In general, an unknown visual-motor function
can be
written as a multidimensional Taylor expansion. In particular we are
interested in the linear approximation:
(1)
where for a sufficiently ``smooth'' f the
residual term
is small for
. J is
called the visual motor Jacobian and is defined as
(2)
The visual motor
Jacobian relates small changes in vision space to small changes in
motor (control) space, e.g., for
,
,
and
we have:
(3)
Thus the visual motor
Jacobian acts as a local model of the function behavior
around
(and
).
Given that we observe an initial perceptual state represented in the
feature vector
and that we wish to bring about the (goal) state
, the best prediction using our linear model J, of the
required change
in robot control signals, is given by
solving the system of equations:
(4)
Executing this move,
i.e. setting
changes the perceived
state to
.
Unless f is a linear model,
is typically
not equal to
.
However, if
is small, and f smooth,
is likely to be smaller than
, and the process can be repeated yielding a sequence
successively closer to
.
What we have is a quasi Newton method for solving a set of nonlinear equations
in
(find
such that
). In a continuous setting we have a similar control law:
(5)
where K is a gain matrix.
These two control laws implement the image based look-and-move and the image based visual servoing.