscope drift of 0.057◦ /h. An about three-hour static data output of an accelerometer is shown in Fig. 1. When the accelerometer starts, its bias changes significantly with rate of change of about 260 g/h, and this change will greatly affect azimuth alignment error. In coarse-alignment, azimuth alignment error is [7] n
∇¯ E g
tan L −
ε¯ nE ωN
,
(2)
C1 C2 . . . Cn
D1 D2 . . . Dn
where is the average accelerometer bias in navigation frame of east component and ε¯ nE is the average gyroscope drift in navigation frame of east component. Unlike fine-alignment based on Kalman filter, in coarsealignment, the rate of change of accelerometer bias will not affect the azimuth alignment error. So in this paper, coarse-alignment method is preferred to realize north-seeking. Comparing with the effect of the gyroscope drift on the azimuth alignment error, the effect of the accelerometer bias on the azimuth alignment error is small and can be neglected. Then Eq. (2) can be simplified as ε¯ nE ωN
.
Because of the initial azimuth angular position of turntable, azimuth alignment error cannot be obtained by the north-seeking result in a single position. North-seeking results in multi positions can be used to compute the azimuth alignment error and the initial azimuth angular position. The following procedure is a typical procedure for calibration. First, the IMU is mounted on a turntable and performs north-seeking in this position. Then the IMU rotates 90◦ around the inside-axis of turntable to the second position and performs north-seeking. After that, 90◦ rotation process is continued and north-seeking at the third and fourth position is performed. The tests described above are repeated for n times and the northseeking result (azimuth) at each position is recorded (shown in Table 1). The north-seeking result at each position should be averaged as follows: A = (A1 + A2 + · · · + An )/n,
(4)
B = (B1 + B2 + · · · + Bn )/n,
(5)
C = (C1 + C2 + · · · + Cn )/n,
(6)
D = (D1 + D2 + · · · + Dn )/n.
(7)
Considering the initial angular position of the turntable ˛, the north-seeking result in the first position can be expressed as A = ˛ + U (εx · cos(A) + εy · sin(A)),
(8)
where U (εx · cos(A) + εy · sin(A)) = (εx · cos(A) + εy · sin(A))/0.01 · U (0.01),
(9)
where U (0.01) is the azimuth alignment error with gyroscope drift of 0.01◦ /h at the alignment latitude and the error can be computed by Eq. (3). εx and εy are the drifts of x and y FOGs in the body frame, respectively. Substituting Eq. (9) into Eq. (8), Eq. (8) can be rewritten as A = ˛ + (εx · cos(A) + εy · sin(A))/0.01 · U (0.01).
(10)
Similarly, the north-seeking result in the third position can be expressed as
n ∇¯ E
U = −
B1 B2 . . . Bn
3. Calibration for drift of FOG in multi-position north-seeking phase
1200
U =
A1 A2 . . . An
1400 1300
U =
7245
(3)
According to Eq. (3), the drift of FOG can be estimated by azimuth alignment error.
C = 180 + ˛ + U (−εx · cos(A) − εy · sin(A)),
(11)
where U (−εx · cos(A) − εy · sin(A)) = −(εx · cos(A) + εy · sin(A))/0.01 · U (0.01).
(12)
Substituting Eq. (12) into Eq. (11), Eq. (11) can be rewritten as C = 180 + ˛ − (εx · cos(A) + εy · sin(A))/0.01 · U (0.01).
(13)
Associating Eq. (10) with Eq. (13), another equation is obtained as εx · cos(A) + εy · sin(A) = (A − C − 180)/2 · 0.01/U (0.01).
(14)
Continue with the second and fourth positions. Similar to Eq. (14), εx · cos(B) + εy · sin(B) = (B − D − 180)/2 · 0.01/U (0.01).
