The trust region method developed above for robot joint space also
works in visual feature space
(since they are
locally linearly related by
,
and the trust region method is not dependent on which norm is used in
eq. 5.3. This allows us to control image velocities, the
same way we controlled joint velocities above. This is useful
since the image velocities are usually more relevant to the task than
the joint velocities.
More importantly, working in image feature space allows us
to do image space trajectory planning. We will briefly describe this
here, but as it ties in more with the vision part of a visual servoing
system than the robot control we direct the reader to
Chapter 7 or [Jägersand and Nelson, 1995], where we describe a
method for high level
visual space task specification, planning, and trajectory generation.
In visual servoing, the
task goals are given in image space. When solving complete tasks,
the image information is also used to plan trajectories (eg. to avoid
obstacles). Given initial visual state
and goal
a sequence of visual space subgoals, or ``way-points''
are generated,
subject to the visual space path planning constraints. The density of
these subgoals controls the maximum speed and acceleration.
In order to get straight line trajectories in visual space a slight
modification of the constraint in eq. 5.3 is used:
.
Last but not least, the introduction of these intermediate way-points
can help us reach the global minimum
of
rather than
a local minimum. This is possible because the sequence of subproblems
may very well be convex on each of
the now much smaller sub domains, while not being convex over
the entire domain needed for solving
directly.
Experimentally we have been able to solve some
difficult manipulation problems with this technique, such as control
in 12 DOF of a nonrigid link (see Section 5.9 or [Jägersand and Nelson, 1994]).
In many of these problems we found that, without the visual space
planning, the control
would terminate in a local minimum, or in a singularity.
In numerical analysis, techniques like this one are
called ``inbäddning'' [Gustafsson, 1991] or homotopy methods [Garcia and Zangwill, 1981],
and are used to improve
the convergence range on difficult problems for a variety of numerical
methods.