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AD-AO5 94 ARMY MISSILE C0OMMAND REDSTONE ARSENAL AL GUIDANCE A-ETC F/G 17/7PERSHING Pl1 INERTIAL MEASUREMENT LIT CALIBRATION TREND STUDY.(U)MAR so H V WHITE

UNCLASSIFIEO ORI/RG0-20O NLEEEEEEEEEEEElElllllEllllEE-EEEllllllE~llEIEEEEIIEIIIIIEEEEEEEEllllllIEIIIEEEIIEIIIEEIIIEEE

111120 I

11111 1.25 1.4 _____]

M :W YPI Yt ) I 'I I H AR I.

TECHNICAL REPORT RG-80-20

PERSHING P11 INERTIAL MEASUREMENT UNITCALIBRATION TREND STUDY

*4-

H. V. White DV., Guidance and Control Directorate4p-- US Army Missile Laboratory

March 1980

LIV Ftedto,r49 Arenrim, A~abomrma 3000.

Approved for public release: distributior, unlimited.

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.EERSHING 111 INERTIAL MEASURF24ENT UNIT CALIBRATION TECHNICAL REPORT

45- . PERFORMING ORG. REIPORT NUMBER

7. AUTHOR(&) S. CONTRACT OR GRANT NUMBE-RVS)

S. PERFORMING ORGANIZATION NAME AND"IDRESS 10. PROGRAM ELEMENT. PROJECT, TASK

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US Army Missile CommandATTN: DRSMI-RGL /Redstone Arsenal, Alabama 35809 -

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Commander March 1980US Army Missile Command 13. NUMBER OF PAGESATTN: DRSMI-RPT _________________________________Redstone Arseal. Alabama 358109 80

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IS. SUPPLEMENTARY NOTES

1S. KEY WORDS (Cothwman r everse side it necessar aid Idmunf by block atiw)

Inertial Measurement Unit (IMU)CalibrationGyroscopeAccelerometer

2ft ARSTACT 'm -NO two -~sol N aesemit md I~&t~~ by block embOe)

tTis report presents an overview of calibration data taken from AdvancedDevelopment Program Inertial Measurement Units. Behavior of calibration

* parameters is examined and worst/best performers are identified. Time trends

a re investigated by the method of linear least squares regression. Continued

surveillance of parameters, determination of the effects of ill-behaved

parameters on alignment and flight performance, and minimization of time

required to perform a calibration are tasks to be performed during theVEngineering Development nrostram.-.

DDI AN S uno"or Nevssis ws*x* UNCLASSIFIEDS9CUJrIY CLASSIFICATION OF THIS PASE (Ohm Deta, Eained!)

'I~~Z j~t~;

UUNCLASSIFIEDg~SCCUmIT CLASSIFICATIO OF THIS PAA@RWfm. Doea~m

Ile

ACKNOWLE DGM ENT

The author thanks Messrs David W. Tarrant and ThomasJ. Snowden of Inertial Systems Development, Guidance andControl Directorate for their contributions to this report.

Acoesion For

DDC TANUnrwnounc edJustification ,

&,:alll aIO/OrDist specilal

iLe

FORWARD

The calibration data compiled and analyzed in thisreport were taken from Advanced Development (AD) ProgramInertial Measurement Units (IMU) during the March 1976 toOctober 1978 time frame. Since that time some requirementsand specifications given in the body of the report andvarious tables and charts have been changed for theEngineering Development (ED) Program based on findingsduring AD and pre-ED work. The criteria and discussionsused in this report are those relative to AD.

During the pre-ED time period three IMUs (serialnumbers 007, 011, and 012) were re-worked to incorporatemany of the improvements identified during AD. Asanticipated the resulting calibration parameter behavior ismuch improved. A future report will document the changesand analyze the performance of the modified units.

I

CONTENTS

Sect ion Page

I. Introduction and object . . . . . . . . . . . . . 5

II. Discussion of Calibration Procedure and Output

III. Discussion of Results . . . . . . . . . . . . . . 7

IV. Conclusions and Recommendations . . . . . . . . . 9

Appendix . . . . . . . . . . . . . . . . . . . . 29

I LLUSTRAT IONS

Figure Page

1. Percent of Parameters in the 0 + 1 Sigma Range . 112. Percent of Parameters in the 0 2 Sigma Range . 113. Percent of Parameters in the 0 3 Sigma Range . 11

A-i. Linear Regression Lines Showing-+ Syx Bounds . . 36A-2. Linear Regression Lines Showing + SB Bounds. . . 37A-3. Linear Regression Lines Allowing for Syx and SB. 38A-4. DFZE . . . . . . . * .* 0 .* * * .* * 39A-5e DSZE * . . 0 . 0 0 . . . . . * . . 40

A-8. KIHE. * * . . . . . . . . * a . . 0 * 43

A-5. DFYE * * * - . e & * * * * o . . . . . 50

A-8. DELTA. . . 9 . 43 ........ 5A-i. DIEA . . . 44 ........ 5A-10. KOZE. * o . . 0 0 0 0 . . . . 4.5

A-11. KSZE . . . . . . . . 0 0 . 0 0 . 0 0 6 . 58A-2 OZHE.o . . . . . . . . . * . . 0 0 * * 0 . 47 5A-25. KSZHE * . a a * . * o . 0 a . . * . . 0 . . 60

A-14. KSXE . . . . . . . . . . . . . . . . . . . . . . 62A-28. OXHE . . . . * . A 0 * . 0 0 . 0 0 . & . . . . . 63A-16. KSXE. . * . . . . . . . . . . 0 0 . . 0 . . 0 . 64A-30. DSEAZ . . . . . 0 . . . . . . 52

A-32. KSYE . * . . .* * . . . . . . . 67A-199 KOYHE . . . * * * * . . 54 . ... 6A-20. KSYHEA . . . . . . . . . . . . . . . . . . . . . 69A-35. DETA . . . . . . . . . . . . . . . . . . . . . 70

A-2 * OZE * . . . . . 0 . . 0 . 0 . . . . .2

$

TABLES

Table Page

1. Calibrated Parameters. . . . . . . . . . . . . . 122. Difference Parameter Limits. . . . . . . . . . . 143. Number of IMU Calibrations and Time Frames . . . 154. IMU S/N 005 Normalized Data, Reference Calibration

Date: June 28, 1977 . . * . * . . . . . . . . . 175. IMU Calibration Data - Parameter Range (Individual

IMUs ) . . . . . . . . . . . . . . . . .. * . *.- 206. IMU Calibration Data - Parameter Range (All IMUs,

All Calibrations).......... . . . 257. IMU Calibration Data - AD'Acceptance Test'CriteriaApplied ..... ..... 26

8. IMU Calibration Data - RM;f 20 FlightParameters . . . . . . . . . . . . . . . . . . . 27

A-1. IMU S/N 001 Statistics . . . . . . . . . . . . . 73A-2. IMU S/N 002 Statistics . . . ... . 74A-3. IMU S/N 003 Statistics . . . . . . . . . . . . . 75A-4. IMU S/N 005 Statistics . . .. . . . . . . . . . 76A-5. Predicted Interval Between Calibrations, X1

(Months) . . . . . . . . . . . . . . . . . ... 77

r-3

I. INTRODUCTION AND OBJECT

The Pershing PII Inertial Measurement Unit (IMU) is afour gimbal, all attitude system incorporating two two-degree-of-freedom dry tuned gyros, one dual axis and onesingle axis force rebalance accelerometer.

