Underwater Robots - Gianluca Antonelli.pdf

3.8 Comparison Among Controllers 59
The ocean current isnot taken into account. Using the controller (3.38)–
(3.39) under the effect ofthe current would lead to an error different from
zero at steady state.
Reduced Controller. The aim of the Authors is to propose already acontroller
that matches the definition of reduced controller that has been given
here. The absence of compensation for the current, however, makes the controller
not appealing for practical implementation. The proposed reduced
controller, modified to take into account atracking problem, is given by:
τ v = K P J T e ( R I B ) ˜η + K D ˜ν + Φ v,P ˆ θ v,P
˙ˆθ
− 1
v,P = K θ Φ T �
v,P ˜ν + ΛJ T e ( R I �
B ) ˜η
(3.40)
(3.41)
where Λ , K P and K D are positive definite matrices, selected at least as
block-diagonal matrices to keep different dynamics for the position and the
orientation.
It is worth noticing that the reduced controller derived isdifferent from
the original controller, that would notreachanull steady state error under the
effect of the current. Also, the regressor Φ v,P embeds the gravity regressor as
proposed by Sun and Cheah but the drawback in the restoring compensation
still exist due tothe not proper update law ofthe parameters.
3.8 Comparison Among Controllers
Foreasy of readings, Table 3.1 reports the label associated with each controller.
The controllers developed by the researchers are quite different one each
other, inthis Section aqualitative comparison with respect to the compensation
performed by the controllers with respect to the persistent dynamic
effects is provided.
3.8.1 Compensation ofthe Restoring Generalized Forces
The controller A is totally conceived in the earth-fixed frame and it is modelbased.
In case of perfect knowledge ofthe restoring-related dynamic parameters,
thus, thereisnot needtocompensatefor therestoring generalized forces.
This is, however, unpractical in areal situation; in case of partial compen-
sation ofthe vector g � RB
,infact, this controller experiences the drawback
discussed inSection 3.2. Asimilar drawback is shared from the controller B
that, again, is totally based on earth-fixed variables, the integral actions,
thus, does not compensate optimally for the restoring action. The controller
F compensates the restoring force in the vehicle-fixed frame; the update
law, however, is not based onthe exact mapping between orientation and
moments resulting inanot clean adaptive action. The effect, however, is as
significant asthe vehicle works with alarge pitch angle. The remaining 3controllers
properly adapt the restoring-related parameters in the vehicle-fixed
frame.