An improved self-alignment method for strapdown inertial navigation system based on gravitational apparent motion and dual-vector Xixiang Liu, Yu Zhao, Xianjun Liu, Yan Yang, Qing Song, and Zhipeng Liu Citation: Review of Scientific Instruments 85, 125108 (2014); doi: 10.1063/1.4903196 View online: http://dx.doi.org/10.1063/1.4903196 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/85/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Inertial Navigation: A Bridge Between Kinematics and Calculus Phys. Teach. 50, 112 (2012); 10.1119/1.3677289 A method for fine positioning of diagnostic packages in inertial confinement fusion experiments Rev. Sci. Instrum. 82, 125113 (2011); 10.1063/1.3669780 Inertial measurement with trapped particles: A microdynamical system Appl. Phys. Lett. 96, 143501 (2010); 10.1063/1.3360808 Wireless Impact Monitoring System for the Return to Flight Mission AIP Conf. Proc. 746, 122 (2005); 10.1063/1.1867126 New miniaturized tunneling-based gyro for inertial measurement applications J. Vac. Sci. Technol. B 17, 2948 (1999); 10.1116/1.590931
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REVIEW OF SCIENTIFIC INSTRUMENTS 85, 125108 (2014)
An improved self-alignment method for strapdown inertial navigation system based on gravitational apparent motion and dual-vector Xixiang Liu,a) Yu Zhao, Xianjun Liu, Yan Yang, Qing Song, and Zhipeng Liu School of Instrument Science and Engineering, Southeast University, Nanjing 210096, China and Key Laboratory of Micro-inertial Instrument and Advanced Navigation Technology, Ministry of Education, Nanjing 210096, China
(Received 11 August 2014; accepted 18 November 2014; published online 10 December 2014) Analysis and simulation results indicate that two problems should be solved when the self-alignment method based on gravitational apparent motion and dual-vector can be used for Strapdown Inertial Navigation System. The first one is how to identify the apparent motion from accelerometer measurement containing random noise and the second is how to avoid the collinear problem between two vectors used in alignment solution. In this paper, a parameter identification and reconstruction algorithm is proposed to solve the first problem and simulation results indicate that proposed algorithm can identify apparent motion from accelerometer measurements effectively; and reconstruction algorithm based on current identified parameters for dual-vector is designed in detail to solve the second problem which can make full use of newest identification and avoid collinear problem completely. Simulation and turntable results show that the proposed alignment method can fulfill self-alignment in a swinging condition and the alignment accuracy can reach the theoretical values determined by the sensor precision. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4903196] I. INTRODUCTION
Initial alignment is one of the key and difficult problems of Inertial Navigation System (INS).1 Compared with the initial attitude, initial velocity and position can be easily acquired with reference navigation system, such as Global Position System (GPS). Thus, initial alignment for INS mainly refers to get initial attitude.2 According to whether external reference navigation information is needed, alignment method can be divided into self-alignment and nonself-alignment methods.2–6 Compass alignment method based on compass effect and zero-velocity alignment based on Kalman-liking filter and analytical method based on dualvector are typical self-alignment method.1, 2 In Strapdown INS (SINS), because the sensor are installed on board directly the measurement data from inertial measurement unit (IMU) are vulnerable to external interference, among which swinging and shaking motion is a typical interference.1 When the above methods are used for SINS to fulfill alignment, compass alignment method can overcame these interference but with a prolonged alignment time;3 zerovelocity alignment method also can do but with a decreased alignment accuracy;4, 5 while analytical method cannot because the Earth rotation rate are completely submerged in measurement noise.6 Recently, anti-disturbance self-alignment method for SINS has become a focus issue.6–14 In 2000, the articles from IxSea Company claimed that a novel alignment algorithm was used in their newest compass product – Octans, which can fulfill alignment within 5 min under any swinging condition.7–9 Considering the protection of commercial interest and technology security, these papers only mentioned that they aca) Author to whom correspondence should be addressed. Electronic mail:
[email protected] 0034-6748/2014/85(12)/125108/11/$30.00
quired geographical north by observing the gravity drift in inertial frame. Based on this alignment mechanism, relevant investigation has been done and many implement methods have been proposed, among which dual-vector alignment method is widely used.6, 11–14 The alignment method based on gravitational apparent motion and dual-vector translates the alignment problem of solving the attitude matrix from body frame to navigation frame to solving the matrix from initial body frame to navigation frame with matrix decomposition. But different from the vectors used in classical analytical method based on dualvector, in this method the gravity vectors at two different moments are used as dual-vector which will be further projected into the initial body frame and initial navigation frame, respectively. By the virtue of these projected vectors and dualvector attitude determination algorithm, the attitude matrix from initial body frame to navigation frame can be acquired and the matrix from body frame and navigation frame can be further solved. Compared with the classical dual-vector method, this new method does not need to separate Earth rotation rate from gyro measurement, thus alignment accuracy will be significantly increased.6–10 However, two problems caused by alignment mechanism should be studied: the first one is how to avoid collinear possibility between two vectors caused by measurement noise and limited alignment time while the second is how to improve constructing accuracy of gravitational apparent motion.6, 11 Reference 11 introduced ill-conditioned matrix theory to study collinear problem and conclude that at least 1 min between two vectors is needed to avoid collinear probability without considering any other errors. References 6 and 7 deduced the theoretical accuracy determined by sensor errors and classified this method as a coarse alignment method because of its low robustness against interferences. References 12 and 13 integrated gravitation apparent motion
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to form velocity and used velocity vectors instead of gravitational vectors participating alignment solution, thus alignment accuracy can be increased because of the smoothing affect to random noise introduced by integration but the alignment time is greatly prolonged to acquire a desired smoothing results. Reference 14 introduced low-pass filter to pre-process the measurement data and achieved some good results. However, it is difficult to find an available low-pass filter for all environments with complex noise. Based on the alignment mechanism of tracking gravitational apparent motion and attitude determination algorithm with dual-vector, an improved self-alignment method for SINS is proposed in this paper. In this method, a parameter identification algorithm is introduced to identify the parameters describing theoretical apparent motion from accelerometer measurement and a reconstruction algorithm is introduced to construct dual-vectors. Simulation and turntable test verified the effectiveness of the proposed improved method. The rest of this paper is organized as follows. In Sec. II, the gravitational apparent motion is detailedly described and the self-alignment method based on gravitational apparent motion and dual vector is designed and simulated. And the causes of low alignment accuracy are also analyzed and summarized. Section III presents an improved alignment method based on parameter identification and reconstruction. In Sec. IV the turntable experiment is carried out and finally the conclusion is given in Sec. V.
Rev. Sci. Instrum. 85, 125108 (2014)
FIG. 1. Apparent motion of local gravity vector. (a) Gravitational apparent motion in inertial frame. (b) The choice of inertial frame.
II. ALIGNMENT MECHANISM A. Gravitational apparent motion in inertial frame
The concept of apparent motion in INS is first used to study the characters of gyros. When the free gyro is observed in navigation frame rotating with the Earth, the track of gyro’s rotating axis in inertial space is defined as apparent motion. But the gravitational apparent motion is defined as the track of gravity at a fix point rotating with the Earth when the observation is processed in inertial frame. Take the point P located in the north Earth illustrated in Fig. 1(a) as an example, where the gravity is g pointing to the Earth center O. In this subsection, the initial Earth frame e0 is fixed as inertial frame ie0 , in which the change of gravity direction is investigated. When the Earth rotates from west to east, the direction of gravity will change both in the azimuth and height and the track of gravity direction is a cone with the vertex located at O. Similarly, when the point located at equator, the track is a circle with the center located at O. If the circle is regarded as a special cone, the gravitational apparent motion can be regarded all as cone without considering the point located at the pole. As shown in Fig. 1, we can see the parameters describing the cone of gravitational apparent motion, such as the conical bottom radius, are independent on the choice of inertial frame and the characters can be summarized as follows: (1) the cone axis is always the rotation axis of the Earth; (2) the cone vertex is located at the Earth center; (3) the conical bottom radius is determined by the latitude where the pointed is located; and (4) 24 h are needed to form a whole cone. Notably, the math-
ematical expression about the cone is corresponding to the selection of inertial frame and different selection will brings different expression, though the parameters about the cone are independent on the selection. For the convenience, Right-Forward-Up, as shown in Fig. 1(b), is defined as body frame b and the initial body frame b0 is fixed as inertial frame ib with the center Ob translating 0 0 to O. In the following parts, the proposed alignment method will be studied in this new inertial frame. B. Self-alignment method based on apparent motion and dual-vector
1. Alignment mechanism
In SINS, the classical analytical method based on dualvector attitude determination algorithm chooses the Earth rotation rate ωie and gravity g as dual vectors and takes the advantage of the measurement values in b frame and theoretical values in navigation frame n to construct the corresponding vectors and matrixes and acquire the attitude matrix C bn . But the method based on gravitational apparent motion and dual-vector attitude determination algorithm, the matrix C bn is decomposed as follows: i
n
C bn (t) = C bi (t)C nb00 (t)C n0 (t), b0
(1)
where C bi (t) is the attitude matrix from b frame to ib0 frame b0 which can be updated with the gyro measurements from IMU; n0 C n (t) is the matrix from initial navigation frame to n frame
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where g n = [ 0 0 −g ]T is the theoretical value of gravity veci
tor in n frame. Unfortunately, the matrix C nb0 (t) in Eq. (4) is an unknown parameter, thus Eq. (4) cannot be used directly. But Eq. (4) can be further expressed as follows: ⎡ ⎤ 0 i i g ib0 (t) = C nb0 (t)g n = C bb0 (t)C bn (t) ⎣ 0 ⎦ , (5) g where C bn (t)([ 0 0 g ]T ) is theoretical values of accelerometer measurement in b frame. Then, the gravitational apparent motion can be constructed as follows: FIG. 2. Self-alignment mechanism based on apparent motion and dualvector.
