a lunar landing guidance system for soft-precision landings-eji

8/3/2019 A Lunar Landing Guidance System for Soft-Precision Landings-Eji

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1964 IEEE TRANSACTIONS ON AEROSPACE AND NAVIGATIONAL ELECTRONICSIn various instances of CW interference, AFSK recep-tion was impossible, QF\AI reception was usually pos-sible with low or moderate error rates. Similarly, AFSKtransmissions were often not received at all during trans-missions through the auroral zone, while preceding orfollowing QF1\1 transmissions were received with ac-ceptable error rates.

X. CONCLUSIONSThe test results indicate that a significant protectionagainst the signal distortion effects of multipath propa-gation can be obtained by a system, such as the QFI\Isystem described, with moderate frequency and timediversity. The instrumentation of the system provideda high degree of immunity against selective interfer-

ence. Both the ability to operate at frequencies far be-low the MIUF under conditions of strong multipathpropagation and the insensitivity to CW interferencemake the described QF1\I technique a valuable to ol forestablishing reliable automated world-wide long-dis-tance air traffic control.ACKNOWLEDGMENTSignificant contributions to the system design weremade by J. Hannum and J. Kerns; to the implementa-tion by D. Thomas, E. Painter and others; and to thetest and evaluation by R. Beams, A. Culbertson andW. Perkins. The cooperation of all who participateddirectly or indirectly in the program is greatly appreci-ated.

A Lunar Landing Guidance System forSoft-Precision LandingsJ. E. VAETH AND M. D. SARLES, MEMBER, IEEE

Summary-A logical development of a novel, minimum-complex-ity guidance system for precise and soft lunar landing is presentedtogether with an evaluation of predominant error sensitivities. Selec-tion of this minimum-complexity system is influenced by its abilityto handle a wide range of initial condition, sensor, propulsion andcontrol system errors with minimum fuel and accuracy penalty.The trajectory control technique allows excellent compromise be-tween sensor requirements (e.g., Doppler and beacon tracking radargimbaling), control system complexity and total fuel usage. Proof ofperformance is given in terms of analog and digital computer simula-tion results plus theoretical correlation. Results include specific appli-cation to the proposed Saturn V Lunar Logistic Vehicle. Design guidesare evolved fo r system synthesis and fo r potential application to futuremissions. Demonstrated performance includes terminal maneuversof 5 km with touchdown displacement errors of less than 200 metersand velocities of less than 4 meters/sec vertically and 1 meter/sechorizontally.

Manuscript received August 26, 1964. This paper was presentedat the 1964 International Convention on Space Electronics and Telem-etry, Huntsville, Ala. Part of the work reported was done undercontract to the Marshall Space Flight Center, Contract No. 85410,NASA.J. E. Vaeth is with the Baltimore Division, Martin Company,Baltimore, Md.M. D. Sarles is with the McDonnell Aircraft Corporation, St.Louis, Mo. He was formerly with the Baltimore Division, MartinCompany.

INTRODUCTIONG UIDANCE techniques and equipments em-ployed to accomplish an accurate and soft lunarlanding must be compatible with lunar approachtrajectories, propulsion system capabilities and accu-racy requirements. An excellent discussion of the prob-lem scope and potential compromise solutions is pre-sented by Digesu.1Lunar landing guidance schemes discussed in the openliterature2'3 emphasize the use of rather sophisticated

steering computations, with resultant fuel and/or accu-racy optimization. Usually not considered, however, arethe required sensor fields of view and/or gimbaling(e.g., Doppler and beacon tracking radar), the effects ofsensor measurement errors and smoothing lags, auto-1 F. E. Digesu, "A Discussion of the Lunar Landing Problem,"presented at the AIAA Guidance and Control Conf., M.I.T., Cam-bridge, Mass.; August 12-14, 1963.2 E. Markson, J. Bryant, and F. Bergsten, "Simulation of mannedlunar landing," Preprint 2482-62, ARS Lunar Missions Meeting,Cleveland, Ohio; July 17-19, 1962.3 B. A. Kriegsman and M. H. Reiss, "Terminal guidance an dcontrol techniques for soft lunar landing," ARS J., vol. 32 , pp. 401-413; March, 1962.

