In this experiment we compared performance of adaptive and
non-adaptive visual controllers in 3 DOF and 6 DOF. The adaptive
controller can adjust its internal model (using the on-line model
estimation)
to environment changes at different times k,
whereas the non-adaptive controller has a fixed model
. The
accuracy of the internal model affects controller performance.
In particular it affects oscillations and the speed of convergence.
At first it may seem that the most relevant evaluation measure for
an adaptive controller is how accurately it can estimate the
visual-motor model. However, accurate model estimation has to be
traded off for efficiency, and ideally one would do this so
the internal model (
) gets estimated only as precisely
as necessary to perform the task at hand. Tuning of this accuracy is
accomplished by varying
and
in
eq. 5.4. For a 3 DOF task we have
found that estimating the partial derivatives
(entries
in the Jacobian)
with
accuracy is sufficient. This finding is in line
with Hosoda and Asada's observations for their visual servoing
controller in [Hosoda and Asada, 1994].
Instead of studying internal aspects of the adaptive and non-adaptive controllers, we wanted to compare how robustly the two controllers react to a suddenly introduced discrepancy between the internal model and the real world. In real world applications this discrepancy can stem from a variety of sources such as errors in initial model estimate, errors in visual data used to adjust the model, and for the non-adaptive case, poor correspondence between model (linearized) and real world (typically non-linear).
The basic experimental
plan was to
add varying amounts of noise to the Jacobian model, and then compare
final endpoint positioning error and trajectory error,
for the adaptive and non-adaptive cases.
(Trajectory error is the average deviation in visual space
from the planned straight line between the initial feature values
and the goal
.)
An initial Jacobian approximation
, valid in the center of
the workspace,
is obtained by executing a sequence of test moves near the middle
of the (visually defined) workspace.
The disturbed Jacobian is defined as
where
indicates the model accuracy and
is a random
matrix with elements drawn from
a uniform distribution on the interval
. This typically (barring some unlikely
cases, such as J = I, the identity matrix)
restricts the gain, but not the direction
(angular transformation)
of the random Jacobian. Each test consisted of 50 runs with varying values
of the model accuracy parameter a.
In Fig. 5.6 we see that for the non-adaptive
algorithm, the percentage of the trials which converged
(
)
quickly falls to zero for model accuracies a below
0.5. The adaptive algorithm has no problem, even with completely
random initial Jacobians.
The trajectory error for the adaptive algorithm increases slightly with high amplitudes of injected noise. This increase comes from a few early erroneous movements, and is quickly eliminated as the model is adapted. For the non-adaptive algorithm, we find a huge increase in trajectory error. This is caused by divergent trials, where the robot goes off in a completely incorrect direction and exhibits long period oscillatory behavior at the transition between convergence and divergence, which occurs near a = 0.25. A typical oscillatory move is shown in Fig. 5.7.
Figure 5.6: Results from 50 runs of an experiment with a disturbed internal
model in the controller. Convergence for an adaptive and
a non adaptive 3 DOF controller (left). Average final endpoint
error per feature in
feature space (right) for
16 tracked features and a 6 controlled DOF problem (right).
Figure 5.7: Left: Visually planned path. Right: Typical oscillatory
path of non-adaptive controller when model accuracy is near the
convergence limit.
When controlling the full 6 DOF of the manipulator we do not get quite
the same robustness against model errors as in the 3 DOF case. Model
errors worse than a = 0.6 yield a significant and increasing proportion
of divergent trials. To get near
convergence we needed to
restrict model accuracy a to be .6 or above. In a test of 50
trials in this
range, two did not converge. One showed oscillatory behavior, and the
other diverged immediately.
In general, average
visual endpoint errors as well as trajectory errors remain low even at
low model accuracies, as can be seen in Fig. 5.6.
A practical method to make 6 and higher DOF control work well without
a new Jacobian, calibrated a-priori, is to bootstrap from
the Jacobian of a 3 DOF
manipulation. For instance assume that during the 3 DOF transportation
move, one feature (e.g. the object centroid) has been tracked in the
two cameras. In order to control the angular DOF's of the object
more visual pose estimation is needed, and we use point features
physically separated on the object surface. Assume k point features on
the objects are found in camera 1 and l in camera 2.
When switching between 3 and 6 DOF mode,
the first three columns of the
DOF Jacobian are filled
from the
DOF Jacobian, the last three with random
numbers.
(21)
This technique is used to switch between the 3 DOF control used for transportation moves, to high DOF control (typically 4-6 DOF for rigid objects) used for the fine manipulation required for alignments and insertions. The bootstrapping works because the visual feature motions are correlated. See Chapter 7 for details.