Underwater Robots - Gianluca Antonelli.pdf

158 7. Dynamic Control of UVMSs
to avoid inversion of the system Jacobian. Further, to avoid representation
singularities of the orientation, attitude control of the vehicle is achieved
through aquaternion based error. To achieve good tracking performance,
the control law includes model-based compensation of the system dynamics.
An adaptive estimate of the model parameters is provided, since they are
uncertain and slowly varying.
Given the dynamic equations in matrix form (2.71)–(2.73):
M ( q ) ˙ ζ + C ( q , ζ ) ζ + D ( q , ζ ) ζ + g ( q , R I B )=Φ ( q , R B I , ζ , ˙ ζ ) θ = Bu,
the control law is
u = B † [ K D s � + Φ ( q , R B I , ζ , ζ r , ˙ ζ r ) ˆ θ ] , (7.22)
with the update law given by
˙ˆθ
− 1
= K θ Φ T ( q , R B I , ζ , ζ r , ˙ ζ r ) s , (7.23)
where B † is the pseudoinverse of matrix B , K θ > O and Φ is the system
regressor defined in (2.73). The vectors s � ∈ R (6+n ) × 1 and s ∈ R (6+n ) × 1 are
defined asfollows
s � ⎡ ⎤
˜ν 1
= ⎣ ˜ν ⎦ 2 +
˙˜q
� − 1
Λ + K D K ⎡ ⎤
B
� R I ˜η 1
⎣ P ˜ε ⎦ =
˜q
˜ ζ + � − 1
Λ + K D K �
P ˜y , (7.24)
s = ˜ ζ + Λ ˜y , (7.25)
with ˜η 1 =[x d − x yd − y zd − z ] T , ˜q = q d − q , ˜ν 1 = ν 1 ,d− ν 1 , ˙˜q = ˙q d − ˙q ,
where the subscript d denotes desired values for the relevant variables.
Λ is defined as Λ =blockdiag { λ p I 3 ,λo I 3 , Λ q } with Λ q ∈ R n × n , Λ > O .
K P is defined as K P =blockdiag { k p I 3 ,ko I 3 , K q } ,with K q ∈ R n × n , K P >
O . K q and Λ q must be defined soas K q Λ q > O .Finally, itis ζ r = ζ d + Λ ˜y
and K D > O .
7.7.1 Stability Analysis
In this Section it will be shown that the control law (7.22)–(7.23) is stable in
aLyapunov-Like sense. Let define the following partition for the variable s
that will be useful later:
⎡ ⎤
s = ⎣ s p
⎦ (7.26)
s o
s q
with s p ∈ R 3 , s o ∈ R 3 , s q ∈ R n respectively.