IMU performance is largely dependent on gyro andaccelerometer parameters which must either exhibit a highdegree of absolute accuracy over the usable life of theIMU, or which must be amenable to a periodic calibration.

The PHI IMU is a calibrated system and as such abso-lute accuracy of gyro and accelerometer parameters is unim-portant. Instrument channel errors are measured in thecalibration procedure and are software compensated so thata high degree of parameter stability is required ratherthan absolute accuracy.

A significant advantage of a calibrated system iscost savings resulting from relaxation of the requirementsto design, fabricate and test for high absolute accuracy atboth the component and system level. Instead, nominalparameter values are built in and are subsequently deter-mined very accurately via the calibration procedure, allow-ing software compensation to be applied during groundalignment and flight modes. Frequency of calibration isdetermined by the stability of the measured parameters.Some of the advantage of a calibrated system is thereforelost if stability is such that frequent and lengthy cali-brations are required.

The object of this study is to compile calibrationdata from all Engineering Model (EM) and Prototype Model(PM) IMUs for:

(1) Identification of least/most stable parameters

(2) Application of acceptance criteria to all runs

(3) Identification of data trends which may be usefulto predict the time interval between calibrations.

The above information should be helpful in determin-ing if the currently proposed calibration frequency of onceper six months is realistic or if the time interval shouldbe increased or decreased. The data should provide visi-bility of parameters which dictate a minimum time betweencalibrations. Application of acceptance criteria to all

5

~MI

calibration results should yield further insight intoparameter behavior and overall system performance. Ananalysis of data trends should also aid in corroboratingparameter stability and time between calibrations.

II. DISCUSSION OF CALIBRATION PROCEDURE AND OUTPUT DATA

Calibration of the PII IMU is an automatic proce-dure1 performed under computer control. The instrumentcluster (azimuth gimbal) is torqued to various positions toorient accelerometers relative to earth's g field and toorient gyros relative to earth's rotational field. Data,in the form of accelerometer and gyro torquing pulses, areacquired in a total of 13 positions. These data areutilized to compute 34 gyro and accelerometer parameterswhich are stored in the IMU Serial Core Memory (SCM) forlater use in the ground alignment and flight modes.Currently, approximately 4 1/2 hours are required toperform a calibration.

The 34 calibrated parameters are shown in Table 1along with definitions and units. Parameters with theletter H as a suffix are used in the ground alignment(gyrocompass) mode only. DELTA YX, DELTA ZY and DELTA ZXare factory/depot calibrated parameters and are measured insubsequent calibrations for goodness checks only. Thesethree parameters along with two other factory/depot deter-mined parameters, HSX and HSY, are stored in a protectedportion of the SCM and theoretically are never updatedexcept at the factory/depot. HSX and HSY are required foraccurate gyrocompassing and are determined by a specialpurpose factory/depot test apart from the standard calibra-tion procedure. Factory/depot determination of more accu-rate values of KIXH and KIYH for gyrocompass purposes hasbeen proposed. This also requires a special purpose fac-tory/depot test. The resulting values would be stored inthe protected portion of the SCM and would be updated onlyat the factory/depot.

A measure of parameter stability is obtained bytaking the difference between the value obtained from a newcalibration and the value obtained from the originalreference calibration. The 34 parameter differences aredefined by adding the letter E as a suffix to each of themeasured parameters, e.g., DFZE = DFZ (NEW) - DFZ (REF).Table 2 shows the proposed one sigma limits placed on each

1. The Singer Company, Kearfott Division; DocumentY256A337, wAlignment Gyrocompass and IMU ParameterCompensation."

6

parameter difference. The specification requires that theIMU meet performance criteria with a calibration frequencyof once every six months. It is therefore of prime impor-tance that parameters remain stable over this interval oftime as a minimum.

The acceptance test criteria utilized during theAdvanced Levelopment (AD) program allowed some degradationof the limits indicated in Table 2. For acceptance, allparameter values were required to lie within +3 sigmalimits. If one or more of the flight parameters exceeded+1 sigma but did not lie outside +3 sigma, the Root MeanSquare (RMS) value of all flight parameters, each normal-ized to its one sigma value, could not exceed a value of1.3 for acceptance. Some flight parameters could thereforeexceed the +1 sigma limits as could all of the groundalignment pirameters but in no case was a value greaterthan +3 sigma acceptable.

III. DISCUSSION OF RESULTS

Calibration data considered in this study were takenfrom a total of 13 IMUs (4 EMs and 9 PMs) over a period of30 months. Calibration sites were the Guidance and ControlDirectorate, Redstone Arsenal, Alabama and Martin MariettaCorporation, Orlando, Florida. The reference calibrationwas ordinarily performed at the Singer Company, KearfottDivision, Little Falls, New Jersey, however, some unitsrequired re-referencing at the other sites, usually due toelectronic card replacement.

Table 3 gives a tabulation of number of calibrationsand timieframes involved for each IMU.

IMUs with serial numbers 001 through 004 are EMs andthe remainder are PMs. Listing of an IMU more than onceindicates that more than one reference calibration wasutilized.

Several of the IMUs as indicated in Table 3, had veryfew calibrations performed, particularly those that wereexpended in AD flights. Only those units with six or morecalibrations over a period of six or more months weresubjected to the trend analysis presented in the appendix.Otherwise, a total of 117 calibration runs, which comprisesthe majority of runs made at Redstone and Martin, wereconsidered for this study.

7

Results from the 10 calibration runs made with IMUserial number 005 during 11.4 months period are presentedin Table 4, as a data sample. The data have been normal-ized by dividing the measured values of a given parameterby the allowable one sigma value given in Table 2.

Table 5 shows the number of times parameter differ-ences feT in the indicated sigma ranges for each IMU. Forexample, the DFZE parameter from IMU serial number 001 waswithin the prescribed +1 sigma limits 18 out of the 26calibrations. The parameter fell between +1 and +2 sigmafour times and between +2 and +3 sigma foui times- None ofIMU serial 001 calibratTon runi yielded a DFZE greater than+3 sigma.

Percentage of calibration runs falling in the varioussigma ranges for each parameter for all IMUs is shown inthe last row of Table 5. A study of this table on an IMUbasis over all parameters or on a parameter basis over allIMUs gives the indication that certain gyro parametersexhibit the least stability whereas certain accelerometerparameters are the most stable. These parameters are dis-played in Figures 1, 2, and 3 which are derived from Table5.

Figure 1 depicts the percentage of parameters fromall calibrations which fell into the 0+1 sigma range.Parameters in the left-most columns of-Fiure 1 are theleast stable with DIXE and DIYE occurring in the 0+1 sigmarange for only 43 percent of the total of 117 measurementsfor each.

The range is expanded to 0+2 sigma in Figure 2 whichdisplays the same 12 parameters in the two lefTit-most col-umns as are indicated in the three left-most columns ofFigure 1.

