Underwater Robots - Gianluca Antonelli.pdf

9.2 Kinematic Control of AUVs 231
the lower-priority (secondary) task is further projected onto the null space of
the higher-priority (primary) task soastoremove those velocity components
that would conflict with it.
Adifferent technique to manage conflicting tasks inthe framework of the
task priority approach has been proposed in [29] (see Section 6.6), where a
fuzzy inference system is used tomerge multiple secondary tasks given to
underwater vehicle-manipulator systems.
Obstacle Avoidance Strategy
In [50] obstacle avoidance has been developed in the frame of task priority
inverse kinematics following the approach of Y. Nakamura [210].
In the following, based on the work developed in 1991 at the Università
degli Studi Napoli Federico II for industrial manipulators [76], the obstacle
avoidance is chosen asthe primary task by selecting:
⎡ ⎤
σ o =
⎢
⎣
� η 1 − η o �
.
� η n − η o �
⎥
⎦ (9.9)
where η o is the coordinate vector of an obstacle, and the other tasks as
secondary tasks. Defining as r i the versor of η i − η o it can be recognized that
the pseudoinverse of J o ∈ IR n × 3 n is simply given by:
J † o =
⎡
⎢
⎣
r 1
0 3 × 1
. . .
0 3 × 1
r n
⎤
⎥
⎦ (9.10)
and the null-space projection matrix, considering the planar case for simplicity,
is given by:
⎡
⎤
⎢
N o = ⎢
⎣
1 − r 2 1 ,x
− r 1 ,xr 1 ,y
O 2 × 2
− r 1 ,xr 1 ,y
1 − r 2 1 ,y
. . .
1 − r 2 n,x
− r n,xr n,y
O 2 × 2
− r n,xr n,y
1 − r 2 n,y
In this approach, the obstacle avoidance task is also used for each vehicle
with respect to the other vehicles ofthe platoon. This prevents from the
possible occurrence of accidents during, e.g., formation changing or obstacle
avoidance. In case of simultaneous proximity ofanobstacle and another vehicle,
the safer behavior is obtained by projecting the secondary-task velocities
in the null space of both the obstacle avoidance tasks. This will simply
zero the required velocity for the vehicle that is commanded to go closer to
another vehicle and the obstacle atthe same time.
The developed theory consider holonomic vehicle, in[49] nonholonomic
vehiclesare explicitlyconsideredascomposing the platoon.Recently,in[278],
⎥
⎦ .