2.1 DS theory

The DS evidence theory is a data decision-level fusion method. It can directly indicate the ‘uncertain’ and ‘unknown’ information, expressed in the basic probability distribution function and retained in evidence synthesis. Evidence theory makes it possible to give credibility to a single element of the hypothesis space and its subset and uses the trust interval to quantify the credibility of the evidence. Hence, the DS evidence theory can solve the problem of multi-level and multi-source data fusion without knowing the prior probability. A mathematician at Harvard University in the United States, A.P. Dempster, proposed the trust and likelihood functions (Petturiti and Vantaggi, Reference Petturiti and Vantaggi2017). Shafer (Aggarwal et al., Reference Aggarwal, Bhatt, Devabhaktuni and Bhattacharya2013) further studied evidence theory to handle uncertainty problems.

In the DS framework, let Θ be a frame of discernment, Θ = {θ ₁, θ ₂,  ⋅  ⋅  ⋅ , θ_N}. Basic probability assignment (BPA) is defined as a real-valued function m, which maps from 2^Θ to [0, 1], called a mass function (Zhou et al., Reference Zhou, Lu and Huang2016). The following conditions are met using function m

(1)\begin{align}\left\{ {\begin{aligned} & {m(\emptyset ) = 0}\\ & {\sum\nolimits_{A \in {\textrm{2}^\Theta }} {m(A)\textrm{ = 1}} } \end{aligned},} \right.\end{align}

The mass function is also called the BPA function. In terms of the specific element A in 2^Θ, m(A) is represented as the degree of supporting evidence. Under the condition of m(A) > 0, subset A is named the focal element, where the focus elements include the part of 2^Θ that the existing evidence concerns.

Because m(A) represents the trust allocation to A and not the trust allocation to a subset of A or the total trust allocation to A, we must add the quantity m(B) of all appropriate subsets B in A to m(A) to obtain total confidence in A. For the given mass function m, the total BPA of a given proposition can be directly obtained using a reliability and likelihood measure. The belief function Bel can be given by

(2)\begin{equation}Bel(A) = \sum\limits_{B \subseteq A} {m(B)} ,\end{equation}

where A∈2^Θ. Bel(A) is the sum of the basic probability distribution functions corresponding to all subsets of A. It is defined as follows for the likelihood function Pl:

(3)\begin{equation}Pl(A) = 1 - Bel(\bar{A}) = \sum\limits_{B \subseteq U} {m(B)} - \sum\limits_{B \subseteq \bar{A}} {m(B)} = \sum\limits_{A \cap B\textrm{ = }\emptyset } {m(B)} ,\end{equation}

where Pl(A) is the sum of the basic credible numbers. One of these two measures (Pl(A) and Bel(A)) is the upper bound, while the other is the lower bound, forming an upper and lower probability interval. In DS theory, we can compute Bel(A) and Pl(A) in the frame of discernment Θ, where Bel(A) and Pl(A) represent the degree of trust that A is true and A is not false, respectively. The belief interval [Bel(A), Pl(A)] illustrates the degree of trust for a hypothesis. By dividing the belief interval, we intuitively observe the representation range of uncertain information, as shown in Figure 2.

Figure 2.

Schematic of the evidence interval

In this research, Bel(A) and Pl(A) have the following features Bel(A) ≤ Pl(A) ≤ Pl(A). As shown in Figure 2, the support evidence interval is represented by [0, Bel(A)], where the upper bound of the support evidence interval is the trust degree Bel(A). However, the likelihood interval is expressed by [0, Pl(A)], in which the upper bound of the likelihood interval denotes the likelihood function Pl(A), and the lower bound of the reject evidence interval [Pl(A), 1]. The [Bel(A), Pl(A)] means a neutral evidence interval, which neither supports nor rejects proposition A.

To address the practical problems, measurement data for the same evidence may come from several data sources, resulting in a non-unique basic probability distribution function (Liu et al., Reference Liu, Pal, Marathe and Lin2018). If the uncertainty is measured and the trust and likelihood functions are used in this case, there is a need to synthesise the basic probability distribution functions from different sensor sources into a basic probability distribution function, that is, Dempster's rule of combination. Therefore, the BPA from multiple data sources is realised.