b i fˆib0 (t) = Cˆ bb0 (t) f˜ (t),
(6)
b
which can be updated with the latitude, Earth rotation rate, n and alignment time. The update for C bi (t)and C n0 (t) are as b0 follows: b b b Cˆ˙ ib0 (t) = Cˆ ib0 (t) ω˜ bib × = Cˆ ib0 (t) ω˜ bi b × , (2)
where f˜ (t) is accelerometer measurement in b frame. According to the relationship of frames defined in Fig. 3, there is f ib0 (t) = −g ib0 (t) without considering the errors caused by measurement and calculation. Thus, the first problem in Sec. II B 1 is solved.
b0
n n e Cˆ n0 (t) = C e00 (t) Cˆ e0 (t)C en ⎡ ⎤ ⎤⎡ 0 − sin L cos L cos(ωie t) − sin(ωie t) 0 ⎢ ⎥ ⎥⎢ 1 0 0 ⎥⎢ sin(ωie t) cos(ωie t) 0⎥ =⎢ ⎣ ⎦ ⎦⎣ 0 0 1 0 cos L sin L
⎡
0
⎢ × ⎣ − sin L cos L
1
0
⎤
0
⎥ cos L ⎦ ,
0
sin L
(3)
where the superscript “∼” and “ ˆ ” denote measurement and calculation data, respectively; ω˜ bi b is the gyro measurement;
3. Calculation for the theoretical gravity in initial navigation frame
The projection of gravity in n0 frame at the moment t can be calculated as follows: ⎡ ⎤ 0 n n gˆ n0 (t) = Cˆ n0 (t)g n (t) = Cˆ n0 (t) ⎣ 0 ⎦ , (7) g n
where Cˆ n0 (t) can be calculated accurately without any error with Eq. (3). That is to say, the projected data in n0 frame is the theoretical one. Thus, the second problem in Sec. II B 1 is solved.
b0
b Cˆ ib0 (0) = I 3×3 ; e is the Earth frame; L is the latitude, and t is alignment time. When vehicle is without translational motion, e C ne (t) and C n00 are both fixed matrixes. With Eqs. (1)–(3), the alignment problem of acquiring i C bn is translated into acquiring the matrix Cˆ nb00 . As shown in Fig. 2, East-North-Up (ENU) is defined as navigation frame. In the alignment method based on gravitational apparent motion and dual-vector, the apparent motion vectors in ib0 frame at the moments t1 and t2 are chosen as dual vectors participating in alignment solution with dual-vector attitude determination algorithm. However, three problems should be presolved: (1) the first problem is how to calculate the gravitational apparent motion in ib0 frame; (2) the second is how to get the theoretical gravity in n0 frame; and (3) the third is how to use the above vectors.
2. Calculation for gravitational apparent motion in inertial frame
According to the analysis in Sec. II A, gravitational apparent motion is the projection of gravity in inertial frame, thus g ib0 at the moment t can be calculated as the follows: ⎡ ⎤ 0 i i (4) g ib0 (t) = C nb0 (t)g n = C nb0 (t) ⎣ 0 ⎦ , g
i 4. Calculation for Cˆ nb00
When the vectors of fˆib0 and gˆ n0 at two moments t1 and i t2 are obtained, C nb00 can be constructed as follows:10 n T i Cˆ nb00 = Cˆ i 0 b0 ⎛⎡ ⎤−1 ( gˆ n0 (t1 ))T ⎜⎢ ⎥ ⎜ ( gˆ n0 (t1 ) × gˆ n0 (t2 ))T ⎥ = − ⎜⎢ ⎦ ⎝⎣ n n n T ( gˆ 0 (t1 ) × gˆ 0 (t2 ) × gˆ 0 (t1 )) ⎡
⎤⎞T
( fˆib0 (t1 ))T
⎢ ×⎢ ⎣
⎥⎟ ⎥⎟ . ⎦⎠
( fˆib0 (t1 ) × fˆib0 (t2 ))T ˆib0
(f
(t1 ) × f
ˆib0
(t2 ) × f
ˆib0
(t1 ))
(8)
T
i Generally speaking, Cˆ nb00 in Eq. (8) cannot satisfy the orthogonality for the calculated errors caused by sensor errors, i thus Cˆ nb00 should be further orthogonalized as follows:10, 12
i
C nb00
o
i i T i − 1 = C nb00 C nb00 C nb00 2 .
(9)
Substitute Eq. (9) into Eq. (1), then initial alignment can be fulfilled. Thus, the last problem in Sec. II B 1 is solved.
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Error (°)
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0.15 0.1 0.05 0 -0.05 -0.1 0
200
Theoretical value 400
Case 1 600
Case 2 800
1000
1200 t (s)
Error (°)
(a) Theoretical value
0
Case 1
Case 2
-0.05 -0.1 0
200
400
600
800
1000
1200 t (s)
Error (°)
(b) 150
10 5 0 -5
100 50 0 0
Case 2 0
200
200
400
400
Case 1
Theoretical value
600
800
600
1000
800
1000
1200
1200 t (s)
(c) FIG. 3. Curves of alignment errors: (a) pitch, (b) roll, and (c) yaw.
5. Alignment accuracy
According to the Ref. 6, the theoretical accuracy with the above method is as follows: φE =
∇N , g
φN = −
∇E + εN t, g
∇ ε t εE − E tan L + U , φU = ωie · cos L g 2
(10)
where ∇ E and ∇ N are the equivalent accelerometer bias in east and north, respectively; εE and εU are the equivalent gyro bias in east and up, respectively. Detailed deduction can be referred to Ref. 6. C. Simulation
1. Simulation setting
The vehicle is assumed without translational movement but with two different swinging motions as defined in Table I. In Case 1, the vehicle is assumed to be static with zero attitude and the sensor is with constant errors as defined in Table II. In Case 2, the vehicle is assumed swinging with the function A sin (2π f · t + η0 ) + θ 0 , in which A and f denote the swinging amplitude and frequency, η0 and θ 0 denote the swinging initial phase and center, respectively. The swinging parameters and sensor errors for Case 2 are defined in Tables III and II, respectively. The longitude and latitude for both cases are as 118◦ E and 32◦ N, respectively. Notably, in order to com-
pare the gravitational apparent motion in two cases, the initial phase and swinging center in Case 2 should be both set as zero to ensure the inertial frame of two cases are the same one. With the above setting, theoretical outputs of IMU can be generated by back-stepping of SINS navigation solution. When the errors in Table II are added into these theoretical outputs, the data sensor outputs can be simulated. To evaluate alignment accuracy, the ideal motions are used as references and the difference between alignment results and references are regarded as alignment errors. At the same time, sensor errors in Table II and alignment time are used to construct theoretical alignment accuracy (the minimized alignment errors) with Eq. (10). In simulation, t1 and t2 are assumed as 0 and t, respectively. The update cycle of sensor data and alignment solution are both as 10 ms, that is to say a single sample solution algorithm is adopt for integration,15 while the update cycle of error is 1 s.
2. Simulation results
The simulation lasts for 1200 s. Alignment results are shown in Fig. 3, in which (a), (b), and (c) denote the alignment errors of pitch, roll, and yaw, respectively; and the
TABLE II. Sensor errors setting. Gyro bias (◦ /h)
Acce bias (ug)
Constant Random (white noise) Constant Random (white noise) TABLE I. Two cases for simulation. Case 1 Case 2
Without swinging motion but with constant sensor errors With swinging motion and constant/random sensor errors
x-axis y-axis z-axis
0.05 0.05 0.05
0.05 0.05 0.05
500 500 500
500 500 500
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TABLE III. Swinging parameters setting.
