Stability and Drift of Underwater Vehicle Dynamics:
Mechanical Systems with Rigid Motion Symmetry
Leonard N. E. and J. E. Marsden
Physica D, 105, (1997), 130-162
Abstract:
This paper develops the stability theory of relative equilibria for
mechanical systems with symmetry. It is especially concerned with
systems that have a noncompact symmetry group,
such as the group of Euclidean motions,and with relative equilibria
for such symmetry groups.
For these systems with rigid motion symmetry,one gets stability but
possibly with drift in certain rotational as well as
translational directions. Motivated by
questions on stability of underwater vehicle dynamics,
it is of particular interest that, in some
cases,we can allow the relative equilibria to have nongeneric values
of their momentum. The results are proved by combining theorems of
Patrick with the technique of reduction by stages.
This theory is then applied to underwater vehicle dynamics. The
stability of specific relative equilibria for the underwater vehicle
is studied. For example, we find
conditions for Liapunov stability of the steadily rising and
possibly spinning,bottom-heavy vehicle, which corresponds
to a relative equilibrium with nongeneric momentum.The results of
this paper should prove useful for the control of underwater vehicles.
