The Kennedy Space Center Robotics Group recently completed development and testing on a novel approach to measure the mass properties of a rigid body. This unique design can measure the payload's weight, mass center location, and moments of inertia about three orthogonal axes. Furthermore, these measurements only require a single torque sensor and a single angular position sensor

Doty, Keith L.

Graham, Todd A.

Wolcott, Kedron R.

English

NASA Technical Reports Server

N94- 33299
INNOVATIVE MECHANISM FOR MEASURING THE MASS PROPERTIES OF AN OBJECT
Kedron R. Wolcott
I-NET Inc.
NASA John F. Kennedy Space Center
Kennedy Space Center, Florida
Todd A. Graham
NASA Kennedy Space Center
Kennedy Space Center, Florida
and
Keith L. Doty
University of Florida
Gainesville, Florida
Abstract
The Kennedy Space Center Robotics Group recently completed
development and testing on a novel approach to measure the mass
properties of a rigid body. This unique design can measure the
payload's weight, mass center location, and moments of inertia about
three orthogonal axes. Furthermore, these measurements only require
a single torque sensor and a single angular position sensor.
1. Introduction
This paper describes the results of KSC's development and testing
efforts. First, a description of the mechanism will be given along with
its principle of operation. Next, experimental results will be discussed,
and a description of the analytic studies will follow. The paper will
conclude with a summary of the results and recommendations for
future study.
2. System Description
The actual mechanism developed and tested by the Robotics and
Automation Group is shown in Figure 1. A schematic representation of
the device is shown in Figures 2-4. U is a shaft whose orientation is
parallel to the hypotenuse of a cube. U can be rotated to any angle O
from an initial position and fixed. A is a shaft rigidly attached to U at
an angle a = 0.9553 rad (54.7°). When O = 0 rad (0°), A is vertically
oriented. B is a circular platter with a center B*. B can be rotated
relative to A about the line OB* to any angle 8 from an initial position
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and fixed. E is the payload and it is rigidly attached to B. The mass
center of E is E*.
The weight of the payload can be calculated by holding O fixed and
first measuring the static torque in U. Then, after moving E with
respect to B a known amount and direction, the static torque in U is
again read. The weight is calculated from the difference in the static
torque readings.
The first mass moment vector of the system is mgr, where mg is the
weight of the payload and r is a position vector from O to E*, as
depicted in Figure 5. This vector can be calculated from three sets of
measurements, where each measurement set consists of samples of the
static torque in U and samples of the static angular position O of the
system. Furthermore, each set of data is taken when the system is in a
different orientation, where a particular orientation of the system is
described by a value of O and 8. Three different orientations, and hence
three data sets, are required to calculate mgr. The position vector r can
be calculated by normalizing mgr with the weight of the test specimen.
The system's total moments of inertia ltzl, Itz2, and Itz3, about three
orthogonal axes parallel to zl, z2, and z3, respectively, can be calculated
by taking three sets of dynamic torque and dynamic position
measurements, one set per axis. The total moment of inertia Itzn
includes: the central moment of inertia of the test specimen E about an
axis parallel to Zn, the tare central moment of inertia about an axis
parallel to Zn, and the parallel axis term md 2, where m is the combined
mass of E and B and d is the minimum distance between the B and
system mass center combination and the axis of rotation.
Figures 6-11 provide an illustration of the three dextral, orthogonal
axes z l, z2, and z3. The first set of dynamic measurements is made by
rotating U in a sinusoidal motion, with 8 = 0 rad (0°), as shown in
Figures 6 and 7. Dynamics measurements are taken for O and for the
torque in U. From this data, the system's total moment of inertia Itzl
about z l, an axis parallel to U, can be calculated. B and the system are
next rotated to 8 = 2z/3 rad (120°), and the process is repeated. Itz3, the
total moment of inertia about z3, can then be determined, as indicated
in Figures 8 and 9. This is again an axis parallel to U as before, but z3 is
perpendicular to Zl. Finally, B and the system are turned to 8 = 4=/3 rad
108
(240°), and the total moment of inertia Itz2 about z2, the third
orthogonal axis, is calculated, as shown in Figures 10 and 11.
The payload's central moments of inertia about the orthogonal axes
can be determined by simple subtraction of the tare terms (system
inertia) from the respective total moments of inertia.
3. System Testing and Results
Extensive testing was done on the prototype shown in the attached
photograph. The test object was an aluminum block, 0.302 x 0.203 x
0.140 m (11.9 x 8.0 x 5.5 in) and weighed 221.5 N (49.8 lb). The torque
sensor was a JR3 3-axis Force Torque Sensor with a full scale (FS)