FiOure 3, in which the range is increased to 0+3sigma, indicates that six (left-most column) of the 12parameters identified in Figures 1 and 2 are the very worstperformers. The six are all gyro parameters, the defini-tions and units of which are given in Table 2.

The most stable parameters are the nine appearing inthe right-most column of Figure 1. They are all acceler-ometer parameters. Definitions and units for these aregiven in Table 2 also.

8

Placement of the parameters in the vertical ordershown in Figures 1 through 3 has no significance.

Data in Table 5 are further condensed in Table 6which provides a summary of frequency of occurrence ofdifference parameters within the indicated sigma ranges forall IMUs and all calibrations. All parameters from allIMUs fall within the prescribed one sigma limits for only75 percent of the total number of data points, however, 91percent lie within 0+2 sigma bounds and 96 percent liewithin 0+3 sigma bounds.

Table 7 presents results of the application of ADacceptance test criteria to all 117 calibration runsconsidered in this study. Fifty-five of 117 runs or 47.0percent passed acceptance criteria, whereas 11 runs or 9.4percent failed all criteria. One or more flight parametersgreater than +3 sigma in combination with larger thanacceptable flight parameter RMS values, accounted for thehighest combination failure category of 24 runs or 20.5percent.

Table 8 lists RMS values of the 20 flight parametersfor each calibration of each IMU. Sixty-five of 117 runsor 55.6 percent passed the RMS criterion of 1.3 sigma. Theremaining runs, with one exception, yielded one or moreparameters greater than +3 sigma and thus failed. Thesingle exception is run Number 9 from serial number 001which failed to meet the RMS criterion of 1.3 sigma eventhough no parameter greater than +3 sigma was measured.Thirteen runs or 11.1 percent met-the RMS criteria eventhough one or more parameters were greater than +3 sigma.

IV. CONCLUSIONS AND RECOMMENDATIONS

This study has taken the initial overview of themajority of calibration data from all AD IMUs.

Identification of parameters which most often exceed-ed three sigma limits has been accomplished. These areDFX, DIX, DIY, DIZ, KTX and KTZ which are all gyro para-meters . Most stable parameters are KIZH, KOY, KOZ, KOXH,KOYH, KOZH, DELTA YX, DELTA ZX and DELTA ZY which areassociated with the accelerometers.

Application of AD acceptance criteria to all runsshows that only 47.0 percent passed the three sigmacriterion and 55.6 percent passed the 1.3 sigma flightcriterion.

9

The Appendix addresses the question of trends byusing the linear least squares regression method. The thtsigma criterion was used to establish a predicted worstcase time between calibrations and in very few cases isthis time less than the AD program required time of sixmonths. However, many of the parameters would fail the sixmonths time criterion when one sigma limits are imposed.

The plotted data in %he Appendix and the correspond-ing tables give some visibklity on trending parameters.Parameters with correlation coefficients near unity aretrending suspects. Gyro parameters KTX, KTY and KTZ andaccelerometer parameters KIX, KIXH, KIY and KIYH show thegreatest propensity toward trending. Conversely, thoseparameters with correlation coefficients near zero are morerandom in nature.

The general conclusion is that many of the parametersneed to be better behaved. There appears to be someproblem associated with test locations as indicated by theplotted data in the Appendix from IMU serial number 005,i.e., note data points taken at Martin (M) compared tothose taken at the Guidance and Control Directorate (A).

It is recommended that close surveillance of thecalibration parameters be continued into the EngineeringDevelopment (ED) program as component and system improve-ments are incorporated. Those parameters which are notwell behaved should be reviewed for their effects onalignment and flight accuracy, and methods for improvingperformance should be defined and implemented.

A factor which requires further investigation isminimization of time required for performing calibration.The current routine has not been time-optimized for fieldusage. Elimination of certain parameters required only ina factory calibration and optimizing the sequence of posi-tions to avoid backtracking between positions are two pos-sibilities for decreasing time.

Additionally, evaluation of the calibration operationunder field-type conditions will be necessary to determinethe impact of environmental factors and unknown initialheading on calibration time.

10

DE LZ V

Ksylf rDFLYXKSZE SXHF OZHE

KIYIIE KIZE KOXC KOYHE

KSYF KIXHF KIXE DSZF I KOXHE IKSXF KIYE KTZE DSYE KOZE

DIYE KTXE KTYE DFZE DSXE KOYEDIXE DI ZE DFXF DFYE DOZE IKIZFIE

40 506o7 0910

PERCENT~-

AFigure 1. Percent of parameters in the 0 +1 sigma range.

DELZYDELZXDELYXKSZHEKSYHEKSX1HE

DELZY KSZEDELZX KSYEDELYX KSXFKSZHE KOZHEKSYHE KOYHEKSXHE KOXHEKOZHE KOZEKOYHE KOYEKOXHE KOXEICOZE KIZHfEKOYE KIYHE

KOXE KIXHEKIZHE IKIZE

KIXHE fKIYE______ KIZE IKIXEKSZE KIYE KTYE

_____ KSYE KIXE KTZE DSZE

E4 KTXE KSXE DSZE E4 KTXE DSYEDIZE KIYHE DSYE DIZE DSXEDIYE KTZE DSXE DIYE DOZE

S DIXE KTYE DOZE DIXE DFZEDFXE DFZE DFYE DFXE DFYE

70 s0 90 160 80 970 100

PERCENT-. PERCENT

Figure 2. Percent of parameters Figure 3. Percent ofin the 0 + 2 sigma parameters inrange. the 0 + 3

sigma range.

-71

TABLE 1. CALIBRATED PARAMETERS

Parameter

KTX X Gyro Torquer Scale Factor(deg/hr per deg/hr)

KTY Y Gyro Torquer Scale Factor(deg/hr per deg/hr)

KTZ Z Gyro Torquer Scale Factor(deg/hr per deg/hr)

DFZ Fixed Drift of Z Gyro (deg/hr)DSZ G Sensitive Drift of Z Gyro (Spin Axis)

(deg/hr per ft/sec2 )DIZ G Sensitive Drift of Z Gyro (Input Axis)

(deg/hr per ft/sec2 )KIY Y Accelerometer Scale Factor (Flight) (Ug/q)KIYH Y Accelerometer Scale Factor (Ground) (Ug/g)KIX X Accelerometer Scale Factor (Flight) (Ug/g)KIXH X Accelerometer Scale Factor (Ground) (ug/g)DOZ G Sensitive Drift of Z Gyro (Output Axis)

(deg/hr per ft/sec2 )

DFX Fixed Drift of X Gyro (deg/hr)KIZ Z Accelerometer Scale Factor (Flight)(ug/g)KIZH Z Accelerometer Scale Factor (Ground)(Ug/g)DFY Fixed Drift of Y Gyro (deg/hr)DSX G Sensitive Drift of X Gyro (Spin Axis)


DSY G Sensitive Drift of Y Gyro (Spin Axis)(deg/hr per ft/sec

2 )DIX G Sensitive Drift of X Gyro (Input Axis)


