Underwater Robots - Gianluca Antonelli.pdf

110 6. Kinematic Control of UVMSs
Singularity-Robust Task Priority Redundancy Resolution
Arobust solution tothe occurrence of the algorithmic singularities is based
on the following mapping [78]:
ζ r = J † �
p ( η , q ) ˙x p,d + I N − J † �
p ( η , q ) J p ( η , q ) J † s ( η , q ) ˙x s,d . (6.6)
This algorithm has aclear geometrical interpretation: the two tasks are
separately inverted bythe use of the pseudoinverse of the corresponding Jacobian;
the vehicle/joint velocities associated with the secondary task are
further projected in the null space of the primary task J p .Similarly to [263],
extension to several tasks for highly redundant systems can be easily achieved
by recursive application of (6.6).
Damped Least-Squares Inverse Kinematics Algorithms
The problem of inverting ill-conditioned matrices that might occur with all
the above algorithms can beavoided by resorting to the damped least-square
inverse given by [209]:
J # ( η , q )=J T ( η , q )
�
J ( η , q ) J T ( η , q )+λ 2 � − 1
I m ,
where λ ∈ IR is adamping factor.
In this case, the introduction of adamping factor allows solving the problem
from the numerical point ofview but, on the other hand, itintroduces
areconstruction error in all the velocity components. Better solutions can be
found with variable damping factors ordamped least-squares with numerical
filtering [196, 209].
Closed-Loop Inverse Kinematic Algorithms
The numerical implementation of the above algorithms would lead to a
numerical drift when obtaining vehicle/joint positions by integrating the
vehicle/joint velocities. Aclosed loop version of the above equations can
then be adopted. By considering as primary task the end-effector position/orientation,
(6.6), as an example, would become:
ζ r = J † ( η , q )(˙x E,d + K E e E )+
+
�
I N − J † �
( η , q ) J ( η , q )
J † s ( η , q )(˙x s,d + K s e s ) , (6.7)
where e E and e s are the numerical reconstruction errors and K E ∈ IR m × m
and K s ∈ IR r × r are design matrix gains to be chosen soastoensure convergence
to zero of the corresponding errors.
If the task considered isposition control, its reconstruction error is simply
given by the difference between the desired and the reconstructed values. In