2900-371-T(t) Report of Project MICHIGAN OPTIMUM CHARACTERISTICS OF SOME MOTION-COMPENSATION SYSTEMS Volume I Theory and Application of Doppler-Inertial Systems W. A. PORTER A. Y. BIAL March 1963 Navogation and Guidance Laborator d tCu C $ S4~ec~ a4 4 744 THe UtNIVe*SITY OF MIC~HIGAN Ann Arbor, Michigan

Institute of Science and Technology The Uni.versity of Michigan NOTICES Sponsorship. The work reported herein was conducted by the Institute of Science and Tcchhology for the U. S. Army Electronics Command under Project MICHIGAN, Contract DA-36-039 SC-78801. Contracts and grants to The University of Michigan for t support of sponsored research by the Institute of Science and Technology are administered through the Office of the Vice-President for Research. Note. The views expressed herein are those of Project MICHIGAN and have not been approved by the Department of the Army. ~4tstrtiution. initi aistiriouon is minacaea a rane end.of this document. Distribution control of Project MICHIGAN documents has been delegated by the U. S. Army Electronics Command tothe office naned below. Please address correspondence concerning distribution of reports to: Commanding Officer U. S. Army Liaison Group Project MICHIGAN The University of Michigan P. O. Box 618 Ann Arbor, Michigan ASTIA Availability. Qualified requesters may 6btain copies of this document from: Armed Services Technical Information Agency Arlington Hall Station Arlington 12, Virginia Final Disposition. After this document has served its purpose, it may be destroyed. Please do not return It to the Institute of Science and Technology. ii

Instiute of Stience. and Technology The Univerity of Michigan PREFACE Project MICHIGAN Is a continuing, long-range research and development program for advancing the Army's combat-surveillance and target-acqistion capabilities. The program is carried out by a ull-time Institute of Science and Technology staff of specialists in the fields of physics, enginoering, mathematics, and psychology, by members of the teaching faculty, by graduate students, and by other research groups and laboratories of The University of Michigan. The emphasis of the Project Is upon research In imaging radar, MTI radar, infrared, radio location, image processing, and pecial investgatlons. Particular attention Is given to all-weather, long-range, high-resolution sensory and location techniques. Project MICHIGAN was established by the U. 8. Army Signal Corps at The University of Michigan in 1953 and has received continuing support from the U.. Army. The Project constitutes a major portion of the diversified program of research conuctd by the Institute of Science and Technology in order to mae available to government and ldwstry the resources of The University of Michigan and to broaden the educational oppormtuntes ot students in the scientific and engineering disciplines. Progress and results described in reports are continually reassessed by Project MICHIGAN. Comments and suggestions from readers are Invted. obeert L. Hess Director Project MICHIGAN HII

lnstitute of Sceinnco und Techn"!ngOy The University of Michigan CONTENTS Notices.................................. ti Preface......................... Lists of Figures and Table........................... Abstract........- * a *............. 1 1. Introduction..........,,. *.......... 1 2. System Representation..............t... 2 2.1. General Block. Diagram 2 2.2. Second-Order/Third-Order Dilemma 8 2.3. Uncoupled Hybridization 11 3. Specific Configuration Comparisons........................ 16 3.1. Parameter Definition of Error-Source Power Spectra 16 3.2. Pure Crosstrack-Velocity Optimization of D1 System 18 3.3. Derivation of loVfS) 18 3.4. Calculation of'v('u 20 3.5. Calculation of a2 21 4. Delayed-Data Operation of a D-I System................... 22. 1. Introduction 22 4.2, Illustrative Calculation 24 4.3. Calculation of o2 for the Nonrealizable Cas 35 4.4. Calculation of the Portion of 2 DDl I Due to the Realizabilty Constraint 36 References...............38 Bibliography....................... 39 Distribution List......................... 40 V

~.,;^1ute of Scionce and Tech.nolngy T h Uniiversity of Michigan FIGURES 1. A General D-1 System:..- *.... 2 2. Alternative D-I Syltem, iFo rm, #.. i.."..... 3. Aternative D..-I, 8ytm, Formr....,..'......,;... 1 4. 4;Second-O erDl System l,*,-......... 5. Seeond-.Order D-I Syst em::.1...,.J....... 9 6. Seqon(dO0rder D-sytm..........Se....... o** 1 7. The Primary Geeral P.l System,.......,.*... 11 8. UncoupledZt. System.................,:. -..........12 9. Delayed,-Dita D.- System.,........-,....... 23 10. Conventional Filter Problem......,... 23 11. Nonrealliable We6iting Functon.. 25 12. Realizable Weighting Function.......................25 13. Pole-Zero Plot...4...'.... *, ~.:..*..34 TABIE L System Mdentificatlon.,....:..............., 6 vi

OPTIMUM CHARACTERISTICS OF SOME MOTION- COMPENSATION SYSTEMS Volume I: Theory and Application of Doppler- Inertial Systems ABSTRACt. A side-looking radar system employing synlhetie-iantenna techniques can obtain azinmuth resolution fitr than that afforded by the radar.beamwidth. The capabilities of such systems depend In part on the accurate sensing of vehicle perturbation from a reference path'and subsequent phase ceompensation forthis motion. This volume deals with the optimization, for such nmotton-comensatlon activIties, of a family of doppler-inertlal systems. By use of the variat!onal calculus techniques of the Wioncr filter theory, functional relationships between optimum system performance and sensor noise errors are developedand systems with and without dynamically induced errors are compated, I INTRODUCTION Over the past several years, high-resolutton radar systems have rapidly advanced In sophistication. y vanced In. Unfortunately, this broad progress has been marred by pockets of technical depression. The technical problem of primary importance today is the development of sophiticated optimum motion-compensation systems with capabilities and performed profiles that remove the primary obstacle to a "really" high resolution system, Although it has been recognized for several years that the inadequacies of the motlon-compensation technology were the main stumbling block to high-resolution performance, curiously enough, no conprehensive analysis of motion-compensation systems has (until the present stuy) been undertaken. Past efforts on such systems were based partly or wholly upon Ideas of system design adapted from navigation applications. These applicaUons did not stress hig-fidelity, instantaneous aceuracy, but were mainly concerned with long-term average accuracy. As a consequence of the ANt'PD- (airborne high-resolution radar) program, new ideas on system design were found to be necessary, Development of new techniques, however, is an arduous task Furthermore, the navigational philosophy of system design has many idiosyncrasies within itself. These have resulted In a multitude of system forms, and hence confusion and delay in the implementation of effective research progress. We still arbitrarily assume noise systems with known spectral forms and rely upon linear system analysis-two assumptions which run contrary to.actual physical realities. The outlook is not completely

Insttftt:, of Science and Technology The University of Michigan bleak, however. Although many new developments are still necessary, some progress has been made. The conclusions presented in Reference 1 and In this report are the first in a series of assaults on the problem. We hope that they will serve to guide present system devellopments and to measure future progress in the research programs that should.follow. 2 SYSTEM REPRESENTATION 2.1.. GENERAL BLOCK DIAGRAM The seemingly endless variety of block diagrams usable for the description of doppler-inertial systems has its.advantages. However, the design engineer many times is faced with such a diverse selection that the possibilities are bewildering. In this,ection, a systematic approach to system representation is presented with the folowing objectives: (1) to emphasize certain aspects which the author feels have not been completely understood; (2) to show:that the motion-conpensation application is compatible with the; concept of generalized system optimization. Figure 1 is a block diagram for the crosstrack channel of a D-I system.'f a sitgle accelerometer is to be used, this figure comprehends most of the possible methods for coupling the doppler and the Inertial systems for motion-compensation applications. The notations used on Figure 1 are as follows: -gs ( gs0 A + IF e -- A 2~ 1 +-.s | —| —-s~ t (Accelerometer)! -- J^R t(s~L- J C L -'G'D..' —-- i.... L.._J 6 r - —' Z x 4 (Gyro) (Doppler Radar) FIGURE,. A GENERAt D-I SYSTEM 2

Instiuta of Science and Technology The University of Michigan EG' cD C accelerometer error, g'yro drift rate, and doppler noise, respoctively 6 - verticality error g,.R gravity field and earth radius, respectively o, accelerometer tilt angle from a level position o sf~ and eo sin Po and cos P, respectively 0 0 0 0 A ^ V, 0 system estimates of along-the-beam velocity and vertitcalty, respectively AB, 0 - truc along-the-beam acceleration and vertical direction, respectively To illustrate the general nature of this block diagram, we will derive the various error equations and, by subsequent identfl cation of the filter transfer functions H1(s), H(s), Hl(s) and H4(s), we will relate this system to the various possible forms. 2.].1. ERROR ANALYSIS. Equations 1, 2i and 3 are derived from signal summation n Figure 1. e() "c)(s). A (s) gcP, a), - O (,V ). -, H()V (a S~) + C (1) -(s)' l1A)^o(s) + ( - (IV I/Rs (2) Sinee V3(s) equals the integral of e1(s), it follows that -.,. v^.v-^-^^ "!"]^ et A ~ iy(() a)o V0(3) L W os 43 B ae,(-)4l'.3oet -.()'. -r,(s) I +Vg sw Il0, if we ubstte Equation 2 and l to tho i e.press i, t follows that VB(s ),2 - (s) [g+s +H - ( ) H() + V( B ) + 5 (a) g sH + n a Ia (a) -I ~ H8 + +- H ()() -- H P H(s(s) (5)( S+~a'~J~ta fl. ~o'],. 3 0 0 3 c

