Adaptive federated filter–combined navigation algorithm based on observability sharing factor for maritime autonomous surface ships

The integrated navigation system ensures maritime autonomous surface ships (MASSs) to safely, efficiently, and autonomously complete various operations in different complex navigation environments. Investigating robust algorithms for integrated navigation is crucial for enhancing the fault tolerance of the system and ensuring the stable and continuous output of the ship’s motion state. However, existing research primarily focused on optimising particular filtering algorithms or examining the foundations of information allocation within a predetermined integrated navigation structure. As such, strategies for enhancing the robustness of the MASS integrated navigation system and the design of subsystems for federated filters in the event of navigation sensor failures have not been sufficiently investigated for complex maritime navigation scenarios. Consequently, this research introduces an observability sharing factor accounting for both system characteristics and state estimation performance in integrated navigation systems, employing nonlinear sampling filtering. Subsequently, a robust integrated navigation framework with distributed federal filter is developed. Within this framework, an adaptive federated filtering integrated navigation algorithm is proposed based on the observability sharing factor to allocate information in federated filtering. Finally, both the theoretical correctness and effectiveness of the algorithm were verified through simulations and real-ship experiments to assist with the development of accurate and fault-tolerant maritime navigation systems.


Introduction
The safe and economical navigation of maritime autonomous surface ships (MASSs) necessitates the stable output of navigation sensors for motion state and precise environmental perception.According to the specifications mandated by the International Maritime Organization, a minimum number of navigation sensors-comprising the inertial navigation system (INS), global navigation satellite system (GNSS), Doppler velocity log (DVL), and compass-must fulfill performance standards (Wang et al. 2017).As a fully autonomous navigational apparatus, inertial navigation exhibits robust anti-interference capabilities and optimal concealment, furnishing real-time information on position, speed, and attitude for maritime vessels (Boudehenn et al. 2023).The strapdown INS (SINS) is a frameless system developed from platform-based inertial systems, incorporating three rate gyroscopes, three linear accelerometers, and a microcomputer.Owing to its simple structure, light weight, cost-effectiveness, ease of maintenance, and high reliability, the SINS has been extensively incorporated into MASS-integrated navigation systems.Contemporary hybrid navigation methodologies employing SINS, GNSS, and DVL sensors can be categorised into three modalities: loose, tight, and ultralight integration.Notably, tight integration apparatuses are limited in number, costly, and substantially dependent on the precision of SINS sensors.Ultralight integration entails a complex GNSSreceiver-hardware code tracking loop (Liu et al. 2023).The present article focuses on the application of the cost-effective and most prevalently utilised loose integration navigation system in MASS operations.However, owing to variables such as navigational conditions and electromagnetic interference, MASS navigation sensors are CONTACT Yuanchang Liu yuanchang.liu@ucl.ac.ukDepartment of Mechanical Engineering, University College London, Torrington Place WC1E 7JE, London, UK susceptible to failure.Moreover, an increase in the number of sensors integrated into these systems elevates both the likelihood and frequency of operational failures.Consequently, the development of robust, fault-tolerant integrated navigation algorithms is crucial for guaranteeing the stable functioning of maritime navigation systems.
The stable ship motion state signal output also has great meaning in the MASS path following (Lakhekar and Waghmare 2023) and collision avoidance (Zaccone and Martelli 2020) tasks.
In the context of MASS integrated navigation systems, the efficacy of filtering algorithms is pivotal for system robustness.The SINS relies on these algorithms for stable output of the ship's motion state.To optimise state estimation in nonlinear integrated navigation systems, extensive research has focused on sampling-based filtering algorithms, such as the unscented Kalman filter (UKF) (Wan and Van Der Merwe 2000), cubature Kalman filter (CKF) (Arasaratnam and Haykin 2009), and particle filter (Gustafsson 2010).However, these algorithms often fall short owing to imprecise noise statistical characteristics, large initial state error metrics, and variations in actual system parameters, rendering the established nonlinear models less accurate than their theoretical assumptions.To this end, adaptive filtering algorithms based on technologies such as Sage-Husa (Yang and Xu 2003), fading factor (Geng and Wang 2008), strong tracking (Lin et al. 2015), Bayesian methods (Liu et al. 2022), and maximum likelihood methods (Song et al. 2020) have been widely studied.
In addition to these adaptive algorithms, researchers have incorporated adaptive factors to improve navigational accuracy.These factors are, in principle, designed based on state discrepancy statistics (Yang et al. 2001) and prediction residual statistics (Yang and Gao 2006).Contrary to conventional methods, H∞ filtering refrains from considering assumptions regarding the statistical characteristics of interference signals and minimises the estimation error under worst-case interference scenarios.The robustness of standard least squares estimation, Kalman filtering, and H∞ filtering with respect to measurement noise drift, delineating the relation between H∞ filtering parameters and the accuracy and robustness of state estimation are comparatively examined (Bolzern et al. 1999).Additionally, the generalised maximum likelihood estimation method (M Estimator) proposed by Huber can effectively address the non-Gaussian noise distribution and outliers, thereby enhancing the resilience of the filtering algorithm (Shi et al. 2023).