(15)
7246
J. Pan et al. / Optik 125 (2014) 7244–7246
Table 2 North-seeking results of the proposed method in eight positions (1 mil = 0.06◦ ). Test number
0◦
45◦
90◦
135◦
180◦
225◦
270◦
315◦
Test 1 (mil) Test 2 (mil) Test 3 (mil) Test 4 (mil) Test 5 (mil) Test 6 (mil) Average (mil)
2.0 1.9 1.8 2.1 2.6 3.0 2.2
751.8 751.6 752.6 752.4 752.1 752.2 752.1
1502.4 1502.3 1502.3 1501.7 1501.6 1501.3 1501.9
2251.5 2250.5 2250.2 2250.3 2250.5 2250.6 2250.6
3000.6 2999.8 3000.2 3000.0 3000.0 2999.7 3000.1
3749.8 3749.6 3749.8 3750.3 3750.4 3750.0 3750.0
4500.2 4501.0 4500.3 4500.4 4499.6 4499.5 4500.2
5250.9 5251.2 5251.4 5251.2 5251.4 5251.3 5251.2
Table 3 Calibration results of the drifts of FOGs. Error source
Group 1 of the proposed method
Group 2 of the proposed method
The traditional Direct method calibration
Drift of x FOG (◦ /h) Drift of y FOG (◦ /h)
−0.016
−0.008
−0.006
−0.014
−0.013
−0.014
−0.021
−0.011
of FOGs are calibrated by the traditional method using two-position alignment based on Kalman filter. Then the drifts of FOGs are calibrated by the proposed method. In multi-position north-seeking phase, the X-, Y- and Z-axes of the IMU point to the east, north and up directions, respectively, as the initial orientation. With 7 times continuous 45◦ rotations around the inside-axis of turntable, north-seeking result (azimuth) in each position is recorded (shown in Table 2). The north-seeking results are divided into two groups. Group 1 includes the north-seeking results at 0◦ , 90◦ , 180◦ and 270◦ positions. Group 2 includes the north-seeking results at 45◦ , 135◦ , 225◦ and 315◦ positions. The initial azimuth angular position of turntable is 1.06 mil, calculated by Eq. (16). After north-seeking, the drifts of FOGs are calibrated by the direct calibration method, and the calibration results are taken as reference. Table 3 presents the calibration results of the drifts of FOGs. The accuracy of the drifts estimates of x and y FOGs are 72.3% and 81.6%, respectively, better than the traditional method, with the accuracy of 42.9% and 52.4%, respectively. 5. Conclusion
Associating Eq. (14) with Eq. (15), the drifts of x and y FOGs can be obtained. In addition, the initial angular position of the turntable can be computed by ˛ = (˛1 + ˛2)/2,
(16)
where ˛1 = (C − 180) − (C − A − 180)/2,
(17)
and ˛2 = (D − 270) − (D − B − 180)/2.
In this paper, an accurate calibration method for drift of FOG in multi-position north-seeking phase is proposed. Considering the effect of the rate of change of accelerometer bias on the drift estimate of FOG in traditional calibration method, in proposed calibration method north-seeking by coarse-alignment is performed in multi positions and the north-seeking results are used to estimate the drift of FOG. The laboratory results prove that the proposed method can accurately identify the drift of FOG, and outperforms the traditional calibration method.
(18)
Calibration for the drift of z FOG is similar to calibration for the drifts of x and y FOGs, with z-axis in the horizontal plane by rotating the outside-axis of the turntable. In this case, z-axis should be changed to x-axis or y-axis and the new frame should meet the right handed rule. 4. Laboratory test An IMU is calibrated by the proposed calibration method as well as by the traditional method for comparison. The IMU consists of three FOGs with a bias stability of 0.02◦ /h and three quartz accelerometers with a bias stability of 50 g. A high-accuracy threeaxis turntable with an accuracy of 3 (1) is used in this experiment. The IMU is mounted on the center of the turntable. First, the drifts
References [1] H. Zhang, Y. Wu, W. Wu, et al., Improved multi-position calibration for inertial measurement units[J], Meas. Sci. Technol. 21 (1) (2010) 015107. [2] L. Camberlein, F. Mazzanti, Calibration technique for laser gyro strapdown inertial navigation system[C], in: Symposium Gyro Technology, Stuttgart, West Germany, 1985, pp. 5.0–5.13. [3] K.-J. Han, C.-K. Sung, M.-J. Yu, Improved calibration method for SDINS considering body-frame drift[J], Int. J. Control Autom. Syst. 9 (3) (2011) 497–505. [4] D.H. Titterton, J.L. Weston, Strapdown Inertial Navigation Technology[M], Peter Peregrinus on behalf of the Institute of Electrical Engineers, London, 2004. [5] R.M. Rogers, Applied Mathematics in Integrated Navigation Systems[M], 3rd ed., American Institute of Aeronautics and Astronautics, Reston, VA, 2007, pp. 238–250. [6] D. Wan, J. Fang, Initial Alignment of Inertial Navigation[M], Southeast University Press, Nanjing, 1998 (in Chinese). [7] G. Yan, On SINS In-Movement Initial Alignment and Some Other Problems[D], Northwestern Polytechnical University, Xi’an, 2008 (in Chinese).
