Abstract
This paper develops a contraction-certified trajectory-tracking and gain-selection framework for fully actuated autonomous underwater vehicles on SE(3). The vehicle dynamics are represented in port-Hamiltonian form with a Rayleigh-type dissipation potential, and a dual potential shaping controller provides an energy-structured rotational–translational cascade. Regional contraction certificates are derived separately for the rotational and translational subsystems. The rotational analysis uses fixed left-trivialised momentum coordinates and retains anisotropic-inertia effects and the complete off-diagonal differential coupling. The translational analysis applies to a general known symmetric positive-definite inertia matrix through an attitude-cover semidefinite programme, with an exact endpoint reduction for isotropic inertia. A scaled composite metric combines the subsystem certificates and guarantees every strict complete-cascade rate below the slower subsystem rate. Large initial attitude errors are handled by an energy-entry phase followed by contraction within a prescribed tube, without controller switching. The four-dimensional gain-selection problem is decomposed into two independent two-dimensional offline searches using bisection and SDP/LMI feasibility tests. Numerical studies on the ODIN AUV quantify the region–gain–rate trade-off and examine small-angle, large-angle, and near-antipodal manoeuvres. The framework certifies complete-cascade rates of 0.042096s−1 and 0.008524s−1 for the 60∘/60∘ and 150∘/80∘ regions, respectively.