AD /-007 056 - DTIC

AD /-007 056DYNAMIC CALIBRATION FOR DELCO'SCAROUSEL VB IMUA.C.Liang,et alAerospace CorporationPrepared for:Space and Missile Systems OperationsAir Force Weapons Laboratory23 September 1974DISTRIBUTED BY:National Technical Information ServiceU. S. DEPARTMENT OF COMMERCE

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U CLASSFIEJD. "---IREMY DOCUMNI ATMg,4 PAM Final.FOR DELCGVMAYN&MI(. CAUIBRAIOTNSA, C. LiiuanJ. L. LeMay J WSeptember 1Q74F04701-74-IL-0075,The Aerospace (orporati-nE-Stgusio, CatLtornix 90245Space and Mssille Systemne OrganiataLionAir For e Sy stems Command904t59,0 "5C.Loz A letes Air ForceRewwOwW1 hoINWON AdV&CV ""M6 aS.IitMr23 September 1974wo- e" OF *"a.E'-AUam04w twS9tufvCUOUnclassifiedApproved lft piblhc release.'1.Filteringl, IS,?tMVt.wUmited.Distributtont tm1Me'IIN,-DAU .calibratiou, Carousel tMU,ItcVWT*¶.-- intervals- Sresults U.00TVThis paper addlresoes the ippticationi of filteringl theory to the problem of IMUdynmicm calibration and, in particular, to the Carousel VB system. Lim theicalibration procedure. accelearometer output* are sumnmed over longl timebefore processiaU. For this reason,, the standard K-3 filiter needs to be modifie d as presented in tho papev. Filter derivation and calibrationfrorn proessing reAl mcolero mnetr data are proenetad heroin.UsNg IIri1

2.INTRODUCTIONt.The Carousel VB IMU is an all-attitude,CAROUSEL VB IMUfour-2. 1 COORDINATE AND TRANSFORMATIONDEFINITIONSgimbal inertial platform in which two orthogonalgyros and accelerometers are mounted on a carouseling platform which rotates at t rpm. TheE-N-U:of the IMU,For a more detailed descriptionand Figure 1.rcfer to Ref. (t)An inertial orthogonal system which coincides with east-north-up at "go-inertial".third set of instruments remains inertial along thecarouseling axis.SYSTEM BEHAVIOR OFThisTurret:(XT. YT' ZT)Orthogonal system fixed to driftingturret (ZT along gravity).Platform-Orthogonal system fixed to car-(Xp Yp Zpousel platform.IMU is currently used in the T IIIC launch vehicle.The calibration process extracts optimum estimates of the IMU parameters for IMU compensa-X-Y, gyro, andaccelerometers are referenced tothis coordinate system. Xp. YPalong the ideal X, Y accelerometertZa longand gyr input X ,Yand gyro input axes, Z along ZTption using accelerometer output data.Because the measurement data are summed overa long time interval compared to the s ,stemdynamics update cycle, the standard extended K-Bfilter has to be reformulated. The filter formu-E:E-N-U --- Turret (3 X 3)lation, as well as filter estimates from process-T:Turret -ing actual IMU output data, are presented.T-sin (eYN)"[where OYNLetP.ous Ails I Z,8,4e./system.0M11"bbe the three Eulerian angles that deE can be expressed as-"9.,nAQN(tiSCCILI U?0Carousel angle: function of time only.o@ .,ne atn n,Co co mo-- COeScase .coso- come sino51* cos (eYN)00fine the turret axes with respect to the E-N-U15N1toPIALO5Platform (3 X 3)cos (eYN)sin (GYN)-osnn sCcosi1coss cmxe,,nmine 4.ln,4,n1sin#comeSCoss coo,!co.-0JAIsREFERENCEFigurei.Carouele(I)VI Gimba) -nd ri r. Final Errnr Analystsl Report, UtIDelco *lectronics. General MStorsWCorp. (15 August 1972),Platform Configuration.-1

2.2CAROUSEL VB MODELINGThere are 29 parameters that characterize theperformance of the IMU. They are gyro drifts (3). local gravity vector in E-N-U system(function of tLme only due to earthrotation)The sensed acceleration as measured by n],egyro unbalances (9). gyro misalignments (4), gyroaccelerometers can be expressed as follows:scale factors (2),accelerometer biases (3). accel-I JiiI4iKJr.f u rZHerometer scale factors (3), and misalignments (5).The drift rates along the turret axes (XT, YT ZT)whereare composed of applied torques, gyro drifts, andweunbalance drifts.Note that all drifts are resolvedinto the turret system.[ TbxbIT ]"' [,]if! . [SF ]1, , fU [-]1'',T(E l IL) ]bJ-b(3)accelerometer biasz[K]0 reciprocal of acceleromet:r scaleT'wXerKLYT]: Tplatform driftrateswZT-[[1"02'32gyro misalign--2ritment matrix0and T is as defined by Eq.[T0M1TSF 2:grU1I.LU31alignments matrixI- Na1.a21 x.y accelerometer mis-(2)&gx gravity vector in E-N-U coordinates IThe integrating accelerometer output overa fixed time interval is simply the integralof the "measured" sensed acceleration overthat interval. This output is used as the"measurement" data for the filter.3gyro unbalancematrixcomponents of E are functions of the EulerTheangles4j, 0. 0. The Euler anglesare governedU3 3 J[R]X z-accolerometer mis-[El as defined in Eq.factorsapplied torque (along x.y gyroinput axes): functions of timeonly.10torquer scalea g g0YJU :00Ii0Z Carousel matrix (see. Eq. (1)]0[TSF' 00hTSF 0[1-31]v 10by the following set of differential equationa-[Ry}: gyro drifts(XTWslno wYT coso)/sinOXT coso - 'JYT sin5Rz .-(WXT sino cosO t wYT cosi.T - Carousel matrix [see Eq. (1)]E - 3 X 3 transformation matrix[see Eq. (2))(sine) zTwhere wXT, WyT,-2-cosO)/WZT are am defined In Eq, (3),

