A Quadratic Interpolation-Based Variational Bayesian Algorithm for Measurement Information Lost in Underwater Navigation

&e main challenges of sequential estimations of underwater navigation applications are the internal/external measurement noise and the missing measurement situations. A quadratic interpolation-based variational Bayesian filter (QIVBF) is proposed to solve the underwater navigation problem of measurement information missing or insufficiency. &e quadratic interpolation is used to improve the observed vector for the precision and stability of sequential estimations when the environment is changed or the measurement information is lost. &e state vector, the predicted error covariance matrix, and the measurement noise matrix are derived based on the variational Bayesian method. Simulation results demonstrate the superiority of the proposed QIVBF compared with the traditional algorithm under the condition of measurement information lost by autonomous underwater vehicles.


Introduction
Ocean has already become the strategic goal of many countries because of its underdeveloped resources, marine environments, and high-tech fields [1,2]. e autonomous underwater vehicles (AUVs) have become one of the important tools for underwater detection, environment survey, and underwater reconnaissance in the ocean [3,4]. e navigation and positioning methods with high accuracy for AUV are the necessary conditions to acquire effective information [5], and also, they are the key technologies to determine whether AUV can work normally and return back safely [6,7]. e more accurate navigation and positioning methods for AUV are the necessary conditions for the underwater vehicle to ensure the correct travel [8,9]. e development of underwater vehicles faces many problems; for instance, the ocean environment is complex and changeable due to all sorts of noise interferences, and the AUV may fail to represent the true sensor measurements in many actual applications, like the presence of ocean current, salt cliffs, and ships around. e measurements are sometimes unobservable or insufficient so that positions calculated are erroneous [10][11][12]. On the other hand, a few navigation methods based on electromagnetic transmitting cannot be used underwater since the signal attenuates quickly underwater [13]. What is more, the localization error of the MEMS system accumulates along with time [14][15][16][17].
ose above weaknesses will cause the measurement missing, which has an adverse impact on the positioning of AUV [18]. However, the debugging and arranging process would waste time and material resources. erefore, the performances of estimation algorithms are vital to working normally for AUV [19,20]. e Kalman filter (KF), which is a widely and classical recursive filter, could provide the optimal state estimates in the linear dynamic system in the scenarios of known noise models and system models [21][22][23][24]. Additionally, the accurate measurements are difficult for the underwater integrated navigation system to describe because the performance of sensors varies with the change in the environment and measurement information probability lost [25,26]. e joint probability density function (PDF) is used to model the information of the AUV. en, the information of the joint PDF is determined by using the variational Bayesian (VB) method [27] in recursive process, which is the optimal. To further improve the capability of the navigation with the missing measurements, some strategies are required.
e recently proposed variational Bayesian-(VB-) based adaptive KF is a deterministic approximate Bayesian method that transforms the solution of the posterior probability density function based on Bayes' theorem into the solution of the functional extreme value [28,29]. It has more ideal approximate estimation results and computational overhead [30,31]. It significantly reduces the difficulty of calculation and makes high-precision filtering possible [32][33][34][35].
Aiming to mitigate the underwater navigation problem of measurement information missing or insufficiency, this paper proposes a quadratic interpolation-based variational Bayesian filter (QIVBF) algorithm. e QIVBF makes better use of the quadratic interpolation (QI) method and the VB method, so that the predicted error covariance matrix and the measurement noise matrix are derived to estimate the state vector more accurately. e quadratic interpolation improves the observed vector for the precision and stability of sequential estimations when the environment is changed or the measurement information is lost. e rest of the paper is presented as follows. In Section 2, the situation of the measurement lost in underwater navigation is described in detail. In Section 3, the QI method is used in measurement missing situations. In Section 4, the VB method is used to estimate the accurately predicted error covariance matrix, measurement error matrix, and state vector. Section 5 shows the simulation in underwater navigation. Section 6 gives the main conclusions of the paper.

Problem Statement
An underwater navigation function with measurement information lost is described as follows [36]: the measurement vector at the time k, with P x,k− 1 and P y,k− 1 representing the position measurements of the AUV at the time k − 1 in x and y directions, respectively, and V x,k− 1 and V y,k− 1 are the velocity measurements of the AUV at the time k − 1 in x and y directions, respectively. e ω k− 1 denotes the process noise, which often follows the Gaussian distribution with zero mean vector and process error covariance Q k− 1 .
Similarly, υ k means the measurement noise, which follows the Gaussian distribution with zero mean vector and measurement error covariance R k . e f k− 1 (·) is the known process function at the time k − 1. e Η k is the measurement model at the time k.
However, the underwater environment is complex and time-varying. Hence, the measurement information is unavoidable lost, which is described as follows: where Y k is the observation vector and pro means the probability.

