Apollo Experience Report Oboard Navigational and Alignment Software

8/8/2019 Apollo Experience Report Oboard Navigational and Alignment Software

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N A S A T ECHN I C A L

APOLLOEXPERIENCE REPORT -ONBOARDNAVIGATIONALAND ALIGNMENT SOFTWAREby Robert T. Suuely, Bedford F. CockreU,and SumuelPinesMannedSpucecrapCenterHouston,exus 77058

NAT I O NALERO NAUT I CSN DP A C EDM I NI ST RAT I O N . W ASHI NG T O N, D . C. MARCH 197


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TECH LIBRARY KAFB, NM

03336681. Report No. 3. Recipient's Catalog No.. Government AccessionNo.

NASA TN D-67414. Title and Subtitle

6. Performing Organization CodeNBOARD NAVIGATIONAL AND ALIGNMENT SOFTWARJIMarch 1972POLLOEXPERIENCEREPORT 5. R e w R Dare

7. Author(s1Robert T. Savely and Bedford F. Cockrell, MSC 8. Performing Organization Report No.Samuel Pines, Analytical Mechanics Associates MSC S-317~ 10. Work Unit No.

9. Performing Organization Name and Address 924-22-20-01-72Manned Spacecraft CenterHouston, Texas 77058 11. Contract or Grant No.

13 . Type of Report and Period Covered12. Sponsoring AgencyNameandAddress Technical NoteNational Aeronautics and Space AdministrationWashington, D.C. 20546 14. Sponsoring Agency Code

I15. Supplementary NotesThe MSC Director waived the use of the International System of Units (SI) orthis Apollo Experience Report, because, in his judgment, use of SI Units would impair the usefulness

of the report or result in excessive cost.~~ ~~ ~~~16. Abstract ~~~ ~ ~The onboard navigational and alignment routines used during the nonthrusting phases of an Apollomission are discussed as to their limitations, and alternate approaches that have more desirablecapabilities are presented. This discussion includes a more efficient procedure for solvingKepler's equation, which is used in the calculation of Kepler's problem and Lambert's problem;and a sixth-order predictor (no corrector used) scheme with a Runge-Kutta starter is recommendedfor numerical integration. Also evaluated are the extension of the rendezvous navigation state toinclude angle biases and the u se of a fixed coordinate system. In general, the discussion reflectsthe advancement of the state of the ar t since the design of the Apollo software.

17. Key Words Suggested byAuthor(s1) 18. Distribution StatementOnboard NavigationOnboard Navigation SoftwareNumerical IntegrationRendezvous Navigation

' Coordinate System19. Security Classif. (of this report) 22. Rice'1. No. of Pages0. Security Classif. (of this page)None 1 4one $3 00

For sale by the National Technical InformationService, Springfield, Virginia 22151


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APO LLO E XPER l ENCE REPORTO N B O A R D N A V I G A T I O N A L A N D A L I G N M E N T S OF TW A REB yRober t T. Save ly ,Bed fo r d F. C o c k r e l l , a n d S a m u e lP i n e s *

M a n n e d S p a c e c r a ft C e n t e rS U M M A R Y

The cur ren t limita tion s of the onboard navigational and alignment software arediscussed, along with more desir able capabilitie s and alternate approach es that areavailable. The onboard navigational and alignment software includes programs forfree-flight prediction, rendezvous navigation , orbital navigation, cislunar navigation,and alignment of the inertial measurement unit and support ro utines (e. g. , lunar andsolar ephemerides and planetary inertial orientation).

I N T R O D U C T I O NThe onboard navigational and alignment routines discussed in this report ar e us edduring the nonthrusting phases of an Apollo mission. The basic objective of the navi-gational system is to mai ntai n esti mate s of the position and velocity of the commandand service module (CSM) and lunar module (LM). Navigation is accomplished byextrapolating the state vector s by mean s of the coasting integration routine and thenupdating this estimate by processing tracking data by means of a recursive navigationalmethod.The CSM guidance and navigation system uses (1)range data from the very-high-frequency (vhf) ranging device and (2 ) angular data from the scanning telescope andfrom the sextant. The LM primary guidance and navigation system uses rendezvousradar (RR) tracking data (angles, range, and range rate). In addition, the LM system

has an alignment telescope for platform alignments.These navigational data are incorporated into the state vector estimates by meansof the measurement-incorporation routine. This routine computes deviations to thestate vectar based on the tracking geometry and the statistics of the state vector history.In this repor t, the limitatio ns of these routin es are discussed, and alternateapproaches with more desirable capabilities are presented. In general, the commentsref lect the advan ceme nt of the st ate of the art since t he des ign of the Apollo software.*Analytical Mechanics Associates.