Assuming that m ₁ and m ₂ are the BPA of two identical frames of discernment, applying the orthogonal sum rule m = m ₁ ⊕ m ₂, the combined output is

(4)\begin{gather}{m_1} \oplus {m_2}(A)\textrm{ = }{K^{ - 1}}\sum\limits_{B \cap C = A} {{m_1}(B){m_2}(C)} ,\end{gather}

(5)\begin{gather}K = 1 - \sum\limits_{B \cap C \ne \emptyset } {{m_1}(B){m_2}(C)} = \sum\limits_{B \cap C = \emptyset } {{m_1}(B){m_2}(C)} ,\end{gather}

where A ≠ ∅︀. K is called the conflict factor (normalisation factor), which evaluates the level of conflict between two pieces of evidence, m ₁ and m ₂. If K = 1, there exists a strong conflict, which cannot be used for composition. In contrast, if K ≠ 1, the orthogonal sum is a new probability distribution function.

2.2 Standard BL algorithm

The many deep learning network parameters mean that the training process of these methods is very time-consuming, and the BL framework is introduced as a fundamental form of modelling. BL network is a classical random vector functional link neural network (RVFLNN) (Vukovi et al., Reference Vukovi, Petrovi and Miljkovi2018). It is different from the primitive RVFLNN, and a set of mapping features displaces the direct connection between the input and output of the BL. The input samples are transformed into feature nodes using feature mapping, while the feature nodes are generated using nonlinear mapping. Therefore, the feature and enhancement nodes are input to the system through the connection matrix. Figure 3 shows the structure of the network.

Figure 3.

Basic structure of the BL

Given a dataset $X = {[x_1^\textrm{T},x_2^\textrm{T}, \cdots ,x_N^\textrm{T}]^\textrm{T}} \in {\textrm{R}^{N \times M}}$ as an input matrix composed of N M-dimensional input vectors, $Y = {[y_1^\textrm{T},y_2^\textrm{T}, \cdots ,y_N^\textrm{T}]^\textrm{T}} \in {\textrm{R}^{N \times C}}$ is the output matrix composed of N C-dimensional output vectors. Z ₁, Z ₂,… and Zn represent the feature node matrix; H ₁, H ₂, … ⋅ and Hn indicate the enhanced node matrix; and W denotes the connection weight matrix.

The feature node Z_i is computed using the function ${\phi _i}(X{W_{ei}} + {\beta _{ei}})$; that is, the i-th group of nodes generated by mapping of input data X is denoted as Z_i. If n groups of feature nodes Z ₁, Z ₂, …, Z_n are generated, then the expression is as follows (Zou et al., Reference Zou, Yu, Li, Lei and Yu2020)

(6)\begin{equation}{Z_i} = {\phi _i}(X{W_{ei}} + {\beta _{ei}}) \in {\textrm{R}^{N \times q}}, \end{equation}

where W_ei and β_ei (i = 1, 2,… ⋅ , n) are weight and offset matrices generated randomly, respectively. q is the number of feature nodes corresponding to each group of feature maps, and ϕ is the linear transformation, which is normalised ( ⋅ ) in this paper. Let Zⁱ ≡ [Z ₁, Z ₂,…, Z_n] be the feature nodes of all input data maps, and then the enhancement nodes H^j ≡ [H ₁, H ₂,…, H_j], j = 1, 2, …, m can be described as

(7)\begin{equation}{H^j} = \xi ({Z^n}{W_{hj}} + {\beta _{hj}}) \in {\textrm{R}^{N \times r}},\end{equation}

where r denotes the number of enhancement nodes corresponding to each group of enhancement transformations. W_hj and β_hj are weight and offset matrices generated randomly, respectively. ξ is a nonlinear activation function set as a hyperbolic tangent function in this paper. All the enhanced node matrices are spliced into a whole, and the total enhanced node matrix is obtained as follows H^m ≡ [H ₁, H ₂,…, H_m]. Hence, the BL model is given by

