Underwater Robots - Gianluca Antonelli.pdf

160 7. Dynamic Control of UVMSs
7.7.2 Simulations
By considering the same UVMS as the previous Section, numerical simulations
have been performed totest the discussed control law.
The full simulated model includes alarge number of dynamic parameters,
thus the symbolic regressor Φ ∈ R (6+n ) × n θ has acomplex expression.
While the simulation is performed via the Newton-Euler based algorithm to
overcome this complexity, the control law requires the expression of the symbolic
regressor. Practicalimplementationof(7.22)–(7.23), then,mightbenefit
from some simplifications. Considering that the vehicle is usually kept still
during the manipulator motion it is first proposed to decouple the vehicle
and manipulator dynamics in the regressor computation. The only coupling
effect considered is the vehicle orientation in the manipulator restoring effects.
Further, it is possible to simplify the hydrodynamic effects taking a
linear and aquadratic velocity dependent term for each degree of freedom.
In this reduced form the vehicle regressor isthe same as if it was without
manipulator. On the other hand the manipulator regressor isthe ground
fixed regressor, except for the computation ofthe restoring forces inwhich
the vehicle orientation cannot beomitted; this means that areduced set of
dynamic parameters has been obtained. Inthis simulation the vehicle parameter
vector is ˆ θ v ∈ R n θ,v with n θ,v =23, the manipulator parameter vector
is ˆ θ m ∈ R n θ,m with n θ,m =13; it can be recognized that n θ,v + n θ,m � n θ .
The regressor implemented in the control law (7.22)–(7.23) has then the
form:
Φ ( q , R B I , ζ , ζ r , ˙ �
Φ v ( R
ζ r )=
B I , ν , ζ r,v , ˙ ζ r,v ) O 6 × 13
Φ m ( q , R B I , ˙q , ζ r,m , ˙ �
ζ r,m )
O 2 × 23
where Φ v ∈ R 6 × 23 is the vehicle regressor and Φ m ∈ R 2 × 13 is the manipulator
regressor. The corresponding parameter vector is ˆ T
θ =[ˆ θ ˆθ v
T
m ] T .The
vectors ζ r,v and ζ r,m are the vehicle and manipulator components of ζ .
In the simulation the initial value of ˆ θ is affected by an error greater then
the 50% of the true value. The control law parameters are
Λ =blockdiag { 0 . 5 I 3 , 0 . 5 I 3 , 3 , 2 } ,
K D =1000I 8 ,
K P =100I 8 .
˙˜y is computed by afiltered numerical time derivative:
˙˜y k = α ˙˜y k − 1 +(1 − α ) ˜y k − ˜y k − 2
2 ∆T
,
with ∆T being the simulation sampling time.
Astation keeping task for the system in the initial configuration was
considered
η i =[0 0 0 0 0 0] T � π
q i =
4
π � T
6
m, rad ,
rad .