Strapdown gyros which employ time modulation torquing techniqu

Beebee, W. S.

Sutherland, A. A., Jr.

English

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THE ANALYTIC SCIENCES CORPORATION
TR-101-5
DESIGN OF STRAPDOWN GYROSCOPES FOR
A DYNAMIC ENVIRONMENT
July 1969
Semi-Annual Report
Prepared under
Contract Number NAS-12-508
for
NATIONAL SPACE AND AERONAUTICS ADMINISTRATION
Electronics Research Cenier
Cambridge, Massachusetts
Prepared by:
William S. Beebee
Arthur A. Sutherland, Jr.
Approved by:
Arthur Gelb
THE ANALYTIC SCIENCES CORPORATION
6 Jacob Way
Reading, Massachusetts 01867
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THE ANALYTIC SCIENCES CORPORATION
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Dr. Herbert Weinstock
Technical Monitor
NAS-12-508
Electronics Research Center
575 Technology Square
Cambridge ,Massachusetts 02139
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ABSTRACT
This report briefly summarizes the work performed during
the period I December 1968 through 31 May 1969 for the NASA
Electronics Research Center under Contract NAS 12-508.
A description of the effort plarmied f ir the remainder of the
contract is also presented.
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TABLE OF CONTENTS
Page
No.
Abstract
	 iii
1. INTRODUCTION	 1
2. SINUSOIDA;_j FREQUENCY RESPONSE
FOR THE TIME-MODULATED GYRO LOOP
	 2
3. MEE.SURING LOOP RESPONSES TO RANDOM INPUTS	 6
4,	 THREE-AXIS SELF-CCP,0TPENSATION	 8
5. GAS BEARING DYNAMICS	 11
6. CONTINUATION OF EFFORT	 13
References	 15
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1.	 IN'T'RODUCTION
The purpose of this docum(:nt is to review briefly the work
performed during the first six months of the present contract. Whenever
a particular item of research is considered to be completed it will be
reported in detail ir_ the form of a technical memorandum. All work will
be discussed in depth in the Final Technical Report.
Section 2 discusses work concerning strapdown gyros which
employ the time-modulation torquing technique. Comparison is made
between the sinusoidal frequency response predicted by describing function
analysis and that obtained from simulation. Section 3 deals with an ex-
tension of that work and the effort discussed in Ref. 1 to cover random
gyro inputs. The fourth section summarizes current investigations related
to compensating strapdown sensors by measuring the environmcnt and
computing corrective action. Section 5 covers the initial steps in an effort
to analyzc the dyna-mics of gyros with gas spin bearings. The final section
of the report outlines the direction of future efforts.
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2.	 SINUSOIDAL FREQUENCY RESPONSE
FOR THE TE\4E- MODULATED GYRO LOOP
Previous efforts have centered on the binary and ternary-torqued
gyro loops. Another type of pulse rebalanced loop, using the time-modulated
torquing scheme, is discussed here. This type of torquing is described in
Section 2.3 of Ref. 1. The time-modulated loop is illustrated in Fig. 2. 1.
Note that the float and torque dynarni-cs are characterized in the same man-
ner as the binary and ternary loops treated earlier.
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Float
Dynamics
Signal
Generator Pulses
Indicating
Angle
Change
Figure 2. 1	 The Time-Modulated Torquing Loop
The time-modulated torquing scheme adds the signal represent-
ing float angle to a sawtooth or other unbiased periodic waveform (dither)
with a frequency considerably higher than gyro inputs of interest. The sum
is then passed through a sampled binary nonlinearity. The result is a
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rebalance torque history which is a square wave composed of positive and
negative torque pulses of unequal duration. Float and torquer dynamics
sharply attenuate the oscillations at the sawtooth frequency, leaving a float
angle which varies slowly with respect to the dither. Consequently, the
float angle can be viewed as a slowly varying quantity with some magnitude,
B(t), which is essentially constant over each sawtooth period. An equiva-
lent gain of the ncnlinearity to the float angle is given in Ref, 2:
NB (B, Ad D) = 
Average Output Over
  
