Underwater Robots - Gianluca Antonelli.pdf

52 3. Dynamic Control of 6-DOF AUVs
• ˙ V ( s v , ˜ θ v ) ≤ 0
• ˙ V ( s v , ˜ θ v )isuniformly continuous
then
• ˙ V ( s v , ˜ θ v ) → 0as t →∞.
Thus s v → 0 as t →∞.Inview of the definition of s v ,this implies that
˜ν → 0 as t →∞;inaddition, due to the properties of the quaternion, it
results that ˜η → 1as t →∞.However, as usual inadaptive control schemes,
it is not possible to prove asymptotic stability ofthe whole state since ˜ θ v is
only guaranteed to be bounded.
Notice that, in [121], further discussion is developed by considering different
choices for the matrix Λ .
In 1997 [89] G.Conte and A.Serrani develop aLyapunov-based control
for AUVs. The designed controller, by introducing arepresentation ofthe
model uncertainties, ismade robust using Lyapunov techniques. It is worth
noticing that this approach does not take into account explicitly for an adaptive/integral
action tocompensate for the current effect. The position error is
represented within avehicle-fixed representation with afeedback term similar
to the vector s v used inthis Section.
Compensation ofthe persistent effects. Following the same reasoning
as previously done for the law A ,itcan be deducted that also in this case
the adaptive compensation cannot guarantee null position error at rest in
the presence of ocean current. In fact, the dynamic model (2.51) shows that
asteady null linear velocity ofthe vehicle with anon-null position error
does not excite acorrective adaptive control action. Again, ocean current
measurement cannot overcome this problem inpractice.
To achieve anull position error in presence of ocean current, an integral
action on body-fixed-frame error variables was considered in[34, 35]. However,
this integral action has the following drawback: let suppose that the
vehicle is at rest inthe left configuration of Figure 3.2 in presence ofawater
current aligned to the earth-fixed y axis; the control action builds the current
compensation term which, in this particular configuration, turns out to be
parallel to the y b axis. If the vehicle is now quickly rotated toright configuration,
the built compensation term rotates together with the vehicle keeping
its alignment tothe y b axis; however, this vehicle-fixed axis has now become
parallel to the x axis ofthe earth-fixed frame. Therefore, the built compensation
term acts as adisturbance until the integral action has re-built proper
current compensation for the right configuration. It is clear at this point that
this drawback does not arise for the controller E and for the controllers A
and B ,since they build ocean current compensation in an earth-fixed frame.
Reduced Controller. Similarly to the other cases, areduced form of the
controller has been derived. An integral action on body-fixed-frame error