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292 IEEE TRANSACTIONS ON AEROSPACE AND NAVIGATIONAL ELECTRONICS Decemberpilot time lags, thrust uncertainties and initial trajec-tory dispersions. For example, fuel optimum programsare characterized by a shallow approach trajectory, withundesirably high attitude (and sensor gimbal) terminalrates needed to accomplish a soft, vertical landing.

STUDY OBJECTIVES AND APPROACHThe analytic study had these objectives, 1) determinelunar landing performance c rit eri a f or major subsys-tems, based on error sensitivities as determined by com-puter simulations and 2) evolve a minimum-complex-ity, implicit guidance scheme. This evolution processwas based on evaluating, by way of comparison with amore sophisticated explicit technique, relative per-formance degradation (if any) with regard to each of thefollowing factors:1) sensor requirements and tolerances2) flight control system complexity and response rate3) trajectory constraints and initial condition toler-ances4) required thrust variability and tolerances5) fuel utilization6) terminal maneuverability7) accuracy of end condition state variables.A similar approach to the solution of the earth re-entry guidance problem proved very effective.4 How-ever, with no lunar atmosphere to provide braking, theneed for simultaneous control of both thrust magnitudeand direction plus the soft landing requirements intro-duce unique and more complex problems.The study approach has emphasized the use of bothdigital and analog computer programs to simulate com-ponent errors effectively, control nonlinearities, enginethrottling, fuel usage and the low-frequency responsecharacteristics of the guidance systems.Two vehicle configurations having radically differentnominal trajectory profiles were evaluated. Perform-ance results and sensitivity trends were quite similar,thus demonstrating the flexibility and mission applica-bility of the implicit guidance scheme. This paper willconcentrate on the proposed Saturn V Lunar LogisticVehicle, fo r which more extensive computer simulationswere conducted.5

DIGITAL COMPUTER STUDIESThe digital studies covered the flight phases fromlunar deorbit until the terminal letdown phase, whichbegins when velocity is reduced to about 25 meters/sec.Initial computer runs determined fuel weight tradeoffsvs trajectory constraints, and a nominal trajectory wasaccordingly selected. Guidance simulations were thenconducted for comparative evaluations of an explicit and4J. E. Vaeth, "Re-entry guidance and flight path control," IRETRANS. ON SPACE ELECTRONICS AND TELEMETRY, vol. SET-6, pp.99-103; September-December, 1960.6"Studies of Guidance Systems fo r Future Space Vehicles,"Martin Co., Baltimore, Md., Final Rept. No. ER 13193, ContractNo. 85410; November, 1963.

an implicit scheme, with emphasis on the latter. MIajorobjectives were to determine performance degradations(fuel and trajectory deviations from nominal) due tovarious forcing functions, thus providing initial condi-tion extremes fo r the analog simulation of terminalmaneuverability and landing.Fuel vs Trajectory Constraints

Tradeoff studies of total fuel usage vs transfer ellipsecentral angle (from lunar deorbit until start of mainbraking) and initial masking angle (Et, defined as theelevation angle from the horizontal plane through thelanding site to the spacecraft at start of main braking)were generated by using a fuel-nearly-optimum, con-stant thrust, digital guidance program fo r the brakingphase of flight. Pertinent results, as summarized inFig. 1, reflect the significance of the specified perform-ance criterion that Ei be 10 0 minimum, which facilitateslock-on to a landing site beacon prior to start of mainbraking. Fig. 1 includes deorbit fuel (from a 185-kmcircular orbit) and indicates that for the stipulated E;of 10 minimum, a transfer ellipse central coast angle(4k) of 600 is nearly optimum. However, total fuel varia-tions fo r central coast angles between 400 and 1000 wereless than 1.0 per cent. Fig. 1 also shows that fuel varia-tions with initial masking angle are much more signif-icant (than with O3). For the case of continuous burn-ing from the 185-km circular orbit, fuel usage would beeven greater than fo r the case with Es= 15. A logicalinference is that required fuel is primarily dependentupon the altitude of braking initiation.In accordance with Fig. 1 results and the specified EBminimum, the nominal trajectory selected fo r detailedguidance evaluation was based on a central coast angleof 600 and an initial masking angle of 10.950.

