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N91-24620
HARMONIC DRIVE GEAR ERROR: CHARACTERIZATION AND COMPENSATION
FOR PRECISION POINTING AND TRACKING
Ted W. Nye and Robert P. Kraml*
ABSTRACT
Imperfections and geometry effects in harmonic drive gear
reducers cause a cyclic gear error, which at a systems level,
results in high frequency torque fluctuations. To address
this problem, gear error testing was performed on a wide
variety of sizes and types of harmonic drives. We found that
although all harmonic drives exhibit a significant first
harmonic, higher harmonics varied greatly with each unit.
From life tests, we found small changes in harmonic content,
phase shift, and error magnitude (on the order of .008 degrees
peak-to-peak maximum) occurred for drives with many millions
of degrees of output travel. Temperature variations also
influenced gear error. Over a spread of approximately 56°C
(100°F), the error varied in magnitude approximately 20
percent but changed in a repeatable and predictable manner.
Concentricity and parallelness tests of harmonic drive parts
resulted in showing alignments influence gear error amplitude.
Tests on dedoidaled harmonic drives showed little effect on
gear error; surprisingly, in one case for a small drive, gear
error actually improved. Electronic compensation of gear
error in harmonic drives was shown to be substantially
effective for units that are first harmonic dominant.
INTRODUCTION
Spacecraft mechanisms of the future are requiring more
precise pointing and tracking features while maintaining
simplicity, long life, and low weight. One key component that
enables space mechanisms to meet these objectives are harmonic
drive gear reducers. Shown in Figure i, harmonic drives are
well known for their high gear reduction ratios, low weight,
small volume, zero backlash, and high efficiency. For the
many beneficial features they possess, there are at least two
undesirable characteristics, namely soft torsional stiffness
and gear error. Soft torsional windup results in undesirable,
low frequency vibration modes in appendages that are
susceptible to excitations from gear error or motor ripple
torque anomalies. At modal frequencies, the cyclic torque
disturbances (or jitter) will resonate payloads and the
spacecraft bus. In this article, we discuss gear error in the
harmonic drive, what it is sensitive to, and what can be done
about it.
* TRW, Space & Technology Group, Redondo Beach, California
237
https://ntrs.nasa.gov/search.jsp?R=19910015306 2018-09-06T21:36:44+00:00Z
WAVE
GENERATOR
Figure 1 Harmonic Drive Section View
Gear error or positional accuracy is the difference
between the theoretical and measured angular position of the
input and output of a gear reducer. Typically, we see
integer values for a gear ratio. What we actually measure is
the integer value plus a small cyclic error imposed upon it.
As mentioned previously, this small cyclic error can cause
substantial disturbances in appendages.
Sasahara et al. in [I]** documented vibrations in an
industrial robot when driven near resonance. Acce!erometer
data presented showed a beating behavior where the amplitude
was influenced by torsional stiffness and circular spline
deformation. They concluded that radial tooth errors and
tooth meshing composite errors are large contributors to
positional accuracy problems. In [2], Ahmadian developed an
expression for geometric error in harmonic drives by use of
parametric curves to model flexspline deformation. He
concluded with a theoretical model that captures th 9 first
order effects of gear error behavior due to geometry. A
brief, applications oriented discussion on positional error is
presented in [3]. Also included is a rough estimating
relationship to quantify the error and figures that show the
nominal behavior over one revolution of travel.
Harmonic drive gear error could be described in two
categories - that caused by internal effects and that caused
by external effects. Internal influences are due to geometry,
tolerances, tooth shape, and material. The last item has an
influence because of machining dimensional stability.
External influences on gear error are due to usage and
environment such as changes over life, temperature, mounting
alignment, incorrect assembly such as dedoidal, over-load
effects such as ratcheting, and how error changes with
external load.
** Numbers in square brackets refer to references.
238
MEASUREMENT TECHNIQUES
Figure 2 shows our test setup used for determining
positional error. Harmonic drives were mounted in assemblies
(actuators) containing a stepper motor and support bearings.
Both input and output shaft positions were measured with 19
bit optical encoders. The test was controlled by a PC/AT
compatible computer via several interface cards to peripheral
devices. Two encoder display readouts were used to visually
compare input and output shaft positions. The stepper motor
was controlled with an indexer/driver arrangement which
allowed the motor to be operated at a selectable power supply
voltage and command rate. For life tests, the actuator
assembly was enclosed in a vacuum chamber to simulate a
spacecraft environment. For thermal tests, the entire
encoder/actuator assembly was enclosed in an oven. The
majority of measurements were performed at room temperature in
a laboratory environment.