Given the increase in the diversity and volume of fusion sensors, a pressing need exists for fault tolerance in maritime multisource sensor information fusion algorithms.Existing research concerning fault-tolerant algorithms for multisource sensors in surface vessels predominantly encompasses federated filtering and interacting multiple models (Wang et al. 2022;Wu et al. 2023;Zhang et al. 2023).Overall, federated filtering is the most extensively employed information fusion fault-tolerant algorithm in integrated navigation systems (Lee et al. 2023).In the event of a navigation subsystem failure, the efficiency of the local filter within the information fusion framework can be adjusted to mitigate the adverse consequences resulting from the malfunction (Qin et al. 2023).Fundamentally, federal filter is employed in two architectures: centralised and distributed fusion (Yue et al. 2020).Centralised filtering requires the relay of measurement information from disparate sensors to a central station for collective processing.This method is associated with several limitations, including a substantial computational burden and compromised fault tolerance, factors that can promote the development of distributed Kalman filtering algorithms (Khan et al. 2014).Regarding distributed federal filters, the principles governing the allocation of information among sub-filters intrinsically influence the accuracy and fault tolerance of the integrated navigation system.Conventional scalar-based federated filter information allocation strategies prove inadequate for estimating the motion state of carriers in complex scenarios.
To address this issue, several researchers have conducted extensive research on the information weight-allocation problem of federated filter sub-filters.An adaptive fault-tolerant federated filtering algorithm based on RBF neural networks, which utilised fault factors as the allocation principle of information weights to achieve automatic isolation and recovery of subsystems when faults occur is proposed (Carlson and Berarducci 1994).Moreover, an adaptive federated Kalman filtering algorithm based on Markov distance is proposed (Gao et al. 2009).When the subsystem works anomalously, the robustness of the integrated navigation system is improved by introducing adaptive fusion and allocation coefficients in the information fusion stage.An INS/GNSS/DVL-integrated navigation algorithm and adopted a new, robust, strong adaptive federal Kalman filter tracking algorithm was proposed (Wang et al. 2017) proposed to improve the robustness of the sub-filter to interference.Xiong et al. (2019) proposed an improved SINS/DVL-integrated navigation subsystem compensation gain filtering algorithm, which improved the accuracy and stability of the navigation and positioning system.
To improve the reliability and positioning accuracy of the integrated navigation system, most of the existing research prioritises the optimisation of specific algorithms or the investigation of information sharing principles within federated filtering frameworks by assuming that the structure of the integrated navigation subsystem has been predefined (Zhou et al. 2023).However, a deficit exists in the theoretical exploration of strategies to enhance the robustness of MASS integrated navigation systems, especially in designing federated filter subsystems when various sensors malfunction in intricate maritime contexts.To rectify these shortcomings, this study introduces a pseudo inverse of the matrix based observability quantification algorithm that serves as the foundational criterion for assessing the state estimation robustness of the integrated navigation system.Currently, three primary methods are used for observability analysis in INSs: eigenvalue of the covariance (Ma et al. 2015), piecewise constant system (PWCS) (Goshen-Meskin and Bar-Itzhack 1992), and the local pseudo-inverse matrix based observability analysis method (Kai et al. 2014;Shen et al. 2023).The first two methods rely on the accuracy of state estimation to determine system observability, but neglect the inherent characteristics of the system.The third method linearises the nonlinear system through Taylor series expansion and conducts an observability analysis on this linearised version.Although this method considers the characteristics of the system and the accuracy of state estimation, it cannot be applied to integrated navigation systems using nonlinear sampling filtering algorithms.To address the aforementioned issues, the present study aims to contribute as follows: (1) A factor incorporating both system attributes and state estimation performance is proposed, aimed at quantifying the observability of state estimation errors within integrated navigation systems based on nonlinear sampling filters.
(2) Capitalising on the sensor array of MASS, we constructed a robust ship integrated navigation framework by employing distributed federated filtering.This framework encompasses two subsystems: SINS/GNSS and SINS/DVL/COMPASS.Within this structure, all states except for position errors were considered common states and participate in the feedback correction of the federated filtering.(3) An adaptive federated filtering integrated navigation algorithm based on observability sharing factor is proposed to augment the robustness of ship integrated navigation.This observability factor serves as the cornerstone for information allocation within federated filtering.
The remainder of this article is organised as follows: the equations for the SINS/GNSS and SINS/DVL/COMPASS integrated navigation systems within MASS are presented in Section 2. The proposed observability factor, the design logic of the robust MASS integrated navigation framework, and the adaptive federated filtering integrated navigation algorithm based on the observability sharing factor are introduced in Section 3. In Section 4, the proposed algorithm is validated through both simulation and historical ship data.The final section offers a comprehensive summary.

