Underwater Robots - Gianluca Antonelli.pdf

112 6. Kinematic Control of UVMSs
6.3 The Drag Minimization Algorithm
In 1999 N. Sarkar and T.K. Podder ([249, 250]) suggest to use the system
redundancy in order to minimize the total hydrodynamic drag. Roughly
speaking the proposed kinematic control generates a coordinate vehicle/manipulator
motion so that the resulting trajectory incrementally reduce
the total drag encountered by the system.
The second-order version of (6.3) is given by
˙ζ r = J † ( η , q )
�
˙x E,d − ˙ J ( η , q ) ζ
�
+
�
I N − J † ( η , q ) J ( η , q )
� ˙ζ a , (6.10)
where, again, the vector ˙ ζ a is arbitrary and can be used tooptimize some
performance criteria that can be chosen, similarly to eq. (6.4), as
˙ζ a = − k H ∇ H ( q ) . (6.11)
The Authors propose the following scalar objective function
H ( q )=D T ( q , ζ ) WD( q , ζ ) (6.12)
where D ( q , ζ )isthe damping matrix and W ∈ IR N × N is apositive definite
weight matrix. Byproperly selecting the weight matrix it is possible to shape
the influence of the drag of the individual components on the total system’s
drag. Apossible choice for W is adiagonal matrix [250].
The method istested in detailed simulations where the drag coefficients
are supposed to be known. As noticed by the Authors, in practical situations,
the drag coefficient need to be identified and this can not be an easy task; it
must be noted, however, that theoretical drag coefficient are available in the
literature for the most common shapes. Asafirst approximation it is possible
to model the vehicle as an ellipsoid and the manipulator arms as cylinders.
The proposed method, thus, even being approximated, provides information
of wider interest.
It is worth noticing that drag minimization has been the objective of
several approaches, even not based on kinematics control, such as, e.g., [160,
161]; in [164], the Authors propose to utilize agenetic algorithm to be trained
over aperiodic motion in order to estimate the trajectory’s parameters that
minimize the directional drag force in the task space.
6.4 The Joint Limits Constraints
N. Sarkar, J.Yuh and T.K. Podder, in 1999 ([252]) take into account the
problem ofhandling the manipulators’ joints limits.
The Authors propose to suitably modifying the weight matrix W in
Eq. (6.2). In detail, W is chosen asadiagonal matrix the entries of which
are related to aproper function. The approach might also take into account