Our test procedure involves these basic steps:
i. Zero out input and output encoders at the starting
position.
2. Make sure fixture was stable.
3. Step the actuator to a new position by a pre-defined
step size.
4. Compare the difference between output and scaled
input, this is gear error.
5. Continue steps 3 and 4 for at least one outputrevolution.
Results were stored on a hard disk, then analyzed and
plotted. This setup has proven to be accurate and repeatable
for many tests involving harmonic drives.
Figure 2 Gear Error Test Station
239
GEAR ERROR CAUSES
Although a detailed analytical study was not performed
with regard to the causes of positional accuracy variations,
discussions with harmonic drive manufacturers have indicated
some qualitative explanations. For the low cycle beating
effect, maximum error is caused by the errors in two parts
combining to produce the total error. As the components
rotate relative to each=other, some errors cancel, which
produce the minimum error.
Most of the first through ninth harmonic effects are
attributed to:
i. Tooth placement errors on the flexspline.
2. Tooth placement errors on th@ circular spline.
3. Out-of-roundness of the circular spline.
4. Variation in wall thickness between the flexspline
pitch diameter and bore.
5. Flexspline out-of-roundness.
6. Bearing outer race out-of-roundness.7. Variation in wall thickness between the outer race
ball grove and the Outside diameter,
8. Lack of concentricity between the flexspline and
circular spline in the assembled position.
9. Lack of squareness between the flexspline and
circular spline in the assembled position.
i0. Fit-up between the flexspline and circular spline
teeth in the assembly. :
Gear error improves as the harmonic drive size increases.Smaller units become much more sensitive to dimensional
tolerances, hence, they are also more likely to contain higher
harmonics. A formula given in [3] for the upper bound of
positional error is:
8 (z)
_7
Where E is the peak-to-peak error in arcminutes and d_
is the pitch diameter in inches. This relationship provides a
good estimate for a commercial unit but substantial
improvement in performance over this has been demonstrated for
flight hardware (typically less than half of the above
prediction) .
240
!
GENERAL CHARACTERISTICS
Harmonic drive positional error is a frustrating
phenomenon due to fact that two seemingly identical units may
have completely different error signatures. Figures 3 and 4
show such behavior, where for similar units, the second
contains harmonics approaching 55 percent of the fundamental.
Without a positional accuracy test, one cannot, for example,
make simple dimensional checks or get indications of error
from other measurements such as friction or stiffness. The
general full rotational behavior of gear error is shown in
Figure 5. Typically, one sees a characteristic beating
amplitude modulated twice per output rotation. There are
sometimes exceptions however, as measured in one unit shown in
Figure 6. Positional error seems to be independent of gear
ratio, but for reference, all units tested in this report had
i00:i reductions. Materials used for the drives in these
tests were primarily 304, 17-4, 15-5, or equivalent stainless
steels.
"Well-behaved" gear error contains few or no higher
harmonics where "poorly-behaved" contains many high frequency
components (superimposed upon the fundamental). This
qualitative label is given because of the ability to digitize
and electronically compensate a given error signature. More
discussion on compensation follows in a later section. The
harmonic content of a typical gear error plot was analyzed
with Fast Fourier Transform software and is shown in Figure 7.
eee6
-o._I0.
-e.e15
i i i ! ! ! _ i i
..............i.........i.........!........._.........i.........i.........i.........i.........!.........i
.............. i.................................................t ........................................
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OUTF'Jr _or_rr_ _0_
.............!.........!.........'_.........',..........i.........i.........i.........i.........}.........i
e _ i i i i i i i ! i
.... [ _ [ " [ i
Figure 3 Well-Behaved
Gear Error
Figure 4 Poorly-Behaved
Gear Error
241
J
Figure 5 Typical Gear Error Figure 6
Over One OutputRevolution
Exceptional Gear
Error Over One
Output Revolution
Harmonic errors occur in factors of 2 of input rotations.
For example in Figure 7, the largest harmonic corresponds to
the first harmonic, occurring twice per input rotation. The
second harmonic occurs every 4 cycles of input rotation. In
many instances we observed a large ninth harmonic, occurring
every 18 cycles per input rotation.
