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8.4 Resolving Redundancy by Extra Constraints for the Kinematics 119

totically.
A very important advantage of self-learning algorithms is their abil-
ity to adapt to different and also changing environments. To demonstrate
the adaptability of the network, we interrupted the learning procedure
after 400 training steps and extended the last arm segment by 150 mm
(l z mm ). The right side of Fig. 8.8 displays how the algorithm re-
sponded. After this drastic change of the robot's geometry only about 100
further iterations where necessary to re-adapt the network for regaining
the robot's previous positioning accuracy.

8.4 Resolving Redundancy by Extra Constraints
for the Kinematics

The control of redundant degrees-of-freedom (DOF) is an important prob-
lem for manipulators built for dextrous operations. A particular task has
a minimal requirement with respect to the manipulator's ability to move
freely. When the task leaves the kinematics problem under-specified, there
is not one possible solution, instead there exists a higher-dimensional so-
lution space, which is compatible with the task specification. The practice
requires a mechanism which determines exactly one solution. Naturally,
it is desirable that these mechanisms offer a high degree of flexibility for
commanding the robot task.

In this section the PSOM will be employed to elegantly realize an inte-
grated system. Important is the flexible selection mechanism for the input
sub-space components and the concept of modulating the cost function, as
it was introduced in Sec. 6.2.
We return to the full 6 DOF Puma kinematics problem (Sec. 8.2) and
use the PSOM to solve the following, typical redundancy problem: e.g.,
specifying only the 3 D target positioning r without any special target ori-
entation, will leave three remaining DOFs open. In this under-constrained
case the solutions form a continuous 3 D space. It is this redundancy that
we want to use to meet additional constraints — in contrast to the discon-
tinuous redundancies by multiple compatible robot configurations. Here
we stay with the right-arm, elbow-up, no-wrist-flip configuration seen in
Fig. 8.7 (see also Fu et al. 1987).
The PSOM input sub-space selection mechanism (matrix P) facilitates