27 , 000K

26, 500-

26, 000K

FUEL WEIGHT(EARTH-L B)25, 500H25, 000H - . (E 10. 950)24,500K24, 000K23, 500_

L1 II 0 0 05 10 15E.- ELEVATION ANGLE FROM BEACON ATBRAKING INITIATION (DEG)i .1 1 SI40 60 80 100

= LUNAR CENTRAL ANGLE, DEORBIT TOSTART OF BRAKING IDEG)Fig. 1 Total fuel consumed from lunar orbit.



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Vaeth and Sarles: Lunar Landing GuidanceSimulation of Implicit SchemeDigital computer evaluation of this technique wasaccomplished by incorporating guidance loops into anexisting trajectory program. As noted in Fig. 2, theguidance loops function to control thrust magnitude anddirection in accordance with sensed trajectory errors,and are sufficiently flexible to simulate various combi-nations of navigation sensors, associated measurementerrors, propulsion characteristics, control limitationsand initial conditions. With reference to the geometryshown in Fig. 3, the necessary navigation intelligenceconsists essentially of R, R and 'YLOs. These can bedetermined from either inertial guidance computationsor more directly (and accurately) from a beacon track-ing radar system.As shown functionally in Fig. 2, the thrust magnitudeand thrust direction control laws are simply

F, = Fref + KT(Rm - Rnom) (1)ac = aref + Ky(YLOS,om - YLO Sm ) (2)where Rnom, YLOSnom, and aref are preprogrammed nom-

inal trajectory curves which are read out as a functionof Rm. These preprogrammed curves are shown in Fig. 4for the proposed Saturn V Lunar Logistic Vehicle, andare based on a constant Fref of 24,000 pounds. The areffunction was evolved by a series of unguided trajectoryruns which resulted in a nominal trajectory whose totalfuel usage was within 0. 5 per cent of the nearly optimumdefined in Fig. 1 (for = 60 and E=t10.95). It isnoteworthy that the curve of 'YLos.Om vs R in Fig. 4 isflat throughout most of the braking phase, and neednot be programmed very accurately (note the Fig. 4approximation actually used). For a smaller lunar logis-tic vehicle configuration, having a much higher thrust-to-weight ratio and lower initial masking angle, theYLOSnom curve (as used very successfully) was a constant10.60 during most of the braking phase of flight.This scheme demonstrated no low-frequency stabilityproblems, which correlates with theoretical predictionsof overdamped response and which was verified byanalog simulation. The adequacy of the a/ac transferfunction approximation was also substantiated theo-retically and by analog results.

-r------_y

h

+ E

S ITEFig. 2-Functional block diagram, digital computer simulation. Fig. 3-Guidance geometry.

1600140012001000

R nom 800-(misec) 60 040 020 00

i- LOSnom

(deg)

R (km)Fig. 4-Nominal control functions.

-101aref -5-(deg)

+5

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294 IEEE TRANSACTIONS ON AEROSPACE AND NAVIGATIONAL ELECTRONICS DecemberSimulation of Explicit Scheme

This technique achieves an adaptive flight-path con-trol by repeated prediction of end conditions from aser ies o f equations derived from the equations of mo-tion of a particle subjected to a central force field andstored in the airborne guidance computer.2 Completeknowledge of the position-velocity state vector is re-quired from the navigation system.A digital computer program for evaluating thisscheme was also developed by incorporating guidanceequations for thrust magnitude and thrust direction con-trol into an existing trajectory program. It was pro-grammed to simulate component errors and time lagsin much the same manner as the implicit program.Presentation of the explicit guidance equations is be-yond the scope of this paper. However, it is noteworthythat simulation results subsequently discussed reflectvery recent (after Ref. 5 publication) improvements tothe explicit computer program. A potential fuel savingof one per cent, compared with the implicit scheme,was demonstrated.Digital Simulation ResultsThe implicit and explicit schemes were evaluated interms of trajectory and fuel deviations from nominal,due to various forcing functions, at a range-to-go of 20km.An altitude error at initiation of the main brakingphase proved more critical than an y other initial con-dition error. Resultant fuel variations are illustrated inFig. 5 fo r the implicit scheme. Results were essentiallythe same with explicit guidance, at an equivalent veloc-