Motion state perception integrated navigation sensors in MASS
The navigation sensors in MASSs are primarily categorised into two types: motion state perception and situational perception sensors (Huy et al. 2023).The latter encompasses sensors such as LiDAR, millimeter-wave radar, visual cameras, navigation radar, electronic charts, and automatic identification systems (AISs).These sensors primarily provide MASSs with environmental context and information on proximate vessels.Although these sensors can contribute to robust motion state outputs through intricate computations, this may compromise the real-time output of the motion states.Considering these factors and facilitating the focused theoretical exploration, this section focuses on the motion state perception sensors required for MASS installation under current regulations.It formulates corresponding error models based on the operating principles and noise attributes of each sensor.
This study adopts a SINS with a high output frequency but with accumulated error as the reference system, and the SINS output is inputted into each local integrated navigation system.Other motion state sensing sensors are used to correct errors in the SINS: The GNSS provides position measurement, DVL provides velocity measurement, and COMPASS provides heading measurement.

Strapdown INS
SINS comprises three gyroscopes and three accelerometers.The origin of the body frame (b-frame) is located at the centroid of the vessel.The Earth-centered Earth-fixed frame (ECEF, e-frame) and the inertial frame (ECI, i-frame) originate at the centre of Earth, the n-frame is denoted as the navigation frame.Accordingly, the SINS error model can be expressed as follows: In Equation ( 1), ε n and ∇ n represent the gyroscope and accelerometer noises, respectively.ω c ab (a, b, c = i, e, n) indicates the local Earth rotation rate in the b-frame with respect to the a-frame expressed in the c-frame.C n b denotes the attitude transformation matrix between the b-frame and n-frame.ρ = [− L, λ cos L, − λ sin L] T , where, L, λ, and h denote the latitude, longitude, and height, respectively (Guo et al. 2023).ϕ n , δV n , and δP n denote the attitude, velocity, and position error vectors, respectively.

