The sequential probability ratio test (SPRT) for 2-IMU FDI  (inertial measuring unit failure detection/isolation) is described. The SPRT is a statistical technique for detecting and isolating soft IMU failures originally developed for the strapdown inertial reference unit. The flowchart of a subroutine incorporating the 2-IMU SPRT is included

Rich, T. M.

English

NASA Technical Reports Server

MC0018lEL!. WGLAS TECt!HICAL SERVICES CO. 
HOUSTCItl ASTROI fAMT I C  S 0 1V I SI C‘f 
DESIG!I NOTE tiO. DC410-007 
A DETAILED DESCRIPTIOC OF Tt!E SECUE!!TIAL PRO%BILITY 
RAT10 TEST FOR E - I K U  F D I  
F- 
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‘D ‘ NISSIO!! PLAFNII.:G, I.1ISSION A M L Y S I S  Af!D SOFT!.!AF!E FOR!WLATIOFI 
24 t-tr(RClI 1976 
APPROVED BY: - 
W .  E. tialrps 
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4 8 8 - 5 6 6 0 ,  Ext. 266 
https://ntrs.nasa.gov/search.jsp?R=19760026188 2020-03-22T13:46:24+00:00Z
-1 - 
1 .o SUt.!%RV 
T h i s  design note describes the sequential prohahility ra t io  test 
(SF91) for 2-IMJ FDI. The SPCT is a s ta t i s t ica l  technique for  de t ec t -  
i n g  and isolatina sof t  ItW failures, originally developed for the 
strapdown inertial  reference u n i t  (EIRU) and l a t e r  adapted a t  Draper 
Labs to redundant qirballed I!IIJ's. T h i s  note docwents the eqiiatiorls 
and describe> the version of SPRT t h a t  w i l l  be used for  analysis a t  
JSC/!?DTSCO. Though some o f  the theory is discussed, a cowlete  
coverage of SPRT theory is  beyond the scope of this report and is  
releaated to the reference rater ia l  cited. The flowchart o f  a sub- 
routine incorporating the 2-IMU SPRT Is included, and is referred 
t o  i n  the text for i l lustration purposes. !!either t e s t  case d a t a  
nor perfomnce evaluation is inclu.:ed, as these will be published 
se pa ra t el y . 
2.0 * I t!TR@DUCTIOlI 
Last Sfptcnker the SPRT a l g o r i t h  vas baselincc! a t  the Level R 
OFT Entry S9R t o  perform the o n h r d  2 and 3 I W  FDI testinn dririno 
shcctle entry. 
SPRT subroutine was added t o  the IMUFDI triple - I N l  simulation 
program on the 3SC llnivac 1110 computer. The version o f  SPRT documented 
i n  th i s  report has evolved o u t  of the devcloprent work to date. Future 
developments i n  SPKT theory are anticipated and vi11 be documented i n  
forthcoming repwts.  The objective of the studies to  date has been t o  
optimize perfcrmance and sensit ivity of the algorithm. I.!ith this approach, 
In ordcr to  bo th  develm and verify the r!ethod, a 
ORIGINAL PAGE L8 
OF POOR QUGLlTyi 
-2- 
the margins between actual performance and external performance 
requirements m y  be easily established. 
- IFIPLE!!E?!TATIOIf DESCRIPTIOII 
T h i s  section describes the fPRT alTorithtn for nerforminr) t h e  
2-1KlJ FDI function w i t h  skewed shuttle IMl's. The SPRT subroutine 
of the ItWFDI program (flowcharted i n  Appendix 1)  is one of several 
versions of the SPRT algorithm that  have been explored i n  t h e  past 
two years. The version described i n  this report is f e l t  to  be both 
the most theoretically straightforward version, and also the version 
most a k i n  to the data tracking t e s t  (References 1 and Z ) ,  thereby 
enabling an accurate comparison betwen the two oethods. The 9raoer 
Labs version (Reference 3) differs  sl ivhtly frorr! t h i s  irtnlemntation 
i n  some areas. 