Amplitude (deg) Cycle (s) Initial phase (deg) Swinging center (deg)
Pitch
Roll
Yaw
7 8 0 0
15 7.5 0 0
5 6 0 0
dotted-dashed and dashed lines are corresponding to Case 1 and Case 2, respectively, while the solid lines denote the theoretical alignment values. The statistics of alignment errors are shown in Table IV. In Fig. 3, the curves in Case 1 are coincide with the theoretical ones and in this condition dotted-dashed lines are overlap by solid lines and the statistics in Table IV also show that the alignment results are highly approximate to theoretical values. The curves in Fig. 3 also show that the alignment for pitch and roll can be fulfilled and alignment for yaw cannot converge to theoretical one; and the statistics also indicate that in this case, the mean and standard covariance of alignment results are all larger than the theoretical ones, among which the error of yaw is larger for several times. At the same time, the curves in Fig. 3 also show that in Case 2, there are large burrs in alignment results throughout the whole alignment process. From the above analysis, it can be concluded that the proposed method can fulfill alignment in an ideal condition, but cannot in a condition with swinging motion and measurement errors which is more in accordance with engineering situation. Thus, some improvements should be done.
possible reasons are as follows: (1) the errors in Cˆ ib0 (t); n (2) the errors in Cˆ n0 (t); (3) the errors in fˆib0 (t) and gˆ n0 (t); and i (4) and the errors in Cˆ nb00 (t). b a. The errors in Cˆ ib0 (t)
The errors from integration and gyro b outputs are the main sources of the error in Cˆ ib0 (t). Considering the integration errors are relatively small when the vehicle is in non-conical motion. The errors of integration can be ignored, thus the investigation focuses on the measurement errors of gyro outputs. Gyro errors can be divided into two types: one is constant error and the other is random one. When the gyro outputs are integrated, the integration errors caused by constant one are gradually increased while those caused by random are randomly presented as the burrs. The maximum integration error caused by the constant error set in Table III is merely 0.0166◦ in 1200 s without considering the coupling effects among three axes. Compared with the rotating angle of the Earth (5.0◦ in 1200 s), this error can be ignored. According to the Wright rules, the maximum random gyro error in Table III can be set as 3σ (0.15◦ /h). Then the maximum integration error from random error is 4.1667 × 10−7◦ with an integration cycle of 10 ms, which can also be ignored. Therefore, the reasons for low alignment accuracy b have no business with the errors in Cˆ ib0 (t).
n b. The errors in Cˆ n0 (t)
According to the Eq. (3), when the n vehicle is in static with an known latitude, the error in Cˆ n0 (t) is mainly caused by timing error which can be ignored, that is n to say, there is no error in Cˆ n0 (t). Further, it can be concluded that there is no error in gˆ n0 (t) described in Sec. II B 3.
3. Further analysis for alignment results
In this section, the reasons caused low accuracy of alignment in Sec. II C 2 are analyzed. From Eqs. (1)–(8), the
i When to construct Cˆ nb00 (t), fˆib0 (t) and gˆ n0 (t) should be first calculated. The investigation
c. The errors in fˆib0 (t) and gˆ n0 (t)
TABLE IV. Statistics of alignment errors (deg). Time (s) Theoretical value
Pitch Roll Yaw
Case 1
Pitch Roll Yaw
Case 2
Pitch Roll Yaw
0–200
201–400
401–600
601–800
801–1000
1001–1200
Mean Std Mean Std Mean Std
0.02865 0.00000 − 0.02727 0.00080 0.20800 0.00040
0.02865 0.00000 − 0.02449 0.00080 0.20939 0.00040
0.02865 0.00000 − 0.02171 0.00080 0.21077 0.00040
0.02865 0.00000 − 0.01893 0.00080 0.21216 0.00040
0.02865 0.00000 − 0.01615 0.00080 0.21355 0.00040
0.02865 0.00000 − 0.01338 0.00080 0.21494 0.00040
Mean Std Mean Std Mean Std
0.02859 0.00000 − 0.02730 0.00074 0.20666 0.00045
0.02863 0.00003 − 0.02474 0.00074 0.20823 0.00045
0.02868 0.00003 − 0.02218 0.00074 0.20979 0.00045
0.02874 0.00003 − 0.01963 0.00074 0.21136 0.00045
0.02884 0.00003 − 0.01707 0.00074 0.21293 0.00045
0.02894 0.00004 − 0.01451 0.00074 0.21450 0.00045
Mean Std Mean Std Mean Std
0.02923 0.03007 − 0.04201 0.02279 10.62012 28.33166
0.02879 0.03025 − 0.04392 0.02233 1.73950 1.82445
0.02907 0.02625 − 0.04200 0.02239 1.11607 0.96228
0.02515 0.02795 − 0.03972 0.02216 0.94730 0.71448
0.02681 0.02852 − 0.03705 0.02256 0.741114 0.53721
0.02979 0.02777 − 0.03500 0.02269 0.60670 0.41590
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FIG. 4. Gravitational apparent motion in inertial frame: (a) axis-xi , (b) axis-yi , and (c) axis-zi . b0
focuses on the error in fˆib0 (t) because gˆ n0 (t) is considered as theoretical values without any error. As shown in Fig. 4, the gravitational apparent motion fˆib0 can be calculated with Eq. (6) and accelerometer measurement f˜b . In Figs. 4(a)–4(c) denote the projections of apparent motion on axes xi , yi , and zi , respectively, b0 b0 b0 while dotted-dashed and dashed lines are corresponding to Case 1 and Case 2. The curves in Fig. 4 indicate that the changes of fˆib0 in Case 1 are smoothly without any burrs. In this condition, the projections can be treated as the true apparent motion due to non-translational and swinging motions without considering the constant sensor errors. But in Case 2, though the changes of projections can present the drift trend of apparent motion, there are large random noises. b Regardless of the errors in Cˆ ib0 (t), these errors in projections are mainly caused by the random noise of accelerometer outputs. Take the projection along yi -axis as an example. During b0 y 1200 s, the projection of fˆ ib0 increased from 0.00479 m/s2 to 0.01926 m/s2 in Case 1 while in Case 2 the maximum and minimum projections are 0.03276 m/s2 and −0.00626 m/s2 , respectively. The random errors in Case 2 are so large that the projection cannot truly represent the changing of gravity. Thus, when these projections are used to full alignment, low accuracy will be inevitably led. Therefore, one important reason for low accuracy should be the errors in fˆib0 .
i d. The errors in Cˆ nb00 (t)
One precondition to use dual-vector attitude determination algorithm for alignment is that these two vectors cannot be collinear. With a longer interval between two vectors, the possibility of collinear will be lower. But as shown in Fig. 4, even though the interval between two vectors is long enough, such as 1200 s, the possibility still exists because of the random noise in fˆib0 . When
b0
b0
the vectors of fˆib0 (t1 ) and fˆib0 (t2 ) are collinear or approximately collinear, the construction for ( fˆib0 (t1 ) × fˆib0 (t2 )) and ( fˆib0 (t1 ) × fˆib0 (t2 ) × fˆib0 (t1 )) will be failure which will further lead to alignment failure. Therefore, the second reason for low alignment accuracy i should be the errors in Cˆ nb00 (t).
III. IMPROVEMENT TO PROPOSED METHOD
As the analysis in Sec. II C 3, low accuracy is evitable with the proposed method in Sec. II A when the vehicle is with a swinging motion and the sensors with random errors. The main reason lies in the random noise in accelerometer outputs, which brings the follows two problems: (1) the calculated apparent motion cannot truly represent the drift of gravity in inertial frame because the true apparent motion is submerged in random noise and (2) there is a high possibility of collinear between two vectors. Thus, the investigation in this section focuses on solving the above two problems.
A. Identification and reconstruction for gravitational apparent motion
1. Theoretical expression of gravitational apparent motion
As analyzed in Sec. II A, the track of apparent motion is an ideal cone and the parameters describing the cone are independent on the choice of inertial frame, but the specific expression is related to the selection of inertial frame. Theoretical expression for gravitational apparent motion can be constructed as follows with Eq. (4): i
i
n
e
f ib0 (t) = −g ib0 (t) = −C nb0 g n (t) = −C nb00 C e00 C e0 (t)C en (t)g n , (11)
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where C M N (t) is the attitude matrix from frame M to frame N. i n Obviously, C nb00 is a fixed matrix; C e00 and C ne (t) are both fixed when the vehicle is without translational motion. Expand Eq. (11) as follows: i
n
i
e
e
f ib0 (t) = C nb00 C e00 C e0 (t)C en (t) f n = C eb00 C e0 (t)C en f n ⎤ ⎤⎡ ⎡ cos(ωie t) − sin(ωie t) 0 a11 a12 a13 ⎥ ⎥⎢ ⎢ a a22 a23 ⎥ ⎢ sin(ωie t) cos(ωie t) 0⎥ =⎢ ⎦ ⎦⎣ ⎣ 21 0 0 1 a31 a32 a33 ⎡ b11 ⎢b21 ×⎣ b31 ⎡
b12 b22 b32
⎤ ⎤ b13 ⎡ 0 b23 ⎥ ⎣ ⎦ 0 ⎦ −g b33
ib0
A11 cos(ωie t) + A12 sin(ωie t) + A13
A11 ⎢ A =⎢ ⎣ 21 A31
A12
⎤
A22 A32
(t) is determined by the accuracy of
3. Simulation for identification and reconstruction
⎤⎡
⎤ cos(ωie t) ⎥⎢ A23 ⎥ ⎣ sin(ω t) ⎥ ie ⎦ , ⎦ A33 1 A13
where K k is the gain matrix; P k is the covariance matrix for state vector; Rk is the covariance matrix for measurement noise; and I k is an unit matrix with the dimension 3 × 3. Similarly, identification for the parameters along xi and b0 yi can be processed. With these identified parameters, gravb0 itational apparent motion can be reconstructed. The accuracy of reconstructed f identification.