DIY G Sensitive Drift of Y Gyro (Input Axis)(deg/hr per ft/sec

2 )DELTA YX Non-Orthogonality Between Y & X Acceler-

ometers (rad)DELTA ZY Non-Orthogonality Between Z & X Acceler-ometers (rad)

KOZ Z Accelerometer Bias (Flight) (ft/sec )KSZ Z Accelerometer Scale Factor Asymmetry

(Flight) (ug/g) 2KOZH Z Accelerometer Bias (Ground) (ft/sec )KSZH Z Accelerometer Scale Factor Asymmetry

(Ground) (ug/g) 2KOX X Accelerometer Bias (Flight) (ft/sec )KSX X Accelerometer Scale Factor Asymmetry

(Flight) (ug/g) 2KOXH X Accelerometer Bias (Ground) (ft/sec )KSXH X Accelerometer Scale Factor Asymmetry

(Ground) (pg/g)

12

TABLE 1. (Concluded)

Parameter

DELTA ZX Non-Orthogonality Between Z & Y Acceler-ometers (rad) 2

KOY Y Accelerometer Bias (Flight) (ft/sec )KSY Y Accelerometer Scale Factor Asymmetry

(Flight) (ug/g) 2KOYH Y Accelerometer Bias (Ground) (ft/sec )KSYH Y Accelerometer Scale Factor Asymmetry

(Ground) (ug/g)

13

& ______________________________

TABLE 2. DIFFERENCE PARAETER UNMITS

Difference

Parameter +1a Max Criteria +3o Max Criteria

DFZE .0250/hr .0750/hrDSZE .0250/hr/g .0750/hr/gDIZE .030/hr/g .090/hr/gKIYE 100 jig/g 300 Iag/gKIYHE 100 isg/g 300 jig/gKIXE 100 ugi/g 300 jig/gICIXHE 100 jig/g 300 jig/gDOZE .020/hr/g .060/hr/gDFXE .025 0/hr .0750/hrKIZE 100 jig/g 300 llg/gKIZHE 100 jig/g 300 uig/gDFYE .0250/hr .0750/hrDSXE .0250/hrig .0750/hr/gDSYE .0250/hr/g .0750/hr/gDIXE .030/hr/g .090/hr/gDIYE .030/hr/g . 090/hr/gDELTYX .00008 rad .00025 radDELTZY .00020 rad .00060 rad

XOZE 300 jig 900 tig

KBZE 5 0 V1 q/9 150 jig/gKOHE30 g 900 jig

KSZHE 50 jig/g 150 jig/g

KOXE 100 jig 300 jigKSXE 50 jig/g 150 jig/gKOXHE 100 jig 300 iugKSXHE 50 jig/g 150 jig/g

DELTZX .00015 rad .00045 radKOYE 100 jig 300 jigKSYE 50 jig/g 150 jig/gKDYI5E 100 jig 300 jigKSYHE 50 jig/g 150 u~g/gKTXE. .00020/hr per 0/hr .00060/hr per 0/hr

KTYE .00020/hr per 0/hr .00060/hr per f/hr

KTZE .00020/hr per 0/hr .00060/hr per 0/hr

14

TABLE 3. NUMBER OF IMU CALIBRATIONS AND TIME FRAMES

IMU S/N NO. CAL. NO. MONTHS TIME FRAME REMARKS

001 26 13.6 3/76 - 5/77

002 9 3.0 2/76 - 5/76

002 18 7.1 5/76 - 12/76

003 6 1.9 9/76 - 11/76

003 9 6.4 1/77 - 7/77

004 7 5.0 9/76 - 2/77

005 10 11.4 6/77 - 6/78

006 3 4.8 5/77 - 10/78

006 4 3.0 12/77 - 3/78 AD FLT 4

007 5 8.7 6/77 - 3/78

008 3 5.3 6/77 - 11/77 AD FLT 2

009 1 1.4 9/77 - 11/77 AD FLT 3

010 1 1.6 8/77 - 9/77 AD FLT 1

0i1 2 3.8 7/77 - 11/77

012 1 6.5 8/77 - 2/78

012 5 3.7 4/78 - 8/78

j 013 7 2.5 2/78 - 5/78 AD FLT 5

TOTALS 117 89.7

15

TABLE 4.

ELAPSEDDATE LOCATION TIME DFZE DSZE DIZE KIYE

(MONTHS)

06/28/77 SKD 0(REF)

08/05/77 MMC 1.27 -.81 1.93 -.73 -. 35

11/08/77 RSA 4.43 .58 1.56 -1.53 -. 58

11/29/77 RSA 5.13 .50 1.48 -.30 -.45

12/07/77 RSA 5.40 -.11 1.78 -1.03 -.43

12/16/77 RSA 5.70 0 1.63 -2.43 -.83

01/03/78 RSA 6.30 .81 1.38 -. 56 -1.42

01/11/78 MiC 6.56 -.24 1.62 -1.45 -1.10 102/27/78 MMC 8.13 .11 1.28 -1.51 -1.55

4,4 03/23/78 MIMC 8.93 -.02 1.23 -1.19 -1.4

06/06/78 MMC 11.43 1.50 2.70 -9.70 -1.8

17

TABLE 4. IMU S/N 005 NORMALIZED DATAREFERENCE CALIBRATION:JUNE 28, 1977

DIZE KIYE KIYHE KIXE KIXHE DOZE DFXE KIZE KIZHE DFYE DSXE DSYE

-. 73 -.35 -.54 -.06 -.38 -.70 .08 -.12 .22 -.58 .16 -.16

-1.53 -.58 -1.10 -.73 -1.28 -1.00 -.40 -1.14 .69 -.74 -.36 .36

-. 30 -.45 -.87 -.56 -1.06 -. 50 -.40 1.08 .64 .32 -.44 .44

-1.03 -.43 -.78 -.14 -1.02 -. 20 -.79 1.12 .64 .39 -.48 .48

3 -2.43 -.83 -1.15 -.65 -1.31 .20 -.56 .76 .39 -.07 -.48 .48

56 -1.42 -1.84 -1.51 -2.02 -1.60 -.38 .65 .29 .63 -. 36 .36

2 -1.45 -1.10 -2.24 -.78 1.55 -. 45 -.36 -.46 -.72 10 .12 -.12

8 -1.51 -1.55 -1.86 -1.28 -1.72 .02 -.72 -.36 -.55 1.14 -.28 .28

3 -1.19 -1.45 -2.70 -1.18 1.12 1.12 -. 74 -. 37 -. 62 1.56 -.36 .36

D -9.70 -1.80 -3.00 -1.80 .80 .56 -.09 -.36 -.56 1.80 .15 -.15

TABLE 4.