The basic ides of t h i s  FDI nethod i s  to  geovetricallv resolve 
a failure direction to one of four uniouely oriented acceleroveters 
or gyros contained i n  a pair of skewed Illl's (Reference 8 ) .  
IMU has one planar ( X Y )  and one sinclle axis (7.) acceleroreter, for 
a total  o f  4 acceleroneter instrrtnents per IT! p a i r .  In adc!ition, 
each IflU has 2 planar gyros ( X Y ,  7R) w i t h  the i t h  redundant (R) 
axis oriented i n  the plane of the XY gyro, 12" frorn the X axis. 
I t  i s  assumed for the 2-Itlll SPRT algori thm t h a t  the ZE gyro can 
f a i l  only along the Z axis, since any failtire w i t h  a component 
along the R axis would be detected by the redundant gyro monitor t e s t  
(Reference 5 ) .  Gyro d a t a  from the I axis can therefore be treated 
Each 
-3- 
as if Zt were sensed by a sinale ax is  Gyro. T h i s  assumption brings 
t h e  gyro set into correspondence w i t h  the accelerometer set (i.e., 
one XY and one Z axis instrunent per I lV) ,  thereby allo\.rin! thc 
same algorithr? t o  be used for both acceleroneter and qvro FDI. 
Ther? are r?inor differences i n  error measurenent calculation an? 
thresholding which will be described subsequently. 
Figure 1 i l lust rates  the functiona; flow of the 2-I!n7 SP2T. 
The method is  applicable to any p a i r  of 1IW's;  hovever, for  the 
sake of notational c lar i ty ,  IMU's f l  and E2 are sincled out i n  t h i s  
report. Major functions are discussed individually a s  follows: 
3.1 Error Heasurenent - Calculation 
T h i s  function takes output data fron the IPW's (CV's,?irhal 
angles) and forns a vector equal t o  the discrepancy (or error) 
between two 1:lLJ's. T h i s  error vector i s  then output i n  the coordinate 
system o f  each 1l:U stable platform. 
3.1.1 GJKO- 
The ayro error neasirrement is the "total relative visa1 icrnTent" 
vector, consisting of the off-diaFona1 elements of the Fatrices Q 
and C as shown i n  Figtire 2.a.  The variables  referred to i n  this 
firjure are described as follors: 
Gjn(n)  = is the Euler angle transformation r a t r i x  fron I'VI 
stable platform !j t o  the navination base, corputed 
us ing  the set  o f  9imbal angles read fror! IIIU Sj a t  
the nth tine step. 
c, 
L tu 
I 
V 
3 
0 
c 
I cu 
ORIGINAL PAGE IS 
OF POOR QU- 
T B12 (n) = BZN(n) Blrl(n) is the gimbal angle derived 
transformation f r o m  IllU 81 to IMU 82 stable 
platforms a t  the nth time step. 
T Q = Gl2(1) 612(n) is the skew syrmetric matrix whose 
off-diagonal elements form the total  relative 
misal ignmcnt vector (oU ) i n  ?!?It ?I coordinates. 
¶ V  ,kt 
T C = Rl2(1) R12(n) is the skew synmtric natrix whose 
off-diagonal elements form the total relative 
misal ignmcnt vector (ox ) i n  IF111 t 2  coordinates. 
rY ,z 
I n  the real time computer the six small misalignment angles, Ou,v,w 
and Ox 
columns of the transformations R12(1) and R12 ( n ) .  
can be comptrted directly as dot  products of rows and 
SYIZ'  
3.1.2 Acceleromcter --- 
The accelerometer error neasurecent i s  the "incremental MY' 
vector formed by differencing I'XJ sensed N ' s  i n  the coordinate systms 
of each IMU cluster,  as shown i n  Fiyure 2.h. The current crimhal 
angle derived transformation, R12(n) ,  is trserl for  a l l  transformations 
to minimize the effects of gyro d r i f t  on accelerometer FI11. 
