A fusing positioning algorithm for small maritime unmanned vehicle

Abstract

Introduction:

The 5th generation (5G) mobile communication and Beidou navigation system (BDS) antennas of offshore small maritime unmanned vehicles (SMUV) are shaken by waves and winds in harsh sea conditions, resulting in poor stability in positioning trajectory tracking and varying degrees of impulse noise in pseudo-range measurements.

Methods:

Therefore, A Kalman filter algorithm combining approximate message passing (AMP) and variational Bayesian (VB) is proposed under a fusion positioning model of 5G and BDS with a high update rate. The AMP algorithm can predict the instantaneous position movement caused by waves, and the VB algorithm smooths the pulse error of trajectory tracking caused by intermittent shielding.

Results:

Experimental results demonstrate that, in 100 positioning epochs, when the impact of waves on SMUV is 20 times, 30 times and 40 times, the root mean square error of the proposed algorithm is 1.19 meter (m), 1.26 m and 1.27 m respectively. Compared with the adaptive variational Bayesian algorithm, the positioning accuracy is improved by 7.75%, 16.4% and 24.4%, which obviously enhances the stability of positioning.

Discussion:

The proposed algorithm can achieve high-precision positioning and tracking, and the performance of the algorithm is better than that of the comparison scheme when the amplitude and frequency of the waves are greater.

1 Introduction

With the extension of 5th generation (5G) mobile communication base station (BS) signals to the ocean, relying on the characteristics of high 5G communication efficiency, offshore (generally ≤ 5 km from the coastline) hydrological environment monitoring (Zhang et al., 2022), port security (Howard et al., 2011), rescue (Matos et al., 2017) and unmanned underwater vehicle reconnaissance (Chen et al., 2020) can be completed by small offshore unmanned carriers such as buoys and unmanned boats. The small maritime unmanned vehicles (SMUVs) have unique advantages such as low cost, high flexibility, and strong stability. With the increasing task difficulty, the precise positioning ability required by an SMUV becomes stronger. The application environment of an SMUV is generally the open ocean; thus, traditional positioning relies solely on satellite signals to determine its location. However, positioning methods such as BeiDou navigation satellite system (BDS), Global Positioning System (GPS), differential GPS, differential BDS, and GPS and BDS fusion have the problems of slow convergence speed and large signal interference by the electromagnetic environment (Sun et al., 2021, 2018; Schmidt et al., 2019). Therefore, BDS positioning can be integrated with 5G positioning in offshore areas.

As shown in Figure 1, owing to the influence of sea waves, the SMUV’s 5G and BDS antennas will shake at different amplitudes, and the signals will be intermittently obscured, resulting in discontinuous positioning results, large errors, and slow convergence. In addition, because waves also affect the motion of the SMUV, it may suddenly accelerate in a certain direction, resulting in a large gap between the motion model used in the Kalman filter (KF) algorithm and the actual motion (Novák et al., 2025; Draycott et al., 2021), which in turn leads to impulsive errors in trajectory tracking.

Figure 1

Global navigation satellite system (GNSS) and 5G joint positioning can be divided into two modes: independent denoising preprocessing of original measurement data and the establishment of GNSS and 5G joint positioning solution model (Wang et al., 2024). In the denoising a preprocessing of GNSS/5G raw measurement data, several algorithms have been proposed such as KF, extended Kalman filter (EKF), unscented Kalman filter (UKF), machine learning algorithm, and deep learning algorithm (Deep et al., 2018; Sharie et al., 2024; Li et al., 2018; Ma et al., 2022; Siemuri et al., 2021; Chen et al., 2022), which independently perform noise filtering after 5G or GNSS distance estimation. In addition, Wang proposed a scheme that uses a dual kF algorithm to asynchronously fuse 5G positioning output results with GNSS output results, which not only filters out noise but also further improves the positioning update rate. However, although the above algorithm improves the positioning accuracy, it also increases the complexity of positioning.

Several scholars have derived (Yin et al., 2018; Lu et al., 2022; Xue et al., 2024) the joint and pseudorange observation equations are derived under the same frequency of 5G and GNSS pseudorange measurement data output. The high update rate of the 5G pseudorange measurement data is not considered; thus, the update rate of the positioning results can only be subject to the update rate of the GNSS pseudorange measurement data. Similar to the single GNSS positioning solution, the least square method (LS) (Li et al., 2022), Newton iteration method (Liu et al., 2013), and particle swarm optimization method (Wang et al., 2023) can also be used to solve the joint pseudorange observation equation, but the positioning result error is large. Therefore, an LSKF (Yue et al., 2022) algorithm has been proposed to perform KF filtering after solving the positioning observation equation, which improves the positioning accuracy but has a better effect only under Gaussian white noise.