ir,:i;uto of Scfi5ce c;^d T.-c hnology The University of Michigan which may be redullctd to ^ 2 2'* VB(s) s + sH ts) 4 = N(.. s) + s VB(s) F + si41(s)eCo'S 2 12l I + Hs3()V.Vt(s) (f6) where the identification Ar. Nfc, s) SE >(s) gcp,1 c(s) F isila(s)l +a n oH(s)i a(s) (7) has been made. Solving Equation 4; for VB(S) yields, 2V () + SH (C +. C.. 1. 1+ Hi....JV 1 A N(c. s) Ct V (s)- 4.- - --- -^+ _ <(8) B [ + sl1 (s) + n H( a +H sI(s) + f ((8 A A Since V (s) is an estimate of VB, we identify the pertinent error In this quantity aS VB(s) - V s);: hence B B B B A A.(c s) EVB(s) t VB(S) - VB(s) 2 sH (]2 + ) - [sH,.) +2 H(s] V (s) + 1si (s){co I + a ci + H ( )l1V s),: 1, - _..., ---- (9) + sH (s) + f (s Noting that VB(s)= cf Vct (s) - S3 V (s) it follows that o 0 Ct 0 oZ A N(e s) EV (s) 2H -2, VB s2) 2 +sH(s}) sH(s) () (10) S + SiliS)) * Q2H4(s) We note that in Equation 10, H is) and li(s), H2 s) and H (s) oocur in particular combination. In an attempt to simplify this notation, let us identify these combinations by the two filter forms H (s) and H (g) which will be defined as H (s)=- H (s)+ 2 H(i) (11) HIs() H (s) +2 -H2(s) (12) 6 3 42 4

tnstitol of Science c,$J TchnoloCy The University of Michigan Through this new notation. Equation 10 reduces to ENV, ) S( (;s) + ca V(s + H.(s)s/V. ((s) E )+ B' 2 -if1 2+2 2 (13 s+ 5' i (s) +n tli.5(s) 2.1.2. RELATED) SYSTEM FORMS. The various forms of system configuration under consideration for purposes of motion compensation can now be identified as i'articular members of the family of systems depicted in Figure 1. This point can be illustrated by use of Table 1, which presents all system forms, ranging from pure inertial with level accclerometer to a completely coupled doppler inertial with tilted accelerometer. The conclusion to be drawn from this dlsplay of "ystems Is simply that FiLure 1 has sufficient mobility to represent this broad scope of systems. A less complicated block diagram would be more convenient, however, provided this mobility is not lost. The present diagram can be simplified by consideration of Equations 11 and 12 of the last seetton,.which show that it is not necessary to have 4 independent filters. As a matter of fact, H (s) adl H (s) in Figure 1 can be omitted if Hj3s) and H4(s) take on the forms of H5(s) and H(ts). Furthermore, If we tre interested In minimizing the error effects of true erosstrack motion, a glance at Equation 13 suggests th6 auxiliary constraint H:(:s) + Hs) + 1." (14) In this case, Figure 1 reduces to either one of the two particular forms depicted in Figures 2 and 3, where Q(s) = H(s).Hs) i?() Since these two forms represent identical systems, the form which leads to the simplest error equations can be selected. This form is shown In Figure 2. We should note here that It is quite similar to the block diagram used in Reference 1, where theoretical optimization studies have been carried out both with and without the constraint of Equation 14. For Figtre 2, the pertinent -rror equation then becomes ^ N. Q(s)l + W2oQ(s)V(s) EVBr() 2 2_. T T - (15) 8 0n(s) where the identification N[(, Q(s)l = s a(s - (1G S) - go(GQ ) (( - ) 16) has been used.

itutJ of Scienc.l atdi T,-rhnology Tha University of Michigan TABlf I. SYSTEM IDENTIFICATION Case Il(s8) Ht( s) 1l() ) l() EKy8 Type 3 0 1 0 0 0 Q...: Pure J' rta St.'_- __. s _ u, (.- _ ( 1 y 0o o ~ 0 I 0 * Pure Inertial +: Self-damped.:;-. gs()+ ( S) I - K) - ylvc Cii Y ~ * 0 c- (a) - - K(G- S -. Self-damnped &: * +,YS * Detuned V r K'IK 0 -1 2 - 2- Doppler mped - +y +n (1++ Detuned V. ~.H~ ( (*s ) -* ( ge f () ) + ct [-sH (S) t f2l -H ())v () V H(s I)0 O () 0. o o G E 2 ~ 1 2 - 4 c C Tilted Self-: a:*::.s + Ht(a) +ntH (a) damped & et ed I f 1 4 | (s) - gc (s) + [sH1(s) + H () ) VI I i l(s) H1s+ (1 +H Ti -te - ie oppler..:....., * -,. "' - __ o.t.:.. -*. - -., - * * 2 -. a 2 + H u)+nl + N H.-a). 1 o a.,. I.,,' *, 3.9) 1 + H 3(s 0 2 T d) * (t + N3(si)1.+ A + H)) a (... Tun) 6.~~~~~~~~~b+.* (a) + a ()+ n1H _ _. E,._' -'..'' -.': -i _ _. 6~~~~~~~~~~~~~~~~~2,(

Institute of Science and Technology The University of Michigan 0(s) _ r —__ —-'_ _ (Accelerometer) ~ _. _ _.. D.. ___J [ Qs) | (Doppler G _ Radar) ~(s) I ^~ —- --- ~- I i Rs., r v+ I/ ct (Gyro) FIGURE 2. ALTERNATIVE D-l SYSTEM., FORM "1 %(s) A- * —-. —- -..i... _._._ (Accelerometer) \: -M r..... c tTa) 1 —+ — I W' Y~t(S)I G() H(s) Radar) Vc,~S~'i~[ 1/8 ----— " L....... —J Vet( ) (Gyro) FIGURE 3. AITERNATIVE D- SYSTEM,, FORM #2 7

Institute of Science:Ind Techni'logy The University of Michigan In the above reference, It was found conventl.at to introduce the filter form I(s) =- ~zq($. ~ ): (17) 2 217) s + -Q(s) In terms of I(s), Equation 15 reduces to EVB( I(= 13- I(s)kl (s) + I(s)eb(s) (18) where,s) " 2 E ) - geo DG(s) - ] oD (s)] (19) b(S s)d(S + 8p YV(s) (20) In partial summary, we have demonstrate(I that a wide range of systems 4a few of which are listed in Table I) can be discussed or described in terms of the generalized notation of Equations 15 and 18 and/or Figures 1 and 2. The. following sections will show that: 1) understandingof these systems is enhanced by this approach; and. 2) more importantly, failure to approach system selection in this manner can lead to false conclusions. 2.2. SECOND-ORDER/THIRD-ORDER DILEMMA An unfortunate side effect of the unsystematic description of D-I systems st the number of ambiguities in the current nomenclature of certain systems. For instance, the use of the terms'second order" and "third order" (to designate the familiar configuration shown in Figure 4) is iacc'rate. This terminology derives from the fact that the characteristic equations of those two specific systems are of order 2 and 3 respectively. However, we shall show in the following sections that there are-a number of possibly second-order systems, each with distinguishing characteristics. Let us begin by defining a second-order system. as a system whose characteristic equation has the 2 2+.. * *':. ( ~ form s + ys + (1 + aQ2. For Instance, the error equation of Figure 4 is given by g s) (s) () + s( y + a )D (s) E (s) (21) s + ys +.(l +a This error equation describes system performance when, as is common practice, the calculated velocity is taken directly from the output of the integrator. However, let us select our output from another spot in the system. For Instance, as shown in Figure 5, the output of the accelerometer may be fed to a 8

ii, 0~ 0 w~~~ ~ ~ i'~:. b - -.! - _ ~~~~~~~~~~~~~~~-. *^~~.~ rr'fn* "Tt ri~.r [ ^ ^ I; I +" r ry ~ k i+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~4 1.+ +.i +~ g.. i!.~ 0)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.~ 0~~~~~~~ -o 1 <$^4"'"0 S 2 pn "-~ l3 T < 3'; *~~o~ + ~t]~~~~~~~~~~~~~~i ~ ~!.:.. I. In I''~'-: ~ I.- -.'~~~ r. ~-~iu rl c~~3 oe I c~~~P I -1c I ~' " I ~ a Ig li jlj~ " u 0~~~~~~~~~~~~~~~~~~~~~~~~~~~,~.) ~ ~ I,.. I..._J' J ~~~~~~~~i5~~~~~~~~~~~~~ ~,...$ m E ~1 0~~~~ C ~ ~ 7 i~ ~.;r V~ a H~~~L

l.ij;-te of Science and Technology The University of Michigan separate integrator, thus yielding a new calculatilo of velocity. lect us denote the error In this calculation by E'v(s). It can be shown that this error is given by the expression -g(l + y/)( (s) + 02( - y/s)r(s) + ( + Y)c (s) E ).s ) (22) 2 + a + (I +a) ( Comparing Equations 21 and 22, we see that the high-frequency doppler noise affects the first output I IT position through a gain s, whereas it affects the second output through a gain Thus the conclusion that second-order systems are necessarily plagued by this high-frequency nioise is unmerited. Note that the second configuration has the disadvantage of an open-loop integrator, which may cause lowfrequency drift problems. However, consider a third example given in Figure 6, where the Equation E" (8) is given by (a + y)~ (s) - g('+./s)c(s) +:n(a - y/s) (s) E". (s..). (23) E"v(s) 2 2 ( s + ys +(1 + a) Clearly this is a second-.order system whose sensitivity to high-frequency doppler noise is as good as that of the second example, and avoids open-loop integrator operation. Va V.EV (Accelerometer)'' (Lopper v I ~,::: ~. i/s iP)4 iRadar) r. (Gyro) - G FIGURE 6. SECOND-ORDER D-j SYSTEM #3 10