ity of 389 meters/sec. It would therefore appear thatfuel reserve provisions are more dependent upon initialaltitude tolerances than upon the sophistication of theguidance computations.The most critical component problem resulted froma 'YLOs measurement error. When using the beacon track-ing radar,'Y LOS, - EeR/R (3)

where Ee is a line-of-sight rate measurement error (forexample, as measured by a rate gyro mounted on thegimbaled tracking radar). For an Ee of 0. 5 X 10radians/sec, YLOSes6.3' at braking initiation but woulddecrease as the R/Rt ratio decreases. This varying YLOS,function was incorporated as a disturbance input intothe implicit guidance program with the following perti-nent results.

1) For Ee= -0.5 X10- radians/sec, a more shallowtrajectory results. This did not affect landing soft-ness, but slightly degraded terminal maneuvercapability. Incremental fuel consumption wasnegligible.2) For e 0.5X10-3 radians/sec, a more archedtrajectory resulted. This loft caused a sizable fuelpenalty (about 4 per cent), but did not degrade

+800-+600 -+400-

FUEL WEIGHT +200DEV IATI ONFROM NOMINAL(EARTH-LB) 0

-200 F-400

-15 -10 -5 0 +5 +10INITIAL ALTITUDE ERROR (km) 15Fig. 5 Fuel deviations vs braking initiation altitude.terminal maneuver and soft landing capability.

An Ee measurement error also proved to be the mostserious for explicit guidance. Trajectory displacementvariations from nominal were less than fo r the implicitscheme, but fuel weight and velocity penalties weregreater, particularly fo r the -Ee case.Other off-nominal errors proved less serious fo r bothguidance schemes. These error sources included thefollowing.1) Gradient errors in measuring R and R of +5 percent.2) Time lags of up to three seconds, both in attituderesponse and in measuring k.3) Initial condition deviations of + 50 meters/secin velocity magnitude, 0.86' in velocity direc-tion and +9 per cent in R,.4) Thrust uncertainties of + 5 per cent.

Typical end conditions at 20-km range-to-go, fo r themore significant error sources, are presented in Table Ifor the implicit scheme. Note that in all cases the valueof 'YLOS (E+yh in Table I) is within 3.4 of nominal.Also, except for the 5 per cent gradient errors in meas-uring R and R, each R value of 20-km range-to-go iswithin 8 meters/sec of nominal. These Rt and 'YLOSparameters, which define the velocity vector with re-spect to the instantaneous line-of-sight to the landingpoint, proved to be the controlling factors in terms ofterminal maneuverability, control system complexityand landing softness.It is noteworthy that all runs in Table I used con-stant KT and K7 gains. Although not considered neces-sary, inmproved performance can be realized by program-ming these gains vs Rm.ANALOG COMPUJTER STUDIES

The analog studies were restricted to implicit guid-ance, and covered the flight phases from approximately400 meters/sec until engine shutdown just prior to soft

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Vaeth and Sarles: Lunar Landing GuidanceTABLE I

DIGITAL SIMULATION RESULTS AT 20-KM RANGE-TO-GO, IMPLICIT GUIDANCE SCHEMEFuel Used(Earth-lb)18,47519,08718,08118,60318,42218,38818,61519,01517,98119,23818,58618,49818,63618,33618,54618,469