SINS/GNSS-integrated navigation system
The traditional SINS/GNSS integrated system uses optimal estimation algorithms to estimate the error states, and it is currently a popular strategy of integrated navigation for surface vessels.Therefore, the continuous-time nonlinear error state equation for SINS/GNSS integrated navigation system can be expressed as follows: where . f c and B c represent the nonlinear coupling system equations for transferring the x t and ω t in Equation (1).The measurement equation of the SINS/GNSS integrated navigation system is expressed as follows: where z sg t and v sg t indicate the measurement and measurement noise vector of SINS/GNSS-integrated navigation system, respectively.

SINS/DVL/COMPASS integrated navigation system
SINS/DVL is regarded as a cornerstone of autonomous underwater vehicle navigation system and has garnered considerable attention (Wang et al. 2020).As an effective technique of suppressing SINS error, the three-dimensional velocities outputted by DVL based on the Doppler frequency shift are independent.Moreover, the errors do not accumulate over time, which can offset the SINS error accumulated over time (Zhu et al. 2022).For surface vessels, COMPASS is a mandatory navigation instrument-it enhances the observability of the SINS/DVL-integrated navigation system in the heading dimension and can be seamlessly integrated into the system.
In this system, the state estimation equation is identical to Equation (2), and the measurement equation can be expressed as follows: where z sdc t and v sdc t denote the measurement and measurement noise of SINS/DVL/COMPASS integrated navigation system.

Robust filtering algorithm for adaptive integrated navigation of MASSs based on observability sharing factor
Analysing the observability results of various motion state errors in the MASS integrated navigation system serves as a valuable guide for algorithmic or structural optimisation.Observability constitutes a key indicator, reflecting both the convergence accuracy and velocity of the observers (Liang et al. 2020).By examining it, the complexity of estimating various states can be assessed online, thereby determining the operational efficacy of sensors and offering a foundation for optimising multi-sensory integrated navigation algorithms.
In this section, we first review the existing theoretical findings on the observability of nonlinear integrated navigation systems.Subsequently, inspired by the research (Shen et al. 2020), a nonlinearsystem-observability analysis methodology based on sampling filters is developed.Thereafter, this method is employed to assess the observability of SINS/GNSS and SINS/DVL integrated navigation subsystems within the context of MASS.Finally, an integrated navigation framework tailored for MASS motion state perception is designed and an adaptive sharing factor robust filtering algorithm is proposed based on these observability analyses.The information weight assignable to each subsystem is thereby established according to the observed metrics.

Sampling filter-based observability factor for integrated navigation system
Sampling-based filtering algorithms such as UKF, CKF, and APF utilise sampling points to approximate the distribution of the system.Assuming that the nonlinear system Equation (2) generates m points (χ k , γ k ) during sampling, where γ k = f (χ k ), k = 1, . . ., m.After linearising the nonlinear system equation, the optimal A k , b k is derived to minimise the linearisation error as follows: where Moreover, to address the aforementioned issues, the minimum values of the following objective functions should be obtained.
These traditional linearisation modes can obtain However, the rank of the matrix rank( (Shen et al. 2020).Therefore, a new method should be developed to linearise Equation (2).Therefore, y k = A k x k + ξ k in Equation ( 5) is reconstructed as follows: which can be expressed as ; the state transition matrix at time k can be denoted as follows: Therefore, there exists local observability of the Mohr-Penrose generalised inverse matrix During the period [k, k + n − 1], the observability of each state can be calculated using the following steps.n denotes the dimension of the system state variable.Approximately, Thus, the following scalar form can be obtained: where The recursive observation noise can be approximated as follows (Shen et al. 2020): Based on the assumption that the measurement matrix can be approximated as a diagonal matrix, it can be approximated as follows: R 0 denotes the covariance of the initial measurement noise.The observability of each state is defined as follows: where E k [(x i ) 2 ] estimates the mean-square deviation of the i-th motion state component.E k [(y i ) 2 ] estimates the mean-square deviation of the i-th measurement components.Substituting Equation ( 11) into ( 12), the observability of each state can be rewritten as follows: This indicator of Equation ( 13) can convert the observability of each state into a dimensionless indicator for evaluation.The computational complexity is relatively small, which is beneficial for real-time evaluation of the observability of various states in integrated navigation.The observability of the system was evaluated based on the proximity between the observed values y i and the predicted values x i , whereas b ij reflects the characteristics of the system.Therefore, this indicator comprehensively considers the impact of external measurements and the nonlinear system features on observability.