ELAP SED

DATE LOCATION TIME DIXE DIYE AYX tYZ KOZ

(MONTHS)

06/28/77 SKD 0(REF)

08/05/77 MMC 1.27 .10 .10 .07 .01 -.3

11/08/77 RSA 4.43 -4.13 -4.13 .84 .03 -.6

11/29/77 RSA 5.13 -1.70 -1.70 1.16 0 -.01

12/07/77 RSA 5.40 -4.60 -4.60 .68 .01 -.1

12/16/77 RSA 5.70 2.80 2.80 1.20 0

01/03/78 RSA 6.30 -2.67 -2.67 .14 .02

01/11/78 MMC 6.56 -.07 -. 07 .05 0 A

02/27/78 MMC 8.13 -. 06 -. 06 .10 .02

03/23/78 MMC 8.93 -1.37 -1.37 .34 .03

06/06/78 MMC 11.43 -1.04 -1.04 .29 .28

18

/

TABLE 4. (Continued),

AYZ KOZE KSZE KOZHE KSZHE KOXE KSXE KOXHE KSXHE AZX KOYE

.01 -.31 -1.47 -. 09 .11 -.34 -.60 -.06 .14 .41 .56

.03 -.60 -1.92 -.25 .69 -1.45 -3.64 -.26 -.72 .54 .65

0 -.07 .80 -.26 .16 -.08 -. 42 -.30 -. 63 1.36 -.62

.01 -.19 .29 -.19 .33 .28 .09 -.21 -.59 1.19 -.47

o -.06 .22 -.15 .03 -.10 -.11 -. 17 -.57 1.20 -. 57

.02 -.06 .79 -.23 .15 -.01 -.32 -.35 -1.08 1.24 -.35

0 -.32 -1.13 -.21 .23 -77 -1.72 .02 -.02 .61 .70

.02 -.19 .21 -.28 -.04 -.61 -1.07 .01 -.33 .68 .64

.03 -.20 .59 -.36 -. 15 -.88 -1.75 -. 08 -.19 .66 .56

.28 -.60 -.60 -.50 .12 -.60 -1.30 -.14 -.34 .70 .40

I~J.

6


ELAPSEDDATE LOCATION TIME KSYE KOYHE KSYHE(MONTHS)

06/28/77 SKD 0(REF)

08/05/77 MMC 1.27 1.20 -.10 .26

11/08/77 RSA 4.43 1.27 -.09 .22

11/29/77 RSA 5.13 -. 91 .16 .71

12/07/77 RSA 5.40 -.94 .15 .52

12/16/77 RSA 5.70 - 1 .09 -.04 .11

01/03/78 RSA 6.30 -.45 -. 18 .22

01/11/78 tMMC 6.56 1.71 -. 05 .08

02/27/78 .tC 8.13 1.00 .10 -.03

03/23/78 X-MC 8.93 1.09 .08 .03

06/06/78 121C 11.43 1.05 .11 .39

19

m-

' III ,, .... .. ....... .. .


-SED KX

M KSYE KOYHE KSYHE KTXE KTYE KTZE

TH S)

[REF)

.27 1.20 -.10 .26 -.70 -.46 1.07

.43 1.27 -.09 .22 .58 .44 2.52

.13 -.91 .16 .71 -.34 -.74 1.95

.40 - 94 .15 .52 .53 .10 2.68

.70 1-1.09 -.04 11 .92 -.65 2.30

.30 -.45 -.18 .22 1.04 4.19 3.10

.56 1.71 -.05 .08 .16 -.34 2.86

.13 1.00 .10 -. 03 1.80 -. 10 3.10

.93 1.09 .08 .03 1.94 .06 3.42

.43 1.05 .11 .39 1.80 .31 3.10

19

TABLE 5. IMU CALIBRATION DATA- PARAMETER RANGE (IND

ENTRIES ARE I REQUENCY OF OCCURRENCE

DFZE DSZE DIZE KIYEIMU SIGMA RANGE SIGMA RANGE SIGMA RANGE SIGMA RANGE SIGS/N +1 +2 +3 > +3 + I+2 +3 :> +3 +1 i+2 +3 > +3 +1 +2 73 >+3 +I- - - - I- - - - - - I- - - I - -

001 18 4 4 0 21 1 0 4 5 8 9 20 6 0 0 23 3ii002 20 6 0 0 27 0 0 0 18 6 2 1 16:11 o0o14 82, 0

003 13 2 0 0 15 0 0 0 13 2 0' 0 13 2 0 i 0 13 2

004 7 0 0 0 7 0 0 0 6 0 0 1 7 0 0 0 6 1005 0'9 10 3 5 1 1 5 5 0 3

55 0 3

006 3 1 2 1 520 4 0 01 1 1 6 1 0 0 4 2

007 1 4 0 0 5 0 o 0 0 0 0 5 5 C 0 0 5 0I t ,

008 3 0 0 0 3 0, 0 0 3 0 ' 0 0 1 2 0 0 0 1

009 1 0 0 0 10o10 0 1 0 0 0 1 0 00 0 0

010 1 o I o o 1 0 0 0 1 0 0 0 1 0 0 0 0

Oil 1 1 0 0 2 0 0 0 2 0 0 0 0 2 0 0 0 ' 1

012 10 0,5 5 1 0 0 3 0 1 2 5 10 0 5 1

013 5 1 06 0 6 0 oi I

TOTALS 83 20 7 6 9916 2 0 64 19 13 21 813 6 0 0 74 2

% 72 17 6 5 85 14 1 0 55 16 11 18 69 31 0 63

-2 I ...

20

RATION DATA - PARAMETER RANGE (INDIVIDUAL IMUs)kRE FREQUZNCY OF OCCURRENCE

DIZE KIYE KIYHE KIXE KIXHE;MA RANGE SIGMA RANGE SIGMA RANGGM A RMA RANGE SIGMA RANGE

-2 -2'+3 >+3 +11+2 f+3>-- +1 4-2 1+3 :Y!+3 +3 +1 +2 +3 +3 +1- I> +

5 5 9 20 6 0 1 0 23 3 0 0 19 7 0 0 24 2 1 0 0

6 2 11611 010 14 8 5f0 27 0 0 0 15 12 0 0

2 3 0 13 13 2 0 0 7 8 0 0 8 7 0 0

0 0 1 7 0 00 6 1 0 0 4 3 0 0 7 0 0 0I

5 1 0 3 4 2 1 6 4 0 0 2 7 1 0

1 . 1 6 1 0 0 4 2 0 1 5 2 0 0 4 2 0 1

0 0 5 5 0 0 5 0 0 0 5 1 0 0 0 4 1 0 0, I i0 3,1 2 020 0 12, 0 0 0 3 0 0 0 3 0

o 01 0i 0 00 0 0 1 0 1 0 0 0 0 1 0 C0

0 0 1 0 0 0 0 0 10 0 0 0.1 0 0 0

0 0 0 0 2 0 0 0I1 01 11 , 0 0 0 0

0 12 5 0 0 0 0 5 0 0 5 I 0

0 0' 0 0 5 0 0 0 0 0I , I _ _ _ 51

913 21 8136 0 0 74 29 11 3 82 32 3 0 70 41 4 2

6 11 18 69 31 1 0 63 25 9 3 70 27 3 0 60 '35 3 2I I ____ _ _ 1I *~*~*f __ ~ ____ ___ -.-. --- __________

TABLE 5. (Continued)

