Swing-Free Movement of Simply Suspended Objects Employing Parameter Estimation Page: 5 of 10
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m - mass of pendulum,
I - length of pendulum
0 , 90 - initial conditions of pendulum,
g - gravity,
T - cable tension, and
w - frequency = V/l,
which produces the nonlinear measurement
Y(ti) = F1(ti) =rng Cos29.+ cosi ,(9)
where 0i = 90coswt + t-sinwti.
Note, using only one channel of force data, F.,
simplifies the development of the estimator algo-
rithm. While Fy and F2 forces are also available
from our force sensor, our algorithm may be ap-
plied on equipment where the force from only one
direction is available.
The weight matrix, W in Eq (4), is a diagonal,
positive definite, constant matrix, and the Jaco-
bian matrix is analytically developed from Eq (9)
by taking the first partial derivatives with respect
to 0, 0, m, and w.
A L 50Y Y 0Y 0)!]
800W80 m C9L
OY(ti) _ OY 0O Y 00i Oh
m [-g sin 2 0i coswti
+ l cos90 (2w29osin2wt, - w 90
- 1i coswti sini )
aY(ti) OY 09 0 0Y O9.
09 - 9 090 00i 090,
OY(ti) _ YY 0 DO00; _
Ow _00i Ow +00lw
mg -sin20i -t; - 2sintt -i
+ t coswti) + Cosoi 2001cos2wt
(9i + 9 5~ 2 292
(02 + - Bti)sin2wt; - 32cos2wt)
- sini ( ti90sinwti - 92 siniot;
+ -t icosti)
Equations (9) and (10), the weight matrix, and
the force sensor measurements are the required
elements of the GLSDC algorithm that provides
the estimate of the frequency (period of oscilla-
tion) to the swing-free trajectory planner. The
computer implementation of the GLSDC algo-
rithm is discussed in the next section,
m - sin 2 0i sinwti
+ icosO, [-w 0sin2wti + 2 0Ocos2wt
l 2 \
- -Oi sinl sinwti )
a-, OL0i a Doi~
---- - 8 i n + -r -i ( =
m 080 m ( Com
g cos29; + I(9 ) cosO,
The adaptive, swing-free trajectory planner is
implemented on a CIMCOR1P XR.6100 robot at
(11) Sandia National Laboratories. A simply sus-
pended object is held by the robot gripper, and a
force-moment sensor (manufactured by J R3 Cor-
poration) capable of transducing the forces in
the three principal axes and the corresponding
moments is used to monitor the oscillations of
the payload. Four different suspended objects
are used to test the adaptive, swing-free plan-
ner: a 22.7kg (50 pound) weight suspended from
(12) a 1.93m (76 inch) cable, a 45.4kg (100 pound)
weight suspended first from a 1.83m (72 inchi) ca-
ble and then from a 0.74m (29 inch) cable, and a
1/5 scale (1m (39 inches) tall) 11.1kg (25 pound)
model of a fuel rod assembly on a pivot joint (see
(13) Figure 4). The readings from the force sensor are
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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/m1/5/: accessed April 18, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.