A new way to estimate the dynamics of an autonomous underwater vehicle using PINNs

A new way to estimate the dynamics of an autonomous underwater vehicle using PINNs Mirco Vangi1,2 and Benedetto Allotta1,2 1Department of Industrial Engineering University of Florence, Florence, Italy 2Interuniversity Center of Integrated Systems for the Marine Environment (ISME), Italy Abstract—The development of autonomous underwater vehicles (AUVs) has recently seen rapid advances in both mechanical design and control systems. Traditional PID controllers are being increasingly replaced by model-based control laws that more accurately capture system dynamics. This paper presents a method for estimating AUV dynamic parameters using PhysicsInformed Neural Networks (PINNs). Unlike conventional neural networks, PINNs embed the physical model directly into training, enabling the extraction of more information from limited data. Tests focused on estimating the added mass and damping terms along the surge motion of the vehicle. To evaluate the estimation, a feedforward action was implemented on the vehicle. By comparing the control action provided by the feedforward and the actual control action of the PID, we obtain an error at steady state of less than 5N, demonstrating an elevate accuracy in the parameters estimation. Index Terms—Underwater vehicle; System Identification; Physical Informed Neural Network I. INTRODUCTION Once regarded as a hostile environment, the underwater domain has become increasingly accessible through the development of specialized platforms for inspection, surveillance, and manipulation. Autonomous Underwater Vehicles (AUVs) enable autonomous inspection, monitoring, intervention, and data collection in marine and riverine settings [1], whereas Remotely Operated Vehicles (ROVs) are teleoperated systems offering high manoeuvrability, ideal for detailed inspection and intervention tasks. Bridging the capabilities of AUVs and ROVs remains an active research challenge in both academia and industry. In this context, the Italian RUVIFIST project introduced the Reconfigurable Underwater Vehicle for Inspection, Free-floating, Intervention and Survey Tasks (RUVIFIST) robot, developed by the Mechatronics and Dynamic Modeling Laboratory (MDM Lab) at Department of Industrial Engineering of the University of Florence (UniFi DIEF). Classified as an Autonomous Underwater Reconfigurable Vehicle (AURV), this vehicle can adapt its geometry and thruster configuration to meet diverse operational requirements, representing a significant step toward multifunctional underwater robotics. The RUVIFIST system is a reconfigurable platform capable of autonomously transitioning between a slender “survey” configuration, optimized for large-scale seabed exploration and a “hovering” configuration,tailored for close-range inspection and intervention [2]. The present study focuses on the identification of the dynamic parameters of RUVIFIST in the “survey” configuration, where the implementation of a FeedForward (FF) can notably enhance motion performance. Future work will extend the analysis to both configurations to enable a comparative evaluation of their dynamic behaviours. (a) (b) Fig. 1: The RUVIFIST vehicle in its two extreme configurations,“survey” in (a) and “hovering” in (b). II. STATE OF THE ART System identification represents a milestone in the modelling and control of dynamic systems, as it enables the derivation of mathematical representations directly from observed data. System identification methods are typically divided into two categories: physics-based and data-driven. Physics-based approaches derive analytical models from hydrodynamics and rigid-body mechanics, while data-driven ones rely solely on measurements, offering flexibility but often lacking robustness and generalization beyond the training domain. For underwater vehicles, nonlinearities and strong coupling across Degree Of Freedoms (DOFs) complicate identification. A common simplification is to linearize the dynamics around operating points and excite one DOF at a time, with parameter estimates typically obtained by solving Linear Programming (LP) problems [3]. Conventional Machine Learning (ML) models often struggle with limited data availability, whereas PhysicsInformed Neural Networks (PINNs) embed governing physical laws directly into the loss function. This enriches the effective information content of the data, allowing convergence to physically consistent solutions and improved generalization, even with few training examples [4]. III. DYNAMIC PARAMETERS ESTIAMTION Regarding the complete dynamics model of a marine vehicle in absence of sea current, the equation describing the forces on the centre of mass is: M˙ν+C(ν)ν+D(ν)ν+g(η)=τ(1) 2025 I-RIM Conference October 17-19, Rome, Italy ISBN: 9788894580570 10.5281/zenodo.17629780 149 where M=MRB +MAis the mass and inertia matrix, arising from the vehicle mass and the fluid accelerated with the vehicle, C(ν)is matrix of Coriolis