On 	Sawtooth Perio d 	 (1)
where Ad is the amplitude of the sawtooth dither signal shown in Fig. 1.
When B < Ad
 the gain is approa.mated by
NB (B, Ad, D)	 DAd  
(2)
Note that, as in the case of the limit cycling binary loop, the equivalent
gain of the nonlinearity to a low frequency sinusoid is independent of the
amplitude and frequency of the sinusoid. For the loop response studies,
the nonlinearity was replaced by the gain NB and the time-modulation gyro
loop viewed as a second-order linear system. If the linearization gain is
computed using Eq. (2) and sampling effects are neglected, the equivalent
damping ratio and natural frequency of the loop are approximately:
_ 1	 CAd	 w _	 Ksg Ktg D2	 KsgKtgD Tf + tg 1	 n	 CAd ^'r_ + f g 1.	 (3)
Note that the above quantities depend on most of the loop parameters shown
in Fig. 2. 1, in addition to the dither amplitude.
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3 mrad of
float angle
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The linear approximation outlined above was checked using an
analog computer simulation of the loop. The set of parameter values
chosen to provide a test case is the same as used for the simulation of the
binary loop in Ref. 1. The sawtooth amplitude, Ad, was 0.83 mrad of
float angle. The results of the study are shown in Fig. 2.2. The analy-
tical response is indicated by the soli- line, with simulation data points
superimposed. There is generally good agreement between the two methods.
Ttg	 = 0.0001 sec
0.1
	