3.2 \!hiteninq Fil  t c r  
A f i rs t  order recursive f i l t e r  i s  used to transform thrt correlated 
IMU error measurenients in t o  a sequence o f  indcpendent samples. The 
theory for using a f i l t e r  i n  this way is whodied i n  the widely known 
result from estimation theory that the residuals o f  an  optimal Kalman 
.e + 
-7- 
f i l t e r  cons t i t u te  a white Gaussian noise sequence (References 6 and 
7). 
by a s ing le  s tate estimator. The outputs o f  the f i l t e r s  are the 
I n  t h i s  SPRT implementation, each ax is  i s  f i l t e r e d  separately 
residuals formed by subtracting the f i l t e r  estimate from the actual 
IMU e r r o r  measurement. The whi tenin9 t r a n s f o r m t i o n  performed by 
t h i s  f i l t e r  i s  required because independancy o f  sarples i s  assumed 
i n  formulating the LLR update equation (Section 3 . h ) .  
3.2.1 F i l t e r  Equations 
F i r s t  pass i n i t i a l i z a t i o n :  
so = 0 
( -DELTAT/T ) = e  
State propagation 
A 
Residual conpu t a t  ion  
n 
rn = Yn - Sn 
S t a t 2  update 
A 
Sn = Sn + K rn 
where Sn = s ta te est inate a t  nth tiiw step 
= residual a t  nth time step ( f i l t e r  output) 'n 
Y, = Iriu er ro r  measurement a t  nth t ime step ( f i l t e r  i npu t )  
Y, = f i l t e r  gain (pre-mission c s t a n t )  
'I = Autocorrelat ion t ime constant (pre-mission constant) 
The above f i l t e r  i s  used f o r  both gyro and accelerometer data; the 
only di f ference being i n  d i f ferent  s ta te update gains (K), which a r e  
l l s t e d  i n  section 4.2. 
-8- 
3.2.2 Pre-mission F i l t e r  Tuning 
The f i l t e r  gains nentioned above were computed pre-mission 
v ia  a tuning procedure derived from rlehrd, Reference 6, which 
bas ica l l y  states t h a t  the op t ima l i t y  o f  the f - i l t e r  can be measured 
by IIOK uncorrelatcd the res iduals  are. I n  a rou t i ne  w r i t t e n  f o r  
the Univac 1110 Demand terminal, the  f i l t e r  gains were var ied 
u n t i l  the average autocovariance of the f i l t e r  res iduals  over 
30 Monte Carlo cycles was zeroed out. The procedure used for 
ca lcu la t ing  the average au tocovariance f o l  lows: 
Step 1. Calculate a residual  Pean for  each of F; axes i n  each 
Honte Carlo Cycle hctveen the times 3n5 and 1205 ( t h i s  
time period contains no vehic le  a t t i t u d e  transients): 
1295 c rt 'axis, cyc le  - w t=3n5 - 1 
by 5 
Step 2. Calculate t h e  f i r s t  autocovariance coef f i c ien t ,  R ( 1 ) ,  
f o r  each o f  6 axes i n  each I?onte Carlo cycle: 
by 5 
Step 3 .  Calculate the averape autocovariancc c o c f f i c i e n t .  
Pave t T ~ o  1- axis=) 5 cyc le=l  E Paxis, cyc le  
-9- 
Repeat steps 1 through 3 on residuals acquired by i terating on the 
f i l ter  gain K u n t i l  a pave o f  approx!mately zero is obtained. 
Uhile computing the gyro g a i n  o f  section 4.2 (rGvR0 = .54) 
the following autocovariances were obtained: 
pave = -.4? 
pmi n 
plXiX 
= -34.7: (X axis, I?!lt t'2, cycle !20) 
= 64.25 (X ax i s ,  It!U f 2 ,  cycle "27) 
While computing the accelerometer Fa in  of section 4.2 (KAcCL = .n04) 
the followina autocovariances were obtained: 
. l Z  - Pave 
' m i  n 
Prte x 
= -52.0Z (2 ax i s ,  I N  %2, cycle 323) 
= 81.9% (2 axis ,  1F.W 62, cycle ?30) 
3.3 LLR Parameter Calculation 
-PI 
The LLR update described i n  section 3.4 reqrrircs two parareters 
to compute the LLR - the classification threshold and the standard 
deviation for  the f i l  tcr  residuals. 