__ ..............i.............i......i......_....._.............!....._......_......i............i......°'_'_.............!............i......i......i.....!.............i.....i......i......i"...........i.......
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0 _ .4 6 8 lO 12 14 I_, 18 28 _ 24 26 _
FF_'E_OUE<Y (CYQ_ES_INPtJT ROTAT_CI',D
Figure 7 Frequency Plot of Harmonic Drive Gear Error
242
Recently there have been developments of a new (IH) toothprofile as discussed in [4]. The new tooth, shown in Figure8, is used with a wave generator plug exhibiting a shape thatallows for a higher quantity of teeth to be engaged without"ticking." Ticking is caused when gear teeth tips prematurelycollide before engagement. The newer tooth profiles (andperhaps very precise machining) seem to have improved thevariation and higher harmonics of gear error for these drives.Figures 9 and i0 show measurements on an IH tooth profileunit. Each IH tooth profile drive tested demonstrates well-behaved gear error. Higher harmonics were present but wereobserved at very high frequencies. The causes of the highfrequency component may be tooth engagement related but aredistinctly different from conventional tooth signatures.
Conventionaltype New type
Figure 8 Comparison of Harmonic Drive Tooth Profiles
.......
OUTPUTI_'_TATIONI
Figure 9 IH Tooth Harmonic
Drive Gear Error
over One Output
Revolution
Figure i0 Detailed View of
IH Tooth Harmonic
Drive Gear Error
243
BEHAVIOR OVER LIFE
For an electronic compensation scheme to be effective,
gear error must be stable in amplitude and phase over life.
Several wear tests have shown this to be generally true.
Figure ii shows gear error before and after a life test of
many million degrees of output travel.
"_ ........._"........."........._.........;.........i.........;.........'_.........L........._.........
_ ! i i i ! i i i
...............! _!.........! _-!........: _....._,".....}- !i.........
-0,eJg 2 ..........._..........
-e._f
Figure Ii Gear Error Change with Life
Data in Table 1 shows the error measured from life tests.
Note that the 8.1 cm (3.2 in) pitch diameter unit used for the
life test turned out to have exceptionally small gear error.
Very slight changes in the error wave forms did occur with
negligible phase shift. Spectral plots shown in Figures 13
and 14 for the 10.2 cm (4.0 in) pitch diameter harmonic drive
illustrated how the spectral components changed slightly over
life. A slight increase in harmonic content was expected due
to generation of wear debris. Wear particles get trapped in
the gear teeth and wave generator, thus they affectconcentricities and clearances. The counter to this effect is
Table 1 Gear Error Changes from Life Tests
Pitch Diameter p-p
Theoretica 1
Gear Error
8.1 cm (3.2 in) .042 deg
i0.2 cm (4.0 in) .033 deg
p-p
Beginning ofLife Gear
Error
P-P End of
Life Gear
Error
.009 deg .017 deg
.028 deg .029 deg
i
|
F
244
removal of material from high and low spots on the gear teethand interface surfaces. We could, on the other hand, expectthis effect to smooth out and reduce the gear error. None-the-less, our only conclusion was that gear error is onlyslightly affected with life.
o._i .......... !...-i-----F---i-----!----i_--_-----_------_-_----_---_...i
_._- ....... i .---.h----_.-----i.-----i-----!-----i..-...i..--..i.-.-..i.-.._------i------i-.-..-!
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Figure 12 Beginning of Life Gear Error Spectral Content
..........!----_[i----!i i-T_:"T '-_i-T"_°'_'_ .....T-T"'"F-'"T"'"_'"'"['_"'"!'""'F""T"T"F""T'- "T""i_: -.......]------6---..!.----._-----L-----!-----i------i---.-.!---.-.i---.-L--..!..-...F---!
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FRE_[_'C_" (CrC_ES/If_r _TATI_
Figure 13 End of Life Gear Error Spectral Content
TEMPERATURE SENSITIVITY
Increasing temperature changes gear error with absolute
shifts and small increases of peak-to-peak errors. Harmonic
content appears to slightly increase with higher temperatures.
Although most of these effects are small, Figures 14, 15, and
16 show the absolute shift and peak-to-peak growth with
temperature. Table 2 shows numerically how these variations
occurred for this 10.2 cm (4.0 in) pitch diameter test unit.