⎢ ⎥ A cos(ωie t) + A22 sin(ωie t) + A23 ⎥ =⎢ ⎣ 21 ⎦ A31 cos(ωie t) + A32 sin(ωie t) + A33 ⎡
The equations of on-line estimation filter based on recursive squares algorithm is as follows:16 ⎧ −1 T T ⎪ ⎪ ⎨ K k = P k−1 H k H k P k−1 H k + Rk , (16) X k = X k−1 + K k (Z k − H k X k−1 ) ⎪ ⎪ ⎩ P k = (I k − K k H k ) P k−1
(12)
i
where aij and bij (i, j = 1–3) are the elements of C eb00 and C en while Aij are the combination of aij and bij . In Eq. (12), the above elements are all fixed values with a fixed inertial frame.
a. Simulation setting The simulation is conducted with the same condition described in Sec. II C. Initial parameters for identification filter are set as follows: X 0 = [ 0 0 0 ]T and P0 = diag[ 10000 10000 10000 ]. R = 1 μg and R = 500 μg are set for Case 1 and Case 2, respectively. In simulation, the reconstructed apparent motion is constructed with the parameters from identification filter and alignment time.
The simulation lasts for 1200 s. The reconstructed motion is shown in Fig. 5, in which (a), (b), and (c) denote the projections along xi , yi , and zi , respectively, b0 b0 b0 and dotted-dashed and dashed lines are corresponding to Case 1 and Case 2, respectively. For the convenience of comparison, the projections from Fig. 4 in Case 1 are redrawn in Fig. 5 with solid lines and used as a reference. The curves in Fig. 5 show that in Case 1 the reconstructed motion is fully coincident with the true one while in Case 2 the reconstructed motion becomes stable after a period of large fluctuation but with a small fluctuation during the following process. This small fluctuation is mainly attributed to the real-time update of measurement information in parameter identification. Compared the reconstructed motions in Case 1 and Case 2, it can be found that: (1) there is a small constant difference between them, but this difference is small enough to be ignored and (2) the changing trend of both reconstructed data are parallel to each other. Therefore, it can be concluded that reconstructed algorithm based on parameter identification can effectively remove the random noise from the calculated apparent motion.
b. Simulation results
2. Identification and reconstruction algorithm for apparent motion
If the parameters Aij describing gravitational apparent motion can be identified from the calculated fˆib0 (t) contain ib0
ing random noise, the theoretical apparent motion f (t) can be reconstructed. Here, the subscript “∩” denote the reconstructed data. Take the identification for A31 , A32 , and A33 describing the motion along zi as an example. During identification, the b0 on-line estimation algorithm based on recursive least square algorithm (RLS) is adopted to avoid large data storage and heavy computation. With RLS, the state vector can defined as X = [ A31 A32 A33 ]T and the measurement vector as Z k zi = fˆk b0 , where k denotes kth recursive step and k = t / t, where t and t denote alignment time and identification period (also alignment solution period). The system equation is as follows: X k+1 = X k .
(13)
The measurement matrix can be constructed as follows with Eq. (12): H k = [ cos(ωie t)
sin(ωie t)
1 ].
(14)
Theoretically, initial alignment can be fulfilled with the
And the measurement equation is as follows: Zk = H k X k + V k , where V k is the vector of measurement noise.
B. Reconstruction algorithm for dual-vector ib0
(15)
ib0
ib0
reconstructed f (0) and f (t). But the accuracy of f (0) is too low to fulfill this task because it is reconstructed with those initial parameters for identification filter. Another
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FIG. 5. Reconstructed gravitational apparent motion based on identified parameters: (a) axis-xi , (b) axis-yi , and (c) axis-zi . b0
ib0
scheme for the selection of dual-vector is using f
(t/2) and
ib0
f
ib0
(0) and f
ib0
ing and updating for f ib0
(t/2) and the second is the accuracy
⎡ ib0
f
ib0
f (t/2) is still lower than that of f (t). Also, there exists the possibility of collinear in this scheme because of the low ib0
accuracy of f (t/2). In this paper, a new reconstruction method for dualvector is proposed in which the current parameters are used to reconstruct the apparent motion. Here the superscript “∪” denotes the reconstructed data of the past moment with cur ib0
rent parameters. That is to say, the reconstructed f ib0
f
(0) and
(t) are used to participate in alignment solution, where
b0
(t) are reconstructed as follows:
ib0
f (t) as described in Ref. 12, but two additional problems will be introduced: the first is the added operation on stor-
b0
⎤
⎢ ⎥ ˆ ⎢ sin(ωie · 0) ⎥ , (0) = A(t) ⎣ ⎦ 1 ⎡
ib0
f
cos(ωie · 0)
cos(ωie t)
⎤
(17)
⎢ ⎥ ˆ ⎢ sin(ωie t) ⎥ . (t) = A(t) ⎣ ⎦ a1
This reconstruction algorithm brings the following two advantages: (1) the first is all the measurement information and the latest identification results can be fully utilized with two vectors, which expand the utilized information at two
FIG. 6. Curves of alignment errors with improved method: (a) pitch, (b) roll, and (c) yaw.
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TABLE V. Statistics of alignment errors of improved method (deg). Time (s) Case 1
Pitch Roll Yaw
Case 2
Pitch Roll Yaw
0–200
201–400
401–600
601–800
801–1000
1001–1200
Mean Std Mean Std Mean Std
0.02851 0.00006 − 0.02732 0.00081 0.20666 0.00045
0.02857 0.00005 − 0.02473 0.00074 0.20823 0.00045
0.02867 0.00003 − 0.02217 0.00074 0.20980 0.00045
0.02874 0.00003 − 0.01961 0.00074 0.21136 0.00045
0.028835 0.00003 − 0.01705 0.00074 0.21293 0.00045
0.02894 0.00004 − 0.01449 0.00074 0.21450 0.00045
Mean Std Mean Std Mean Std
0.02820 0.01041 − 0.02765 0.02269 − 5.00769 27.67993
0.02840 0.00998 − 0.02410 0.02234 0.21231 0.02576
0.02848 0.00992 − 0.02183 0.02253 0.20735 0.00013
0.02854 0.00981 − 0.01953 0.02229 0.20787 0.00010
0.02863 0.00984 − 0.01635 0.02231 0.20926 0.00010
0.02874 0.00983 − 0.01412 0.02251 0.21083 0.00010
moments to the whole alignment process and (2) the second is the possibility of collinear is completely avoided for the same set of parameters are used in these two vectors. C. Simulation for the improved method
1. Simulation setting
The simulation is conducted with the same conditions described in Sec. II C and parameter for identification filter is as those in Sec. III A. 2. Simulation results
The simulation lasts for 1200 s. Alignment results are shown in Fig. 6, in which (a), (b), and (c) denote the alignment errors of pitch, roll, and yaw, respectively, and the dotteddashed and dashed lines are corresponding to Case 1 and Case 2 with improved method, respectively. The statistics of alignment results are shown in Table V. The curves in Fig. 6 and statistics in Table V all indicate that with the improved alignment method, initial alignment accuracy can reach theoretical alignment one. Compared the curves in Fig. 6 with those in Fig. 3, it can be found that in Case 2 with the improved method, the level alignment results oscillate with constant amplitudes showing no burrs.
These constant amplitudes are mainly caused by the change of equivalent sensor errors in navigation frame. At the same time, the variances of level alignment errors in Table V are smaller than those in Table IV. The curves of yaw also indicate that the alignment accuracy increases with the increased alignment time. When the alignment lasts for 100 s, the mean of alignment result is approaching the theoretical one while the variance will further decrease with alignment time and identification accuracy. Obviously, the above results illustrate an interesting phenomenon: alignment accuracy is related to alignment time which is similar to that in classical methods, such as compass alignment method. The main reasons for this phenomenon lie in: (1) with RLS, the identification accuracy will be increased with the added measurement information and (2) solution accuracy based on dual-vector attitude determination method will be increased with the augmented intervals between those two vectors. Thus, it can be seen as a normal phenomenon decided by its principles. From the above analysis, it can be concluded that initial alignment can be realized by the improved method with parameter identification and reconstruction when the vehicle is with swing motion and the sensors with random errors. IV. TURNTABLE TEST A. Test setup
Inner frame SINS
intermediate frame
1. Turntable and SINS
The turntable used in this test is shown in Fig. 7. In this turntable, the rate controlling accuracy is ±0.0005◦ /s and angle controlling accuracy is ±0.0001◦ . Additionally, angle information can be provided via serial communication port as a response to external time-synchronization signal. In this test,
outer frame
TABLE VI. Swinging parameters for turntable.
turntable
Amplitude(deg) Cycle (s)
Inner frame
Intermediate frame
Outer frame
15 7.5
7 3.6
5 9.2
FIG. 7. Turntable and SINS.
