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8.4 Resolving Redundancy by Extra Constraints for the Kinematics 121

model this goal as a “discomfort” term in the cost function E and demon-
strate how to incorporate extra cost terms in the standard PSOM mecha-
nism.

c
j

Figure 8.9: “Discomfort” cost function

mid

c j j
j j for each joint

max
min
j j
θ angle i. A target value of zero, will
j
attract the best-match towards the joint
θ j-min θ j-max range center mid j .

Fig. 8.9 shows a suitable cost function term, which is constructed by
a parabola shaped function c j j for all joint angles . c j j is zero
at the interval midpoint mid j and positive at both joint range limits. The
15-dimensional embedding space X is augmented to 21 dimensions such
that all training vectors w become extended by the tuple c . If the c
corresponding p k in the selection matrix P are chosen as zero, the PSOM
provides the same kinematics mapping as in the absence of the extension.
However, when we now turn on the new P elements (p ), and set

the input components to zero (x ), the iterative best-match proce-
dure of the PSOM tries to simultaneously satisfy the constraints imposed
by the kinematics equation together with the constraints c j . The latter

Figure 8.10: Series of intermediate steps for optimizing the remaining joint angle
mobility in the same position.