Underwater Robots - Gianluca Antonelli.pdf

58 3. Dynamic Control of 6-DOF AUVs
The function f ∈ IR 6 collecting the elements f i is basically asaturated function.
Guidelines in its selection are given in [38].
The proposed controller is given by:
τ v = K P J T e ( R I B ) f ( ˜η ) − K D ν + Φ v,R ( R I B ) ˆ θ v,R
(3.38)
where K P and K D are positive definite matrices, selected as scalar in [280]
and
˙ˆθ
− 1
v,R = K θ Φ T v,R ( R I �
B ) ν + α J T e ( R I �
B ) f ( ˜η )
(3.39)
where α>0isascalar gain.
It can be noticed that the proportional action is similar to the action
proposed by Fossen and Balchen in Section 3.5. The derivative action is, on
the contrary, based onthe vehicle-fixed variable ν that is different from the
derivative action developed by Fossen and Balchen based on the term J T e ˙˜η .
Finally they both have an adaptive action that is already reduced to the sole
restoring terms in [280].
It is worth noticing that the Authors propose scalar gains K P , K D and
α for the controller. There is amain problem related with the fact that the
position and orientation variables have different unit measures; using the
same gains might force the designer to tune the performance to the lower
bandwidth.
Compensation ofthe persistent effects. The controller proposed by the
Authors compensates for the gravity inthe vehicle-fixed frame. However,
the update law isbased onthe transpose of the Jacobian, and not onits
inverse; the mapping, thus, is not exact and acoupling among the error
directions is experienced. This has as aconsequence that this control law
is not suitable for the restoring force compensation. Itisworth noticing,
moreover, that, differently from the industrial manipulator case, inversion of
the Jacobian is not computational demanding since, being J e asimple (6×6)
− 1
matrix (see eq. (2.19)), its inverse J e can be symbolically computed; from
acomputational aspect, thus, there is no difference in using J T − 1
e or J e .By
visual inspection of J T − 1
e and J
e it can be observed that the difference is in
the rotational part J k,o ,being the transpose of arotation matrix equal to
its inverse. Inparticular, using the common definition that roll pitch anyaw
means the use of elementary rotation around x , y and z in fixed frame [254],
the corresponding matrix J k,o is not function of the yaw angle, incase of null
pitch angle it is J T − 1
k,o = J k,o ,and it is singular for apitch angle of ± π/2; close
to that singularity the numerical difference between J T − 1
k,o and J k,o increases.