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TABLE VII. Sensor precision of IMU. Gyro Constant errors Random errors
Accelerometer
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REVIEW OF SCIENTIFIC INSTRUMENTS 85, 125108 (2014)
An improved self-alignment method for strapdown inertial navigation system based on gravitational apparent motion and dual-vector Xixiang Liu,a) Yu Zhao, Xianjun Liu, Yan Yang, Qing Song, and Zhipeng Liu School of Instrument Science and Engineering, Southeast University, Nanjing 210096, China and Key Laboratory of Micro-inertial Instrument and Advanced Navigation Technology, Ministry of Education, Nanjing 210096, China
(Received 11 August 2014; accepted 18 November 2014; published online 10 December 2014) Analysis and simulation results indicate that two problems should be solved when the self-alignment method based on gravitational apparent motion and dual-vector can be used for Strapdown Inertial Navigation System. The first one is how to identify the apparent motion from accelerometer measurement containing random noise and the second is how to avoid the collinear problem between two vectors used in alignment solution. In this paper, a parameter identification and reconstruction algorithm is proposed to solve the first problem and simulation results indicate that proposed algorithm can identify apparent motion from accelerometer measurements effectively; and reconstruction algorithm based on current identified parameters for dual-vector is designed in detail to solve the second problem which can make full use of newest identification and avoid collinear problem completely. Simulation and turntable results show that the proposed alignment method can fulfill self-alignment in a swinging condition and the alignment accuracy can reach the theoretical values determined by the sensor precision. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4903196] I. INTRODUCTION
Initial alignment is one of the key and difficult problems of Inertial Navigation System (INS).1 Compared with the initial attitude, initial velocity and position can be easily acquired with reference navigation system, such as Global Position System (GPS). Thus, initial alignment for INS mainly refers to get initial attitude.2 According to whether external reference navigation information is needed, alignment method can be divided into self-alignment and nonself-alignment methods.2–6 Compass alignment method based on compass effect and zero-velocity alignment based on Kalman-liking filter and analytical method based on dualvector are typical self-alignment method.1, 2 In Strapdown INS (SINS), because the sensor are installed on board directly the measurement data from inertial measurement unit (IMU) are vulnerable to external interference, among which swinging and shaking motion is a typical interference.1 When the above methods are used for SINS to fulfill alignment, compass alignment method can overcame these interference but with a prolonged alignment time;3 zerovelocity alignment method also can do but with a decreased alignment accuracy;4, 5 while analytical method cannot because the Earth rotation rate are completely submerged in measurement noise.6 Recently, anti-disturbance self-alignment method for SINS has become a focus issue.6–14 In 2000, the articles from IxSea Company claimed that a novel alignment algorithm was used in their newest compass product – Octans, which can fulfill alignment within 5 min under any swinging condition.7–9 Considering the protection of commercial interest and technology security, these papers only mentioned that they aca) Author to whom correspondence should be addressed. Electronic mail:
[email protected] 0034-6748/2014/85(12)/125108/11/$30.00
quired geographical north by observing the gravity drift in inertial frame. Based on this alignment mechanism, relevant investigation has been done and many implement methods have been proposed, among which dual-vector alignment method is widely used.6, 11–14 The alignment method based on gravitational apparent motion and dual-vector translates the alignment problem of solving the attitude matrix from body frame to navigation frame to solving the matrix from initial body frame to navigation frame with matrix decomposition. But different from the vectors used in classical analytical method based on dualvector, in this method the gravity vectors at two different moments are used as dual-vector which will be further projected into the initial body frame and initial navigation frame, respectively. By the virtue of these projected vectors and dualvector attitude determination algorithm, the attitude matrix from initial body frame to navigation frame can be acquired and the matrix from body frame and navigation frame can be further solved. Compared with the classical dual-vector method, this new method does not need to separate Earth rotation rate from gyro measurement, thus alignment accuracy will be significantly increased.6–10 However, two problems caused by alignment mechanism should be studied: the first one is how to avoid collinear possibility between two vectors caused by measurement noise and limited alignment time while the second is how to improve constructing accuracy of gravitational apparent motion.6, 11 Reference 11 introduced ill-conditioned matrix theory to study collinear problem and conclude that at least 1 min between two vectors is needed to avoid collinear probability without considering any other errors. References 6 and 7 deduced the theoretical accuracy determined by sensor errors and classified this method as a coarse alignment method because of its low robustness against interferences. References 12 and 13 integrated gravitation apparent motion
85, 125108-1
© 2014 AIP Publishing LLC
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to form velocity and used velocity vectors instead of gravitational vectors participating alignment solution, thus alignment accuracy can be increased because of the smoothing affect to random noise introduced by integration but the alignment time is greatly prolonged to acquire a desired smoothing results. Reference 14 introduced low-pass filter to pre-process the measurement data and achieved some good results. However, it is difficult to find an available low-pass filter for all environments with complex noise. Based on the alignment mechanism of tracking gravitational apparent motion and attitude determination algorithm with dual-vector, an improved self-alignment method for SINS is proposed in this paper. In this method, a parameter identification algorithm is introduced to identify the parameters describing theoretical apparent motion from accelerometer measurement and a reconstruction algorithm is introduced to construct dual-vectors. Simulation and turntable test verified the effectiveness of the proposed improved method. The rest of this paper is organized as follows. In Sec. II, the gravitational apparent motion is detailedly described and the self-alignment method based on gravitational apparent motion and dual vector is designed and simulated. And the causes of low alignment accuracy are also analyzed and summarized. Section III presents an improved alignment method based on parameter identification and reconstruction. In Sec. IV the turntable experiment is carried out and finally the conclusion is given in Sec. V.
Rev. Sci. Instrum. 85, 125108 (2014)
FIG. 1. Apparent motion of local gravity vector. (a) Gravitational apparent motion in inertial frame. (b) The choice of inertial frame.
II. ALIGNMENT MECHANISM A. Gravitational apparent motion in inertial frame
The concept of apparent motion in INS is first used to study the characters of gyros. When the free gyro is observed in navigation frame rotating with the Earth, the track of gyro’s rotating axis in inertial space is defined as apparent motion. But the gravitational apparent motion is defined as the track of gravity at a fix point rotating with the Earth when the observation is processed in inertial frame. Take the point P located in the north Earth illustrated in Fig. 1(a) as an example, where the gravity is g pointing to the Earth center O. In this subsection, the initial Earth frame e0 is fixed as inertial frame ie0 , in which the change of gravity direction is investigated. When the Earth rotates from west to east, the direction of gravity will change both in the azimuth and height and the track of gravity direction is a cone with the vertex located at O. Similarly, when the point located at equator, the track is a circle with the center located at O. If the circle is regarded as a special cone, the gravitational apparent motion can be regarded all as cone without considering the point located at the pole. As shown in Fig. 1, we can see the parameters describing the cone of gravitational apparent motion, such as the conical bottom radius, are independent on the choice of inertial frame and the characters can be summarized as follows: (1) the cone axis is always the rotation axis of the Earth; (2) the cone vertex is located at the Earth center; (3) the conical bottom radius is determined by the latitude where the pointed is located; and (4) 24 h are needed to form a whole cone. Notably, the math-
ematical expression about the cone is corresponding to the selection of inertial frame and different selection will brings different expression, though the parameters about the cone are independent on the selection. For the convenience, Right-Forward-Up, as shown in Fig. 1(b), is defined as body frame b and the initial body frame b0 is fixed as inertial frame ib with the center Ob translating 0 0 to O. In the following parts, the proposed alignment method will be studied in this new inertial frame. B. Self-alignment method based on apparent motion and dual-vector
1. Alignment mechanism
In SINS, the classical analytical method based on dualvector attitude determination algorithm chooses the Earth rotation rate ωie and gravity g as dual vectors and takes the advantage of the measurement values in b frame and theoretical values in navigation frame n to construct the corresponding vectors and matrixes and acquire the attitude matrix C bn . But the method based on gravitational apparent motion and dual-vector attitude determination algorithm, the matrix C bn is decomposed as follows: i
n
C bn (t) = C bi (t)C nb00 (t)C n0 (t), b0
(1)
where C bi (t) is the attitude matrix from b frame to ib0 frame b0 which can be updated with the gyro measurements from IMU; n0 C n (t) is the matrix from initial navigation frame to n frame
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where g n = [ 0 0 −g ]T is the theoretical value of gravity veci
tor in n frame. Unfortunately, the matrix C nb0 (t) in Eq. (4) is an unknown parameter, thus Eq. (4) cannot be used directly. But Eq. (4) can be further expressed as follows: ⎡ ⎤ 0 i i g ib0 (t) = C nb0 (t)g n = C bb0 (t)C bn (t) ⎣ 0 ⎦ , (5) g where C bn (t)([ 0 0 g ]T ) is theoretical values of accelerometer measurement in b frame. Then, the gravitational apparent motion can be constructed as follows: FIG. 2. Self-alignment mechanism based on apparent motion and dualvector.
