Swing-Free Movement of Simply Suspended Objects Employing Parameter Estimation Page: 7 of 10
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Average Residual Residual
Measured Calculated Oscillation Oscillation
Test Object Lcngth Period Period w/measured w/calculated
_(m) (sec) (sec) (degrees) (degrees)
22.7kg payload 1.93 2.79 2.80 - 0.05 0.9 .
45.4kg payload 1.83 2.71 2.77 0.05 0.10
45.4kg payload 0.74 1.75 1.76 0.15 0.31
11.4kg Fuel rod
assembly model 1.0 1.64 1.66 0.14 0 .27
45.4kg payload, no
oscillation damping 1.93 - - 6.8 -
Table 1: Measured and Calculated Periods of Test Objects
percent of the measured values, and although the
resulting residual oscillations for the calculated
periods are about double the value of those with
the measured periods, both are over 95% smaller
than the oscillation that occurs with an uncon-
trolled move (last entry in Table 1). The results
in Table I show that the nonlinear least squares
estimator selected predicts parameters which al-
low 95% oscillation damped moves and a second
order method is not necessary for our conditions.
Thus, using the batch estimator to determine
the necessary parameters for oscillation damped
moves was successful. However, a few precautions
must be taken to insure success with this method.
For example, the initial guess for the initial an-
gle must be larger than the guess for the initial
angular velocity; we used nonnegative values for
both. Plus, an error in the initial guesses of up to
15% of the mass and up to 20% of the length of
the swinging object can be tolerated. The force
sensor's coordinate frame must be aligned with
The batch estimator is based on a model which
expects input data which is symmetrical about
the weight of the suspended object, and cannot,
calculate the desired parameters accurately if the
input force data is skewed. Also, the force data is
expected to be reasonably continuous. Our algo-
rithn checked to see if each force data point read
was similar to the previous readirg, to eliminate
a large noise spike, if one occurred. If the force
reading was unacceptable, another reading was
taken. If a number of sampling periods passed
between acceptable readings, the estimator would
not converge as closely as usual (i.e., our perfor-
mance function value was higher at the end of
the final iteration), and the value fur the period
was a few percent further away from the expected
value. Another effect to note is that noise in
the force readings will be more significant with
a lighter object, where the range of force read-
ings is smaller, than for a heavier object. In our
experiments, we used a force sensor with a range
of 500 pounds and a resolution of 0.24 pounds.
Thus, noise can have a larger effect with small
ranges of force data. A noise error of just one
least significant bit (0.24 lb) is a significant per-
centage (2%) of the range of our data for a typi-
cal test of our lightest test object. Ilowever, even
with these potential errors the I:,ich, nonlinear
least square estimator determine the model pa-
rameters which are sufficient to formulate 9"V
oscillation damped trajectories.
The batch, nonlinear least square estimator ac-
curately predicts the parameters of a variety of
simply suspended objects. The resulting oscilla-
tion damped moves using the predicted periods
were successful. Thus, moving simply suspended
objects with unknown parameters without in-
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Werner, Jill C.; Robinett, Rush D. & Petterson, Ben J. Swing-Free Movement of Simply Suspended Objects Employing Parameter Estimation, report, June 1, 1990; Albuquerque, New Mexico. (https://digital.library.unt.edu/ark:/67531/metadc1193169/m1/7/: accessed April 23, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.