## Stability and Drift of Underwater Vehicle Dynamics:
Mechanical Systems with Rigid Motion Symmetry

### N.E. Leonard and J.E. Marsden

Mechanical and Aerospace Engineering, Princeton University Technical
Report 2075

In *Physica D*, Volume 105, June 1997, p. 130-162.
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.

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