![](bg2.png)
position to the next position on the linear track, the
positional changings of some measurement points on
the rigid object are only caused by the linear track.
The very high reapeatability of modern industrial
robots makes it possible to use the robot as such a
rigid object. The inaccuracy in moving to one point
repeatedly is in most of the cases more then 10 times
less than the inaccuracies caused by the linear track.
In (7) there is an overview about the absolute and
repeat accuracy of the used robot.
One important step in the process of identifying
the linear track is the determination of the needed
number of sampling points on the track. This value
is depending on the maximum error which has to be
identified and varies with each individual linear track.
A special procedure developed before is here used
again and is in certain ways optimised. For this,
the robot is moved along the linear track with its
Tool-Center-Point (TCP) in one constant position.
During the robots movement it is measured by
the external measurement system in a continious
measurement mode. This creates measurement data
from a theoretical straight line movement disturbed
by the inaccuracies of the linear track on the one
hand, the robots own vibrations and the measurement
inaccuracies on the other hand.
So if we sperate the theoretical path from the mea-
surement results we get:
h
k
=
p
∆x
2
+ ∆y
2
+ ∆z
2
− s
k
(1)
with
s
k
= k · v · dt , k = 0, 1, . . . , N − 1 (2)
Taking this function h
k
we make the discrete fourier
transform on it and get the spectrum of the non-
linearities of the track. One special aspect not con-
sidered before is the spectral leakage effect of time
delimited functions. Considering the finite number of
measurement values taken in a finite time intervall,
the number of the discrete function values of h
k
is
N − 1 as can be seen in (1). In respect to this, there
is a convolution with the spectrum of the rectengular
function in the frequency domain.
So:
H(f
n
) =
Z
∞
−∞
h(s)e
−j2πf
n
s
ds ∗ G(rect
n
) (3)
and
G(rect
n
) = T
p
· si(πfT
p
) , T
p
=
1
f
n
(4)
This convolusion causes new spectral components
which are not included in the measured signal
whereby the original spectrum can not be analysed
properly. The solution for this problem is finding a
adequate window function, where the leakage effect
is lower, so that the important spectral components
can be detected properly.
The appropriate window function, a bartlett window
was found in some special analyses and after some
experiments in comparing different window functions
on the measurement values the discrete fourier trans-
form gets now an improved spectrum of the linear
track scan depicted in figure 2:
−0.005 0 0.005 0.01 0.015 0.02 0.025
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
spectrum of h(t) with bartlett window
frequency [1/mm]
amplitude [mm]
Figure 2: Spectrum of the track scan.
Due to the windowing three different frequency peaks
can be seen clearly. The first peak with an amplitude
of 0.7 mm at a frequency of 0.0003/mm represents the
maximum error caused by the track which was mea-
sured and as you can see it is at the lowest frequency.
The other two peaks at 0.003/mm and 0.006/mm are
caused by the robots vibration and by the measure-
ment system.
To get the needed number of sampling points for an
identification of the linear track the new spectrum of
the windowed function is used, considering the sam-
pling theorem. With this new results the number of
sampling points can be reduced by 25%.
3 POSITIONING ERRORS DUE
TO SPLINE LINEARISATION
The benefit of the interpolation of the identified track
coordinate systems via cubic splines is a continous
description of the linear track in 6 dimensions
(x, y, z, α, β, γ). As already mentioned the single
track coordinate systems are identified in the sam-
pling positions depending on the frequency spectrum
after the continous track scan. So what we get after
calculating the track coordinate systems is:
T
R
i
W
= f
i
(α(l
i
), β(l
i
), γ(l
i
), t
x
(l
i
), t
y
(l
i
), t
z
(l
i
))
(5)
ICINCO 2006 - ROBOTICS AND AUTOMATION
470