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An efficient Cartesian coordinate transformation method is described in theappendix written by Paul F. Flanagan of the NASA Manned Spacecr aft Cente r andSamuel Pines of Analytical Mechanics Associates.

D I S C U S S IO N

N a v i g a t i o n a I SystemThe basic objective of the navigational system (maintaining estimates of positioand velocity vectors for he CSM and LM) is accomplished by extrapolating the statevector by means of the coasting integration routine. The procedure used by this routine is to extrapolate the state vec tor by mean s of Encke's method of diffe rentia l accerations, in which only deviations from conic motion are integrated numerically. Tapproach is sound and represents the current state of the art. However, subtle im-provements are now available for determining the conic motion and numerically inte-grating the deviations.Even with accurate state vector extrapolation, initial-condition errors grow tointolerable size. Thus, it is necessary to periodically obtain additional data to modithe state vectors. These modifications are computed from navigational data obtainedfrom sensor measurements. The nature of the measurement sensors requires signifcant crew interface. The introduction of human errors into the navigational systemcan be minimized in future projects by using a more automated sensor system; thisstep not only would improve basic navigational data but also could partially eliminateor reduce the need for premis sion s chedul ing of sightings.When a measurement is made, the best estimate of the state vector is extrapo-lated to the measurement t ime. From this estimate, it is possible to compute anesti mate of the quantity measured. When this computed measurement is comparedwith the actual measured quantity, the difference is used to update the state vector.This difference o r deviation corrects the state as a linear multiplier of a weighting

vector (which in tu rn is a function of th e geometry, the assumed measurement uncertainties, and the state uncertainties). The major problem for this area of navigationhas been error-transition-matrix saturation. This matrix allows reasonable correc-tions in the beginning but rapidly reduceshe allowable correction untila point isreached when no useful corr ections are permitted. This problem has been circumveed by constant reinitialization of this matrix, requiring extensive premission naviga-tional analysis to determine when this event should occur. Nonnominal navigation couldhave reduced the effectiveness of this scheduling.The Apollo guidance computer was designed with a 15-bit word length. This cputer is a fractional machine (all numbers in the computer are less than one); thus, tos tore or usea number with a true v alue gr eater th an one, suitable scaling is necessaA s a resul t , parameters with a large dynamic range are scaled s o as to optimize the

capabil ities of th e computer. Invariably, however, the extremes of these numbers acompromised. This limitation has produced e r r o r s of as much as 240 feet down randuring coasting integration. The down-range er r o r s have the same effect as increasthe e r r o r s on the rendezvous tracking data. In the following paragraphs, the variousrouti nes of the navigational system, their limitations, and recommendations for im-provements are discussed.2


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Coasti ng I ntegration RoutineThe coasting integration routine is the subroutine that- hen given initial time,position, and velocity coordinates - omputes the position and velocity of t he vehicl eat a specified time. This time may be either before or after the time of the particu larinitial state vector. This is the basic routine used in the navigational and guidanceprograms; therefore, the speed and accuracy of th e onboard p rogram ar e limited by

the perfo rmance of this routine.Coasting integration is accomplished by using Encke's method of dif feren tialacce lerations. With this technique, the motion of the vehic le is assumed to be domina-ted by the conic orbit that would result if the spacecraft were in a central force field.Then, only the deviations from conic motion are inte gra ted num eric ally . The natu re of

this method necessitates a discussion of coasting integration in two parts, namely,conic solution and numerical integration of dist urbing accelera tions.Conic method. - The conic method is used in the solution of five conic problems.Three of the conic problems (Kepler, time theta , and time radius) are initial-valueproblems, and two (Lambert and reentry) are boundary-value problems (ref. 1). Thenumerical integration formulation requires considerable computer time. For example,

