Inspired by the new alignment algorithm of xSea Company, several Gravitational Apparent Motion (GAM)-based coarse alignment methods have been proposed [
4,
7,
8,
9]. The alignment problem is converted from determining the attitude matrix between the body frame and the navigation frame to determining the matrix between the body inertial frame and the navigation inertial frame by the GAM-based alignment method. It has been proven that the GAM-based alignment method has the same theoretical alignment accuracy as the conventional methods [
7,
8]. In practical applications, however, because the accelerometer data are applied to calculate the GAM directly, some non-negligible errors, especially random noise from accelerometers, are brought into the alignment process [
8,
10]. In response to this problem, Xu J.et al. [
6] proposed to adopt velocity vectors by integrating gravitational acceleration to participate in alignment calculation. Nevertheless, in the case of linear velocity interference, the performance of this method will be poor without the reference of external velocity sensors. Chang L. et al. [
3] also employed gradient descent optimization to determine the initial attitude matrix according to the characteristics of GAM. The shortcoming of this method is that gradient descent optimization has strict requirements on the objective function and step-size, which limits its wide application in practice. Besides, Liu X. et al. [
10] recognized and reconstructed the GAM by analyzing the general expressions of apparent motion. However, the reconstructed system was unable to maintain the complete observability of the whole coarse alignment process. Based on the different frequency characteristics of noise, Xu X. et al. [
11] and Sun F. et al. [
9] employed the low filter to filter random noises. Meanwhile, Xu X. et al. [
12] filtered the high-frequency noises of the measurements with the designed Real-time Wavelet Denoising (RWD). However, due to the instability of external conditions, it is difficult to determine the parameters of the above filters in practical applications. An alternative denoising method was presented by Huang called Empirical Mode Decomposition (EMD) [
13,
14], which is totally adaptive. To surmount the defect of mode mixing and end effect in conventional EMD, Complementary Ensemble Empirical Mode Decomposition (CEEMD) was proposed in [
15]. Then, the Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) was proposed as a more effective method [
16]. However, the improvement of the filtering accuracy of CEEMDAN is based on longer decomposition time, which makes it restrictive in practical applications. Focusing on the problem of effective IMF selection, Ayenu-Prah et al. [
17] and Duan et al. [
18] adopted a Correlation-based method (EMD-COR) to determine the effective IMFs. Wang Y. et al. [
19] employed EMD-COR to complete the initial alignment of SINS. However, the relevant modes were selected based on prior information, actually. The performances of some methods will be verified later in this paper. To solve this imperfection, a method based on Consecutive Mean Squared Error (CMSE) was proposed in [
20]. This method does not need a threshold and can adapt to most conditions, but it will perform poorly because of the local minimum. On the basis of the above studies, Komaty obtained effective IMFs according to the similarity measure between the Probability Density Functions (PDF) of IMFs and the original signal [
21]. Yang proposed EMD Interval Thresholding (EMD-IT) based on the probability density function with the order of time complexity comparable to EMD, but having higher accuracy [
22]. The limitation of the above two methods is that the mode mixture of EMD has not been fundamentally solved.
Given the problems above, this paper introduces a GAM-based self-alignment method by a novel adaptive filter called effective IMF selection based on CEEMD-
. The main superiority of this method is to select IMFs self-adaptively without any prior information. Measurement signals of the accelerometer with noises are decomposed into IMFs through CEEMD. For the acquisition of relative IMFs, their PDFs are estimated by the kernel density estimator, followed by self-adaptive separation of the main signal and harmful noise. The final reconstructed signal will be applied to calculate GAM in self-alignment. The remainder part of this paper is organized as follows.
Section 2 provides the general alignment algorithm based on GAM and the corresponding simulation. Then, the improved denoising method by CEEMD-
and reconstructed gravitational apparent motion vectors are presented in
Section 3. Moreover, simulations, the turntable test, and the ship experiment are carried out to verify the effectiveness of the proposed algorithm in
Section 4, whilst the conclusions are given in
Section 5.