Underwater Robots - Gianluca Antonelli.pdf

210 8. Interaction Control of UVMSs
thruster’s characteristics. This implies that the use of the desired configuration
in the kinematic errors computation in(8.8) would lead to coupling
between the force and motion directions, since the Jacobian matrix is computed
with respect to aposition different from the actual position. In(8.8),
thus, the real positions will be used tocompute the errors.
• Force control tasks require accurate positioning ofthe end effector. On the
other hand, the vehicle,i.e., the base of the manipulator, is characterized by
large position errors. In (8.8), thus, it could be appropriate to decompose
the desired end-effector velocity inaway to involve the manipulator alone
in the fulfillment ofthe primary task. Let us define J p,man( R I B , q )asthe
Jacobian ofthe manipulator, it is possible to rewrite (8.8) as
�
0 6 × 1
ζ d =
J # �
+
p,man [ ˙x p,d + ˙x c + Λ p ( x p,d − x p )]
+(I − J # p J p ) J # s [ ˙x s,d + Λ s ( x s,d − x s )] . (8.16)
Notice that the same properties of (8.8) applies also for (8.16). Aphysical
interpretation of (8.16) is the following: it is asked the manipulator tofulfill
the primary task taking into account the movement ofits base. At the same
time the secondary task is fulfilled, with less strict requirements, with the
whole system (e.g., the vehicle must move when the manipulator isworking
on the boundaries ofits workspace).
• UVMSs are usually highly redundant. If a6-DOF manipulator isused this
means that 12 DOFs are available. Itisthen possible to define more tasks
to be iteratively projected on the null space of the higher priority tasks. An
example of 3tasks could be: 1)motion/force control of the end effector, 2)
increase manipulability measure of the manipulator, 3) limit roll and pitch
orientation of the vehicle. See also Chap. 6.
• To decrease power consumption it is possible to implement IKalgorithms
with bounded reference values for the secondary tasks. Using some smooth
functions, or fuzzy techniques, it is possible to activate the secondary tasks
only when the relevant variables are out ofadesired range. Forthe roll and
pitch vehicle’s angles, for example, it might beconvenient to implement an
algorithm that keeps them in arange, e.g., ± 10 ◦ .See also Chap. 6.
• Force/moment sensor readings are usually corrupted by noise. The use of
aderivative action in the control law, thus, can be difficult toimplement.
With the assumption of africtionless and elastically compliant plane it can
be observed alinear relation between ˙ f e and ˙x .The force derivative action
can then be substituted by aterm proportional to ˙x .The latter will be
computed by differential kinematics from ζ that is usually available from
direct sensor readings or from the position readings via anumerical filter.
8.4.6 Simulations
To prove the effectiveness of the proposed force control scheme several simulations
have been run under Matlab c� / Simulink c� environment. The