(8)\begin{gather}\begin{aligned} Y & = [{Z_1},{Z_2}, \cdots {Z_n}|{\xi ({Z^n}{W_{{h_1}}} + {\beta_{{h_1}}}), \cdots ,\xi ({Z^n}{W_{{h_m}}} + {\beta_{{h_m}}})} ]{W^m}\\ & = [{Z_1},{Z_2}, \cdots {Z_n}|{{H_1}, \cdots ,{H_i}, \cdots {H_m}} ]{W^m}\\ & = [{Z^n}|{{H^m}} ]{W^m}, \end{aligned}\end{gather}

(9)\begin{gather}{W^m} = {[{Z^n}|{{H^m}} ]^\textrm{ + }}Y,\end{gather}

where a new variable A = [Zⁿ|H^m] is introduced to express all nodes generated by input, and the final estimated output (Equation [8]) of the system can be written as $\hat{Y} = AW$. Because $\{ {W_{ei}},{\beta _{ei}}\} _{i = 1}^n$ and $\{ {W_{hj}},{\beta _{hj}}\} _{j = 1}^m$ remain unchanged after being randomly generated, the goal is to find a suitable connection weight parameter W so that the difference between $\hat{Y}$ and Y is as small as possible. λ denotes the regularised parameter. So the optimisation problem is modelled as

(10)\begin{equation}\mathop {\arg \min }\limits_W :||{Y - \hat{Y}} ||_2^2 + \frac{\lambda }{2}||W ||_2^2,\end{equation}

Since the optimisation problem is convex, the partial derivative of Equation (10) with respect to W can be directly calculated. Hence, the optimal connection weight W can be expressed as

(11)\begin{equation}W = {(A{A^\textrm{T}} + \lambda I)^{ - 1}}{A^\textrm{T}}Y,\end{equation}

3.1 SINS/GPS integration scheme

The state and measurement models of the SINS/GPS integration scheme can be defined as

(12)\begin{align}\left\{ \begin{aligned} \dot{x}(t) & = F(t)x(t) + G(t)w(t)\\ y(t) & = H(t)x(t) + v(t)\end{aligned} \right.,\end{align}

where F and H denote the state transfer and measurement transition matrices, respectively. G is the system distribution noise matrix of process noise vector w. y and v represent the measurement information and its corresponding noise. The state vector x of the SINS/GPS integration includes the following parameters

(13)\begin{equation}x(t) = {[\begin{array}{@{}lllll@{}} {\delta {\varphi ^\textrm{T}}}& {\delta {v^\textrm{T}}}& {\delta {p^\textrm{T}}}& {{\varepsilon ^\textrm{T}}}& {{\nabla ^\textrm{T}}} \end{array}]^\textrm{T}},\end{equation}

where $\delta \varphi = {\left[ {\begin{array}{@{}lll@{}} {\delta {\varphi_E}}& {\delta {\varphi_N}}& {\delta {\varphi_U}} \end{array}} \right]^T}$, $\delta v = {\left[ {\begin{array}{@{}lll@{}} {\delta {v_E}}& {\delta {v_N}}& {\delta {v_U}} \end{array}} \right]^T}$, and $\delta p\, = \,{\left[ {\begin{array}{@{}lll@{}} {\delta L}& {\delta \lambda }& {\delta h} \end{array}} \right]^T}$ denote the attitude (east, north, and up), speed (east, north, and up), and position (latitude, longitude, and height) error vectors, respectively. $\varepsilon \, = \,{\left[ {\begin{array}{@{}lll@{}} {{\varepsilon_x}}& {{\varepsilon_y}}& {{\varepsilon_z}} \end{array}} \right]^T}$ and $\nabla \, = \,{[\begin{array}{@{}lll@{}} {{\nabla _x}}& {{\nabla _y}}& {{\nabla _z}} \end{array}]^T}$ represent the gyro constant drifts and accelerometer zero errors, respectively. The state transfer matrix F and G can be expressed as (Liu et al., Reference Liu, Fan, Chen, Jian, Liang and Ding2017)