Institute of Science and Technology The University of Michigan By this time it is apparent that further examples can he concocted. For example, we may extract Information from any of Figures 4, 5. amn1 i; directly ahead of the "division by R." and thus derive additional possibilities for second-order systems. Similar ambiguities can be found in inertial self-damped systems, where the system errors are functions of the particular damping and output positions. In light of these examples, we can conclude that by failing to approach the system from a general view, r.e may reach false conclusions about osytcm lmitatiagr;. 2.3. UNCOUPLED HYBRIDIZATION So far we have concentrated on D-I systems which are coupled by signal mixing within the system framework. This section will show that equivalent operations can be performed without disturbing the autonomies of the two component systems. Let us begin with the generalized system representation in Figure 7. In Figure 7 there are two possible outputs —if we exclude the possibility of using an open-loop integrator and ignore direct readout from the doppler radar itself..Equations 24 and 25 describe the signals at these points. 2 2!-I 2 Vit()' V(s) + ( + R s)')1 f C(s)) (- g( 1 - Q(s)I"(s (24) V~(s) = V(s) +(s2 + 2Q(s))l' SQ($).();- g1($)<G(s) | - QW Ifc(s)}' 25) (yoA 1 2r L — (Accelerometer) | (I>'Flcr iiar) | Q(S) (Gyro)', FIGURE 7. THE PRIMARY GENERAL D-I SYSTEM

Institute of Science.'d Tcxhnology The University of Michigan We denote the errors in these signals as E (S) I s2 +21Q(s)1 ( gS)-g(s) - Q(s)IC(S)2 (26) V i Ev (s) S I+ Q(S I sQ(s)(s) - gQ(s)^(s) +' s2 - Q(s)1 c(s) (27) In comparing Eqilations 26 and 27. we note that apparently the first position would be much less sensitive to the high-frequency noise contributed by Co(s). However, the optimum Q(s) would not be the same for both velocity output positions; that is, optimizing Equations 26 and 27 will give different Q(s) forms for minimum error. Indeed it will shortly be shown that these respective minimum errors can be identical or not, according to how the optimization is conducted. We shall use the termn "uncoupled hybridization" to designate those D-I systems which function without intercoupling and whose hybridization is carried out by simple signal mixing in an output channel. This hybridization is represented in Figure 8. o Pure.Inertial A + r —, V __,_j (Doppler(Accelerometer rR —ad FICUIE 8. UNCO!PLED D-I SYSTEM ~~~~12 ~ (Gyro) V + V D (Doppler Radar) FIGURE 8. UNCCOtUPLED D-I SYSTEM 12

Institute of Science and Technology The University of Michigan We shall use the (leslgnatlon V (s) and V (s) fow the velocity output of a pure inertial and a pure doppler system. The system otttput is computed by mixing signals V1(s) and VD(s). It can be shown that the two signals are given by sa (s) - ge (8) V (s) V(s) + 2 ~ (28) + VD(s) V(s) + ef(s) (29) Hence the output V (s) can be computed. to be 0 fs() - gc ( V(s) 5 V() + Y()8> [ 2 + - Y IC <(8)l (30) We shall designate the error in V as EV (s). which is defined by 0 V Ey ()- Y(s): n ~ + I - Y ) (31) To show that this open-loop hybridization produces the same. signal as the closed-loop configuration in Figure 7, let Os rewrite Equations 26 and 27 as follows: C (s) - gc (s)) EV (8) = Y (S) 11 - Yl (s)i (s) (32) 1 L *2,n! J where 2 +2 s2+.2 Y(- (a 2) (33) 2 2 and EV (s) - Y2(s)- V S 2' + i.2. - Y.JD (34) 2 2 + + 1 where 2 2 (a + n )Q (S) Y2(s)= 2.2... Because the filters and error sources appear In Identical fashion In Equations 31, 32, arn 34, the optimlzation process will lead to identical minimum errors. Althoiugh the optimum filters Yo(s), Y (s), and 13

in-titute of Science and Technology The University of Michigan Y 2() will be Identical, it is apparent from Equations 33 and 35 that the optimum filters Qoi(s), given by Equation 33, and Q2(s), given by Equation 35,- will differ. In fact, equating Y (s) adl Y2(s) shows that Q l(s) - + - (6) We will now show that if the optimization process is performed from the intercoupled equations, the resulting minimum errors given by Equations 2({ and 27 will differ. In Reference 1, a filter I(s) waa defined as I(s) 2 2 (37) s + Q(s) By use of this filter, Equations 2(6 and 27 may be written EV (8) (s) lD( +I''(s2 (s) - ) (38) 1 2 12 2 R 21 E2(s) = (s) f c(S) - G(8) - CD(Sj + [1 - ((S)31) (~ ) If I(s) i-s optimized without constraints in Equations 38 and 39, the resulting minimum errors will be identical. (This is obvious, since we have merely rewritten Equations 26 and 7.). Howeverf if we constrain the poles of I(s) to the left.half of the complex s plane, the different forms of the noise sources in Equations 38 and 39 cause different minimum errors. The crux of this argument is that the optimum filters I (s), given by Equation 32, and I (s), given by Equation 39, are optimized, in the cofstrained case, through a process which uses the separation of a polynomial into left-half and right-half plane parts. This operation is "nonlnear" and does not commute with the operation of substitution. To illustrate, let us perform a portion of the calculations. (Reference 1 gives detailed definitions of the following quantities.) (s)0 ( -s) 3- g 2 [2 (8 a () + (s2 + 2)2di (40) 2(s)2(-s) 2 s- 2f(s) + (s2 + 2)2a (; (41) 14,- - + + (s 14

Institute of Science and Ttchnology The University of Michigan hence I R2 ~ff i2 n ^,(s) ^('s)' i^-2+ v() (42) rA ) ) V(-8) (44) A (sR-' -s Now we deflne the combinations: ^(s )= (s)+ %baA(") - -*f24 %() - + (2 + Z*( (4> +2X=a) "' + *8s) (8s2 +QDl() t4 By comparlng Equations 45 and 46, we can see that 2 4 (s) i 2 s) - ) (-s) 2(, - v( )-) (47) 2is S b2R2) 2 Now the optimum system formss are determined by the equations F1 1 lov= - r.7l. — l (4e) o^~~A-< ^,A^<^J5'L 1^(49) oA ZA{s (.3 Substituting 43, 44, and 47 in 49, we have'2s 2 ["."4 (50) oA 2 1 2 (50 A n vf.L v 3ihp lb v( ihp 15

t!:i tule of Science'!.:d Technology The University of Michigan which redtices to FS4+ (s)1!onls) = I';... (52) 4 (s) Lv lhp Hence if the condition of Equation 52 Is to be consistent with the expression 2'oA( 1 2 WI( (53) then' ( s).., - s) 2 2,AT I -'~.; (54) * t-hp L -hp which Is clearly not always true. 3 SPECIFIC CONFIGURATION COMPARISONS We turn our attention now to specific calculations of the available accuracy of several of the system eonfigurations mentioned in Sections 2, 3. and 4. To do this, we will select a standard set of error sources. (representing actunl hardware reasonably well) and carry out optimization for the various configurations studied. The resultant maximum system performance can then be computed for each configuratlon. 3.1. PARAMETER DEFINITION OF ERROR-SOURCE POWER SPECTRA The discussion of the power-spectra functions. 4(w,). D(w). and 4 ( [) [2. 3, 4, 5] Indicates that,G will usually consist mainly of low-frequency content, and 4)<w) mainly of high-frequency content. Generally, % (w) contains both low and high frequencies. In the following example, the units used will be those of the mks system: *G(a) is measured In (rad/ sec) /unit bandwidth; 4 (w) is measured in (m/sec ) /unit bandwidth; and 4D(w ) Is measured in (m/sec) / unit bandwidth. The power spectra used in this calculation will be 2 -12 0.15rw a x 10 <G() e2 --- 2- - (rad/sec) /(rad/sec) (55) 16G 2 2 16

itstitute of Science and Tec!Enology The University of Michigan _ a %() =,,D2 (m/scc/''d/scO (5.) 4~ (D -re 2 x 1 8 (m/sec2)/(rd/sec) (57) In more familiar terminology, it can be shown that Equation 55 represents a typical gyro-drift-rate spectrum with a stanlard deviation of 0.1 o degrees/hour and a correlation time of 1/w seconds. Equation 56 represents a velocity'.;cO6r with a white-noise error whose variance/unit bandwidth is a (m/sec). Equation 57 represents an accelerometer with a variance/unit bandwidth of a 104 (m/sec)2. or roughly a 10 g's. a.. In this as ln other problems. the analyis is considerably simplified:ly introducing dimensionless parameters and variables. Thi. particular catculttlon will be slnplifed by three such identifications: 6 - / (dimensionless) (58) /A D 4a12(/E;) (dimensionless) (59) W- S/f4 (dimensionless) (60) At this point the definition of Equation 59 is far from obvious; suffice to say that,len the quantity /V(s)$v(-s) is expressed in terms of the variable W, it assumes an easily factorable formn The variable change of Equation 60 can, of course, be spelled out in more detail. If the complex variable W has the real and imaginary parts W - t + j3, and the complex variable s has the real and imaginary form s a + jW, then Equation 60 is equivalent to the two equations I' -,A/A and t = a/A, To provide a range of parameter-consideration which realistically represents actual gyros, accelerometers. and velocity radars, the parameters will be assumed to be within the following limits: O.i i n a (sec2) (61.) I _ _ 3 (? m/sec) (612 0.1l~aa0.3 (r lOf4m/see (63) 0.3 ~a < 10 (teaths of degrees/hour) (61 14 17