the return-to-earth maneuver-guidance program (P-37) requires approximately 15 to30 minutes to compute a solution.Because much of th e comput er time is spent solving Kepler's equation, which isused in the calculation of Kepl er's prob lem and Lamber t's probl em, a more efficientprocedure (such as that proposed in ref. 2) would have been useful. To solve thisequation, it is necessary to sum a special trigonometric series. The proposed formu -lation would modify the method used for generating and summing the series. Theseries currently used requires special procedures for integration over 2n (and addi-tional terms for special cases) and is slower to converge than the proposed method.In addition, the procedure described in ref. 2 for solving Lambert's problem is moreefficient. This method avoids slowly converging inverse trigonometric series andwould decreas e the requir ed compu ter time. Other advant ages of the proposals forsolving Kepler's problem and Lambert's problem are an improvement in storage re-qui rements , the accu racy of the computations, the reliability of the convergence,and computer time.Numerical integration method. - The complexity of the numerical evaluation ofthe equations of motion is the largest single item affecting the machine time requiredfor each integration step. The integration method used is a third-order Runge-Kuttascheme developed by Nystrom. This method requires three entries into the lengthyderivative routine for each integration step. For a rapid computing cycle, it is impera-tive to minimize t h e number of entries into the derivative routine.Backward difference schemes have the advantageof requir ing only one entry per

integration step for prediction methods anda maximum of two entries per integrationstep for predi ctor-co rrecto r methods. Another advant age of backward differenceschemes is t h a t they enable the use of a much larger integration step than the third-order Runge- Kuth scheme now in use. The limiting considerations of the step sizea r e (1) h e truncation error in the Taylor series used to evaluate the derivative and

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(2 ) the machine digit word length, which controls the round -off error. The best stepsize is provided when the magnitude of the tr uncat ion err or is the same as that of theround-off error.Because the magnitude of the truncation erro r decreases with the number ofterms, the limitation on the step size is the correlation of the number of t erm s andthe round-off er ro r. The maj or sour ce of round-off er r o r is the bias in th e round-of

digit of the integration coefficients. These coefficients can be kept as exact integersthe number of terms in the backward table s chosen as shown in the following table.

Computing word length,digits56

Backward differencepredic tor {no corre ctor used) ,number of terms67

7 789 89

10 1 011 11

For the numerica l integr ation by the onboard computer with eight-digit arithmetic, a sixth-order predictor (no correct or used) scheme (ref. 3 ) is recommended.The reference 3 predictor is self-started, but for multiple starts during a mission, aRunge-Kutta starter is recommended. Although an eighth-order scheme could be ac-commodated while still maintaining the integration coefficients as exact integers (seepreceding table), the computer space limitations led to the sixth-order recommenda-tion. Thus, a Runge-Kutta starter would be required only for build ing the table. A l-though the backward difference scheme will prove efficient over a relatively longcomputing arc, the scheme cannot be recommended for use in a Kalman sequentialpoint-by-point update mode with observations.For the navigation update mode, it is recommended that a less complex methodbe considered to predict the state to the next observation with an extra error sourceadded to the W-matrix to account for unmodeled errors in the dynamic model.The err or tran sitio n m atri x. The position and velocity vectors maintained bythe computer with numerical integration are only estimates of the true values . A s pof the navigational technique, it is also necessary to maintain statistical data in the

computer to aid in processing navigational measurements. To accomplish this, a corelation matrix is defined that represents probable uncertainties in the state and thecorrelation of these uncertainties. For convenience, a matrix cal led the error t ransition matrix {or square root of the correlatio n matrix ) is defined, and this matrix mustbe maintained by the computer.

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Extrapolation of the error transition matrix in theApollo guidance computer ismade by dir ect numeric al in tegratio n of as many as nine vector differential equations.This method is costly because most of the time spen t in this matrix propagation s inthe derivative routine (that portion of the logic that numerically evaluates the integra-tion equations (equations 2.2.35 of ref. l ) ) .Reference 2 contains an alternate method for state error propagation. This ana-

lytic method, sometimes called the mean conic method, estimates errors at futuretimes by introducing predicted errors at the initial time and propagating these errorsalong the average conic. This average or mean conic is defined by the initial and finalstate estimates and by the transfer time. The method involves no numer ical integr a-tion or iteration and r equire s the defin ition of only four indexed analytic partials.