DOZE DFXE KIZE KIZHEIMU SIGMA RANGE SIGMA RANGE SIGMA RANGE SIGMA RANGE

! + 2 +1>±-3-- + >S/N +1 +3 +2

001 22 3 0 1 10 5 2 9 25 1 0 0 22 2 2 0 19

002 25 2 o o 27 0 0 0 1 11 2 3 27' 0 0 027

003 15 0 0 0 7 1 4 3 14 1 14 1 0 8

004 7 0 0 0 7 0 0 : 0 0 0 7 0 0? 0 4

005 8 2 0 0 10 0 0 0 7 3 0 0 10 0 0 0 7

006 3 3 1 0 7 0 0 42 1 0 7 0 0 0 7

007 5 0 0 0 1 2 1 2 5 0 0 0 1 2 0 2 1

008 2 I 1 0 0 1 0 2 0 3 0 0 0 3 1 0 0 0 3

009 0,1 0 0 0 0o0 1 0 o ~ 0 o o0 0 0 1

o~o o1, o~o 10o1010 1.0 ojo o o o010 0 1 0 0 1 0 0 0 1 0 0 0 1' 0 0 0 1

011 2 0 0 0 0 0 1 1 2 0 0 0 2 0 0 0 0

012 6 0 0 0 6 0 0 0 5 1 0 0 4 2 0 0 6

013 6 10 011 32 0 2 2 3 0170 0 017

TOTALS 101 14 1 1 79 11 12 15 87 21 6 3106 6 3 2 91

86 12 1 1 68 9 10 13 74 18 5 3 90 5 3 2 78

21

S/


KIZE KIZHE DFYE DSXE DSYESIGMA RANGE SIGMA RANGE SIGMA RANGE SIGMA RANGE SIGMA RANGE

+1 1+2 1±3 }>43 +1 +2 3]+3 +1 +2 +3 1+3 +1 '+ 1+ >+3 +1 +2 _3>+

25 1 0 0 22 2 2 0 19 6 1 0 25 0 0 1 25 0 011 11 2 3 27 0 0 0 27 0 0 0 26 i 1 0 0 26 1 '0 0

1 2I ' ,V

14 1 0 0 14 00 80 4 3 : 0 9 5 0 1 9 5 0 1

7 0 0 0 7 0 0 0 4 3 0 0 7 0 0 0 7 0 0 1 0

7 3 0 0 10 0 0 0 7 3 0 0 10 0 0 0 0 0 01 04 2 1 0 7 i 0 0 0 7 0 0 0 7 0 0 0

5 0 0 0 1 2 0 2 1 0 1 i 3 1 3 0 1 1 3 0 13 0 0 0 3 0 0 0 3 0 0 0

S0 3 I 0 0 0 3 i 0 0

1 1 0 0 0 1 0 0 0 1 :0 0 0

1 0 0 0 1 0 i 0 0 1 1 o0 oio0 1 0 0 0 1 0 0 0

i o o I I

2 0 0 0 2 0 0 ! 0 0 0 0 2 2 0 0 0 2 0 0 00 1 , : I5 1 0 0 4 02 0 6 i0 0 0 5 1 0 0 5 1 i0 0

2 2 3 0 7 0 0 0 7 0 0 0 7 0 0 0 7 0 0 0

87 21 6 3 106 6 3 2 91 16 5 5 104 10 0 310 0 0 3

74 18 5 3 90 52 4 4 89 8 0 3 89 1 0 3


DIXE DIYE DELTYX DELTZYIMU SIGMA RANGE SIGMA RANGE SIGMA RANGE SIGMA RANGE SIG

S/ 1 2+3 1>+3 +1 +2 :t3 [> 3 +1 1+2 1+3 >_ +1+_3>+ 1+

001 10 7 8 1 10 7 8 1 23 3 0 0 26 0 0o1 0 26t 0I 1 I , '002 17 5 3 2 17 5 3 2 27 0 0 0 27 0 0 0 27

003 4 10 1 0 4 10 1 0 13 2 00 15 i0 0 0 14 1

004 2 0 1 4 2 0 1 4 2 0 0 7I 0 0 0 7 0

253 23 2 0 01000 0i0

007 3 i 4 0112I

00 1 o:I'10 1 0 I',0 l006 2 10 40 1 0 4 7 0 0 0 7 0 0 7 0

007 04 0 0 1 5 0 5 0 0 0 5 0

012 5 0 1 0 5 0 0 6 00 0 5 10,0 6 0

013 3 3 1 0 3 3 1jO 0 0 0 7 0

TOTALS 50 34 17 16 50 3417 16 107 10 0 0 1 1 0 0 114 3

432914 14 432914 14 919 0 099 1 0 0 97 3

22

/ i 0 { 0 2 i


DELTYX DELTZY KOZE KSZE KOZHEGMA RANGE SIGMA RANGE SIGMA RANGE SIGMA RANGE SIGMA RANGEJ+3 > +3 k'1 +1 +2 !+ f>+3 +1 +2 jt3±~I 3r+3 3

3 0 0 26 0 0 0 26 0 1 0 0 7 7 12 0 26 0 0 0

0 0 0 27 0 0 0 27 0 0 0 24 3 0 0 27 0 0 0

2 0 0 15 1 0 0 0 14 1 0 0 121 3 0 0 15 0 0 0

2 0 0 7! 0 0 7 0 0 I 51 1 0 7 0 0 0

2 0 0 1 0 00 100 00 0 7i3 0 0 10 0 0 0

o10o0 7 0 0 0 7 0o 3 4 0 0 7 0

0 0 0 5 0 00 5 0 00 4 1 100 3 10 1

1 0 0 3 0 0 0 1 2 0O0 1 0 0 2 30000 0 0 , 0 0

0 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 01 0

0 0 0 2 0 00 0 12 00 203 0 0 0

12 . 1 0 i 0 100

0 0 0 5 1 0 0 6 0 0 0 51 00 6. 0 0 0

0 0 0 7 0 0 0 7 0 0 0 7 0 0 0 7 0 0 0

00 116 1 0 0 114 3 0 0 752713 2115 1 0 1

0 0 99 1 0 0 97 3 0 0 64 23 11 2 98 1 0 1

-~ - ____ - - - - - __ _ - __


KSZHE KOXE KSXE KOXHEIMU SIGMA RANGE SIGMA RANGE SIGMA RANGE SIGMA RANGE SIS/N +1 ,T+2 +2 :+3 +3 + +7 1>+3 _ :

001 10 12 4 7 0 0 17 1

002 27 0 0 21 6 0 0 10 8 6 3 26 1 0 0 1

003 13 2 0 0 14 j1 0 0 7 6 2 0 15 0 15

004 6 1 0 0 5 2 00 3r2 2'0 6 1 0 0

005 10 0 0 0 9 1 0 0 5 4 0 1 10 0 0 0 9

006 6 1 0 0 3 4 0 0 7 0 0 0 6 1 0 0 4

007 2 1 0 2 41 0.0 2 3 0i 0 5 0 0 0 4

008 2 1 0 0 3 0 0 0 2 1 0 0 3 0 0 0 2

0090 1 0 0 0 0 0 10 1 0 0 0

010 1 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0

011 2 0 0 0 2 0 0 0 2 0 0 0 2 0 0 0012 6 0 0 0 5 0 1 0 2 3 1 0 5 0 1 0

013 7 0 0 0 7 0 0 0 6 1 0 0 7 0 0 0

TOTALS 93 18 4 2 99 17 1 0 63 36 14 4 113 3 1 0 83

79 15 3 2 85141 0 54 31 112 3 96 3 1 0 71

23

f


KSXE KOXHE KSXHE DELTZX KOYESIGMA RANGE SIGMA RANGE SIGMA RANGE SIGMA RANGE SIGMA RANGE

+1 ++ > 3 +1+2 :t3 1> +3 +1 [+2T+31> ±3t ++3 TTT+3-l±2Jt>+__________ _ I___ ____3 _ . I__ j+_ I_+ _ +_ + __ ,__ _ .- +I - -, - -, +. +_ I_ +_