3.3.1 Residual Threshold 
The SPRT alpori thm requires thresholds separating residuals 
indicative of fa i lure  operation from residuals encountered Crtirinn 
nominal operation. As i n  a l l  threshold type tes t s ,  i t  i s  desirahle 
to  have the threshold ti7htly f i t t ed  for crrtatcr sensit ivitv,  yet  
s t i l l  h i g h  enoucth so t h a t  no fa lse  alarms are prodriced rindcr noq ina l  
operation. The failure residual thresholds used i n  this version 
o f  SPRT were gcnerated by multiplyin? a base fa i lure  threshold 
(Figure 3) by a dynamic scaling factor which accounts for thc f i l t e r  
3 t.a 
2 1-3 
k 
Figure 3.a) 
1.5 €4 
.o t-4 
IO00 IW $00 
b r ,  ~ e c .  from htv Irtrrfacr 
Ffgure 3.b) Incremental AV Differences 
-11- 
effects of bias  attenuation and a t t i t u d e  zansients .  The base 
thresholds, specified i n  section 4.4,  are the sane clnscly f i t t ed  
thresholds used i n  the data tracking tes t  (Reference 1) .  The 
scaling factor r , derived frori Reference 8 ,  i s  calculated by: 
where: 
-At /T  i n  which Q = e 
K = f i l t e r  g a i n  
pNB = percentage attributed to vehicle 
a t t i tude transients (Section .5) 
j i s  set  equal to n whenever the greateqt. 
gimbal angle change exceeds the limit 
AGim (section 4.5)  
The final threshold i s  then 
B W  = r ( t )  T ( t )  
which i s  plotted i n  Fiqurc! 3. 
3.3.2 Residual Standard  Devisti,.r 
I n  addition t o  the residual threshold, the SPRT requires 
a residual standard deviation charactcritinq thc "spread" of 
-1 2- 
nominal da ta .  Standard deviat ions were computed a t  several  t i n e  
s l o t s  by the formula 
Residual standard deviat ions \rere found t o  be f a i r l y  constant  a t  
the values given i n  sect ion 4.5 .  
3.4 Log Likelihood - Pittin l lpda ter  -
The l ikel ihood r a t i o  i s  t.he heart of the SPPT a l ? o r i J h ,  
being proportiona: t o  the probabi l i ty  t h a t  a f a i l u r e  has C.L :red. 
The - loci of the  l ikcl ihood r a t i o  is used t n  avoid conmta t ion  of  
the EXP function ai: each t i v  point .  Lori l ikel ihood r a t io ;  ( l - l .n 's )  
are comFuted sequent ia l ly ,  F O ~ C  1 in! the Til t c r  rp s idx i l s  3s i-*!p?.:n:!mt 
Gaussian samples (not necessar i ly  zero m a n ' .  usin? the residual  
thresholds and sinnas conputed i n  sec t ion  3 . 3 .  
