To validate the mechanism and model of the bias error, a simulation experiment is set up in this section.
3.1. Simulation Conditions and Model Establishment
In the simulation, the static misalignment angle between SINS and MINS was [0.2°0.2°0.2°]
T. The dynamic misalignment angles were treated as three independent second-order Markov processes whose parameters are illustrated in Equation (16). The parameters
, and
used to depict the dynamic misalignment angles are listed in
Table 1. The parameters were identified from the real measurement data.
According to the principle of hydroelasticity, both the linear (or angular) motion and the dynamic linear (or angular) deformation are the response of a ship to the sea wave loads [
29,
30]. Therefore, the ship’s inertial angular velocity, the dynamic lever-arm acceleration, and the acceleration measured by SINS can be treated as second-order Markov processes. The second-order Markov model of the ship’s inertial angular velocity, the dynamic lever-arm acceleration, and the acceleration measured by SINS can be expressed as follows:
where all the symbols are defined in the previous section and the parameters
, and
in Equations (51)–(53) are identified from our real measurement data, experimental experience, as well as the works from others [
15,
32]. The parameters
, and
are listed in
Table 2,
Table 3 and
Table 4, respectively. Finally, the accelerometer data of MINS were generated according to Equation 1.
The additional acceleration data generated by the static lever-arm was also generated according to Equation (5). The static lever-arm between SINS and MINS was set as [5m 20m 1m]T. However, this additional acceleration caused by was completely compensated in our simulation.
According to Equations (16), (51)–(53), every model in each direction is driven by a Gaussian white noise with unit variance. In order to investigate the bias error caused by the cross-correlation between the dynamic lever-arm acceleration
and the acceleration
measured by SINS, the driven noises of different models in each direction are listed in
Table 5.
With the change in the weight factor ξ, the correlation coefficient without lag between the driven noise of SINS acceleration data in the X- (or Z-) direction and the driven noise of the dynamic lever-arm acceleration in the Y-direction also takes a value from 0 to 1. Thus, the correlation coefficient without lag between and (or and ) takes a value from 0 to 1 because the second-order Markov model can be seen as a linear system.
In the simulation, all the accelerometers of MINS were assumed to be ideal. The accelerometer signals of SINS are subjected to the bias error vector
and random walk (white noise) vector
Given the assumed conditions, the error parameters of the accelerometers are listed in
Table 6, where SD denotes the standard deviation of the white noise.
The generated data were used to carry out the TA based on OLS or KF.
3.2. Result and Analysis
The total data length for TA was set to 600 s with a sampling frequency of 20 Hz. First, fix all the parameters presented from
Table 1,
Table 2,
Table 3,
Table 4,
Table 5 and
Table 6 except for the weight factor
ξ in
Table 5. Then, as the value of
ξ changes, a number of simulations were performed to investigate the alignment performance under different relevancies between
(or
) and
. The KF-based alignment results of when
(or
) and
were uncorrelated, i.e.,
ξ = 0, are shown in
Figure 3a.
The average alignment error caused by the dynamic lever-arm was almost zero at the end of 100 s in KF-based TA. When
(or
) and
were partly correlated, i.e.,
ξ = 0.5, the average alignment error caused by the dynamic lever-arm in the yawing angle was −0.6 mrad, as shown in
Figure 3b. These results show a preliminary conclusion that the bias error is related to the cross-correlation coefficient between the SINS acceleration data and dynamic lever-arm acceleration. However, it is interesting that the error of the roll was still very small, as can be seen in
Figure 3b, which shows that the cross-correlation coefficient (or
ξ) is not the only factor that determines the bias error.
In order to ensure the reliability and stability of the simulation results, all the following results were obtained by averaging over 100 independent trials in the presence of randomly generated noise signals. Similarly, the bias error of KF-based TA in each trial was calculated by averaging the estimation errors. The OLS-based TA errors are also provided to verify the reasonability of explaining the bias error mechanism through OLS theory. Furthermore, the bias error k was calculated to validate the correctness of the bias error prediction model shown in Equation (50).
To further the investigation of the cross correlation effect on the alignment accuracy, the weight factor
ξ was increased from 0 to 1 with a step length of 0.2. The average bias errors caused by the dynamic lever-arm are shown in
Figure 4, where the vertical lines indicate the related standard deviations. It can be seen from
Figure 4a that the bias error increased from −0.01 mrad to −0.9 mrad as the correlation coefficient between
and
increased from 0 to 1. This result coincides with the first inference about Equation (50). However,
Figure 4b shows that the bias error in the pitching angle was almost unaffected as the cross-correlation coefficient changed. A reason for this finding is that the gravity in
increased the autocorrelation of
more than that of
, so the cross-correlation coefficient had little influence on the bias error in the pitch direction according to Equation (50). This result also provides an indirect proof that the bias error is related to the amplitude of
as the analysis in
Section 4. Both the bias error caused by the dynamic lever-arm in KF-based TA and OLS-based TA were in accordance with the bias error
k, as shown in
Figure 4. This result indicates that the basic principles of OLS and KF are almost the same. Therefore, our analysis of the mechanism of the bias error and the mathematic expression of bias error
k are reasonable.
Next, set
ξ = 0.5 and fix other experimental conditions except the amplitude of
. The influence of the amplitude of
on the alignment accuracy was then investigated. As presented in Equation (50), the amplitude of
is n proportional to the square root of
.
Figure 5 depicts the corresponding bias error caused by the dynamic lever-arm as the amplitude of
changes. It shows that as the amplitude of
increased from 0.8 to 3.2 mm/s
2, the bias error in yawing angle increased from −0.6 to −2.4 mrad. This result fits the second inference about Equation (50) well.
Figure 5b shows that the bias error in pitching angle changed slowly as the amplitude of
increased. Similarly, the reason for this finding is that the gravity in
makes the autocorrelation of
much higher than that of
, which was discussed through
Figure 4b.
Finally, a simulation was carried out to investigate the influence on the alignment accuracy caused by the amplitude of
. In this simulation, we set
ξ = 0.5 and fixed other experimental conditions except for the amplitude of
.
Figure 6 depicts the corresponding bias error caused by the dynamic lever-arm as the amplitude of
.
increased simultaneously by one to four times.
Figure 6a shows that as the amplitude of
grew from 0.6 to 2.4 m/s
2, the bias error in yawing angle decreased from −0.6 to −0.2 mrad. This result matches the third inference for Equation (50) well.
Furthermore, it can be inferred from
Figure 6a that the bias error in yawing angle can be reduced when the ship is maneuvering, such as accelerating or turning. However, as illustrated in
Figure 6b, it was hard to obtain the bias error in lower pitching angles, even though the amplitude of
was increasing. The reason for this is that the gravity in
makes the bias error in the pitching angle already very low, which can be deduced from Equation (50).
In summary, bias error can be observed in standard TA, especially in the estimation of yawing angle, when there is a correlation between the SINS acceleration data and dynamic lever-arm acceleration. The simulation results showed that the bias error in TA is related to three factors: the amplitude of
, the amplitude of
, and the relationship between the two signals. The bias error is exacerbated when that coefficient or the amplitude of the dynamic lever-arm acceleration increases, whereas the accuracy of TA can be improved by increasing the linear acceleration of the ship, which means that the amplitude of SINS acceleration
increases. This conclusion agrees with Equation (50) in
Section 4, suggesting that the prediction model of the bias error is reasonable. Furthermore, the coupling effects in the KF-based and OLS-based TA were almost the same, which demonstrates the reasonability of the process to analyze the mechanism of bias error in KF-based TA through the OLS theory.