147
Correcting for Flexibilities
In this section we present simulations obtained for a manipulator
with flexible links. Again, as in the last section, the manipulator is
to move over a nominal trajectory. Here we wish the hand to actually
follow the nominal trajectory and this is to occur even when a distur
bance force is introduced which causes bending in the links. The scheme
to accomplish this task was discussed in Chapter Six and requires that
the joint angles track reference signals which are computed to offset
link flexibilities.
Figure 7-22 shows the way in which the bending of a link is
modeled. The constants Kv>p, KVjy, K^p, and K^y are taken to be the
same for each link and are given as follows
Kv>p = 2000 m/N
Kv y = 800 m/N-m
K^p = 800 m/N
K^y = 240 rad/N-m
It is assumed that the bending which occurs in the links is a result of
the vertical force acting downward on the hand.
The results of the simulations are given in figures 7-23 through
7-25. Here we consider only a constant disturbance and hence the
internal model system is constructed using integrators. Figure 7-23
shows the actual and desired hand trajectories which occur when a 50 N
disturbance is applied and no compensation is used (i.e., corrective
reference signals are not supplied). It is seen that a significant
amount of error is present. In Figure 7-24 we then supply