Underwater Robots - Gianluca Antonelli.pdf

6.5 Singularity-Robust Task Priority 113
the vehicle position, but, for seek of simplicity, let us consider only the joints
positions by defining
n� 1 q i,max − q i,min
H ( q )=
c i ( q i,max − q i )(q i − q i,min)
i =1
with c i > 0and the subscript max and min that obviously denotes the two
joint limits. This function ([252]) inherits the concepts developed in [44].
Its partial derivative with respect to the joint positions isgiven by
∂H( q )
∂qi
= 1 ( q i,max − q i,min)(2q i − q i,max − q i,min)
c i ( q i,max − q i ) 2 ( q i − q i,min) 2 .
The elements of the weight matrix are then defined as
� �
�
W i,i =1+ �
∂H( q ) �
�
� ∂qi
� ,
in fact, it can be easily observed that the element goes to1when the joint
is in the center of its allowed range and goes toinfinity when the joint is
approaching its limit.
As afurther improvement itispossible to relate the weight also to the
direction of the joint bydefining
⎧ � � � �
�
⎪⎨
1+ � ∂H( q ) � �
�
� ∂qi � if ∆ � ∂H( q ) �
�
� ∂qi � > 0
W i,i =
� �
⎪⎩
�
1 if∆�∂H( q ) �
�
� ∂qi � ≤ 0
In Section 6.6, the joint limits are part of anumber oftasks handled with
afuzzy approach. In [163], B.H. Jun, P.M. Lee and J. Lee propose afirst
order task priority approach with the optimization ofspecific cost functions
developed for the ROV named KORDI. The joints constraints are taken into
account also.
6.5 Singularity-Robust Task Priority
To achieve an effective coordinated motion ofthe vehicle and manipulator
while exploiting the redundant degrees of freedom available, G.Antonelli
and S.Chiaverini, in[20], resort tothe singularity-robust task priority redundancy
resolution technique. The velocity vector ζ r is then computed as
shown in (6.7).
In the case of aUVMS, the primary task vector will usually include the
end-effector task vector, while the secondary task vector might include the
vehicle position coordinates. This choice isaimed at achieving station keeping
of the vehicle as long as the end-effector task can be fulfilled with the sole
manipulator arm. It is worth noticing that this approach isconceptually