Underwater Robots - Gianluca Antonelli.pdf

188 7. Dynamic Control of UVMSs
• ˙ V ≤ 0;
• ˙ V is uniformly continuous.
then
• ˙ V → 0as t →∞.
Thus s 0 0 ,...,s n n → 0 as t →∞.Due to the recursive definition of the vectors
s i i ,the position errors converge tothe null value as well. In fact, the
vector s 0 0 → 0 implies ν 0 0 → 0 and e 0 → 0 ( λ p,0 > 0and λ o,0 > 0). Moreover,
since the rotation matrix has full rank, the vehicle position error ˜η 1 too
decreases tothe null value. Convergence tozero of s i i directly implies convergence
to zero of s q,1 ,...,sq,n ;consequently, ˙˜q → 0 and ˜q → 0 .Asusual in
adaptive control technique, it is not possible to prove asymptotic stability of
the whole state, since ˜ θ i is only guaranteed to be bounded.
Remarks.
• Achieving null error of n +1 rigid bodies with 6-DOF with only 6+n
inputs is physically coherent since the control law iscomputed bytaking
into account the kinematic constraints of the system.
• In [325] the control law performs an implicit kinematic inversion. This
approach requires to work with a6-DOF robot for the stability analysis of
aposition/orientation control of the end effector. In case of aredundant
robot, the Authors suggest the implementation ofanaugmented Jacobian
approach in order to have asquare Jacobian to work with. However, the
possible occurrence of algorithmic singularities is not avoided (see also
Section 6.2). On the other hand, ifthe kinematic control is kept separate
from the dynamic loop, as in the discussed approach, it is possible to
use inverse kinematic techniques robust to the occurrence of algorithmic
singularities.
7.9.2 Simulations
Dynamic simulations have been performed toshow the effectiveness ofthe discussed
control law based onthe simulation tool described in [23]. The vehicle
data are taken from [145] and are referred to the experimental Autonomous
Underwater Vehicle NPS Phoenix. For simulation purposes, asix-degree-offreedom
manipulator with large inertia has been considered which ismounted
under the vehicle’s body. The manipulator structure and the dynamic parameters
are those of the Smart-3S manufactured by COMAU. Its weight is
about 5%ofthe vehicle weight, its length is about 2m stretched while the
vehicle is 5m long. The overall structure, thus, has 12degrees of freedom.
The relevant physical parameters of the system are reported in Table 7.5.
Notice one of the features of the controller: by changing the manipulator
does not imply redesigning the control but itsimply modifies the Denavit-
Hartenberg table and the initial estimate ofthe dynamic parameters. As a