Underwater Robots - Gianluca Antonelli.pdf

8.4 External Force Control 209
that the integral action of the force controller guarantees anull force error
atsteady state while anon-null position error x d − x is obtained. This
implies that, in the direction where anull desired force is commanded, the
manipulator reacts tounexpected impacts with asafe behavior. Moreover, in
the directions in which adesired force is commanded, the desired position is
overcome by the controller.
If f e,d /∈ R ( K ), i.e., the direction of f e,d is not parallel to n ,the controller
is trying to interact with the environment in directions in which the
environment cannot generate reaction forces. Adrift in that direction is then
experienced.
8.4.4 Loss of Contact
Due to the floating base and the possible occurrence of external disturbances
(suchas, e.g., ocean current), it can happen that the end effector loses contact
with the environment during the task fulfillment. In such acase the control
action might become unsuitable. In fact, from (8.8) itcan be noted that the
desired force is interpreted as amotion reference velocity scaled by the force
control gain. One way to handle this problem is the following: if the force
sensor does not read any force value in the desired contact direction, the
integrator inthe force controller is reset and (8.8) is modified asfollows:
ζ d = J # p { H [ ˙x p,d + ˙x c + Λ p ( x p,d − x p )] +(I − H ) ˙x l } +
( I − J p J # p ) J # s [ ˙x s,d + Λ s ( x s,d − x s )] , (8.15)
where H ∈ IR m × m is adiagonal selection matrix with ones for the motion
directions and zeros for the force directions, and ˙x l is adesired velocity at
which the end effector can safely impact the environment. Basically, the IK
is handling the motion directions in the same way as when the contact occurs
but, for the force direction, areference velocity isgiven in away to obtain
again the contact. Notice that ˙x c is not dropped out in case of loss of contact
because it guarantees from unexpected contacts in the direction wheremotion
control is expected (i.e., the directions in which the desired force is zero).
Matrix H can be interpreted as the selection matrix of ahybrid control
scheme. Notice that this matrix is used only when there is no contact at the
end effector and the properties of robustness discussed above still hold.
8.4.5 Implementation Issues
The implementation ofthe proposed force control scheme might benefit from
some practical considerations:
• The vehicle and the manipulator are characterized byadifferent control
bandwidth, due to the different inertia and actuator performance. Moreover
limit cycles in underwater vehicles are usually experienced due tothe