rnstitute of Science arid Technology The University of Michigan Froms these limits it may he derived that 0.1 < (dimensionless) (i2) 0.12i) o0] A l (dlimclnsionless) (;22) 3.2. PURE CROSSTRACK-VELOCITY OPTIMIZATION OF D-I SYSTEM In this section, we shall optimize the configuration of Figure 2 with f.,.that is, purely orosstrack0 velocity sensing. In Reference 1, it is shown that the symbolic form for the optimum l(s) is given by IoV()-= [- t"I/..i......... lo4f ) GV~s) hp (63) 4~(s) j~'C ~j.)..' vlhp where 4aaV(s/) + 4b^v/j:= ) + i s )4(s/) (G4) and, (s), -,s) - 1A4) g2%(s/j) -s2 (S/ (2 + l 2)2 s/j) (65) In the combination of spectra in Equation 65, 4 (w) could almost be eliminated by inspection. At low frequencies, 4i ({) is weighted with an w. thus reducing its size in comparison with G(4, which Is largest 2 4 G at low frequencies. At high frequencies, W b (w) Is compared with w %4 and thus again is negligible. It should not be surprising if 4 (4( plays a very minor role in determining I.(). 3,3. DERIVATIQN OF I (s) By means of Equations 63, 64, and 65, the symbolic specification of I (S) has been obtained; and through Equations 55, 56;, and 57, it has been connected'with the physical characteristics of the sensors. This cluster of information will now be reduced through a series of substitutions so that the final form of I V(s) results. The work will begin with the calculation of' (s). which proceeds in a straightforward manner. By substituting Equations 55, 56, and 57 into Equation 6;5, we find that 2[.l 2 2 - 8 2 2 ( r~usg u) <r. *,o ~ n 1 89V(-s); =~^ —'- z - s - A!~ + (a ) f ( 18

Institute of Science and Technology Tha n tivrrsity of Michigan When the noWllinmctnionl parameter nlotations of Equations 58, 59, and 0;O are intmlruced, this relationsip is transformed thus: 0. - a22 22 -8 422 2 2'0 C' 2 2 2 2 v w2 2 3 1<2 2 By removal of a common factor and straightforwnrd manipulation, the relationship reduces to 2 6 (W)V(}W) - T2 2 w2 -W ) (fi8) The fact that the six.zeroes of Equation a8 lie exactly on the unit circle results from the frequencyscale transformations of Equations 59 and 4;0. t should be clear that these transformations were evolved from the development of Equation 68 In the s notation and are used here merely for simplicity. Equation 68 is easily factored into the following right-lialf and eft-half plane parts:.. - *W+.6 3.. 3V:*w (W -6 Equation 64 can also be expressed in terms of the W variable: aaV( + baVy(W ) (1 + ^2W)D.(2 71) The optimum filter. I vW), follows directly from Equation 63: oV 2 H1/2W I v(W)J -' --- (72) oY AyW (W) -',1)(4w w+ IL The bracketed part of Equation 72- has no left-half plane poles. Now, however, the bracketed function approaches zero as W approaches nfinity, a development which leads to the equivalent expanded form,1/2 K W +K IW+K I -f + 2 1 0 (3) o(V yW - l)yOy-W + 19

Institute of Science and Technology'i'h Urtiveriity of Michigan where tD is taken as the left-half plane part of tils exlpression. Thus I/2' ovW'() A (W (7 )) Equation 74 may be used to express this result in the original variable notation Io (s) r (75 (S + Sr)(s~ + * hs *Qf"-A') 3.4. CALCULATION OF 4V(w) In this section, the optimum filter I (s) will be substituted back into the error equation which it minimizes. The resultant system-error power spectrum will be found and the standard deviation of this error calculated. To do this, recall that V() -,oV(.)-'lov(-J)' aV- +'ov-J')-.o.0V abtV( + oV^OW) - I1 V(H + V(4) I +' O) bbV(4 +,..v0 )-,aVov ov' bv. By rearrangement the equation lbecoftes V oVO oV( -j3,aaV) abV( %baV() b+ bV -I 0o)lv (w) +, bV()) - IoV' (w)a + 4 ( + 4aaV' ( (76) oV aaV 9bvaV baV J aaVz ) It is possible to attack Equation 76 directly with a barrage of substitutions for the various symbolic functions, but the process can be simplified considerably by noting the following factors. Equation 74 states that I (vO ) aD/tVOW) and Io (-jw) = D/e^v(-ja. Furthermore, the term inside the first parenthesis of Equation 76 is by definition v(JOW)v(-jiw). In addition we note that 4a (w) a and i a(W)?- ba(w). With these facts in mind, we can reduce Equation 76 to 2t \ i 4 ) = +D2-1201 o + (77) 2V D0oV oV' J 07 20

r..t;tute Qf Science and Technology The University of Michigan The fIllowing cntculatlons, like those preceding. are nost easily handled with the terminology of the W variable. If Equation 7.1 is used, then lPquatlon 77 can be shown to reduce to 4n2f + l - 264,) + W -4 (26., + 3-4 2)+n0 43 6A;I *v(W) 2ra -- (78) 2 3.5. CALCULATION OF OV The square of the standard deviation of the system velocity-measurement error can be obtained by integrating Equation 78 from zero to infinity.'Theikntegrationa facilitated by making the change of variable 3o -; hence dw i fl4d. In terms ot ther tvariable, t he ntegral becomei <, 2 -. 4 3 -1 -2 2 -'i3- 2 ^^43' -3 2 v D(79) J 2 (7V D p.00 1. By dividing Equation 79 into three integrals and applying the approprate integral given Inbe useful:ation *x dx r "..' **** "... By dividing Equation 79 tnto three integrals and applying the appopriate integral gen in Equations 80,, 80, and 803, the evaluation of a beco me a2 ^ + -2 I W 3 )4 -?S ) 4a^ (81).. WA a ~ 2. ~+ 1 3( By combining terms, Equation 81 may be simplified to the expression (7a-2(i4? j1 4 2)j (82) "v OrD.,4 2 Since 6 < 1 and A > 1, a good first-order approaimation of this result wotld be 2v -jryao2 ~(83) 21

Institute of Science and Technology Th e Univer ity of M c higa n If the origimnl dfinition of ution of tn 59) is used, V 2='2d G D2)' (84 A graph of Equation 84 and a discussion of this result are provided in Volume II. The sequel includes the computation and comparison of standard deviations for many other configurations. The variety to be encountered makes mandatory the adoption of a standard subscript notation which conveys the identification of error being described. Since all errors to be minimized are velocity-measurement errors, we shall omit "V" as a subscript. The notation to be used is 2 The first group will use either DI or I to indicate doppler-inertial or inertial system, respectively. The second group will use NE or E to indicate nonexact (dependent upon the motion itself) or exact (independent of motion), respectively. The third group will indicate the amount of filter delay time allowed: 0 - no delay; a0 - infinite delay; T - T seconds delay. To illustrate, Equation 84 becomes S.224S f DI, E'O 8.24' Q 2D \2, (842) 4 DELAYED-DATA OPERATION OF A D-I SYSTEM 4.1. INTRODUCTION In this section. we shall consider the feasibility of using a delayed-data motion-sensing system for motion compensation. The operative rationale is that a system may be able to derive compensating signals of considerably improved fidelity for the high-resolution radar system if this derivation need not be Instantaneous. This means, however, that the radar information must also be delayed by an equal time interval; hence it is more than merely possible that the increased complexity would far overshadow any gain in system performance. In addition, we canl derive an increase in precision, basing our design on a knowledge of vehicle perturbations. We begin this discussion by considering i> e simplest of the many possible block diagrams. Figure 9 depicts a D-I system whose output is fed to a linear interpolator. The ideal interpolator would yield as an 22