R e n d e z v o u s N a v i g a t i o n (P-20)Rendezvous navigation has been excellent in the Apollo Program; however, iflarge down-range errors occur in the locationof the non-updated vehic le, the CSMrendezvous navigation is degraded. The CSM rendezvous navigation degradation occursbecause the capability to solve for angle biases in the CSM fil ter was not provided, due

to the small sextant biases; however, t h e sextant line-of-sight measurements are madewith respect to iner tial space and a down-range error. In the non-updated vehicle,position appears as an angle bias. Because rendezvous navigation is a relative prob-lem, it is sufficient t h a t the LM and CSM state vecto rs be accur ate relative to eachother , even though the iner tial accu racy of each state is poor. The fact t h a t a relativeproblem is being solved enabled the software designers to design the program to updateor solve for the state vecto r of only one of the two vehic les. Another way to accountfor the downtrack error is to solve for the state vec tor of bo th vehicles; however, th i sprocedure requires too much computer time and is not necessary because the relativeproblem can be solved inall cases i f the solution for the angle biases s part of thesolution vector.Anomalous simulation results. - During the navigational planning and analysis forApollo 10, two interes ting results were obtained from simula tions of the onboard ren-dezvous navigation of the CSM and LM .1. Although the LM RR is less accurate in its me as ur em en ts of line of sigh t andrange than is the CSM sextant and vhf ranging, the LM guidance computer (LGC) oftenobtained a super ior deter mination of the relative state (producing a more accuraterendezvous).2. The method of immediately obtain ing a set of "good" es ti ma te s of the RRangle bi ase s (by rolling the LM 180 between the first two data sets) resulted in infe-rior accuracy compared to the method of allowing these estimates to be determinedthroughout the rendezvous sequence.Explanation of results. - A s a re su lt of r econsi deratio n of the problem, the anom-alous results were attributed toa down-range error in the state vectorof the non-updated vehicle.

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For the onboard formulations in both vehicles, t is assumed that the state of onvehicle is perfectly known; therefo re, sensor residuals are used to correct the othervehicle state vector. Because line-of-sight measurements are made with respect toinertial space in a near-ci rcular orb it, down- range err ors in the positi on of the non-updated vehicle act as angle biases in the filter. (If all navigational measurements andrendezvous maneuvers were made with respect to the local (orbital plane) coordinates,su ch er ro r s would not be significant; however, the mechanization of s uch measu re-ments is not feasible. ) This apparent bias "drives" the filter away from the correc tsolution, resulting in state estimates with poor error-propagation characteristics.

During the design of t he filters for onboard navigation, the RR angle biases werexpected to be rather large; therefore, they were included in the LGC solution vector.Apparent angle biases caused by down-range e r r o r s are attributed by the LGC to physical angle biases and do not severely degrade the state vector of the updated vehicle.The command module compu ter (CMC) does not provide such an attribution for thesebiases; therefore, they degrade the state vector solution. Rolling the LM to quicklydetermine the biases degrades the solution because the filter "closes out" subsequentsignificant updates to the angle biases, and the down-range error is not recovered assuch a bias. (Reinitializing the filter in order to reopen it to further bias updates,although retaining the bias estimates, would theoretically provide excellent performance; however, the RR angle biases have tended to be small enough in practice thatsuch a procedure is not worthwhile. )

Command module computer change. - Studies were undertaken to evaluate theusefulness of expanding the CMC solution vect or to solve for the dow n-range error ,either explicitly o r as angle biases; favorable results were obtained. Simulations in-dicated that the ap proach was valuabl e and also led t o slight modification of theoriginal equations.Examination of the equations and the existing CMC pr og ram str uct ur e l ed tonewesti mate s of additional storage requirements, namely, 50 words of fixed memory andtwo or three words of er asable memory. A Pr og ra m Change Request (PCR) waswritten for consi deration by the Apollo Software Control Board (SCB).During the evaluation of t he proposa l, the contractor for the design of th e prim aguidance, navigation, and control system pointed out that, in the Apollo CMC prog rama reallocation of erasable memory assignments would be required because the rendez-vous targeting routines sharederasable memory with cells used for W-matrix storagein the 9-D mode. The SCB did not approve the PCR primarily because the CMC ren-dezvous navigation program is used as a backup capability on lunar flights. Of coursein earth orbit , down-range errors are less important; but the error that does occur,when taken in combination with the inertial measurement unit (IMU) and sextant biasesproduces poor accuracy for some rendezvous profiles.