and centripetal term D(ν)=Dl(ν)+Dq(ν)is damping matrix, due to viscous drag, g(η)is the vector of gravitational forces and moments and τis the vector of control inputs. Since AUVs typically operate at low speeds, quadratic damping Dq(ν)is negligible, leaving only the linear component Dl(ν). The proposed architecture, shown in Fig. 2, combines Long Short-Term Memory (LSTM) and Fully Connected (FC) layers. The LSTM layer captures temporal dependencies, as instantaneous data are insufficient to estimate system parameters. This is followed by two FC stages separated by a Tanh activation, which introduces nonlinear behavior into the model. Fig. 2: PINN structure used for parameters estimation. In addition to the standard network weights, the model incorporates trainable physical parameters representing the added mass matrix MAand the linear damping matrix DlThese parameters are defined as learnable entities within the network architecture and are updated simultaneously with the network weights through back-propagation. The overall cost function comprises two components: one enforcing consistency with the dynamic model and another minimizing the mean squared error between predicted and measured velocities. To validate the estimation on the real vehicle, we implemented an onboard FF control system in parallel with the GS-PID controller already implemented on the vehicle [2]. IV. RESULTS AND CONCLUSIONS Experimental tests with the vehicle surfaced were performed in Bilancino Lake, Barberino del Mugello (FI), Italy. During these test the FF terms were computed but not fed into the control action of the system, to assess their fitness in a comparison to the actual PID control output was performed. The results show that, at steady state, the estimation of the FF terms are comparable with the output of the PID control system (around 5 N of difference), meaning that the elements of the the matrix Dlare correctly estimated. It is important nothing that in the first test the difference between the FF terms and the Proportional Integral Derivative (PID) are greater than in the second one, this is reasonable caused by the surface currents and the wind, since the two test were performed in different directions. Regarding the transient response, the contribution of the FF terms is significantly greater than that of the PID. However, since the PID is intentionally slow for stability reasons, the dominant FF contribution enhances the overall system efficiency. Future work aims to test the system with only the FF to validate this improvements in the efficiency. In conclusion, the proposed PINN structure provides highly accurate estimates of the vehicle’s dynamic parameters, simplifying underwater 0 20 40 60 80 100 120 140 160 180 Time [s] sec 0 20 40 60 80 Thrust [N] Surge direction contributions of PID and Feedforward terms FF contribution PID speed contribution 0 20 40 60 80 100 120 140 160 180 Time [s] sec 0 0.2 0.4 0.6 0.8 Speed [m/s] Actual and demanded surge speed. Vehicle Surge Speed Surge Demanded Speed Fig. 3: Results for the FF terms and PID terms on surge motion. vehicle system identification and consequently expanding its operational autonomy and potential applications. 400 450 500 550 600 Time [s] sec 0 5 10 15 20 25 Thrust [N] Surge direction contributions of PID and Feedforward terms FF contribution PID speed contribution 400 450 500 550 600 Time [s] sec 0 0.1 0.2 0.3 0.4 Speed [m/s] Actual and demanded Surge speed Measured Speed Demanded Speed Fig. 4: Vehicle motion with only FF control system. Figure 4 shows the surge speed evolution with the only FF control system enabled with the estimated values of the added mass and drag terms. From this test we can see that the estimation of the added mass is very accurate since the acceleration phase percetly follow the desired speed profile, exposing an mean error of 0.02 m/s. Future work will focus on acquiring data for the vehicle’s hovering configuration, enabling estimation of the dynamic parameters for both configurations and the development of a more efficient control system. REFERENCES [1] A. Bucci, A. Topini, E. Topini, L. Bartalucci, G. Liverani, M. Vangi, G. Lazzerini, L. Cecchi, A. D. Valle, N. Secciani, A. Ridolfi, and B. Allotta, “Rami 2023: The experience of the unifi robotics team with feelhippo auv,” in OCEANS 2024 - Halifax, pp. 1–6, 2024. [2] M. Vangi, E. Topini, G. Liverani, A. Topini, A. Ridolfi, and B. Allotta, “Design, development, and testing of an innovative autonomous underwater reconfigurable vehicle for versatile applications,” IEEE Journal of Oceanic Engineering, pp. 1–18, 2025. [3] C. Martinelli, A. Coraddu, and A. Cammarano, “Strongly nonlinear multidegree of freedom systems: Experimental analysis and model identification,” Mechanical Systems and Signal Processing, vol. 218, p. 111532, 2024. [4] M. Raissi, P. Perdikaris, and G. E. Karniadakis, “Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,” Journal of Computational physics, vol. 378, pp. 686–707, 2019. 150