Underwater Robots - Gianluca Antonelli.pdf

3.4 Vehicle-Fixed-Frame-Based, Model-Based Controller 51
3.4 Vehicle-Fixed-Frame-Based, Model-Based Controller
Several controllers have been proposed in literature which are based on
vehicle-fixed-frame error variables. Among them, the controller proposed in
1994 by O. Fjellstad and T. Fossen [121]; full DOFs experimental results have
been reported by G. Antonelli et al. in [34, 35].
Let consider the vehicle-fixed variables:
� � B
R
˜y I ˜η 1 =
˜ε
˜ν = ν d − ν ,
where ˜η 1 = η 1 ,d − η 1 ,being η 1 ,d the desired position, and ˜ε is the quaternion
based attitude error,
s v = ˜ν + Λ ˜y , (3.10)
with Λ =blockdiag { λ p I 3 ,λo I 3 } , Λ > O .
ν a = ν d + Λ ˜y . (3.11)
Reminding the vehicle regressor Φ v ∈ R 6 × n θ,v defined in (2.54) and the
corresponding vector of dynamic parameters θ v ∈ R n θ,v ,the control law is
given by:
τ v = Φ v ( R I B , ν , ν a , ˙ν a ) ˆ θ v + K D s v , (3.12)
where K D is a(6 × 6) positive definite matrix. The parameter estimate ˆ θ v
is updated by
˙ˆθ
− 1
v = K θ Φ T v ( R I B , ν , ν a , ˙ν a ) s v , (3.13)
where K θ is asuitable positive definite matrix of appropriate dimension.
The stability analysis isachieved by considering the following Lyapunov
candidate function:
V = 1
2 s T v M v s v + 1
2 ˜ θ T
v K θ ˜ θ v > 0 , ∀ s �= 0 , ˜ θ v �= 0 (3.14)
the time derivative ofwhich, by applying the proposed control law, is given
by
˙V = − s T v [ K D + D RB] s v
(3.15)
It is now possible to prove the system stability inaLyapunov-Like sense
using the Barbălat’s Lemma. Since
• V is lower bounded