The significance of the absolute error shifts are moot due to
the fact that the peak-to-peak oscillations are primarily
responsible for dynamic disturbances that concern us. The
cause of the error change is unknown, however, similar testson harmonic drives for stiffness show substantial reductions
with increasing temperatures approaching 66oC (150OF).
245
• • b _Q
Figure 14
OU_='UT ROT_T _ON __S_
Gear Error Comparison at 52°C (125°F)
......... : .......................................................................................
F
.....iiiiiii{ii/_iiiiiiiilliiii[iiiiiiiiiiiilli[iiiiiiiiii
÷e.e,e
-ee_s
............................................i.......................................i.........[..........
Figure 15 Gear Error Comparison at 24°C (75°F)
; : ', 1 ......
OU'_UT IPOTAT|O_4 (_G)
ilii,iiiiiiiiiiiiiiiiii. ..;..i...i..._._..._...!...!.._.!.j../...:._.!.:..._....:..: ./._.:__!..:._:..:
(_u'rmuT _)TAT_ON _0_
Figure 16 Gear Error Comparison at -i ° (30°F)
246
Table 2 Temperature Effects on Gear Error
Temperature P-P P-P Measured 5th HarmonicTheoretical Gear Error Polar Fourier
Gear Error Coefficient
52oC (125°F) .033 deg .028 deg .00040
24°C (75°F) .033 deg .026 deg .00035
-IoC (30°F) .033 deg .023 deg .00030
ALIGNMENT SENSITIVITY
Positional error tests were run on a 8.1 cm (3.2 in)
pitch diameter unit to determine the importance of concentric
and perpendicular mountings. In the first test, several
flexspline mounting hubs were intentionally made off-center by
a given displacement. Gear error results from this effect are
shown in Figure 17. Sensitivity to concentric alignment
appears to be relatively small for tolerances that are typical
for aerospace assemblies. Commercial products with coarser
tolerancing however, could be affected with this runout. The
data supports using care in determining concentricity for
mounting assemblies.
The second test involved modifying the perpendicularity
of the circular spline with respect to the flexspline. Figure
18 shows how gear error increased with mounting angle error.
The same conclusion can be drawn as previous, that
perpendicularity must be carefully controlled.
e,s_ ................................................................................................
e.e34 ...................................................... i ........................................
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eels .................... _................................................................................
...................i.............................................................................i
......................!..............................................................................i@e12s
_xs_I_ _ OffSET _I_
Figure 17 Flexspline and
Circular SplineOffset
e e_ .......................................................................................
e eTe .................................................................................................
e e3_ ....................................................................................................
ee_ ..................................................................................................
!:= ............;...................i...................!...................!........_-ii ® : ol _ i.....!i i!!!i)i
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.....................i.....................................}........................................
e._ e_,se e,_e e._s_ e. L,ee e_e
C_RCUL_ Sm.I_IE rlLr _DE_I
Figure 18 Flexspline and
Circular Spline
Perpendicularity
247
OR|GIN_L PAGE IS
0_, POOR _RJALITY
EFFECTS OF RATCHET AND DEDOIDAL
Ratcheting is an over-load condition where, due to a high
output load, the harmonic drive will slip one or more teeth.
Many times this ratcheting will leave the harmonic drive in
the dedoidaled condition. Dedoidal is commonly labeled to a
harmonic drive where the wave generator and flexspline gears
are no longer concentric with the circular spline. Rather
than the oval wave generator and flexspline teeth being
centered in the circular spline, they are slipped over to one
side. This effect is more commonly encountered during
incorrect assembly. Symptoms of dedoidal are large variations
in starting friction torque and a significantly reduced wearlife.
Tests of gear error changes after high loads and ratchet
were performed on 3.6 cm (1.4 in) and 8.1 cm (3.2 in) pitch
diameter harmonic drives. The smaller units had large gear
error prior to testing and did not exhibit the typical
modulated two lobed beating pattern as shown for one set of
data in Figure 19. Dedoidal conditions did not significantly
change the error signatures as seen in Figure 20, but
surprisingly, slightly improved the error amplitude. To
ratchet our units, the flexspline and circular spline were
fixed, and the wave generator was turned until tooth skippingoccurred. Each unit was ratcheted at least four times in
separate tooth locations with each test resulting in a
dedoidaled condition. After ratchet, the gear error again did
not seem significantly affected. Figures 21 and 22 show the
amplitude and characteristics were comparable, but the
dedoidal caused an absolute shift. Tests on the larger units
were done in the same manner as the smaller units, but were
not torqued to a ratchet condition. Gear error changes were
again small. We concluded that gear error is not an effective
means to verify the load history of a harmonic drive, nor can
it be used to determine dedoidaled conditions.