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Accurate calibration for drift of fiber optic gyroscope in multi-position north-seeking phase Jianye Pan ∗ , Chunxi Zhang, Yanxiong Niu, Zhe Fan School of Instrumentation Science and Optoelectronics Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
a r t i c l e
i n f o
Article history: Received 13 December 2013 Accepted 20 July 2014 Keywords: Inertial measurement unit Fiber optic gyroscope Accelerometer bias North-seeking Calibration
a b s t r a c t This paper presents an accurate calibration method for drift of fiber optic gyroscope (FOG) in multiposition north-seeking phase. To avoid the effect of the rate of change of accelerometer bias on the drift estimate of FOG, north-seeking by coarse-alignment is performed in multi positions and the northseeking results are used to compute the drift of FOG. The laboratory test results prove that the proposed method can not only achieve multi-position north-seeking, but also accurately calibrate the drift of FOG. The comparison with the traditional method highlights the superior performance of the proposed method. © 2014 Elsevier GmbH. All rights reserved.
1. Introduction Inertial navigation is entirely self-contained and can provide information including position, velocity and attitude at a high rate. It is now widely used as the main navigation means in missiles, aircraft, robots and other autonomous vehicles [1]. Calibration for gyroscope drift is very important in inertial navigation technologies. A 1 nm/h class strapdown inertial navigation system (SINS) for fighter aircraft typically requires a stability and a calibration accuracy of 0.005◦ /h for gyroscope drifts [2]. The calibration techniques for gyroscope drifts are divided into two categories according to their measurement data types. The first category is direct calibration methods, which compare raw sensor data measured on the turntable with projections of reference gravity and earth rate [3,4]. This calibration approach requires not only rotating around inside-axis of turntable, but also rotating around outside-axis of turntable. The other category of calibration methods is using two-position alignment based on Kalman filter [5,6]. It uses the velocity and attitude indications of SINS to estimate calibration coefficient. Such methods are relatively robust as to instrumentation errors because they have the advantage of using a navigation algorithm that can determine the initial attitude and the amount of change
∗ Corresponding author. E-mail address: [email protected] (J. Pan). http://dx.doi.org/10.1016/j.ijleo.2014.07.126 0030-4026/© 2014 Elsevier GmbH. All rights reserved.
in angular position. However, in these methods gyroscope drift estimate is affected by the rate of change of accelerometer bias. Before the SINS being used in actual, to validate the accuracy of north-seeking in different azimuth directions, multi-position north-seeking test is usually performed. Multi-position northseeking test can be described as: for example, an IMU mounted on a turntable, with 7 times continuous 45◦ rotations around the inside-axis of turntable, north-seeking result (azimuth) in each position is recorded. The accuracy of north-seeking directly depends on the accuracy of gyroscope drift. In this current work, an accurate calibration method for drift of FOG in multi-position north-seeking phase is proposed, not only achieving multi-position north-seeking, but also calibrating the drift of FOG. To avoid the effect of the rate of change of accelerometer bias on drift estimate of FOG, north-seeking by coarse-alignment is performed in multi positions. After that, the drifts of FOGs are computed by the north-seeking results. The paper is organized as follows. In Section 2, the effect of the rate of change of accelerometer bias on the drift estimate of FOG is discussed. In Section 3, north-seeking by coarse-alignment is performed in multi positions and the north-seeking results are used to estimate the drift of FOG. In Section 4, the laboratory test results of the calibration as well as accuracy verification are reported. In Section 5, the conclusion is discussed.
J. Pan et al. / Optik 125 (2014) 7244–7246 2000
Table 1 North-seeking results in four positions.
1900
Test number Azimuth (0◦ ) Azimuth (90◦ ) Azimuth (180◦ ) Azimuth (270◦ )
Accelerometer outputs (μg)
1800
Test 1 Test 2 . . . Test n
1700 1600 1500
1100
0
2000
4000
6000 Time (s)
8000
10000
12000
Fig. 1. Static data output of an accelerometer.
