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78 Extensions to the Standard PSOM Algorithm

Furthermore, this option can be used to resolve redundancies. Such
an example is presented later in the context of the 6 D robot kinematics in
Sec. 8.4.
There the situation arises that the target is under-specified in such a
way that a continuous solution space exists which satisfies the goal specifi-
cation. In this case, the PSOM will find a solution depending on the initial

starting condition s (usually depending on the node with the closest ref-

erence vector a ) which might appear somehow arbitrary.
In many situations, it is of particular interest to intentionally utilize
these kinds of redundancies, in order to serve auxiliary goals. These are
goals of second priority, which might contradict a primary goal. Here the
PSOM offers an elegant and intrinsic solutions mechanism.
Auxiliary goals can be formulated in additional cost function terms
and can be activated whenever desired. The cost function terms can be
freely constructed with various functional forms and are supplied during
the learning phase of the PSOM. Remember, that the PSOM input sub-
space selection mechanism (p k ) facilitates easy augmentation of the em-
bedding space X with extraneous components, which do not impair the
normal operation.
For example, for positioning a robot manipulator at a particular posi-
tion in the 3 D workspace, the 6 degress-of-freedom (DOF) of the manipu-
lator are under-constrained. There are infinite many solutions – but, how
to encode some sort of “reasonableness” within the current situation? In
Sec. 8.4, different secondary goals are discussed: e.g. finding the shortest
path, or finding an arm configuration being advantageous for further ac-
tions. They can be elegantly implemented using the same PSOM.

6.3 The Local-PSOM

In section 3.2 the distinction between local and global models was intro-
duced. As it turns out, the PSOM approach can be viewed as an interme-
diate position. It is a global model, but in the vicinity of reference points
it behaves locally. Still, to overcome the difficulties inherited from high-
degree polynomial basis functions, we can look for additional control over
the locality of the PSOM.
The concept of Local-PSOMs precisely addresses this issue and allows