Underwater Robots - Gianluca Antonelli.pdf

36 2. Modelling of Underwater Robots
Starting from the equation (2.52) let first consider the current as an external
disturbance τ v,C constant inthe body-fixed frame, itispossible to
write:
M v ˙ν + C v ( ν ) ν + D RB( ν ) ν + Φ v,R ( R I B ) θ v,R + Φ � v,C θ v,C = τ v
that can be rewritten as:
M v ˙ν + C v ( ν ) ν + D RB( ν ) ν + Φ v,P � ( R I B ) θ v,P = τ v
with the use of the (6 × 10) regressor:
Φ v,P � ( R I B )=
� R B I z O3 × 3 I 3 O 3 × 3
0 3 × 1
S
�
R B �
I z
O 3 × 3
I 3
�
.
(2.55)
On the other side the current can be modeled as constant inthe earth-fixed
frame and, merged again with the restoring forces contribution, gives the
following
M v ˙ν + C v ( ν ) ν + D RB( ν ) ν + Φ v,P ( R I B ) θ v,P = τ v
with the use of the (6 × 9) regressor:
Φ v,P ( R I B )=
� O 3 × 3 R B I O 3 × 3
S
�
R B �
I z
O 3 × 3
R B I
�
.
(2.56)
It is worth noticing that the two regressors have different dimensions.
In order to extrapolate the minimum number of independent parameters,
i.e., the number of columns of the regressor, it is possible to resort to the
numerical method proposed by Gautier [135] based onthe Singular Value
Decomposition.
Model (2.56) can by rewritten inasole regressor of proper dimension
yielding:
Φ v,T ( R I B , ν , ˙ν ) θ v,T = τ v . (2.57)
2.8 Kinematics of Manipulators with Mobile Base
In Figure 2.3 asketch ofanUnderwater Vehicle-Manipulator System with
relevant frames isshown. The frames are assumed tosatisfy the Denavit-
Hartenberg convention [254]. The position and orientation ofthe end effector,
thus, is easily obtained by the use of homogeneous transformation matrices.
Let q ∈ IR n be the vector of joint positions where n is the number of
joints. The vector ˙q ∈ IR n is the corresponding time derivative. Let define
ζ ∈ IR 6+n as