_i	 I	 I	 I I I_ 1 1	 1	 1
?0	 100
Frequency (11z)
Figure 2.2	 Closed Troop Gain and Phase Shift
for the Time-Modulated Gyro
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The uacompensatcd gyro loop response illustrated in Fig. 2.2
is not generally representa.tiv^: of the performance sought in strapdown
gyros. However linear compensation acting on the output of the signal
generator can be used to share she response of a gyro using time-modulation
torquing. Furt;;ei-:..ore, synthesis of the cor_?pensation network is easier
to perform for this gyro rebalance te^hniqu n than for the binary or ternary
torqued instruments.
The research discussed above illustrates that a simple analytical
technique can provide accurate descriptions of the response of a nonlinear
system to sinusoidal excitations. As in the case of the binary and ternary
loops, simulation results support the validity of the describing function
approach for analyzing the time-modulated loop. Furthermore, there is
sound theoretical basis for expecting this same analytical technique to en-
able computation of the gyro response to random and other input signal
forms.
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3.	 MEASURING LOOP RESPONSES TO RANDOM INPUTS
Reference 1 contains describing function treatments of the bi-
nary and ternary loops for single sinusoid inputs. It is shown that the
assumption of small signal inputs to t_ne binary loop permits a closed loop
sinusoidal transfer function to be written which is equivalent to that of a
linear second-order system. The ternary loop requires a more complex
treatment and generally retains its nonlinear character. Reference 2
indicates that the "linearizing' effect of the binary loop limit cycle on the
nonlinearity will cause the describing function representation of the closed
loop to be independent of the shape of the input signal waveform. Approxi-
mation of the binary loop as a linear system should also 1-e valid for
random inputs. The activity presently underway seeks to verify this state-
ment by analog simulation, and more generally, to investigate the responses
of the binary, ternary and time-modulation loops to random irip lits when a
variety of input spectra a-re assumed.
To date, random input responses have been measured for the
binary and ternary loops with broad band,-Adth inputs employed. Some
difficulty has been encountered in accurately measuring the power density
spectra in oived. For the present, a method which gives approximate
results has been chosen. Two sixth-order Butterworth filters are tuned
at different break frequencies and the random signal to be measured is
applied to each filter input. An eleAronic squarer and low-pass filter are
used to measure the average power output of each filter. The diff(.,ence
is approximately equal to the power contained in the band between the two
filter break frequencies. This method is limited in accuracy bE ause the
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filters involved do not have ideal cut-off characteristics. Response mea-
surements have been obtained for the binary loop which shows general
agreement with what would be expected from the previously derived sinu-
soidal response characteristics.
Future random input studies are to be implemented using a
cciinmercially-ma;iufact,ired spectrum anal%Tzer. Arrangements are cur-
rently being made at NASA- ERC for the selection and purchase of such
ar_ instrument.
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4.	 THREE-AXIS SELF-COMPENSATION
One area of current investigation is the removal of motion-
induced errors from strapdown inertial naviga'.ors. When gimballed
single-degree-of-freedom gyros are used, many errors cannot be signi-
ficantly reduced through parameter selection. It was proposed in Ref. 3
that the angular rates measured! by a gyro triad be used to generate com-
pensation torques for nulling error torques on the individual gyros. The
accuracy and :stability c.f such a compensation scheme is difficult to assess
analytically because of the complexity of the interconnected information
loops and the presence of time-dependent gains. Thus, a simulation study
has been undertaken on an analog computer to investigate these p-roperties
and draw conclu--ions about the value of such a scheme.
Initial efforts have centered on compensation of the cross-
coupling error of a single gy ro. Two gyros are 	simulated, with the instru-
ment sensitive axes assumed to be at right an gles to each other. One gyro
(A) experiences angular rates about its spin and input axes whic',
 generate
a cross-coupling error. The other gyro (B) measures the angular rate
about the spin axis of the first in order to provide information or compen-
sation. The cross-coupling error torque appears as the product of the
,spin-s 9 me and the float angle of A, as shown in Fig. 4. 1, The compen-
sation scheme takes the output pulse train of gyro B, multiplies it by the
float angle of gyro A and applies a correction torque pulse to A.
Ignoring the dynamics of the three torque generators, the compen-
sation torque, Mc, is given by
Mc(t) = Ktg ao(t)x(t)	 (4-1)
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(A) GYRO TO BE COMPENSATED
Figure 4, 1	 Block Diagram of Cross-Coupling Error
Compensation of One Gyro
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and the crosscoupling error torque, Me, is
Me(t) = H ao(t) ws(t)
	 (4-2)
These two torques oppose each other in the compensated gyro giving a net
torque
Mc(t) - Me(t ) = aa (t) (Ktg x(t) - Hws(t))	 (4-3)
In Ref, I it was demonstrated that, at least when the input ws is a sinusoid
in a certain range of frequencies, the fundamental component of the torquer
current xf(t) is related to the input to gyro B by
xf(t ) = Ktg w s(t)	 (4-4)
(The current work reported in Section 3 seeks to extend the validity of Eq.
(4-4) to the case where random inputs are present, ) As 2. consequence of
Eq. (4-4)
Me(t) - Me(t ) = ao(t ) Ktg xh(t )	 (4-5)
wher-- xh (t) rep. esents harmonics of the torquer current. If xh(t) and cro(t)
are uncorrelated and one of them is an unbiased variable the average value
of the expression in Eq. (4-5) will be zero.
Simulation studies of this compensation have assumed that the
input and spin axis rates are in-phase sinusoids. Thus fax, the compensa-
tion has proven successful in removing constant cross-coupling errors from
the compensated gyro. With sinusoidal inputs of amplitude 0. 1 rad/sec on
both axes, the compensation is effective at frequencies up to at least 100Hz.
