Underwater Robots - Gianluca Antonelli.pdf

2.4 Hydrodynamic Effects 25
τ 1 = m ˙ν 1 + m ˙ν 2 × r b C + m ν 2 × ν 1 + m ν 2 × ( ν 2 × r b C ) .
Equation (2.29) is written inaninertial frame. Itispossible to rewrite
the angular momentum in terms of the body-fixed velocities:
k Ω = R I �
B I b C ν 2 + m ν 1 × r b �
C . (2.38)
Derivating (2.38) one obtains:
τ I 2 = ω × R I �
B I b C ν 2 + m ν 1 × r b �
C + R I �
B I b C ˙ν 2 + m ˙ν 1 × r b �
C ,
that, using the relations above, can be written in the form:
τ 2 = I b C ˙ν 2 + ν 2 × ( I b C ν 2 )+m ν 2 × ( ν 1 × r b C )+m ˙ν 1 × r b C .
It is now possible to rewrite the Newton-Euler equations of motion ofa
rigid body moving in the space. Itis:
M RB ˙ν + C RB( ν ) ν = τ v , (2.39)
where
�
τ 1
τ v =
τ 2
�
.
The matrix M RB is constant, symmetric and positive definite, i.e.,
˙M RB = O , M RB = M T RB > O .Its unique parametrization isinthe form:
�
m I 3 − m S ( r
M RB =
b C )
m S ( r b C ) I �
,
O b
where I 3 is the (3×3) identity matrix, and I O is the inertia tensor expressed
b
in the body-fixed frame.
On the other hand, itdoes not exist aunique parametrization of the
matrix C RB, representing the Coriolis and centripetal terms. It can be demonstrated
that the matrix C RB can always beparameterized such that it
is skew-symmetrical, i.e.,
C RB( ν )=− C T RB( ν ) ∀ ν ∈ IR 6 ,
explicit expressions for C RB can be found, e.g., in [127].
Notice that (2.39) can be greatly simplified if the origin ofthe body-fixed
frame is chosen coincident with the central frame, i.e., r b C = 0 and I O is a b
diagonal matrix.
2.4 Hydrodynamic Effects
In this Section the major hydrodynamic effects on arigid body moving in a
fluid will be briefly discussed.
The theory offluidodynamics is rather complex and it is difficult todevelop
areliable model for most of the hydrodynamic effects. Arigorous analysis