Analysis of MASS integrated navigation subsystem based on observability factor
The application of the aforementioned nonlinear system observability quantitative analysis theory to the MASS SINS/GNSS and SINS/DVL/COMPASS integrated navigation systems serves as a fundamental framework for constructing federated filtering architectures.
Initially, the criterion for observability is determined by examining if the term O k O T k in Equation ( 8) is equal to n (i.e. a full rank).If a full rank is achieved, the entire state is observable and the federal filter can receive entire information from the subsystem.In contrast, if only a part of states can be observed, the federal filter should not receive non-observable states.Otherwise, it will affect the estimation accuracy of the integrated navigation.
After calculation and verification, the SINS/GNSS-integrated navigation system comprised Equations ( 2) and (3), with an observable matrix of a full rank, indicating that the optimised filtering algorithms can be employed to improve the accuracy of SINS/GNSS state estimation.Although the rank of the observability matrix of the SINS/DVL/COMPASS integrated navigation system is 14, the longitude error is not observable.Therefore, relying solely on optimising the filtering algorithm in SINS/DVL/COMPASS integrated navigation offers limited effectiveness in improving the positioning accuracy of MASS.Consequently, in designing a federal filter, the position error estimation from the SINS/GNSS system ought to be incorporated as a distinct component, while other states may be included in a common component for feedback correction (Figure 1).Distributed federated filtering offers advantages such as design flexibility, computational efficiency, and enhanced fault tolerance. Notably, T and X b k and X c k denote the position errors of the SINS/GNSS system and other motion state errors in the published components of SINS/GNSS and SINS/DVL/COMPASS, respectively.In particular, X c k = [δV, δ , δε, δ∇] T and X b k = [δP] T .P 1 k and P 2 k denote the covariance produced by subfilters 1 and 2, respectively, during the optimal estimation time.Although during the process of correcting subfilters through feedback from the main filter, P c k and P b k represent the feedback covariance corresponding to subfilters 1 and 2. ϒ k denotes the information sharing factor of federated filtering algorithm.The complete calculation process of the aforementioned filter and the setting of ϒ k are discussed in the upcoming section.

Adaptive integrated navigation algorithm based on observability sharing factor
For a system in operation, a higher observability enables a more accurate estimation of its error.Thus, this section introduces an adaptive information sharing distributed federated filtering approach based on observability metrics.In conventional distributed federal filters, the comprehensive computational procedure outlined in Figure 1 proceeds as follows.
The β i (i = 1, 2) is an information sharing factor.Moreover, according to Carlson and Berarducci (1994), β 1 + β 2 = 1.Therefore, the global optimal motion state estimation Xc k+1 and its covariance matrix P c k can be derived based on the aforementioned principles of information sharing.Upon updating the subfilters, the global optimal information fusion result can be obtained as follows: Subsequently, an observability-based information-sharing factor is proposed to refine the extant distributed federated filtering approach.The optimised information-sharing factor considers the form of a dynamic diagonal matrix, enabling each public motion state in the subsystem to acquire a time-dependent information-sharing scalar ϒ ci m,k .The information-sharing factor scalar for the m-th motion state error is defined at k in two sub-integrated systems SINS/GNSS and SINS/DVL/COMPASS as follows: The information-sharing factor diagonal matrix with n motion states at time k of the i-th subfilter can be described as follows: where at any instant k, there exists 15) can be written as During the reverse information allocation procedure, a feedback information allocation algorithm is employed depending on the normalisation of the eigenvalues of the state covariance matrix.This algorithm constructs a bounded regularisation matrix exhibiting both positive definiteness and symmetry, articulated as follows: The normalised eigenvalues of the Pc k matrix k of the i-th subfilter are denoted as ρ i 1,k , ρ i 2,k , . . ., ρ i n,k .Thereafter, the feedback sharing factor scalar for the m-th state is stated as follows: The information feedback matrix of the i-th subfilter is expressed as The state estimation and covariance feedback reset of the adaptive federated filtering are stated as follows: The flowchart of the complete algorithm is illustrated in Figure 2.

