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plane [4, 5] can be applied, to the extent that the properties outlined in Section 4.2.3
are satisfied.
4.3.3
More General Camera Configurations
Until now, we have only considered a single end-effector mounted camera. How-
ever, it is important to note that everything we have said above can be applied to
the case of a fixed camera observing an independently moving manipulator, with
suitable adjustment of the form of the image Jacobian. Furthermore, we note that
for either configuration, observing both the end-effector and the external features
defining the task creates an endpoint closed-loop control law which has well-known
robustness against camera calibration error [8, 10, 13, 14]. Likewise, methods for
estimating the image Jacobian on line [15] can, in principle, be applied practically
without change.
As a final generalization, we could also add a second observing camera. It is
well known [10, 14] that the relationship between control velocities and changes in
observed image errors are expressed by “stacking” the individual image Jacobians
for each camera, now expressed in a common coordinate system. Furthermore, the
estimate of z (depth) in the Jacobian becomes trivial using triangulation from the
two cameras. If we return to our list of examples, the following comments apply.
Pure translation . In the case of feature points and pure translation, the stacked
Jacobian matrix spans the entire space of robot motions (except for points along
the baseline of the two-camera systems), and therefore it is not possible to define
interesting virtual fixtures other than point targeting.
If we consider the case of following a line, however, then when we stack the two
Jacobians as before and apply the general vision-based virtual fixture rule, we arrive
at a control law that effectively creates a prismatic joint that permits motion strictly
along a line in space.
General motion . For general motion, we see that the “stack” of two Jacobians
for feature-point servoing creates a spherical joint: the preferred DOF are motion
on a sphere about the observed point while maintaining direction to the point (2
DOF) and rotation about that line of sight (1 DOF). This may, at first, seem counter-
intuitive since the stacked Jacobian has 4 rows. However, due to the epipolar con-
straints of the camera, one of these constraints is redundant and thus the Jacobian
spans only 3 DOF. If we add a second point, we further reduce the DOF by 2,
with the remaining allowed motion being rotation about the line defined by the two
observed points. A third observed point completely determines the pose of the ob-
served (or observing) system, and so virtual fixturing once again reduces to motion
to a target pose.
In the case of placing a point on a line, the image constraints now create a con-
straint on two positional DOF. It is, however, still possible to rotate in any direction
(with the constraint that the rotation preserved distance to the observed point) and
to move along the line. Placing a second point on the line reduces this to 2 DOF
(rotation about the line and translation along it).
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