Page 70 - Rapid Learning in Robotics

56 The PSOM Algorithm

1
x
2
0.5

100
50
0 0 0.5 1
-1 -0.5
Steps
Adaptation
x
1
Figure 4.9: In contrast to the adaptation rule Eq. 4.14, the modified rule Eq. 4.15
is here instable, see text.

4.5 Implementation Aspects and

the Choice of Basis Functions

As discussed previously in this chapter, the PSOM utilizes a set of basis
functions H a s , which must satisfy the orthonormality and the division-
of-unity condition. Here we explain one particular class of basis functions.
It is devoted to the reader also interested in the numerical and implemen-
tation aspects. (This section is not a prerequisite for the understanding of
the following chapters.)
A favorable choice for H a s is the multidimensional extension of the
well known Lagrange polynomial. In one dimension (m d ) a is from

n
a set A fa i j i
g of discrete values and Eq. 4.1 can be written as
the Lagrange polynomial interpolating through n support points a i i : w
n
X
w s l s A w w l n s A w n l k s A w k (4.16)
l s A

k
where the Lagrange factors l i s A are defined as
n
Y s a j
l i s A (4.17)
a i a j
j i j
If m , Eq. 4.16 and 4.17 can be generalized in a straightforward
fashion, provided the set of node points form a m-dimensional hyper-