Page 64 - Rapid Learning in Robotics

50 The PSOM Algorithm

alternative set of input vectors an alternative input subspace I f
g is
specified.

1

0.5

1
0
0 0.5
0.5
0
(a) (b) (c) (d)

Figure 4.7: (a:) Reference vectors of a SOM, shared as training vectors by
x
a PSOM, representing one octant of the unit sphere surface (x x ,
see also the projection on the x base plane). (b:) Surface plot of the map-
ping manifold M as image of a rectangular test grid in S. (c:) A mapping
x x x x obtained from the PSOM with P diag , (d:) same PSOM,
but used for mapping x x x x by choosing P diag .

As another simple example, consider a 2-dimensional data manifold

in IR that is given by the portion of the unit sphere in the octant x i
(i
). Fig. 4.7, left, shows a SOM, providing a discrete approximation

to this manifold with a -mesh. While the number of nodes could be
easily increased to obtain a better approximation for the two-dimensional
manifold of this example, this remedy becomes impractical for higher di-
mensional manifolds. There the coarse approximation that results from
having only three nodes along each manifold dimension is typical. How-
ever, we can use the nine reference vectors together with the neighborhood
information from the SOM to construct a PSOM that provides a much
better, fully continuous representation of the underlying manifold.

Fig. 4.7 demonstrates the PSOM working in two different map-
ping “directions”. This flexibility in associative completion of alternative
input spaces X in is useful in many contexts. For instance, in robotics a
positioning constraint can be formulated in joint, Cartesian or, more gen-
eral, in mixed variables (e.g. position and some wrist joint angles), and one
may need to know the respective complementary coordinate representa-
tion, requiring the direct and the inverse kinematics in the first two cases,
and a mixed transform in the third case. If one knows the required cases