Verification
To validate the effectiveness of the proposed algorithm, a specific ship model was employed for simulation, the parameters of which are shown in Table 1.The simulated SINS, produced by XSENS, is an amalgamation of satellite and inertial navigation capabilities, capable of generating high-frequency measurement data pertinent to the ship's motion state.The detailed specifications of the SINS unit employed for the simulation are listed in Table 2.For the external measurement simulations, a positioning error of 5 m and a velocity error of 0.4 m/s were incorporated into the simulated GNSS signal.This reflects the positioning error range of 5-10 m typically observed in commercial GNSS receivers utilised for general and submeter level navigation.The ship's motion trajectory is depicted in Figure 3. Assuming the initial position of the ship as (38.818°N, 121.879°E), the propeller maintains a constant speed of rotation, and under natural conditions of 3-m/s western wind and 1-m/s crossflow, the ship moves toward 000°at an initial speed of 15.5 kt.Thereafter, the heading is fine-tuned between 0 and 800 s, the rudder angles are adjusted to 10°and −10°, respectively, between 1000, 1400, and 2000 s, and the rudder angles are adjusted to 25°and −25°, respectively, between 2000, 2500, and 3000 s.The simulated motion trajectory diagram is illustrated below referring to the research (Guo et al. 2023).For the SINS/DVL/COMPASS system, the DVL colour noise is generated arbitrarily according to the research (Wang et al. 2017), which was derived from actual ship measurements.The amplitude of the Gaussian white noise was set to 1.For the COMPASS noise simulation, zero-mean Gaussian white noise with an amplitude of 1°and the constant error of 0.5°were added to the simulated COMPASS signal.Ships navigate in complex marine environments and are subject to a series of challenges.The GNSS system is vulnerable to satellite errors, signal propagation inaccuracies, receiver malfunctions, physical damage, and signal masking, all of which impair the precision of position and velocity measurements.Similarly, the SINS is influenced by factors such as temperature, which can induce drift in the gyroscope or accelerometer.GNSS faults predominantly manifest as step faults, while SINS faults usually exhibit a gradual nature.Given these considerations, and for the sake of theoretical comprehensiveness and operational efficacy, the CKF was selected as the representative sampling filtering algorithm in the subsystem for empirical validation.The simulation results are discussed in the following sections.