b i fˆib0 (t) = Cˆ bb0 (t) f˜ (t),
(6)
b
which can be updated with the latitude, Earth rotation rate, n and alignment time. The update for C bi (t)and C n0 (t) are as b0 follows: b b b Cˆ˙ ib0 (t) = Cˆ ib0 (t) ω˜ bib × = Cˆ ib0 (t) ω˜ bi b × , (2)
where f˜ (t) is accelerometer measurement in b frame. According to the relationship of frames defined in Fig. 3, there is f ib0 (t) = −g ib0 (t) without considering the errors caused by measurement and calculation. Thus, the first problem in Sec. II B 1 is solved.
b0
n n e Cˆ n0 (t) = C e00 (t) Cˆ e0 (t)C en ⎡ ⎤ ⎤⎡ 0 − sin L cos L cos(ωie t) − sin(ωie t) 0 ⎢ ⎥ ⎥⎢ 1 0 0 ⎥⎢ sin(ωie t) cos(ωie t) 0⎥ =⎢ ⎣ ⎦ ⎦⎣ 0 0 1 0 cos L sin L
⎡
0
⎢ × ⎣ − sin L cos L
1
0
⎤
0
⎥ cos L ⎦ ,
0
sin L
(3)
where the superscript “∼” and “ ˆ ” denote measurement and calculation data, respectively; ω˜ bi b is the gyro measurement;
3. Calculation for the theoretical gravity in initial navigation frame
The projection of gravity in n0 frame at the moment t can be calculated as follows: ⎡ ⎤ 0 n n gˆ n0 (t) = Cˆ n0 (t)g n (t) = Cˆ n0 (t) ⎣ 0 ⎦ , (7) g n
where Cˆ n0 (t) can be calculated accurately without any error with Eq. (3). That is to say, the projected data in n0 frame is the theoretical one. Thus, the second problem in Sec. II B 1 is solved.
b0
b Cˆ ib0 (0) = I 3×3 ; e is the Earth frame; L is the latitude, and t is alignment time. When vehicle is without translational motion, e C ne (t) and C n00 are both fixed matrixes. With Eqs. (1)–(3), the alignment problem of acquiring i C bn is translated into acquiring the matrix Cˆ nb00 . As shown in Fig. 2, East-North-Up (ENU) is defined as navigation frame. In the alignment method based on gravitational apparent motion and dual-vector, the apparent motion vectors in ib0 frame at the moments t1 and t2 are chosen as dual vectors participating in alignment solution with dual-vector attitude determination algorithm. However, three problems should be presolved: (1) the first problem is how to calculate the gravitational apparent motion in ib0 frame; (2) the second is how to get the theoretical gravity in n0 frame; and (3) the third is how to use the above vectors.
2. Calculation for gravitational apparent motion in inertial frame
According to the analysis in Sec. II A, gravitational apparent motion is the projection of gravity in inertial frame, thus g ib0 at the moment t can be calculated as the follows: ⎡ ⎤ 0 i i (4) g ib0 (t) = C nb0 (t)g n = C nb0 (t) ⎣ 0 ⎦ , g
i 4. Calculation for Cˆ nb00
When the vectors of fˆib0 and gˆ n0 at two moments t1 and i t2 are obtained, C nb00 can be constructed as follows:10 n T i Cˆ nb00 = Cˆ i 0 b0 ⎛⎡ ⎤−1 ( gˆ n0 (t1 ))T ⎜⎢ ⎥ ⎜ ( gˆ n0 (t1 ) × gˆ n0 (t2 ))T ⎥ = − ⎜⎢ ⎦ ⎝⎣ n n n T ( gˆ 0 (t1 ) × gˆ 0 (t2 ) × gˆ 0 (t1 )) ⎡
⎤⎞T
( fˆib0 (t1 ))T
⎢ ×⎢ ⎣
⎥⎟ ⎥⎟ . ⎦⎠
( fˆib0 (t1 ) × fˆib0 (t2 ))T ˆib0
(f
(t1 ) × f
ˆib0
(t2 ) × f
ˆib0
(t1 ))
(8)
T
i Generally speaking, Cˆ nb00 in Eq. (8) cannot satisfy the orthogonality for the calculated errors caused by sensor errors, i thus Cˆ nb00 should be further orthogonalized as follows:10, 12
i
C nb00
o
i i T i − 1 = C nb00 C nb00 C nb00 2 .
(9)
Substitute Eq. (9) into Eq. (1), then initial alignment can be fulfilled. Thus, the last problem in Sec. II B 1 is solved.
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Error (°)
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0.15 0.1 0.05 0 -0.05 -0.1 0
200
Theoretical value 400
Case 1 600
Case 2 800
1000
1200 t (s)
Error (°)
(a) Theoretical value
0
Case 1
Case 2
-0.05 -0.1 0
200
400
600
800
1000
1200 t (s)
Error (°)
(b) 150
10 5 0 -5
100 50 0 0
Case 2 0
200
200
400
400
Case 1
Theoretical value
600
800
600
1000
800
1000
1200
1200 t (s)
(c) FIG. 3. Curves of alignment errors: (a) pitch, (b) roll, and (c) yaw.
5. Alignment accuracy
According to the Ref. 6, the theoretical accuracy with the above method is as follows: φE =
∇N , g
φN = −
∇E + εN t, g
∇ ε t εE − E tan L + U , φU = ωie · cos L g 2
(10)
where ∇ E and ∇ N are the equivalent accelerometer bias in east and north, respectively; εE and εU are the equivalent gyro bias in east and up, respectively. Detailed deduction can be referred to Ref. 6. C. Simulation
1. Simulation setting
The vehicle is assumed without translational movement but with two different swinging motions as defined in Table I. In Case 1, the vehicle is assumed to be static with zero attitude and the sensor is with constant errors as defined in Table II. In Case 2, the vehicle is assumed swinging with the function A sin (2π f · t + η0 ) + θ 0 , in which A and f denote the swinging amplitude and frequency, η0 and θ 0 denote the swinging initial phase and center, respectively. The swinging parameters and sensor errors for Case 2 are defined in Tables III and II, respectively. The longitude and latitude for both cases are as 118◦ E and 32◦ N, respectively. Notably, in order to com-
pare the gravitational apparent motion in two cases, the initial phase and swinging center in Case 2 should be both set as zero to ensure the inertial frame of two cases are the same one. With the above setting, theoretical outputs of IMU can be generated by back-stepping of SINS navigation solution. When the errors in Table II are added into these theoretical outputs, the data sensor outputs can be simulated. To evaluate alignment accuracy, the ideal motions are used as references and the difference between alignment results and references are regarded as alignment errors. At the same time, sensor errors in Table II and alignment time are used to construct theoretical alignment accuracy (the minimized alignment errors) with Eq. (10). In simulation, t1 and t2 are assumed as 0 and t, respectively. The update cycle of sensor data and alignment solution are both as 10 ms, that is to say a single sample solution algorithm is adopt for integration,15 while the update cycle of error is 1 s.
2. Simulation results
The simulation lasts for 1200 s. Alignment results are shown in Fig. 3, in which (a), (b), and (c) denote the alignment errors of pitch, roll, and yaw, respectively; and the
TABLE II. Sensor errors setting. Gyro bias (◦ /h)
Acce bias (ug)
Constant Random (white noise) Constant Random (white noise) TABLE I. Two cases for simulation. Case 1 Case 2
Without swinging motion but with constant sensor errors With swinging motion and constant/random sensor errors
x-axis y-axis z-axis
0.05 0.05 0.05
0.05 0.05 0.05
500 500 500
500 500 500
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TABLE III. Swinging parameters setting.
