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In this paper, we propose a Lagrangian mechanics framework for modeling the motion of a submerged vehicle operating near a free surface in calm water. By augmenting the system Lagrangian used to derive Kirchhoff's equations for a rigid body moving in an unbounded fluid, we directly incorporate the free surface into the derivation of the equations of motion. This is done using a free surface Lagrangian, which accounts for the energy stored within the free surface due to the motion of a submerged vehicle. The system Lagrangian then enables us to derive the six-degrees-of-freedom nonlinear equations of motion using the Euler–Lagrange equations. The model structure is similar to standard maneuvering models for surface ships, though additional complexities are present since the hydrodynamic parameters are shown to depend on the vessel position and orientation relative to the free surface. We also examine a fundamental tradeoff between model simplicity and model fidelity: we choose to model only the instantaneous fluid response and neglect the hydrodynamic memory effects. Without the memory effects, the proposed model does not incorporate wave-resistance forces. However, the simplified representation of the hydrodynamics does enable a control-oriented representation of the nonlinear radiation forces and moments commonly associated with added mass and free surface suction effects. [ABSTRACT FROM AUTHOR]

• A semi-recursive vehicle model is validated in modeling and accuracy. • Constant and adaptive time-step algorithms for real-time simulation are presented. • An accurate real-time simulation of a vehicle is completed in an affordable laptop. • A benchmark problem for the multibody dynamics community is put forward. Semi-recursive formulations and their various versions have made it possible to describe complex nonlinear systems such as vehicles precisely while still solving the relevant equations of motion in real time. An optimal combination of an efficient multibody formulation and a fast numerical time integration scheme are needed to accurately simulate complex systems in real time. This paper introduces a double-step semi-recursive multibody formulation and analyzes its performance with high-order numerical time-integration algorithms for real-time simulation. The Runge-Kutta, Gill, Runge-Kutta-Fehlberg, Adams-Bashforth-Moulton, and adaptive time step Runge-Kutta numerical methods are explained and compared. Results are verified against a commercial multibody software solution. A 15-degree-of-freedom sedan vehicle model serves as a benchmark to verify the theoretical results. The results highlight the differences between the numerical algorithms and suggest appropriate approaches for a nonlinear vehicle dynamics model, particularly for cases where simulation times are long. [ABSTRACT FROM AUTHOR]