Human-machine Cooperative Manipulation with Vision-based Motion Constraints - Visual Servoing via Advanced Numerical Methods - page 58

3 )where
the camera is aligned with the robot base frame. In this case, the relationship be-
tween the motion of the camera and the image motion of a fixed point in space is
given by the well-known image Jacobian relationship [14]:

To keep things simple, first consider the case of pure translation ( v ∈ ℜ

h = J v

, (4.15)

v ) ∈ ℜ

2

where h =( u 3.

It is again well-known [3, 14] that the rows of J span the (two-dimensional) space
of motions that create feature motion in the image, and therefore

, is the image location of a feature point, and J is 2 ×

J

is the (one-
dimensional) space of motions that leave the point fixed in the image. Consider,
thus, creating a virtual fixture by defining

D = J (4.16)

in (4.9). From the discussion above, it should be clear that this will create a virtual
fixture that prefers motion in any direction except along the viewing direction. While
it would seem we are done at this point, there is one minor issue: the image Jacobian
depends on the depth of the estimated point. However, if we consider the form of
the Jacobian in this case, we see it can be written thus:

⎡
⎣

⎤
⎦

1
z

0 − u
z

10

= 1
z

− u

J = . (4.17)

1

z − v

01 − v

0

z

As such, we see that the term involving z is a scale factor and, as noted earlier, our
projection operators are invariant over scaling of their argument. Thus, we have our
first result as follows.
Image plane translation . If we restrict v to be pure translation and choose an
image location h =( u v ) ,

then implementing (4.9) using
1. D = J creates a virtual fixture that prefers motion in the plane normal to the
viewing direction defined by h and the camera optical center ;
2. D =

,

J creates a virtual fixture that prefers motion along the viewing direction

defined by h and the camera optical center.

As a special case, choosing u = v = 0 yields a virtual fixture parallel to the image
plane, which, as is obvious from (4.17), is the camera x - y plane.
It is important to note that the image plane virtual fixtures defined above can
be implemented both with and without feedback. That is, if we simply choose a
fixed image location ( e.g. the origin), then the camera will always move in a plane
orthogonal to the line of sight through the chosen location. On the other hand, if we
choose to track a feature over time, then the motion will always be orthogonal to the
line of sight to that feature.
The other possibility, with a single feature, is to maintain the visual cue at a
specific image location. For example, suppose the goal is to center an observed