3.FILTER FORMULATION,to be estimated and the Euler angles which define H n-i On-''"the orientation of the IMU (turret system). Equation (5) describes the dynamics of the attitude; theHS -nu''" vx (Hn0 I "n-11O 1(Hec)nn nnn-t""n-k Hn-!In-t On-2 " "" n-k Iwhere and3Eachm9Ise. measurement noisesec, whereas each n step ismeasurement matrix relating accel-MV ELU.UN-YN'states at each cycle (t sec)aeprceseoceernn-ent C).nf 2m-nkln-k4n On On-nI3erometer counts (velocities) to the0n-I11-n kZ- Ink# On *n-. .*n-kZ- nt*n1O-kZan-k-t fn On. . *n-k Z 14- )0.4NFL VN1 - 0. * 6. OnEtNE [y. v']I nm Rn.nmmnkthesystem model can be written an follows:2! ON -XN-I -N(7)whereON On On- V'"(8)n-k I ."30(Sn , S . t - nt '"("n On On.(seconds) with the accelerom eters accum ulating-NV.n. :*"n-k-2'("'"'{ n On% 1.ometer data are processed once every 135 stepsN "N N-1n"(1n"lnan "In 1 "n On -" n-11 on-1{ In the factory calibration problem , the acceler-Therefore,0., c ,".z", .% RN[VN.N]LV* Y"'I:nHn are updated over I!-sec intervals.n(ik)On-)I I1n-t On-I "n-)n-21ln15"tpsIIw ithcounts over the entire 035 sec.The cTr-are no longer uncorrelated.8nsdec)at - relation functions become"o'1n state transition matrix over one cyclewith9paametrsad 3 ttitde erorsn"oetr-The state and measurement noise termsnunt state noise 13)H-IUN step isz t meaourements (3-dimensional vector)n-I HnI n- Hcontainec) states (32-dimensaonal vectorBoth29 parameterd35Both 0nseVeN "n n Untn (Hbn 0on Hen-1) un-on-nHn90 sa0nd mauen-k tU-n -k nn-" ''n n-k i nk The linearized system can be describedasv( I -w sen nist-YENequation for the filter. Equations (4) and (5) arethen linearized for the purposes of computing fil-XOn-k inkIO-parameters are constants with white Gaussianstate noise. Equation (4) is the mppaarerentter gains.n-k 1n- I,"noNThe "states" of the filter are the 29 parameters%- 1n-1%," "n-1 *n-1.'"nkt)nk:l''n-k. 'lnz1'(2

IetwithNNE[2NIZi'ZK-N/KPN :[X,-RN/N)(-XNP E(L(.,Given the system of Eq. (2).K (0NNNNN- . XN/N)']the filtering equations-and (12),respectively.The covariance update equations becomeZN.I-- [XNIZ'N(18KN is the filter gain with MN and RN defined inEqs. (10).--E-NIZI.1 8N[NN HN PN-[ HN R N]()can be deriv-ed as follows:-N/NHN MN)HN NN.jI/N,.13)LetE[-XNPN---N/N) Q-N "- 2N/N)butN-f N-t/N-txN -xN/NON--N-IThen Eq. (13) can be written as-NKNHRN INV*fN'N'-l--N/Nl-N/N-i E[--XNZN]E[NZ"I Z(14)01. ""-NN(NNBut,-E[((NO '*NN-where -N'{HNIXN1(N.I/N 1)PN-S0N PN-where MNMN VN}(t11). and (12).Ef- **1HN EL N NNUYNJ* 0N PN(15)N -N'NN(-XNH * (."HN( -N- -11 * P"AHN PN- Combining Eqs.(14),- (*N MNMN (RN are as defined in Eqs. (10).respectively.in Eqs. (7) through (12),that the subscripts n areThe criteria for filter perforrrnnceH *N EL- *-V *of real time.**,HN RNwere the convergence of filter covarlance and themeasurement residual. The residuals were un-(15).and (16).the filteringbiased and of low magnitude (about 0.surement magnitude).5.NLN "9Plots of filter estimates of representative IMU parametersare shown in Figure 2. The data werei/N-I)-Ntaken on 5 June 197Z from'Cirousel V IMU pro *,duction unit No. 2. The abcissa oi ihe plots is in2N-1 I-tNY1cycles, where each cycle is equivalent to 2. 2;5 ze'.-The NNRequation can be written asN/NRK'NKN1-sec steps; the subscripts N are 135-sec steps.4. FILTER PERFORMANCEis as defined in Eq. (10).E(INEquations (17). (18), and (19) constitute the complete set of filtering equations, keeping in mind.HN MNFurthermore.' N N-tfN-t N IPKNNHN Q KE[XZ[ON2N-I/N-I-30'1of mea-CONCLUSIONoptimum filter derived in this -. per nroA,"z-,N--1N-I/N-11(17)uptinum IMU parameter '-;.,aates and showedshorter ,-.urgence time than suboptimal filters,Best Available Copy)

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AD /-007 056 DYNAMIC CALIBRATION FOR DELCO'S CAROUSEL VB IMU A. C. Liang, et al Aerospace Corporation Prepared for: Space and Missile Systems Operations . (eYN) 0 M1 "[ 0 0 1 "where OYN Carousel angle: function of time only. P.ous Ails I Z, Let 4e. 8, b be the three Eulerian angles that de-15N1toPIAL fine the turret axes with respect to the E .