Amplitude (deg) Cycle (s) Initial phase (deg) Swinging center (deg)
Pitch
Roll
Yaw
7 8 0 0
15 7.5 0 0
5 6 0 0
dotted-dashed and dashed lines are corresponding to Case 1 and Case 2, respectively, while the solid lines denote the theoretical alignment values. The statistics of alignment errors are shown in Table IV. In Fig. 3, the curves in Case 1 are coincide with the theoretical ones and in this condition dotted-dashed lines are overlap by solid lines and the statistics in Table IV also show that the alignment results are highly approximate to theoretical values. The curves in Fig. 3 also show that the alignment for pitch and roll can be fulfilled and alignment for yaw cannot converge to theoretical one; and the statistics also indicate that in this case, the mean and standard covariance of alignment results are all larger than the theoretical ones, among which the error of yaw is larger for several times. At the same time, the curves in Fig. 3 also show that in Case 2, there are large burrs in alignment results throughout the whole alignment process. From the above analysis, it can be concluded that the proposed method can fulfill alignment in an ideal condition, but cannot in a condition with swinging motion and measurement errors which is more in accordance with engineering situation. Thus, some improvements should be done.
possible reasons are as follows: (1) the errors in Cˆ ib0 (t); n (2) the errors in Cˆ n0 (t); (3) the errors in fˆib0 (t) and gˆ n0 (t); and i (4) and the errors in Cˆ nb00 (t). b a. The errors in Cˆ ib0 (t)
The errors from integration and gyro b outputs are the main sources of the error in Cˆ ib0 (t). Considering the integration errors are relatively small when the vehicle is in non-conical motion. The errors of integration can be ignored, thus the investigation focuses on the measurement errors of gyro outputs. Gyro errors can be divided into two types: one is constant error and the other is random one. When the gyro outputs are integrated, the integration errors caused by constant one are gradually increased while those caused by random are randomly presented as the burrs. The maximum integration error caused by the constant error set in Table III is merely 0.0166◦ in 1200 s without considering the coupling effects among three axes. Compared with the rotating angle of the Earth (5.0◦ in 1200 s), this error can be ignored. According to the Wright rules, the maximum random gyro error in Table III can be set as 3σ (0.15◦ /h). Then the maximum integration error from random error is 4.1667 × 10−7◦ with an integration cycle of 10 ms, which can also be ignored. Therefore, the reasons for low alignment accuracy b have no business with the errors in Cˆ ib0 (t).
n b. The errors in Cˆ n0 (t)
According to the Eq. (3), when the n vehicle is in static with an known latitude, the error in Cˆ n0 (t) is mainly caused by timing error which can be ignored, that is n to say, there is no error in Cˆ n0 (t). Further, it can be concluded that there is no error in gˆ n0 (t) described in Sec. II B 3.
3. Further analysis for alignment results
In this section, the reasons caused low accuracy of alignment in Sec. II C 2 are analyzed. From Eqs. (1)–(8), the
i When to construct Cˆ nb00 (t), fˆib0 (t) and gˆ n0 (t) should be first calculated. The investigation
c. The errors in fˆib0 (t) and gˆ n0 (t)
TABLE IV. Statistics of alignment errors (deg). Time (s) Theoretical value
Pitch Roll Yaw
Case 1
Pitch Roll Yaw
Case 2
Pitch Roll Yaw
0–200
201–400
401–600
601–800
801–1000
1001–1200
Mean Std Mean Std Mean Std
0.02865 0.00000 − 0.02727 0.00080 0.20800 0.00040
0.02865 0.00000 − 0.02449 0.00080 0.20939 0.00040
0.02865 0.00000 − 0.02171 0.00080 0.21077 0.00040
0.02865 0.00000 − 0.01893 0.00080 0.21216 0.00040
0.02865 0.00000 − 0.01615 0.00080 0.21355 0.00040
0.02865 0.00000 − 0.01338 0.00080 0.21494 0.00040
Mean Std Mean Std Mean Std
0.02859 0.00000 − 0.02730 0.00074 0.20666 0.00045
0.02863 0.00003 − 0.02474 0.00074 0.20823 0.00045
0.02868 0.00003 − 0.02218 0.00074 0.20979 0.00045
0.02874 0.00003 − 0.01963 0.00074 0.21136 0.00045
0.02884 0.00003 − 0.01707 0.00074 0.21293 0.00045
0.02894 0.00004 − 0.01451 0.00074 0.21450 0.00045
Mean Std Mean Std Mean Std
0.02923 0.03007 − 0.04201 0.02279 10.62012 28.33166
0.02879 0.03025 − 0.04392 0.02233 1.73950 1.82445
0.02907 0.02625 − 0.04200 0.02239 1.11607 0.96228
0.02515 0.02795 − 0.03972 0.02216 0.94730 0.71448
0.02681 0.02852 − 0.03705 0.02256 0.741114 0.53721
0.02979 0.02777 − 0.03500 0.02269 0.60670 0.41590
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Rev. Sci. Instrum. 85, 125108 (2014)
FIG. 4. Gravitational apparent motion in inertial frame: (a) axis-xi , (b) axis-yi , and (c) axis-zi . b0
focuses on the error in fˆib0 (t) because gˆ n0 (t) is considered as theoretical values without any error. As shown in Fig. 4, the gravitational apparent motion fˆib0 can be calculated with Eq. (6) and accelerometer measurement f˜b . In Figs. 4(a)–4(c) denote the projections of apparent motion on axes xi , yi , and zi , respectively, b0 b0 b0 while dotted-dashed and dashed lines are corresponding to Case 1 and Case 2. The curves in Fig. 4 indicate that the changes of fˆib0 in Case 1 are smoothly without any burrs. In this condition, the projections can be treated as the true apparent motion due to non-translational and swinging motions without considering the constant sensor errors. But in Case 2, though the changes of projections can present the drift trend of apparent motion, there are large random noises. b Regardless of the errors in Cˆ ib0 (t), these errors in projections are mainly caused by the random noise of accelerometer outputs. Take the projection along yi -axis as an example. During b0 y 1200 s, the projection of fˆ ib0 increased from 0.00479 m/s2 to 0.01926 m/s2 in Case 1 while in Case 2 the maximum and minimum projections are 0.03276 m/s2 and −0.00626 m/s2 , respectively. The random errors in Case 2 are so large that the projection cannot truly represent the changing of gravity. Thus, when these projections are used to full alignment, low accuracy will be inevitably led. Therefore, one important reason for low accuracy should be the errors in fˆib0 .
i d. The errors in Cˆ nb00 (t)
One precondition to use dual-vector attitude determination algorithm for alignment is that these two vectors cannot be collinear. With a longer interval between two vectors, the possibility of collinear will be lower. But as shown in Fig. 4, even though the interval between two vectors is long enough, such as 1200 s, the possibility still exists because of the random noise in fˆib0 . When
b0
b0
the vectors of fˆib0 (t1 ) and fˆib0 (t2 ) are collinear or approximately collinear, the construction for ( fˆib0 (t1 ) × fˆib0 (t2 )) and ( fˆib0 (t1 ) × fˆib0 (t2 ) × fˆib0 (t1 )) will be failure which will further lead to alignment failure. Therefore, the second reason for low alignment accuracy i should be the errors in Cˆ nb00 (t).
III. IMPROVEMENT TO PROPOSED METHOD
As the analysis in Sec. II C 3, low accuracy is evitable with the proposed method in Sec. II A when the vehicle is with a swinging motion and the sensors with random errors. The main reason lies in the random noise in accelerometer outputs, which brings the follows two problems: (1) the calculated apparent motion cannot truly represent the drift of gravity in inertial frame because the true apparent motion is submerged in random noise and (2) there is a high possibility of collinear between two vectors. Thus, the investigation in this section focuses on solving the above two problems.
A. Identification and reconstruction for gravitational apparent motion
1. Theoretical expression of gravitational apparent motion
As analyzed in Sec. II A, the track of apparent motion is an ideal cone and the parameters describing the cone are independent on the choice of inertial frame, but the specific expression is related to the selection of inertial frame. Theoretical expression for gravitational apparent motion can be constructed as follows with Eq. (4): i
i
n
e
f ib0 (t) = −g ib0 (t) = −C nb0 g n (t) = −C nb00 C e00 C e0 (t)C en (t)g n , (11)
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where C M N (t) is the attitude matrix from frame M to frame N. i n Obviously, C nb00 is a fixed matrix; C e00 and C ne (t) are both fixed when the vehicle is without translational motion. Expand Eq. (11) as follows: i
n
i
e
e
f ib0 (t) = C nb00 C e00 C e0 (t)C en (t) f n = C eb00 C e0 (t)C en f n ⎤ ⎤⎡ ⎡ cos(ωie t) − sin(ωie t) 0 a11 a12 a13 ⎥ ⎥⎢ ⎢ a a22 a23 ⎥ ⎢ sin(ωie t) cos(ωie t) 0⎥ =⎢ ⎦ ⎦⎣ ⎣ 21 0 0 1 a31 a32 a33 ⎡ b11 ⎢b21 ×⎣ b31 ⎡
b12 b22 b32
⎤ ⎤ b13 ⎡ 0 b23 ⎥ ⎣ ⎦ 0 ⎦ −g b33
ib0
A11 cos(ωie t) + A12 sin(ωie t) + A13
A11 ⎢ A =⎢ ⎣ 21 A31
A12
⎤
A22 A32
(t) is determined by the accuracy of
3. Simulation for identification and reconstruction
⎤⎡
⎤ cos(ωie t) ⎥⎢ A23 ⎥ ⎣ sin(ω t) ⎥ ie ⎦ , ⎦ A33 1 A13
where K k is the gain matrix; P k is the covariance matrix for state vector; Rk is the covariance matrix for measurement noise; and I k is an unit matrix with the dimension 3 × 3. Similarly, identification for the parameters along xi and b0 yi can be processed. With these identified parameters, gravb0 itational apparent motion can be reconstructed. The accuracy of reconstructed f identification.
