Page 89 - Rapid Learning in Robotics

Chapter 6

Extensions to the Standard PSOM

Algorithm

From the previous examples, we clearly see that in general we have to ad-
dress the problem of multiple minima, which we combine with a solution
to the problem of local minima. This is the subject of the next section.

In the following, section 6.2 describes a way of employing the multi-
way mapping capabilities of the PSOM algorithm for additional purposes,
e.g. in order to simultaneously comply to auxiliary constraints given to
resolve redundancies.

If an increase in mapping accuracy is desired, one usually increases the
number of free parameters, which translates in the PSOM method to more
training points per parameter axis. Here we encounter two shortcomings
with the original approach:

The choice of polynomials as basis functions of increasing order leads
to unsatisfactory convergence properties. Mappings of sharply peaked
functions can force a high degree interpolation polynomial to strong
oscillations, spreading across the entire manifold.

The computational effort per mapping manifold dimension grows

as O Q m n for the number of reference points n along each axis

. Even with a moderate number of sampling points along each pa-
rameter axis, the inclusion of all nodes in Eq. 4.1 may still require
too much computational effort if the dimensionality of the mapping
manifold m is high (“curse of dimensionality”).

J. Walter “Rapid Learning in Robotics” 75