Page 71 - Rapid Learning in Robotics

4.5 Basis Function Sets, Choice and Implementation Aspects 57

lattice in S, i.e. the Cartesian product of m one-dimensional node point

sets A A A A m, with f a a a n g. Here, a i denotes

A

the i-th value that can be taken by the -th component ( g)of f m
a along the -th dimension of s s s m s T S. Fig. 4.10

illustrates
an example with n , n , and n .
1 a 3
1 a 2
s 1 a 2
3
1 a 3
S 3 a 1
s
1 s
2
2 a 2 a 2 a 2 a
1 2 3 4
Figure 4.10: Example of a m dimensional mapping manifold S with a
3 4 2 node set A S in orthonormal projection. Note the rectangular struc-

ture and the non-equidistant spacing of the a i .

With this notation, we can write H a s H a s A as
m
Y
H a s l i s A (4.18)

using the one-dimensional Lagrange factors and the notation

a a i a i a i a i T A

m

m
In Fig. 4.3 (p. 46) some basis functions of a m dimensional PSOM
with
equidistantly chosen node spacing, are rendered. Note, that the PSOM al-
gorithm is invariant to any offset for each S axis, and together with the
iterative best-match finding procedure, it becomes also invariant to rescal-
ing.
derived from the
Comparative results using (for n ) a node spacing
Chebyshev polynomial are reported in Sec. 6.4.
The first derivative of (4.1) turns out to be surprisingly simple, if we
write the product rule in the form
n

X f k x

f x f x f n x f x f x f n x x f k x

x f k x
k
(4.19)