Error SourceNoneAHj = 15 kmAHi = -15 kmA 0.0 960Aoi =- 0.960,Ay = - 0.860Ayi = 0.860zAVi=50 m/secA/v 50 mr/secE 0.5 X 10-3 rad/secE, -0.5 X 10-3 rad/secTa and TT= 3 secThrust Error = 5 per centThrust Error= -5 per centR.m gradient error=-5 per centRm gradient error= -5 per cent

landing. Major objectives were to evaluate the signifi-cance of control system parameters an d nonlinearities,and to demonstrate system capabilities in terms of sta-bility, maneuverability, accuracy and landing softness.M1aneuver initiation was assumed to be based on a com-puted offset command, such as from TV sensor data.A functional block diagram of the analog computermechanization is shown in Fig. 6. An x-y coordinateframe was used for guidance loop mechanization (referto Figs. 2 and 3) in order to facilitate x-direction maneu-vers. Detailed descriptions of computer mechanizationand results5 are beyond the scope of this paper, but cer-tain features warrant discussion.Attitude Control Loop

Initial analog runs concentrated on vehicle attitudeand engine gimbal dynamics. Pertinent results are asfollows.1) Attitude loop stability required proper selectionof K5 an d rL (see Fig. 6) consistent with enginegimbal rate limit (6L). Increasing 6L allowed K5 toincrease and TL to decrease, thus improving atti-tude loop response rate.2) Large initial condition errors in 0. and 0 required

a further reduction of K5 because of the enginegimbal angle limit (6L). This pointed out the de-sirability of utilizing a maneuver phase guidancescheme which is compatible with and/or whichminimizes (or limits) initial condition errors andOc magnitudes.3) Thrust misalignment and cg errors resulted pri-marily in small steady-state and 0 errors. Nodynamic degradation due to fuel change resultingfrom a motions (C 1 in Fig. 6) was apparent.4) Optimum response rate of the attitude loop canbe realized by varying K5 inversely as FIIJ. Vary-ing K5 inversely with F, (limited) proved to bequite adequate.

Fig. 6-Functional block diagram, analog computer simulation.Guidance LawsThe laws defined by (1) and (2) proved very ade-quate throughout the maneuver flight phase. The KTand K, gains were again maintained constant, alongwith Fref = 24,000 pounds constant and ar,f =O. Anoffset maneuver command (Axmaii) was introducedsimply by adding it into the computations for Rm, Rm

and 'YLOSmEqs. (1) and (2) proved inadequate during the ter-minal letdown phase, particularly after large maneu-vers. This was remedied simply by modifying (1)and (2), when Am


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296 IEEE TRANSACTIONS ON AEROSPACE AND NAVIGATIONAL ELECTRONICS DecemberManeuver Capability

Typical maneuver traces are shown in Fig. 7, andterminal conditions at engine cutoff (as printed out bya digital voltmeter) are presented in Table II. Theseresults demonstrate a maneuver envelope of +5 and-3 km, from an initial altitude of 16 km. Note theexcellent terminal conditions in Table II fo r all runsexcept the -4 km maneuver (which was not compatiblewith initial conditions). Also note that maneuver fuelpenalty never exceeded 1.3 per cent of total fuel usedfrom deorbit. Pertinent parameters for these runs in-cluded a 10 per cent larger than nominal value of initial|z|; 8L=8/sec; O limits of -450 and +150; 2. 0 percent errors in measuring x, x, z and z and 2.0 per centthrust error.From an initial altitude of 13.4 km, the maneuvercapability reduced to +5 km and -2 km. Also signif-icant was the maneuverability with the worst initial

6 4 2 0 -2 -4 -6 -8 -1 0

x~~~~~~~~~~1 z{km!

