Human-machine Cooperative Manipulation with Vision-based Motion Constraints - Visual Servoing via Advanced Numerical Methods - page 59

point h in the image. Since our objective is at the origin, we can define a control law
of the form

J h

− ,

(4.18)
where J is evaluated at the origin. It is possible to show that this law will converge
for any feature starting and remaining in the image [16]. Furthermore, the preferred
directions of motion in this case are

u =

J J [ J ] J h = 0

and so it follows that u = −

only when h = 0 we can apply the general virtual fixturing

rule. At this point, we can state a useful specialization for the rest of this chapter.
General vision-based virtual fixtures . Suppose we are supplied with an error term
e = e ( x )

. Since u = 0when h = 0 ,

and define u = WJ e where W is
a symmetric, positive definite matrix of appropriate dimension ( e.g. W =( J J ) + ).
Then the general virtual fixture rule can be applied with preferred directions

. Let S = { x | e ( x )=0 }, let J = ∂ e / ∂ x ,

J

provided u so computed converges to S under a nominal plant model.
There is an interesting variation on this. Suppose we choose no preferred di-
rection of motion ( i.e. D = 0

) In this case, the first term of (4.13) disappears and
the preferred direction in (4.9) is simply u

.

Thus, the result is a virtual fixture that
guides the robot to a target position (compare with the rules at the beginning of Sec-
tion 4.2.3) and then becomes isotropic. Note, however, that by definition u is always
orthogonal to the line of sight, so the camera prefers to maintain a constant distance
to the point during motion.
To press home these points, consider a final problem: to place a specific image
location on an observed line, and to facilitate motion along the line. Following the
development in [10], suppose we observe a fixed line l

.

3

∈ ℜ

wherethethreecom-
ponents of l can be thought of as the normal vector to the line in the image, and the
distance from the origin to the line. This vector is also parallel to the normal vector
to the plane formed by the optical axis and the line as it appears in the image plane.
We also furnish a distinguished image location h

,

3

∈ ℜ , expressed in homogeneous

coordinates. We can then define e = h

· l to be the image-plane distance between the

point and the line.
First, we note that the image Jacobian (relative to e )isnowsimply
L = l J
0

3

∈ ℜ

(note that the z dependence in L is once again a non-issue). As we would now expect,
L represents non-preferred directions of motion, as it spans the space of motions
that change the distance from the point to the line. As a result, choosing preferred
directions as L in (4.9) would prefer camera motion within the plane encoded

by l .

In order to actually place the designated point onto the line, we note that the
control law

u = L e

(4.19)