State Model and Measurement Models of a Linear System.
For linear systems, the state transfer model is [37] where F can be established from system transfer models; W is the system input noise, assumed to be white with zero mean. e state vector is defined as X � δL δλ δh δV E δV N δV U ϕ E ϕ N ϕ U ∇ bx ∇ by ∇ bz ε bx ε by ε bz M x M y M z ] T with δL, δλ, and δh denoting the latitude, longitude, and height position error in ENU axes, respectively; δV E , δV N , and δV U are the east, north, and upward velocity errors in ENU axes, respectively; ϕ E , ϕ N , and ϕ U represent the heading, pitch, and roll errors in ENU axes, respectively; ∇ bx , ∇ by , and ∇ bz set as the biases of the accelerometer projections onto the ENU axes; ε bx , ε by , and ε bz for the gyroscope drift projections onto the ENU axes, respectively; M x , M y , and M z are the measured magnetic field vector in the ENU axes, respectively. e linear measurement model is defined as where V INS E , V INS N , and V INS U are the estimated velocities by DR (Dead Reckoning) along the east, north, and upward direction, respectively; V DR E , V DR N , and V DR U represent the measurement velocity of INS (Inertial Navigation System) along the east, north, and upward direction, respectively. φ Gyro and φ Mag are the heading measured by gyroscopes and magnetometers, respectively; θ Gyro and θ Acce denote the pitch measured by gyroscopes and accelerometers, respectively; c Gyro and c Acce describe the roll measured by gyroscopes and accelerometers, respectively; H is the observation matrix; V is a white Gaussian measurement noise with zero mean value.

CUKF Models.
Generally, for underwater navigation, the state and measurement models of multisensor fusion are nonlinear [38,39]. How to derive a nonlinear sequential state estimation from equations (3) and (4) is the main task. One can draw the nonlinear state and measurement models as follows: where X k and Z k represent the state vector and measurement at time step k, respectively. f(·) is n-dimensional nonlinear function for state transition and h(·) is the mdimensional nonlinear measurement function. W k and V k are zero mean process noise and measurement noise with covariance matrix Q k and R k , respectively. rough discretization, it is straightforward to convert the process and measurement equations in (12) and (13) to the nonlinear process function f(·) and measurement function h(·) in (5) and (6). e degree of nonlinearity in the practical underwater navigation is much bigger than the theoretical analysis, especially for MEMS-grade sensors [40,41]. It may not achieve the desired effect for traditional UKF, and this paper proposes a new algorithm of cubature unscented Kalman filter (CUKF) considering the trade-off among different aspects to improve further accuracy, real-time, and stability. e procedure for implementing the CUKF can be summarized as follows: Step 1: Sample. e n-dimensional random variable X k with mean X k and covariance P k is approximated by sigma points selected using the following equations: where a ∈ R is a turning parameter denoting the spread of the sigma points around X k− 1 and is often set to a small positive value. is parameter only affects errors caused by more than second-order matrices. e ( which is real symmetric positive definite matrix obtained through Cholesky resolution for LP k− 1 .
e sample points are composed of sigma points set χ i , i � 0, . . . , 2L, and the corresponding weight with these sample points is described as Step 2: e first prediction. Each point is instantiated through the process model to yield a set of transformed samples e predicted mean and covariance are computed as Step 3: e first update, with the process of measurement update, is as follows: e measurement vector is e first predicted measurement vector is e first covariance matrix of the innovations is e first cross covariance matrix between the predicted state estimate errors and innovations is e gain matrix is e updated state estimates are e covariance matrix of errors in the updated state estimates is Mathematical Problems in Engineering 3 Step 4: Cubature points generation Update the covariance matrix of error Generate the cubature points Calculate the propagated cubature points Step 5: e second prediction. e predicted mean and covariance are calculated as Step 6: e second update. Factorize Generate the cubature points Calculate the propagated cubature points: e second predicted measurement e second covariance matrix of the innovations  Mathematical Problems in Engineering Step 7: Compute gain matrix, estimate state, and its covariance matrix secondly: Step 8: Repeat Steps 1 to 7 for the next sample.

