Underwater Robots - Gianluca Antonelli.pdf

228 9. Coordinated Control of Platoons of AUVs
where η 1 ,i is the 3dimensional position of the i th vehicle, and
v =
⎡
⎢
⎣
˙η 1 , 1
.
˙η 1 ,n
⎤
⎥
⎦ .
By defining as σ ∈ IR m the task variable to be controlled, it is:
with
σ = f ( η 1 , 1 ,...,η 1 ,n) =f ( p ) (9.1)
˙σ =
n�
i =1
∂ f ( p )
∂ η 1 ,i
˙η 1 ,i = J ( p ) v , (9.2)
where J ∈ IR m × 3 n is the configuration-dependent task Jacobian matrix 1 .
An effective way to generate motion references for the AUVs starting
from desired values σ d ( t )ofthe task function istoact at the differential
level by inverting the (locally linear) mapping (9.2); infact, this problem has
been widely studied in robotics (see, e.g., [262] for atutorial orChapter 6
for anUVMS’s application). Forlow-rectangular matrices —which isusually
the case in platoon formation control, being 3 n � m —the problem admits
infinite solutions and such redundancy is often exploited to optimize some
criterion. Atypical requirement istopursue minimum-norm velocity, leading
to the least-squares solution:
v d = J † �
T
˙σ d = J J T J
� − 1
˙σ d . (9.3)
At this point, the vehicle motion controllers need reference position trajectories
besides the velocity references; these can be obtained by time integration
of v d .However, discrete-time integration of the vehicles’ reference
velocities would result in anumerical drift of the reconstructed vehicles’ positions;
the drift can be counteracted by aso-called Closed Loop Inverse
Kinematics (CLIK) version of the algorithm, namely,
� �
��
t = t k
(9.4)
p d ( t k )=p d ( t k − 1 )+v d ( t k ) ∆t , (9.5)
�
†
v d ( t k )=J ˙σ d + Λ ( σ d − σ )
where t k is the k -th time sample, ∆t is the sampling period, and Λ is asuitable
constant positive definite matrix ofgains. It must be remarked that the loop
is closed onalgorithmic quantities at the input of the motion controllers
and does not involve measurement ofquantities at the output of the motion
controllers; in other words, the dynamics of the vehicles’ motion controllers
is out of the loop in (9.4).
1 The symbol J has been used in Equation (2.68) for aspecific Jacobian matrix,
in the remainder of the Chapter, however, it will denote ageneric task Jacobian
matrix.