O r b i t a l N a v i g a t i o n ( P - 2 2 )Although a limited capability to perform orbital navigation exists in the softwareit ha s not been a requirement in the Apollo Program. However, P-22 was used togather time-ma rk data for Manne d S pace Fligh t Net work (MSFN) orbital navigationevaluation, unarmapping,anddescent argeting. (A new pr og ra m (P-24),which

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removed the liabilities of P-22, was added to the software for Apollo 14 and subsequentflights. The liabilities of P-22 are the lac k of acquisition assistance and the five-marklimitation. )

Cislunar Navigation (P-23)Cislunar navigation was satisfactory for the Apollo missions. A problem oc-curred on the Apollo 14 mission when the incorrect lunar horizon was used as a target.Procedures have been developed to eliminate this error, which occurred as a result ofthe moon's being nearly fully su nlit. Software modifications such as solutions for thealtitude of the horizon locator and the sextantbias could be evaluated.

Iner tia l Measurement Unit AlignmentThe IMU alignment progr ams could possibly b e improved in the rea of transfor-mation computation. The current 3 X 3 matrix product-transformation procedure couldbe replace d by the vector operation discussed in the appendix to this report. However,the only significant problem is in the area of the hardw are-software incapability expe-rien ced in the LM. The LM IMU alignm ent system is sufficiently accurate for a safelanding but has only marginal accuracy for precision landing.The LM alignment problem resulted from the fact that the sighting instrument(the alignment optical telescope (AOT)) is much less accurate than the IMU. The sys-tem could have been improved by a better instrument (such as the star t racker or thesextant) o r by increasing the number of stars tracked and by averaging data. That is,instead of sighting on only two stars and defining the reference stable member matrix(REFSMMAT), three to five stars could have been sighted in an iden tical manner(three marks per star). The stars would have been properly chosen as a function oftheir relative position with respect to one another. Some simple type of averaging(such as a least squares approximation for definingREFSMMAT) that would minimize

the error in this transformation could then be accomplished. This averaging techniquecould be used in the onboard computer to determine the LM position on the moon.

Support Routi nesCoordinate systems. - An additional difficulty encountered in the manned spaceflight program is the use of unconventional coordinate systems. The system based onthe X-axis through Greenwich at midnight on the day of launch was used in ProjectMercury and in the Gemini Program. The Nearest Besselian Year (NBY) sys tem wasused in the Apollo Program. The primar y advan tage of the NBY coord inate system isthat the transformation from earth-fixed coordinates to inertial coordinates is a rota-tion instead of the 3 X 3 transformation required for the standard fixed-coordinate sys-

tems. However, in retrospect, this small saving in computer time is offset by thecomplex effort required to convert the ephemeris data to the NBY syst em and by thenecessity to remake the computer "ropes" for each Besselian year. This revision isnecessary because the fixed memory contains data that are NBY dependent. In general,the us e of a fixed-coordinate system would simplify the overall software problem. The

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fixed-coordinate system should probably be the 1950.0 system, in which the basiclunar-solar-planet ephemerides are located.The NBY system was established for the following reasons:1. The pre cess ion of the pole of the ear th can be neg lected when computing theearth-oblateness accelerations.2. The transformation to earth-fixed coordinates may use small-angleapproximations.3 . The NBY coor dinate syst em is widely used and is one for which information

is available from the Nautical Almanac Office.Although these reasons seem valid, they are not decisive. The effects of prece ssi onmay be accounted for easily, the small-angle approximations are valid over extendedperiods, and the Nautical Almanac Office does not publish the data until after they areneeded (necessitating an annual request to that office for prepublication data).

Modifications needed for use of a constant system. - The following are the Apollguidance computer modifications that would be required i f a fixed-coordinate systemwere adopted.

Lunar-solar ephemerides: To use the lunar-solar ephemerides, no problemexists for the CMC because the data are stored entirely in an erasable memory; thegeneration program may be used with little o r no change. For the LGC, a minorchange would improve the accuracy and extend the usefulness to several years for thelunar ephem erides. The root mean squar e of the erro r in loca tion of the moon wasapproximately equal to the diameter of the moon (0.5"), and the maximum error durina 1 year span was greater than 1 (17 milliradians). A plot of the lunar ephemerideserror was made for in-plane and perpendicular-to-plane locations. The in-plane errowas an irregular function but seemed to have a period of 1 4 . 5 to 15 .0 days. The errorperpendicular to the plane was a regular function with a period of 32 days.The error in the lunar orbit plane could be correctedy an additional term in thexpression for the longitude of the moon. The error perpendicu lar to the orbit planecan be corre cted by modifying the earth-mean unit vector to include a f i n a l Y-axisrotation. These additional terms correspond approximately to the truncated terms ofBrown's series for the posi tion of t he moon.Earth orientation: The small-angle approximations for earth orientation arevalid for several years; to use the true earth pole in the oblateness perturbation com-putation, a small change to cos @ is required (requiring one word of fixed memory inboth CMC and LGC, but four words of erasable memory in the LGC). The change inthe oblateness computation could probably be omitted from the LGC, however.The diurnal rotation of the earth b uilds up an error in theApollo guidance com-puter of approximately 30 ft/yr becaus e of round-off er r o r s . If this amount is con-sidered excessive, moving constants to erasable memory would permit sufficientcorrection.