(14)\begin{gather}F(t) = \left[ {\begin{array}{@{}lllll@{}} {{0_{3 \times 3}}}& {{D_1}}& {{D_2}}& {C_b^n}& {{0_{3 \times 3}}}\\ {{0_{3 \times 3}}}& {{D_3}}& {{D_4}}& {{0_{3 \times 3}}}& {C_b^n}\\ {{0_{3 \times 3}}}& {{D_5}}& {{D_6}}& {{0_{3 \times 3}}}& {{0_{3 \times 3}}}\\ {{0_{6 \times 3}}}& {{0_{6 \times 3}}}& {{0_{6 \times 3}}}& {{0_{6 \times 3}}}& {{0_{6 \times 3}}} \end{array}} \right],\end{gather}

(15)\begin{gather}G(t) = \left[ {\begin{array}{@{}ll@{}} {C_b^n}& {{0_{3 \times 3}}}\\ {{0_{3 \times 3}}}& {C_b^n}\\ {{0_{9 \times 3}}}& {{0_{9 \times 3}}} \end{array}} \right],\end{gather}

where

\[\begin{array}{@{}l@{}} {D_1} = \left[ {\begin{array}{@{}ccc@{}} 0& { - 1/{R_M}}& 0\\ {1/{R_N}}& 0& 0\\ {\tan L/{R_N}}& 0& 0 \end{array}} \right],\\ {D_2} = \left[ {\begin{array}{*{20}{c}} 0& 0& 0\\ { - {\omega_N}}& 0& 0\\ {\dot{\lambda }\sec \textrm{L + }{\omega_N}}& 0& 0 \end{array}} \right], \end{array}\]

\[{D_3} = \left[ {\begin{array}{@{}ccc@{}} {\dot{L}\tan L\textrm{ - }{v_U}/{R_M}}& {2{\omega_U} + \dot{\lambda }\cos L}& { - 2{\omega_N} - \dot{\lambda }\cos L}\\ { - 2{\omega_U} - \dot{\lambda }\sin L}& { - {v_U}/{R_M}}& { - \dot{L}}\\ {2({\omega_N} + \dot{\lambda }\cos L\textrm{)}}& {2\dot{L}}& 0 \end{array}} \right],\]

\[\begin{array}{@{}llll@{}} {D_4} & = \left[ {\begin{array}{@{}ccc@{}} {2{\omega_N}{\omega_N} + \dot{L}{v_E}{{\sec }^2}L\textrm{ + 2}{\omega_U}{v_U}}& 0& 0\\ { - (2{\omega_N}{v_E} + \dot{\lambda }{v_E}\sec L)}& 0& 0\\ { - 2{\omega_U}{v_E}}& 0& 0 \end{array}} \right],\\ {D_5} & = \left[ {\begin{array}{@{}ccc@{}} 0& {1/{R_M}}& 0\\ {\sec L/{R_N}}& 0& 0\\ 0& 0& 1 \end{array}} \right],\\ {D_6} & = \left[ {\begin{array}{@{}ccc@{}} 0& 0& { - \dot{L}/{R_M}}\\ {\dot{\lambda }\tan L}& 0& { - \dot{\lambda }/{R_N}}\\ 0& 0& 0 \end{array}} \right], \end{array}\]

\[C_b^n = \left[ {\begin{array}{@{}ccc@{}} {\sin \mu \cos \psi - \sin \mu \sin \theta \sin \psi }& { - \cos \theta \sin \psi }& {\sin \mu \cos \psi + \cos \mu \sin \theta \sin \psi }\\ {\cos \mu \sin \psi + \sin \mu \sin \theta \cos \psi }& {\cos \theta \cos \psi }& {\sin \mu \sin \psi - \cos \mu \sin \theta \cos \psi }\\ { - \sin \mu \cos \theta }& {\sin \theta }& {\cos \psi \cos \theta } \end{array}} \right]\]

where R_N and R_M are the meridian radii of curvature and transverse. ω_N = ω_ecosL, ω_U = ω_esinL, where ω_e is the Earth's rotation rate, and θ, μ, and ψ are the pitch, roll, and heading angles, respectively. The measurement vector y can be updated by subtracting the GPS measurement value from the integrated navigation system output data. This is given by

(16)\begin{equation}y(t) = {[\begin{array}{@{}cc@{}} {{{(v - {v_{\textrm{GPS}}})}^\textrm{T}}}& {{{(p - {p_{\textrm{GPS}}})}^\textrm{T}}} \end{array}]^\textrm{T}},\end{equation}

where v _GPS and p _GPS indicate the output speed (east, north, and up) and position (longitude, latitude, and height) vectors derived from GPS, respectively.

