Underwater Robots - Gianluca Antonelli.pdf

2.2 Rigid Body’s Kinematics 17
The matrix J k,o ∈ IR 3 × 3 can be expressed in terms of Euler angles as:
⎡
J k,o ( η 2 )= ⎣ 1 0 − s ⎤
θ
0 c φ c θ s ⎦ φ , (2.3)
0 − s φ c θ c φ
where c α and s α are short notations for cos( α )and sin(α ), respectively. Matrix
J k,o ( η 2 )isnot invertible for every value of η 2 .Indetail, it is
− 1
J k,o ( η 2 )= 1
c θ
⎡
⎣ 1 s φ s θ c φ s θ
0 c φ c θ − c θ s φ
0 s φ c φ
⎤
⎦ , (2.4)
that it is singular for θ =(2l +1) π
2 rad, with l ∈ IN,i.e., for apitch angle
of ± π
2 rad.
The rotation matrix R B I ,needed in (2.1)totransform the linearvelocities,
is expressed in terms of Euler angles by the following:
R B ⎡
⎤
c ψ c θ s ψ c θ − s θ
I ( η 2 )= ⎣ − s ψ c φ + c ψ s θ s φ c ψ c φ + s ψ s θ s φ s φ c ⎦ θ . (2.5)
s ψ s φ + c ψ s θ c φ − c ψ s φ + s ψ s θ c φ c φ c θ
Table 2.1 shows the common notation used for marine vehicles according
to the SNAME notation ([272]), Figure 2.1 shows the defined frames and the
elementary motions.
2.2.2 Attitude Representation by Quaternion
To overcome the possible occurrence of representation singularities it might
be convenient to resort to non-minimal attitude representations. One possible
choice isgiven by the quaternion. The term quaternion was introduced by
Hamilton in1840, 70 years after the introduction ofafour-parameter rigidbody
attitude representation by Euler. In the following, ashort introduction
to quaternion is given.
By defining the mutual orientation between two frames ofcommon origin
in terms of the rotation matrix
R k ( δ )=cosδ I 3 +(1 − cosδ ) kk T − sinδ S ( k ) ,
where δ is the angle and k ∈ IR 3 is the unit vector of the axis expressing the
rotation needed to align the two frames, I 3 is the (3 × 3) identity matrix,
S ( x )isthe matrix operator performing the cross product between two (3×1)
vectors
⎡
S ( x )= ⎣ 0 − x ⎤
3 x 2
⎦ , (2.6)
x 3 0 − x 1
− x 2 x 1 0
the unit quaternion is defined as