Underwater Robots - Gianluca Antonelli.pdf

6.5 Singularity-Robust Task Priority 117
Asecond simulation, starting from the same initial system configuration,
considers anend-effector trajectory that cannot be tracked bysole manipulator
motion. Therefore, the vehicle must be moved toallow the manipulator
end-effector to track its reference trajectory. Also in this simulation, alignment
ofthe vehicle fore aft direction with the ocean current ispursued.
The desired end-effector trajectory is astraight-line motion starting from
the same initial location asinthe previous simulation and lasting at the final
location
x E =8. 00 m ,
y E =9. 00 m ,
ψ E =0. 78 rad .
The path is followed according toaquintic polynomial interpolating law with
null initial and final velocities and acceleration and aduration of 10 s. The
other task variables and gains are the same as in the previous simulation;
remarkably, the desired values of the vehicle position variables are coincident
with their initial value also in this case.
The simulation results are reported in Figure 6.5 and 6.6. Itcan be recognized
that the primary task is successfully executed, inthat the end-effector
location and vehicle orientation achieve their target. On the other hand, the
vehicle moves from its initial position despite the secondary task demands
for station keeping. Remarkably, the obtained vehicle reference trajectory is
smooth.
To show generality ofthe proposed approach asecond case study has been
developed. Adrawback of the previous case study might bethat the manipulator
arm is almost completely stretched out when the end-effector trajectory
requires large displacements going far from the vehicle body. Nevertheless,
this is related to our choice tokeep the position of the vehicle constant and
to align the fore aft direction with the ocean current. To overcome this drawback,
adifferent choice ofthe tasks tobefulfilled isnecessary. Inparticular,
the task of vehicle re-orientation might bereplaced with the task of keeping
the manipulator arm in dexterous configurations. To this aim, it would be
possible to use atask variable expressing amanipulability measure of the
manipulator arm [312]. In this simple case, it is clear that arm singularities
occur when q 2 =0;therefore, the use of q 2 as manipulability task variable
would reduce the computational burden of the algorithm.
To implement the proposed approach in this second case study, both the
end-effector position+orientation and the second manipulator joint variable
are thus considered asprimary task, i.e.
x p =[x E y E ψ E q 2 ] T ,
and assecondary task the vehicle position, i.e.
x s =[x y] T .