Underwater Robots - Gianluca Antonelli.pdf

42 2. Modelling of Underwater Robots
2.9.1 Linearity inthe Parameters
UVMS have aproperty that iscommon to most mechanical systems, e.g., serial
chain manipulators: linearity inthe dynamic parameters. Using asuitable
mathematical model for the hydrodynamic forces, (2.71) can be rewritten in
amatrix form that exploits this property:
Φ ( q , R I B , ζ , ˙ ζ ) θ = τ (2.73)
with Φ ∈ R (6+n ) × n θ ,being n θ the total number ofparameters. Notice that n θ
depends on the model used for the hydrodynamic generalized forces and joint
friction terms. For asingle rigid body the number of dynamic parameter n θ,v
is anumber greater than 100 [127]. For an UVMS it is n θ =(n +1) · n θ,v ,
that gives an idea of the complexity ofsuch systems.
Differently from ground fixed manipulators, in this case the number of
parameters can not be reduced because, due tothe 6degrees of freedom
(DOFs) of the sole vehicle, all the dynamic parameters provide an individual
contribution to the motion.
2.10 Contact with the Environment
If the end effector of arobotic system is in contact with the environment,
the force/moment atthe tip ofthe manipulator acts onthe whole system
according to the equation ([254])
M ( q ) ˙ ζ + C ( q , ζ ) ζ + D ( q , ζ ) ζ + g ( q , R I B )=τ + J T w ( q , R I B ) h e , (2.74)
where J w is the Jacobian matrix defined in (2.67) and the vector h e ∈ IR 6 is
defined as
� �
f e h e =
µ e
i.e., the vector of force/moments at the end effector expressed in the inertial
frame. If it is assumed that only linear forces act on the end effector
equation (2.74) becomes
M ( q ) ˙ ζ + C ( q , ζ ) ζ + D ( q , ζ ) ζ + g ( q , R I B )=τ + J T pos( q , R I B ) f e
(2.75)
Contact between the manipulator and the environment is usually difficult
to model. In the following the simple model constituted by africtionless and
elastically compliant plane will be considered. The force at the end effector is
then related to the deformation of the environmentbythe following simplified
model [79] (see Figure 2.5)