Zhuang (Zhuang et al., 2023) proposed a positioning algorithm based on the EKF and UKF to address the nonlinear problem of positioning. The nonlinear approximation makes the positioning result deviate. The intermittent shadowing caused by waves can be abstracted as heavy-tailed non-Gaussian noise during pseudorange measurements, and the instantaneous position movement caused by waves on an SMUV can be regarded as periodic pulse acceleration in the motion model. Student’s t KF (Huang et al., 2017), variational Bayesian-KF (Huang et al., 2017), Robust adaptive filtering (RAF) (Tong et al., 2021), and other algorithms have been proposed to address the problem of heavy-tailed, non-Gaussian measurement noise, but they cannot completely resolve the issues caused by ocean waves. Therefore, the BDS/5G fusion model proposed in this paper is based on the characteristics of SMUV motion. On the one hand, multirate data information fusion is realized by relying on the replication operation of low-complexity data before the positioning solution. On the other hand, the output of the positioning result is no longer limited to the satellite pseudorange measurement rate; instead, the 5G pseudorange measurement rate is used to produce the positioning result, improving trajectory tracking accuracy. On this basis, a KF algorithm combining approximate message passing (AMP) and variational Bayesian (VB) is proposed, which not only solves the problem of impulsive deviation between the original motion model and the actual motion caused by waves but also solves the problem of heavy-tailed, non-Gaussian measurement noise. The main contributions of this article are as follows:

• A high-speed updated asynchronous fusion BDS/5G positioning model was established.

• An improved KF algorithm is proposed to estimate the sparse pulse acceleration of the motion model caused by waves using AMP, thereby improving position prediction accuracy. VB is used to address the intermittent masking of signals caused by SMUV sloshing, thereby improving stability of positioning and trajectory tracking.

• In the motion simulation of SMUVs, wave induced acceleration under varying sea conditions, and motion states such as powered and unpowered operation are considered. The estimated positioning trajectory is closer to the actual trajectory than that obtained using comparison algorithm.

2 BDS/5G hybrid positioning model

In this paper, a BDS/5G hybrid positioning observation model is established to realize the pseudorange fusion of BDS pseudocode measurements and 5G time of arrival (TOA) measurement. Afterward, according to the difference between the update rate of the 5G TOA pseudorange data and the update rate of BDS pseudorange data, asynchronous fusion is carried out to improve the positioning update rate.

2.1 BDS/5G hybrid positioning observation model

The BDS is a member of the GNSS. Therefore, it also performs pseudorange measurements through pseudocode; that is, the space satellite transmits pseudocode to the ground terminal through broadcasting. The ground terminal completes the calculation of the electromagnetic wave propagation time by capturing and tracking some columns. The pseudorange can be expressed as follows:

In Equation 1, n represents the number of available satellites, n = 1,2···N, and N is the total number of available satellites. ρ⁽ⁿ⁾ represents the pseudorange measured by the pseudocode between the ground terminal and the satellite, c represents the propagation velocity of the electromagnetic wave, and δ is the clock difference between the receiving terminal and the satellite. I, T, and ϵ are the ionospheric error, tropospheric error, and other measurement errors, respectively. r⁽ⁿ⁾ is the geometric distance between the satellite and the receiving terminal, which can be expressed as follows:

In Equation 2, is the Earth Centered Earth Fixed (ECEF) coordinate of the first satellite and (x, y, z) is the ECEF coordinate of the receiving terminal. δ, I, and T can be corrected by navigation messages and calibration models. The calibrated pseudorange can be expressed as follows:

Therefore, the BDS positioning observation equation is obtained by Equations 2, 3 into Equation 1, which can be expressed as follows:

The principle of 5G TOA pseudorange measurement is similar to that of satellite pseudocode ranging. The signal propagation time estimation is estimated through signal synchronization, rather than using broadcast pseudocode. Therefore, it can be considered a ground pseudolite, and the observation equation can be expressed as follows:

In Equation 5, k represents the kth available 5G BS, ; is the clock difference between the 5G BS and the receiving terminal; represents the other measurement errors; is the geometric distance between the 5G BS and the receiving terminal, and is the pseudorange between the 5G BS and the receiving terminal. The pseudorange can be expressed by correction as follows:

where . Combining Equation 4 and Equation 6, the mixed observation equation can be obtained as follows:

Equation 7 shows that the establishment of the hybrid observation model can be completed only by the SMUV receiving equipment. In addition, owing to the requirements for the clock consistency in the space-ground network, the clock difference between the satellite-SMUV and the 5G-SMUV can be considered negligible and effectively the same.

