Underwater Robots - Gianluca Antonelli.pdf

130 6. Kinematic Control of UVMSs
• Manipulability .Since the fuzzy approach tries to move the manipulator
alone, its manipulability has to be checked. Acomputationally limited
measure of the manipulability can be obtained by checking the minimum
singular value of the Jacobian [195, 77]. Since the manipulability function
is strongly non-linear it is possible to adopt the following approach: when
close to asingular configuration, the system tries to reconfigure itself in a
dexterous configuration. The task, thus, is anominal manipulator configuration
whose Jacobian is given by:
J s 1 =[O 6 × 6
I 6 ];
• Mechanical limits. Due to the mechanical structure, each joint has a
limited allowed range. In case ofareal-time trajectory, avoidance of such
limits is crucial. Forthis reason the minimum distance from amechanical
limit is considered as another secondary task. Notice that the Jacobian is
the same of to the previous task:
J s 2 =[O 6 × 6
I 6 ];
• Vehicle attitude. Asfor the previous cases, the vehicle attitude (roll and
pitch angles) has tobekept null when possible. The Jacobian is then given
by
J s 3 =
� �
0 0 0 1 0 0 0 0 0 0 0 0
.
0 0 0 0 1 0 0 0 0 0 0 0
Due to the simple structure ofthe matrices, the pseudoinversion of the
secondary tasks is trivial: J †
s 1 = J T s 1 , J †
s 2 = J T s 2 , J †
s 3 = J T s 3 .
As shown in Section 6.6, the above tasks are activated by fuzzy variables.
The fuzzy inference system has 3inputs, namely: ameasure of the robot
manipulability, ameasure of the distance from the joints limits and ameasure
of the vehicle attitude. Hence, 3linguistic variables can be defined that can
take the values:
manipulability = { singular, not singular}
joint limits = { close, not close}
vehicle attitude = { small, not small}
The output isgiven by the linguistic variables β and the 3 α i ’s. The latter
can take the following values:
α i = { high, low } .
The linguistic variable β ,named motion can take the following values:
motion = { vehicle, manipulator } .
As an example, the membership function ofthe linguistic variable joint limits
is reported inFigure 6.18.
The FIS outputs are considered at two different levels. The variable motion
( β in (6.15)) can be considered at ahigher level with respect to the