Accordingly, the measurement transition matrix can be expressed as

(17)\begin{equation}H(t) = \left[ {\begin{array}{@{}llll@{}} {{0_{3 \times 3}}}& {{I_{3 \times 3}}}& {{0_{3 \times 3}}}& {{0_{3 \times 6}}}\\ {{0_{3 \times 3}}}& {{0_{3 \times 3}}}& {{I_{3 \times 3}}}& {{0_{3 \times 6}}} \end{array}} \right],\end{equation}

3.2 SINS/GPS fusion method

Assuming that SINS and GPS correspond to the two different evidence sources, the set Θ can be defined as

(18)\begin{equation}\Theta = \{ \textrm{SINS,GPS}\},\end{equation}

where SINS and GPS components are not related, the three-dimensional (3D) speed and position information provided by SINS and GPS are independent. The power set 2^Θ is specified using

(19)\begin{equation}{2^\Theta } = \{\emptyset, \{ \textrm{SINS}\} ,\{ \textrm{GPS}\} ,\{ \textrm{SINS} \cup \textrm{GPS}\}\} ,\end{equation}

If the given mass functions m ₁(SINS) and m ₂(GPS) are available, the DS combination rule is used to compute the degree of confidence from each segment with SINS and GPS, defined as m _SINS and m _GPS according to the new and used accessible evidence and the value of conflict K (Bhatt et al., Reference Bhatt, Babu and Chudgar2016).

(20)\begin{equation}\begin{cases} {{m_{\textrm{SINS}}} = \dfrac{1}{K}\sum\limits_{\textrm{SINS} \cap \textrm{GPS} = \textrm{SINS}} {{m_1}(\textrm{SINS}) \cdot {m_2}(\textrm{GPS})} }\\ {{m_{\textrm{GPS}}} = \dfrac{1}{K}\sum\limits_{\textrm{SINS} \cap \textrm{GPS} = \textrm{GPS}} {{m_1}(\textrm{SINS}) \cdot {m_2}(\textrm{GPS})}} \end{cases},\end{equation}

Referring to Equation (5), the evidence conflicts can be defined as

(21)\begin{equation}K = 1 - \sum\limits_{\textrm{SINS} \cap \textrm{GPS} = \emptyset } {{m_1}(\textrm{SINS}) \cdot {m_2}(\textrm{GPS})} ,\end{equation}

Further development can be expressed as

(22)\begin{equation}K = 1 - ({m_1}(\textrm{SINS}) \cdot {m_2}(\textrm{GPS}) + {m_1}(\textrm{GPS}) \cdot {m_2}(\textrm{SINS})),\end{equation}

According to Table 1 and the rule of combination, the confidence of the combination for SINS and GPS data can be expressed as follows.

(23)\begin{align}\left\{ {\begin{aligned} {m_{\textrm{SINS}}} & = \{ {m_1}(\textrm{SINS}) \cdot {m_2}(\textrm{SINS}) + {m_1}(\textrm{SINS} \cup \textrm{GPS}) \cdot {m_2}(\textrm{SINS})\\ & \quad + {m_1}(\textrm{SINS}) \cdot {m_2}(\textrm{SINS} \cup \textrm{GPS})\} /K\\ {m_{\textrm{GPS}}} & = \{ {m_1}(\textrm{GPS}) \cdot {m_2}(\textrm{GPS}) + {m_1}(\textrm{SINS} \cup \textrm{GPS}) \cdot {m_2}(\textrm{GPS})\\ & \quad+ {m_1}(\textrm{GPS}) \cdot {m_2}(\textrm{SINS} \cup \textrm{GPS})\} /K \end{aligned}} \right.,\end{align}

Table 1.

DS theory combination rule

According to DS fusion theory, if GPS works normally, the data derived from SINS and GPS equipment can be effectively fused.