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Lunar orientation: No problem exists with use of lunar orientation data, becauseall residuals may be collected intodhe lunar "libration vector, ' ' which is part of theerasable load.Star table: Most of the stars in the star table have small proper motions; thevectors are accurate to 5 seconds of arc over perio ds of * 3 years. Occasional up-dating would be required.Transformations. - A more efficient Cartesian coordinate transformation methodthat precludes matrix manipulations is discussed in the appendix.

CONCLUDING R E M A R K SApollo mission experience has shown that the basic onboard nav igational andalignment software is adequate for lunar landing missions. However, since its incep-tion, the advancement of the state of the art has presented alternate approaches withmore desirable capabilities. These new techniques involve more automated sensorsystems; improved mathematical functions for numerical integration, conic calcula-

tions, and state error propagation; a larger solution state during rendezvous naviga-tion; averaging techniques for star sightings; a more standard basic coordinate system;and an improved transformation algorithm.

Manned Spacecraft CenterNational Aeronautics and Space AdministrationHouston,Texas,December14,1971924-22-20-01-72

REFERENCES1. Anon. : Guidance System Operations Plan for Manned CM Earth Orbital and LunarMissions Using Program COLOSSUS 2D (COMANCHE 72). Section 5, GuidanceEquations (Rev. 9). Report R-577, Massachusetts Institute of TechnologyInstrumentationLaboratory , Nov. 1969.2. Pines, S. : Formulation of the Two-Body Problem for the On-Board Computer.Repor t 68-14, Analytical Mechanics Associates, June 1968.3. Pines, S. ; Lefton,L. ; Levine, N. ; andWhitlock, F. : AVariableOrderInterpo-lation Scheme for Integrating Second Order Differential Equations.Repo rt 69 -16, Analytical Mechanics Associates, July 1969.

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A P P E N D I XA N EFFIC IENT M ETHOD FOR CARTES I A N

C O O R D I N A T E T R A N S F O R M A T I O N SB y P a u l F. F l a n a g a n * a n d S a m u e l P i n e s T

M a n n e d S p a c e c r a f t C e n te rI NTRODUCT IO N

An efficient alternate method used to perform Cartesian coordinate transfo rma-tions is derived. This method replaces the 3 X 3 matrix product-transformation proce-dure by a simpler vector operation that requires less computer storage and lesscomputer execution time. Also, for applications in which interpolation routines areapplied to stored transformation matrices, this method is more efficient because itrequir es the interpo lation and sto rage of on ly four elements, which results in a moreaccurate final computation.

C AR T ES I A N C O O R D I N A TE T R A N S F O R M A T I O N S B Y M E A N SOF R IG ID VECTOR ROTATIONS

A 3 x 3 transformation matrix A is given that causes the Cartesian coordinatesystem I to be transformed to the Cartesian coordinate system 11. Thus, any vectorF$ in the original coordinate system is carried into the second coordinate system by

The same transformation can be achieved by a rigid rotation of F$ about a unitvectorN hroughsomeangle w. Theresultantvectorequation is

% = F$ cos w + N?(1 - cos w)N + sin w N3

*For mer ly of NASA Manned Spacecraft Center, current ly with Analy tical Mechan-tAnalytical Mechanics Associates.

ics Associates.

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The skew-symmetric terms are given by

A skew sym = sin w Nx

The trace of A is given by the trace of its symmetric part; thus

Because N is a unit vector, equation (4a) is obtained.A rigid rotation about N through an angle w greater han 7~ may be replacedby a rigid negative rotation about the same vector N through 277 - w; the refore , theangle w can be restricted to the fi rst o r second quadrants, and equation (4b) results.To obtain the components of the vector N, note that the skew- symmet ric part ofA is given by

A - A2

TA skew sym =

This expression must be equal to the skew-symmetric part of A(N, w) so that

TA - A = sinw NX2It follows that because N is a unit vector, equation (4c) is obtained.

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Apollo Experience Report Oboard Navigational and Alignment Software