⎢ ⎥ A cos(ωie t) + A22 sin(ωie t) + A23 ⎥ =⎢ ⎣ 21 ⎦ A31 cos(ωie t) + A32 sin(ωie t) + A33 ⎡
The equations of on-line estimation filter based on recursive squares algorithm is as follows:16 ⎧ −1 T T ⎪ ⎪ ⎨ K k = P k−1 H k H k P k−1 H k + Rk , (16) X k = X k−1 + K k (Z k − H k X k−1 ) ⎪ ⎪ ⎩ P k = (I k − K k H k ) P k−1
(12)
i
where aij and bij (i, j = 1–3) are the elements of C eb00 and C en while Aij are the combination of aij and bij . In Eq. (12), the above elements are all fixed values with a fixed inertial frame.
a. Simulation setting The simulation is conducted with the same condition described in Sec. II C. Initial parameters for identification filter are set as follows: X 0 = [ 0 0 0 ]T and P0 = diag[ 10000 10000 10000 ]. R = 1 μg and R = 500 μg are set for Case 1 and Case 2, respectively. In simulation, the reconstructed apparent motion is constructed with the parameters from identification filter and alignment time.
The simulation lasts for 1200 s. The reconstructed motion is shown in Fig. 5, in which (a), (b), and (c) denote the projections along xi , yi , and zi , respectively, b0 b0 b0 and dotted-dashed and dashed lines are corresponding to Case 1 and Case 2, respectively. For the convenience of comparison, the projections from Fig. 4 in Case 1 are redrawn in Fig. 5 with solid lines and used as a reference. The curves in Fig. 5 show that in Case 1 the reconstructed motion is fully coincident with the true one while in Case 2 the reconstructed motion becomes stable after a period of large fluctuation but with a small fluctuation during the following process. This small fluctuation is mainly attributed to the real-time update of measurement information in parameter identification. Compared the reconstructed motions in Case 1 and Case 2, it can be found that: (1) there is a small constant difference between them, but this difference is small enough to be ignored and (2) the changing trend of both reconstructed data are parallel to each other. Therefore, it can be concluded that reconstructed algorithm based on parameter identification can effectively remove the random noise from the calculated apparent motion.
b. Simulation results
2. Identification and reconstruction algorithm for apparent motion
If the parameters Aij describing gravitational apparent motion can be identified from the calculated fˆib0 (t) contain ib0
ing random noise, the theoretical apparent motion f (t) can be reconstructed. Here, the subscript “∩” denote the reconstructed data. Take the identification for A31 , A32 , and A33 describing the motion along zi as an example. During identification, the b0 on-line estimation algorithm based on recursive least square algorithm (RLS) is adopted to avoid large data storage and heavy computation. With RLS, the state vector can defined as X = [ A31 A32 A33 ]T and the measurement vector as Z k zi = fˆk b0 , where k denotes kth recursive step and k = t / t, where t and t denote alignment time and identification period (also alignment solution period). The system equation is as follows: X k+1 = X k .
(13)
The measurement matrix can be constructed as follows with Eq. (12): H k = [ cos(ωie t)
sin(ωie t)
1 ].
(14)
Theoretically, initial alignment can be fulfilled with the
And the measurement equation is as follows: Zk = H k X k + V k , where V k is the vector of measurement noise.
B. Reconstruction algorithm for dual-vector ib0
(15)
ib0
ib0
reconstructed f (0) and f (t). But the accuracy of f (0) is too low to fulfill this task because it is reconstructed with those initial parameters for identification filter. Another
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Rev. Sci. Instrum. 85, 125108 (2014)
FIG. 5. Reconstructed gravitational apparent motion based on identified parameters: (a) axis-xi , (b) axis-yi , and (c) axis-zi . b0
ib0
scheme for the selection of dual-vector is using f
(t/2) and
ib0
f
ib0
(0) and f
ib0
ing and updating for f ib0
(t/2) and the second is the accuracy
⎡ ib0
f
ib0
f (t/2) is still lower than that of f (t). Also, there exists the possibility of collinear in this scheme because of the low ib0
accuracy of f (t/2). In this paper, a new reconstruction method for dualvector is proposed in which the current parameters are used to reconstruct the apparent motion. Here the superscript “∪” denotes the reconstructed data of the past moment with cur ib0
rent parameters. That is to say, the reconstructed f ib0
f
(0) and
(t) are used to participate in alignment solution, where
b0
(t) are reconstructed as follows:
ib0
f (t) as described in Ref. 12, but two additional problems will be introduced: the first is the added operation on stor-
b0
⎤
⎢ ⎥ ˆ ⎢ sin(ωie · 0) ⎥ , (0) = A(t) ⎣ ⎦ 1 ⎡
ib0
f
cos(ωie · 0)
cos(ωie t)
⎤
(17)
⎢ ⎥ ˆ ⎢ sin(ωie t) ⎥ . (t) = A(t) ⎣ ⎦ a1
This reconstruction algorithm brings the following two advantages: (1) the first is all the measurement information and the latest identification results can be fully utilized with two vectors, which expand the utilized information at two
FIG. 6. Curves of alignment errors with improved method: (a) pitch, (b) roll, and (c) yaw.
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Liu et al.
Rev. Sci. Instrum. 85, 125108 (2014)
TABLE V. Statistics of alignment errors of improved method (deg). Time (s) Case 1
Pitch Roll Yaw
Case 2
Pitch Roll Yaw
0–200
201–400
401–600
601–800
801–1000
1001–1200
Mean Std Mean Std Mean Std
0.02851 0.00006 − 0.02732 0.00081 0.20666 0.00045
0.02857 0.00005 − 0.02473 0.00074 0.20823 0.00045
0.02867 0.00003 − 0.02217 0.00074 0.20980 0.00045
0.02874 0.00003 − 0.01961 0.00074 0.21136 0.00045
0.028835 0.00003 − 0.01705 0.00074 0.21293 0.00045
0.02894 0.00004 − 0.01449 0.00074 0.21450 0.00045
Mean Std Mean Std Mean Std
0.02820 0.01041 − 0.02765 0.02269 − 5.00769 27.67993
0.02840 0.00998 − 0.02410 0.02234 0.21231 0.02576
0.02848 0.00992 − 0.02183 0.02253 0.20735 0.00013
0.02854 0.00981 − 0.01953 0.02229 0.20787 0.00010
0.02863 0.00984 − 0.01635 0.02231 0.20926 0.00010
0.02874 0.00983 − 0.01412 0.02251 0.21083 0.00010
moments to the whole alignment process and (2) the second is the possibility of collinear is completely avoided for the same set of parameters are used in these two vectors. C. Simulation for the improved method
1. Simulation setting
The simulation is conducted with the same conditions described in Sec. II C and parameter for identification filter is as those in Sec. III A. 2. Simulation results
The simulation lasts for 1200 s. Alignment results are shown in Fig. 6, in which (a), (b), and (c) denote the alignment errors of pitch, roll, and yaw, respectively, and the dotteddashed and dashed lines are corresponding to Case 1 and Case 2 with improved method, respectively. The statistics of alignment results are shown in Table V. The curves in Fig. 6 and statistics in Table V all indicate that with the improved alignment method, initial alignment accuracy can reach theoretical alignment one. Compared the curves in Fig. 6 with those in Fig. 3, it can be found that in Case 2 with the improved method, the level alignment results oscillate with constant amplitudes showing no burrs.
These constant amplitudes are mainly caused by the change of equivalent sensor errors in navigation frame. At the same time, the variances of level alignment errors in Table V are smaller than those in Table IV. The curves of yaw also indicate that the alignment accuracy increases with the increased alignment time. When the alignment lasts for 100 s, the mean of alignment result is approaching the theoretical one while the variance will further decrease with alignment time and identification accuracy. Obviously, the above results illustrate an interesting phenomenon: alignment accuracy is related to alignment time which is similar to that in classical methods, such as compass alignment method. The main reasons for this phenomenon lie in: (1) with RLS, the identification accuracy will be increased with the added measurement information and (2) solution accuracy based on dual-vector attitude determination method will be increased with the augmented intervals between those two vectors. Thus, it can be seen as a normal phenomenon decided by its principles. From the above analysis, it can be concluded that initial alignment can be realized by the improved method with parameter identification and reconstruction when the vehicle is with swing motion and the sensors with random errors. IV. TURNTABLE TEST A. Test setup
Inner frame SINS
intermediate frame
1. Turntable and SINS
The turntable used in this test is shown in Fig. 7. In this turntable, the rate controlling accuracy is ±0.0005◦ /s and angle controlling accuracy is ±0.0001◦ . Additionally, angle information can be provided via serial communication port as a response to external time-synchronization signal. In this test,
outer frame
TABLE VI. Swinging parameters for turntable.
turntable
Amplitude(deg) Cycle (s)
Inner frame
Intermediate frame
Outer frame
15 7.5
7 3.6
5 9.2
FIG. 7. Turntable and SINS.
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Rev. Sci. Instrum. 85, 125108 (2014)
TABLE VII. Sensor precision of IMU. Gyro Constant errors Random errors
Accelerometer