the LLF! f o r  each a x i s  i s  spli t  i n t o  two srih-I.LE's, one fo r  pos i t i ve  
In t h i s  ioplcmcntation 
and one f o r  n q a t i v e  f a i l u r e s ,  s ince  e i t h c r  po la r i ty  i s  assuned 
equal ly  l i ke ly .  The L.LR eqriations follo;.:: 
-1 3- 
LLR Reset 
-_I_ 
+ + If An < 0 then A, = 0 
I f  A i  < 0 then A i  = 0 
Take the biggest: 
+ An = max (An, A i )  
3.5 LLR Threshold Calculation -
The LLR fa i lure  decision threshold is calculated according t o  
Reference 9 ,  using the residual thresholds and s i q s  from section 
3.3, by the folloninq eqiation: 
W( t )  = I n  (ZER*(t)/ALPi!A T*) 
3.6 -- Sinrlle Ar.is/lrual !!xis FDI Locic 
T h i s  FDI function perfoms a threshold t e s t  an the 6 error 
neasurenent channels for each ir,stnir.ent tyw,  both acceleroceter 
* and gyro. tach instrment (2DOF, IDOF) is tested ineependently 
for  evidence of failure. For example, if  ei ther the X o r  the Y 
gyro rneasurenent from IXU 81 is  out o f  tolerance, the fai lure  has 
bcen detected 5n the XVl tyro. Figure 4 shows the logic by which 
i n d i v i d u a l  instrunont fa i lure  detection t e s t  are conbincd to i s o -  
late the failure t o  a particular instrunent. An Il-IU failtirr? has 
teen detccte: when a t  least  - one o f  the four instruments of either 
type, gyro or accclcroveter, i s  out of tolerance. 
inplenentation, a t  least  - tw simultaneous threshold crossincq out 
o f  the four instrurents is required tvforc an PI! fa i lure  detection 
i s  registered. A single threshold crossing would be sufficicnt,  
I n  the IV[!FDI 
-14- 
F *+F 3+ F 
+ 
1 
Fl+l 
i L 
’ IZ,Ih . F2+1 . .. 
Figure 4. Single Axis/Dual Axis FDI Logic 
3 IX2i’T or 1Y213T 
ORIGINAL PAGE IS 
OF POOR Q U m  
r 
. F 3 4  
.. lZ,lZT F4+l 
1 
Ft+F + F $F 3+F 
b A 
3 3 IFU $ 1  
F j I  and F4f F9 I LUElE 
-1 5- 
but is also potentially more susceptible t o  false alarms. A failurc! 
can be isolated when exactly I- three of four instr*nents sinul taneously 
show out-of-tolerance measurments. The 1W where the fai lure  has 
been detected i n  b o t h  instrments is the unfailed Ih!!J, since two 
detections would be indicative o f  a simultaneous fai lure  - an assumed 
impossibility. Consequently, the II!V i n  which the fai lure  has been 
detected on only one instrument neasurericnt is the failed I!.!U. The 
case of - four simultaneous threshold crossing5 (not covered i n  the 
flowchart o f  Figure 4 )  is an ahnomal condition and should be flagged 
to the crew. 
4.0 RESULTS 
The followin9 paragraphs present a croup o f  constants, pertinent 
to  the 2-1!1U SPRT alCorithm, t h a t  were used i n  the I’’!IFDf prooram 
to generate the resiilts contained i n  Reference In. These constants 
apply t:, the reference m’ssion 3C entry trajectory. 
4.1 Time Step 
DELTAT = 5 sec. 
4.2 Fi l te r  Constants 
Autocorrel a t i  on t ine cmstant : 
= 120 sec. - ‘GYRO - *ACCL 
Fil ter  gains:  
l;CVRO = ,54 
K*ccL = .os4 
-1 6- 
4.3 
4.4 
4.5 
Start  - Stop Tines - 
= 0 (400,000 ft.) 
= 1945 (Touchdown) 
Tstar t  
Tend 
Rase Failure Th-esholds 
Gyro: 
_I_ 
3 
i =o 
TG(t) = Ci ti (radians) 
Co = 3.7000 E-4 
C, = 3.4967 E-6 
C p  =-1.1786 E-10 
C3 =-5.8263 E-13 
Accel erone ter : 
for Wt<1145, -- 
3 
i =o 
TA(t) = Ci ti ( Inlsec) 
Cg = 2.830r) E-5 
c1 = 4.3593 E-8 
C2 =-5.3665 E-11 
C3 = 9.7743 E-14 
for t>llh5, 
T A ( t )  = TA(t=1145) 
- LLR Constas: 
Gyro : 
PNB = *I5 
cG = 2.4 E-4 radians 
-1 7- 
Pccel erometer : 
P1,B = 08 
= 1.2 E-5 Y./sec 'A 
4.C Plean Tine t'etwen False Alarms - 
T=5000 sec 
'lean False Alarn Rate: 
ALPP'=DELTAi/T=l 0-3 
5 .0  - REFEREIICCS 
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8 .  