Fig. 7-Terminal maneuver trajectories, implicit guidance.TABLE II

ANALOG FINAL CONDITIONSRun No. ((m/sec) x(m) x(m/sec) ((deg) (deg/sec) Fuel Used(Earth-lb) AXman(km)

251 -3.35 -16 +0.5 -0.81 +0.015 6236.5 0252 -3.35 +1514 +0.55 -1.09 +0.045 6262.0 -1.5253 -3.35 -1506 +0.45 -0.55 -0.01 6218.0 -1.5254 -3.35 -2920 +0.50 -0.91 +0.02 6269.0 -3.0255 -3.35 +3002 +0.55 -1.145 +0.05 6336.0 +3.0256 -3.3 +3950 +0.45 -0.67 0 6476.0 +4.0257 -88.5 -3910 -63.65 -37.705 -3.705 4988.0 -4.0258 -3.3 +4946 +0.45 -0.605 0 6559.0 +5.0

condition deviations noted in Table I (Runs 18 and 19fo r Ee = 0. 5 X1 0- 3 radians/sec). Total maneuvercapability was reduced from 7 to 4 km for the shallowapproach case, but increased to 9 km fo r the more loftedtrajectory ( Ee positive).Stability CriteriaAn approximate but useful stability criterion fo r thethrust direction guidance loop can be inferred from Ap-pendix I.

JaX1rLR >> (1 + K,). (4)mlThis criterion points out an obvious limitation whencontrolling vehicle flight path by thrust vector control,in that the initial thrust deflection must produce ashort-duration thrust component which first changesthe flight path in the wrong direction. The large magni-tude of the J/1 ratio fo r the vehicle configuration simu-lated, in conjunction with the desire to minimize ao-rLbecause of and limits, restricts guidance loop re-sponse rate. However, this response rate restrictionproved to be of secondary importance when comparedto the effects of the necessary 0, limits when Axman islarge. For example, decreasing the 0, limit from +45 to+15 fo r the downrange maneuver showed a more sig-

nificant improvement than increasing 5L from 8/secto 30/sec.The (4) criterion also explains the need fo r switchingthe guidance law from (2) to (2a) when R approacheszero. This criterion wa s verified by numerous runs.For example, note in Fig. 8 (which is a transient re-sponse plot of Run 255 in Fig. 7 and on Table II) the0. 5 radian/sec 0 oscillation that had started just before(2) was switched to (2a) at Rm,20 metersjsec, afterwhich the 0 oscillation damps out. The expected tend-ency toward instability when R


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1964 ~~~~Vaeth and Sarles: Lunar Landing Guidance27

x 400 M/LlINE j2 E

~25 M/SECILINE 2.5 MISEC/LINE 0.25 MISEC/LINE00.50 /LINE 8L8 SEC

w 250 LBILINE

Fig. 8-Analog transient response, 3-km maneuver.CONCLUSIONS AND MISSION APPLICATIONS

Based on comparative performance capabilities andcomponent design criteria, as determined by computerstudies, implicit guidance appears quite adequate forimplementing the lunar landing guidance functionthroughout the braking and maneuver flight phases.E-xcellent accuracy, maneuverability and initial condi-tion flexibility were demonstrated by both digital andanalog simulations of a relatively simple implicitscheme. A wide range of simulated initial condition,sensor, propulsion and control system errors werehandled with very small accuracy and fuel penalties,with modest requirements for thrust and angle of at-tack variations, without the need fo r gain programming,and with ideal trajectory parameters fo r accomplishinga terminal maneuver and soft landing. There was noneed fo r mode switching (changing of control param-eters, reference curves, gain settings, etc.) at transitionfrom the braking to the maneuver phase of flight. Thisallows maneuver initiation over a wide range of descentaltitudes, or not at all should the TV sensor malfunc-tion or indicate no need for maneuvering.The implicit guidance scheme, fo r which design guideshave been evolved and substantiated, is applicable to awide range of vehicle thrust-to-weight ratios, terminalapproach trajectories and to manual control using suita-bl e displays. Moreover, it offers excellent potential fo rfuture Mars landing missions, particularly in view ofthe suspected lack of sufficient atmosphere fo r aero-dynamic braking. The implicit scheme is ideally suitedfor maintaining adherence to a programmed nominaltrajectory despite large atmospheric density uncer-tainties.

analog computer simulation (refer to Figs. 2, 3 and 6) .Note that 0 in Fig. 9 is the component of vehicle ac-celeration normal to the instantaneous line-of-sight tothe landing site.The system characteristic equation (denominator ofthe 'YLOSm,YLO5nom, closed-loop transfer function) wasdetermined as follows.1) The G/O, linear transfer function was evolved fromFig. 6 by assuming that 8/8, = 1 (cw, in Fig. 6 large)and that C1-=A,O=O0. Thus

