Human-machine Cooperative Manipulation with Vision-based Motion Constraints - Visual Servoing via Advanced Numerical Methods - page 60

h to the observed line [10]. Hence, we apply (4.13)

will move the feature point

L using D = and u as defined above (in short, another application of the general

vision-based virtual fixture rule).

4.3.2 Controlling the Viewer: General Case

In moving from pure translation to general robot motion, almost nothing changes
from the previous section other than the increase in dimensionality. In the case of full
motion in SE (3), the image Jacobian becomes 2 × 6 and has the following general

form:

⎡ ⎤

1 2 + u 2
1

1
z

0 − u
z

− uv
1

− v

⎣ ⎦ .

(4.20)

1 2 v 2 1

z − v z −

− uv

1

0 u

1

As shown in [12], the kernel of the image Jacobian given in (4.20), is spanned by
the four vectors

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

u
v
1
0
0
0

0
0
0
u
v
1

uvz dz

0

⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

( u 2 + 1) z
vz
−

−

− udz

uv

(4.21)

1
0
u

( u 2 + 1) z
u

−

where d =( u 2 + v 2 + 1). As such, we can see that the kernel spans motions that
include: 1) motion along the line of sight; 2) rotation about the line of sight; 3)
motion on a sphere linking the point with the camera; and 4) linear combinations
thereof. Note that 3) spans 2 DOF.
If we reconsider all of the cases of the previous section, we see that by using
the full Jacobian, we achieve the same virtual fixtures, albeit in a larger space of
allowed motions. In particular, choosing D = now prefers motions on a sphere

about the observed point, together with rotation about and translation along, the line
of sight. It is, however, important to note that the distance from the camera to the
observed point no longer “drops out” of the system as in the case of pure translation.
Therefore distance must be estimated, for example by using adaptive schemes as
outlined in [26].
At this point, we redirect the reader to [3], where it is observed that regulating the
motion of the camera through invariants defined on observed features or measures
thereof creates a broad family of “virtual linkages” between the camera and the
world. In effect, applying the constructions as laid out above to these control laws
creates a corresponding family of virtual fixtures. Likewise, more recent “hybrid”
approaches to control that seek to produce more reliable converge of visual servoing
methods [23] and/or control other properties of the motion of features in the image

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