9. 
10. 
1. tl. Rich, "A Detailed Description o f  the nata Trackino Test 
for. 2-Il.lU FDI", !!DTSCO Desipn !!ote Po. t'lO?lr)-005, 12 Jan. 1976. 
T. It. Rich, "Performance Results o f  the Data Trackin!: Test f o r  
2-ItlU FDI", f.lDTSCO !.!orking Paper Ho. E?1&-8A-026, 27 Fcb .  197C. 
P I .  n. Adam, "SPRT Documentation f o r  Level C F S R " ,  CSDI. ISS 
E. C. Solov and 3. P. Thibodeau, "Failure Detection and Isolation 
i.l~?tllods for  9-dundant Girilml led Iner t ia l  fleasurevent Units", 
ATAA Paper 3 .  73451, 20 Aug. 1375. 
T. I!. Rich, "Iner t ia l  lleasurcnent U n i t  Pedundant Axis CITE 
Sensi t ivi ty  Analysis", YDTSCO Desiqn f.!ote M. 1.4-8-QC11, 
4 Feb .  1975 
R.  K. tlehra, "On the Identification of Variances and Adaptive 
Kalrian Fii lcring",  ICEE Trans. Auto. Control, Sol. AC-15, 
ApviT 1970. 
T. Ki  i l a th ,  "Thrl 1I:novations Approach t o  Detection and Estimation 
Thcory", Proc. o f  the IEFE, Vol. 53, 'tay 107". 
'4. R .  Ada s. "Iripact of If:tl 'kv Vase Frrors on V!lt F D I " ,  CSnL 
ISS $7': ?!.:, 16 9cc. 1075. 
J .  V .  hdrrison and T .  T.  Chien, "Failure Jsolation For fi r7init-allv 
!!edrll,dant lncr t ia l  Censor Svster", I E W  Trans.  ?rrosrace and 
Clc,tronic Zysters, Vol. . W - l l ,  'fay lQ75. 
Ratio T t s t  f o r  ?-Itill FDI", f lDTECO Oesioti Vote t!o. DV.lO-nn8, 
31 f'arch 1976. 
75-308, 27 Oct. 1975. 
tl. Rich, "Pcrforriancc Pcsults o f  the ccnriential Prnl-ability 
APPEHDIX 1 
Flowchart o f  the  SPRT Subroutine 
I: : :: : : : : ........ .. " 9 . 9 9 .  ........ ........ LJ uuuuvuuu
-18- 
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SUFROUTIIIES CALLFD RY SPRT 
I )  WlTX11 (!I1 , M2, ?13) 
Matrix transposed tires a rutrix. 
f13 = f.1, T !*I* 
2) D:4XtlT (??,  f12, M3) 
tlatrix tines matrix transposed 
3) ONTXV (?Il, Vz, V3) 
Matrix transposed tines a vector 
4)  D!4XV (Il l ,  V2,  V 3 )  
!.latrix t i m s  a vector 
5 )  DVSUP (Ql, Y2, V j )  
Vector suhtrac t ion 
Thc above suhrautines assme t h a t  a l l  v e c t t r s  and mtrices are 
diriensioncd ( 3 )  and ( 3 , 3 ) ,  respectively. 
6 )  TII.’,ESl a x !  TI tXX! are use? by the driver pro?rm to provitk a 
surmary o f  detection and isolation t ines  a t  the end of a rulti  
tlonte Carlo cycle run. 
7)  DVSCLP, (SCALAR,* V 2 *  V 3 )  
Scalar tines a vcctw 
-28- 
8) DVAOD (VIS V2$ V3) 
Vector Addition 
v3 = VI + v2 


A detailed description of the sequential probability ratio test for 2-IMU FDI

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