Underwater Robots - Gianluca Antonelli.pdf

7.8 Output Feedback Control 165
It is worth pointing out that the computation of the desired acceleration
˙ζ d requires knowledge ofthe actual angular velocity ν 2 ;infact, inview of
˙R B
I = − S ( ν 2 ) R B I ,itis
⎡
˙ζ d = ⎣
R B I ¨η 1 ,d − S ( ν 2 ) ν 1 ,d
R B I ˙ν I 2 ,d − S ( ν 2 ) ν 2 ,d
¨q d
⎤
⎦ .
Hence, it is convenient to use in the control law the modified acceleration
vector defined as
⎡
⎤
a d = ⎣
R B I ¨η 1 ,d − S ( ν 2 ,d) ν 1 ,d
R B I ˙ν I 2 ,d − S ( ν 2 ,d) ν 2 ,d
¨q d
⎦ ,
which can be evaluated without using the actual velocity; the two vectors are
related by the equality
˙ζ d = a d + S PO( �ν 2 ,d) ζ d ,
where S PO( · )=blockdiag { S ( · ) , S ( · ) , O n × n } ,and �ν 2 ,d = ν 2 ,d − ν 2 .
Hereafter it is assumed that � ζ d ( t ) �≤ζ dM for all t ≥ 0.
Atracking control law isnaturally based onthe tracking error
⎡ ⎤
e d = ⎣
B
�η 1 ,d
�ε d
�q d
⎦ , (7.34)
where �η B
1 ,d = η B 1 ,d − η B 1 , �q d = q d − q and �ε d is the vector part of the unit
quaternion � Q d = Q − 1 ∗Qd .
It must be noticed that aderivative control action based on(7.34) would
require velocity measurements in the control loop. In the absence of velocity
measurements, asuitable estimate ζ e of the velocity vector has to be considered.
Let also η B 1 ,e and Q e denote the estimated position and attitude ofthe
vehicle, respectively; the estimated joint variables are denoted by q e .Hence,
the following error vector has to be considered
⎡ ⎤
e de = ⎣
B
�η 1 ,de
�ε e
de
�q de
⎦ , (7.35)
where �η B
1 ,de = η B 1 ,d − η B 1 ,e, �q de = q d − q e ,and �ε e
de is the vector part of the
unit quaternion � − 1 Q de = Q e ∗Qd .
In order to avoid direct velocity feedback, the corresponding velocity error
can be defined as
⎡
R B I ˙ �η 1 ,de − S ( ν 2 ,d) �η B ⎤
�ζ de = ⎣
˙�ε e
de
˙�q de
1 ,de
⎦ ,