Page 130 - Rapid Learning in Robotics

116 Application Examples in the Robotics Domain

2. What is the influence of standard and Chebyshev-spaced sampling
of training points inside their working interval? When the data val-
ues (here 3 per axis) are sampled proportional to the Chebyshev ze-
ros in the unit interval (Eq. 6.3), the border samples are moved by a
constant fraction (here 16 %) towards the center.

Tab. 8.2 summarizes the resulting mean deviation of the desired Carte-
sian positions and orientations. While the tool length l z has only marginal
influence on the performance, the Chebyshev-spaced PSOM exhibits a sig-
nifcant advantage. As argued in Sect. 6.4, Chebyshev polynomials have ar-
guably better approximation capabilities. However, in the case n both
sampling schemes have equidistant node-spacing, but the Chebyshev-spacing
approach contracts the marginal sampling points inside the working inter-
val. Since the vicinity of each reference vector is principally approximated
with high-accuracy, this advantage is better exploited if the reference train-
ing vector is located within the given workspace, instead of located at the
border.

Figure 8.7: Spatial dis-
tribution of positioning
errors of the PUMA
robot arm using the
6 D inverse kinematics
transform computed
with a 3 3 3 3 3 3
C-PSOM. The six-
dimensional man-
ifold is embedded
in a 15-dimensional
r a n -space.

The spatial distribution of the resulting deviations is displayed in
r
Fig. 8.7 (of the third case in Tab. 8.2). The local deviations are indicated