LECTURE 2. STABILITY AND UNDERWATER VEHICLE DYNAMICS 3630.1,:::. .0
0.1 1
0 10 20 30 40 0 10 20 30 40
0.1~^0 .0 N^0
0.1 1
0 10 20 30 40 0 10 20 30 40
150 60100 40
{2 .0 "'
50 200
10 20 30 40 0 10 20 30 40Figure 2.6. The angle /3 and the position b drift.One can verify the hypothesis on the second variation by using an argument
on the Poisson reduced space P / S as in the energy-Casimir and energy-subcasimir
method.
In Leonard and Marsden [1997] a more delicate energy criterion is given for
non-generic μa which is relevant for and applied to the rising, spinning vehicle.
Dynamics of an underwater vehicle. Now we give a few more details on
why the dynamics of the underwater vehicle problem can be viewed as Lie-Poisson
dynamics; i .e., as generalized Euler equations.
Assume the underwater vehicle is a neutrally buoyant, rigid body submerged
in an infinitely large volume of incompressible, inviscid, irrotational fluid which
is at rest at infinity. The dynamics of the body-fluid system are described by
Kirchhoff's equations, where we assume the only external forces and torques acting
on the system are due to buoyancy and gravity.
Consider the group W, the semidirect product of S0(3) with two copies of IR^3 ,
i.e., W = S0(3)@(IR^3 x JR^3 ), where we take the action of S0(3) on IR^3 x JR^3 to be
R · (b, w) = (Rb, Rw). The group multiplication in Wis
(R, b, w)(R', b', w') =(RR', Rb'+ b, Rw' + w),
while the Lie algebra of W is tu = so(3) @JR^3 x JR^3.
Let (II, P , f) E tu * correspond, to angular and linear components of the impulse
of the system and the direction of gravity in body coordinates. The Poisson bracket
used is the Lie-Poisson bracket on tu*. Let n and v, respectively, be the angular
and translational velocity of the vehicle in body coordinates. From the Lagrangian