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122 Application Examples in the Robotics Domain

attracts the solution to the particular single configuration with all joints in
mid range position. Any further kinematics specification is usually con-
flicting, and the result therefore a compromise (the least-square optimum;
jIj ). How to solve this conflict?

To avoid this mis-attraction effect, the auxiliary constraint terms p k p k t

1. should be generally kept small, otherwise the solution would be too
strongly attracted to the single mid-point position;

2. should decay during the gradient descent iteration. The final step
should be done with all extra terms c j weighted with factors p k zero

(here p ). This assures that the final solution will be – without
compromise – within the solution space, spanned by the primary
goal, here the end-effector position.

To demonstrate the impact of the auxiliary constraints the augmented
m PSOM is engaged to re-arrange a suitable robot arm configuration.
The initial starting position is already a solution of the desired end-effector
positions and Fig. 8.10 and Fig. 8.11 show intermediate steps in approach-
ing the desired result. Here, the extra cost components were weighted in a
fixed ratio of 0:0.04:0.06:1:1:0.04 among each other and weighted initially
by 0.5 % with respect to the components (see Eq. 8.3). During interme-
r
diate best-match search steps all weights gradually decay to zero. The
stroboscopic image (Fig. 8.11d) shows how the arm frees itself from an ex-
tremal configuration (position close to the limit) to a configuration leaving
more space to move freely.

It should be emphasized that several constraint functions can be simul-
taneously inserted and turned “on” and “off” to suit the current needs.
This a good example of the strength of a versatile and flexible input se-
lection mechanism. The implementation should care that any in-active
augmentations (with p ) of the embedding space X are handled effi-
ciently, i.e. all related component operations are skipped. By this means,
the extraneous features do not impair the PSOM's performance, but can
be engaged at any time.