A Maneuvering Model for an Underwater Vehicle Near a Free Surface—Part I: Motion Without Memory Effects

Abstract

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.