Underwater Robots - Gianluca Antonelli.pdf

108 6. Kinematic Control of UVMSs
ζ r = J † ( η , q ) ˙x E,d , (6.1)
where ˙x E,d is the end-effector task defined in (2.68) and
J † ( η , q )=J T �
( η , q ) J ( η , q ) J T � − 1
( η , q ) .
This solution corresponds to the minimization of the vehicle/joint velocities
in aleast-square sense [254], i.e., of the function:
E = 1
2 ζ T ζ .
Notice that subscript r in ζ r stands for reference value ,meaning that those
velocities are the desired values for the low-level motion control of the manipulator
(see also Figure 6.1, where aclosed-loop inverse kinematics, detailed
in next Subsections, is sketched). It is possible to minimize aweighted norm
of the vehicle/joint velocities
E = 1
2 ζ T Wζ
leading to the weighted pseudoinverse:
J †
W = W − 1 �
T
J JW − 1 J T � − 1
. (6.2)
With this approach, however, the problem ofhandling kinematic singularities
is not addressed and their avoidance cannot be guaranteed.
˙x E,d
x E,d
e E K E
+
+
IK
k ( · )
ζ r � J − 1 ( · )
� �� �
Kinematic Loop
η r , q r
+ Motion
− Control
UVMS
� �� �
Dynamic Loop
Fig. 6.1. Kinematic and dynamic loops. The block labeled e E is defined by equation
(6.8)
Augmented Jacobian
Another approach toredundancy resolution is the augmented Jacobian [107].
In this case, aconstraint task is added to the end-effector task as to obtain
asquare Jacobian matrix which can be inverted.
The main drawback of this technique is that new singularities might arise
in configurations in which the end-effector Jacobian J is still full rank. Those
η , q