torque reading of 211.9 N.m (1875 in.lb) and an accuracy of +--1% FS
about the axis of concern. The angle O was measured by a Rotary
Variable Differential Transducer.
The results of the testing are given in Table 1. The determination of the
weight and mass center location was conducted with static
measurements, and the determination of the moments of inertia was
done through dynamics measurements.
Table 1: Experimental Results
Measurement Typ
Weight
m_r
Itzl
Accuracy
4.9%
not measured
not measured
Repeatability
not measured
+3.5%
-I-10%
The prototype was not configured to easily measure the weight of
the payload, as per the procedure outlined in the System Description.
However, one weight measurement was conducted, to experimentally
verify the procedure. The system was held at a fixed O - 0 rad (0°), and
the static torque in U was measured with the payload in an initial
position. Next, the payload was moved 0.076 + 0.0016 m (3_.0 + 1/16 in)
in a known direction and the torque in U was again determined. From
there, the weight of the specimen was calculated, and that value
compared to the known weight. Since only one experiment was
conducted, the repeatability issue was not addressed.
109
The second row of Table 1 provides the repeatability results for the
first mass moment vector mgr. The numerical value for mgr was
calculated in 30 experiments, and the minimum-norm, least-squares
result of those experiments was used as the standard for comparison.
As mentioned above in the System Description, each experimental
calculation of mgr takes three sets of measurements. Consequently, 30
experiments would normally require 90 data sets. For the sake of
efficiency, the 30 experiments were constructed using permutations of
30 measurement sets--10 sets taken at each of three different
orientations of the mechanism. The three orientations were:
(o,s)= [(+35°,0°),(-35°,120°),(-35°,240°)I
Each set of data was made from 3000 samples of the static torque in U
and 3000 samples of the position O. The result listed in Table 1 is the
largest difference between the 30 calculated values of mgr and the
standard value. The accuracy issue was not addressed since it was
believed that benefits-to-effort ratio would not be favorable for this
first-generation prototype.
The third row of Table 1 lists the repeatability results for the total
moment of inertia, Itzl. The repeatability result was resolved from
repeating the same experiment 10 times. In all cases, 8 = 0 rad (0°).
For each experiment, the system was first tilted at an angle O such that
the effects of gravity were minimized. Next, the system was manually
oscillated about U at a frequency of approximately 8 Hz and 5000
samples of the dynamic torque in U and 5000 samples of the dynamic
angular position O were taken. From that data, Itzl was calculated. The
minimum-norm, least-squares fit to the results of the 10 experiments
was used as the standard. The repeatability value was the largest of
the differences between each of the experiments an_ the standard
value. Again, the accuracy was not addressed for the reason given
above.
4. Analytic Studies
Analytic studies were made to model the mechanism's static and
rigid body dynamic characteristics, and these studies were used to
develop techniques for data analysis. The initial study was performed
using Kane's Method of Dynamic Analysis. A redundant analysis was
conducted with a Lagrangian Formulation.
110
The dynamics for determining the payload's moments of inertia are
x=_g+ltznO (1)
where x is the dynamic torque measured in U, Zg is the gravity torque
-- which results from the payload's mass center being offset from the
axis of rotation, and Itzn is the total moment of inertia. To determine
Itzn, the following equation was used:
(2)
where X = 5000 represents the number of samples taken.
The numerical attributes of this approach made it necessary to
simultaneously minimize xg and maximize O. By initially tilting the
system to a particular value of O such that the system was "balanced",
i.e., xg- 0 N.m, and oscillating the system about that point with only
small displacements, the effects of zg could be kept at a minimum.
Furthermore, since the amplitude of the oscillation was small, the
frequency had to be very high in order to maximize O. Thus, the
system was jogged as fast as possible by hand, which was at a
frequency of approximately 8 Hz.
The analytic studies also provided a very important insight into the
measurement of mgr: a system configuration was determined that
optimized the numerical characteristics of the mgr calculation. If this
calculation is made with the system in the optimum configuration, the
accuracy of the mgr measurement is equal to the accuracy of the torque
sensor used to collect the data. Consequently, this system is capable of
measuring mgr to an accuracy of +_0.1%, the accuracy of many
commercially available torque sensors. The optimum configuration
follows a function of 8, O, and a. The optimal selections for 8 are at u
rad., u + 2_r/3 rad, and u + 4x/3 rad, where u is an arbitrary initial angle.
These selections for 8 are independent of O and a. The optimal values
111
for O and a, however, are not independent of each other; rather, they
