Underwater Robots - Gianluca Antonelli.pdf

y
z
2.8 Kinematics of Manipulators with Mobile Base 37
y b
x
x 1
Fig. 2.3. Sketch ofanUnderwater Vehicle-Manipulator System with relevant frames
⎡
ζ = ⎣ ν ⎤
1
ν ⎦ 2 ;
˙q
It is useful to rewrite the relationship between body-fixed and earth-fixed
velocities given in equations (2.1)-(2.2) inamore compact form:
⎡
ζ = ⎣ ν ⎤ ⎡
1
ν ⎦ 2 = ⎣
˙q
R B ⎤ ⎡ ⎤ ⎡ ⎤
I O 3 × 3 O 3 × n ˙η 1 ˙η 1
⎦ ⎣ ˙η ⎦
2 = J ⎣ k ˙η ⎦
2 , (2.58)
˙q
˙q
z b
y 1
O 3 × 3 J k,o ( R B I ) O 3 × n
O n × 3 O n × 3 I n
where O n 1 × n 2 is the null ( n 1 × n 2 )matrix and the matrix J k,o ( R B I )interms
of Euler angles has been introduced in (2.3).
z 1
x b
z n
y n
x n