'1ir^ u. of rt: r -;.?'';,:'', y *' Se Uta U i',,,, * t f,.* I ~ S i g an output. the tirlue velocity nt a time T itl t,,ll.s'. o. Hence, we may juldge..;r interpl:itor's pc:rfotrnanct byy examiningi the dliffcrelncl iatween Its ouk:;.'t.tI.(i tht' idlhal output. This,'rItence has iOcen dte;ig-nated E (s). 0 E (s) is given by'0 Eo () - vl(s) I(s) - cT I T(5 (85) The system of Figure 9 is identical In form to that of Figure 10, where X the signal, n - the noise, and E (s) takes the form o E (a) = n(s)ll(s) + (II - e )X(s) (8i;) 0 Thus, comparing Equations 85 and 8(;, we see that E (t) = n(t) -- the rtoise input to the system; and X(t) = V(t) = the signal input to the system. In the actual case, E is the error in the system's measurement of V and hence (as has been dilscus..'td in Section 1) is a function of ibt i-arious mixing configurations chosen. To illustrate the techniques involved In this problem, let us make a simple introductory calculation. V- D-I System H(s) T Interpolator + E =.Error -sT Ideal System FIGURE 9. DELAYED-DATA D-I SYSTEM n 0 FIGLe- lI FIGUilE 10. CONVENT'ION\AL FIITER11 Pi3O-2EM3 23

h;i,.;s:',,,f 4,"S a.nce and T.,-o!,oy The University of Michigan 4.2,ILLUSTRATIVE CA ICULATION Let us find by means of the Wiener filter theory'lhe optimum rcalizable Il(s), for estimation of X(t - T). Our optimizing criteria is an integrated squaretd crror. WVle assume for this example that the signal and noise;re imndependlent,.vth,'e-p.cctive power spectra given by,XX 2 (87) 1+ U 2, (W) (88) Hence,i)(s) = 4.(w) + 4bnl (W) = 1 (89) at all w. We assume here that (1(w.) is f;ctor.ble i::to exclusive left-half and right-half parts and hence is expressible as (1(w)' —- 4> (4o), (w) (90) where 41, (cw) contains all its poles and zeros of 45(s/j) in lhp,I'(as) cortains all its poles and zeros of,(s/j) in rhp i(w) Is factorable in this sense if f. 7 < c (91) +-o l(+ Following standard techniques for optirnization, we arrive at the conditions ) (w) - (X+) 0 (92) rri (W)() 1- i (Tcxx^ n 1 H(w) -. —- F - (t) (93) where F - the Fourier t'xnmsform -1 F the inverse transform u(t) = unit step function 24

ltisiitut of Sci,:nco. vnd fre.hnoloyy "'%e Uri'.-rs-ity Af Michijan In this exnmple P(") Ideal filter * e j (3J) 4* (. ). I (,sa ) {(i~:1 (952) For Irnependent signal and noise we have 4X(X+n)( " X+xx and M.+ _.L —-'e —- (96) From the Fourier transformation calculus, It can be sbown that which Is shownI.n Figure 11. L(t)u(t) Is shown In Fig ure 12 L(t) FIGURE 11. NONREALIZABLE ~WEIGHTING FL'.CTIOX u(t)L(t) FIGURE 12. REALIZABLE' W!TIGHTING FatNCTlO 25

Inr.it.jte of Sc;knce and Technology The University of Michigan The impulse response of the optimum rcalizable filter associated with Equation 92 is give,: by h(t) = eT u(t) (98) The transfer function HI(jw) of this system is determined by JWT H(tw) h(t)e' dt (99) which In light of Equation 98 is.(w, -Io t-TI-jw1 T ~I().=' f= e e' dt (100) *. ** JQ.... Evaluation of this integral results In -WTe -T -jwT e -e e flow) ( - (101) =(I. -jw) (1 + jw) Equation 101 represents a stable transfer function since the singularity at jO=" 1 is removable. This s ev.ident if we expand the numerator in a Taylor series as follows: e -e I-iwT -j- 3 4 jkl-T+ 7- 49. (102) 2: 3 4 ] 2' a: 4! or e-)JWT -T _ T(1J _W) T 2-4l_ 1- -w.. e =-c 2T(1- I -+ (1 +-jT)T2 + -( ) 1+...: (103) Therefore jwT0T 2 3 ~ e -e _T T_' 2 Tr 2 (eiWTO&T)J'2:' t + 4w + 0 *1 11 0 _________ =jw 3: 4 Jw3i1 0'&'~ 1+.. (104) -sT -T Ifet jw — s. Then c -e is analytic in the whoi: complex plane, and therefore can be represented by e -e a power series about s = 0. If we let -sT -T 1-s r- — 26Fs (105) 26

Institute of Science and Technology The University of Michigan then by e.pansion fbout the orgifn 0 where a w a:I $)a v^}(107) The optimum realizable filter has a transfer functlon that can be put In the following form.-.jT c, H Ts)a * 3(108) The minimum realizable standard deviation can te calculated from the expression r' * + A |L 12|2 (109) *''~~~~~ ~. 2V 2'where 9 = minimum realizable standard deviation ct 2 O =-minimum nonrealizable standard deviation Using Equation 97, f L(t d110) Also, o is given by CoO In our exampie |jp|2 |eITj.=: hence we ve a2 2 = l @ 7*'(1122 27

Institute of Science and Technology The University of Michigan Evaluating this integral and using Equation 110 results in -2T 2 e o (T) = + (1- + xr/4) (113) cr 2 Equation 113 has a classic form for this type of problem. a (T) is composed of a constant, (1 + f/4), representing the minimum u for filters with poles in either half of the complex plane, and a contribution -2T due to the realizability constraining e /2. Since T is the amount of delay ia the filter, Equation 113 2 shows that the increase in accuracy, that is, decrease in a (T), is asymptotic with T, as anticipated. Now returning to the V-I system case, where we have from previous workl Ey(s) = (1s)-) - ) (s) + I(s) b(s) 14) From Equation 85 it can be concluded that: Wo) = How) W(-Jw) EV + (HW)- e ir (- - e j) () (115) Obtaining {EV(w) from Equation 114 and substituting In Equation 115 results in 4(w) = HO()H (-J) tlIOw) - 11tI(-jW)- Iltav( + II(-Jw- 11 0)4abV^( + (lO^ - Ii-JW) V + IOw(-W bbV( -jwT +JwT + H(jUo) - e J)H(-W) - e () (116) Equation 116 represents the power spectrum of the output error of the interpolation of Figure 9. The rms values of this error are given by 2 2 J (w (017). o' j-oo To keep calculations to a minimum, we assume that I(s) has been chosen in advance possibly to minimize E (s) itself. The job that remains, as in the foregoing example. is the selection of optimum H(s). 4.2.1. NONREALIZABLE CASE, If we do not require physical realizability for H(s), we can use the techniques of the previous work-which lead to the result (/) w T -juf (118) where 4v () = power spectrum of the signal and I EV(W) power spectra of noise (ipenpedenl t of signal). 28

Institute of Scienc:! Technology The University of Michigan 1*,ti.2. REALI, ABLE CASE. Following the llustrative' calculation V<~) *v (W) +,4V(W ) (119) Again we hnve nssumed that 4V(tI) is to Ib factorlzed Into 4V(w) and V4'(w). that Is, 4V('N(O ) " (120) where fV(w) containa lef-half plane poles only, and 4;(w) contains right-hall plane poles only. The optimum, H(c), then takes the form r I I 1-T uW HOUW) - F 1F (T i u(T) (121) - ~~(~ ~ y:..4 <( W.) / /. "vsx L \~v,;V. where u(t) Is the unit step function. As before, we use the notations -1 *T(S. WoJt l ~r2 ~ J| L(tI 2dt (123) (7 2(t) J.[ mw )( j ds + fJ L (r* T)Idr (124) The first trm Ln Equttion 124 Is a constant independent of T; the second part depends, of course, on T. We will assume the D I syFtcm to be affected by a ranidom gyro drift rate,'(w ), a random accelerometer nole. 4 (w), and a random doppler error. 4(w). To see how a (T) varies with T, we take the realistic error source power spectra defined i Equawtions 55-.2 of Section 3.1. In Section 3.3, we have shown that the optimum I(s) form for these error sources is V o(,).}... n( + 22 _ 2) Or (+A) c + a+ fnl ) 9 29

lt:.titute of Science and Technology The Urniv.rsity of Michioan and Section 3.4 shows the minimum t y(w) to be 2 2f2 (92 +1 - 26S) + w2i4 (264, + < A3 -.2 + 1663 _ ( EV(w z 6D - ( 78) Finally we will assume velocity-perturbation power spectrum to be of the form 2K 2 v -)= v 3 v2(125) T + +et Adding Equations 78 and 125 and setting w 1, we find that 2F+( w 4 $2(24 2 2)+2Q4- 2)+n63 K6 2 (126) The combined numerator. N(?), of this expression has the form N(w = AA +B Bi +C.i +D1 (127) where 2(2(2 2 2 A =2drn2 2i(22 +1 - 26A).+ K K 2 (128).2 3 2 I 2 222 B1 = 2)aD Y4(26A + 63 - 22 + 2 22(2A + - 2) (129). D. 2C, 2 33 2 4 3 2 cl 21 oD2 naa3-(A ) + 2,r~,,,..(2,a + - 2A.. (1360 6D [ =G i,2,A3 )] (131)23 30

Institute of Science and Technology The University of Michigan.......... i,,,,........ -. — ": hence t TL+ AW+> B CW + D (V1 1 v1^,,J'' e(i' r. r(132) For' large, B C D N4u~ w)4(^ ^1 (133) For w smill. /AW + B + C) + D (134) At high and low frequency, the B and C terms can be neglected. By comaring the relative sizes of A; B. C, and D, we can conclude that B1 and C are negligible atall frequencies. Equation 132 reduces to *( ~A)(W +1) Now we define 6r u t?6^6 (136f) 1 =:1DA (136) For the range of parametera selected for this calculation (Equations 55-62), /3 can be approximated by 6, 860 2tW 266 o~:_ AI+ -- (13 3) 2-' Kv and therefore Equation 135 becomes ( +v)(W +1)J 31

'.-.I'.j Cf SIunce c.!d Technoul!)y The University of Michigan Now Equation 137 can be fiactorlzed as follows: 6 +P - (c) - 31)(w - 32)(w - X 4)(w, - t ()(t - P) - ) (138) iwhere i p(cs 30 + sin( ) 30~)-1( i ) (139) 2 - i (140) 1331 2 ) (141) 34,( i- 2_ ) (142) 3 = -.j. (143) 65 -' 2' a,; = ( -^ I; (144) And... w6 +y (- y1)(wY2-)(u )3)(O- -) - YW)( -y>6) (145) where the roots y through ry correspond to P/ throigh t, respectively, except for the appropriate magnltude factor. Finally, w2 + l (I + )(> - j) (146) In terms of this notation, Equation 137 can be separated into. (s-8J11)(S-3132)(8 s143 and K (5 JY )(s iv )(s i/33)(..+.1) (147) (v - Y1)(s - jY2)(s - JP )(8 + 1) anid i-)v~ K(s - jy4)(s - 5y)(8 - jS)(s - 1) (148) 32

