1. Introduction
Uncrewed underwater vehicles (UUVs) and their associated core technologies have advanced significantly, supporting various applications ranging from oceanographic research to exploration and surveillance. Fundamental improvements in hydrodynamic design, propulsion, control, docking, and navigation are needed because these missions become more diverse and complex. Among these challenges, the accurate and rapid estimation of the state of a UUV in the underwater environment is particularly crucial because it directly affects mission success and the meaningful interpretation of collected data. A variety of methods have been explored to improve UUV state estimations. Specifically, research inspired by the lateral line system (LLS) found in fish has attracted growing attention. Chambers et al. (2014) introduced a data-driven methodology that enables the simultaneous recording of three-dimensional pressure signals during motion. They examined flow features using a sensor array placed inside a fish-shaped head. Levi et al. (2014) proposed a strategy for an airfoil-shaped UUV to estimate flow properties using a bio-inspired, multi-modal artificial lateral line (ALL). They revealed these capabilities through feedback control, using a robotic prototype equipped with a multi-modal ALL, highlighting the potential of distributed flow sensing and closed-loop control. Strokina et al. (2016) used an ALL array to estimate the bulk flow velocity and angle, achieving accurate measurements even under river conditions. Their approach yielded an average velocity estimation error of 14 cm/s. Wang et al. (2016) used an ALL to evaluate the performance of a swimming robot. They proposed a nonlinear predictive model based on pressure analysis and motion kinematics, offering improved evaluation capabilities. Xu and Mohseni (2016) introduced a sensor system that leverages differential pressure sensors for high-precision measurements. Their results confirmed the ability of the system to estimate the hydrodynamic forces and detect nearby walls by analyzing the pressure distribution. Ali et al. (2017) examined the optimal three-dimensional design strategies for characterizing dipole sources using ALLs. They explored strategies to overcome the design limitations, examined cooperative ALL swarms, and analyzed the trade-offs between the sensor quantity and characterization accuracy. Yen et al. (2018) focused on measuring the hydrodynamic pressure of a robotic fish during movement near a straight wall. They utilized pressure data for motion feedback control, enhancing maneuverability. Liu et al. (2018) explored improvements in ALLs for underwater applications using a cylindrical carrier, examining biomechanics, developing optimal sensor distributions, and validating their system through experiments. Their work provided effective methods for estimating the flow velocity and identifying obstacles. Zheng et al. (2018) used an ALL using the hydrodynamic characteristics of a reverse Kármán vortex street-like wake. They revealed the efficiency and feasibility of applying ALLs for local sensing in nearby underwater robotic platforms. Subsequently, Zheng et al. (2020) investigated state estimations in a swimming robotic fish under various motions using a pressure variation (PV) model combined with regression analysis. They proposed a trajectory estimation method and achieved minimal estimation errors. Fish use their LLS to perceive subtle variations in flow velocity and pressure, enabling them to navigate, forage, and evade obstacles with exceptional agility. Building on this biological model, artificial lateral line systems (ALLSs) have become a promising approach for underwater sensing and navigation in UUVs. By mimicking the capacity of LLSs to detect and interpret nuanced flow phenomena, ALLSs offer an effective alternative or complement to existing sensing and navigation technologies, having the potential to enhance UUV state estimations. A previous study (Kim et al., 2024) developed a foundational pressure variation model (PVM) based on regression analysis of dynamic pressures calculated through numerical simulations. This model could estimate motion states, including straight, turning, and gliding motions, by analyzing pressure differences across a sensor array. Nevertheless, its reliance on motion-specific coefficients limited its adaptability to multi-motion scenarios, and its prediction accuracy was within 15%. For state estimations of UUVs, equipment such as DVLs and INSs are commonly used. On the other hand, DVLs rely on the signals reflected from the seabed, leading to reduced reliability in deep waters or complex terrains and susceptibility to environmental factors, high cost, and significant power consumption. INSs, however, suffer from drift errors that accumulate over time, require complex sensor calibration, and become increasingly expensive and intricate as the precision demands rise. These limitations highlight the need for an alternative approach. The enhanced PVM addresses this need by providing a cost-effective, robust solution. The improved PVM reduces long-term operational challenges and enhances the prediction accuracy in diverse underwater environments by minimizing the dependency on external environmental conditions and incorporating unified coefficients.