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9.	 GAS BEARING DYNAMICS
The presence of rotor bearing compliance in a gyroscope can
greatly effect the performance of a sensor loop containing the instrument.
Reference 1 contains an analysis based on a damped mass--spring model
of the compliance, an assumption appropriate to ball or roller bearing rotor
supports. However, if a gas bearing is used, generally nonlinear and
frequency dependent compliance terms appear and the analysis becomes
more complex. The paragraphs below describe research directed towards
modelling the gas rotor bearing in such a way as to permit analysis of the
gyro's sinusoidal response characteristics.
The technique employed in analyzing the sinusoidal response of
the gas bearing gyro makes use of work reported in Ref. 4. There it is
shown that the stiffness parameters can be determined and used to compute
a compliance matrix describing the rotor displacement in response to a
sinusoidal force of constant amplitude and direction. The matrix is written
in phasor notation with a particular whirl ratio and whirl sense assumed.
This matrix can reportedly be used to compute by superposition the motor
motion in response to any nLmber of any such forces of the same frequency
or different frequencies.* The only necessary assumption is that the net
rotor ,-Usplacement is small compared to the nominal rotor-bearing clear-
ance. Making use of this superposition property, any relative motion
*Telephone conversaLon of May 23, 1969 with Dr. C.H.T. Par. of Mechanical
Technology, Inc. , Latham ; New York.
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between rotor and gimbal can be calculated as a function of the gyro input
and Output axis torques. If sinusoidal excitation of a particular frequency
is assumed, the gain and phase shift between the gyro float angle and the
net torque applied about the output axis can be computed. Consequently,
the describing function techniques detailed in Ref. 1 can again be used to
obtain closed loop frequency responses for the gas bearing gyro, the major
difference being a frequency dependent rotor bearing stiffness.
Implementation of the above analysis depends on the availability
of data for computing the entries in the compliance matrix. The radial
and tangential stiffnesses can be generated via a digital computer program
which is currently av-Ulable. * Alternatively, the compliance matrix may
be evaluated directly from vibration tests data of the gas bearing gyro of
interest. In the latter case a particular gyro, the Hamilton Standard
RI ll_70, is c>rrently being studied. It is expected that test data for this
instrument will be made available by November, 1969. In the meantime
a study will be performed using the stiffness coefficients presented in
Ref. 4.
*From Mechanical Technology, Inc. , Laiham, New York.
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6.	 CONTINUATION OF EFFORT
During the remainder of the present contract the investigation
of gyro loop responses to random inputs will be concluded. The result
will be a series of comparisons between the spectral densities observed
at the output of a simulated gyro with those predicted from some simpli-
fied (quasi-linear) description of its transmission characteristics.
Initially the gains to sinusoidal inputs described in Ref 1 and Section 2 of
this report will be used to project random input transmission characteris-
tics. It is anticipated that modifications to these analytic techniques will
be made as further experience dictates. Once good agreement between
analyses and simulations is obtained,linear compensation networks will be
designed to produce more satisfactory closed loop gyro responses.
The simulation study of self-compensation of gyro dynamic
Errors will continue, with anisoinertia and output axis inertia effects given
particular attention. Sorre simulation work will also be performed regard-
ing compensation using additional sensors or by applying corrections to the
gyro outputs.
Ana-,-, sis of the closed loop response characteristics of gas-
bearing gyros will continue. This effort will increase when the Hamilton
Standard bearing test data becomes available. A brief study of the laser
gyro will be conducted to determine what kinds of strapdown system errors
may be generated when these instruments are used to measure angular
motion.
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A set of equations will be written, describing the constant errors
generated in single-degree-of-freedom gyros and pendulous acceler-
ometers by a vibration environment. These equations will be incorporated
in a computer program to generate inertial sensor errors, given vibration
density spectra. Two system-generated errors, pseudo-coning and pseudo-
skulling, will a'.so be treated. The vibration-induced errors will then be
converted into position, velocity and attitude errors in a strapdown naviga-
tor fixed to the earth.
In addition to the deterministic errors computed in the above
manner, sensitivity coefficients will be found to permit calculation of the
covariance of the navigator errors arising from random coefficients of
dynamic errors. The study of vibration-induced errors will be enhanced
by comparison of predicted errors with real data gathered during heli-
copter ground tests. At the end of the present contract a considerable
body of knowledge regarding accurate prediction of vibration-induced strap-
down system errors will have been ge^lerated.
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REFERENCES
1. Sutherland, Arthur A. , Jr. , and Bee-bee, Wilt: in S.
?design of Strapdown Gyi oscope^ for a Dynamic Environment ,
Interim Scientific Report II, The Analytic Sciences Corp. ,
TR-101-4, January 1969 (NASA CR-1396).
2. Gelb, Arthur and IanderVelde, Wallace E. , Multiple-Input
Describing Functions and Nonlinear System Design, McGraw-
Hill Book Co. , Inc. , August 1968.
3. Gelb, Arthur, and Sutherland, Arthur A. , Jr. , Design of
Strapdown Gyroscopes for a Dynamic Environment, Semi-
Annual Report, The Analytic Sciences Corp. , TR-101-1,
7 July 1967.
4. Keating, W.H. and Pan, C. H. T. , Design Studies of an
Opposed-Hemisphere Gyro Spin-Axis Gas Bearing, Paper No.
6G u_ S-22, ASME Lubrication Symposium, Las Vegas,
Nev. , June 17-20, 1968.
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Design of strapdown gyroscopes for a dynamic environment Semiannual report, 1 Dec. 1968 - 31 May 1969

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