Underwater Robots - Gianluca Antonelli.pdf

54 3. Dynamic Control of 6-DOF AUVs
industrial manipulators [254]. The current compensation ˆv is updated by the
following
˙ˆv = W − 1 − 1
J e ( R I B ) s e (3.20)
with W > 0and the dynamic parameters are updated by
˙ˆθ
− 1
v = K
θ Φ T v ( R I − 1
B , ν , ν a , ˙ν a ) J e ( R I B ) s e , (3.21)
where, again, K θ ,isapositive definite matrices ofgains ofappropriate dimensions.
It is worth noticing that with this control the model considers the current
as an additive disturbance, constant inthe body-fixed frame (see Subsection
2.4.3).
The stability analysis isdeveloped by defining as Lyapunov candidate
function
V = s T e M � v s e + 1
2 ˜ θ T
v K θ ˜ θ v + 1
2 ˜v T W ˜v (3.22)
that is positive definite ∀ s e �= 0 , ˜ θ v �= 0 , ˜v �= 0 .
By applying the proposed control law, and following the guidelines
in [125], it is possible to demonstrate that
˙V = − s T e [ K D + D � RB] s e ≤ 0 (3.23)
It is now possible to prove again the system stability inaLyapunov-Like
sense using the Barbălat’s Lemma. Since
• V is lower bounded
• ˙ V ( s e , ˜ θ v , ˜v ) ≤ 0
• ˙ V ( s e , ˜ θ v , ˜v )isuniformly continuous
then
• ˙ V ( s e , ˜ θ v , ˜v ) → 0as t →∞.
Thus s e → 0 as t →∞.Inview of the definition of s e ,this implies that
˜η → 0 as t →∞.However, the vectors ˜ θ v and ˜v are only guaranteed to be
bounded.
Compensation of the persistent effects. Since the adaptive compensation
term Φ v ( R I B , ν , ν a , ˙ν a ) ˆ θ v operates invehicle-fixed coordinates, the
control law issuited for effective compensation of the restoring moment.
On the other hand, despite the error vector s e is based on earth-fixed
quantities, the current-compensation term ˆv is built through the integral