To determine whether to give more weight to SINS or GPS more efficiently, the confidence measure of SINS or GPS mass function can be evaluated using DS theory; that is, to provide a precise navigation solution, more weight is assigned to SINS or GPS.

(24)\begin{align} \left\{ \begin{aligned} & {m_1}(\textrm{SINS}) = \dfrac{1}{{(2\pi )}^{\frac{1}{2}}{C_{\textrm{SINS}}}} \cdot exp [-0.5{{(\textrm{SINS} - {\mu_{\textrm{SINS}}})}^{\textrm{T}}}C_{\textrm{SINS}}^{-1} (\textrm{SINS} - {\mu_{\textrm{SINS}}})]\\ & {m_2}(\textrm{GPS})= \dfrac{1}{{{{(2\pi )}^{{\frac{1}{2}}}}{C_{\textrm{GPS}}}}} \cdot exp [ - 0.5{{(\textrm{GPS} - {\mu_{\textrm{GPS}}})}^{\textrm{T}}}C_{\textrm{GPS}}^{- 1}(\textrm{GPS} - {\mu_{\textrm{GPS}}})] \end{aligned}\right.,\end{align}

where μ _GPS and μ _SINS denote the average values given by GPS and SINS measurements, while C _GPS and C _SINS indicate the covariances derived from GPS and SINS measurements. This research assumes that the distribution satisfies Gauss's central limit theorem. The sliding window method is used to calculate the covariance and mean of SINS and GPS data in the actual experiments (Wang et al., Reference Wang, Xu, Yao and Zhang2021).

(25)\begin{gather} \begin{cases} {\mu_{\textrm{GPS}}} = \dfrac{{\sum\nolimits_{j = 1}^N {{\mu_j}} }}{N}\\ {C_{\textrm{GPS}}} = \dfrac{{\sum\nolimits_{j = 1}^N {{{({\mu_j} - {\mu_{\textrm{GPS}}})}^2}} }}{{N - 1}} \end{cases},\end{gather}

(26)\begin{gather} \begin{cases} {\mu_{\textrm{SINS}}} = \dfrac{{\sum\nolimits_{i = 1}^N {{\mu_i}} }}{N}\\ {C_{\textrm{SINS}}} = \dfrac{{\sum\nolimits_{i = 1}^N {{{({\mu_i} - {\mu_{\textrm{SINS}}})}^2}} }}{{N - 1}} \end{cases},\end{gather}

where μ_j and μ_i represent GPS and SINS information at the i-th and j-th moment, while μ_GPS and μ _SINS denote the mean of GPS and SINS information, respectively. N represents the dimension of the sliding window, and N = 8 was selected during sea trials.

If GPS is working normally, the velocity and position information provided by the SINS and GPS equipment can be effectively fused according to DS fusion theory.

(27)\begin{equation} \left\{\begin{array}{@{}l@{}} {{v_{DS}}\textrm{ = }{v_{\textrm{SINS}}} \cdot {m_{\textrm{SINS}}} + {v_{\textrm{GPS}}} \cdot {m_{\textrm{GPS}}}}\\ {{p_{DS}}\textrm{ = }{p_{\textrm{SINS}}} \cdot {m_{\textrm{SINS}}} + {p_{\textrm{GPS}}} \cdot {m_{\textrm{GPS}}}} \end{array} \right.,\end{equation}

where v _DS and p _DS are the velocity and position data after fusion.

3.3 Virtual GPS construction of USV based on the DS–BL framework

GPS has become an important part of integrated navigation because of its accurate information on 3D position and velocity. However, the function of GPS mainly relies on the external environment and the number of available satellites. Also, under some environmental conditions, the update rate of GPS could be affected, and the GPS signal may be blocked or seriously interfered with. Hence, it is quite significant to introduce a novel alternative navigation approach to construct a virtual GPS and realise the function of unmanned navigation when GPS is unavailable.

In this research, the virtual GPS parameters estimation of the USV based on the proposed DS–BL method can be broken down into two main steps. First, implement the model training. Under the condition that a GPS signal is available, on the one hand, the navigation output is obtained using the extended KF (EKF) method based on the SINS/GPS integration. However, the velocity and position information derived from SINS and their corresponding GPS are fused using DS fusion rules. The fused high-precision velocity and position data, as well as the velocity and position data of the SINS, are used as the input of the BL network. This step is called the training process, where the BL framework implements the learning and training of sensor information. The errors of velocity and position of SINS are selected as BL output, as illustrated in Figure 4.

