Page 678 - The Mechatronics Handbook

FIGURE 21.23 Learning feed-forward controller for repetitive motions.

where L is the transfer function of the learning filter. The superscript k denotes the kth repetitive motion.
The signal U F should converge to a feed-forward signal that compensates for all repetitive errors. An
example of a situation where such errors are present is, for instance, a CD player that has to compensate
for the eccentricity of the disk.
A variation on this idea and even more straightforward is the learning feed-forward controller (LFFC)
setup of Fig. 21.23. When the feed-forward signal would be perfect, the output of the controller would
be zero. This implies that this output can be used as a training signal for a neural network. An adaptive
B-spline network enables learning of complex nonlinear characteristics. Also support vector machines
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have been used to implement the learning feed forward (De Kruif and De Vries ). The input of the
B-spline network is the time t. It is reset each time a new motion starts. This is called a time-indexed
LFFC. Instead of the time, also the reference signal and its derivatives—obtained from a path genera-
tor—could be used as index for the network (path-indexed LFFC). The advantage of this structure is
that after proper training the LFFC can successfully be used for nonrepetitive motions as well. Velthuis 19
has given a stability analysis for time-indexed as well as path-indexed LFFC-controllers. The stability
analysis is relatively easy for the time-indexed case. For the path-indexed case it is more complex and
some heuristics are required to guarantee a stable system. The main issue is that the number of B-splines
should not be too large. On the other hand a sufficiently dense B-spline distribution is desired for an
accurate approximation of the nonlinear process. LFFC has successfully been applied to compensate for
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cogging in an industrial Linear Motor (Otten et al. ) and for compensation of (Coulomb) friction of a
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linear motor used in a flight simulator (Velthuis ). It has also been applied to the tracking control of
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the mobile robot described in the section Design of a Mobile Robot (Starrenburg et al. ).
The application to cogging compensation of a linear motor will be described in a little bit more detail.
Such a motor is a commonly used element in assembly machines. Even with the best magnets and accurate
assembly the error could not be made smaller than 100 µ, with a PID controller in combination with
nonlearning feed-forward control. The design goal was to improve the maximally achievable accuracy
from 100 µ to less than 10 µ. Figure 21.24 shows a picture of a linear motor.
According to the structure of Fig. 21.23 the linear motor is controlled by means of a PID controller, while
a B-spline neural network is present to learn the inverse motor model, including the nonlinearity due to
cogging. Cogging occurs in DC motors with permanent magnets. It causes more or less sinusoidal shaped
forces that depend on the position of the translator with respect to the stator. If these forces really had a
sinusoidal shape, they would be easy to compensate for by means of a feed-forward compensator. However,
this would require magnets with completely similar magnetic properties and very accurate spacing of the
magnets. An alternative is to design a controller that learns the disturbance pattern and compensates it by
means of a learning feed-forward compensator. An additional advantage is that such a system can also be
used to compensate for other nonlinear effects, such as friction. This has also been demonstrated in a part
of a flight simulator (a control stick) where friction forces spoil the feeling of a realistic simulation especially
at almost zero speed. Figure 21.25 shows that learning is almost completed after six training cycles.
Learning feed-forward control is an attractive method to compensate for nonlinearities that are present
in mechatronic systems, such as cogging and friction. The use of B-spline neural networks results in fast
convergence, relatively low computational effort, and a good generalizing ability. Because of recently
obtained results with respect to the stability of such systems, robust control systems can be designed.

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