Underwater Robots - Gianluca Antonelli.pdf

208 8. Interaction Control of UVMSs
common inexternal force control approach, the motion controller guarantees
perfect tracking, yielding J p ζ d = ˙x .Wefinally choose Λ p = λ p I m .Thus,
(8.10) can be rewritten as
˙x = ˙x c + λ p ( x p,d − x ) . (8.11)
In view of the above assumptions the vector ˙x c belongs to R ( K ); it is
then simple to recognize that the motion component ofthe dynamics along
the normal direction to the surface is decoupled from the motion components
lying onto the contact plane. Therefore, it is convenient to multiply (8.11) by
the two orthogonal projectors nn T and I − nn T (see Figure 2.5), that gives
the two decoupled dynamics
nn T ˙x = ˙x c + nn T λ p ( x p,d − x ) (8.12)
( I − nn T ) ˙x =(I − nn T ) λ p ( x p,d − x ) (8.13)
Equation (8.13) clearly shows convergence ofthe components of x tangent
to the contact plane to the corresponding desired values when λ p > 0.
To analyze convergence of(8.12), bydifferentiating (8.12) and by taking
into account (2.76) and (8.9) we obtain the scalar equation
(1 + k f,v k )¨w +(λ p + k f,p k )˙w + k f,i kw = k f,i n T ( f d + k x ∞ ) , (8.14)
where the variable w = n T x is used for notation compactness and k ,as
defined inthe modeling Chapter, is the environment stiffness.
Equation (8.14) shows that, with aproper choice ofthe control parameters,
the component of x normal tothe contact plane converges to the
value
w ∞ = n T ( 1
k f e,d + x ∞ ) .
In summary, the overall system converges to the equilibrium
x ∞ = nn T ( 1
k f e,d + x ∞ )+(I − nn T ) x p,d ,
which can be easily recognized to ensure f e,∞ = f e,d.
8.4.3 Robustness
In the following, the robustness of the controller to react against unexpected
impacts or errors in planning desired force/position directions is discussed.
The major drawback of hybrid control isthat itisnot robust to the
occurrence of an impact in adirection wheremotion control hasbeen planned.
In such acase, in fact, the end effector is not compliant along that direction
and strong interaction between manipulator and environment is experienced.
This problem has been solved by resorting to the parallel approach [79] where
the force control action overcomes the position control action atthe contact.
The proposed scheme shows the same feature asthe parallel control. In
detail, ifacontact occurs along amotion direction, we can see from (8.8)