2.2 High update rate measurement fusion scheme

The update rate of the 5G pseudorange measurement data is approximately 10∼100 Hz, and the update rate of the BDS pseudorange measurement data is approximately 1∼5Hz. The update rate of the BDS pseudorange measurement data is much lower than that of 5G. With respect to the traditional BDS/5G positioning scheme, only the pseudorange measurement data when the BDS and 5G are approximately the same frequency are sampled, and some 5G pseudorange measurement data are ignored. It is reasonable to ignore the data in the positioning of cars, drones, etc. due to their higher movement speeds. However, the SMUV movement speed is lower, so the 5G pseudorange measurement data with a high update rate should be fully utilized. As shown in Figure 2, in this paper, a high update rate measurement fusion scheme for short-term BDS pseudorange measurement data delay is proposed. In this scheme, the BDS pseudorange measurement data are delayed for a short time until the next BDS update. During this period, only the 5G pseudorange measurement data are updated. The updated 5G pseudorange measurement data and the BDS data during this period are merged into mixed pseudorange measurement models to obtain new positioning results. In addition, on basis of the difficulty of moving the position of the 5G BS, this paper preprocesses the pseudorange measurement data of the same 5G BS at two adjacent moments. When there is a large gap between the two moments, the pseudorange data of the BS and the receiving terminal measured at the next moment are discarded to reduce the heavy-tailed effect of the measurement noise.

Figure 2

3 BDS/5G hybrid positioning algorithm

Through the model construction and asynchronous data fusion in Chapter 2, we can obtain an overdetermined nonlinear equation set, which contains four unknowns, namely, the three-dimensional coordinates of the receiving terminal and the clock difference between the sending and receiving ends, denoted by L = [x, y, z, δ]^T. Therefore, it is necessary to find a suitable algorithm to suppress the noise caused by the environment and obtain the optimal solution of the equation.

3.1 Newton iteration method

The Newton iteration method is a classical algorithm for solving observation equations. The number of iterations of the Newton iteration method is generally set according to experience, and is expressed as j. Let the formula on the left-hand side of (Equation 8) be assumed as a function:

Then, the first-order Taylor expansion of h at the current estimation point L is carried out to obtain Equation 9.

Where, ΔL = L_j − L_j−1; set L₀ = [0, 0, 0, 0]^T; and H(L_j−1) is the Jacobi matrix, expressed as follows:

Thus, substitute Equation 10 into Equation 9 to get:

Here, the observation pseudorange , and then update L_j = L_j−1 + ΔL. The newly obtained L is brought into Equation 10 and Equation 11 until the end of the iteration, and is obtained.

3.2 KF localization algorithm based on joint AMP and VB

The positioning at this stage can be determined using the method in Section 3.1, but the results are still affected by waves and have large errors. The proposed algorithm is an improvement of the KF algorithm, which also includes three steps: initialization, time prediction, and Kalman update. Therefore, we use the result as the input and initialization of the proposed algorithm, i.e., , and . Afterward, the algorithm enters the prediction stage, and the formula is

In Equation 12, F is the motion state transition matrix, P is the state estimation covariance matrix, and whose initial value depends on the observation, and Q is the predicted noise covariance matrix. In this step, we add AMP sparse estimation to estimate whether acceleration occurs in a certain direction in the next position prediction. The formula is

In Equation 13, H is the observation matrix, α is the ratio of the observed number and the unknown number, ⟨⟩ represents the average of the vector fraction, η(·) is a sparse estimation function, is determined by processing noise variance , is the derivative of η(·). Afterward, the VB is used to reduce the influence of heavy-tailed, non-Gaussian measurement noise. The KF gain is first obtained:

In Equation 14, R is the noise covariance matrix, and λ₁ and λ₂ are the VB gain update parameters, which can be written as follows:

In Equation 15, and are the degrees of freedom, n_x is the number of state variables, is the number of observed variables, and

In Equation 16, tr represents the trace of the matrix. The covariance matrix expressed by Equation 12 can be rewritten as Equation 15:

Finally, the predicted value is obtained by Equation 18.

