Simulation Fitting Method for Setting Motion Parameters of Robotic ...

In which, , each represents the head and tail of the rod; J represents moment inertia, and in
this situation where rigid even rod rotate surrounding its end, 。
Then:
(2) For both head and tail free rods.
According to Newton second law:
And work energy relationship:
(4)
(5)
Then we get:
(6)
(7)
(8)
If we simulate w ith equal step, in one step, rod’s shift is proportional to and its rotate angle to .
4.2.2 Analyze the link
An obvious fact is that the link must not be broken, so the tail of a former rod should keep
pace with the head of the latter one. In this way, the plain shift of the latter
one will cause rotation
on the former one. Since the shift in every step is much smaller than the rod’s length, the rotation
angle is proportional to of the latter one. Suppose ∆ is the modifying rotation angle caused
by this effect, then:
∆ ∆ /
(9)
In which ∆ represents the plain shift of the latter rod; represents the length of the former. So
far we have calculated all crucial parameters of the model.
4.3 Experimental modeling and simulation
According to the physical model above, it’s easy to program and simulate with equal step
method in Matlab. When the precision comes to certain range, result is got. Flowchart of
simulation program is shown as Figure 7. The result of simulation is shown as Figure 8.
With SFM, we can repeat the simulation and get link shape at other time points one by one
and finally get every shape.
5 Comparison and Analysis
JLM performs well when the second derivative of the curve varies slightly, that is when curve
is close to straight line. Nevertheless, in general, accumulated error is unavoidable, especially
when the curve is concave or convex function.
SFM avoids that kind of error theoretically, and gets similar amplitude with the original curve.
Of course, different factors always violate each other. When the curve is close to straight line, the
link we get doesn’t stick tightly to the curve; instead, there is a small gap, which, in fact, can be
neglected. Besides, the computational complexity of SFM is a little higher than that of JLM.
Table 1 gives 2 kinds of error by the two methods. Linear error (err) is the integral of