Instit:?t of Science and T:;ohnoloqy T he UJniversity of' ichigan Now, to get.(t) fromt Equiation 122, we need K (s - 1(Yi(- v( - J,;)5 (s + _s)(s" JO )s (119) Equation 149 can be put in partial fraction form: V (.)] [A+B +D.e isT (5 ^*rU)8* t.s j- A ^ B,+ + O 50~ If we let 2 2(1 A -..... (15 x then = tLAA, and the constants A, B, C, and D are given:b the approxmate xpression: A. K (152 B — a K(A* - 1)(2,- J O.;(153) h RK2 v -.. C' ^ IA( 1 - A)+.(154) Since n is of the order 10'3. the dominant residue in Equdatio 150 is A. The functi< of Equatlon 150 has i poles as shown in Figure 13. We can find L(t) by contour Integration, the results of dhlch are t(t) ~F.T.. 4(t-T) Ji3(t-T) Be 4 +Ce J i. t-T +e'e t<T ( -l(s 33

Insi'tute of Science and Technology Tho University of Michigan -_l 15 I x 14 FIGURE 13. POLE-ZERO PLOT Applying Equation 95 (restated here), a.-.St H(>w) =- F' 3't) (95) + + V We arrive at (s -JY)1 s -l2(9 JY 3)( + 1) K i (s - Jo)(s - Jp X)s - j ]) / ip T\ ( i 15T\ / j\1 -sT /est e4\ /st "5 \ /st j6TB e +1 -e _e- -e x L+ e B SD s (157) Equation 157 reduces to 2 2 F-T st T s - I2f-e e -2 1I H(-) -- +C J+ X \i+ 1 S /+J C - / \D ] (158) 34

nr'!utfe of S'r;ice ond T i\nology The Univar.ity of Michigan Since the R, C, and D te rms are snimll conmared to the term A, a good approximation for H(s) is s + (t2A)(l +'i)32 + (Qn l+ +|( 3eta L\V/ JL /'~ j 2u 2 \/3 1/2 H(s). 4. -,.-. /(159) 2.where the j's and 7's have been eliminated. Finally, we apply Equation 124 to get a (t). As stated before, the first part of this equation is that due t6 the nonrealizable filter and the wecond part is' due to the realliabitity constraint. 4<3. CALCULATION OF a FOR THE NONREALIZABLE CASE Now, by uslug:Equations 78, 125, and 132 we find that vT )+.EVs I 2 K 21+)+2Q426 - + c.,J3- +. 160) Neglecting B1 and C1 as before, we got f1 2 (*V L 2bDK t2[n + I- 26T2a + (26A + 4A - 22 + a3' - y~~~~- 240 to get~(161) Using the definite Integratl in Equations 801e 802, and S03, and r.oting that A1 is approximately equal to K,* wc write Equation 161 as F 144~# 4(2\ ~ 2 2`4 2 333 4T (E 2 2 (2A + I.- 26A) - 2 + 24 " -s ) "VT (i3 Neglectin" B Y aL.,~f o(re 5/w ( 1) 35

Institute of Science and T.chlf:o` )iy The University of Michigan Substituting for 1) in Equation 162 gives F TV E"V.. 2 2" T (:i)P4Qn + 2w/ *'j ^'2 — I3- 2 2) 3 + a(24 +.6, _ 22.'.. 3(A3 6) 3 3/ 2, o\1r as/ 2 5/6 (163) 6Tl3 2A1 2j * 23eD2K+K 2 2. 2:r K 2 2 3 -, 2 2 rD K 2 1- 26 (2+a a-2b) (34-6) Fvi).24K ZLK 2) 3~ 2 J ( 2 +2' / (-2r/ (164) 1.4. CALCULATION OF THE PORTION OF a. DUE TO THE REALZABILITY CONSTRAINT 1,, NE 2 The contribution to a due to the realizabllity constraint is of the form DI, NE r~. 1 | t. T) t, where L(t, T) is given by Equation 156, repeated here: jP (t-T) AP 0(S-TY Jo(t-T) L(t, T)' Be + C + De t <T (15(;) Using Equations 142, 1.13, 144,'153, and 155 for illumination of constants f.; 5, fl;, B. C, and D, respectively, and also usang the facts that f =* j* and D - B* we reduce Equation I15 to the form 2 (t-T) L(t, T) K^(l A)(I ) s A(t- T + 2R11 4 (16T5)) 36

Inslitute of Science and Techno!ogyl The University of M5ichignn.,.... where the Re letnotes the olerantion of taking the real part. After evaluation of the real part In Equation 165, e e can show that l.(t. T) ln.. ~...2.AA(-TT) Le.T) =KL( - ^A)(l ^Z4J'+ 24QnA^^K i y (,1 0,707 co at - T) +.866 sna (t - T)l (l;5)g2 where it - (6AA5 and where.' _ ~^(t-T-I':. L(t T) ) KVA1tI - A){AJ ()iI ff yg T4 mi0T* l - n } (166). A bold (but in'this case, accurate) approximation for L(t, T) s given by L(t, T) K ^A(1 - A {( ela T) K fl4(1* (A (167)...... -.A{T) } where 0 = tanl j/2 Employing the approximation (x + y) < 2x + 2y, a bound on L (t, T)i s given by 4AA(t-T)_ L (t, T) < 2K 2A 2 (l- A}'2O-' j + / coi - n } {) (t.).2K 1(1-A). - IiT} S (168) For purposes of establllt.g an uppr.boumid w.will rqaplae co ((Pt - 0) - by unity and iheoe obtain J L(t.T)dt<2K 2n^21A +. J e 1 dt (169) which when evaluated yields J~ L (t - T)dt 2 e-(41 v'AL2 (170)-: (41)A)/(1 +).'i 037 37

;:2:;-itse of Si:;(rc- cir d Technology The University of Michigan which finally results in ^. "* - w^:^-)^)~~. 4 L T)d = t +se (171) Finally, the minimum realizable standard deviation for filters with delay time T is found by combining Equations 171 and 164-which results In* 2 2 2ir 2K 22 3 DI, NE.T 34A j I,2 27 ^ 1/66 I 5/6 K -YK 2 + 264 + 2A 22 2&K + 2 32lK r7 + 2K D |v V' v v / v ID D/..a K 2nl2'li!1+2 4. 4- (172): Volume II of this report will tnclude a discussion of the theory of a pure inertial system as applied to motion compensation, and a simmary of the results of the present volume, REFERENCES. W. A. Porter, A Generalized Statistically Optimum Veloctty-lnertil Sytem, Report Nuom ber 2900-236-T, The University of Michigan, Institute of Science and Technology, Ana Arbor, Michigan, April 1961. 2. F. B. Berger, "The Nature of Deppler Velocity Measurement," IRE Trans., September 1957, Vol. ANE-4. 3. R. H. Cannon, "Kinematic Drift of Single-Axis Gyroscopes," Am. Soc. Mech. Engrs., New York, December 1957, Paper Number 57-A-72. 4, H. Goldstein, "Sea Echo," In C. R. Crand and B. S. Yaplee (ed.), Propagation of Short Radio Waves, McGraw-Hill, New York. N. Y.. Vol. 13, Radiation Laboratoriy Series, 5... E. Goodman and A. R. Robinson, "Thermal Drift of Floated Gyroscopes," J. App' Mec., 1957, Vol. 24, p. 506. 38

Institu*t of. cnnce and Technology The University of Michigan BIBUOGRAPHY Berger. F. B.. "The Design of Airborne )oppler Velocity Measuring Systems," iRE Trans., December 19t7, Vol. ANE-4. Davenport, W. B. and W. L. Rnoot, An Intrduction to the Theory of Random Signals and Noise. McCGrw-Hill, New York, N. Y., 1'58,. )woretzky, L. 1f. and A. Edwards, "Introduction to Doppler-inertial System Design," J. Am. Rocket Soc., December 1959; Isaacs, R.. Noise Error in Inertial Guidance Systems Report Number RM-483, The RAND Corp.., Santa Monica, California, October 1. 1950 (CONFIDENTIAL). James, H. M., N. R. Nichols, qnd R. S. Phillips, Theory of Servomechanls, MOcraw-HII, New York. N. Y., 1947. Johnson, F., "The Synthesis of Velocity Inertial Navigation Systems," Proc. Natl. Efeetronlcs Conf., 1959. Lanzos, C. The Variational P.rincples of Mechanics, University of Toronto Press, Toront, 1 957Tr7?46. Press, Toront tLanning, J. H.. and It. H. Batten, Randmm Processes in Automatic Control, McGraw-BHil, New York, N. Y., 1956. - Le, Y. W. Statstical Theory of Communications, John Wiley & Sons, lno. New York, N. Y., 196a0. Porter, W. A., and L. F. Kazda,'Optimization of a Generalized Veloclty-Inerttal System," IRE Trans., June 1961. Vol. ANE-8. Strell, i.. Mathematical Theory of Inertial and Ddppler-Inertlal Levettng, Intemat Technical Memorandum Number' 619, Oeneral Precision Laboratory., Pleasntvlle, N. Y*, November 14, 1957. Tsien, H. S., Engineering Cybernetics, McGraw-Hill, New York, N. Y., 1954. Vowels, R. E.. "The Applicationof Statistical Methods to Servomechaniasm*" Australian J. Appl. St.1 i95., Vol. 4, p. 469-488..Wiener, Nt. Extrapolation, Interpolation and Smoothing of Stationary Time Series, Wiley, New York. N. Y., 1949. 39