Lagrange Interpolation Polynomial.
When the measurement information is lost, the Lagrange interpolation polynomial (LIP) can be used to calculate the predicted observed vectorY ⌢ k . e QI method is used because the Runge phenomenon needs to be avoided [42,43].  Figure 1. e QI polynomial is described as where Variational Bayesian measurement update initialization: Variational Bayesian measurement estimation: for l � 0: NVB-1 e parameters of P k|k− 1 : e expectations of P k|k− 1 and R k are calculated: e state vector X (l+1) k|k and error covariance matrix P (l+1) k|k : Output: X k|k , P k|k .  (4) Mathematical Problems in Engineering Hence, Y ⌢ k can be factored as follows: where Y ⌢ k is the predicted observed vector at the time k and t k means the time of k.

e Variational Bayesian Approach.
In order to estimate the state vector X k|k with the measurement information lost, the joint PDF PDF(X k|k , P k|k− 1 , R k , Y 1: k− 1 , Y ⌢ k ) is considered to calculate the optimal solutions of X k|k , P k|k− 1 and R k . However, this PDF cannot be obtained, so the VB method is used to find some independent PDFs q(·), which are used to approximate PDF(X k|k , P k|k− 1 , R k , Y 1: k− 1 , Y ⌢ k ). Set Θ � X k||k , P k|k− 1 , R k . en, the optimal solutions of Θ, which is satisfied, are as follows: of the Θ except for κ. C κ is a constant about κ. In Bayesian statistics theory, the covariance matrix is defined to follow the inverse Wishart (IW) distribution. e distributions of P k|k− 1 and R k are set in where IW(P k|k− 1 ; m k|k− 1 , M k|k− 1 ) is the variable P k|k− 1 that follows IW distribution with m k|k− 1 degree of freedom (DOF) and  Mathematical Problems in Engineering Hence, PDF(Θ, e optimal solution is defined as follows: When choosing κ � P k|k− 1 , the parameters of P k|k− 1 are given by using

Mathematical Problems in Engineering
where A (l) k � P k|k− 1 + (X (l) k|k − X k|k− 1 )(X (l) k|k − X k|k− 1 ) T . When choosing κ � R k , the parameters of R k are given by using where B (l) k is calculated as follows: en, P k|k− 1 and R k at l+1th step are calculated as follows, respectively: When choosing κ � X k|k , q(X k|k ) is modelled as follows: where N( X k|k ; X (l+1) k|k , P (l+1) k|k ) denotes the variable X k|k that follows Gaussian distribution with X (l+1) k|k mean vector and P (l+1) k|k error covariance matrix, and the measurement update is written as follows: where the Kalman gain K (l+1) k at the l+1th step is calculated by using (41) at time k.
After fixed-point iterationN VB , the VB approximate of PDF is derived by e pseudocode is presented in Table 1. If the measurement information vector is lost, the LIP method is used to calculate the predicted observed vector Y ⌢ k . en, the predicted state vector X k|k− 1 and predicted error covariance matrix P k|k− 1 are calculated by using the state vector X k|k , error covariance matrix P k|k and process noise matrix Q k .

Simulations and Results
e proposed QIVBF is compared with the traditional KF in a simulation with measurement missing scenarios, and the root mean squared error (RMSE) is defined to compare the performances of these algorithms.
where (P x,k , P y,k ) and ( X t k , Y t k ) are the predicted position and reference position, respectively. e parameters are set in Table 2.
According to the navigation system, the accuracy of the methods is compared, and the above position estimation errors are analyzed. From Figures 2 and 3, the proposed QIVBF performs closer to the reference than the traditional KF. e proposed QIVBF results in a much better estimation accuracy than the existing filter on handling the problems of underwater navigation in the scenarios of measurement information missing. In particular, KF can easily lose tracking of the reference values by reason that the accumulated errors of iteration. On the other hand, the quadratic interpolation improves the observation vector for the precision and stability of estimation iterations when the environments change, or the measurement information is lost. erefore, the results demonstrate that the proposed QIVBF outperforms the traditional KF in the condition of measurement information missing. e navigation accuracy and anti-interference ability of the KF and QIVBF are compared by calculating the rootmean-square error (RMSE) of the position estimation. e RMSE has a good reflection of the measurement precision. It can be seen from Figure 4 that the QIVBF performs better than the traditional KF. When the measurement information is missing, the traditional KF cannot estimate the accurate Kalman gain, which leads to the filter divergence. However, the proposed QIVBF uses the QI method to estimate the lost observed vector, which can help estimate the accurate information about cross error covariance matrix. en, the VB method can help calculate the accurate state vector and error covariance matrix. e RMSE of QIVBF is lower than KF. Hence, the performance of QIVBF is better than KF.

Conclusions
e paper proposes a quadratic interpolation-based variational Bayesian filter for underwater navigation when measurement information is lost, where the quadratic interpolation method is used to calculate the observed vector when the measurement information is lost. en, the state vector together with the predicted error covariance matrix and the measurement noise matrix is estimated based on the VB method. Simulation results demonstrated the effectiveness of the proposed QIVBF algorithm as compared with the traditional algorithm in the case of measurement information missing. e idea of the proposed QIVBF can be extended to design a nonlinear filter for the non-Gaussian noises with measurement information lost. In future work, we plan to apply multiple datasets of underwater navigation measurements, so that the position and attitude can be estimated based on the fusion of various data sources for better accuracy.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.