Simulation
MASS sailing in complex marine environments and GNSS signals are susceptible to satellite errors, signal propagation errors, receiver errors, physical damage, and masking, which affect positioning and velocity measurement accuracy.The SINS is susceptible to temperature and other factors that can cause drift in the gyroscope or accelerometer.Among them, GNSS faults are mostly step faults, whereas SINS faults are generally gradual faults.Thus, a step-position fault of 6 m and a step velocity deviation of 1 m/s are added in the time intervals, which are added in the time intervals of 410-420, 510-520, and 610-620 s at the straight motion stage.Moreover, the step faults were added at 1410-1420, 1510-1520, and 1610-1620 s with a 10°rudder angle and at 2510-2520, 2610-2620, and 2710-2720 s with a 25°rudder angle.In the linear motion stages of 700-800 and 800-900 s, the 10°rudder angle steering stage of 1700-1800 and 1800-1900 s and the 25°rudder angle steering stage of 2700-2800 and 2800-2900 s as well as the gyroscope gradual faults of w k = w k + (0.036 • /h) × t and w k = w k + (0.18 • /h) × t were added, respectively.At this instant, the simulation results are stated as follows.To avoid singular values in matrix O k O T k , the initial state error of the system is adequately increased, with a position error of 1 m, an initial velocity error of 1 m/s, and an initial attitude error of 0.1°in the integrated navigation system.
The superiority of the proposed algorithm is demonstrated using a robust Sage-Husa adaptive robust filter (Qiao et al. 2022) (adaptive adjustment covariance measurement matrix R k and state estimation covariance matrix P k ) for comparative analyses.The results of calculating the observability of the MASS SINS/GNSS system are displayed in Figure 4, where the degree of observability is denoted as Doo.In the event of a GNSS fault, the ARCKF improved its observability of the state error estimation at the GNSS fault period.This finding indicates a connection between observability and estimation ability, signifying that the variations in observability can reflect those in the estimation performance of filtering algorithms over time.As the ship turns, the observability of ARCKF relative to CKF increases significantly.As such, this is consistent with the cognitive understanding that the state error of the integrated navigation system can be estimated more accurately when the carrier undergoes a significant turn.In case of a gradual fault in SINS, the algorithm of adjusting covariance P k considerable increases the observability of the system.Therefore, optimising the filtering algorithm in the subsystem is beneficial for improving the fault tolerance of SINS/GNSS systems.
Velocity faults were introduced with the same value as GNSS during the corresponding time period of DVL.As depicted in Figure 5, within the framework of SINS/DVL/COMPASS systems, the ARCKF did not considerably enhance the observability of the CKF in the presence of faults.Regarding the heading estimation error, the observability under conditions of the SINS faults does not deviate significantly from that under normal conditions.Therefore, incorporating the external measurements more effectively improved the robustness of the SINS/DVL/COMPASS-integrated navigation systems than adjusting the state error covariance matrices P k or R k .
In case of faults in the GNSS, a comparative assessment is undertaken between the adaptive federated filter-integrated navigation algorithm, predicated on an observability sharing factor, and the state estimation results of the ARCKF, as depicted in Figure 6.The position estimation accuracy of the adaptive federated filter is inferior to ARCKF, largely because position is not subject to adaptive adjustment as a private state.Conversely, the algorithm exhibits enhanced fault tolerance in velocity estimation when contrasted with ARCKF.This is attributed to the SINS/DVL/COMPASS system's capacity for directly observable, accurate velocity measurements, which ameliorate the GNSS faults through weighted assignments based on observability.Additionally, the observability of roll and heading state errors is significantly improved because of the presence of external heading measurements in the SINS/DVL/COMPASS subsystem, which effectively counteracts the divergence in heading estimation errors.In summary, the adaptive federated filter-integrated navigation algorithm offers notable advantages in the realm of observability.
As depicted in Figure 7, in the event of faults in the DVL, the adaptive federated filter-integrated navigation algorithm ameliorates the deleterious impact of large position estimation errors caused by unobservable longitude errors within SINS/DVL/COMPASS systems.Owing to the directly observable heading error of the system, the system's estimation accuracy remains largely unaltered.The proposed algorithm surpasses ARCKF in terms of accuracy of both roll and heading estimations.In terms of velocity estimation, the presence of directly observable measurements within SINS/GNSS enables the proposed algorithm to deliver higher estimation accuracy, effectively compensating for DVL-related faults.This algorithmic approach thus endows the system with increased robustness in comparison to employing ARCKF in subfilters.With respect to the position error, the proposed algorithm outperforms ARCKF in terms of estimation accuracy, thereby reducing the vulnerability of the system to DVL faults.

