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.

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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.

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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.

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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.

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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)

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Figure 15 Gear Error Comparison at 24°C (75°F)

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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.

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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

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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