Underwater Robots - Gianluca Antonelli.pdf

2.9 Dynamics ofUnderwater Vehicle-Manipulator Systems 41
� � �
τ v B v
τ = =
τ q
O n × 6
O 6 × n
I n
�
u = Bu, (2.72)
where u ∈ IR p v + n is the vector of the control input. Notice that, while for
the vehicle ageneric number p v ≥ 6ofcontrol inputs is assumed, for the
manipulator it is supposed that n joint motors are available.
It can be proven that:
• The inertia matrix M of the system is symmetric and positive definite:
M = M T > O
moreover, it satisfies the inequality
λ min( M ) ≤�M �≤λ max( M ) ,
where λ min( M )(λ max( M )) is the minimum (maximum) eigenvalue of M .
• Forasuitable choice ofthe parametrization of C and ifall the single bodies
of the system are symmetric, ˙ M − 2 C is skew-symmetric [67]
ζ T � ˙ M − 2 C
which implies
˙M = C + C T
�
ζ =0
moreover, the inequality
� C ( a , b ) c �≤C M � b ��c �
and the equality
C ( a ,α1 b + α 2 c )=α 1 C ( a , b )+α 2 C ( a , c )
hold.
• The matrix D is positive definite
D > O
and satisfies
� D ( q , a ) − D ( q , b ) �≤D M � a − b � .
In [255], it can be found the mathematical model written with respect
to the earth-fixed-frame-based vehicle position and the manipulator endeffector.
However, it must be noted that, in that case, a6-dimensional manipulatorisconsideredinorder
to have square Jacobian to work with; moreover,
kinematic singularities need to be avoided.
Reference [174] reports some interestingdynamic considerations about the
interaction between the vehicle and the manipulator. The analysis performed
allows to divide the dynamics in separate meaningful terms.