In our framework the active robot agent specifies its actions in terms of
desired
visual perceptions
. We need a control system capable of turning
these goal perceptions into motor actions (in our system, joint movements).
On the low level (excluding visual trajectory planning etc.) these
movements are reactive.
A simple control law, occuring in some form in most visual servoing research
(e.g. [Conkie and Chongstitvatana, 1990, Hager et al., 1995, Hosoda and Asada, 1994]) is
(17)
where K is a
gain matrix.
In a discrete time system running at a fixed cycle frequency
(at or below the 60Hz video frequency), the gain K
turns into a step length
:
, where
is the (least squares) solution to the (over determined) system
This popular controller however has major deficiencies. Even for a
convex problem (
in eq. 4.1 is convex) it is not
guaranteed to be
convergent [Dahlquist and Björck, 1995]. There is also the problem of selecting
an
so that the method converges, and does so reasonably
quickly.
Previous work has overcome these problems by only
making a single, small distance move within a relatively smooth and well scaled
region of f. A fixed
, giving convergence within the
small region can often be found by trial and error.
For complete tasks this is not a viable solution, as moves may be
over a large part of the robot workspace, and using a single,
experimentally found
is likely to be inefficient, as well as requiring a lot of
trials to find that
.