Stability of a bottom-heavy underwater vehicle

We study stability of underwater vehicle dynamics for a six-degree-of-freedom vehicle modeled as a neutrally buoyant, submerged rigid body in an ideal fluid. We consider the case in which the center of gravity and the center of buoyancy of the vehicle are noncoincident such that gravity introduces an orientation-dependent moment. Noting that Kirchhoff's equations of motion for a submerged rigid body are Hamiltonian with respect to a Lie-Poisson structure, we derive the Lie-Poisson structure for the underwater vehicle dynamics with noncoincident centers of gravity and buoyancy. Using the energy-Casimir method, we then derive conditions for Lyapunov stability of relative equilibria, i.e. stability of motions corresponding to constant translations and rotations. The conditions reveal for the vehicle stability problem the relevant design parameters, which in some cases can be interpreted as control parameters. Further, the formulation provides a setting for exploring the stabilizing and destabilizing effects of dissipation and externally applied control forces and torques.