Underwater Robots - Gianluca Antonelli.pdf

5. compute the quaternion Q by the following:
[ ε η ] T = 1
2 [ c 1 c 2 c 3 c 4 ] T .
Quaternion from Euler Angles
2.2 Rigid Body’s Kinematics 21
The transformation from Euler angles toquaternion isalways possible, i.e.,
it is not affected by the occurrence of representation singularities [127]. This
implies that the use of quaternion to control underwater vehiclesiscompatible
with the common use of Euler angles toexpress the desired trajectory of the
vehicle.
The algorithm consists in computing the rotation matrix expressed in
Euler angles by(2.5) and using the procedure described in the previous subsection
to extract the corresponding quaternion.
2.2.4 6-DOFs Kinematics
It is useful to collect the kinematic equations in 6-dimensional matrix forms.
Let us define the vector η ∈ IR 6 as
� �
η 1 η =
(2.17)
η 2
and the vector ν ∈ IR 6 as
� �
ν 1
ν = , (2.18)
ν 2
and by defining the matrix J e ( R I 6 × 6
B ) ∈ IR
J e ( R I � B
R I
B )=
O 3 × 3
O 3 × 3
J k,o
�
, (2.19)
where the rotation matrix R B I given in (2.5) and J k,o is given in (2.3), it is
ν = J e ( R I B ) ˙η . (2.20)
The inverse mapping, given the block-diagonal structure of J e ,isgiven by:
− 1
˙η = J e ( R I � �
I
R B O 3 × 3
B ) ν =
− 1 ν , (2.21)
J
− 1
where J k,o
O 3 × 3
k,o
is given in (2.4).
On the other side, it is possible to represent the orientation by means of
quaternions. Let us define the vector η q ∈ IR 7 as
⎡
η q = ⎣ η ⎤
1
ε ⎦ (2.22)
η