2. Advancement of Pressure Variation Model
2.1 Test Model
The autonomous underwater vehicle (AUV) used in this study is a streamlined REMUS (Remote environmental monitoring units) model, with primary dimensions of 1.345 m in length and 0.191 m in diameter. Fig. 1 shows the AUV structure and the arrangement of 0.01 m diameter pressure sensors (depicted as red circles with numerical labels) on its front and sides. Each sensor was designated and positioned as indicated in the lower left corner of Fig. 1, with R, L, T, and B denoting the sensors on the right (starboard), left (port), top, and bottom sides, respectively. Previous studies have reported that placing sensors at locations where pressure variations are significant due to speed changes improves the accuracy of the speed and drift angle predictions for AUVs. In particular, pressure sensors placed along the parallel middle body of the hull have minimal impact on speed and drift angle estimation. Based on this finding, the sensors were concentrated on the forebody of the hull, where the geometric changes are more prominent and pressure variations are more sensitive to motion. In addition, considering the physical size of each sensor and the minimum spacing required for practical fabrication, 11 pressure sensors were installed strategically on the AUV forebody to optimize functionality and manufacturability.
2.2 Overview of Underwater Vehicle State Estimation
The state estimation process for underwater vehicles proposed in previous research (Kim et al., 2024) is as follows:
(1) Pressure data acquisition: Pressure data were obtained based on AUV motion using computational fluid dynamics (CFD).
(2) Regression coefficient estimation: The regression coefficients were estimated using the acquired pressure data. These coefficients are vital for constructing the PVM.
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(3) AUV State Estimation: This includes two steps:
Estimating the speed of the AUV using the sum of dynamic pressures.
-
Estimating the vertical and horizontal drift angles based on the speed obtained from step A and the dynamic pressure differences.
These two steps constituted the PVM. It was assumed that the angular velocity was measured using an inertial sensor.
(4) Performance validation: The effectiveness of the PVM was tested by estimating the motion in various scenarios, validating its performance.
2.3 Limitations of Existing Models
The previous PVM was an adaptation of the model proposed by Zheng et al. (2020). The model used regression analysis of dynamic pressure to estimate the motion states such as velocity, drift angle, and angular velocity. This equation, derived from the unsteady Bernoulli equation, accounts for dynamic interactions in the underwater environment. Eq. (1) expresses the form of the model presented by Zheng et al. (2020).
where Pd, U, θ, ω, and C represent the dynamic pressure, velocity, pitch angle, sway angular velocity, and the regression coefficients determined through experimental data, respectively. Although this model revealed potential, several limitations were identified:
(1) Motion-specific coefficients: The PVM required different regression coefficients for each motion pattern (e.g., straight, turning, and gliding), necessitating motion pattern identification before state estimation, which increased the complexity and reduced adaptability.
(2) Simplified dynamic variables: The previous PVM overlooked the key dynamic variables essential for underwater motion. In particular, it did not consider the influence of pitch angle θ on the dynamic pressure changes, limiting its effectiveness in complex, real-world scenarios.
Therefore, based on the model proposed by Zheng et al. (2020), the authors’ team developed the previous PVM that considered the actual motion of underwater vehicles by incorporating the vertical and horizontal drift angles α and β, heave angular velocity q, while excluding the pitch angle θ, which was deemed to have low influence.
Additionally, the yaw angular velocity was denoted as r, which is consistent with maritime maneuvering problems. The PVM was divided into pressure-sum and pressure-difference models, as shown in Eq. (2) to (4):
where U is the velocity; α and β are the vertical and horizontal drift angles; q and r are the heave and yaw angular velocities (Fig. 2); C and D are the regression coefficients derived through analysis. The regression coefficients were derived based on the data obtained through numerical analysis, and the final speed and drift angles were estimated using the pseudo-inverse matrices of Eq. (5)–Eq. (7).
(5)
(6)
(7)
Although this model demonstrated potential, it was developed to achieve a prediction accuracy of 15%. Despite meeting this target, several limitations were identified:
(1) Trial-and-error model development:
- Relied on a trial-and-error approach to determine the model structure.
- Showed a lack of a logical foundation for the model.
(2) Simplified slip angle estimation:
- The slip angle estimation model did not account for the coupling effects between the angular velocity and slip angle.
- This simplification increased state estimation errors during complex motions such as slipping turns and spiral movements.
These limitations highlighted the need for a more robust and unified approach to enhance the PVM, paving the way for developing the improved model presented in this study. The current research aimed to establish a predictive framework that addresses these deficiencies by refining the state estimation matrices, incorporating nonlinear effects, and unifying model coefficients for broader applicability.
2.4 Proposed Model Enhancements
The improved PVM incorporates the following advances:
(1) ) Analytical approach for PVM structuring: A Taylor series expansion, correlation analysis, and regression analysis were conducted to determine the structure of the PV model based on a logical foundation.
(2) Unified coefficients: A single set of coefficients applicable across all motion types to streamline calculations.
(3) Nonlinear interactions: Inclusion of terms accounting for correlations between the angular velocity and drift angle.
A Taylor series expansion, correlation analysis, and regression analysis were conducted to determine the structure of the PV model based on a logical foundation. Based on the second-year research results, the PV can be expressed as a function of the motion variables of the underwater vehicle u, v, w, q, r in Eq. (8).
(8)
Assuming that α and β are sufficiently small, the velocity components in each axis direction can be approximated as shown in Eq. (9). Consequently, the PV corresponding to the motion of the underwater vehicle is expressed as Eq. (10).
(10)
The terms related to vertical and horizontal plane motions have similar values because the test model has a slender body and exhibits vertical and horizontal plane symmetry. Therefore, the results of the Taylor series expansion can be formulated as shown in Eq. (11). Furthermore, this study does not consider motion in which the yaw and heave angular velocities coexist. The last term in Eq. (10) could not be identified at the current stage, so it was excluded from consideration.
(11)
The following single and complex motion patterns (Fig. 3) were defined to validate the PV model.
The statistical significance of the PV model established through regression analysis was examined to assess the relative influence of each term on PVM. A z-test was used for regression analysis. The z-test calculates the z-value, a test statistic that follows a normal distribution assuming the null hypothesis is true. The p-value, or significance probability, was then determined based on the location of the z-value in the normal distribution. Statistical decisions were made by comparing the p-value with the significance level (α). A p-value smaller than the significance level means that the null hypothesis is rejected in favor of the alternative hypothesis. Tables 1, to 9 summarize the significance probabilities at representative pressure sensor locations under different motion conditions. Regression coefficients with a p-value greater than 0.005 were removed to simplify the PVM (Benjamin et al., 2018). Subsequently, the p-value validation was conducted again to ensure statistical significance.
Coefficients with a significance probability greater than the threshold value were considered to have minimal impact on the speed and angle of attack estimation and were then removed from the PV model. As a result, the final structure of the PV model was expressed as Eq. (12). The pressure sum model for speed estimations and the pressure difference model for angle of attack estimation are formulated as follows, and the final derived regression coefficients are summarized in the Appendix (Tables A1 to A7). Here, q and r were assumed to be measured by an inertial sensor:
First, the pseudo-inverse matrix of the pressure sum model was calculated (Eq. (16)) to estimate the velocity U. Here, the subscript of each coefficient denotes the regression equation coefficient number, while the superscript represents the pressure sensor number:
(16)
Based on the previously estimated velocity, the drift angle of the vehicle was calculated using the following matrix of Eq. (17) and (18). Tables A1 to A7 list the derived coefficients.
3. Numerical Analysis
The coefficients for the PVM were determined by conducting numerical simulations under the conditions specified in Table 10. The flow analysis of the underwater vehicle in specific motion scenarios (Fig. 4) was performed using the commercial software STAR-CCM+ ver. 13.06.012. The diffusion and convection terms in the governing equations were discretized with second-order accuracy, and the SIMPLE (Semi-implicit method for the pressure-linked Equations) algorithm was applied for velocity and pressure analysis. The k-ω SST (Shear stress transport) turbulence model was used along with wall functions, considering the boundary conditions of the submerged body (Kim et al., 2024).
The computational domain followed the ITTC (2011) guidelines, ensuring a sufficient size to prevent backflow and reflection. The velocity inlets were set at the upstream, top, bottom, and sidewalls, with a pressure outlet downstream. An elliptical overset region allowed the smooth execution of various motions. Trimmed cell meshing was used for the volumes, while a surface remesher ensured high-quality surface meshes. The AUV featured a 10-layer prismatic boundary layer, with volumetric control refining trajectory conditions. The y+ value remained below 1 for accuracy.