2. Effect of the rate of change of accelerometer bias on the drift estimate of FOG In fine-alignment based on Kalman filter, azimuth alignment error is [7]
∇ nE +
n
∇E
g
tan L −
εnE +
ωN
ε nE
n
−
∇ N /g ωN
,
(1)
T
where ∇ n = [ ∇ nE ∇ nN ∇ nU ] is the accelerometer bias vector in navigation frame (the subscripts stand for north, upward and east n
n
n
n
T
velocity components, respectively), ∇ = [ ∇ E ∇ N ∇ U ] is the rate of change of accelerometer bias vector in navigation frame (the subscripts stand for north, upward and east velocity components, respectively), εnE and ε nE are the east gyroscope drift and the rate of change of east gyroscope drift in navigation frame, respectively, g is the gravity, L is the latitude and ωN is the earth rotation rate of north component. This result shows that azimuth alignment error depends on not only the gyroscope drift and accelerometer bias, but also on the rate of change of accelerometer bias. From Eq. (1), rate of change of accelerometer bias of 1 mg/h corresponds to gyroscope drift of 0.057◦ /h. An about three-hour static data output of an accelerometer is shown in Fig. 1. When the accelerometer starts, its bias changes significantly with rate of change of about 260 g/h, and this change will greatly affect azimuth alignment error. In coarse-alignment, azimuth alignment error is [7] n
∇¯ E g
tan L −
ε¯ nE ωN
,
(2)
C1 C2 . . . Cn
D1 D2 . . . Dn
where is the average accelerometer bias in navigation frame of east component and ε¯ nE is the average gyroscope drift in navigation frame of east component. Unlike fine-alignment based on Kalman filter, in coarsealignment, the rate of change of accelerometer bias will not affect the azimuth alignment error. So in this paper, coarse-alignment method is preferred to realize north-seeking. Comparing with the effect of the gyroscope drift on the azimuth alignment error, the effect of the accelerometer bias on the azimuth alignment error is small and can be neglected. Then Eq. (2) can be simplified as ε¯ nE ωN
.
Because of the initial azimuth angular position of turntable, azimuth alignment error cannot be obtained by the north-seeking result in a single position. North-seeking results in multi positions can be used to compute the azimuth alignment error and the initial azimuth angular position. The following procedure is a typical procedure for calibration. First, the IMU is mounted on a turntable and performs north-seeking in this position. Then the IMU rotates 90◦ around the inside-axis of turntable to the second position and performs north-seeking. After that, 90◦ rotation process is continued and north-seeking at the third and fourth position is performed. The tests described above are repeated for n times and the northseeking result (azimuth) at each position is recorded (shown in Table 1). The north-seeking result at each position should be averaged as follows: A = (A1 + A2 + · · · + An )/n,
(4)
B = (B1 + B2 + · · · + Bn )/n,
(5)
C = (C1 + C2 + · · · + Cn )/n,
(6)
D = (D1 + D2 + · · · + Dn )/n.
(7)
Considering the initial angular position of the turntable ˛, the north-seeking result in the first position can be expressed as A = ˛ + U (εx · cos(A) + εy · sin(A)),
(8)
where U (εx · cos(A) + εy · sin(A)) = (εx · cos(A) + εy · sin(A))/0.01 · U (0.01),
(9)
where U (0.01) is the azimuth alignment error with gyroscope drift of 0.01◦ /h at the alignment latitude and the error can be computed by Eq. (3). εx and εy are the drifts of x and y FOGs in the body frame, respectively. Substituting Eq. (9) into Eq. (8), Eq. (8) can be rewritten as A = ˛ + (εx · cos(A) + εy · sin(A))/0.01 · U (0.01).
(10)
Similarly, the north-seeking result in the third position can be expressed as
n ∇¯ E
U = −
B1 B2 . . . Bn
3. Calibration for drift of FOG in multi-position north-seeking phase
1200
U =
A1 A2 . . . An
1400 1300
U =
7245
(3)
According to Eq. (3), the drift of FOG can be estimated by azimuth alignment error.
C = 180 + ˛ + U (−εx · cos(A) − εy · sin(A)),
(11)
where U (−εx · cos(A) − εy · sin(A)) = −(εx · cos(A) + εy · sin(A))/0.01 · U (0.01).
(12)
Substituting Eq. (12) into Eq. (11), Eq. (11) can be rewritten as C = 180 + ˛ − (εx · cos(A) + εy · sin(A))/0.01 · U (0.01).
(13)
Associating Eq. (10) with Eq. (13), another equation is obtained as εx · cos(A) + εy · sin(A) = (A − C − 180)/2 · 0.01/U (0.01).
(14)
Continue with the second and fourth positions. Similar to Eq. (14), εx · cos(B) + εy · sin(B) = (B − D − 180)/2 · 0.01/U (0.01).