4 Numerical simulation

In this Chapter, we set the parameters of the simulated waves in the simulation and carry out the dynamic and nondynamic SMUV positioning simulations under these parameters.

4.1 Simulation parameter setting

Missions of SMUVs in the offshore sea are considered open application scenarios; thus, compared with vehicles and drones under urban environmental conditions, they can receive more satellite and 5G BS signals. Therefore, we assume that we can receive four satellite signals and four 5G BS signals at the same time, and their ECEF coordinates are described in Equations 19, 20:

Periodic pulse acceleration is added to the motion model to simulate the influence of waves on the motion of the SMUV, and the motion of the SMUV is modeled as follows:

In Equation 21, obeys Gaussian white noise, σ² is the variance, and the value is 0.01. G is the acceleration state transition matrix, β is the pulse acceleration amplitude, given as β = diag([1.2,0.8,0.1])m/s², and ν are periodic pulse vectors. In addition, the masking effect of ocean waves on 5G signals is abstracted because the measurement noise is heavy-tailed and non-Gaussian. In the case of interference, the measurement noise matrix variance is amplified by one hundred times with a probability of p; that is, the added measurement noise obeys the following distribution.

In Equation 22, is Gaussian white noise variance, = 0.2, and p = 0.05. Since we preprocess the data during data fusion and propose data with a large gap between adjacent moments, the impulse error variance here is 10 . In addition, the detailed parameters of the KF algorithm combined with AMP and VB are initialized, where . P changes with the iteration of the algorithm, so its initial value is set by experience, and the initial value of this paper is diag ([0.01, 0.01, 0.01, 0.01, 0.01, 0.01]). In this work, the noise variance is determined according to the irregular acceleration of waves affecting SMUV motion. The specific settings are as follows:

In Equation 23, ; the variance in , is the signal bandwidth.

4.2 Simulation of SMUV positioning

When the positioning update frequency is consistent with BDS, the frequency of the wave is fixed. Through Monte Carlo simulation, the error between the positioning result and the actual position of each update is compared and evaluated by Equation 24.

Moreover, we compare the proposed algorithm with the LS, EKF, UKF, RAF, LSKF, and adaptive variational bayesian (AVB) (Xue et al., 2024) algorithms, as shown in Figure 3. Under the same update frequency of positioning results, the positioning results of each time of the proposed algorithm are better than those of the other algorithms. Obviously, the proposed algorithm has strong anti-interference effects. In order to prove that the rmse results obtained in this paper are not errors caused by noise, we perform t-test on rmse of different algorithms, and the results are shown in Figure 4. The P values obtained were significantly less than 0.05, and there were significant differences. To reflect the tracking accuracy of the proposed algorithm, we draw a trajectory map of the positioning tracking and compare it with the algorithm, as shown in Figure 5. The algorithm proposed in this paper does not exhibit sudden large errors, and achieves the close tracking of actual positioning data. In addition, this paper also simulates the positioning error caused by different wave frequencies under the condition of a high update rate, and evaluates the performance via the root mean square error (rmse).

Figure 3

Figure 4

Figure 5

The positioning errors under different wave frequencies are shown in Figure 6. The increase in wave frequency does not cause deterioration, and the positioning rmse remains stable at approximately 1.1 meter (m), whereas the positioning rmse of the LS, EKF, UKF, RAF, LSKF and AVB remains around 2 m, and is strongly affected by the environment. With increasing the influence frequency of waves on the SMUV, the rmse tends to increase.

Figure 6

5 Conclusion

In this paper, the pseudorange measurement data are preprocessed first, and the observed pseudoranges of the same 5G BS at adjacent times are correlated. BSs with large pseudorange changes are removed from the pseudo-range measurement equation to reduce the non-Gaussianity of the measurement noise. Afterward, using the short-term delay method, the BDS pseudorange measurement data is kept unchanged, and the 5G positioning data are updated to improve the update rate of the positioning result output. In this paper, a KF algorithm combining AMP and VB is proposed. The AMP algorithm is used to estimate the pulse acceleration generated by waves on the SMUV in a certain direction, reducing the error between the preset motion model and the actual SMUV motion and further improving the positioning accuracy. In addition, the VB algorithm is used to improve the sensitivity of the 5G antenna to SMUV sloshing, and further reduce the influence of heavy-tailed, non-Gaussian measurement noise on the positioning results. Compared with the LS, EKF, UKF, RAF, LSKF and AVB methods, our method has significant advantages for the diverse simulation setups considered in the paper.

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