Underwater Robots - Gianluca Antonelli.pdf

2. Modelling ofUnderwater Robots
“We have Einstein’s space, de Sitter’s spaces, expanding universes, contracting
universes, vibrating universes, mysterious universes. Infact the pure
mathematician may create universes just by writing down an equation, and
indeed, ifheisanindividualist hecan have an universe ofhis own”.
J.J. Thomson, around 1919.
2.1 Introduction
In this Chapter the mathematical model of UVMSs isderived. Modeling of
rigid bodies moving in afluid or underwater manipulators has been studied
in literature by, among others, [137, 156, 157, 174, 182, 189, 203, 242, 255,
256, 285, 286], where adeeper discussion of specific aspects can be found.
In [224], the model of two UVMSs holding the same rigid object isderived.
2.2 Rigid Body’s Kinematics
Arigid body is completely described by its position and orientation with
respect to areference frame Σ i ,O− xyz that it is supposed to be earth-fixed
and inertial. Let define η 1 ∈ IR 3 as
⎡
η 1 = ⎣ x
⎤
y ⎦ ,
z
the vector of the body position coordinates in aearth-fixed reference frame.
The vector ˙η 1 is the corresponding time derivative (expressed in the earthfixed
frame). If one defines
⎡
ν 1 = ⎣ u
⎤
v ⎦
w
as the linear velocity ofthe origin ofthe body-fixed frame Σ b ,Ob − x b y b z b
with respect to the origin of the earth-fixed frame expressed in the bodyfixed
frame (from now on: body-fixed linear velocity) the following relation
between the defined linear velocities holds:
G. Antonelli: Underwater Robots, 2nd Edition, STAR 2, pp. 15–44, 2006.
© Springer-Verlag Berlin Heidelberg 2006