(15)
7246
J. Pan et al. / Optik 125 (2014) 7244–7246
Table 2 North-seeking results of the proposed method in eight positions (1 mil = 0.06◦ ). Test number
0◦
45◦
90◦
135◦
180◦
225◦
270◦
315◦
Test 1 (mil) Test 2 (mil) Test 3 (mil) Test 4 (mil) Test 5 (mil) Test 6 (mil) Average (mil)
2.0 1.9 1.8 2.1 2.6 3.0 2.2
751.8 751.6 752.6 752.4 752.1 752.2 752.1
1502.4 1502.3 1502.3 1501.7 1501.6 1501.3 1501.9
2251.5 2250.5 2250.2 2250.3 2250.5 2250.6 2250.6
3000.6 2999.8 3000.2 3000.0 3000.0 2999.7 3000.1
3749.8 3749.6 3749.8 3750.3 3750.4 3750.0 3750.0
4500.2 4501.0 4500.3 4500.4 4499.6 4499.5 4500.2
5250.9 5251.2 5251.4 5251.2 5251.4 5251.3 5251.2
Table 3 Calibration results of the drifts of FOGs. Error source
Group 1 of the proposed method
Group 2 of the proposed method
The traditional Direct method calibration
Drift of x FOG (◦ /h) Drift of y FOG (◦ /h)
−0.016
−0.008
−0.006
−0.014
−0.013
−0.014
−0.021
−0.011
of FOGs are calibrated by the traditional method using two-position alignment based on Kalman filter. Then the drifts of FOGs are calibrated by the proposed method. In multi-position north-seeking phase, the X-, Y- and Z-axes of the IMU point to the east, north and up directions, respectively, as the initial orientation. With 7 times continuous 45◦ rotations around the inside-axis of turntable, north-seeking result (azimuth) in each position is recorded (shown in Table 2). The north-seeking results are divided into two groups. Group 1 includes the north-seeking results at 0◦ , 90◦ , 180◦ and 270◦ positions. Group 2 includes the north-seeking results at 45◦ , 135◦ , 225◦ and 315◦ positions. The initial azimuth angular position of turntable is 1.06 mil, calculated by Eq. (16). After north-seeking, the drifts of FOGs are calibrated by the direct calibration method, and the calibration results are taken as reference. Table 3 presents the calibration results of the drifts of FOGs. The accuracy of the drifts estimates of x and y FOGs are 72.3% and 81.6%, respectively, better than the traditional method, with the accuracy of 42.9% and 52.4%, respectively. 5. Conclusion
Associating Eq. (14) with Eq. (15), the drifts of x and y FOGs can be obtained. In addition, the initial angular position of the turntable can be computed by ˛ = (˛1 + ˛2)/2,
(16)
where ˛1 = (C − 180) − (C − A − 180)/2,
(17)
and ˛2 = (D − 270) − (D − B − 180)/2.
In this paper, an accurate calibration method for drift of FOG in multi-position north-seeking phase is proposed. Considering the effect of the rate of change of accelerometer bias on the drift estimate of FOG in traditional calibration method, in proposed calibration method north-seeking by coarse-alignment is performed in multi positions and the north-seeking results are used to estimate the drift of FOG. The laboratory results prove that the proposed method can accurately identify the drift of FOG, and outperforms the traditional calibration method.
(18)
Calibration for the drift of z FOG is similar to calibration for the drifts of x and y FOGs, with z-axis in the horizontal plane by rotating the outside-axis of the turntable. In this case, z-axis should be changed to x-axis or y-axis and the new frame should meet the right handed rule. 4. Laboratory test An IMU is calibrated by the proposed calibration method as well as by the traditional method for comparison. The IMU consists of three FOGs with a bias stability of 0.02◦ /h and three quartz accelerometers with a bias stability of 50 g. A high-accuracy threeaxis turntable with an accuracy of 3 (1) is used in this experiment. The IMU is mounted on the center of the turntable. First, the drifts
References [1] H. Zhang, Y. Wu, W. Wu, et al., Improved multi-position calibration for inertial measurement units[J], Meas. Sci. Technol. 21 (1) (2010) 015107. [2] L. Camberlein, F. Mazzanti, Calibration technique for laser gyro strapdown inertial navigation system[C], in: Symposium Gyro Technology, Stuttgart, West Germany, 1985, pp. 5.0–5.13. [3] K.-J. Han, C.-K. Sung, M.-J. Yu, Improved calibration method for SDINS considering body-frame drift[J], Int. J. Control Autom. Syst. 9 (3) (2011) 497–505. [4] D.H. Titterton, J.L. Weston, Strapdown Inertial Navigation Technology[M], Peter Peregrinus on behalf of the Institute of Electrical Engineers, London, 2004. [5] R.M. Rogers, Applied Mathematics in Integrated Navigation Systems[M], 3rd ed., American Institute of Aeronautics and Astronautics, Reston, VA, 2007, pp. 238–250. [6] D. Wan, J. Fang, Initial Alignment of Inertial Navigation[M], Southeast University Press, Nanjing, 1998 (in Chinese). [7] G. Yan, On SINS In-Movement Initial Alignment and Some Other Problems[D], Northwestern Polytechnical University, Xi’an, 2008 (in Chinese).