Underwater Robots - Gianluca Antonelli.pdf

J T k,oqJ k,oq = 1
4 I 3 ,
that allows to invert the mapping (2.10) yielding:
ν 2 =4J T � �
˙ε
k,oq .
˙η
2.2 Rigid Body’s Kinematics 19
For completeness the rotation matrix R B I ,needed to compute (2.1), in
terms of quaternion is given:
R B ⎡
I ( Q )= ⎣ 1 − 2(ε 2 2 + ε 2 3 ) 2( ε 1 ε 2 + ε 3 η ) 2( ε 1 ε 3 − ε 2 η )
2(ε 1 ε 2 − ε 3 η ) 1− 2(ε 2 1 + ε 2 3 ) 2( ε 2 ε 3 + ε 1 η )
2(ε 1 ε 3 + ε 2 η ) 2( ε 2 ε 3 − ε 1 η ) 1− 2(ε 2 1 + ε 2 ⎤
⎦ . (2.11)
2 )
2.2.3 Attitude Error Representation
Let now define R I B ∈ IR 3 × 3 as the rotation matrix from the body-fixed frame
to the earth-fixed frame, which isalso described by the quaternion Q ,and
R I d ∈ IR 3 × 3 the rotation matrix from the frame expressing the desired vehicle
orientationtothe earth-fixedframe,whichisalso described by the quaternion
Q d = { ε d ,ηd } .One possible choice for the rotation matrix necessary to align
the two frames is
T
�R
I
= R B R I d = R B I R I d ,
where R B I = R I T
B . The quaternion Q � = { ˜ε , ˜η } associated with R � can
be obtained directly from � R or computed bycomposition (quaternion product):
� Q = Q − 1 ∗Qd ,where Q − 1 = {−ε ,η} :
˜ε = η ε d − η d ε + S ( ε d ) ε , (2.12)
˜η = ηηd + ε T ε d . (2.13)
Since the quaternion associated with � R = I 3 (i.e. representing two aligned
frames) is � Q = { 0 , 1 } ,itissufficient to represent the attitude error as ˜ε .
The quaternion propagation equations can be rewritten also in terms of
the error variables:
˙˜ε = 1
2 ˜η ˜ν 2 + 1
2 S ( ˜ε ) ˜ν 2 , (2.14)
˙˜η = − 1
2 ˜ε T ˜ν 2 , (2.15)
where ˜ν 2 = ν 2 ,d − ν 2 is the angular velocity error expressed in body-fixed
frame. Defining
� �
˜ε
z = ,
˜η
the relations in (2.14)–(2.15) can be rewritten inthe form: