Underwater Robots - Gianluca Antonelli.pdf

2.7 Underwater Vehicles’ Dynamics in Matrix Form 35
• the matrix C v ( ν )isskew-symmetric, i.e., C v ( ν )=− C T v ( ν ) , ∀ ν ∈ IR 6 .
It is possible to rewrite the dynamic model (2.51) in terms of earth-fixed
coordinates; in this case, the state variables are the (6 × 1) vectors η , ˙η
and ¨η .The equations ofmotion are then obtained, through the kinematic
relations (2.1)–(2.2) as
where [127]
M � v ( R I B ) ¨η + C � v ( R I B , ˙η ) ˙η + D � RB( R I B , ˙η ) ˙η + g � RB( R I B )=τ � v , (2.53)
M � − T
v = J
− 1
e ( R I B ) M v J e ( R I B )
�
− 1
C v ( ν ) − M v J e ( R I B ) ˙ J ( R I �
B )
C � − T
v = J e ( R I B )
D � − T
RB = J
− 1
e ( R I B ) D RB( ν ) J e ( R I B )
g � − T
RB = J e ( R I B ) g RB( R I B )
τ � − T
v = J e ( R I B ) τ v .
− 1
J e ( R I B )
Again, the current can be taken into account byresorting to the relative
velocity or, introducing an approximation, considering the following equations
of motion:
M � v ( R I B ) ¨η + C � v ( R I B , ˙η ) ˙η + D � RB( R I B , ˙η ) ˙η + g � RB( R I B )=τ � v − τ � v,C ,
where τ � v,C ∈ IR6 is the disturbance introduced by the current. It is worth
noticing that the earth-fixed and the body-fixed models with the introduction
of the current asasimple external disturbance implies different dynamic
properties. In particular, this is true if, in case of the design of acontrol
action, the disturbance is considered asconstant orslowly varying.
2.7.1 Linearity inthe Parameters
Relation (2.51) can be written by exploiting the linearity inthe parameters
property. It must be noted that, while this property isproved for rigid bodies
moving in the space [254], for underwater rigid bodiesitdependsonasuitable
representations of the hydrodynamics terms. With avector of parameters θ v
of proper dimension it is possible to write the following:
Φ v ( R I B , ν , ˙ν ) θ v = τ v . (2.54)
The inclusion of the ocean current isstraightforward by using the relative
velocity asshown in Subsection 2.4.3. However, it might beuseful toconsider
also the regressor form ofthe two approximations given by considering the
current asanexternal disturbance. In particular, it is of interest to isolate
the contribution of the restoring forces and current effects, those are the sole
terms giving anon-null contribution tothe dynamic with the vehicle still and
for this reason will be defined as persistent dynamic terms.