Underwater Robots - Gianluca Antonelli.pdf

32 2. Modelling of Underwater Robots
while in terms of quaternion is represented by:
⎡
2(ηε2 − ε 1 ε 3 )(W − B )
⎢
− 2(ηε1 + ε 2 ε 3 )(W − B )
⎢
( − η
g RB ( Q )= ⎢
⎣
2 + ε 2 1 + ε 2 2 − ε 2 3 )(W − B )
( − η 2 + ε 2 1 + ε 2 2 − ε 2 3 )(y G W − y B B )+2( ηε1 + ε 2 ε 3 )(z G W − z B B )
− ( − η 2 + ε 2 1 + ε 2 2 − ε 2 ⎤
⎥
⎦
3 )(x G W − x B B )+2( ηε2 − ε 1 ε 3 )(z G W − z B B )
− 2(ηε1 + ε 2 ε 3 )(x G W − x B B ) − 2(ηε2 − ε 1 ε 3 )(y G W − y B B )
.
By looking at (2.45), it can be recognized that the difference between
gravity and buoyancy ( W − B )only affects the linear force acting onthe
vehicle; itisalso clear that the restoring linear force is constant inthe earthfixed
frame. Onthe other hand, the two vectors of the first moment ofinertia
W r B G and B r B B affect the moment acting onthe vehicle and are constant in
the vehicle-fixed frame. Insummary, the expression of the restoring vector is
linear with respect to the vector of four constant parameters
θ v,R =[W − B xG W − x B B yG W − y B B zG W − z B B ] T
through the (6 × 4) regressor
i.e.,
Φ v,R ( R I B )=
� R B I z O3 × 3
0 3 × 1
S
g RB( R I B )=Φ v,R ( R I B ) θ v,R .
(2.46)
�
R B �
�
I z
, (2.47)
In (2.47) S ( · )isthe operator performing the cross product. Notice that, alternatively
to (2.45), the restoring vector can be written in terms of quaternions;
however, this would lead again to the regressor (2.47) and tothe vector of
dynamic parameters (2.46).
2.6 Thrusters’ Dynamics
Underwater vehicles are usually controlled by thrusters (Figure 2.2) and/or
control surfaces.
Control surfaces, such asrudders and sterns, are common incruise vehicles;
those are torpedo-shaped and usually used in cable/pipeline inspection.
Since the force/moment provided by the control surfaces is function
of the velocity and it is null in hovering, they are not useful to manipulation
missions inwhich, due to the manipulator interaction, full control ofthe
vehicle is required.
The relationship between the force/moment acting on the vehicle τ v ∈
IR 6 and the control input of the thrusters u v ∈ IR p v is highly nonlinear. It
is function of some structural variables such as: the density ofthe water;
the tunnel cross-sectional area; the tunnel length; the volumetric flowrate
between input-output of the thrusters and the propeller diameter. The state