.l:tihut of Science and Techrlogy The University of Michigan PROJECT MICHIGAN DISTRIBUTION LIST 5 1 March 1963- Effective Date Copy No. Addr*esW Copy No. AddremNW Commmanding General. 512 I )tir,.or.'U. S. Naval RItecarth C altrattr U.S. Army Comult Surveitlance Agency Wastshi1ton 2, S. ) C. i124 N. Ilkhiand Street Arlington 1, VirgnATTN: C t.l ~owmand 53hral" Conmandlng (Xfler U.S. Army tyletronakn Command U.f.. Navy OrdnAtc taboretry.Fort 1Mo1iuoaolh. Now C.. loroep ATTN: AMSb L RDAT r i-iw C *omoanu~dloK otlc CImrnrsdh (S Oft~r. & rDrector Commandlng Wicrer U.S. Nlvy lec'trowIoa lAboratOry U.S. Army lectrontc. R & D Laboratory San Dt Ctslri4 ruortt M sooA~U(~L;`Nn u tb.Dieo $2. CUlfurfr~ ATTN: SEt.RA/ADT ATTN: Utf 34 Command,.g OGenral 24 Cuomosendtts et3sra~ 55 Coarn der, U.S, NaraI Ordehanrc tabtory U.,S ArWy 6t1edronlr~ Prowkl Ground Siver Sprinog. Maryland Fort Huacfiau, Artlru SlvSrtn,. M.;ryfhcu ATTN: Technical Ul brryAT: Tehlcat trary 3.A T0 ASTIA (TIPCA) A rli ogl mn k 11. Station CGodery,ltelltgenw & Mapplio R & D Anc Artitt li rgtnt Port nlovolr. Virginialto. VrStk~ (S2) ATTN: dltllgenocr tlv~iloim' 1-.84 Commander (56) ATTN; HerlArth Anoltyj tvll ft WrtgMt-Pattter*m ATFr Ohto (37) ATTN: Phiutmraommltry Dtvtlo ( A1 A b (3J) ATTN: Stratrec S.yrrm tVito ( ATTN: )ASR-I (ErNOGMSSO) (80(4)'ATTN: AS (ASAPfR-fI *&$-(.) ATTO: ASnD(AnsRGE*-l) 5 Dtrctor. U.S. Army Cold:ROtl Resrerch & Engtioring taboraory RP.oxarch 2 Delopni tLabo Rratory ome) AADveom PASt S Hanover. NewIrtimpda - e. C Atll Ntneti~icr A:gort 42 CosinmadtoanHg(QlVc 249 E. StrMe. N.. U.S. Army Rtsearth Ol(lcc(tNwhm) Waolasm 2,m 0. C, Doxo ClM, Iute SaUon1 ATTN: OCR *AW lltoam Durham. lorth CaroHiMa ~r~earch & D, Plv~elent Liboratr7 ftS ) ATTil: AI&TSc o ATTN: Chef tloiormnitUa Proceso~lng Ca 9S 1 StlnttYLct Thnuieal h lrmitioq.rilit 42 Asottnot CComimandant tht~ta. Manytuad U.S. Art y Atr Dec S ATTN: AARe.rtatlt ort 4 llsreet N. WT F4ort ibs..Taoo nit Ntt t Aerwattlr Sport Aed;irltratlto 44. CommtAant McrednStaco Croft CWenter Fort yeleolr, VlrgTaecAca"bn i. ora IJSK;.hM. tArtmty e Oiet,. ATTN: teSSY-. ig 45 Commandlts Ofter o Coumbrt Surveltoace ProJert U. S. Army lnttrlltrwce Combat Dcvrelopment y AerouticaA tasortry, e ncarpoctred Fortl.Hotabtrd P. To.tox t Btaltimore I. Moarylad ArtUntgi a 10, tVrginIa 4? U.S. Army R esrkb Latoon (t. leTTo:. Tex b MIT-Linooob Laborstory JOt Recb Anatlytt Oporstios Lexlnogon 73, Ma~sacltuetta 53 Arlsngton Road Departmeot od the Stry Dai. C ATTN.: F:SSY-, 17th & Contttullt Ave. ON.W.TT Cf. ffiontlonnd Crontrol Syln. te Walhlngton 25, D. C. U.(48 S. A rmy: Cdtlellfee Combt r10.0velopme Agency Correll Aeroauttcal laboratory, Incorporated F(40)ort. Aor Co ^0 445 1enee "rt''40 Baltimore to., Iftrylfd'Arlingto IO rstnls'ATTN: Tec:!nical Labrary.' 4 Commanding ORfcer. U.S. Army Ec t ronic PAainetch UR# P. O.; 11ox 205 "'prtmo~l of lw P~n ~ W mllrto Tefo Cormratt UWur, t,3n View, C17tMorn ATTN: 1,cstronkdenwe-tdboratorle Santa monte Cilliffr'v 47 U.S. Armny Rnsearch Liaiotn OffC*ic'MIT-Lincol Laboratclry lo Research Aat.-I C rtoratlotor 41J-49.Office of Naval'Research h Bet~hesda,. y ATTN: Cude 1640.Iwfalo It. New Yor* "40

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AD - L.UCXASSIFIED AD UNCLASSIFIED This votum* deals with the pttmizatioa (for such aotLonI- DESCRIPTORS This volum, deals with the optitriuton,for such motion- DESCIUPTORS Ipe-satlon actlvltles) of a family of dopplr-lrwrtltl ay*- Aclertn compensatlon actlvltes) of a family of doppler-iaertlay ya* Acet, tins. y use of the vartiatlonl calculus techalqus of the ni 1tems. By use of th variatlaonl calculus techatmua. of theU Wi 1cHr f tlryt fruncinal"r relationsthips 1iwVwe iKtluntm %%. fAulter O1irl *,r fuctiwoal relationships l 0tabwNe tImdt polr ptimmt*on evsiteirn pnl~rfnurmaenif nsisd s nir. ae rr'wer are.$ypiquumdd hkwrlla r yts timona w ss? ae reaaeiiein rtia svritio N*4 symom OA #~r4 *frvtw~~t d(yMmalv~y ~Wftto Pwrwr#i VA~,,. ovito<~ws *n awl ~i(kwl (~y",at~i~~4y ~rotwI~~ orroioft1 w"',lvl<(h Solmmfcf~~~~~lT*'~Vat~wlrM Yams po it, al ait lyt~al l lata ro alyr wiy^Kri-...wmIOIPrVl~.~u 4WCLASSYIED UNCLASSIFIED + ADUCWSW AD UCLASSITFI) This volume dadla with the ept(!isanom (for sueb eMoUo- DESCIPTO This volume dteals with the ptisalzation (for sach mation-.EKPTOitS ee*upeassU<e activities) of a family of doppler-hiertial sy-. AClIOMB01emaatlon actlviUtes of a fmily of doppter-taertisAal Acceleration W mo s a. l r uMse ofliteisonltal oul~di* etdskoiq, of lw - texa. By us of the varatisotlw calculus techaiques of theDo ler at on inertial nasc~vt~atioa - --— i errors arec~s Wea~rpni~ir Utotrr)fmcto a fralftttonrie~plbmweeop teF~ clWubtore (Uirrvu) oio~ f~kmcto*l~ r~afionlpp.rLa b ~wnotiuyperMlto Aaral~nttoc~ ~cclarostmparsd,.rarr, ~J ulr mT< Itb~ rud-^orrrl aloiMmr# tscla~~td LASIFIED | rxa~ d ymtorifl a" *rUKrrmro CLSare ~Wrb~~M~i u~r kncrsarlra~t~arhl~~kt~ao rllwl, ~ ~ pp~l etwity f "M rcr uvigCllo vycLAwraMOUCYW O

