Steady motion of underwater gliders and stability analysis

The steady motion of underwater gliders (UGs) is key to deep-sea exploration because it represents the working pattern of the glider and has essential dynamic features embedded. Generally, there are two types of UG steady motion, the longitudinal section motion (LSM) and the three-dimensional spiral motion (SM). Although many studies have been done on the LSM mode, little is known about the important and general SM mode. The core of this challenge is that traditional formulation of the dynamics equations describes the UG’s motion in configuration space with three Euler angles, which cannot allow the periodic SM mode to be expressed as a fixed equilibrium point. This brings difficulties for the solution of SM and the analysis of its stability. Noting that the UG has symmetrical shape and that gravity and hydrodynamic forces remain constant in the SM motion, we identify a Lie group action from the UG’s configuration space, under which the Lagrangian function and force field have invariance. These properties allow us to establish a reduced dynamics (RD) model using the theory of geometric mechanics. Then the SM mode can be solved as a fixed equilibrium point in a reduced quotient space. With the RD model and parameters from Petrel II, we achieve equilibrium solutions for steady motions and find that these equilibrium solutions are asymptotically stable according to Lyapunov stability theory. Furthermore, noting that the hydrodynamic parameters play a vital role in determining the SM mode, we identify and define two dimensionless indicators to evaluate their influence on the equilibrium solution. In summary, this paper provides a viable approach to study the steady motion of UGs, which can potentially guide UG design and control.