1. Introduction
Recently, the importance of the polar region, especially the Arctic Ocean, in traffic, economic, military, and scientific research has received increasing attention. Many researchers are devoted to polar science. In order to better explore the polar region, many challenging problems need to be solved. Polar navigation is the primary issue to be addressed. No matter what kind of carrier (e.g., the autonomous underwater vehicle (AUV)), precise navigation information is essential for the performance of tasks in the polar region. Considering the flexibility and concealment of AUVs, they can break through the season and terrain constraints of the polar environment for detection activities, improve data acquisition efficiency, and realize intelligent polar environment detection [
1]. Thus, AUVs play an important role in polar ocean exploration, and AUV related navigation research is receiving increasing focus [
2,
3,
4]. The reliability and accuracy of AUV navigation systems is critical for AUVs accomplishing tasks autonomously.
However, compared to the navigation method used in non-polar regions, the polar navigation of AUVs faces some difficulties. First, due to the peculiarities of polar geography, the traditional strapdown inertial navigation system (SINS) mechanism in the local-level geographic frame loses efficacy in the polar region [
5]. The reason is that the meridian converges rapidly as the latitude increases, and the north reference definition thus becomes meaningless. Moreover, because the command angular velocity includes the tangent of latitude, calculation overflow and error amplification are inevitable in the polar region. A lot of work has been devoted to solving this problem. The main idea is to redefine the navigation frame used in the polar region in order to solve the calculation overflow problem and the north reference failure issue. Among the redefined navigation frames, the grid frame and transverse frame are generally used for polar navigation [
6,
7]. The grid frame adopts the intersection of the grid plane, which is parallel to the Greenwich plane and the tangent plane as grid north, thus the meridian rapid convergence in the polar region is avoided. The SINS mechanism in the grid frame has been widely adopted by many aircrafts because of its adaptability with aerial flight charts. The transverse frame redefines the transverse earth frame and the transverse local-level geographic frame through special rotation from the traditional earth frame and local-level geographic frame. The polar region in the traditional local-level geographic frame becomes adjacent to the equator in the transverse local-level geographic frame. Its north reference definition and position representation method also vary with changes in the navigation frame. The transverse frame is widely used in marine navigation. Yao et al. [
8] proposed a transverse frame navigation algorithm and verified the scope of application of the transverse frame, demonstrating that the transverse frame at the mid-latitudes has accuracy as consistent as the traditional local-level geographic frame. However, the traditional transverse frame definition is based on the spherical earth model which has an error in principle.
Secondly, because of the complexity of the natural polar environment, the use of common navigation systems (e.g., the global navigation satellite system (GNSS) or the geomagnetic navigation system) is limited in the polar region, especially for the underwater polar environment. Though the SINS, which is the major navigation equipment for AUVs, has the advantages of autonomy and concealment, its navigation errors (caused by inertial sensor errors) accumulate over time due to its dead reckoning principle. Thus, the SINS cannot work alone for extended periods of time. To overcome this problem, an aided navigation system must be introduced to assist the SINS in achieving high-precision navigation. Considering navigation accuracy and the concealment requirement, the Doppler velocity log (DVL) is ideal equipment for assisting an AUV’s SINS [
9,
10]. The DVL transmits ultrasonic waves from the hull to the seabed or water layer and receives the reflected signal, with which it can calculate the speed of the AUV in real-time according to the Doppler effect. It can provide real-time precise external speed observation for the SINS, and the navigation computer can fuse the navigation information from the SINS and the DVL through a filter algorithm, such as the Kalman filter, the extended Kalman filter, the unscented Kalman filter, and the particle filter.
As a whole, the DVL-aided SINS has become the mainstream solution for current underwater navigation [
11], and it is also an important way of realizing polar navigation with AUVs. Zhang et al. [
12] designed a DVL-aided SINS navigation algorithm in the transverse frame, which can effectively suppress the increase of azimuth misalignment angle in the polar region. However, due to the influence of ice layers, the underwater acoustic environment in the polar region is more complicated than the non-polar region. Moreover, the other unknown factors, e.g., the unknown polar ocean current, also influence the working status of DVLs. The operation of the DVL may be unstable, and the abnormal values may pollute the DVL measurement values. Additionally, the mentioned phenomenon is more common in the polar region than in the non-polar region, which can significantly affect the accuracy of the DVL-aided SINS. Aiming at solving this problem, an adaptive Kalman filter or robust Kalman filter is usually adopted [
13,
14,
15,
16,
17]. The main idea of the adaptive Kalman filter is to adjust the measurement noise covariance matrix or the system noise covariance matrix, which can further modify the Kalman filter gain and improve filter performance. Many studies have been published. Sun et al. [
18] proposed an adaptive Kalman filter based on the interactive dual model. They derived a method to dynamically estimate the noise covariance matrix, which enhances the accuracy of the navigation system in the Arctic region. However, the algorithm does not fundamentally solve the problem regarding failure of the local-level geographic frame in the polar region. Though the algorithm proposed by Zhang et al. [
12] shows effectiveness in polar DVL-aided SINSs, the algorithm does not consider the influence of the environment on DVL measurement noise, so its performance in terms of robustness is more or less poor. Chang [
19] proposes a robust Kalman filter algorithm, with the Mahalanobis distance between the measurement vector and its one-step prediction as a criterion, which can resist the influence of outliers on the filter results and adaptively estimate the measurement noise covariance value. The algorithm has good performance in terms of improving navigation accuracy in the presence of observation outliers.
