Underwater Robots - Gianluca Antonelli.pdf

50 3. Dynamic Control of 6-DOF AUVs
τ � v = K 1 ¨η d + K 2 ˙η + K 3 + K 4 ˙˜η + K 5 ˜η =
5�
K i φ i , (3.9)
i =1
where ˜η = η d − η ,the gains K i ∈ IR 6 × 6 are computed as
where
K i = ˆγ i s e φ T i
� s e ��φ i �
i =1,...,5 ,
s e = ˙˜η + σ ˜η with σ>0 ,
and the factors ˆγ i ’s are updated by
˙ˆγ i = f i � s e ��φ i � with f i > 0 i =1,...,5 .
Please notice that φ 3 = k ∈ IR 6 ,i.e., apositive constant vector.
Experimental results in 6-DOFs onthe use of (3.9) are reported in [215,
216, 323]; these have proven the effectiveness ofthis controller starting from
the surface (with null initial gains) where asmooth version of the controller
has been implemented.
The stability analysis can be performed using Lyapunov-like arguments
starting from the function
V = 1
2 ˜η T ˜η + 1
5� 1
( γ i − ˆγ i )
2
2
f
i =1
i
that, differentiated with respect to the time yields anegative semi-definite
scalar function; details can be found in [323].
Recently, in[323], Zhao and Yuh propose amodel-based version of this
control law. The eventual knowledge ofthe vehicle dynamics is exploited by
implementing a disturbance observer in charge of partiallycompensatefor the
system dynamics. Interesting experimental results with the vehicle ODIN are
reported where the tuning of the gains has been achieved on the single DOFs
independently. This has been made possible in view of the specific shape
of the vehicle, close to asphere, that reduces the coupling effects and the
difference among the directions. It is worth noticing that the considerations
below are valid also for this version of the controller.
Compensation ofthe persistent effects. Since the tracking errors are
defined in the earth-fixed frame, similar considerations to those developed
for the law A can be done with respect to compensation of both restoring
generalized forces and ocean current effects.
Reduced Controller. Being the sole non-model-based law, the controller
presented in [320] is characterized by an error behavior much different from
that obtained by the other considered controllers and afair comparison is
made difficult. Forthis reason, areduced controller has not been retrieved.