4. Results and Discussion
With the final structure of the PV model determined, the performance of underwater vehicle state estimation for each motion was evaluated using the motion estimation model. The pressure data at each sensor was obtained under the predefined conditions of speed (Uset) and drift angle (αset, βset,) through numerical analysis. Table 11 lists the estimation results of speed and drift angles during straight motion. The speed estimation achieved an accuracy within 1%, and the drift angles were also well estimated.
Tables 12,–14 present the estimation results of the speed and drift angle during gliding motion. The estimation error increases slightly under relatively low-speed conditions (1.0 kn) or when the drift angle exceeds 20°. Nevertheless, it remained within the target error range of 15%, and the accuracy was within 2% under all other conditions.
Tables 15,–17 present the speed and drift angle estimation results for turning motion. The estimation error increased slightly under lower-speed conditions (1.0 kn) or when the drift angle surpassed 20°. Nevertheless, it remained within the target error threshold of 15% while maintaining an accuracy of within 5% under all other conditions.
Tables 18 and 19 show the speed and drift angle estimation results during gliding motion with drift angles. When the drift angle exceeded 20°, the errors increased, but the speed estimation error remained within 15%, meeting the target performance. In other cases, the error remained within 5%, showing good accuracy. Similarly, drift angle estimation errors increased for drift angles over 20°, due likely to speed estimation errors. Excluding these cases, the error rate remained within 6%, indicating overall good performance.
Table 20 lists the speed and drift angle estimation results during turning motion with drift angles. The speed estimation results showed that the error rate remained within 3%, showing overall good performance. In addition, the estimation accuracy improved as the angular velocity and drift angle decreased. For drift angle estimations, the vertical drift angle was estimated accurately at the set value of 0, while the horizontal drift angle maintained an error rate within 8%, indicating generally good estimation performance. Table 21 presents the speed and drift angle estimation results during spiral motion. The speed estimation results showed a slight increase in error under large-angle conditions, but the error rate remained within 6%, indicating overall good performance. Similarly, the errors in drift angle estimations increased slightly under large-angle conditions, as observed in the speed estimation model. Nevertheless, the error rate for the vertical drift angle remained within 9%, and the horizontal drift angle was estimated accurately at the set value of 0°.
Table 22 lists the estimation results of speed and drift angle during spiral motion with drift angles. The speed estimation results show that the error rate remained within 5%, excluding large-angle conditions. Similarly, the drift angle estimation results indicated a significant increase in error under low-speed large-angle conditions, reaffirming the need to incorporate large-angle conditions in the formulation of the PVM model.
Compared to the previous PVM, the model structure and the accuracy of the speed and drift angle estimations improved. On the other hand, the prediction accuracy decreased under conditions with large drift angles or high angular velocity. The operating range of the vehicle must be defined, and if conditions exceeding 20° need to be considered, the PVM should be restructured without assuming small angles and incorporating trigonometric functions in the formulation process.
5. Conclusions
5.1 Summary of the Study
This study developed an improved PVM to enhance the state estimation of UUVs. By leveraging pressure sensor data and CFD simulations, the model accurately estimated the velocity and drift angles without relying on DVLs or INSs, which are expensive and environmentally dependent. Enhanced PVM incorporated the unified regression coefficients and nonlinear effects, improving the adaptability across diverse motion scenarios. The validation results confirmed that the model achieved high accuracy, maintaining estimation errors within the acceptable thresholds.
5.2 Contributions of the Study
The key contributions of this study include the following:
(1) Enhanced state estimation model – The improved PVM refined existing approaches by introducing a unified coefficient framework, reducing the reliance on motion-specific parameters.
(2) Robust and cost-effective approach – Unlike DVL and INS, the proposed method minimized the dependency on external conditions, making it more suitable for real-world underwater applications.
(3) Validated accuracy through CFD simulations – The effectiveness of the model was demonstrated through extensive CFD-based validation, confirming its reliability in various motion scenarios.
5.3 Future Research Directions
Future research should focus on extending the applicability of the model by
(1) Addressing large-angle motions – Improving prediction accuracy for scenarios with high drift angles or complex maneuvers.
(2) Experimental validation – Conducting real-world experiments to validate performance beyond numerical simulations.
(3) Developing a real-time data processing module – Enhancing the PVM by implementing a real-time processing system for efficient in situ state estimation, enabling faster and more reliable UUV navigation.
By advancing the UUV state estimation, this study contributes to developing more autonomous and efficient underwater navigation systems with potential applications in marine robotics and ocean exploration.
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