are related by the equation
3 sin2(O) sin2(a)= 1 (3)
For example, in the model presented in the System Description,
cx = 0.9553 rad (54.7°), so the optimum value for O is +_/4 rad (.+.45°).
Therefore, for optimum numerical characteristics in the calculation of
mgr, the orientation of the payload must be at O = +_/4 rad (.t_45°), and
8=u, 8=u+2x/3, and 8=u+4x/3.
The model was also used to study the merits of alternate
configurations of the geometry, such as that shown in Figure 12. This
configuration, in particular, minimizes the amount of tilt, i.e. O, at which
the payload must be positioned. In this concept, a = '_/2 rad (90°), so
from the equation, the optimum value for 0 = 0.6155 rad (35.26°).
5. Lessons Learned
The results emphasize the difficulty in determining the moments of
inertia. While it is theoretically possible to measure the moments of
inertia with this design, steps were necessary to achieve even
repeatability results of +10%, such as the extreme care taken to
minimize the effects of gravity: tilting the system until it was
"balanced" about U and shaking it at _8 Hz. A torque sensor that
matched the measured torque more closely would substantially
improve the results, since the maximum torque read during the testing
was +33.9 N,m (+300 in-lb), only 16% of the FS torque.
Friction was also more of a problem than anticipated. Originally, it
was believed that the friction forces would not affect results
appreciable since they would induce negligible torques when compared
with the torques necessary to drive the system. However, friction and
stiction significantly influenced the "balance point" of the system.
Instead of a true point, there was a balance range of +0.0873 rad (+ 5°).
Consequently, the effects of '_g were not minimized to the greatest
extent possible. Replacement of the roller bearings with air bearings
would be one possible solution to this problem.
Measuring an object's weight with this approach has not been
rigorously tested but only basic feasibility determined. The results
listed above in Table 1 could likewise be greatly improved with a more
112
appropriate torque sensor, but mechanically simpler and more accurate
methods may prove to be more practical.
This project has demonstrated the ease with which modern
prototyping can be done. The mechanical design of the actual
mechanism was carried out using Intergraph, so blueprints could be
generated from an initial concept in a matter of minutes instead of days.
The data acquisition system was developed with the National
Instruments LabVIEW, which allowed for the necessary acquisition
software to be written in 2 days--by an engineer, not a programmer--
and the electrical hardware setup to be finished in a single day. Finally,
the analysis was done with the numerical package MATLAB, a program
that readily allowed for the manipulation of literally over hundreds of
thousands points of data. Essentially, the power and the ease-of-use of
commercially available equipment now allows for the physical testing of
a concept in a remarkably short period of time.
6. Conclusions & Recommendations
The mechanism holds significant promise for the measurement of
the mass center location of an object. As delineated earlier, an optimal
combination of payload orientation exists that allows for the calculation
of the mass center to an accuracy equivalent to the accuracy of the
torque sensor used in the implementation. Consequently, a
mechanically simpler configuration, like that shown in Figure 12, could
be built that would measure the mass center location of a payload with
an accuracy of 0.1%, the accuracy of many commercially available
torque sensors.
Additional work needs to be done to refine torque measurement
techniques and the mechanism design to enable accurate measurements
of the mass moments of inertia of an object. The challenge experienced
with the prototype device was that the dynamic torque was roughly an
order of magnitude less than the gravity torque. Innovative methods
for correcting this problem need to be developed in order for
measurement of mass moments of inertia to be pursued any further.
Additionally, mechanism design changes should be made to eliminate all
aspects and effects of friction (e.g., air bearings).
In conclusion, the testing done on the prototype confirmed: 1) the
feasibility of accurately measuring an object's center of mass, and 2) the
difficulty in measuring moments of inertia of a payload. A derivative of
the prototype design, used in conjunction with a device that can
113
accurately measure the weight of the payload, could yield a system that
has the capability of accurately and easily measuring the mass center of
a payload.
7. Acknowledgments
The authors would like to thank William Jones, James Spencer,
Richard Bennett, Rhonda McNulty, and Ronald Remus for their
engineering support, leadership and contributions. Thanks are also
given to the NASA Summer Faculty Program and I-Net, the Engineering
Support Contract.
114
Figure 1. Prototype Mechanism
115
QFigure 2
Figure 3
116
®Figure 4
Figure 5
117
Figure 6
Figure 7
118
Figure 8
!
ad
Figure 9
119
Zl Z3 Zl
Figure 10
8 = 4n _ rad
Figure 11
120
Figure 12
0
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