Underwater Robots - Gianluca Antonelli.pdf

148 7. Dynamic Control of UVMSs
The equations of motion can be written as:
− 1
¨η = M 11 ( τ � v − n � ) − M 12 ( τ q − n q ) (7.5)
− 1
¨q = M 22 ( τ q − n q ) − M T 12 ( τ � v − n � ) (7.6)
where the nonlinear terms of the equations of motion have been collected in
a(6+n )dimensional vector:
� �
� n
n =
n q
with n � ∈ IR 6 and n q ∈ IR n .
In [67] the following controller is assumed for the vehicle:
τ � v =
and for the manipulator
τ q = ˆ M q
�
�
ˆM
v + ˆ ��
M qq ¨η d + k vv ˙˜η
�
+ k pv ˜η + ∆ τ � v , (7.7)
�
¨q d + k vq ˙˜q
�
+ k pq˜q
(7.8)
where, as usual, the symbol hat: ˆ· denotes anestimate, positive definite in
this case, of the corresponding matrix and the tilde: ˜· represents the error
defined asthe desired minus the current variable. The scalar gains are chosen
so that:
k vv =2ξω0 v ,
k pv = ω 2 0 v ,
k vq =2ξω0 q ,
k pq = ω 2 0 q ,
i.e., they are defined by the damping ratio and the natural frequency ofthe
linearized model. The bandwidth ratio
ε = ω 0 v
ω 0 q
� 1
is given by the closed-loop bandwidths the the vehicle and the manipulator
and itissmall due tothe dynamics of the two subsystems. The additional
control action ∆ τ � v can be chosen indifferent ways, in [69] apartial singular
perturbed model-based compensation and arobust non-linear control have
been proposed.
Asingular perturbation analysis can be carried out when ε is small. It
can be demonstrated the the manipulator is not affected by the slow vehicle
dynamics and that, after the fast transient, the approximated model has an
error O ( ε ). On the other side, the vehicle dynamics is strongly affected by