Human-machine Cooperative Manipulation with Vision-based Motion Constraints - Visual Servoing via Advanced Numerical Methods - page 56

necessary to adjust the preferred direction to move the tool tip toward it. Noting that
P ( x ) is the (signed) distance from the plane, we define a new preferred direction as
follows:

− / − <

k d <

.

(4.10)
The geometry of (4.10) is as follows. The idea is to first produce the projection
of the applied force onto the nominal set of preferred directions, in this case

D c ( x )=[(1 k d ) n f f k d [ n ] x ] 0 1

At
the same time, the location of the tool tip is projected onto the plane normal vector.
The convex combination of the two vectors yields a resultant vector that will return
the tool tip to the plane. Choosing the constant k d governs how quickly the tool is
moved toward the plane. One minor issue here is that the division by

n .

is undefined
when no user force is present. Anticipating the use of projection operators (which
are scale invariant, as noted earlier), we make use of a scaled version of (4.10) that
does not suffer this problem:

f

D c ( x )=(1 − k d ) n f − k d f [ n ] x 0 < k d < 1 . (4.11)

We now apply (4.9) with D = D c . Noting that the second term on the right hand side
could also be written

f k d P ( x ) n ,

it is easy to see that, when the tool tip lies in the plane, the second term vanishes. In
this case, it is not hard to show, using the properties of the projection operators, that
combining (4.11) with (4.9) results in a law equivalent to a pure subspace motion
constraint. One potential disadvantage of this law is that when user applied force
is zero, there is no virtual fixture as there is no defined preferred direction. Thus,
there is a discontinuity at the origin. However, in practice the resolution of any force
sensing device is usually well below the numerical resolution of the underlying
computational hardware, so the user will never experience this discontinuity.
With this example in place, it is not hard to see its generalization to a broader set
of control laws. We first note that another way of expressing this example would be
to posit a control law of the form

u = − ( n · x ) n = − [ n ] x , (4.12)

and to note that assigning v = u would drive the manipulator into the plane. This is,
of course, exactly what appears in the second term of (4.11). If we now generalize
this idea, we can state the following informal rule.
General virtual fixture rule .Given:

1. a surface S SE (3) (the motion objective) ;

2. a control law u = f ( x

⊆

, S ) where by setting v = u ,

lim
t

x ∈ S

→ ∞

(the control law moves the tool tip into S ) under a nominal plant model ;