-~- + ~ AD Div. 19/4 UNCLASSIFIED AD Div. 19/4 UNCLASSIFIFD Inst. of Science and Technology, Unlv. of Mich.. Ann Arbor L. Title: Project MICHIGAN Inast. of Science and Technology. Unlv. of Mich., Ann Arbor L Title: Project* MIHIGAN OPTIMUM CHARACTERISTICS OF SOME MOTION-COM- ILn. Porter, W. A.. Bilat, A. Y. OPTIMUM CHARACTERISTICS OF SOME MOTION-COM- I. Porter, W. A., Bll~1. A. Y. PENSATION SYSTEMS. Volume L Theory and Application of IIm. U. S. Army Electronle PENSATION SYSTEMS. Volume I: Theory and Application of m. U. S. Army Electrtnics Doppler-Iertial Systems, by W. A. Potter. A. Y. 8ilal. Mar. Command Doppler-Inertial Systemsa, by W. A. Porter. A. Y. BIal. Mar. Command 63. 39 p., incl. illus.. table, 5 refa. IV. Contract DA-36-039 SC-781801 63. 39 o.. ncl. Ullus.. table. 5 refs. fV. Contract DA-S36-0; SC-78601 (Report No. 2900-371lT(I)) V. Project No. 3D5801001 (Report No. 2900-371-T(M) V. Project No. WD50<1001 (Contract DA-35-030 SC-78601) Unclassified report (Contract DA-36-039 SC-T8801) Unclassified report (Project 305801001) (Project 3D5601001) A side-looking radar system employing synthetic-antenna tech- A side-lookint radar system employing synthetic-antenna techniques can obtain azimuth resolution finer than that afforded niques can obtain atirmuth resolution finer than that afforded ty the rtadar beamwidtk. The capabilitIes of Such syatem by t9barduar beamwidth. The capabilittes of such systemas depend in piarton tihe accurate eosing alff vehicle perturbation depend i part on the accurate sensing of vehicle perturbation from a reference path and subsequent phase oompenation for from a rfrence path awB subsequent phase copansation for this unties.ir slt motUo. lover) (over) Armed Sorvices Armed Servicc. Technical Information Aency Technical Inforni'lrton Agency UN.CLASSIFIED UNCLASSIFIED + + + AD Div. 19/4 UNCLASSIFIED AD Dlv. 10/4< UNCLASSIFIED. Inast. of Science and Technology, Univ. of Mich.. Ann Arbor L Title- Project MICHIGAN Inst. of Science and Technology, Unti. of Mich.. Ann Arbor L Title: Project MICHIGAN OPTIMUM CHARACTERISTICS OF SOU. MOTION-COM- IL Porter, W. A., Bilal, A. Y. OPTIMUM CHARACTERISTICS OF SOME MOTION-COM- n. Porter, W. A.. Bilal, A. Y. PENSATION SYSTEMS, Volume L: Theory and Application of i1. U. S. Army Electronics PENSATION SYSTEMS. Volume I Theory and Application of m. U. S. Army Electfonics Doppler-Inertial Systems, by W. A. Porter, A. Y. 11sa1l. Mar. Command Doppler-lnertial Systems, byW. A. Porter. A. Y.-Bilal. Mar. Command $,3. 39 p., Incl. Illus., table, S ref. IV. Contract DA;-36-039 SC-78801 63. 39 p.. incI. Inlus.. table. S refa. IV. Contract DA-36-039 SC-76801 (Report No. 2900-371-T(l)) V. Project No. 305801001 (Report No. 24)0-371-T(l)) V. Project No. 3Ds801001 (Contract DA-36-039 C50-78801) Unclassified report (Contract DA-16-039 SC-78501) Unclassified report (Project 31)4801001) (Project 3D5801001) A side-looking radar system employing synthetic-antenna tech A side-looking radar system employing synthetic-antenna technlquea can obtain altmuth resolution finer than that afforded niques can obtain azimuth resolution finer than that afforded by the radar besamwkith. Th capabilltis ot suach systems by the radar beamwidth. The capabilities of such systems depend In pa-t oa the acsurat sen"itng of vehicle perturbation depend In part on tha accurate sening of vehicle perturbation from a reference path sand asubsequent phase oompeasation for from a reference path sn4d subsequent phase compensation for this amoUo, thi mouon. <owve< (over~ Armed Servlces Armed Servicca Technical Information Agency Technical Information Apmncy UNCLAS&IFIZE UNCLASSIIED + + +

+4 + + AD Div. 19/4 UNCLASSIFIID AD Dlv. 19/4 UNCLASSIFIED Inst. of Science and Technology. Univ. of Mich.. Ann Arbor.L Title: Project MICHIGAN naLt of Science and Technology, Univ. of Mich.. Aqn Arbor I. Title: Projelt MICHIGA OPTIMUM CHARACTERISTICS OF SOME MOTION-COM- I Porter, W. A., Bilal, AY.. OPTIUM CHARACTERISTICS OF SOME MOTION-COM- L. Porter, W. A., Bilal. A. Y. PELNATION SYSTEMS. Volume L Theory and Application of i. U. S. Army Electronics PENSATION SYSTEMS. Volume: Theory and Application of IL U. S. Army Electronics Dopplr-Inertial Systems, by W. A. Porter, A. Y. Bial. Mar. Command Doppler-Inertial Systems, by W. A. Porter, A. Y. Bl1al. Mar. Commandu 63. 3 p.. Icl. Ulus., table. 5 refs. V. Contract DA-36-039 SC-78601 63. 39 p.. inc. illus.. table. S refs. IV. Contract DA-3G-039 SC-78i 1 (Report No. 2900-371-T(I)) V Project No. 3DS6 I0010 (Report No. 2900-371-T'()) V. Project No. 3D5801001 (Contract DA-36-039 SC-78801) Unclassified report (Contract DA-36-039 SC-76?80 Unclasified repon (Project 3DS301001) (Project 3D5S01001) A side-looking radar sytem employing syntheUc-antenna tech- A idc-looking rndar system employing ythetic-aUnerarn teckniques can obtain azimuth resolution finer than that afforded niques can obtain azimuth resolution finer than hat afforded by the radar beamwidth. The capabillles of uch. systems by the ra4ar beamwidth. The capabllittes of such systems depend in part on the accurate sensing of ehicle perturbatio depend In part on the accurate nlagt of vhfcle perturbation from a reference path and subsequert phase compenation for from a reference path and subequeat phase compenation for this motion. this moUton. (over) (otver Armed Services Armed S.ervces achical Information Agency Technical Informatlon Agency. - CLASSIFIED UNCLASSIFIED.+ + + AD Dlv. 19i,4 U. CLASSIFIED AD Div. 19/4 CN CLASSIFIED Ibut. of Science and Technology, Univ. of Mich., Ann Arbor I Title: Project MICHIGAN Inst. of Science and Technuol gy. Univ. of Mich.. Ann Arbor L Title: Project MICHIGAN OPTISL'M CHARACTERISti'CS OF SOME MOTlON-COM-. I Porter, W. A., Bilal, A. Y. OPTIMUM CHARACTERISTICS OF SOME ):OTIOr-COM- I. Porter. W. A., Bilal. A. Y. PEN.SATION SYSTEMS, Volume L Theory and Application of II. U. S. Army Electronic PENSATION SYSTEMSS, Volume:.Theory and Application of T11. U. S Army Electronics Dappler-lnertal Systems, by W. A Porter, A, Y. YBll. Ma. Cornmmand Doppler-Inertial Systems, by W. A. Porter. A. Y. Ilial. Mar. Com.mand 3..3 p.. incl. illus.. tbe, 5 refs. I. Crtsct DA-3-039 SC-7801 63. 39p.. Inc. ttllw.. ta-e. S rf..IV. Contract DA-3G-039 SC-76801 (Report No. 2900-371-T(I) V.;'roect No. 3D$801001 (Report No. 2900-371-T()) V. Project No. 3D580101'Ceotract DA-36-039 SC-78801) Unctasifid report (Cotract DA-X3-039 SC-7801) n Uaclaified report (Project 3D5801001) (Project 3D301001) A side-looking radar system employing sythet-antenna t ch- A side-looking radar system employing synthetic-artenna techniques can obtain asimuth resolution finer than tht afforded. -que can obtain aaimuth reolution finer than that afforded by the radar beamwidth. The capabilities of such system by the radar beamwidth. Th. capabilities of such systems d4ped in part oa the acsorate sening of vehicle perturbatio depend a part on te accurate sensing of vehicle perturbatUo from a fenwe pathB aal soeqeut phat oonluatieo br fo. reference path ind wub phase compensaton for lau moUtior. thl motion. - (ovor ~ (over) Armed ervice Armed Services Tecltical Informstion Agency Technical Ifornation A^cencv -UNCLAS3SFIE:D U. UN'CLAilFIl) ++ +

AD UNCLASSIFIED AD I UNCLASSIFIED Thla volume deals with the optimization (for such motion- DEISCRIPTORIS This volume deals with the optimization (for such motion- DESCRIPTORS conipensation activities) of a family of Joppler-lncrtlal sys- Ac eration compensation activities) of a family of doppler-inertial sys- Acceleration tcms. By use of the variational calculus techniques of the 1 vii tcums. By use of the variational calculus techniques of theDp l navigatlon Wiener filter theory, functional relationships between optimum rWiener flt.r theory, functional relationships between optimum,.,,, i ~ ofgtcr aystems.,,oppler ays terns ^ ^K system prrformance and sensor noise errors are developed, system performance and sensor noise errors are developed,. yse Inortial navigation Inertial navigation and systems with anc without dynamically Induced errors are l navaonand systems with and without dynamically induced errors arene navigaion Velocity Velocity compared. compared. UNCLASSIFIED I UNCLASSIFIED PAD UNCLASSIFIED AD UNCLASSIFIED This volume deals with the optimization (for such motion- DESCRIPTORS This volume deals with the optimiza -r such motion- DESCRIPTORS compensation activities) of a family of doppler-inertial sys- eati compensation activities) of a family.iler-inertial sys-A.Acceleration Acceleration tems. By use of the variational calculus techniques of the tems. By use of the variational ca' chnlques of theer aton - Dopcypler sysigtem n sno rrr Doppler- sysiitems Wiener filter theory, functional relationships between optimum Doppler t Wiener filter theory, functional relationai..ps between optimum Doppler navigation. " -..Doppler systems.r rDo'^ole" 6Svten*8 system performance and sensor noise errors are developed. system performance and sensor nolse errors are developed, sInertial navigation Inertial navigation and systims with and without dynamically induced errors are Velocity and systems with and without dynamically induced errors are Velocity compared. compared,. UNCLASSIFIED UNCLASSIFIED

UNCLASSI FIED UNCLASSIFIED