Figure 4.

Training process of SINS/GPS integration

Figure 5 shows the process of predicting GPS. The trained BL framework estimates the velocity and position according to the SINS speed and position information in case of GPS signal loss to replace the GPS output and generate a virtual GPS. Accordingly, the data of the speed and position of SINS are used as inputs to the BL framework, while the prediction of velocity and position (v _BL, p _BL) are applied as the measurement output (virtual GPS) of the BL framework. The velocity and position errors between the virtual GPS and the SINS are regarded as the extended KF output. Therefore, the SINS/GPS integrated navigation function is completed.

Figure 5.

Prediction process of SINS/GPS integration

Based on these analyses, Table 2 gives the detailed implementation flow of the DS–BL algorithm.

Table 2.

Process of the DS–BL algorithm

4.1 Simulation and analysis

A simulation experiment of 1,500 s was designed to verify the method's validity, including speed-changing and course-changing of USV. Table 3 summarises the characteristics of the sensors. Figure 6 shows the ship's trajectory, where the sample rate required by the gyroscope and accelerometer was 50 Hz, and the updating rate derived from the GPS was 1 Hz. Figure 7 shows the velocity curve of the USV.

Figure 6.

Trajectory of USV

Figure 7.

USV velocity result curve

Table 3.

Some parameters of simulation test

The design of the simulation experiment can be divided into two main sections to show the performance of the proposed virtual GPS method. In Figure 4, when the GPS signal was available, the SINS/GPS integration performed the training step between 0 and 1,000 s. However, in Figure 5, when the GPS was in an outage, the virtual GPS was constructed between 1,000 and 1,200 s.

The DS method was applied to fuse the speed and position information of GPS and SINS, respectively, in the training process to achieve a more precise output of velocity and position. Figures 8 and 9 show the confidence results of the velocity and position of SINS and GPS computed using Equations (21) and (22). Therefore, since the GPS can provide the precise and reliable output in the entire training process, the position and velocity derived using GPS have a good confidence measure.

Figure 8.

Result of the velocity confidence measure of SINS and GPS

Figure 9.

Result of the position confidence measure of SINS and GPS

When the GPS outage lasts 1,000 s, speed and position information are anticipated to replace the velocity and position data provided by GPS, allowing the navigation function to be completed. The DS–ANN method was introduced as a comparative approach to demonstrate the improvement attained with the proposed DS–BL approach. Figures 10 and 11 depict the velocity and position prediction curves of different algorithms in three directions between 1,000 and 1,200 s, containing the true, GPS measured values, DS–ANN predicted values, and DS–BL predicted values. From Figures 10 and 11, it can be observed that the velocity and position information predicted using the proposed DS–BL approach are better than the DS–ANN approach in three directions. The predicted 3D velocity $\left( {\begin{array}{@{}lll@{}} {{v_E}}& {{v_N}}& {{v_U}} \end{array}} \right)$ and position $(\begin{array}{@{}lll@{}} {{p_E}}& {{p_N}}& {{p_U}} \end{array})$ values were superior to the measured value of GPS, while the prediction results were near the true value.

Figure 10.

Velocity curves of various methods

Figure 11.

Position curves of various methods

To fully demonstrate the effectiveness of the DS–BL approach, the evaluation parameters that reflect the degree of dispersion of the errors derived from velocity and position were introduced, in which the standard deviation (STD) can be defined as.

