Underwater Robots - Gianluca Antonelli.pdf

7.8 Output Feedback Control 167
where the matrix L p = blockdiag { l pP I 3 ,lpOI 3 , L pQ} is diagonal positive
definite. The matrix L v =blockdiag { L vP ,lvOI 3 , L vQ} is symmetric and po-
sitive definite, and
A ( � Q e )=blockdiag
�
I 3 , E ( � �
Q e ) / 2 , I n .
The estimated quantities η 1 ,e and q e are computed byintegrating the corresponding
estimated velocities ˙η 1 ,e = R I B ν 1 ,e and ˙q e ,respectively, whereas
theestimated orientation Q e is computed from the estimated angular velocity
ν I 2 ,e = R I B ν 2 ,e via the quaternion propagation rule.
Implementation issues. Implementation ofthe above controller-observer
scheme (7.38),(7.40), requires computation of the dynamic compensation
terms. While this can be done quite effectively for the terms related to rigid
body dynamics, the terms related tohydrodynamic effects are usually affected
by some degree of approximation and/or uncertainty. Besides the use of
adaptive control schemes aimed at on-line estimation ofrelevant model parameters,
e.g. [22, 197, 314], it is important to have an estimate ofthe main
hydrodynamic coefficients.
An estimate of the added mass coefficients can be obtained via strip
theory [127]. Arough approximation of the hydrodynamic damping isobtained
byconsidering only the linear skin friction and the drag generalized
forces.
Another important point concerns the computational complexity associated
with dynamic compensation against the limited computing power typically
available on board. This might suggest the adoption of acontrol law
computationally lighter than the one derived above. Areasonable compromise
between tracking performance and computational burden is achieved
if the compensation of Coriolis, centripetal and damping terms is omitted
resulting in the controller
u = B † � M ( q ) a r + K v ( ζ r − ζ o )+K p e d + g ( q , R I B ) � , (7.41)
with the simplified observer
�
˙z = M ( q ) a r − � L p + L v A ( � �
Q e ) Λ e e e + K p e d
ζ = M − 1 ( q )(z − L v e e ) − Λ e e e ,
(7.42)
The computational load can be further reduced if asuitable constant
diagonal inertia matrix � M is used in lieu of the matrix M ( q ), i.e.,
u = B † � � Mar + K v ( ζ r − ζ o )+K p e d + g ( q , R I B ) � , (7.43)
with the observer
�
˙z = Mar
� − � L p + L v A ( � �
Q e ) Λ e e e + K p e d
�ζ = � − 1
M ( z − L v e e ) − Λ e e e ,
(7.44)