2.1. The Traditional Body-Frame Velocity-Aided In-Motion Attitude Determination Alignment
In this paper, we denote the initial frame by
, the Earth frame by
, the navigation frame by
, the body frame by
. Furthermore, the non-rotating inertial frames are denoted by
and
, which are identical to the navigation frame and the body frame at time 0 respectively. The time-varying navigation frame and the time-varying body frame are denoted by
and
, with time
. With the attitude matrix decomposition technique, the attitude matrix
can be written by [
30]:
where the
is the initial attitude matrix between
-frame and
-frame at time 0, and it is also a constant matrix. Moreover, the other two matrixes
and
can also be updated in real time as follow [
32]:
where the initialization of
and
are all set to the identity matrix;
is gyroscopes outputs from the inertial measurement unit (IMU);
. Since the coarse alignment is usually achieved with a short period of time, the change of vehicle position can be neglected and we have
. Then, the
and
can be calculated by
where
is the initial latitude of vehicle;
denotes the angular rate of Earth’s rotation;
denotes the Earth radius;
and
are the East velocity and North velocity, respectively. Moreover, the vehicle velocity expressed in navigation frame
is usually calculated by
where
is the calculated value of strapdown attitude matrix, and is usually acquired in the process of the coarse alignment.
According to the analysis above, the SINS attitude alignment is also transformed into the solution of constant initial attitude matrix
. With the traditional
-based IMADA, then,
can be determined by calculating the solution of the following observation equation [
33]:
where
is the accelerometer outputs;
,
is the local gravity. For the solution of the observation equation, which is the known Wahba’s problem, the recursive Davenport’s q-method is applied here. As a result, the quaternion representation of the constant matrix
would be the largest positive eigenvalue of the matrix
as follow:
With the above description, the
-aided IMADA can be implemented. Since the
(the angular rate of Earth’s rotation expressed in
-frame) is difficult to obtain, however, the item
of the observation vector
in Equation (6) is usually negligible. Consequently, the
is calculated approximatively as follows:
As a result, the principled model errors (namely, the approximationerror
) would occur in
-aided IMADA, thereby influencing the alignment accuracy. Moreover, it is easy to see that those influences are increasing with the vehicle velocity
. In the process of coarse alignment, on the other hand, the SINS attitude matrix usually would have the lower accuracy. Further, the process attitude matrix is employed to calculate navigation-frame velocity
from Equation (5) and update the rotation matrix
from Equation (3), which would also result in the large calculation errors, thereby influencing the alignment performance [
31].
In order to verify the influence of the principled model errors of
-based IMADA on the alignment accuracy, the Monte Carlo simulation experiments with the different vehicle velocities are carried out here. Total of 50 groups of 120 s simulation data are generated respectively from the SINS simulator [
31]. The main simulation parameters are shown in
Table 1. The sample frequency of IMU is 100 Hz. The vehicle sails at the different speed values of 20 m/s, 60 m/s, 80 m/s, respectively, and the generated external aiding velocity information
is added intentionally by the Gaussian white noise of standard deviation 0.03 m/s. With the above
-based IMADA, then, the curves of mean absolute deviation (MAE) and standard deviation (STD) of 50 alignment errors are shown in
Figure 1 and
Figure 2. The statistics of 50 heading alignment errors are also shown in
Table 2.
From
Figure 1 and
Figure 2, the MAE and STD curves of three-axis alignment errors are all convergent with the time. As a result, the
-aided IMADA can solve the in-motion coarse alignment problem. However, it is also easy to see that the alignment accuracy would degrade with the vehicle velocity. The MAE and STD of 50 alignment results with the different speeds of 20 m/s, 60 m/s, 80 m/s are 1.5125°, 2.0287, 2.3574 and 1.8170°, 2.4259°, 2.8069°, respectively. From
Table 2, on the other hand, the statistic characteristic also becomes worse with the velocity. The maximum of absolute value of 50 heading alignment errors are 4.0657°, 5.5640°, 6.3852°, respectively. This is coincident with the above analysis about the
-based IMADA. Because of the omitted item
, the presented principled model errors would influence the alignment accuracy, and this degradation aggravates with the vehicle speed. Then, the worse statistic characteristic would result in a lower reliability for SINS alignment system. This would be disadvantageous for the practical engineering applications. As a result, the principled model errors arising in the traditional
-based IMADA must be eliminated, especially for high-speed vehicle.
2.2. The Ttraditional MultistageIn-Motion Attitude Determination Alignment and the Aegradation Phenomenon
As mentioned and demonstrated above, the principled model errors and the calculation errors of the traditional
-aided IMADA would degrade the alignment accuracy and influence the reliability of the initial alignment system. As a result, those two error sources should be removed to improve the performance of the initial alignment system. In [
31], then, the
-aided IMADA without the principled model errors is employed tactfully to solve the inherent designed defections of the
-based IMADA. The multistage in-motion attitude determination alignment (MIMADA) with two different velocity models is presented. By integrating the
-based IMADA and the
-based IMADA, as well as the multiple repeated alignment process, the MIMADA can eliminate effectively the principled model errors and the calculation errors, thereby improving the alignment performance.