GEAR ERROR COMPENSATION
Gear error compensation is accomplished by measuring the
error of a given harmonic drive and then modifying an input
motor voltage appropriately to effect an error cancellation.
Figures 23 and 24 show the before and after result of
compensation.
248
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ORIGINAL PAGE 15
OF POOR QUALITY249
,,._- i' ' i .... ' -i : _--_-_-- _ .... i- +
-e+e_6 i , ........
(_lr_JT 5_CjPttlO_ IIjEOI+::_ _++.:+:
Figure 2 3 Uncompensated Gear
Error+
+_m+I_+_+_+_:_+_+_+_+_+_+_P_`_f__:em+---+---_--4---_---p---[--+f---[---f+-[+--[---[--+[[r[:i::
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....+++y++++++y+y++++++++_ i i i i i ! !_ ! i ! i ! i i ! ! ! !
Figure 24 Compensated Gear
Error
To implement this scheme, a sine wave excited stepper
motor was driven in a variable frequency fashion. This means
for one sine wave that would cause 15 degrees of motor output
rotation for example, the voltage waveform was artificially
shortened or lengthened to effect an output position
compensation. Figures 25, 26 and 27 show graphically the
compensation effect. Compensation is implemented with a look-
up table (and linear interpolation algorithm) that contains
sufficient entries to cancel as many higher harmonic terms as
desired. For our 15 + stepper system with i00:i gear ratio, we
chose one compensation voltage point for every 15 degrees of
motor shaft rotation, or 24 entries per revolution. Thus, our
compensation table contained 2400 entries.
CORR£SPONDS TO 15+ MOTOR S_ ROTATION
\
v_ ANGLE
N
3
Figure 25 Nominal Motor Input Voltage
250
MPENSAONPO
COMPENSATION POINT
Figure 26 Compensated Motor Input Voltage
o_
F
_ t T] lr] IFI_FTI rT] i i i i i i i , i i i i , i_ COMPENSATED
TIME
Figure 27 Actuator Output Displacement with Time
The need for "well-behaved" gear error that is first
harmonic dominant is now evident. Canceling the higher
harmonics is required above the Nyquist frequency. The
alternative is a much larger compensation or look-up table.
The Nyquist frequency of our system was 12 cycles per
revolution of the input shaft, which meant we could cancel up
to the sixth harmonic. Some harmonic drives contained a
strong ninth harmonic that, with our chosen frequency, could
"alias" to below the Nyquist frequency. Higher harmonics such
as this were therefore filtered out from the compensation
table.
251
This compensation scheme has worked well provided the
harmonic content of gear error does not become too large. The
total positional error, which translates into a torque ripple
when gimballing a payload, can be typically reduced as much as
80 percent over a non-compensated system. The idea has been
implemented on flight hardware to increase pointing/tracking
accuracy and reduce on-orbit vibrational disturbances.
SUMMARY
Harmonic drive gear error was studied for the effects of
unit size, tooth form, life, temperature, alignment and over-
load conditions• We verified that gear error decreases with
larger size and always demonstrates a significant first
harmonic• Life tests showed the error to be fairly stable in
amplitude and phase for many million degrees of output travel•
Our tests showed gear error increased slightly with
temperature and misalignments between components. Overload
conditions resulted in little change to gear error signatures,
including tests run in the dedoidaled condition. Finally, a
compensation scheme was presented that takes measured error
anomalies for the gear reducers and electronically correctsfor them via the motor driver.
.
•
•
•
REFERENCES
Sasahara, M., Zhang, Y., and Hidaka, T., "Vibration
of a Strain Wave Gearing in an Industrial Robot,"
ASME Proceedings of the 1989 Power Transmission and
Gearing Conference, 1989, pp. 789-794•
Ahmadian, M., "Kinematic Modeling and Positional
Error Analysis of Flexible Gears," Clemson
University research report under grant No. RI-A-86-
8, 1987.
Harmonic Drive. Precision Reduct__ign Gear_.nq,
Designer's Manual, Harmonic Drive Division, Quincy
Technologies, Inc.
Harmonic .Drive Gearinq, S Series Design Brochure
based on the IH Tooth Profile, No. HI-I0-1988, HD
Systems, Inc.
252