Underwater Robots - Gianluca Antonelli.pdf

6.2 Kinematic Control 111
case of the orientation, however, care in the definition of such error is required
to ensure convergence tothe desired value. In this work, the quaternion attitude
representation isused [246]; the vector e E for the task defined in(2.68)
is then given by [60, 80]:
�
�
η ee1 ,d − η ee1 ,r
e E =
, (6.8)
η r ε d − η d ε r − S ( ε d ) ε r
where Q d = { η d , ε d } and Q r = { η r , ε r } are the desired and reference attitudes
expressed by quaternions, respectively, and S ( · )isthe matrix operator
performing the cross product.
The obtained ζ r can then be used tocompute the position and orientation
of the vehicle η r and the manipulator configuration q r :
� � t
�
� η r ( t )
q r ( t )
=
=
0
� t
0
� �
˙η r ( σ )
dσ +
˙q r ( σ )
− 1
J k ( σ ) ζ r ( σ ) dσ +
�
η (0)
q (0)
� �
η (0)
. (6.9)
q (0)
As customary in kinematic control approaches, the output ofthe above
inverse kinematics algorithm provides the reference values to the dynamic
control law ofthe vehicle-manipulator system (see Figure 6.1). This dynamic
control law will be in charge of computing the driving forces, i.e., the
vehicle thrusters and the manipulator torques. The kinematic control algorithm
is independent from the dynamic control law aslong asthe latter is
avehicle/joint space-based control, i.e., itrequires as input the reference
vehicle-joint position and velocity. In the literature number ofsuch control
laws have been proposed that are suitable to be used within the proposed
kinematic control approach; aliterature survey is presented inthe next Chapter.
Forthe seek of simplicity, inthe Figure, only the primary task is shown.
Remarkably, all those inverse kinematics approaches are suitable for realtime
implementation. Of course, depending on the specific algorithm, adifferent
computational load is required [78].
Transpose of the Jacobian
Asimple algorithm, conceptually similar to the closed loop approach, isgiven
by the use of the transpose of the Jacobian. In this case, the joint velocities
are given by [254]:
ζ r = J T K E e E ,
where adirect relationship between the reference joint velocities and the
end-effector reconstruction error is obtained.