To be more specific, the traditional
-based IMADA is first carried out to acquire a final value of initial constant attitude matrix
, which would be a higher accuracy value. In the second-level alignment process, then, the first final value
is utilized to calculate the stapdown attitude matrix
by updating Equation (1). Hence, the calculated errors of stapdown attitude matrix can be decreased as compared with the traditional method with the process value of the constant attitude matrix
, which is updated recursively with the IMADA and has worse accuracy. As a result, the influence of the calculation errors on the
-aided IMADA can also be decreased, thereby improving the alignment performance. Further, the real-time navigation-frame velocity
can also be obtained with the body-frame velocity
from Equation (5) and have higher calculation accuracy. Meanwhile, the
-based IMADA can also be implemented and can be applied to avoid the principled model errors of traditional
-based IMADA. With the latest final value of higher accuracy constant attitude matrix, moreover, the
-based IMADA is conducted repeatedly to further decrease and gradually remove the calculated errors. The more details about MIMADA can be found in [
31]. According to the above description, the MIMADA can eliminate the principled model errors and the calculation errors of the traditional
-aided IMADA and has superior performance.
For the
-based IMADA, on the other hand, the observation equation of attitude determination is also presented as follows [
13]:
From Equation (10), the in-motion coarse alignment with the
-based IMADA would be implemented and have no principled errors. Then, the MIMADA proposed in [
31] can also be implemented to solve the existing inherent problem of the traditional
-based IMADA, thereby improving the alignment performance.
In order to verify the superior performance of MIMADA, the car-mounted experiment was carried out. The experimental equipments and the main parameters are shown in
Figure 3 and
Table 3, respectively. Here, the integrated navigation system with both the differential GPS and the navigation-grade fiber optic gyroscope (FOG) SINS served as the attitude benchmark for the in-motion coarse alignment and provided the external aided velocity information
. A total of 120 s coarse alignment test is executed and operated in Harbin (126° E,46° N), Heilongjiang Province, China. Here, only the second-level alignment for MIMADA is carried out. If not explicitly stated, the follow MIMADA experiments are all achieved by the second-level alignment. With the traditional
-aided IMADA and the multistage IMADA, the three-axis alignment errors are shown in
Figure 4.
In
Figure 4, the three-axis alignment errors are all convergence with time. As a result, the MIMADA proposed in [
31] can be applied to achieve the in-motion alignment for body-frame velocity-aided SINS. Moreover, the alignment errors (roll, pitch, and heading) in 120 s with both the IMADA and MIMADA algorithms are −0.0262°,−0.0062°,0.3920°, and −0.0278°,0.0032°,0.0506°, respectively. Hence, the MIMADA can improve the alignment accuracy. This is because that the MIMADA removes the principled model errors and the calculation errors, thereby eliminating the existing inherent defections of the
-based IMADA and improving the alignment performance. Therefore, the MIMADA would have superior performance as described above.
Nonetheless, it should be still remarkable that the performance of the second-level and the higher level alignments with MIMADA would depend on the first-level alignment performance significantly. Since the performance of the
-based IMADA would mainly be influenced by the accuracy of the navigation-frame velocity
according to Equation (10). In MIMADA, however,
would only be obtained by the first final value of initial constant attitude matrix
and the body-frame velocity
. If the first-level alignment performance is worse, therefore, the degradation of the alignment performance would possibly arise in MIMADA to a certain extent. With the same experimental equipments above, for example, another 120 s coarse alignment test is also conducted. A degradation phenomenon would occur and the alignment results are shown in
Figure 5.
From
Figure 5, it is obvious that the alignment accuracy would be worse, not improving, when the MIMADA is applied to achieve coarse alignment. The heading alignment errors with two algorithms in 120 s are 1.15° and 6.265°, respectively. As a result, it is contrary to the designed expectation of the MIMADA. Further, the MIMADA proposed in [
31] might be less appropriate for practical engineering application.
Without loss of generality, other 30 groups 120 s coarse alignment tests are also carried out to further illustrate the degradation phenomenon of the traditional MIMADA. With same experimental equipments above, the total 60-min test data are collected. The test trajectory is shown in
Figure 6. Then, those test data are divided successively into 30 groups 120 s data segments. The in-motion coarse alignments with the two algorithms are all conducted. The 30 differences of absolute value of alignment errors between the MIMADA and traditional
-aided IMADA in 120 s are all shown in
Figure 7. The subscript 1 denotes alignment results with the MIMADA. In
Figure 7, obviously, the difference values greater than 0 mean that the alignment results are degraded with MIMADA. Whereas, the difference values less than 0 mean that the alignment results are improved. Moreover, the statistics of the 30 alignment results are also shown in
Table 4.
From
Figure 7, it is easy to see that the degradation number for alignment performance with MIMADA would be more than the improvement number. From
Table 4, the degradation number of heading alignment would be 20 in 30 alignment experiments. Even the maximum value of degradation of alignment accuracy is 5.1148°. Then the performance of the subsequent fine alignment might be influenced seriously because of the poor coarse alignment accuracy. Moreover, the mean and variance of the differences of absolute value of heading alignment errors are 0.5523° and 1.0928°. As a result, the statistic characteristics of alignment accuracy with MIMADA would be bad, thereby decreasing the reliability of alignment system.
This is because that the performance of MIMADA would mainly depend on the first level alignment performance as analyzed above, thereby resulting in the degradation phenomenon. Thus, the final alignment accuracy would be influenced. As a result, the first level alignment accuracy must be guaranteed strictly. Otherwise, the MIMADA might be less appropriate for practical application because of the degradation phenomenon. Because of unavoidable random noises of the sensors, however, the first-level alignment accuracy with the traditional -aided IMADA is usually difficult to guarantee and control. Moreover, the higher requirement for the first-level alignment accuracy may cost more alignment time, which is contradiction to the rapidness of the initial alignment. Therefore, a new solution for MIMADA is expected to be proposed to guarantee the statistic characteristics.