In this paper, a new transverse integrated navigation method based on the earth ellipsoidal model is proposed. Compared to the traditional method, the proposed method reduces the error model based on the earth ellipsoid model and has higher accuracy in relation to long-endurance navigation. In addition, a DVL-aided SINS using a robust Kalman filter based on the Mahalanobis distance is proposed in order to cope with the complex environment in polar regions. The ship experiments and semi-physical simulation experiments verify that the algorithm can improve the positioning accuracy of the navigation system at mid-latitudes and in the polar region and resist the impact of DVL outliers on positioning accuracy. The structure of this paper is as follows:
Section 2 introduces the definition of the transverse frame and designs a corresponding SINS mechanism based on the earth ellipsoidal model.
Section 3 analyzes the system dynamic model of the DVL-aided SINS and designs a respective measurement model. In
Section 4, a robust Kalman filter with Mahalanobis distance as a criterion is proposed to resist the influence of outliers on the filter results and adaptively estimate the measurement noise covariance value. In
Section 5, the ship and semi-physical experiments used to evaluate the proposed algorithm are described. Finally, conclusions are drawn in
Section 6.
2. Transverse Frame Definition and SINS Navigation Mechanism
The transverse frame can be used as navigation frame in the polar region. The meridian convergence at high latitudes has no effect on it due to the fact that the polar region in the traditional local-level geographic frame becomes the region adjacent to the equator in the redefined transverse frame. This section introduces the definition of the transverse frame, including the transverse earth frame and transverse local-level geographic frame, as well as the transformation relationship between them. Further, based on assumption of the ellipsoidal earth model, the SINS mechanism in the transverse local-level geographic frame is designed.
2.1. Transverse Frame Definition
There are several different transverse frame definitions, and the differences between them are small. The transverse frame defined in this paper is the same as the one defined in Yao et al. [
8]. As shown in
Figure 1, the transverse earth frame is denoted by the
e’ frame. The
e’ frame can be defined through two sequence rotations of the earth-centered earth-fixed frame, i.e.,
e frame. Namely, the
e frame is rotated −90° around its
xe axis, then rotated −90° around the intermediate
ze axis. The direction cosine matrix (DCM) between them can be written as follows:
The Greenwich meridian plane becomes the transverse equator in the transverse earth frame. The intersection point of the 90° E meridian with the equator is the transverse north pole N’, and the intersection point of the 90° W meridian with the equator is the transverse south pole S’. The transverse parallel is parallel to the transverse equator. The semi-ellipse passing through the transverse north pole N’, the transverse south pole S’, and the north pole N is the transverse 0° meridian. The transverse local-level geographic frame is defined as the t-frame. Its definition is as follows: the origin is at point P0, which is the projection point of the AUV position point P on the local-level plane; the xt axis is along the tangent of the transverse parallel and toward to the transverse east; the zt axis is along the upward direction, which is vertical to the local-level plane, and coincides with the zg axis of the local-level geographic frame; the yt axis, xt axis, and zt axis constitute a right-handed orthogonal frame. The xt axis, yt axis, and zt axis are respectively denoted by Et, Nt, and Ut. The projection of point P on the transverse equator is denoted by point M. The intersection point of the normal PP0 and the transverse equator is denoted by point Q. The transverse longitude λt is defined as the included angle between the transverse equator and the PQM plane. The transverse latitude Lt is defined as the included angle between the line PQ and the transverse equator, namely, the included angle between the line PQ and the line MQ.
Moreover, in this paper, the i frame, b frame, d frame, and the g frame respectively represent the inertial frame, the body frame of the AUV, the body frame of the DVL, and the local-level geographic frame, with tri-axes denoted by Eg, Ng, Ug (xg axis, yg axis, and zg axis). The b frame and d frame both adopt the ‘right-front-up’ definition.
The DCM from the
e frame to the
g frame is written as
where
L is the traditional latitude and
λ the traditional longitude.
Similarly, the DCM from the
e’ frame to the
t frame is written as
According to the chain rule, the DCM from the
g frame to the
t frame can be written as
By geometric trigonometry, the relationships between the transverse latitude and transverse longitude and the traditional latitude and traditional longitude are as follows:
where
represents the element in the second row and third column of the matrix
.
represents the element in the first row and second column of the matrix
. The inverse transformation relationship between them can also be obtained by a similar method. Note that the height definition
ht, no matter whether in the transverse local-level geographic frame or the local-level geographic frame, is the same and is defined as the vertical distance from the horizontal plane. The height’s positive direction is upward.