17 7 2 0 26 0 0i 0 17 9 0 0 26 0 0 0 23 3 0' 0

10 8 6 3 26 1 0 0 15 11 1 0 27 0 0 0 26 1 0 0

'2 0 15i 0 0 10 1 0 0 13 1 1 0 14 1:0 0

3 2 2 0 6 1 0 0 3 2 2 0 7 0 0 0 6 0 1 010 0 1 0 0 0

5 4 01 1i00 6 4 0 0 100 0.0

7 :0 0 0 6 1 0 0 4 3 i0 0 7 0 0 0 7 0 0 0

2 3 0 0 5 0 0 0 4 1 0 0 4 1 0 0 1 2 1 1

2i1 O 0 3 0 :0 2 io i o 3100 0 2 1.0 0

o 00 0 0 1 0 0 0 1 0 0 0 1 0 0 0

20 0 0 1 0 0 0 1 0 0 1 0 0 0 1 1 0 0 0

2 0 0 0 2 00 0 0 0 2 0 00 2 0 0 0

2 3 1 0 5 0 1 0 6 O 0 00 6 0o0 0 5 1 ol0

6 10 0 7 0 0 0 5 2 0 0 7 0 0 0 7 0 0 0!'I 1-63 36 114 4 113 3 1 0 83 31 3 0 110 6 1 0 105 9 2 1

5 3 o 3 96 3 1 0 7 2- 3 0 9 5 1 8 o2 -

54 31 112 3 96 3{ 1 0 7 2 6 3 O0 9 4 5 0 90 821

--- __ II - . 51


KSYE KOYHE KSYHE KTXEIMU SIGMA RANGE SIGMA RANGE SIGMA RANGE SIGMS/N +1 +21+3 >+3 +1 1+2 1+3 .>+3 +1 1+2 +3 >+3 +1 +2 +3

001 12 12 2 0 25 0 1 0 22 3 0 1 8 5 4

002 17 5 1 22 5 0 0 12 15 0 0 14 2

003 9 5 1 0 15 0 0 0 15 0 0 0 10 ;2

004 3 3 1 0 6 0 1 0 5 2 0 0 4 1,

1005 4 6 0 10 1 0 1 0 0 0 6 4

006 3 3 1 0 7 0 0 0 7 0 0 0 6 1 i

007 1 21 1 5 0 0 0 1 40 0 2008 0i 1o0 13 0 0 010 3 0 0 1 0

009 1 0 0 i0 0 0 0 0 1 0 0 0 1 0

010 1i 0 0 01 1 0 0 0 1 0 0 0 0 !

011 2 o0o!o 210100 0i0Oil~ 2 2 0 i0 i 0 0 0I I I i. *

012 3 3 0 0 5 1 0 0 5 1 0 0 3 2

013 7 0 0 0 0 010 5 2 0 0 24

TOTALS 64 40 10 3 109 6 2 0 86 30 0 1 57 21

55 34 8 3 93 5 2 0 74 26 0 1 54 20

24


KSYHE KTXE KTYE KTZENGE i SIGMA RANGE SIGMA RANGE SIGMA RANGE SIGMA RANGE'>+3 +2 '+3 +3 +1 r+2 +3 >+3 +1 +2 +3 '>+3 +1 T+2 1+3 >+3

+2 +3..._-_-_. .__- I- --

22 3 0 1 8 5 4 3 6 9 4 1 19 1 0 00 12 15 0 0 14 217 20310 1 i

S 0 15 0 0 0 10 2 1 :1 12 0 1 2 0 14 0 0 0

0 5 2 0 0 4 1 0 0 4 1 0 t 5 0 0 00 i10 0 0 0 6 4 0,0 9010 1 0 2 4

0 7 0 0 0 6 1 0 0 4 3 0 0 5 2 0 1 0

I0 I1 4 0.0 2 01023 11 3 4 . 00

0 0 0 3 0 0 1 0 2 0 0 0 3 0 2i 0 1 0

o 0 0 0 1 0:00 1 00 0 0 00

0 1 0 . 0 0 0! 0 1 0 O 0 1 11 0 0 0I 0I

0 2 00 0 0 0 0~ 2 020

0 5 1 ° 0 3 210 1 3 011 2 3 20 1

• 0 5 2 0 0 214 1 0 5 2 0 0 2 2 2 1

0 86 30 0 1 57 21 9 18 65 19 11 10 73 10 13

0 74 26 0 1 54 20 9 17 62 18 1 0 10 70 10 9 12

- - -~ - -- J-

TABLE 6. IMU CALIBRATION DATA - PARAMETER RANGE(ALL IMUs, ALL CALIBRATIONS) ENTRIES AREFREQUENCY OF OCCURRENCE

PARAMETER +lo +20 +3y >+3a

DFZE 83 20 7 6DSZE 99 16 2 0DIZE 64 19 13 21KIYE 81 36 0 0KIYHE 74 29 11 3KIXE 82 32 3 0KIXHE 70 41 4 2DOZE 101 14 1 1DFXE 79 11 12 15KIZE 87 21 6 3KIZHE 106 6 3 2DFYE 91 16 5 5DSXE 104 10 0 3DSYE 104 10 0 3DIXE 50 34 17 16DIYE 50 34 17 16DELYX 107 10 0 0DELZY 116 1 0 0KOZE 114 3 0 0KSZE 75 27 13 2KOZHE 115 1 0 1

KSZHE 93 18 4 2KOXE 99 17 1 0KSXE 63 36 14 4KOXHE 113 3 1 0KSXHE 83 31 3 0DELZX 110 6 1 0KOYE 105 9 2 1KSYE. 64 40 10 3KOYHE 109 6 2 0KSYHE 86 30 0 1KTXE 57 21 9 18KTYE 65. 19 11 10KTZE 73 10 9 13

TOTALS 2971 637 182 151 TOTAL

75 16 5 4 3941

25

0W S.ZP 'IVDL io C0 . ... no. SIM

rz TO N/sIS ____ OCN3dX3

i-~ o o 00 00

H 10 N/S I

pOW 8t

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SE4 To N/ 3ON3dxA

k0 0 N/S 13CN X

o OW CIS~'IVDE Z0o Ila (IV

~ 800 N/S moo___ OJ~N3dX3

A- OW VDS 000000 0L0 N/ 0

s~ow0.IE 0SIVO v o00.0(I900 N/S H< -____ aamadxa-sow~ 8* '100k ~vo E 11000900 N/S _____

E SOW VIT1 oSN13 01 0.00M.00. 0.0. 0

Soo N/S -C__ =___ _ _ _

t-E 0..0 o r

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sowN/ 0.00. 0 07 1

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r1 SOW 6*

g. SIM 9 0. 00 00COO NIS __ _ __ __ _ _ _ _ _

SOW9 C CCDC oo81x~9 M0.010,0.0.0. .0.0.0. 00000 00 0 . 0

ZOO N/S _____ ____ ___

SOWsow ____

SIVO o oo 06

ZOONI

[10 N/s ..... , 2 GO1O:4dx'

SON Lf-

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11O N/s -.