Experimental results and discussion
To assess the effectiveness of the adaptive federated filter-integrated navigation algorithm predicated on an observability sharing factor, real-ship data validation is performed using historical data (Guo et al. 2023) from the unmanned boat experiment at the Lingshui  Port.As depicted in Figure 8, the historical trajectory of the experimental unmanned surface vessel (USV320) comprises both straight lines and turns.The experimental site is situated within the Maritime Training and Engineering Practice Center of the Dalian Maritime University and encompasses an area exceeding 40,000 square meters.This bay is minimally influenced by tidal variations, and the seawater exhibits a relatively sluggish flow rate.The experimental unmanned vessel is outfitted with various sensors and antennas including a GNSS receiver, an Mti-G-710 strapdown inertial navigation system, LiDAR, camera, 4G communication antenna, and 5.8G WIFI antenna.Notably, the vessel was not equipped with a DVL.To emulate DVL data, noise was introduced based on GNSS receiver velocity measurements, as validated through real-ship experiments in extant research (Wang et al. 2017) The total duration of the experimental data was 437 s, during which a SINS gradual fault of 0.18°/h was induced between 101 and 150 s, a GNSS speed step fault of 1 m/s between 201 and 230 s, and a DVL step fault of 1 m/s between 301 and 330 s.The output of the Mti-G-710 system after integrated navigation serves as the benchmark for validation.
Figure 9 depicts the observability of state errors.Regarding the roll estimation error, the SINS/DVL/COMPASS subsystem exhibits the highest degree of observability upon the implementation of the adaptive federated filter-integrated navigation algorithm.With respect to heading errors, the SINS/DVL/COMPASS subsystem demonstrates greater observability than SINS/GNSS systems.Concerning the observability of velocity errors, GNSS faults inserted between 201 and 230 s reveal that the eastern velocity error observability based on the ARCKF algorithm is maximal in the SINS/DVL/COMPASS subsystem, followed by SINS/GNSS.Note that the applied ARCKF is same as the algorithm in Section 4.1.In terms of northern velocity errors, the observability in the SINS/GNSS federated filtering subsystem surpasses that of the other algorithms.During the 301-330-s period, when a DVL fault is introduced and the heading of the vessel is oriented due north, the northern velocity observability of the adaptive federated filter-based SINS/GNSS subsystem is elevated.In summary, when the eastern velocity of the ship is high, the federated filtering SINS/GNSS and SINS/DVL/COMPASS subsystems manifest a superior level of observability compared to the ARCKF algorithm.
As observed in Figure 10(a), the presence of DVL faults triggers a marked divergence in the position error within the SINS/DVL/COMPASS subsystem.This divergence intensifies with the ship's maneuverability.Regarding speed errors, the empirical data reveal larger discrepancies than those generated in the simulations.This divergence can be attributed to two factors: the overly idealised simulation conditions and the inherently superior calculation strategy of the Mti-G-710-a device that consolidates magnetometers, temperature sensors, and pressure sensors-relative to the strapdown inertial navigation calculation method employed herein.
Considering Figure 10(b), when GNSS faults are introduced between 201 and 230 s, the velocity estimation accuracy of the SINS/GNSS subsystem falls short of that attained by the federated  filtering algorithm based on observability.Similarly, following the introduction of DVL faults between 301 and 330 s, the performance of the ARCKF algorithm, when executed solely by the SINS/DVL/COMPASS subsystem, is less robust than that of the adaptive federated filter-integrated navigation algorithm based on observability sharing factor.

Figure 1 .
Figure 1.MASS-integrated navigation framework based on distributed federated filtering.

Figure 2 .
Figure 2. Adaptive integrated navigation algorithm based on observability sharing factor.

Figure 3 .
Figure 3. Ship motion trajectory and simulation system interface.

Figure 4 .
Figure 4. Observability analysis for the motion state of an autonomous ship SINS/GNSS-integrated navigation system: (a) Observability of attitude: (b) Observability of velocity: (c) Observability of position.

Figure 6 .
Figure 6.Vessel adaptive federated filtering-integrated navigation system with GNSS faults: (a) Estimation of position and position error: (b) Estimation error of velocity: (c) Estimation of position and position error.

Figure 7 .
Figure 7. of vessel adaptive federated filtering-integrated navigation system with DVL faults: (a) Estimation of position and position error: (b) Estimation error of velocity: (c) Estimation error of attitude.

Figure 8 .
Figure 8. Test site of USV 320 and the schematic of the experimental route.

Figure 10 .
Figure 10.USV320 Federated navigation system state estimation error: (a) Estimation error of position: (b) Estimation error of attitude: (c)Estimation error of velocity.

Table 1 .
Parameters of ship simulation model.