According to the above defined transverse local-level geographic frame, there is an angle, σ, between the transverse north reference and the traditional north reference. The DCM from the
g frame to the
t frame can also be written as
According to Equations (4) and (6), substituting Equations (1)–(3) and (5) into Equation (4) yields
2.2. Transverse SINS Mechanism Based on the Earth Ellipsoidal Model
Generally, the transverse SINS mechanism adopts the spherical earth model regarding the flexibility of mathematical calculation. However, the earth is closer to an ellipsoid, and there is an error in principle related to the spherical assumption. Aiming at solving this problem, the earth ellipsoidal model is adopted in this paper. In fact, under assumption of the ellipsoidal earth model, the transverse north reference can be obtained by rotating the grid north reference −90° around the upward direction, which makes it easier to transform between them.
The attitude update equation in the transverse local-level geographic frame is written as
where
is the DCM from the
b frame to the
t frame,
is the angular velocity of the
b frame relative to the
i frame as measured by the gyro assembly,
is the symmetric matrix of vector
, and
is the angular velocity of the
t frame relative to the
i frame, which can be expressed as
with
where
is the angular velocity of the
e’ frame relative to the
i frame,
is the angular velocity of the
t frame relative to the
e’ frame,
is the earth rotation rate,
Rx is the radius of the curvature along the
x axis,
Ry is the radius of the curvature along the
y axis,
τ is the twist rate of the ellipsoidal,
is the transverse east velocity, and
is the transverse north velocity. The right side of Equation (10) can be written as
where
is the earth semi-major axis radius and
is the eccentricity of the earth.
The velocity update equation in the transverse local-level geographic frame is written as
where
is the AUV’s velocity,
is the specific force measured by the accelerometer assembly, and
is the gravity vector.
The position evolution in the transverse local-level geographic frame is expressed as the following position DCM differential equation and height differential equation:
When the AUV sails into the polar region, the navigation frame is switched from the
g frame to the
t frame. Therefore, the navigation parameters should also be projected in the corresponding navigation frame, and the transformation relationship is as follows:
Similarly, when the AUV sails out of the polar region, the inverse transformation relationship can be obtained.
3. Kalman Filter Model Design for a DVL-Aided SINS in the Transverse Frame
With the assist of the DVL, the SINS error can be restrained, and the polar navigation accuracy can thus be maintained. Generally, a Kalman filter is used as the information fusion method to fuse the information from the DVL and the SINS. This section introduces the system dynamic model and measurement model of the DVL-aided SINS, and corresponding models are designed in the transverse local-level geographic frame.
3.1. System Dynamic Model Design
The system dynamic model of the DVL-aided SINS includes two parts: the SINS error equation and the DVL error equation. The SINS error equation includes the attitude error equation, the velocity error equation, and the position error equation. These equations have similar styles to the corresponding equations projected in the traditional local-level geographic frame. These equations can be obtained by perturbing Equations (8), (12), and (13).
The SINS attitude error equation projected in the
t frame is as follows:
with
where
is the position error angle related to the position DCM
(its components are
,
, and
, respectively),
is the gyro assembly error including gyro bias
and gyro noise
, and
is just a simplified representation. Note that the Equation (18) ignores the earth ellipse and the influence of distortion.
and
are the equivalent representational forms of position error.
The SINS velocity error equation projected in the
t frame is as follows:
where
is the accelerometer assembly error including accelerometer bias
and accelerometer noise
.
The gyro and accelerometer biases can be modelled by a first-order Markov process as
where
and
are the correlation time of the Markov process and
and
represent the zero-mean Gaussian white noise.
The SINS position error equation projected in the
t frame is as follows:
where the position error angle and the height error are adopted to describe position error. Moreover, perturbing Equation (3) and rearranging both sides of it yields
Equation (22) demonstrates that the elements of the position error angle vector are linearly related:
Therefore, only two error angles are required to describe the error in the position DCM. In this paper, and are selected, as Equation (18) shows.
For simplicity, the DVL error equation includes the installation error equation and the scale factor error equation. The output of the DVL in the
b frame can be expressed as follows:
with
where
is the estimated installation error matrix,
is the installation error of the DVL relative to the AUV’s body frame,
is the true installation error matrix,
is the scale factor error of the DVL,
is the true AUV velocity projected in the
d frame, and
is the DVL measurement noise. Since the projection of forward velocity measured by the DVL does not receive the effect of the roll angle, the roll angle error can be ignored. The DVL error parameters can be taken as constants, namely,
At this point, the system error states are listed as follows:
The system dynamic model is written as
where
is the system state matrix,
is the system noise matrix, and
is system noise. These matrices can be determined by Equations (15), (19)–(21), and (26).
3.2. Measurement Model Design
The difference between the SINS velocity output and the DVL velocity output is used as the Kalman filter measurement vector, which can be expressed as
where
is the SINS velocity computed value (which is the sum of true velocity
and velocity error
) and
is the computed attitude DCM, which can be written as
Substituting Equations (25) and (30) into Equation (29) yields
The measurement model can be written as follows:
where
is the measurement vector,
is the measurement matrix whose components can be determined according to Equation (32), and
is measurement noise.