(28)\begin{equation}\textrm{STD = }\sqrt {\frac{{\sum\nolimits_{i = 1}^N {{{({x_i} - \bar{x})}^2}} }}{N}} ,\end{equation}

As shown in Table 4, the calculation results show that the mean velocity errors in the east, north, and up directions of the proposed DS–BL approach are reduced by 38 ⋅ 1%, 55 ⋅ 3%, and 94 ⋅ 5%, respectively, compared with the DS–ANN method. Also, the mean position errors of the proposed DS–BL method are reduced by 19%, 22 ⋅ 6%, and 60%, respectively, compared with the DS–ANN method. Furthermore, compared with the DS–ANN approach, the proposed DS–BL method reduces velocity errors by 84 ⋅ 4%, 24 ⋅ 7%, and 95 ⋅ 9%, while the proposed DS–BL method reduces position errors by 36%, 25%, and 60%. Table 5 lists the training network parameters and testing time compared with the ANN. It can be observed that the training time of BL is much quicker than that of ANN, mainly because BL can expand more ‘neurons’ horizontally in real time during the learning process to improve the training efficiency.

Table 4.

Mean and STD velocity and position errors between two different algorithms

Table 5.

Comparison of training and testing times

4.2 Analysis and comparison of USV in sea trial

To evaluate the effectiveness of the proposed approach in a real aquatic environment, the sea trial was conducted in Dalian, Liaoning Province, in October 2020. Figure 12 shows the instruments used in this experiment, including the IMU and GPS sensor. The output frequencies were 1 and 200 Hz in GPS devices and IMU equipment, respectively, and Table 6 shows the other detailed parameters.

Figure 12.

Experimental equipment of USV

Table 6.

Some parameters of the sea trial

As shown in Figure 13, a large quantity of actual data (2,500 s) was collected during the sea trial process to achieve better training model parameters. Figure 14 shows the experimental trajectory of the USV before explaining the validity of this approach, and Figure 15 illustrates the speed curves of the USV during sailing. The whole time of the USV sailing lasted for about 1,000 s.

Figure 13.

Training trajectory of USV in the test

Figure 14.

Actual trajectory of USV

Figure 15.

Velocity curves of USV during sailing

For the first 0–800 s, the GPS was in normal condition, and the GPS signal was then interrupted between 800 and 900 s. Therefore, the integrated system was in training mode between 0 and 800 s. The velocity and position information of GPS and SINS were fused during the training process using the DS approach, and Figures 16 and 17 show their corresponding confidence measure. The velocity and position provided using GPS show better performance than SINS for the confidence measure.

Figure 16.

Result of the velocity confidence measure of SINS and GPS during sea trial

Figure 17.

Result of the position confidence measure of SINS and GPS during sea trial

The velocity and position outputs derived from GPS were interrupted between 800 and 900 s. Hence, the speed and position data were predicted to replace the GPS-derived speed and position output and thus complete the navigation function. Figures 18 and 19 depict the velocity and position prediction curves of different algorithms in three directions between 800 and 900 s, including the true, GPS measured, DS–ANN predicted, and DS–BL predicted values. From Figures 18 and 19, it can be observed that the velocity and position information predicted by the proposed DS–BL approach are better than the DS–ANN approach in the three directions. The predicted 3D velocity and position values are superior to the measured values on GPS.

Figure 18.

Velocity curves of different algorithms during sea trial

Figure 19.

Position curves of different algorithms during sea trial

Table 7 shows the mean and STD of the velocity errors in the east, north, and up directions. The mean velocity errors in the three corresponding directions of the proposed DS–BL method were reduced by 15 ⋅ 8%, 92 ⋅ 9%, and 54 ⋅ 9%, respectively, compared with the DS–ANN method. Also, the mean position errors of the proposed DS–BL method were reduced by 18 ⋅ 4%, 23 ⋅ 5%, and 57 ⋅ 1% compared with the DS–ANN method. Additionally, when compared with the DS–ANN approach, the proposed DS–BL method reduced velocity errors by 18 ⋅ 4%, 12 ⋅ 4%, and 49 ⋅ 5%, while the proposed DS–BL method reduced position errors by 29 ⋅ 2%, 31 ⋅ 4%, and 69 ⋅ 2%.

Table 7.

Mean and STD velocity and position errors between two different algorithms during sea trial

Table 8 lists the training and testing time comparison between DS–ANN and DS–BL. It can be observed that the total time of BLS was much quicker than ANN, and the navigation solutions produced using DS–BL were almost close to real-time navigation. The problem of real-time navigation performance can be solved in the future by adopting better hardware devices.

Table 8.

Comparison of training and testing time between DS–ANN and DS–BL