0 1 TrLS (5 )Go JS2 JTLS'I1calTLS + -FK5Fl K5FZ

2) The line-of-sight feedback loop E in Fig. 9 wasassumed negligible. This was substantiated by de-riving the characteristic equation to include the Efeedback, and comparing the resulting roots.3) The transfer function fo r 6/0 in Fig. 9 is simply'6 JS20 Fl (6 )

The characteristic equation (without E feedback) istherefore

JS5 - I-S4 (T,ry L)K5FZ+SF T+K5FZ RiM ]

APPENDIX ISTABILITY CONSIDERATIONS, THRUSTDIRECTION GuIDANCE Loop

Fig. 9 is a simplified functional representation of thethrust direction guidance scheme as mechanized in the

+ S2 [alrL + 1-7y - J(1K7)+ s[I + _(1 + Ky)] F(+K7 0. (7)

II....... ...........I................ I................... I.....................u t

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0

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IEEE TRANSACTIONS ON AEROSPACE AND NAVIGATIONAL ELECTRONICS

Fig. 9 Functional block diagram, thrust direction guidance loop.

= 30 meters/sec. Case 2a includes the E feedback terms;note the negligible change (from Case 2) in the low-frequency roots.APPENDIX I I

STABILITY CONSIDERATIONS, THRUSTMAGNITUDE GUIDANCE LooPFig. 10 is a functional representation of the thrustmagnitude loop used in the implicit guidance simulation( se e F ig . 2).The open-loop transfer function can be readily ob-tained by breaking the loop at FC to give

KTG 1 +-open loop = S2M(1 TTS) (8)

TABLE I IIROOTS OF CHARACTERISTIc EQUATION, IMPLICIT GUIDANCE SCHEME*

TL (sec) Ky0.22 4.00.22 4.00.22 4.00.11 4.00.155 2.00.22 4.0

R(m/sec)91.53030303030

R(m)1220305305305305305

r.' (sec)1.01.00*1 .01.01.0

Roots-1.3; -1.97 +0.626j; -0.150 +0.382j-1.35; -2.14 +0.83j; +0.048 0.626j-1.48; -2.57; -0.088+0.86j-1.42; -4.31.6j; -0.031+0.87j-1.37; -2.92 +0.95j; -0.11 0.60j-1.35; -2.14 +0.83]; +0.045 0.639j

* With r.Y=O, one root concels with numerator root of closed-loop transfer function. Assumed parameters, M1= 1.3X101 slugs; J= 1 X 105slug-feet2; I=8.0 feet; F=24,000 pounds; a,=lO.THROTTLE SERVO

FREF = 24, 000 lB

R I ..

Rj (P O SIT IVE) g SIN (E)Fig. 10-Functional block diagram, thrust magnitude guidance loop.

Note that the S2 and SI terms in (7) become negativeas A approaches zero, which signifies unstable operation.This was verified by correlation of analog computerresults with the (7) roots (as extracted by digital com-putation). These roots are presented in Table III fo rflight conditions wherein R is small. Case 1 in TableII I i nd ic at es s ta bl e operation at R =91.5 meters/sec.However, Case 2 denotes a low-frequency divergencewith A decreased to 30 meters/sec. Case 5 (with -r = 0)and Cases 10 and 16 (with faster attitude loop response)indicate stable but poorly damped response at A

where G is the slope of the nominal R vs R curve at themeasured value of R, and TT represents the time la g inthe throttle servo. From the standpoint of stability,it is significant that -r T could represent the combinedthrottle servo time la g plus the radar time lag in smooth-ing the measured value of R. Excessively large valuesof TT would require the use of compensation networks.Eq. (8) indicates that adequate stability margins canbe provided by satisfying the following conditions:1 KTTT M

0.5G _ rTT2KTG

a lunar landing guidance system for soft-precision landings-eji

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