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E-4 S10 NI £5~x

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SOW~ ~~ 0£~~'900v s - -4 9 O3N X

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sow 6T n ,N

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SON 9T' Ln lo m afl '4-

SOW z* , -,i n t

N/ SI ! 4rZOO___ NIS___ _______ _ _ _ _ _____ ____

SO 9"E ai~ ~ In, -. ~ w) In.4~4~ -4a l nI aU o -a D c o m mc

SI 9 a r c 4 -4c n L mm 4r I*-w%2a7n%. . . . . . .. . .

-~ APPENDIX

I CALIBRATION TREND ANALYSIS

1 29

One method that is widely used to examine data fortrending effects is linear regression. 2 A straight lineof the form

Y = A + BX (1)

is fitted to the data such that the sum of squares of ver-tical deviations of the observed data from ti is line issmaller than the corresponding sum for any other straightline. The regression line is therefore a least-squareestimate of the unknown true line. Equation (1) is thefamiliar slope/intercept form with

NEXY-EXZYB X2- 2 (2)NEX (EX)

and

ZY-BEXA = N (3)

in which

N = number of data point pairs

X = independent variable

Y = dependent variable (observed data, each valuepaired with a corresponding value of X)

Z = summation of indicated variables over all datapoints from 1 to N.

After determination of A and B it is a simple matterto use Equation (1) to provide an estimate of Y for a ndwspecified value of X. Alternately, X can be calculated forany value of Y:

Y-Ax = -- (4)

2. US Naval Ordnance Test Station; Statistics Manual byCrowe, E. L., David, F. A., and Maxfield, MargaretW., China Lake, California, 1955, Chapter 6.

31

ROO -X

Equation (4) provides an optimistic estimate of Xbecause the actual Y data is scattered about the regressionline. The scatter is measured by

S2 N-1 $2 B2S 2 (yx = - y (2 (5)

and the positive square root is sometimes called thestandard error of the estimate.

In Equation (5)

$ 2 NZy_ (-ry)2

y N(N-l)

and

2 NEX 2- (EX) 2x N(N-I) (7)

which are the Y and X variances about their respectivemeans.

If the Y data corresponding to the various Xs isnormally distributed about the true regression line, twolines drawn parallel to the calculated regression line atvertical distances +" S x envelop 68 percent of the data.These lines are defTneg by

Y = (A+Syx)+BX (8)

and

Y = (A-S yx)+BX (9)

and are illustrated in Figure A-1. If it is desired toestimate a maximum X for an assumed value of Y, e.g., Y1in Figure A-i, the conservative approach would be to useEquation (8) or the top line of Figure A-1 to account forsome of the variation about the calculated regression line.Thus X1 would be the conservative estimate.

32

Further conservatism may be applied to the estimateby consideration of the standard error of the slope whichis given by

SS yx (10)B = x AN -l

Equation (10) allows for variations of the slope Band is used to define two lines given by

Y = A+(B+SB)X (11)

and

Y A+(B-SB)X (12)

which are illustrated in Figure A-2. Again X1 would bethe conservative estimate for a given value of Yl.

Allowance for variation of slope and variation ofdata about the line of chosen slope can be made by combin-ing elements of Equations (8) and (11) and Equations (9)and (12) to give

Y= (A+S) + (B+S)X (13)

and

Y= (A-S yx ) + (B-SB)X (14)

Plots of Equations (13) and (14), as shown in FigureA-3, provide a funnel about the original regression TineTOi should enclose practically all of the observed data.

Equation (13) provides the most conservative estimateof X for any of the cases considered when the slope B is

positive as indicated in Figure A-3.

Similarly, when the slope is negative, Equation (14)provides the conservative estimate.

33

no

The criterion for the estimation-of a conservative

value of X when Y - ±Yl is thus established as

X= +Y-(A+S) (15)

in which the sign of Y1 , Syx and SB is taken the sameas the sign of the slope B.

In the case of the PII IMU calibration trendanalysis, an estimate of X (months) is sought which indi-cates when +Y1 = +3 sigma since the +3 sigma limitsspecify the-unqualified requirement for a new calibrationunder AD ground rules.

The linear regression line as given in Equation (1)was established for each parameter from all IMUs havingsix or more calibrations over a period of six or moremonths (serial numbers 001, 002, 003 and 005). Equation(15) was used to calculate a conservative estimate of X -

X1 (months) which would be expected to elapse before aparameter exceeded its +3 sigma limits thus giving anindication of the amount of time that may be allowedbetween calibrations.

Figures A-4 through A-37 show data obtained for eachparameter from IMU serial num-ber 005 and the correspondingregression lines given by Equation (1). Also shown are thelines specified by Equations (13) and (14).

Other statistics computed from calibration data fromthose IMUs analyzed are:

(1) Correlation coefficient:

R = BS /S (16)x y

with B given by Equation (2) and Sy and Sx given by therespective square roots of Equations (6) and (7).

(2) Root mean square error of the dependent variable:

Y 1iS) = (EY2/N)h (17)

34

with symbol definitions the same as those given forEquations (2) and (3).

(3) Average error of the dependent variable:

Y(AVG) = EY/N (18)

with symbol definitions the same as those given forEquations (2) and (3).

Computed values of all statistics for each calibra-tion parameter from IMUs with serial numbers 001, 002, 003and 005 along with conservative estimate, X, and estimate,XZ , from the original regression line, are compiled inTables A-1 through A-4 respectively.

Resulting values of the conservative estimate, Xl,are summarized in Table A-5 which also shows the values ofX1 for each parameter averaged over the four IMUsanalyzed.

35

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VI 78

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Defense Technical Information

Cameron Station

Alexandria, Virginia 23144 12

CommanderUS Army Materiel Development

and Readiness CommandATTN: DRCRD 1

DRCDL I

5001 Eisenhower Avenue

Alexandria, Virginia 22333

The University of TennesseeDepartment of Electrical Engineering

ATTN: Dr. J. C. Hung

Knoxville, Tennessee 37916

NASA Johnson Space CenterEG5ATTN: Mr. Malcolm JonesHouston, Texas 77058

US Army Materiel Systems Analysis ActivityATTN: DRXSY-MP

Aberdeen Proving Ground, Maryland 21005

IIT Researach InstituteATTN: GACIAC

10 West 35th Street

Chicago, Illinois 60616

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Mr. Bond 1

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unclassifieo eeeeeeeeeeeel pershing mar h v white … · the 34 calibrated parameters are shown in...

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