Underwater Robots - Gianluca Antonelli.pdf

230 9. Coordinated Control of Platoons of AUVs
σ a = f a ( η 1 , 1 ,...,η 1 ,n) = 1
n
n�
i =1
η 1 ,i = p , (9.6)
whose Jacobian matrix J a ∈ IR 3 × 3 n can be easily derived [28].
Task Function for Platoon Variance
Together with the platoon average position, it is of interest considering the
variance of all the vehicles’ positions as asynthetic data on their spreading
around the average position. It is clear that by controlling these two task
variablesitispossible to influencethe shape of theplatoon. The task function
for platoon variance σ v ∈ IR 3 is defined as
σ v = 1
n
n�
i =1
( η 1 ,i − p ) 2 . (9.7)
In [28] the Jacobian J v is provided; moreover, different tasks, such as
keeping the platoon in abounded geometrical area or in arigid formation,
are discussed.
Merging Several Tasks
Different task functions may beused at the same time by stacking the corresponding
single-task variables inanoverall task vector. As aresult, the
corresponding single-task Jacobians are stacked too in an overall task Jacobian,
and the inverse solution (9.3) acts bysimply adding the partial vehicles’
velocities that would be obtained if (locally atthe current configuration) each
task were executedalone. It is simple to recognize that this approach is poorly
effective since conflicting tasks would generate counteracting partial vehicles’
velocities.
Apossible technique to handle this problem has been proposed in [50],
which consists inassigning arelative priority tothe single task functions,
thus resorting to the task-priority inverse kinematics introduced in[196, 210]
for ground-fixed redundant manipulators. Nevertheless, as discussed in [78],
just in case of conflicting tasks itisnecessary to devise singularity-robust
algorithms that ensure proper behavior of the inverse velocity mapping. For
this reason, according to [78], G. Antonelli and S.Chiaverini propose to
modify the CLIK solution (9.4) into
v d ( t k )=J † �
�
p ˙σ p,d + Λ p ( σ p,d− σ p ) + � I − J † p J p
� J † s
�
�
˙σ s,d + Λ s ( σ s,d− σ s ) , (9.8)
where the subscript p denotes primary-task quantities, the subscript s
denotes secondary-task quantities, and Λ p , Λ s are suitable positive definite
matrices. The extension to ageneric number of tasks can be easily derived.
Equation (9.8) has anice geometrical interpretation. Each task is projected
onto the vehicles’ velocityspace by the use of the correspondingpseudoinverse,i.e.,
as if it were actingalone; then,before addingthe two contributions,