Underwater Robots - Gianluca Antonelli.pdf

154 7. Dynamic Control of UVMSs
Let consider the Lyapunov function candidate
V = ˜ε T ˜ε . (7.19)
The time derivative of V is:
˙V =2˜ε T ˙˜ε = − ˜ε T ˜η ν 2 − ˜ε T S ( ˜ε ) ν 2 . (7.20)
Plugging (7.17) into (7.20) and taking Λ o = λ o I 3 with λ o > 0, gives
˙V = − ˜ηλo ˜ε T ˜ε − λ o ˜ε T S ( ˜ε ) ˜ε = − ˜ηλo ˜ε T ˜ε .
which isnegative semidefinite with ˜η ≥ 0. It must be noted that, in view
of (7.18), ˜η is anot-decreasing function oftime and thus it stays positive
when starting from apositive initial value.
The set R of all points ˜ε where ˙ V =0is given by
R = { ˜ε = 0 , ˜ε :˜η =0} ;
from (2.7), however, it can berecognized that
˜η =0 ⇒ �˜ε � =1
and thus ˙˜η>0inview of (7.18). Therefore, the largest invariant set in R is
M = { ˜ε = 0 }
and the invariant set theorem ensures asymptotic convergence tothe origin.
Manipulator joint error dynamics. The manipulator joint error dynamics
on the sliding manifold is described bythe equation
− ˙q + Λ q ˜q = 0
whose convergence to ˜q = 0 is evident taking Λ q > O .
7.6.2 Simulations
Dynamic simulations have been performed inorder to show the effectiveness
of the discussed control law. The UVMS simulator was developed in the
Matlab c� / Simulink c� environment.
Forthis simulations, the vehicle data are taken from [145]; they refer to
the experimental Autonomous Underwater Vehicle NPS Phoenix. Atwo-link
manipulator with rotational joints has been considered which ismounted
under the vehicle body with the joint axes parallel to the fore-aft direction;
since the vehicle inertia along that axis is minimum, this choice increases
dynamic coupling between the vehicle and the manipulator. The length of
eachlink is 1m,the center of gravityiscoincidentwith the center of buoyancy
and itissupposed tobeinthe geometrical center of the link; each link is not
neutrally buoyant. Dry and viscous joint frictions are also taken into account.
As for the control law, implementation of(7.10) was considered; however,
it is well known that the sign function would lead to chattering inthe system.