Underwater Robots - Gianluca Antonelli.pdf

6.2 Kinematic Control 109
singularities, named algorithmic singularities, occur when the additional task
does cause conflict with the end-effector task.
Asimilar approach, with the same drawback, isthe extended Jacobian
approach.
Task Priority Redundancy Resolution
By solving (2.68) in terms of aminimization problem of the quadratic cost
function ζ T ζ the general solution [191] is given:
ζ r = J † �
( η , q ) ˙x E,d + I N − J † �
( η , q ) J ( η , q ) ζ a , (6.3)
where N =6+ n and ζ a ∈ IR 6+n is an arbitrary � vehicle/joint velocity vector.
It can be recognized that the operator I N − J † �
( η , q ) J ( η , q )
projects
ageneric joint velocity vector in the null space of the Jacobian matrix. This
corresponds to generating an internal motion ofthe manipulator arm that
does not affect the end-effector motion.
Solution (6.3) can be seen in terms of projection of asecondary task,
described by ζ a ,inthe null space of the higher priority primary task, i.e., the
end-effector task. Afirst possibility istochoose the vector ζ a as the gradient
of ascalar objective function H ( q )inorder to achieve alocal minimum [191]:
ζ a = − k H ∇ H ( q ) , (6.4)
where k H is ascalar gain factor. Another possibility istochose aprimary
task x p,d ∈ IR m m × (6+n )
and acorrespondent Jacobian matrix J p ( q ) ∈ IR
˙x p,d = J p ( q ) ζ .
and todesign asecondary task x s,d ∈ IR r and acorrespondent Jacobian
matrix J s ( q ) ∈ IR r × (6+n ) :
˙x s,d = J s ( q ) ζ .
for which the vector of joint velocity isthen given by [195, 210]:
ζ r = J † p ˙x � �
p,d + J s I N − J † �� † �
p J p ˙x s,d − J s J † p ˙x �
p,d . (6.5)
However, for this solution too, the problem ofthe algorithmic singularities
still remains unsolved. In this case, it is possible to experience an � algorithmic
singularity when J s and J p are full rank but the matrix J s I N − J † �
p J p
looses rank. Extension of the approach to several tasks for highly redundant
systems can be achieved bygeneralization of (6.5), as described in [263].